Heterogeneous Firms in International Trade · 2020. 11. 20. · sional CDCP described previously in three ways. First, if each agent has a diﬀerent optimal solution to the CDCP,

Heterogeneous Firms in

International TradeHilary (Rowan) Shi

A dissertation

presented to the faculty

of Princeton University

in candidacy for the degree

of Doctor of Philosophy

recommended for acceptance

by the Department of

Economics

Advisor: Stephen Redding

September 2020

© Copyright by Hilary (Rowan) Shi, 2020.

All rights reserved.

Abstract

This dissertation consists of three chapters exploring how producer

heterogeneity and international trade interact.

Chapter 1 proposes a new method to solving a class of complex

discrete choice problems with agent heterogeneity. While this class

of discrete choice problems have many applications in international

trade, the chapter takes the firm’s plant location problem as its leading

example. It then shows how to solve for the policy function of the

problem, which maps heterogeneous firm types in the optimal plant

location strategy for a firm of that specific type. The chapter concludes

with an exposition of several other discrete choice problems examined

by the literature to which the policy function method is applicable.

iii

Chapter 2, written with Kathleen Hu, builds a quantitative spa-

tial model to investigate corporate tax equalization in the European

Union. Though member states set their rates independently, an EU-

wide policy has become a recent topic of political discussion. Since

profits of EU-based firms are taxed in the production location, current

tax rate variation could create firm-level incentives to open facilities

in low tax countries. These would be eliminated in a tax rate equaliza-

tion. The model is estimated and used to perform a counterfactual ex-

periment where tax rates are harmonized, operationalizing the policy

function method from the previous chapter. The counterfactual pre-

dicts substantial redistribution within the EU, in particular industrial

activity being relocated from currently low-tax to high-tax countries.

Chapter 3 investigates the impact of import liberalization policy in

Brazil, uncovering differential establishment-level outcomes depen-

dent on initial employment level. Large establishments were more

likely to survive than their smaller counterparts. Conditional on sur-

vival, they were less likely to shrink than similar small establishments.

These patterns hold in both traded and nontraded industries, with in-

dustrial composition shifting as labor is systematically reallocated

iv

from small to large establishments. Finally, the chapter shows that

the standard heterogeneous trade model naturally extended to incor-

porate a nontraded sector cannot replicate these nontraded sector pat-

terns. A model featuring both final and intermediate input demand

for nontraded varieties is proposed and qualitatively consistent with

the empirical patterns.

v

Acknowledgements

This dissertation has benefited immeasurably from the consistent sup-

port and encouragement of my advisors, Stephen Redding, Eduardo

Morales, and Oleg Itskhoki. I am extremely grateful for their never-

ending generosity of time and thoughtful feedback.

Material in this thesis has also been greatly bettered by careful con-

sideration and comments from International Economics Section Work-

shop participants, especially Esteban Rossi-Hansberg, Richard Roger-

son, and Fabien Eckert, as well as my classmates Charly Porcher, An-

ders Humlum, and Mathilde Le Moigne. Many conversations with

Ben Lund also improved this dissertation.

My graduate studies would not have been the same without Chrissy

vi

Ostrowski, Kathleen Hu, Ben Shields, and Nico Fernandez-Arias. A

special mention must be paid to Chrissy for her unwavering friend-

ship and enthusiasm. Thank you to Christian Wolf, who has been an

endless source of positivity.

Finally, this dissertation is dedicated to my parents. They have

always nurtured and believed in my inquisitivity. I would not be

where am I or who I am today if not for them.

vii

Contents

Abstract iii

Acknowledgements vi

1. Combinatorial discrete choices with agent heterogeneity 1

1.1. A simple example . . . . . . . . . . . . . . . . . . . . . 16

1.1.1. The problem . . . . . . . . . . . . . . . . . . . . 16

1.1.2. Squeezing the optimal set . . . . . . . . . . . . 23

1.2. General solution method . . . . . . . . . . . . . . . . . 32

1.2.1. Formal overview of CDCPs . . . . . . . . . . . 32

1.2.2. Solution method . . . . . . . . . . . . . . . . . 39

1.2.3. Method advantages . . . . . . . . . . . . . . . 55

viii

1.3. CDCPs in trade . . . . . . . . . . . . . . . . . . . . . . 60

1.3.1. Plant location decision . . . . . . . . . . . . . . 61

1.3.2. Input sourcing . . . . . . . . . . . . . . . . . . 69

1.3.3. Export market entry . . . . . . . . . . . . . . . 79

1.3.4. Idiosyncratic fixed costs . . . . . . . . . . . . . 88

1.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 96

2. European Union corporate tax equalization 100

2.1. Corporate taxation in the EU . . . . . . . . . . . . . . 110

2.2. Model framework . . . . . . . . . . . . . . . . . . . . . 112

2.2.1. Consumers . . . . . . . . . . . . . . . . . . . . 112

2.2.2. Nontraded sector . . . . . . . . . . . . . . . . . 116

2.2.3. Traded sector . . . . . . . . . . . . . . . . . . . 117

2.3. General equilibrium . . . . . . . . . . . . . . . . . . . 129

2.3.1. Optimal firm behavior . . . . . . . . . . . . . . 129

2.3.2. Aggregate conditions . . . . . . . . . . . . . . 131

2.4. Computation and estimation . . . . . . . . . . . . . . 142

2.4.1. Calibration and estimation . . . . . . . . . . . 143

2.4.2. Computing the firm’s problem . . . . . . . . . 158

ix

2.5. Counterfactual experiment . . . . . . . . . . . . . . . . 172

2.5.1. Results . . . . . . . . . . . . . . . . . . . . . . . 173

2.5.2. Discussion . . . . . . . . . . . . . . . . . . . . . 197

2.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 198

3. Reallocation among nontraded producers: Brazil’s im-

port liberalization 203

3.1. Import liberalization in Brazil . . . . . . . . . . . . . . 211

3.2. Empirical approach . . . . . . . . . . . . . . . . . . . . 212

3.2.1. Data . . . . . . . . . . . . . . . . . . . . . . . . 213

3.2.2. Liberalization exposure in local labor markets 215

3.2.3. Gross job flows . . . . . . . . . . . . . . . . . . 218

3.3. Empirical results . . . . . . . . . . . . . . . . . . . . . 221

3.3.1. Effects at the local labor market level . . . . . . 221

3.3.2. Effects at the establishment level . . . . . . . . 231

3.4. Baseline model . . . . . . . . . . . . . . . . . . . . . . 243

3.4.1. Theoretical framework . . . . . . . . . . . . . . 244

3.4.2. Equilibrium . . . . . . . . . . . . . . . . . . . . 260

3.4.3. Liberalization counterfactual . . . . . . . . . . 266

x

3.5. Model with intermediate inputs . . . . . . . . . . . . . 273

3.5.1. Theoretical framework . . . . . . . . . . . . . . 274

3.5.2. Equilibrium . . . . . . . . . . . . . . . . . . . . 284

3.5.3. Liberalization counterfactual . . . . . . . . . . 293

3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 301

A. Combinatorial discrete choices 304

B. European Union corporate tax equalization 311

C. Brazil’s import liberalization 317

References 336

xi

Chapter 1.

Combinatorial discrete choices with agent

heterogeneity

Many problems in economics, especially in international trade and

industrial organization, feature optimization over combinatorial dis-

crete choices which tend to be high dimensional and difficult to com-

pute.1 For example, a classic question in international trade is the

multinational firm’s optimal selection of countries in which to oper-

ate plants. This decision effectively encapsulates a collection of dis-

1A recent burgeoning literature is precisely interested in solving these types ofproblems. They are formalized in Arkolakis and Eckert [2017], which provides asolution method and accompanying theory to address games played over thesechoices and other settings with a small number of agents. To draw quantitativeconclusions, Antràs et al. [2017] incorporates the solution method proposed in Jia[2008] to a numerically discretized version of a general equilibrium model.

1. Combinatorial discrete choices 2

crete binary choices: for each country where it is possible to open a

plant, should the firm establish one there or not? If the binary choices

are interdependent, that is, opening a plant in one location affects

the decision of whether to open a plant in another, then they cannot

be considered separately. The multinational is thus left comparing

each combination of possible choices on the individual binary deci-

sions. Recasting the problem as selecting the optimal set of locations

in which a multinational firm operates plants, it is easy to see that the

space of possible strategies balloons given the combinatorial nature of

the problem. In particular, if there are N available locations, the firm

must decide among 2N possible sets. The plant location problem is

not the only one this of type. Selecting partners from whom to source

inputs, export markets to enter, and which goods to produce as a mul-

tiproduct firm have all been formulated this way. Without a strategy,

solving combinatorial discrete choice problems (CDCPs) quickly be-

comes computationally burdensome or even unfeasible due to the

choice space’s large size and discrete nature.

The relevance of solving these types of problems is highlighted by

the large body of work in the international trade literature emphasiz-


ing the importance of heterogeneous firms. In particular, studies in

several settings have consistently concluded that multinational firms,

importers, exporters, and firms offering many products all tend to be

large and to account for disproportionate shares of an economy’s total

sales, employment, and other aggregate variables. Therefore, under-

standing how to solve the CDCPs underlying the behavior of these

“global” firms is an important research agenda, since their activity can

have aggregate implications.

However, a single firm’s CDCP solution is often not enough to make

aggregate conclusions. To do so, heterogeneous firms each solving

CDCPs must be embedded in a quantitative general equilibrium. This

type of exercise exacerbates the challenge of solving the high dimen-

sional CDCP described previously in three ways. First, if each agent

has a different optimal solution to the CDCP, it must now be solved

at every point along the heterogeneous distribution. In the plant lo-

cation context, suppose firms vary by productivity, where firms of

different productivity choose different plant locations. The optimal

set of plant locations would then need to be solved for each level of

productivity to arrive at aggregate variables like labor demand. In


the scenario where heterogeneity follows a continuous distribution,

there is an infinite number of CDCPs, one for every point within the

support. Thus, introducing heterogeneity could greatly multiply the

number of CDCPs to be computed. Next, nesting a block of hetero-

geneous agents, each solving a CDCP, within a general equilibrium

magnifies the computational difficulty. The collection of CDCPs must

be repeatedly and accurately calculated as the general equilibrium

routine searches for a fixed point on aggregates. Lastly, estimating

parameters of the general equilibrium model potentially creates an-

other layer of repetition because the fixed point for general equilib-

rium must be solved for each possible parameter vector. All in all,

computing a quantitatively-estimated general equilibrium model in-

corporating heterogeneous agents facing CDCPs could imply solving

them an enormous number of times.

The goal of this chapter is to provide a novel way to solve combi-

natorial discrete choice problems for a distribution of heterogeneous

agents, enabling such a block to be feasibly included into a quantita-

tive general equilibrium model. It is applicable to problems where

the task is to find the optimal subset of a defined superset. The super-


set could be interpreted as the list of all available locations for plants,

sourcing partners for inputs, or any similar collection of binary pos-

sibilities. In turn, for the plant location problem, the optimal subset

would be the set of locations in which a firm should optimally estab-

lish plants. A similar interpretation of the optimal subset follows for

other binary choice problems, including optimal sourcing partners or

export market entry.

The main innovation of this solution method is to find the policy

function mapping an agent’s type to their optimal CDCP solution.

Broadly, it does so by imposing two monotonicity conditions that arise

naturally in many economic settings. These are discussed in more de-

tail below. Then, this structure is leveraged to compute cutoffs within

the range of heterogeneity at which elements are removed or added

into the optimal set. With heterogeneous productivity firms making

plant location decisions, for example, these would be marginal pro-

ductivities corresponding to firms indifferent between opening plants

in a certain location or not. The intuition of how to find these cutoffs

is provided below.

In solving the problem this way, the solutions to a possibly large


number of CDCPs implied by allowing for agent heterogeneity are

succinctly summarized by a list of cutoffs and the marginal decision

at each. If there are multiple dimensions of heterogeneity, the mono-

tonicity conditions can be extended, allowing for a general method

that identifies boundaries within the type space along which agents

remain indifferent between including an element or not.

This approach is attractive for a number of reasons. The first is con-

ceptual. Since it calculates the policy function over the entire range

of heterogeneity without needing the distribution of agents across

the type space, the algorithm can handle distributions of any shape,

whether discrete or continuous. It is consequently highly versatile.

There are also substantial computational time gains. In a way that

will become clear when the solution’s intuition is discussed, it is much

faster to compute the policy function as a set of indifference points

compared to finding optimal sets at multiple discrete agent types. Fi-

nally, the algorithm avoids discretizing continuous type spaces, elim-

inating inaccuracy that would arise from interpolation between grid

points. Aggregations based on the algorithm’s policy functions are

instead exact. Taken together, the algorithm is well suited for solving


CDCPs faced by heterogeneous agents within quantitative models,

opening the door to computing general equilibrium feasibly. More-

over, being able to numerically solve for general equilibrium allows

for models to be estimated by matching data on aggregate variables,

which tend to be more readily available than detailed microdata.

Chapter 2 demonstrates both of these advantages by evaluating the

aggregate effects of a corporate tax equalization across the European

Union. The chapter builds a structural model whose key ingredient is

a block of firms choosing where to open plants. They weigh the bene-

fits of better market access, cheaper production, and lower tax burdens

against the fixed costs of opening plants. Crucially, the firms vary in

productivity, which influences these trade-offs. Therefore, each firm

faces a CDCP dependent on their type. To make progress, the method

in this chapter is applied to compute the entire distribution of firm

behavior. Then, the model can be estimated by matching aggregate

moments with EU-wide data. On the other hand, if the distribution

of behavior is calculated by solving CDCPs one by one for a discrete

number of agent types, then aggregation is both slow and inexact.

Solving general equilibrium is potentially infeasible with this strat-


egy, precluding both drawing aggregate conclusions and estimating

by matching aggregate moments.

To isolate the cutoffs discussed above, the chapter imposes struc-

ture on the CDCP by making two monotonicity assumptions on the

marginal value derived from including an item into a set. First, it

must be either (weakly) monotonically decreasing or monotonically

increasing in the set. In the first case, the gains from including an

element into a set will fall as the set grows. For example, if a multina-

tional firm engages in export platform foreign direct investment (FDI),

the additional market access provided by any single plant declines

as the multinational opens more and more plants. In other words,

the elements loosely can be interpreted as substitutes for each other.

Conversely, in the monotonically increasing case, an item’s additional

value to any set rises as the set grows. In this case, each individual

item is more beneficial when part of larger groups. The plant loca-

tion problem with vertical FDI could be an example of this structure,

if each facility carries out complementary production tasks. The ele-

ments can accordingly be interpreted as being complements for each

other. For that reason, the two cases are respectively referred to as the


substitutes and complements case in what follows. Either is sufficient

to satisfy monotonicity in the set.

The second restriction is (weak) monotonicity along dimensions

of heterogeneity, which implies that the marginal value of an addi-

tional element’s inclusion into any set will increase as the agent’s type

increases. This assumption specifies how the heterogeneous type in-

teracts with the interdependent binary decision payoffs. Returning

to the leading example of firms selecting production locations while

varying in productivity, monotonicity in type translates to higher pro-

ductivity firms benefiting more from opening extra plants than their

low productivity counterparts, all else equal. With multiple dimen-

sions of heterogeneity, it is sufficient if an element’s marginal value in-

creases in every characteristic. The text focuses on the monotonically

increasing case without loss of generality, since the problem can be

reformulated in terms of 1/z if the marginal value is instead decreas-

ing in a dimension of heterogeneity z. Ultimately, both assumptions

appear naturally in CDCPs derived from economic questions since

they discipline the marginal value derived from an additional item.

Along with the structure imposed above, the proposed solution


leverages a series of necessary conditions on the optimal set. Observe

that, if an element is part of the optimal set, it must add weakly posi-

tive marginal value in that set. Put another way, removing it from the

set cannot increase payoffs. In the plant location example, if a plant

is optimally opened in a specific area, it cannot be that closing this

plant increases profits. Likewise, if an element does not appear in

the optimal set, then it cannot add positive value if included. That is,

profits cannot be increased by establishing a plant where optimally

no plants are open. Thus, signing an element’s marginal value to the

optimal set seals its fate. If it is positive, then the element should be

included, while if it is negative, the element should be excluded. The

algorithm proceeds by signing these marginal values using the two

monotonicity assumptions described earlier to rule out many possible

combinations from the choice space.

Of course, these sufficient conditions require the optimal set to

calculate marginal values, precisely the unknown object being com-

puted. However, monotonicity in the set guarantees that an element’s

marginal value from inclusion in the optimal set can be bounded from

above and below for any given type z. To demonstrate this statement,


fix an agent type z and first consider the substitutes case. Suppose

that a subset of the optimal set is known. Such a set can always be

specified since the empty set trivially qualifies. This subset will get

more out of any extra element compared to the optimal set, because

it contains fewer elements that can serve as substitutes. Then, any

element’s marginal contribution to the optimal set is bounded from

above by its marginal value of inclusion in the subset. The lower

bound can be identified in a similar way. Suppose a superset of the

optimal set is known, where the full set of items can always serve as a

natural starting point. Then, an element’s contribution to this super-

set is smaller than its marginal value from inclusion in the optimal set,

since the superset will contain more elements that are substitutable.

Thus, the element’s marginal value from inclusion in the subset and

superset provide upper and lower bounds on its marginal value from

inclusion the optimal set, respectively. A reverse argument holds in

the complements setting. In that setting, elements add more value in

large groups, so the subset and superset provide corresponding lower

and upper bounds on any element’s marginal value of inclusion into

the optimal set. Therefore, in both the substitutes and complements


case, monotonicity in the set ensures lower and upper bounds on an

item’s additional contribution to the optimal set.

The horizontal FDI plant location problem where plants are sub-

stitutes can provide intuition. The prospect of opening a plant when

the firm does not open any other plants is very attractive, since this

plant affords a large increase in market access. Conversely, the added

market access from opening the same plant when the firm also con-

siders opening plants everywhere else is much smaller. These are

both extreme cases. The marginal value of the plant, evaluated at the

firm’s optimal choice of locations, will therefore lie within these two

bounds.

The original goal was to determine the sign of an item’s marginal

value of inclusion into the optimal set. By combining the lower and

upper bounds derived above with the monotonicity in type condition,

the sign can be determined for large portions of the type space. To

see how, fix an element j. Given monotonocity in type, its maximal

marginal value also increases in agent type because the upper bound

is itself a marginal value of inclusion. If there is a threshold type for

which the upper bound crosses zero, then monotonicity in type en-


sures that it is negative for types below. As a result, these types receive

negative marginal reward from including the element in considera-

tion to their optimal sets. Following the argument established above,

the firms of these productivities should optimally exclude it. In the

plant location scenario, this conclusion establishes that firms with

productivity lower than some threshold should not open plants in

the specific location considered. Therefore, conclusions can be drawn

for large sections of the type space precisely by harnessing the dis-

cipline imposed by monotonicity in type. This argument creates the

considerable computational speed gains referenced earlier.

A similar thread of logic can be made using the lower bound, which

is also increasing in agent type due to monotonicity in type. Likewise,

suppose there is a threshold at which the lower bound crosses zero.

Then, the element’s marginal value of inclusion into the optimal set

is guaranteed to be positive for all agents with a higher type. These

agents will therefore optimally include the element in considerations

in their optimal sets.

This chapter builds on previous work studying CDCPs. A foun-

dational paper is Jia [2008], which proposes a solution method for


games with two players over binary choices featuring complementar-

ity. Arkolakis and Eckert [2017] generalize this approach for games

with a small number of agents, characterized by either substitutability

or complementarity on the set. In particular, this literature has not

structurally incorporated any agent heterogeneity within the CDCP.

However, agent heterogeneity is naturally featured in several quan-

titative studies with CDCPs. For example, Tintelnot [2016] studies

the quantitative implications of multinational behavior, building a

structural model linking a firm’s productivity to its plant location de-

cision. Likewise, Antràs et al. [2017] studies the CDCP represented

by input sourcing decisions of heterogeneous firms. Computational

exercises in both proceed by discretizing a continuous productivity

distribution to apply a single-type algorithm at each grid point. In

particular, Antràs et al. [2017] operationalize the method introduced

in Jia [2008]. In these analyses, several general equilibrium variables

are held constant. They also rely on detailed firm-level microdata for

estimation. The CDCP represented by the optimal sourcing strategy

is further explored in Antràs [2016], which comprehensively studies

the sourcing decisions of multinational firms while focusing on the


complements case.

The key contributions of this chapter to the literature is to develop

an approach explicitly solving for the policy function as a function

of the agent’s type, enabling straightforward and precise aggregation

over CDCP solutions of a distribution of heterogeneous agents. It

is flexible enough to allow for a continuum of agent types over an

arbitrary number of dimensions heterogeneity. Further, it is applicable

to problems featuring either substitutablility or complementarity.

The remainder of the chapter is organized as follows. Section 1.1

introduces the leading example of a CDCP, the plant location problem.

In particular, the section highlights the problem’s important structural

features, and the intuition of the solution method. Section 1.2 provides

a formalization of the class of problems studied and provides general

sufficient conditions to identify when the algorithm can be applied

before describing it in detail. Section 1.4 concludes.


1.1. A simple example

Before presenting the formal method in Section 1.2, this section is ded-

icated to outlining an illustrative example of the problems addressed

in this chapter. It considers firms heterogeneous in productivity, each

which must establish plants in order to produce and serve destination

markets. Section 1.1.1 formalizes the problem and highlights features

of its structure that are used in the solution method. For simplicity,

this section only outlines easily interpreted sufficient conditions. Gen-

eralized necessary conditions are deferred to Section 1.2. Section 1.1.2

then discusses the general intuition used by the solution routine.

1.1.1. The problem

This example considers a firm maximizing profits by choosing where

to establish affiliates that serve as production facilities and export

platforms. The available countries are Germany (G) and Romania (R),

with locations interacting in a way that will be defined formally below.

This section presents the problem with only two countries for now to

remain as simple as possible while building conceptual intuition.


Firms are heterogeneous in productivity, z, and take aggregate con-

ditions as given when deciding where to locate plants. The profit

function

π : P(G, R)× Z× Υ→ R ,

is defined over the power set of available locations, the range of pro-

ductivity Z ⊆ R, and the range of aggregate vectors Υ. In this setting,

possibilities for the chosen set of a firm are , G, R, or G, R,

corresponding to the actions of opening plants nowhere, only in Ger-

many, only in Romania, or in both countries.

The key feature of the formulation presented here is that the objec-

tive function can also vary by the firm’s productivity and an aggre-

gate vector υ ∈ Υ, which includes all relevant aggregate variables,

like wages and prices.2 The vector of aggregates υ is assumed to be

fixed at an arbitrary value for the rest of the discussion below. Conse-

quently, the task at hand is to find the policy function J ∗(·; υ). This

policy function is a set-valued function taking firm productivity as its

argument and returning the optimal set of countries in which a firm

2In what follows, vectors are represented with boldface variables.


should open plants, dependent on the aggregate vector.

Plants in a country j ∈ G, R should be added only if they increase

profits. Thus, the key difference considered is

Dj(J ; z, υ) ≡

π(J ; z, υ)− π(J \ j; z, υ) if j ∈ J

π(J ∪ j; z, υ)− π(J ; z, υ) if j 6∈ J.

Following the existing CDC literature, this difference is interpreted

as the marginal value function, describing the marginal value from an

element j’s inclusion in a set J .3 For example, the difference given

by DG(; z, υ) = π(G; z, υ)−π(; z, υ) corresponds to, for a firm

with productivity z, the profit difference between choosing to open a

plant in only Germany compared to opening no plants anywhere.

Because the marginal value function specifies the set J to which

j is included, it leaves room for interdependence among the choices.

For example, in the previous example, the value of opening a plant in

Germany is contextualized with the fact that there are no plants in any

3Note that, J represents a set while J ∗(·; υ) represents a set-valued function.Functions will always be denoted this way.


other locations. This information is provided by the first argument

. In contrast, DG(R; z, υ) evaluates how profits would change if

a plant were opened in Germany, conditional on the firm also estab-

lishing a plant in Romania. If these two valuations are different, the

contribution of Germany changes based on the choice made for the

other binary decision, Romania, which reflects interaction between

these two choices. Likewise, since z is the second argument, firm pro-

ductivity could also influence j’s marginal value in J . For example,

DG(; z, υ) varying with z could generate a nontrivial distribution

of optimal behavior over the range of firm productivity. The solution

relies on the marginal value function exhibiting a specific structure

over its two arguments, described below.

Monotonicity in set

The first structural assumption builds on the single crossing condi-

tions introduced in Arkolakis and Eckert [2017]. These restrict how

plant locations interact with each other in the marginal value function

denoted by Dj(·; z, υ). To illustrate the intuition, fix a firm productiv-

ity z and consider the value of adding a plant in Germany. There are


two possibilities for Romania: either the firm opens a plant there or

not. Since the plant locations interact, the marginal value of having a

plant in Germany depends on the decision for Romania.

Substitutability among plants asserts that opening a plant in Ger-

many adds more value if there are no plants in Romania. The follow-

ing inequality holds in this case for all z.

DG(; z, υ) = π(G; z, υ)− π(; z, υ)

≥ π(G, R; z, υ)− π(R; z, υ) = DG(R; z, υ) (1.1)

For every productivity level, this inequality ensures that a plant in

Germany increases profits by more if there is no plant in Romania.

A profit function satisfying this condition could emerge from export-

platform horizontal FDI models where each plant produces the same

product. With no affiliates, the firm cannot serve either the German or

Romanian markets. Then, opening a German plant would be valuable

since they could serve both the domestic consumers as well as those in

Romania through trade. If there is instead a plant in Romania to begin

with, both markets can already be served by them. Adding a plant in


Germany may lower costs of serving German customers, increasing

variable profits, but gains are smaller relative to the first scenario.

Intuitively, the condition can be interpreted as corresponding to the

case in which the plants are substitutes for each other.

On the other hand, reversing the weak the inequality in equation

(1.1) defines the complements case. In this case, a plant in Germany is

more valuable when considered in tandem with a plant in Romania.

Profit functions emerging from vertical FDI models, where production

facilities perform complementary tasks, could satisfy this restriction.

Without any other affiliates, a single one in Germany must perform

all duties associated with production. The resulting increase in profits

gain may be smaller than if there already were a plant in Romania.

In that scenario, the Romanian plant is performing all tasks before

the introduction of a German plant. After it is opened, both locations

could specialize on separate portions of the production process. The

additional value of opening the German plant is then larger if a Ro-

manian plant is also established. In this setting, the condition can be

interpreted as corresponding to the case in which plants are comple-

mentary.


Discussion here concentrated on the marginal value of opening a

plant in Germany to focus on the interpretation of the monotonicity

in set condition. Of course, for the condition to hold either as the sub-

stitutes or complements case, parallel statements also must be true

when assessing the marginal value of opening a plant in Romania.

For concreteness, the rest of this simple example assumes the substi-

tutes case, in accordance with the application in Chapter 2. However,

problems characterized by either type of monotonicity in set can be

solved by the generalized method presented in Section 1.2.

Monotonicity in type

The second assumption disciplines how firm productivity affects a

location’s marginal returns. For this assumption, fix a set J and con-

sider two firms with differing productivities z1 < z2. Without loss

of generality, assuming monotonicity in type involves assuming that

marginal returns increase in productivity, or

DG(J ; z1, υ) ≤ DG(J ; z2, υ) . (1.2)


In words, opening an extra plant in Germany has a larger benefit for

a more productive firm than for a less productive firm. The problem

can be redefined in terms of 1/z if the inequality goes the other way.

1.1.2. Squeezing the optimal set

Observe that the objective function is continuous in productivity, but

could be discontinuous as elements are added to J . If J ∗(z; υ) is the

optimal set of countries in which to open a plant for a firm with pro-

ductivity z, it will also be optimal for firms with productivity in a small

enough neighborhood of z. The optimal policy function J ∗(·; υ) then

separates the productivity range into intervals, each associated with

a distinct optimal set. Figure 1.1 provides an example of how J ∗(·; υ)

might look. The strategy in this chapter is therefore to solve for these

intervals and their associated optimal sets. For the remainder of this

section, the υ is omitted for notational brevity when unambiguous.

More concretely, consider a firm with productivity z. Initially, the

optimal choice on each binary decision is unclear. In other words, for

each country, it is unknown whether the firm should open a plant


−∞ z1 z2 z3 ∞ R G G, R

Figure 1.1.: An example of the policy function. At each cutoff zn, theoptimal set of countries J ∗(·; υ) changes.

there. The goal is to progressively determine z’s best choice for each

binary decision, narrowing down the possibilities for its optimal set.

Formally, define “subset”- and “superset”-valued functions, mapping

z respectively to

L(z) = j ∈ J known to be included in optimal set

U(z) = j ∈ J not known to be excluded from optimal set. (1.3)

Crucially, L(z) ⊆ J ∗(z) ⊆ U(z), so the subset is the smallest the

optimal set could be, while the superset is the largest. The solution

method squeezes J ∗(·) iteratively by including elements into L(·)

while excluding elements from U(·). During the iteration, monotonic-

ity in type will be evoked to update the subsets and supersets for

many values of productivity at once.

To begin, fix z and consider Dj(J ∗(z); z), a country j’s marginal


value to the optimal set. Suppose it is negative. Then, it must be the

case that j 6∈ J ∗(z). If it were, using the definition of marginal value

implies that π(J ∗(z); z)− π(J ∗(z) \ j; z) < 0, or that the firm is

better off removing it. This statement contradicts the fact thatJ ∗(z) is

the optimal set. Then, it must be that j should not be included for firms

with productivity z. Conversely, suppose Dj(J ∗(z); z) > 0. In this

case, it must be that j ∈ J ∗(z). A similar logic can be applied to arrive

at a contradiction if it were not. As a result, the sign of a country j’s

marginal value to J ∗(z) is directly linked to whether or not it should

be optimally included by a firm with productivity z. The squeezing

strategy described below attempts to sign Dj(J ∗(z); z) by bounding

it from above and below with the marginal value of j’s inclusion into

L(z) and U(z).

Note that ⊆ J ∗(z) ⊆ G, R for all z. Thus, setting L(z) =

and U(z) = G, R is a valid initialization. Starting with the subset,

right away some firms can be identified as better off excluding Ger-

many. To do so, consider the productivity value zLG such that

DG(; zLG) = π(G; zL

G)− π(; zLG) = 0 . (1.4)


Monotonicity in type guarantees there is at most one productivity

level for which firms are indifferent between opening a plant in Ger-

many or opening no plants at all.

Suppose there is exactly one. Then,

z < zLG : DG(J ∗(z); z) ≤ DG(; z) ≤ DG(; zL

G) = 0 . (1.5)

Assuming the substitutes case establishes the first inequality. It says

that, for any firm productivity z, Germany’s inclusion into the optimal

set can be (weakly) bounded above by its marginal value when added

to L(z) = . The reason is that L(z) is a (weak) subset of the optimal

set. If the firm opens a plant only in countries specified by L(z), there

are fewer plants to serve as substitutes to those opened in Germany

than if it had acted according to J ∗(z). Therefore, DG(L(z); z) repre-

sents an upper bound on the value obtained by opening a German

plant. The productivity level zLG in (1.4) is precisely associated with

firms for whom this best-case scenario is zero. The second inequality

thereby captures that it is non-positive for those of lower productivity

by monotonicity in type. Overall, the chain of inequalities (1.5) estab-


lishes that any firm ranking below zLG in productivity cannot optimally

open a plant in Germany.

Notably, this cutoff is conservative, as it may be the case that a plant

in Germany will turn out to be suboptimal for firms with higher pro-

ductivities. However, it is impossible to conclude that Germany is

excluded for firm productivities above zLG with the current informa-

tion.

Now, consider the superset U(·) = G, R. A mirror argument can

be used to infer whether some firms are better off including Germany.

This time, define zUG as

DG(G, R; zUG) = π(G, R; zU

G)− π(R; zUG) = 0 (1.6)

if it exists. A firm with this productivity receives no added value from

Germany’s inclusion in the full set, and would earn equal profits if

it were removed. Using both monotonicity conditions arrives at the


conclusion

z > zUG : DG(J ∗(z); z) ≥ DG(G, R; z) ≥ DG(G, R; zU

G) = 0 .

(1.7)

Observe this time that U(·) = G, R (weakly) contains the optimal

set. The relative contribution of a plant in Germany must then be

(weakly) lower if the firm opens a plant in both countries than if it acts

optimally. Again, this statement derives from the fact that plants are

substitutes for each other. The productivity zUG is specifically defined

in (1.6) to identify when Germany’s additional value in this “extreme”

scenario is zero. Firms with productivity above zUG are such that the

lower bound on Germany’s contribution is still positive as a result.

Expression (1.7) combines these conclusions, ultimately implying that

firms exceeding zUG in productivity must optimally open a plant in

Germany.

To summarize the inferences up to this point, Germany should

be excluded for firms with productivity below zLG and included for

firms with productivity above zUG . If the subsets and supersets were


to be updated at this point, G would be removed from U(·) in the

range (−∞, zLG]. Likewise, G would be added into L(·) over the range

[zUG , ∞). Observe that zU

G necessarily exceeds zLG by combining the

monotonicity assumptions. Then, for all firms with productivity in

the interval (zLG, zU

G), the status of the Germany binary decision is still

unclear.

Before they are, however, the same process can be repeated for Ro-

mania. Undertaking this analysis results in two additional cutoffs,

zLR < zU

R , with

DR(; zLR) = 0 , DR(G, R; zU

R ) = 0 .

Following the exposition above, all firms with productivities below zLR

should not open a plant in Romania while those above zUR should. Us-

ing these results, L(·) and U(·) can be updated to incorporate the con-

clusions concerning both Germany and Romania so far. The real line is

thus partitioned into five subintervals defined by zLG, zU

G , zLR, zU

R. Ad-

ditionally, the pair L(·) and U(·) are constant within each subinterval,

but at least one changes from one subinterval to the next. Figure 1.2


−∞ zLR zL

G zUG zU

R ∞

L(·) :U(·) :

R

G, R

G, RG

G, RG, R

Figure 1.2.: An example of subsets, supersets, and resulting subinter-vals after one iteration.

displays a possible outcome of the described procedure.

The logic outlined in this section can now be applied in turn to

every generated subinterval. Since the subsets and supersets have

been weakly narrowed down in each, the updating process has more

information for the second iteration. However, observe that it must

be applied to each subinterval separately, because it crucially uses the

subset and superset to compute indifference points. Thus, it is only

defined over productivity ranges where both L(·) and U(·) are con-

stant. Ultimately, the solution method involves iteratively repeating

this logic until the subset and superset functions no longer change

when updated. Describing the full algorithm in depth is covered in

Section 1.2.

Discussion of the simple example concludes for now with the key

observation that the optimal sets are not necessarily nested as produc-


tivity increases. That is, z1 < z2 does not imply that J ∗(z1) ⊆ J ∗(z2).

Consider the case presented in Figure 1.2 as an example. Within

the second subinterval (zLR, zL

G), there could be a value of productiv-

ity where Romania is optimally added. Next, examining the fourth

subinterval, (zUG , zU

R ), reveals that Romania is not necessarily in the

optimal set for firms of these productivities. Suppose the extreme

case holds where in fact none of these firms should open a plant in

Romania. Then, Romania is only added back into the optimal set

for firms exceeding zUR in productivity. There is thus a range of pro-

ductivity where firms open a plant in Romania, then another range

later on where they do not, followed by a third range where they

will again. Monotonicity in type implies that more productive firms

gain more from extra plants, so it is tempting to conclude that they

have larger optimal sets. However, substitutability is a force in the

opposite direction, since extra elements are less beneficial in larger

sets, implying smaller optimal sets. Nested optimal sets consequently

only necessarily occur in the complementarities case. In that scenario,

both monotonicity in set and type work in the same direction. Higher

productivity firms derive more value from additional plants, imply-


ing more plants, leading to even higher marginal value from plants

from complementarity.

1.2. General solution method

This section formalizes the optimization problems considered in the

chapter before presenting the algorithm. Section 1.2.1 begins by defin-

ing a combinatorial discrete choice problem and recasts the two as-

sumptions introduced above in a general setting. They turn out to

be weaker statements of more standard super- and submodularity

conditions arising in economic settings, so these sufficient conditions

are also discussed. Section 1.2 details a generalized version of the

solution method, defined over a multidimensional type space. Then,

Section 1.2.3 discusses its main advantages as well as highlighting

nested solution methods.

1.2.1. Formal overview of CDCPs

The solution method’s description begins with a formal definition of

a CDCP parameterized by agent heterogeneity. Building on existing


formulations of CDC problems, agents maximize a function

π : P(J)×RNz × Υ→ R

over its first argument. It is defined over the power set of a known set

J, all possible vectors of types RNz ⊆ RN, and all possible vectors of

aggregates Υ. The second argument allows for the objective function

to vary with agent type, a key focus of the analysis in this chapter.

While the body of the chapter focuses on the problem of finding

an optimal subset, which can be cast as a series of 0-1 decisions, Ap-

pendix A shows that the problem can be extended to include multiple

discrete levels for each decision. Conditional on the objective function,

a CDCP can be written as

maxJ⊆J

π(J ; z, υ)

where J is a choice of set, z summarizes individual-level variables

which are not choice variables, and υ collects aggregate variables. In

the plant location example, π represented a firm’s profits, J the set of


countries where the firm opens a plant, z is a scalar denoting the firm’s

productivity, and υ is the vector containing aggregates such as wages.

In most applications, the z argument will summarize heterogeneous

characteristics across agents. Given this interpretation, z is referred

to as the “type vector” for this section.

Since the CDCP varies in the type vector, the optimal solution can

also vary by agent type. As before, the aim is therefore to findJ ∗(·; υ),

the policy function for each value of type vector conditional on a vec-

tor of aggregates. To this end, the marginal value function representing

the additional benefit derived from j’s addition to J is defined as

Dj(J ; z, υ) ≡

π(J ; z, υ)− π(J \ j; z, υ) if j ∈ J

π(J ∪ j; z, υ)− π(J ; z, υ) if j 6∈ J. (1.8)

Introducing heterogeneity and solving for the policy function in terms

of agent type, instead of solving the CDCP for a single value of agent

characteristics, is the key innovation compared to the work developed

in Arkolakis and Eckert [2017]. A deeper discussion is deferred to

Section 1.2.3, after the full solution method has been presented.


Next, the two assumptions on the marginal value function intro-

duced in Section 1.1 are generalized. More easily interpreted sufficient

conditions are also discussed for each.

Monotonicity in set

The monotonicity in set conditions are similar to the single crossing

formulations presented in Arkolakis and Eckert [2017], here general-

ized to allow for agent heterogeneity. To illustrate these assumptions,

consider how much an extra element j would generate if added in-

sto two different sets, J1 ⊂ J2. Then, the monotonicity condition

imposes that, for all types z and vectors of aggregates υ, either

Dj (J1; z, υ) ≥ Dj (J2; z, υ) , or (MS-S)

Dj (J1; z, υ) ≤ Dj (J2; z, υ) . (MS-C)

The first condition (MS-S) asserts that the value of adding an element

j to any set falls as the set grows. Conversely, (MS-C) implies that the

extra element j will become more valuable as the original set grows.

As discussed in Section 1.1, CDCPs where elements are substitutes


for each other are broadly linked to property (MS-S), while (MS-C)

generally arises settings where elements complement each other.

A sufficient condition for (MS-S) is submodularity of the discrete

choices, shown in Arkolakis and Eckert [2017]. In this context, sub-

modularity on π(·; z) asserts that for any two sets J1 and J2,

π (J1 ∪ J2; z, υ) + π (J1 ∩ J2; z, υ) ≤ π (J1; z, υ) + π (J2; z, υ) .

Similarly, supermodularity of the object function π(·; z) can be de-

fined by reversing the weak inequality in the above expression. Then,

supermodularity of the objective function is sufficient to guarantee

(MS-C).

Monotonicity in type

The next condition specifies how the value of an additional element j

interacts with types z. To satisfy monotonicity in type, the space

Zj(J ; υ) =

z ∈ RNz | Dj(J ; z, υ) = 0


must define an (N− 1)-dimensional surface partitioning RNz into two

cells, Zj(J ; υ) and Zj(J ; υ). In addition, the division must be mean-

ingful in the sense that

Dj(J ; z, υ) < 0 if z ∈ Zj(J ; υ)

Dj(J ; z, υ) > 0 if z ∈ Zj(J ; υ)

. (MT)

This statement must hold for anyJ ⊆ J and provides a generalization

of the condition from the single dimensional case. This condition im-

poses structure on interactions between the objective function’s first

and second arguments. Now, there are multiple points z for which

agents are indifferent to j’s inclusion into J . However, it is still the

case that these indifference points divide the type space RNz into two

distinct halves. In turn, the halves directly coincide with the values

of z for which j’s inclusion into J respectively creates positive or

negative additional value.

Asserting that Dj(J ; z) is monotonically increasing in each dimen-

sion of z is a simple sufficient condition. Formally, if z = [z1, z2, ..., zN],


it is sufficient that∂Dj(J ; z, υ)

∂zi≥ 0

for each i. Observe that if instead the marginal value is decreasing

in any dimension, the problem can be recast to satisfy the original

condition. At its core, this sufficient condition thus only requires that

Dj(J ; ·, υ) is (weakly) monotonic in every characteristic zi. For exam-

ple, a plant location problem where heterogeneity arises from varying

idiosyncratic plant fixed costs could satisfy this sufficient condition.

A concrete example of this problem is discussed in Section 1.3.

If there exist some dimensions of heterogeneity that do not satisfy

monotonicity in type, the chapter’s solution method can still make

progress over the dimensions that do. For example, suppose that

monotonicity in type holds for the first M < N dimensions of RNz , if

the remaining ones are fixed at any value. Then, redefine the objective

function as

π(J ; ·, ·, ·) : P(J)× Z× X× Υ→ R

where Z is the subspace of RNz spanning the first M dimensions, and

X is the subspace spanning the rest. The method can then solve for


J ∗(·; x, υ) over Z, at a specific value x ∈ X.

1.2.2. Solution method

This section describes the generalized solution method for a CDCP

defined over a type space. To do so, a generalization of the updating

procedure described previously is presented first. It is then incorpo-

rated into an iterative routine for updating the subsets and supersets,

which is part of a larger branching procedure. Finally, the full solu-

tion method is summarized, including the inner iterative routine for

updating the subsets and supersets as well as the outer branching

procedure.

Iterative step

This section generalizes the procedure used to update subsets and

supersets during Section 1.1’s illustrative example. Fix a vector of ag-

gregates υ ∈ Υ. For these aggregates, the algorithm takes a connected

region Z ⊆ RNz of the type space as well as subset and superset L and


U.4 It returns a partition of Z into smaller connected regions, each

with their own (L, U) set pair. The analog in Section 1.1’s single di-

mensional case would be intervals, each with their own (L, U) pair, as

depicted in Figure 1.2. The iterative step updates L(·) and U(·) by re-

peating the process on each resulting connected region until updating

has no effect. Since the rest of the discussion in this section applies for

a fixed vector of aggregates υ, this indexing is dropped for notational

brevity when unambiguous.

Specifically, consider an initial region Z associated with a subset

and superset (L, U). Assume that these sets sandwich the optimal

set over the region so that for every z ∈ Z, L ⊆ J ∗(z) ⊆ U. This

assumption is not binding since it is met trivially by the empty set

and full set respectively. Notice that L and U are not parameterized

by z, and are therefore constant for all z ∈ Z. Then, U \ L contains

the elements for which the optimal binary decision is still uncertain.

The updating step is defined on these decisions as follows.

Algorithm 1 (Updating step). Given a triple (Z, L, U),

4As before, L and U represent sets, while L(·) and U(·) represent set-valued func-tions defined over portions of the type space.


(i) For each element j ∈ U \ L:

a. Divide RNz into two cells Zj(L) and Zj(L) where

z ∈ Zj(L) ⇔ Dj(L; z) < 0

z ∈ Zj(L) ⇔ Dj(L; z) > 0.

Monotonicity in type (MT) guarantees that this partitioning ex-

ists.

b. Similarly, divide RNz into two cells Zj(U) and Zj(U) where

z ∈ Zj(U) ⇔ Dj(U; z) < 0

z ∈ Zj(U) ⇔ Dj(U; z) > 0.

(ii) With substitutability (MS-S), define L(·) and U(·) over Z as:

L(z) = L ∪ j ∈ U \ L | z ∈ Zj(U)

U(z) = U \ j ∈ U \ L | z 6∈ Zj(L).

Otherwise, with complementarity (MS-C), define L(·) and U(·) over


Z as:L(z) = L ∪ j ∈ U \ L | z 6∈ Zj(L)

U(z) = U \ j ∈ U \ L | z ∈ Zj(U).

(iii) Given the newly defined L(·) and U(·), partition Z into cells Zee so

that for every Ze, L(·) and U(·) are both constant. Let these associated

constant subsets and supersets be Le and Ue for each Ze respectively.

(iv) Return a set of triples (Ze, Le, Ue)e.

Algorithm 1 provides the formal generalization of the logic de-

scribed in section 1.1. Part (i) uses monotonicity in type (MT) to divide

the characteristic space for each uncertain decision j, according to the

provided L and U. Then, part (ii) uses monotonicity in set, (MS-S)

or (MS-C), to construct subsets and supersets by either adding items

into L or excluding them from U. Finally, the originally given space

Z is partitioned into a collection of smaller regions Zee, each with

constant (Le, Ue), in anticipation of the next iteration.

The proposition below establishes that Algorithm 1 indeed moves

the routine closer to the optimal set, without wrongful inclusions or

exclusions. Proofs are provided in Appendix A.


Proposition 1. Suppose that (Z, L, U) are so that, for all z ∈ Z, L ⊆

J ∗(z) ⊆ U. Then, for each Ze returned by Algorithm 1, z ∈ Ze implies

L ⊆ Le ⊆ J ∗(z) ⊆ Ue ⊆ U.

Though short, Proposition 1 states two key results. The first is that

the updated subsets and supersets are correct. Alternately put, they

truly sandwich the optimal set. The second is that the updating step

weakly incorporates more information. The new subsets contain the

original subsets while the new supersets are contained in the original

supersets. Then, through iteration, Algorithm 1 can close in on the

optimal set in a way that takes advantage of the CDCP’s structure.

The iterative step is now explicitly defined.

Algorithm 2 (Iterative step). Given an initial triple (Z, L, U):

(i) Define two collections, C and S. The set C will contain triples to be

iterated on (“Continue”) while the set S will contain triples where

iteration has completed (“Stop”).

(ii) Set C = (Z, L, U) and S = .

(iii) Iteratively, for each element c ∈ C:


a. Remove c from C.

b. Apply Algorithm 1 to c, which will return a collection R of triples

(“Returned”).

c. If R = c, then c cannot be updated further. Thus, it should

be added to S. Otherwise, add the elements from R into C for

further updating.

(iv) Stop when C is empty. Return S, a collection of triples (Ze, Le, Ue)e.

In particular, Zee will partition the characteristic space.

The outcome of Algorithm 2 will effectively be two functions, L(·)

and U(·), summarizing the subsets and supersets. They are defined

over the type space so that for any z ∈ Z, L(z) ⊆ J ∗(z) ⊆ U(z).

This result follows directly from Proposition 1. The next proposition

confirms that this iteration stops in finite time.

Proposition 2. With any initialization, Algorithm 2 repeats step (iii) a

maximum of |J| times.

The argument follows a similar structure to Arkolakis and Eckert

[2017], but is extended to allow for agent heterogeneity and to solve


explicitly for the policy function. For every agent type z, there are at

most |J| binary decisions to determine, with each instance of step (iii)

fixing at least one. If no decisions are determined, then the returned

subsets and supersets will not have changed. In this case, the algo-

rithm has converged by step (iii)c., so it stops. Thus, it can continue

repeating a maximum of |J| times, if each time only determines one

binary decision.

Effectively, the iterative step reduces the dimensionality of a CDCP,

returning a smaller problem. To fix ideas, return to the plant location

problem with heterogeneity. For each type vector z, L(z) contains

countries in which firms of this type should open a plant. For ex-

ample, if Germany is contained in L(z), then the binary decision for

Germany is clear. The firm will optimally open a plant there. Like-

wise, U(z) excludes countries where a plant should not be opened. If

Romania is not an element in this superset, then the binary decision

on Romania has also been determined. Overall, the iterative step uses

both monotonicity conditions to draw conclusions for some of the

binary choices to progress towards the optimal sets.

Nevertheless, the decision of whether or not to open a plant is still


undetermined for every country contained in U(z) but not in L(z).

The iterative step’s output remains a CDCP over these. However,

by Proposition 1, the resulting CDCP is (weakly) reduced compared

to the original in the sense that there are fewer binary decisions to

determine. For some type vectors, it may be that L(z) = U(z). In

those cases, the reduced CDCP is trivial, with no undetermined binary

decisions left. Then, the optimal set has been found. Conversely, in

areas of the type space where L(z) ⊂ U(z), a nontrivial CDCP is left.

Observe that Algorithm 2 therefore takes one CDCP and returns

a series of simpler ones, some even trivial. Why not simply reapply

Algorithm 2 to the nontrivial ones? For these returned CDCPs, the

iterative step will not simplify them further. To see why, observe that

the simplified CDCPs were themselves returned from Algorithm 2.

However, by step (iii)c., these are precisely the instances where the

updating step, Algorithm 1, makes no progress. Therefore, for the

reduced CDCPs that are nontrivial, another approach must be taken.

The next section addresses this issue with a branching step.


Branching step

The branching step builds on the reasoning provided by Arkolakis

and Eckert [2017], which outlines the procedure for the case of a single-

type. Since the solution algorithm presented in this chapter solves for

the policy function over the type space, the intuition must be extended

to cover the entire policy function.

Before describing the branching step formally, return to Figure 1.2.

It illustrates a possible outcome of the updating step in the simple

example. For firm productivities within (zLG, zU

G), neither the subset

nor superset were updated. In other words, the binary decisions on

both Germany and Romania remain undetermined, posing exactly

the challenge described at the end of the previous section.

Falling back on the brute force strategy is a possible but unattractive

option. Doing so would entail exhaustively guessing all possibilities

for both the Germany and Romania decisions, then comparing across

combinations. This approach does not take advantage of the CDCP’s

structure afforded by the monotonicity conditions. Consider instead

only guessing for the decision of whether or not to place a plant in


Germany. The solution method now creates two branches. One cor-

responds to the scenario where a plant is opened in Germany, while

the other represents the one where they are not. In either case, the

decision on the Romanian plant remains undetermined. Rather than

guessing for this one as well, as in the brute force method, the up-

dating step can now be reapplied. Each branch will then summarize

“locally optimal” behavior, conditional on the guess of whether or

not a German plant is established. The objective function must then

be compared across branches to find the globally optimal solution.

Reapplying Algorithm 2 in each branch therefore uses the monotonic-

ity assumptions to draw conclusions for as many choices as possible,

minimizing the number determined by guessing.

In practice, the two triples below would be defined, on which Al-

gorithm 2 is reapplied.

((zL

G, zUG), , R

) ((zL

G, zUG), G, G, R

)

While the intervals are the same in both, the subsets and supersets

are not. They reflect the guesses for the Germany decision. The first


triple corresponds to the branch where no plant is opened in Germany.

In that case, the subset contains no countries, while the superset con-

tains only Romania. This definition of the subset and superset rules

out opening a plant in Germany. On the other hand, the second rep-

resents the opposite case. A German plant is assumed to be opened,

so the subset contains Germany. Since the Romanian plant can still

be opened, the superset contains both countries. This subset-superset

pair likewise disallows the option of not opening a plant in Germany.

Then, Algorithm 2 can be iterated on both of these branches.

Figure 1.3 shows a possible outcome from this exercise. The top

row represents the policy function if no plant is opened in Germany,

J ∗(· | G → 0). Under this assumption, zG→0R emerges as the cut-

off value separating firms that should open a plant in Romania from

those that should not. Likewise, J ∗(· | G → 1) summarizes optimal

behavior if instead a German plant is opened. This branch identifies

a different productivity cutoff zG→1R regarding the Romanian plant.

The policy function in each branch must now be compared across

branches. Concretely, there are three relevant intervals. Each has its

own pair of optimal set candidates, one from each branch. As an


zLG zG→0

R zG→1R zU

G

J ∗(· | G → 0):J ∗(· | G → 1):

G

RG

RG, R

Figure 1.3.: A possible outcome from applying the recursive step once.

example, these are G and R for firms with productivity within

(zG→0R , zG→1

R ). Compared to the brute force strategy, where there are

four possible combinations to compare, the branching approach has

half the number of combinations.

In general, consider a triple (Z, L, U) emerging from Algorithm 2

where L ⊂ U.5 Selecting some undetermined item j ∈ U \ L, create

two branches. Proceed assuming that j is included in the optimal set

along one branch, while the other branch supposes it is not. In each

branch, reapply Algorithm 2 accordingly to fix as many of the remain-

ing choices as possible with the economic logic from the previous

section. Of course, in either branch, it may be that Algorithm 2 still

leaves some undetermined for some intervals. In those intervals, new

branches must be created in the same way. Formally, the branching

5The e superscript is dropped since these are considered one at a time.


step is defined as follows.

Algorithm 3 (Branching step). Given a triple (Z, L, U) that cannot be

further updated by Algorithm 2, but with L ⊂ U:

(i) Define two collections, C and S. The set C will contain triples for

recursion (“Continue”) while the set S will contain tuples where re-

cursion has completed (“Stop”).

(ii) Set C = (Z, L, U) and S = .

(iii) For each triple c ∈ C:

a. Remove c from C.

b. Let c = (Zc, Lc, Uc). Then, select an item j ∈ Uc \ Lc and create

two new triples

(Zc, Lc, Uc \ j) (Zc, Lc ∪ j, Uc) .

c. Apply Algorithm 2 to each, returning corresponding collections

Rj→0 and Rj→1.


d. For each r ≡ (Zr, Lr, Ur) ∈ Rj→0, check if Lr = Ur. When

they are equal, add r into S. If they are not, add r to C. Proceed

similarly for all r ∈ Rj→1.

(iv) Stop when C is empty. S will be a collection of tuples (Ze, Le = Ue ≡

J e)e. Form an “augmented policy function” returning the optimal

sets from the recursion’s branches:

J ∗(z) = J e | z ∈ Ze .

(v) Partition the original Z so that J ∗(·) is constant within each cell. For

each cell, determine J ∗(·) by using the objective function to globally

select among the sets in J ∗(·).

Note that, Algorithm 3 creates at most 2|U|−|L| branches with any

initialization. This case corresponds to a worst-case scenario where

applying the iterative step in each branch accomplishes nothing. Then,

every remaining decision j ∈ U \ L in the reduced CDCP must be de-

termined by guessing, ultimately reducing in this worst-case scenario

to brute force.


Full solution method

Following the intuition from Section 1.1, a generalized updating rou-

tine has been defined with Algorithm 1. It is the fundamental building

block of iterative Algorithm 2 and, in turn, branching Algorithm 3.

With these now specified, the solution method can be laid out in its

entirety.

Algorithm 4 (Full method). To find the policy functionJ ∗(·) for a CDCP

defined over P(J)6 and type space RNz :

(i) Perform Algorithm 2 on (Z, , J), returning R.

(ii) For each triple (Zr, Lr, Ur) ≡ r ∈ R:

(a) Check if Lr = Ur. If they are equal, set J ∗(z) = Lr for all

z ∈ Zr.

(b) If Lr ⊂ Ur, use Algorithm 3 on (Zr, Lr, Ur) for J ∗(·) over Zr.

(iii) Return J ∗(·) : Z→P(J).

6If some decisions in J are predetermined for some prior reason, then the problemcan be redefined over the smaller set of unknown elements.


At its core, the algorithm harnesses the intuition outlined in Sec-

tion 1.1 to update subsets and supersets, thereby ruling out sets that

are not optimal at each z. This intuition is formalized and generalized

in Algorithm 1. From there, Algorithm 2 simply embeds the updating

step in a standard iterative loop which converges to a fixed point on

the pair of functions L(·) and U(·). However, it is possible that this

fixed point falls short of defining the CDCP’s solution. Algorithm 3

includes the iterative loop in a larger branching procedure to handle

these cases.

Let the fixed points emerging from Algorithm 2 be L∞(·) and U∞(·).

In the worst-case scenario, Algorithm 2 could make no progress for a

region of the type space. That is, there could be some Z where L∞(z) =

and U∞(z) = J for z ∈ Z. Similarly, the worst-case scenario of the

recursion defined in Algorithm 3 simply creates a branch for every

undetermined decision. Taken together, the overall worst-case for

the full solution method amounts to computing the optimal policy

function by brute force over Z. In the simulation exercises presented in

Appendix B, such worst case scenarios does not occur in conventional

settings.


1.2.3. Method advantages

This text is the first to formulate the monotonicity in type condition

in the context of CDCPs. This condition is crucially leveraged when

solving explicitly for the policy function in scenarios with heteroge-

neous agents. The main conceptual contribution of this approach is

therefore to map an entire type space of arbitrary dimension to CDCP

solutions. No assumptions are placed on the distribution of agents

across types, allowing it to either be continuous or discrete. Moreover,

CDCPs can be solved exactly and quickly across the type space given

the structure afforded by monotonicity in type, opening the door to

general equilibrium analysis.

Since the method developed in this chapter imposes no restrictions

on the type distribution, it is more flexible than the existing meth-

ods described in Jia [2008] and Arkolakis and Eckert [2017]. These

were developed for use in settings with a small number of agents,

and therefore solve the CDCP for a single agent type at a time. The

key obstacle is that direct applicability in a heterogeneous agent set-

ting only occurs when there is a discrete distribution of firms over


types. In the firm’s plant location problem described in Section 1.1,

this restriction would amount to only allowing a finite number of

firm productivities z1, ..., zM. Then, the single-type method could

solve the CDCP for each possible productivity level zm. On the other

hand, a continuous support of firm productivities, common in the

international trade literature, must be discretized before using this

method. However, discretization is unattractive because it introduces

computational inaccuracy. If there is little guidance ex ante on where

grid points should be placed or the correct interpolation method be-

tween them, then aggregations over the firm distribution would be

inaccurate.

In contrast, the method developed here remains agnostic on the

distribution of firms. By simply mapping the type space to associated

optimal sets, it neither presupposes discrete supports nor does it create

discretization error for continuous ones. Additionally, in the special

case where agents are identical so the type space is a single point,

the solution method presented in this chapter collapses to the single-

type method. It is consequently broadly applicable, whether type

distributions are trivial, discrete, or continuous.


The key innovation enabling this flexibility is the monotonicity in

type condition. Incorporating an arbitrary number of heterogeneous

dimensions requires specifying a condition disciplining the CDCP’s

interaction with agent type. Otherwise, there is no way to relate an

agent’s type with their optimal solution to the CDCP. The monotonic-

ity in type condition (MT), while allowing for this structure, is also

relatively general and satisfied by existing heterogeneous agent CDCP

models in the literature.

The monotonicity on type condition not only enables this concep-

tual advantage, but can also bring dramatic improvements in compu-

tational speed compared to the alternative of solving CDCPs using a

single-type method at multiple points within a type space.7 The rea-

son is that the single-type methods also use monotonicity in set condi-

tions to make a squeezing argument. For the plant location example,

fixing productivity at a certain z, the single-type method would check

each country’s marginal value to the empty set at z. If it were negative

for any country, then that country would be discarded from consid-7A numeric comparison of computation speed and accuracy is explored in Sec-

tion 2.4.2, within the context of an estimated quantitative model of the EuropeanUnion.


eration. Similarly, if the marginal value of any country’s inclusion

into the full set were positive, then a plant would be opened there.

The process would update subsets and supersets, then iterate. While

this method may be feasible if there is a small finite number of types,

it cannot be generalized over a continuous support. However, het-

erogeneous agent models are the central motivation for the solution

method developed here.

The crucial observation is that only the sign of an element’s marginal

value to subsets and supersets is necessary, so calculating its precise

value for each z creates unnecessary calculations. For example, sup-

pose the empty set is chosen as the initial subset for each type z when

applying the single-type method. The strategy would then proceed by

evaluating the marginal benefit from the inclusion in the empty set of

an element like Germany, simply to check the sign. This calculation

would then be carried out for every point in the type space. Con-

versely, with the policy function approach, monotonicity in type is

leveraged to sign marginal values for entire intervals of the productiv-

ity range without repeated calculations. Finding indifference points,

like the cutoff zUG from the simple example, is equivalent to signing


the marginal value of Germany’s inclusion into the empty set for all

productivity values. Monotonicity in type implies that it is negative

for all productivities below the indifference cutoff and positive for all

above. Therefore, as long as finding these threshold productivities is

straightforward, there are considerable speed gains from eliminating

redundant calculations that would occur when repeatedly applying

the single-type method.

Finally, with continuous type spaces, the solution allows for both

speed and accuracy. In contrast, discretization methods entail an

accuracy-speed trade-off, governed by the density of the grid defined

over the type space. Section 2.4.2 in Chapter 2 provides a numeri-

cal illustration of this trade-off in a quantitative model. Interpolation

approximation is necessary to aggregate a continuous distribution

over a collection of discrete grid points. Then, sparser grids are as-

sociated with fast compute times but inaccurate aggregations, while

the opposite is true for dense grids. However, the proposed solu-

tion method leverages monotonicity in type to solve the exact policy

function quickly, eliminating the need to choose between accuracy

and speed. Both are necessary for computing general equilibrium,


since the entire distribution of firm behavior must accurately be solved

many times while searching for aggregates. Therefore, this method

enables the general equilibrium nature of the analysis in Chapter 2,

including an estimation strategy matching on aggregate variables. In

comparison, existing studies which have applied single-type meth-

ods to a discretized grid. Notably, Tintelnot [2016] and Antràs et al.

[2017] employ this strategy to quantitatively evaluate CDCPs for a

continuous distribution of firms. They not only hold many general

equilibrium variables constant in counterfactuals, but also rely on

detailed micro-data for parameter estimation.

1.3. CDCPs in trade

This section presents a collection of common questions in the interna-

tional trade literature which give rise to CDCPs. In particular, models

for plant location decisions, input sourcing decisions, and export mar-

ket entry are formulated. A discussion follows after each model is

described, showing how the CDCPs arising in each naturally obey

the monotonicity conditions from Section 1.2.1. Since the problems


are varied, they provide examples of both substitutability and com-

plementarity among items.

1.3.1. Plant location decision

The firm’s plant location problem with export platform FDI can give

rise to a CDCP featuring substitutability. This example has been con-

ceptually used as the context for the chapter thus far. Since a structural

model of this type is precisely presented in Chapter 2, the interested

reader is referred there. This section instead explores a firm’s location

CDCP featuring complementarity, which arises from a vertical FDI

structure.

Production comprises of a set of necessary tasks, which can be sep-

arated from each other and carried out in different locations. A firm

with multiple plants therefore chooses the location to complete each

task. The problem’s central characteristic is that establishments differ

in productivity task by task. Given task-level productivity dispersion

and resulting task specialization, the firm regards the establishments

as complements. As a result, when choosing the location of its estab-


lishments, the firm faces a CDCP with complementarity among its

items.

Demand

Consumers are assumed to have preferences over a continuum of

differentiated goods, which are imperfectly substitutable at a constant

elasticity of ρ. Indexing the goods by ω, total expenditure on any

given good is

p(ω)q(ω) = X(

p(ω)

P

)1−ρ

P =

(∫Ω

p(ω)1−ρdω

) 11−ρ

where Ω summarizes all goods available, P defines the aggregate price

index, and X is total expenditure. While X is an aggregate variable,

its determination has no effect on the salient features of the firm’s

problem discussed below. As such, its determination is not specified

here.


Firm problem

Each firm produces a unique good ω, so that every ω corresponds

with a single firm. In what follows, the problem of a single firm is

considered, so the ω is omitted when clear. Since there is a continuum

of firms, they compete monopolistically. Optimal pricing behavior is

therefore set to a constant markup ρ/(ρ− 1) over marginal costs.

Production consists of a sequence of stages. In each stage, input

from the previous stage is transformed into the input needed for the

next stage. For example, the production chain of a bakery firm may

first require wheat seeds as a fundamental input. The first stage con-

verts wheat seeds to harvested wheat by farming. Next, the wheat is

transformed by a milling technique into flour, which in turn becomes

bread when combined with water and baked. In this example, there

are three transformative tasks: farming, milling, and baking. Each

converts its input into a different output. The first and last steps aside,

the output for one is the input for the next, creating a production chain.

For the rest of this section, the input for the first stage will be called

the “fundamental input”, while the output from the final stage is the


final good.

Index the tasks by n = 1, . . . , N, and the available production loca-

tions by j = 1, . . . , J. These production locations could be the set of

countries in which the firm can open establishments. Suppose that

N ≥ J, implying that there are more tasks than countries. Each pro-

duction location j can complete task n by converting one unit of its

input into aj(n) units of its output. For simplicity, define these output

functions as

aj(n) = 1 + 1[j = n]λj λj > 1 .

Each country is adept at a unique task, with tasks and countries appro-

priately labeled so that country 1 is good at task 1, country 2 is good

at task 2, and so on. This pattern of productivity may reflect each

country’s inherent factor endowments, local knowledge, or other fun-

damentals. For example, a country’s sunny weather could make it

especially good at farming.

If a task is carried out in a country j that is not good at, its input is

simply converted into its output one for one. However, if completing


a task in the country with the productive advantage, one unit of input

to is transformed into 1 + λj units of output. Here, λj measures the

extent to which country j has a productive advantage in task n = j

compared to the other countries. Of course, there could be some tasks

that all countries perform equally well because there are weakly more

tasks than countries.8

The firm must choose where to complete each task n, selecting

among its production locations. It is easy to see that the firm prefers

to assign task n to the establishment in j = n if it has one. Otherwise,

it is indifferent where the task is completed. Following this logic, the

firm with plants in the set of countries J can convert one unit of fun-

damental input into

a(J ) = ∏j∈J

(1 + λj) (1.9)

units of final output. This function summarizes what will be called a

set J ’s “production capability”. Choosing the price of the fundamen-

8It is possible for N < J, but that leaves some countries without a specialty. As itwill become clear below, firms will have no incentive to operate in these areas.


tal input as the numeraire, the firm’s unit cost is simply 1/a(J ).

Finally, to open a plant in country j, the firm must pay a flat cost

f j/z, where z describes the firm’s productivity in opening plants. For

example, the fixed cost may represent legal and administrative costs

associated with registering a business in a new country or the cost of

ensuring compliance with the country’s regulations. Then, z could

be interpreted as the firm’s efficiency at completing these regulatory

procedures. Incorporating these fixed costs, the firm’s overall profit

function is

AXPρ−1a(J )ρ−1 − ∑j∈J

f j

z

where A is a constant. Maximizing profits with respect to J defines

the firm’s CDCP. Accordingly,

Dj(J ; z, υ) = AXPρ−1[

a(J ∪ j)ρ−1 − a(J \ j)ρ−1]−

f j

z

encapsulates the marginal value of including a plant j into the setJ . It

captures the increased variable profits afforded by the cost reductions

from an additional plant in j, but trades it off against the fixed cost


necessary to establish the plant. Here, the vector of aggregates rele-

vant to the firm decision υ = X, P contains aggregate expenditure

and the price index.

Discussion

The first observation to make is that plants are complements in the

firm’s production capability given by (1.9). In other words, the gain

to production capability when adding a plant to J increases as J

grows. The complementarity arises from the vertical nature of pro-

duction, since the output from one step is used as input for the next.

Then, as the firm opens more plants, each additional plant grows more

attractive from a cost perspective. Fundamentally, adding a plant in j

is valuable because the firm can then extract λj more output from step

n = j. The ability to produce this extra λj is more valuable if the firm

has other productive specialized plants, regardless of whether they are

upstream or downstream from j. Suppose they are upstream. Then,

they increase the input for task j, so the extra λj factor generates more

output than if the firm did not have the specialized plant upstream.

Likewise, if the firm has productive specialized plants downstream


from j, then the additional λj increases input for the downstream tasks.

The downstream plant can therefore get more out of its own special-

ization. The plant in j therefore increases total output by more than if

the firm did not have the specialized downstream plant.

As long as ρ > 1, so that there is substitutability between goods in

consumer preferences, complementarity among plants persists in the

firm’s profit function. The reason is that substitutability in demand,

even imperfect, implies that consumers are price sensitive with down-

ward sloping demand curves. As a result, the firm gains market share

if it decreases prices. Since plants are complementary in reducing

costs and prices are a constant markup over prices, they are also com-

plements in reducing prices. In turn, they are complements in raising

profits. Given this logic, the firm’s plant location problem features

monotonocity in set of the complements type (MS-C).

Next, the CDCP satisfies the monotonicity in type (MT) condition

since the marginal value function is increasing in the firm’s adminis-

trative capability z. As these grow, the fixed cost of establishing plants

in new countries monotonically declines. Profit gains from opening

new plants necessarily declines as z grows, since the new plants con-


tribute to variable profits as before but cost less to set up. In fact, it is

easy to find the administrative productivity level z∗j (J ; υ) at which

firms are indifferent between adding j toJ . Rearranging the marginal

value function,

z∗j (J ; υ) =f j

AXP1−ρ

[a(J ∪ j)ρ−1 − a(J \ j)ρ−1

]−1

solves the indifference condition closed form. Then, the solution

method outlined by Section 1.2.2 is easily applicable and can be useful

if solving this plant location problem for a distribution of heteroge-

neous firms.

1.3.2. Input sourcing

Another extensive margin discrete choice that has been studied in

the trade literature has been the firm’s input sourcing problem. In

particular, Antràs et al. [2017] and Antràs [2016] examine a firm’s entry

into import markets. In this setting, firms requires a unit continuum of

inputs, each of which can be sourced from import markets. However,

the variable cost of importing a specific input from one import market


may differ from the variable cost of importing the same input from

another import market. Conditional on having entered a set of import

markets, the firm therefore chooses to source each input from the

import market offering it at the lowest variable cost. Then, in the

first stage, the firm must decide which import markets to enter. This

decision is the firm’s CDCP.

The model is sketched below, followed by a discussion of its im-

portant features. In particular, whether or not the import markets are

substitutes or complements depends on the structural parameters of

the model. A more detailed discussion is provided once the model

has been fully specified. Regardless, monotonicity in set is satisfied in

this setup. In addition, the sourcing problems in these works exhibit

monotonicity in type, since firm are heterogeneous in productivity. In

other words, higher productivity firms derive a larger benefit from

access to an additional import market compared to lower productivity

firms.


Demand

Consumers are described first. Households exhibit a love for variety,

aggregating differentiated goods with a constant elasticity of substitu-

tion ρ. Indexing goods by ω, the utility function follows the standard

form (∫Ω

q(ω)ρ−1

ρ dω

) ρρ−1

where Ω contains all goods sold and consumed. Then, if the house-

hold has a total budget of X, household expenditure on any single

good ω can be determined by utility maximization as

p(ω)q(ω) = X(

p(ω)

P

)1−ρ

P =

(∫Ω

p(ω)1−ρdω

) 11−ρ

where P is the aggregate price index. The household budget X is an

aggregate variable whose determination is orthogonal to the firm’s

CDCP described below.


Firm problem

A continuum of firms each hold the blueprint to make one unique

good ω, so that they compete monopolistically with each other. This

section describes the problem of a single firm, so the ω is omitted in

what follows.

Production requires a unit continuum of intermediates indexed by

m, which are imperfect substitutes for each other. They are combined

to form the final good with constant elasticity of substitution s. Firms

source these inputs from input partners, which each produce the en-

tire continuum of intermediates. However, the cost of sourcing a par-

ticular input m may vary from input partner to input partner. Dif-

ferences in costs may reflect the capability of each input partner j to

produce a specific intermediate m. For example, in Antràs et al. [2017],

the input partners are interpreted as import markets.9 Then, disper-

sion in the cost of sourcing input m from import partner to import

partner could reflect differences in comparative and absolute advan-

tages across countries. Accordingly, let the cost of sourcing input m

9Similarly, the input partners could be interpreted as firms if studying firm net-works.


from partner j be denoted cj(m).

Suppose the firm has linkages with a set of input providersJ . Then,

for each input m, it simply chooses the provider j among its set J

offering it at the lowest price. In other words, the cost of sourcing

input m for the firm is c(m) ≡ minj cj(m). Putting it together, the

overall cost of producing one unit of the final good can be written

c =1z

(∫ 1

0c(m)1−sdm

) 11−s

where z is the firm’s innate productivity.

The firm must also choose the set of input suppliers J with which

to establish partnerships. Establishing a partnership with j entails

a fixed cost f j, but allows the firm to source inputs from supplier j.

The firm cannot source from suppliers that it does not have a partner-

ship with. To characterize this problem sharply, suppose that the in-

verse input-partner costs 1/cj(m)m are drawn independently from

a Fréchet distribution with shape θ and scale Tj once partnerships are

established. The associated CDF is F(x) = e−(x/T)−θ . Broadly, the

shape parameter θ governs the draw variance, while the scale Tj gov-


erns the mean. Then, countries with a higher scale Tj offer inputs at a

lower price on average.

Because the costs cj(m) are not drawn until after partnerships are es-

tablished, the firm does not observe the realized draws while making

its decision. However, since there is a unit continuum of intermedi-

ates, a law of large numbers argument implies that realized overall

outcomes will equal expected outcomes. In particular, if the firm has

linkages with suppliers J , then its cost of production is

c =1z

Γ(

θ + 1− sθ

) 11−s[

∑j∈J

Tθj

]− 1θ

≡ 1z

Γ(

θ + 1− sθ

) 11−s

Θ(J )−1θ

where the properties of the Fréchet distribution have been used.10

The Θ(·) is similar to terms found in many gravity models using the

Fréchet distribution.11 In this setting, it represents a set J ’s “cost ef-

fectiveness potential”. In particular, it combines the overall efficiency

Tj of each partner with which the firm establishes a connection.

10A similar cost function emerges in Rodríguez-Clare [2010], which studies produc-tion fragmentation and off-shoring with a Leontief production function.11Eaton and Kortum [2002] provides a deeper discussion on the Fréchet distribu-tion.


Like many other trade models employing the Fréchet distribution,

the addition of more partners mechanically increases the cost effec-

tiveness potential of a set of establishments J , thus lowering unit

costs. However, the central observation is that it does so in a concave

way, since the exponent |1/θ| < 1. This concavity implies that input

partners are substitutes in terms of unit cost savings. Intuitively, part-

ners can all supply every intermediate, with the firm choosing among

them for each intermediate.

In addition, the higher the θ, the less substitutability there is be-

tween partners in terms of unit costs. Since θ inversely governs the

dispersion of Fréchet draws, a low θ corresponds to high variance cjt

draws. The partners are therefore less substitutable to the firm in this

case, because the firm selects the least expensive partner to supply

each input. With very different cost draws, the firm cannot substitute

a low-cost supplier very well with a high-cost supplier. However, the

firm does not choose partners in order to minimize unit costs, but to

maximize profits. It is therefore still possible for the input sourcing

problem to feature complementarities among the partners, as long as

they are complements in generating profits. To this end, firm profits


are described next.

Since it is in a monopolistic competition environment, the firm will

set a constant markup ρ/(ρ− 1) over unit costs. This pricing strategy

generates variable profits AX(zP)ρ−1Θ(J )ρ−1

θ , where A is a constant.

Finally, incorporating the fixed costs necessary to establish a sourcing

relationship, the firm’s objective function is written as

maxJ

AX(zP)ρ−1Θ(J )ρ−1

θ −∑J

f j (1.10)

with associated marginal value function

Dj(J ; z, υ) = AX(zP)ρ−1[Θ(J ∪ j)

ρ−1θ −Θ(J \ j)

ρ−1θ

]− f j .

In this example, the aggregate vector collecting the relevant aggre-

gates for the firm’s CDCP is υ = (X, P).

Discussion

As long as ρ− 1 > θ, the exponent on Θ(J ) exceeds 1, corresponding

to convexity on J ’s cost effectiveness potential Θ(J ). In this case, an


additional partner j will increase variable profits by more as J grows.

In other words, adding partners is relatively less valuable when the

firm’s set of partners is small, but relatively more valuable as the set

grows. On the other hand, if ρ − 1 < θ, then variable profits grow

with cost effectiveness potential in a concave way. Then, partners add

more value when the firm does not have many partnerships.12

How can the input partners be complements in generating profits

when they are substitutes in terms of unit cost? The main reason is

curvature in consumer demand. The firm’s market share is propor-

tional to p1−ρ, where p is the price charged by the firm. Then, if the

firm’s price is high, small changes to price do not win very much more

market share. However, when the firm has a low price and commands

a large market share, the same price reduction will increase market

share by more. When consumer demand is price-sensitive enough, i.e.

when ρ− 1 > θ, this force dominates. In this case, partnerships are

complements in firm profits. Thus, the firm faces a CDCP satisfying12As observed in Antràs et al. [2017], readers familiar with Eaton and Kortum[2002] may expect that ρ − 1 < θ is necessary. However, the parallel restrictionin this exposition is s− 1 < θ, since unit costs require calculating the (s− 1)th mo-ment of a Fréchet distribution. There is consequently no restriction on ρ in relationto θ.


complementary monotonicity in set (MS-C). The higher the ρ, or the

more price-sensitive consumers become, the higher the complemen-

tarity. Likewise, when ρ− 1 < θ, then different goods are not very

substitutable in demand. Consumers are not price-sensitive enough

to outweigh the fact that input partners are substitutes in reducing

unit cost. In this case, the firm views the partners as substitutes in

generating profits. Then, the CDCP satisfies the substitutes version

of monotonicity in set (MS-S).

Finally, examination of Dj in this setting confirms that the CDCP

obeys monotonicity in type (MT). Since variable profits are linear in

zρ−1, profit gains from unit cost reductions are magnified for more

productive firms. It is straightforward to solve for indifference cut-

offs with this functional form. For example, to find the productivity

z∗ of firms that are indifferent between entering into German import

markets G and entering no import markets at all, rearrange the ex-

pression for the marginal value of including Germany in the empty


set as follows.

z∗ =1P

(fG

AX

) 1ρ−1 [

Θ(G)ρ−1

θ −Θ()ρ−1

θ

] 11−ρ

This expression gives a simple closed form way to find the produc-

tivity of the indifferent firms, with other indifference cutoffs defined

similarly.

The input sourcing problem outlined in this section is therefore

monotonic in set, with complementarity between partners if ρ− 1 > θ

and substitutability otherwise. It is also monotonic in type, with indif-

ference cutoffs easily solved in closed form. The algorithm outlined

in this chapter is readily applicable, and could be valuable if studying

a block of these heterogeneous firms in a general equilibrium environ-

ment.

1.3.3. Export market entry

Finally, this section lays out an example of market entry decisions

generating CDCPs with substitutability among the markets. These

markets could be a collection of export markets as well as the home


market. Broadly, entering each market entails a fixed cost in exchange

for increased market access, in a similar setup to Melitz [2003]. How-

ever, the firm faces decreasing returns to scale production, so serving

more destination markets increases marginal costs. In this environ-

ment, the increasing nature of marginal costs is the source of the sub-

stitutability among markets, as the firm balances increased market

access against increased marginal costs.

A similar intuition is explored in Almunia et al. [2020] as a way to ex-

plain the fact that Spanish exports grew dramatically during the Great

Recession. The paper shows that Spanish firms with more adversely

impacted domestic demand increased exports relatively more than

firms with better domestic shocks. Cushioning the domestic demand

drop with a “vent-for-surplus” suggests substitutability between the

home and export markets. Finally, the study argues that increasing

marginal costs are consistent with results emerging from the empirical

analysis. The structural analysis there shares this key element with

the model below, as well as heterogeneity arising from firm-specific

demand shifters.


Demand

Consumer demand in each market is first outlined. In each market k,

households value variety, aggregating goods with a constant elasticity

of substitution ρ. Indexing goods by ω, their utility is

(∫Ωk

ξ(ω)1ρ qk(ω)

ρ−1ρ dω

) ρρ−1

where Ωk collects the set of goods available for sale in k and ξ(ω) is a

demand shifter for good ω. This demand shifter is described further

as part of the exposition below of the firm problem. Then, using the

properties of the CES function, utility maximization implies that the

quantity demanded of any single good is

qk(ω) = ξ(ω)XkPk

(pk(ω)

Pk

)−ρ

Pk =

(∫Ωk

ξ(ω)pk(ω)1−ρdω

) 11−ρ

where Pk is the CES-accurate price index and Xk is total expenditure

in k. Total expenditure Xk can be specified with a model-dependent

aggregate condition independently from the firm problem described

below.


Firm problem

Firms each produce a single differentiated good and therefore each

correspond to one ω. They compete monopolistically in each market

they enter. In what follows, the problem of a single firm is discussed,

so the ω indexing is omitted for notational brevity.

If the firm were to serve a market k, it would earn ξXk(pk/Pk)1−ρ in

revenues. Variation in ξ is therefore the source of firm heterogeneity

in this model and could represent a firm-wide factor like global brand

appeal. Finally, suppose the firm has a convex total variable cost

function C(·), in accordance to decreasing returns to scale production.

Conditional on serving a set of marketsK, the firm’s problem is simply

to choose prices pk for each destination market k ∈ K. Total variable

profits in case this would be

∑k∈K

ξXk

(pkPk

)1−ρ

− C(Q), Q = ∑k∈K

ξXkPk

(pkPk

)−ρ

.

Note that the firm takes into account how its own prices affect its quan-

tity demanded in each market k. Given this setup, the firm optimally

chooses a constant markup ρ/(ρ− 1) over its marginal cost C′(Q). Be-


cause the production technology features decreasing returns to scale,

variable profits are positive. Then, total production is defined implic-

itly by

Q = ∑k∈K

ξXk

(ρ

ρ− 1C′(Q)

Pk

)1−ρ

.

For concreteness, suppose the variable cost function takes the form

Q1+d, where d > 1 parameterizes the strength of decreasing returns

to scale in production. Then, total variable profits can be solved for

explicitly in terms of aggregates as follows.

[ρ(1 + d)

ρ− 1− 1]

Q1+d, Q =

[∑

k∈Kξ

XkPk

(ρ

ρ− 11 + d

Pk

)−ρ] 1

1+ρd

.

Note that, since production is decreasing returns to scale, variable

profits are positive.

Having solved the firm’s problem given entry in a set of markets

K, now the optimal set to enter must be determined. Suppose each is

associated with a fixed cost of entry fk. Putting everything together,


the firm must choose K maximizing total profits, given by

Aξ1+d

1+ρd Θ(K)1+d

1+ρd − ∑k∈K

fk, Θ(K) = ∑k∈K

XkPρ−1k

where A is a positive constant. Here, the Θ(·) function summarizes

the profit potential of a set of markets K. It combines a market’s size,

given by total expenditures Xk, as well as its competitiveness, mea-

sured by the aggregate price index Pk. Unsurprisingly, markets that

are large or not very competitive on price increase profit potential by

more than markets that are small or have low prices on average. Simi-

larly, adding a market to Kmechanically increases its profit potential,

since the firm simply has access to more consumers.

Accordingly, the marginal value of including a market k into K is

given by the difference

Dk(K; ξ, υ) = Aξ1+d

1+ρd

[Θ (K ∪ k)

1+d1+ρd −Θ (K \ k)

1+d1+ρd

]− fk .

Observe that, in this case, the vector of relevant aggregates υ collects

all expenditures Xkk and price indices Pkk, which are necessary


to calculate profit potential. The marginal value function captures

how entering the additional export market increases variable profits

through increased market access, while also entailing a fixed cost of

access fk.

Discussion

The key exponent in Dk(·; ·, υ) is (1 + d)/(1 + ρd), which raises both

the set’s profit potential and the firm’s demand shifter. Observe that

this exponent is between 0 and 1. How does it affect the relationship

between profit potential and the marginal value of adding a market

k? Since it is below 1, profits are concave in profit potential. All else

equal, adding a new export location therefore increases variable prof-

its by more when K contains few elements compared to many. Intu-

itively, market entry is only valuable to the firm because it can access

more consumers. However, the firm must produce and sell to those

additional consumers to earn profits. When marginal costs are in-

creasing in total quantity produced, then producing to earn profits

in one market will increase the marginal cost of production for all

other markets, decreasing the firm’s profitability elsewhere. More-


over, marginal costs are convex, growing more quickly as production

increases. When the firm serves many markets, production is high,

so the added production for a new market is extremely costly. Conse-

quently, the market entry problem presented here is a CDCP featuring

substitutability among markets, satisfying (MS-S).

The exponent can in fact be interpreted as inversely summarizing

the degree of substitutability among the export destinations. In the

extreme case where the exponent is 1, there is no interdependence

among export destinations — entering an export market k has the

same effect on profits regardless of the choices made for all other ex-

port markets. In other words, there is no substitutability between

markets. On the other hand, as the exponent approaches 0 from the

right, then variable profits flatten very quickly, with additional market

access increasing variable profits by very little as the markets become

more substitutable.

Further, the exponent is determined by ρ and d, which are the struc-

tural parameters governing the elasticity of demand and the convex-

ity of marginal costs. Then, markets are more substitutable when ρ

is high, corresponding to scenarios where demand is very elastic. In


this case, the firm earns lower profits when entering and producing

for any given market, all else equal. Especially when weighed against

the disincentive of higher marginal costs for all markets, entering new

markets is much less attractive compared to when ρ is high. In a simi-

lar vein, markets appear more substitutable to the firm when d is high,

since marginal costs grow at a fast rate.

In addition, the marginal value of including k in K is increasing

in the firm’s demand shifter ξ. This shifter is the firm’s idiosyncratic

characteristic, and so the problem satisfies monotonicity in type (MT).

Intuitively, the demand shifter increases the firm’s market share, all

else equal. Then, the benefit of entering any market will be corre-

spondingly higher for a firm with high ξ compared to an otherwise

identical firm with low ξ. Moreover, the marginal value function is lin-

ear in ξ(1+d)/(1+ρd), so it is possible to find indifference cutoffs closed

form. In particular, the value ξ∗k (K; υ) leaving firms indifferent be-

tween entering market k when they participate in markets K is

ξ∗k (K; υ) =

(fkA

) 1+ρλ1+λ

[Θ (K ∪ k)

1+λ1+ρλ −Θ (K \ k)

1+λ1+ρλ

]− 1+ρλ1+λ

,


derived by rearranging the marginal value function.

To summarize, the market entry problem presented here satisfies

both the monotonicity set (MS-S) and monotonicity in type (MT) con-

ditions necessary to apply the algorithm. Furthermore, there is a

closed form expression for indifference conditions, so it can be easily

operationalized.

1.3.4. Idiosyncratic fixed costs

In the previous examples, agent heterogeneity was one dimensional,

describing either firm productivity or brand appeal. They each also

feature fixed costs of adding an element into the set, which are the

focus of this section. In particular, this section analyzes a setting with

multidimensional heterogeneity in the form of idiosyncratic fixed

costs. This type of heterogeneity is common in quantitative exercises

from the international trade literature, popularized by Eaton et al.

[2011].

While all the examples above feature fixed costs and can therefore

be extended to allow for idiosyncratic fixed costs, the input sourcing


problem will be used as an example here. This problem is chosen

to serve as the example since it has many elements in common with

the quantitative model from Antràs et al. [2017], which incorporates

idiosyncratic fixed costs in the numerical analysis. However, the dis-

cussion below readily translates to the other problems described thus

far.

Recall that the firm’s objective is to choose a set of input sourcing

partners J to maximize total profits given by (1.10). The firm does so

by trading off the added potential cost benefit of the extra partner with

a fixed cost of establishing the relationship. In doing so, the firm takes

into account its own idiosyncratic productivity z, which magnifies the

extra partner’s potential cost benefits. Consider a slight modification

to the firm’s problem, where the fixed costs f j are also firm-specific.

The firm’s maximization problem is to choose J maximizing

AX(z(ω)P)ρ−1Θ(J )ρ−1

θ −∑J

f j(ω) .

The ω notation is reintroduced to index the firm, to explicitly de-

note which variables are idiosyncratic. This maximization problem is


equivalent to

maxJ

AXPρ−1Θ(J )ρ−1

θ −∑J

f j(ω)z(ω)1−ρ

where profits have been divided through by the firm specific produc-

tivity. Letting hj(ω) = − f j(ω)z(ω)1−ρ, the firm’s relevant charac-

teristics can be summarized by the vector h(ω). Each hj(ω) can be

interpreted as the inverse fixed cost of establishing a partnership with

j, adjusted by firm-wide productivity. There are therefore |J| dimen-

sions of heterogeneity in this CDCP, one for the adjusted fixed cost of

each possible partner j.

The marginal value function in this case can now be written

Dj(J ; h, υ) = AXPρ−1[Θ(J ∪ j)

ρ−1θ −Θ(J \ j)

ρ−1θ

]+ hj

where the ω indexing has now been omitted. The CDCP still satisfies

monotonicity in type by the same argument as in Section 1.3.2. The

marginal value function is also increasing in each hj, satisfying the suf-

ficient condition for monotonicity in type with multiple dimensions.


Then, it is possible to apply the full algorithm from Section 1.2.2.

To solve for the firm types indifferent between adding a partnership

with j when it considersJ , find the vectors h∗ so that Dj(J ; h∗, υ) = 0.

Since there are |J| dimensions of heterogeneity but only one indiffer-

ence condition, there are |J| − 1 degrees of freedom. In other words,

the indifferent firm types lie on a (|J| − 1)-dimensional plane which

divides R|J|+ in two. Since the problem exhibits monotonicity in type,

firm types on one side of the plane will have positive marginal value

from including j inJ while firms types on the other will have negative

marginal value.

A key observation to make is that, even though it is possible for the

marginal value function to incorporate all entries of the type vector in

general, only hj appears with the fixed cost formulation. That is, when

considering the marginal value of adding j to J , only hj is directly

relevant while all other hj′ are not. Intuitively, the marginal value

of establishing a relationship with sourcing partner j trades off its

implied variable profits gains with its fixed cost. The fixed cost of

establishing other relationships does not enter this trade-off explicitly.

The marginal value of including j in J can therefore be written more


simply with hj as an argument instead of the full type vector h. Let it

be denoted by Dj(J ; hj, υ).

Now, finding the firm types indifferent to adding j to J can be ac-

complished by finding h∗j so that Dj(J ; h∗j , υ) = 0. Then the (|J| − 1)-

dimensional plane summarizing the firm types indifferent to adding

j to J takes a simple form. It is given by a particular value of h∗j ,

with the adjusted fixed costs of other partners hj′ taking on any value.

Firms with hj > h∗j derive a positive marginal value for j when added

to J , while firms with hj < h∗j will receive a negative marginal value.

Notably, when the marginal value function only involves a single di-

mension of heterogeneity at a time, the indifference conditions can

once again be expressed as simple cutoffs. Moreover, in this case with

fixed costs, the marginal value function is easy to rearrange since it is

linear in hj. Then,

h∗j (J ; υ) = −AXPρ−1[Θ(J ∪ j)

ρ−1θ −Θ(J \ j)

ρ−1θ

]

defines the firm types indifferent between adding j to J .

For concreteness, Figure 1.4 shows an example of the outcome after


(0, 0)hL

G hUG

hG

hLR

hUR

hR

G

GG

R

G, R

G, RG

RR

G, RR

G, RG, R

Figure 1.4.: An example of subsets, supersets, and resulting subinter-vals after one iteration.

Supersets are printed above subsets for each region of the type space. Firmsabove hU

G should enter Germany while those below hLG should not. Similarly,

firms above hUR should enter Romania while those below hL

R should not.

one algorithm iteration with substitutability between two possible

import markets, called Germany and Romania. The type space is two

dimensional, as firms vary in hG and hR. The inverse fixed cost for

entering the German import market is graphed along the horizontal

axis while the inverse fixed cost for entering the Romanian import

market is shown on the vertical axis. The type space is divided into

nine regions, each with constant superset and subsets. The supersets

are displayed above the subsets.


Recall the iteration begins with a constant subset and superset

G, R for the entire type space. They are updated by including el-

ements into the subset and removing them from the superset. Be-

gin with Germany and identify the firm types indifferent between

entering the German import market and entering none. These firms

have the adjusted fixed cost of entering Germany hG = hLG so that

DG(; hLG, υ) = 0. For all firms with hG < hL

G, they definitely de-

rive negative marginal value from Germany since import markets

are substitutes. As a result, G is removed from the superset for all

regions to the left of hLG. Likewise, identify the firm types indifferent

between participating both German and Romanian import markets

compared to only participating in Romanian import markets. These

have hG = hUG where that DG(G, R; hU

G , υ) = 0. Following the logic

of the algorithm, all firms with hG > hUG derive positive marginal

value from entering German import markets when also participating

in Romanian markets. They definitely include G in their optimal set,

so G is added to the subset for all regions to the right of hUG .

A similar argument holds for entering the Romanian import market.

All firms with hR < hLR should not enter while all firms with hR > hU

R


should. Accordingly, R has been removed from the superset for the

bottom row of regions and added to the subset for the regions in the

top row.

Finally, observe that the regions are rectangles when there are two

dimensions of heterogeneity. In particular, each region has a con-

stant lower and upper bound on hG. Likewise, they maintain con-

stant lower and upper bounds on hR. For example, the lower left

region is summarized succinctly by the two conditions hG ∈ [0, hLG]

and hR ∈ [0, hUG ]. The reason for this simple shape is because the

marginal value function only includes one dimension of heterogene-

ity at a time, allowing the indifference conditions to maintain their

cutoff structure. The same general shape will appear if there are more

than two countries. Regions will be defined parsimoniously by a set

of |J| conditions hj ∈ [h`j , hrj ], one for each country in j. Therefore, the

shape can easily be stored numerically as a 2|J|-length vector, further

simplifying the application of Section 1.2.2’s method.

In summary, this example illustrates how the method from this

chapter can be easily applied to a CDCP characterized by idiosyncratic

fixed costs. Given the way fixed costs enter the profit function, the


marginal value function of j’s inclusion to J only incorporates the

fixed cost of adding j. Crucially, the fixed cost of adding any other

element j′ 6= j does not appear. As a result, indifferent firm types

are simply identified as cutoffs values. The type space is therefore

divided into cells with constant left and right boundaries along each

dimension given by these cutoffs, allowing them to be stored easily

during computation.

1.4. Conclusion

A major challenge in international trade and industrial organization

is solving combinatorial discrete choice problems in settings with het-

erogeneous agents and a high dimensional state space. This chapter

has presented a unified strategy for solving CDCPs in such heteroge-

neous agent settings based on explicitly solving for the policy function

in terms of an agent’s type. This approach flexibly handles any dis-

tribution of agents across types, an arbitrary dimensional type space,

and either substitutable or complementary discrete choices.

Moreover, it is both faster and more accurate than methods cur-


rently employed in the literature. Overall, these advantages render it

especially suitable for aggregating agent behavior in models featuring

heterogeneous agents facing CDCPs. Calculating these aggregates are

in turn necessary when computing or estimating quantitative general

equilibrium models. As this algorithm explicitly solves for the agent’s

policy function, it is straightforward to embed within a general equi-

librium fixed point problem. The algorithm’s computational speed

also lends itself to inclusion within parameter estimation routines,

where it is especially useful for identification strategies targeting ag-

gregate moments.

The strategy explicitly solving for the policy function relies on two

structural assumptions governing the combinatorial discrete choices

and how they interact with agent type. These properties have natu-

ral economic interpretations, so arise in many economic settings. The

first requires the individual discrete choices to interact in a way where

they are either substitutes to each other or complements. Respectively,

submodularity or supermodularity among the decisions are sufficient.

The second asserts that the value of any decision is monotonically re-

lated to agent type. Intuitively, this condition assumes a type of super-


modularity between the choices and the agent type. Both conditions

are satisfied by the discrete choice problems investigated across the

literature.

The method is applicable to a wide array of questions in interna-

tional trade and industrial organization. This chapter presents models

arising in CDCPs when firms heterogeneous in efficiency or brand ap-

peal choose locations for their plants, input partners with which to

form partnerships, or export markets to enter. In each of these, the two

necessary structural assumptions are naturally satisfied in an econom-

ically intuitive way. For example, plants interactions are complemen-

tary under a vertical FDI structure with production chains. They are

therefore supermodular in the profit function. Likewise, since more

efficient firms can produce more with the same inputs, there is su-

permodularity between plants and firm efficiency. A similar intuitive

argument holds for input partners and export markets. Moreover, the

algorithm is easily applicable in practice for each of these problems.

The reason is that profits are simple functions of the differentiated

firm characteristics, even after introducing idiosyncratic fixed costs

as is common in the quantitative heterogeneous firm trade literature.


Finally, the unifying element in the problems discussed above is

that the firms must make a number of extensive margin decisions

which interact to determine total profits. In the plant location prob-

lem, each country in which it is possible place a plant represents a

single extensive margin decision: should a firm open plants there or

not? Therefore, the method explored in this chapter is also applicable

to analyzing similar problems over a collection of extensive margin

decisions such as a multigood firm’s decisions on which products

to produce, or the portfolio choice problem of an investor that must

choose among asset classes.

Chapter 2.

European Union corporate tax equalization

with Kathleen Hu

This chapter applies the method outlined in Chapter 1 to the policy

problem of evaluating a corporate tax equalization in the European

Union. One of the main goals of the EU is to reduce economic frictions

between its member states. However, while the area’s economies are

integrated by free trade and movement of people, member states cur-

rently determine their tax policies independently. As a consequence,

there is considerable dispersion in corporate tax rates across the union,

as depicted in Figure 2.1. For example, a gap exceeding 20 percent-

age points separated the highest and lowest tax rates of the region in

2014. Corporate tax variation among member states has sparked pol-

icy debate within the union, with high tax countries concerned that

2. EU corporate tax reform 101

EU multinational firms may be incentivized to move production into

low tax areas. Relocating production within the EU from high tax to

low tax countries could be attractive for multinationals based in the

EU because a series of tax treaties imply that their profits are largely

taxed only in the country of production. If multinational firms opened

production affiliates in low tax countries, their profits would then be

taxed at a low rate while goods could still be sold tariff-free anywhere

within the EU. high tax countries could lose industrial activity and

jobs as a result.

Moreover, cross-member FDI is of particular importance within the

EU. In 2014, production by the foreign affiliates of companies based in

other EU member states accounted for 22.5% of total EU manufactur-

ing turnover. Understanding the behavior of these multinationals is

therefore important for understanding aggregate economic outcomes.

If tax rates are taken into consideration when EU firms establish plants,

then their variation among members countries could affect ultimate

production location choices. Resulting relocation into low tax coun-

tries could imply fewer jobs, lower wages, and lower tax revenue for

the remaining countries. Perhaps for this reason, policy leaders in


Figure 2.1.: There is high variance in member-set corporate tax rateswithin the EU.

large high tax countries, notably Germany and France, have pushed

for a EU-wide corporate tax policy harmonization. Unsurprisingly,

low tax countries like Hungary have resisted this proposal.

In this chapter, the effects of a tax rate equalization policy is as-

sessed using a quantitative spatial general equilibrium model. It fea-

tures rich motives for foreign direct investment (FDI) such as market

access, production capability, and tax rates dispersion, as well as com-


peting motives for concentrating production. These are namely fixed

costs of plant establishment and losses associated with arms-length

production, like communication costs. Multinational activity arises

endogenously as firms choose any subset of countries in the EU in

which to open plants. The solution method presented in Chapter 1 is

applied to compute the optimal plant location strategy as a function

of firm productivity. The model is then estimated using aggregate

moments from the EU as well as key multinational shares. As a coun-

terfactual exercise, taxes are unified at the rate where EU-wide tax

revenues match the total amount from the estimated model. Once the

differences in relative tax rates are eliminated, industry relocates to

countries with formerly high rates. Domestic economic activity ex-

pands in these countries, as they enjoy more firms, higher real wages,

and more workers. All together, these countries host 6.8% more firms,

0.8% more workers, and between 0.3% to 1.6% higher real wages in

the counterfactual equilibrium. A mirror effect occurs in the remain-

ing countries, with 9% fewer firms, 0.9% fewer workers, and between

0.8% to 4.6% lower real wages.

The chapter is related to the literature on multinational production,


summarized by Antràs and Yeaple [2014]. Its main innovation in

relation to this body of work is computing general equilibrium in

a model where heterogeneous firms engage in export platform FDI

under fixed costs of plant establishment. Applying the algorithm from

Chapter 1 maintains numerical tractability even with this rich plant

location problem.

Since this literature is vast, a selection of most closely related papers

is highlighted here. First, this chapter builds in spirit on Helpman et al.

[2004], which establishes significant empirical differences between

firms which engage in FDI and those that do not. A structural model

featuring firm heterogeneity and selection into FDI is proposed to ra-

tionalize these empirical regularities. However, the paper does not

examine the sets of FDI locations chosen by different firms in detail,

nor does the model allow for export platform FDI. These simplifica-

tions limit the analysis of the interactions among plants in a firm’s

decision of how to serve each market. The model in this chapter will

deliver a parallel result that the least productive firms will not engage

in FDI at all. However, it additionally delivers predictions of the full

set of locations a firm will be active in, taking into account the in-


teractions among locations. Less productive multinational firms will

open a different set of production locations than their more productive

counterparts.

On the other hand, Ramondo and Rodríguez-Clare [2013] incorpo-

rate export platform FDI while estimating the gains from trade and

multinational production in a Ricardian model based on Eaton and

Kortum [2002]. However, the model there does not include fixed costs

of establishing a foreign presence, so firms choose whichever produc-

tion location has the lowest variable cost. As a result, firms do not face

as strong a proximity-concentration trade-off. However, this trade-off

has been emphasized by the empirical literature of FDI.1 In contrast,

the model in this chapter includes fixed costs, so firms must weigh

a location’s lower production costs against the cost of establishment

there, resulting in lower extensive margin FDI. Arkolakis et al. [2018]

also develop a Ricardian model of multinational production without

fixed costs of FDI to investigate the multinational firm’s decision of

where to locate production facilities separate from their country of1For example, this literature is referenced by Helpman et al. [2004] in motivating

the development of a model where firms balance transport costs against plant-levelreturns to scale when choosing between exporting or FDI.


ownership. Since the model does not include fixed costs, firms simi-

larly do not face as strong a proximity-concentration trade-off. How-

ever, the paper’s goal is mainly to develop a theory explaining why

countries differentially specialize in production activities and head-

quarter activities. Following the insights from this paper, the model

in this chapter also allows a country’s production capability, which

applies to all plants within its borders, to be distinct from the average

productivity of firms born there.

With a model most closely related to the one in this chapter, Tintel-

not [2016] analyzes affiliate location and production choices of Ger-

man multinationals within a set of twelve countries. The model de-

scribes heterogeneous firms engaging in horizontal FDI. It crucially

allows for export platform FDI while also incorporating a fixed cost of

multinational production, creating a complex discrete choice decision

for the firm’s location problem. In this chapter, a similar firm decision

is expanded to incorporate corporate taxation, then placed within a

general equilibrium framework. The choice of production locations

is also extended to 27 possibilities to represent the EU, exponentially

increasing the number of possible sets in which a firm can place plants.


Applying the solution algorithm from Chapter 1 maintains tractability

and allows for computing general equilibrium across all 27 countries.

Since firms engaging in multinational production tend to be large,

this paper is also part of the vast literature on heterogeneous firms

in trade. Pioneered by Melitz [2003], this literature emphasizes the

importance of firm heterogeneity in the context of international trade.

More specifically, Melitz and Redding [2014] reports that firms engag-

ing in a broad class of global activities, such as exporting, importing,

or multinational production, all tend to be larger and more produc-

tive than the average firm. As a result, globalization can favor these

large firms, spurring within-industry reallocation towards them. This

within-industry compositional change can in turn contribute to ag-

gregate effects. The model in this chapter features heterogeneity in

behavior across firms in line with this literature, with more produc-

tive firms selecting different sets of production locations from less

productive firms. The least productive firms optimally choose not to

participate in FDI at all.

This chapter is also related to the literature analyzing the effect of

tax incentives on the production location choices of firms. For exam-


ple, Devereux and Griffith [2003] and Buettner and Ruf [2007] empiri-

cally establish that tax incentives are linked to FDI and multinational

production. This chapter differs because it incorporates a structural

model of the multinational firm’s FDI decision, which is then used to

evaluate the effects of counterfactual policies. On the other hand, Fa-

jgelbaum et al. [2018] also uses a structural model to evaluate spatial

misallocation across states in the U.S. attributable to tax rate disper-

sion. While this analysis includes many different taxes, including

corporate taxes and income taxes, firms in their model are single-

location. They therefore do not incorporate the trade-offs faced by

multi-location firms, such as establishing plants both for production

capability concerns as well as market access.

Finally, the macroeconomic literature also explores taxation in open

economies. In particular, Mendoza and Tesar [1998] and Mendoza and

Tesar [2005] study capital taxation in this setting. Both build dynamic

models with transition paths, from which this chapter abstracts. Men-

doza and Tesar [1998] highlights two main effects once international

finance markets are introduced. When one country lowers capital

taxes, households build up the domestic capital stock by borrowing


internationally. Compared to a closed economy model, the transition

costs are smoothed. In return, the domestic country’s trade deficit

is larger in the long run, effectively transferring abroad a portion of

the gains from lower domestic tax rates. Mendoza and Tesar [2005]

instead examines the public policy game played by EU members on

capital tax rates. It finds that, if countries compete on capital tax rates

while adjusting labor tax rates to maintain fiscal solvency, the Nash

equilibrium does not necessarily result in a “race to the bottom”. Their

results are consistent with current levels of EU tax rates. While both

these studies include analyses of dynamics from which this paper

abstracts, the production sector is kept relative simple with a repre-

sentative firm in perfect competition. By contrast, this chapter focuses

on the effect of tax rate variation across countries on the cross sectional

behavior of heterogeneous firms.

The remainder of the chapter proceeds as follows. Section 2.1 de-

scribes in more detail the current state of corporate taxes in the EU.

The structural framework is then presented in Section 2.2. A key fea-

ture of the model is that firms internalize tax differences when decid-

ing where to locate production. This plant location decision represents


the central feature of the model linked to the tax policy analysis, so its

salient features are highlighted. Section 2.3 then covers aggregation

and general equilibrium in the model. A discussion of numerical com-

putation and estimation follows in Section 2.4, with counterfactual

exercises conducted in 2.5. Finally, Section 2.6 concludes.

2.1. Corporate taxation in the EU

While the EU has an integrated market and currency union, corporate

taxation is individually controlled by member states. As a result, sig-

nificant tax rate dispersion persists across the countries. In 2014, the

lowest marginal tax rate in the EU was 10% in Bulgaria, compared to

the highest rate of 34% in Belgium. Table B.1 in Appendix B reports

tax rates for all 27 EU countries considered in this study.2

Corporate taxation can either be territorial, on profits made within

the borders of a country, or residential, on worldwide profits of firms

registered in the country. In practice, an extensive set of deferral poli-

cies and double taxation treaties among EU states effectively eliminate2Due to its small size and data limitations, Malta is excluded from the analysis.

Since empirical exercises use aggregate EU data from 2014, the UK is included.


taxation on profits made in other EU countries. In addition, EU-wide

directives waive withholding tax on interest, royalties, and repatri-

ated profits across borders within the EU. The model thus proceeds

with the reasonable approximation that profits made by an EU firm

are taxed in the country of production only. In most EU countries, cor-

porate tax is assessed at a flat rate, so the average tax rate is equal to

the marginal tax rate. While a handful of countries provide reduced

rates for small firms, the top marginal statutory tax rate is used to

capture the incentives for multinational enterprises, which tend to be

large.

In contrast, external tax policy with non-EU countries is not as sim-

ple. As a result, it is not necessarily the case that profits are taxed

in the location of production if a EU-based firm establishes a plant

in a non-member country. Likewise, for firms based outside of the

EU but operating plants within the EU, profits earned by these plants

may not be taxed at the host country’s rate. For this reason, only EU

countries are included in the analysis of this chapter. Accordingly,

the structural model allows for the 27 EU countries studied but does

not include the rest of the world. The chapter consequently abstracts


from interactions between the EU and any other countries, focusing

on interactions among EU member states.

2.2. Model framework

This section lays out ingredients for the structural model, which forms

the basis for the chapter’s later numerical policy analysis. Consumers

are presented in Section 2.2.1, followed by the nontraded sector in Sec-

tion 2.2.2. These are kept simple in order to focus on the traded sector,

laid out in Section 2.2.3. This section characterizes the firm’s plant

location decision, which represents the key component to the model

and policy question. In what follows, a firm’s country of origin will

be indexed by i, a plant’s location with j, and a destination markets

using k.

2.2.1. Consumers

Consumers live in markets indexed by k corresponding to countries in

the EU. The economy therefore contains 27 destination markets, one

for each EU country studied. Consumers have Cobb-Douglas prefer-


ences over the traded basket and nontraded goods. The traded basket

has weight β < 1 and thus receives this share of consumption expen-

diture, while nontraded goods receive the remaining consumption

expenditure.

The traded basket is comprised of a continuum differentiated trad-

able goods, aggregated with a constant elasticity of substitution ρ.

Given that a consumer in country k demands cTk of the traded basket,

demand for a variety ω with price pTk (ω) is

qTk (ω) = cT

k

(pT

k (ω)

PTk

)−ρ

PTk =

[∫ΩT

k

pTk (ω)1−ρdω

] 11−ρ

where PTk is the aggregate price index for the traded basket in country

k. In the definition of PTk , all varieties available in k are gathered by

ΩTk . There is a single nontraded good, which is simply sold in market

k at a perfectly competitive price of PNk .

Consumers are either workers or absentee capitalists. In addition to

the consumption goods discussed above, workers also value housing.

In particular, the representative worker living and working in country


k has utility

uk

((cN

k

)1−β (cT

k

)β)η

h1−ηk

where cNk is the quantity of nontraded good, cT

k is the quantity of

the traded basket, hk is the quantity of housing, and uk is a country-

specific utility shifter. The utility shifter can be interpreted as measur-

ing amenities in country k.

Workers have two sources of income, which are wages and a lump

sum rebate of total tax revenues raised in their country of residence.

They inelastically supply one unit of labor, so their total income is

wk + Rk/Lk since total tax revenues Rk are divided equally among all

workers in k. Indirect utility of a household in k is thus

ΨukLk

(wkLk + Rk(

PNk

)1−β (PTk

)β

)η

H1−ηk ≡ V (2.1)

where Ψ is a constant and Hk is k’s fixed stock of housing. The hous-

ing market is assumed to be perfectly competitive in deriving this

expression, with housing rents paid into a global portfolio which is

described below.


Given the European Union’s policies promoting free movement

of people across national borders, workers are assumed to be freely

mobile across countries. Then, workers move to regions with high

indirect utility, equalizing it across all k. For example, countries with

high amenities, high wages, or low prices will be attractive places to

live. Therefore, workers will move there, contributing to congestion

in the housing market and decreasing tax revenue per worker, all else

equal. In general equilibrium, workers will therefore move so that

indirect utility is equalized in all countries. Populations Lk in each

country k are endogenously determined this way.

On the other hand, absentee capitalists in each country are assumed

to be massless so they do not contribute to congestion. In other words,

they only purchase consumption goods and do not consume hous-

ing. Capitalists receive a fixed share bk of a global portfolio as in-

come, which consists of total rents paid to housing as well as firm

profits. Put together, their expenditure on consumption goods to-

tals bk ∑j(Πj + (1− η)

(wjLj + Rj

)), where Πj represents the profits

made by firms born in j. Since there is free entry in each country, to

be further detailed when firms are described, there will be zero total


profits in equilibrium.

Gathering expenditure by both workers and capitalists together,

total sales of traded goods in k can be summarized as

XTk = βη (wkLk + Rk) + βbk ∑

j

(Πj + (1− η)

(wjLj + Rj

)). (2.2)

Observe that expression (2.2) represents a balance of payments con-

dition. However, despite balance of payments, trade balance will not

hold in general in this model. The presence of capitalists who con-

sume goods locally in k but have income from all countries introduces

a wedge into trade balance, which will be fully characterized in Sec-

tion 2.3 once aggregate equilibrium has been fully described. The

fixed portfolio holdings bk allow the model to match aggregate EU

data on both expenditures and production during the estimation in

Section 2.4.1.

2.2.2. Nontraded sector

A perfectly competitive nontraded sector in each country uses lin-

ear technology in labor with country-specific productivity αk. Conse-


quently, the price of the nontraded good is

PNk = wk/αk . (2.3)

Workers are freely mobile across the traded and nontraded sectors,

resulting in a single equilibrium wage for each country. This sector

is kept parsimonious to focus attention on the traded sector, but in-

cluded to discipline the traded sector’s general equilibrium effect in

counterfactual exercises.

2.2.3. Traded sector

The traded sector is populated by a continuum of firms. They each

produce a single differentiated product, competing monopolistically

in every destination market k. For simplicity, there are no fixed costs

of market entry, so all firms serve all markets.

There is free entry in each country i, where firms must hire f ei units

of labor to develop the blueprint for its differentiated good ω. Once

this cost has been sunk, firms observe their core productivity z(ω),

which is drawn from a fixed distribution Gi. This core productivity


represents the firm’s innate production ability, which will affect all

its plants. For example, a firm may have a particularly efficient man-

agement philosophy which is applied at all its plants. Every firm is

endowed with a plant in their country of birth, but can also open

them in foreign countries at a fixed cost of f j in the foreign country’s

labor. Thus, fij = 1[i 6= j] f j will be used to denote the labor costs of

establishment to be paid.

In the model, firms face two main decisions after entry. First, they

select a set of countries J ⊆ J in which to establish plants. Next, they

choose prices, quantities, and shipments from each of their plants

j ∈ J for each market k. These two decisions will be referred to

respectively as the plant location and sourcing decisions. This section

characterizes the sourcing decision first, before moving on to discuss

the plant location decision.

Sourcing

The sourcing decision is made once plant locations are chosen, so J

is already fixed. Then, conditional on the established set of plants,

markets can be reached through exporting, supplying from a local


plant (horizontal FDI), or shipping from a plant in a third country

(export platform FDI).

Each plant has its own idiosyncratic productivity, which is observed

once plants are opened and fixed costs are sunk. These are distinct

from the firm’s core productivity, summarizing the individual char-

acteristics of the plants. For example, each plant may have access to

different local technologies, with some being more productive than

others. Unit costs are wj/ϕij(ω) at a plant j with idiosyncratic pro-

ductivity ϕij(ω), where wj is the local unit cost of labor.

At this stage, firms choose a sourcing strategy for each destination

market. Since production is constant returns to scale at every plant

and sales in one destination market do not affect demand in another,

the firm can determine its sourcing strategy for the destination mar-

kets separately. In addition to the unit costs of production above,

firms also face variable iceberg costs of trade djk ≥ 1 and arms-length

production γij ≥ 1. That is, djk units of output must be shipped to

ensure one unit arrives in market k from a plant in j. Likewise, input

requirements are magnified by γij if a firm born in i produces in j.

These costs are meant to capture the various challenges a multina-


tional firm may encounter when producing in a different country, like

costs of communication. Taken together, a firm born in i will have

marginal cost wjγijdjk/ϕij(ω) of serving market k from its plant in

j. The sourcing problem for each market k amounts to choosing the

plant j(k) ∈ J generating the highest variable profits.

These variable profits are determined as follows. Given the CES

monopolistic competition market structure, prices will be set at a con-

stant markup ρ/(ρ − 1) over marginal costs. In addition, variable

profits are taxed at a flat rate τj in the country of production. Then, a

firm born in i serving market k from plant j would earn

1ρ(1− τj)XT

k

(ρ

ρ− 1wjdjkγij

ϕj(ω)PTk

)1−ρ

in variable profits. To solve its sourcing decision, the firm selects j(k)

among its plants that maximizes this amount for each market k. Note

that both marginal cost considerations and tax burden enter into this

decision.

In fact, tax rate dispersion, not the presence of taxes, affects the firm’s

sourcing decision. If tax rates were constant across all locations, then


the firm’s sourcing strategy would be identical to its optimal behavior

in the absence of taxes. Specifically, it would choose the same plant j to

serve each market k, produce the same quantity, and charge the same

price as if there were no taxes. On the other hand, the variation in tax

rates has the potential to change which plant is most profitable to serve

k from. Suppose the firm would choose its plant in j to serve market

k if there were no tax rate dispersion but instead chooses j′ 6= j with

the tax rate dispersion. In that case, the firm will produce a different

amount and charge a different price based on the marginal cost of

production in j′.3 Therefore, the direct effect of tax rate dispersion in

the sourcing decision is to potentially distort which plant is selected to

serve a market, which in turn changes its pricing and quantity choices

in that market.

Until this point, firms have been restricted to opening only one plant

in each country. However, since marginal costs are constant, the firm

problem can instead be reinterpreted to allow for multiple plants in

3Of course, if the firm could transfer its taxable income from one location to theother, it could separate where goods are produced from where profits are taxed.Due to lack of data availability, income shifting is beyond the scope of this analysisand disallowed in the model.


any country. In this case, the fixed cost fij would represent a firm’s one-

time cost to establish a production presence in a foreign country j. For

example, these could be the fees of officially registering the firm with

tax authorities. With this interpretation, plant establishment would

carry no further fixed costs. Then, without loss of generality, the paper

continues assuming there is at most one plant per country.

Plant location

The discussion above determines firm choices once plants are estab-

lished. Now, taking a step back, consider how firms should choose

the set of locations in which to place plants J .

Since they do not observe idiosyncratic plant productivities ϕij(ω)

until after fixed costs are sunk, firms select J by maximizing the ex-

pected value of profits, taken over possible realizations of these plant

productivities ϕij(ω). With the local technology interpretation, this

assumption means that firms do know the exact quality of the local

technology available to the plant before opening. It then does its best

given the known distribution of local technology quality.

To characterize this expectation sharply, idiosyncratic plant pro-


ductivities are assumed to be drawn from a Fréchet distribution with

shape θ and scale z(ω)1/(ρ−1)Tj. Loosely, the Fréchet’s shape inversely

governs dispersion while its scale is related to the mean. While θ is

common across countries, the scale contains a country-specific com-

ponent that is multiplied by the firm’s core productivity. As an impli-

cation, firms with higher core productivity will have more productive

plants on average. For example, firms with a better management style

may be able utilize any quality local technology more efficiently than

firms with a bad management style. Likewise, plants opened in coun-

tries with high Tj will be more productive. Then, local technology

in countries with high Tj would be more efficient, holding firm type

constant.

Using the Fréchet properties popularized by Eaton and Kortum

[2002], expected profits are

πi(J ; z) = Az ∑k

XTk(

PTk

)1−ρΘik(J ; υ)

ρ−1θ −∑

j1[j ∈ J ]wj fij , (2.4)


where

Θik(J ) = ∑j

1[j ∈ J ](1− τj

) ρ−1θ(wjdjkγij/Tj

)−θ︸︷︷︸κijk

and A is a constant. The firm’s plant location problem therefore con-

sists of selecting the set of locationsJ that maximizes expected profits

(2.4). Multinational firms therefore emerge endogenously when a firm

finds it optimal to choose a set with more than one location.

The function Θik(·) describes the (variable) profit potential of a set

J . Observe that it is obtained by adding together the κijk terms asso-

ciated with each country j in the set J , which incorporate the char-

acteristics of a country salient for profitability. Namely, they are the

base cost of production wj/Tj, trade market access djk, losses from

arms-length production γij, and the country’s tax rate τj. The first

three capture marginal cost considerations while the last represents

tax burden, so these terms specify exactly how firms trade off a plant’s

expected marginal cost with its associated tax rate. In particular, the

Θik(·) expression here is similar to the market access terms found in


the gravity literature, with the key difference that it is a function of the

chosen set J .4 Observe that Θik(J ; υ) also varies by the market k and

origin i, due to bilateral costs of trade and arms-length production

respectively. Each set is consequently valued differently depending

on the firm’s country of origin as well as the destination market in

consideration.

There are several properties of the expected profit function of note.

First, expected variable profits increase mechanically as a new plant

j is added. Recall that for every market k, the firm simply selects

the plant delivering the highest variable profits. Then, the addition of

another plant to this choice set can only increase the expected value of

the maximum. If its realized profitability is lower than the maximum

from the original set J , it is simply not chosen. Accordingly, the

additional plant’s value is higher if its marginal cost or tax rate is

lower, increasing the expected maximum by more since it is more

likely to exceed the maximum from the original set.

Next, the profit potential terms are raised by the exponent (ρ− 1)/θ.

In this context, it is necessary to have ρ− 1 < θ for the expectation4Section 2.3 will show that gravity will not exactly hold for aggregate trade flows.


over ϕij(ω)s to exist. Then, inspection of the expected profit function

(2.4) reveals that it is concave in each profit potential term. A direct

implication is that there are interactions among the plants, which turn

out to be substitutes for each other. To clarify this point, consider the

change in profits caused by j’s inclusion into J .

Dij(J ; z) = Az ∑k

XTk(

PTk

)1−ρ

(κijk +

(∑

j 6=j′∈Jκij′k

)) ρ−1θ

−(

∑j 6=j′∈J

κij′k

) ρ−1θ

− wj fij (2.5)

This difference will be referred to as the marginal value of including a

location j in a set J . Because J is an argument to the marginal value

of j, it is necessary to know the firm’s entire plant location strategy to

know the impact from adding a plant in the location j. This interde-

pendence highlights the interactions among the plants in the firm’s

decision.

In particular, as the profit potential of the original set J decreases,

concavity implies that a plant j’s marginal value rises. Broadly, when


the original set is not expected to be profitable for market k so that

Θij(J ) is low, the additional plant will likely be the one offering max-

imal profits. Then, its presence in the set increases expected prof-

its substantially. On the other hand, if the original set already has

high Θik(J ) so that it is quite profitable in expectation, for example

through low tax rates, an additional plant will not improve expected

profits as much. In this way, the plants are substitutes since only one

is ultimately chosen to serve each market. The chosen plant in turn

cannibalizes potential profits from the others.

In fact, it can be shown that the ex ante probability j is chosen

amount the set of plants J to serve market k is pijk(J ) = 1[j ∈

J ]κijk/Θik(J ). This expression also resembles those from the gravity

literature and provides another way to see the substitutability among

plants. From the probability expression, it is clear that a specific

plant’s chance of serving a market is decreasing in the profit potential

of the others available to the firm.

Finally, observe that the exponent (ρ− 1)/θ is the key model compo-

nent governing the substitutability of the plants. Concavity in profit

potential terms, and thus substitutability among plants, intensifies


when θ increases or ρ decreases. Since these are structural parameters,

there is an economic intuition for their role in determining the degree

of substitutability among the plants. The dispersion of plant produc-

tivity draws is inversely related to θ, so when θ is high, plants are

more similar when θ is high. Likewise, ρ parameterizes the elasticity

of consumer demand, so that consumers are less responsive to the

price of goods when ρ is low. In turn, it is less important for the firm

to lower marginal costs when ρ is low, so the productivity advantages

of one plant over another make little difference. Both comparative

statics decrease the dispersion in potential variable profits afforded

by different plants, rendering them more substitutable.

Profit potential terms are weighted by the relative sizes XTk (PT

k )ρ−1

of the corresponding markets then summed to arrive at the total ex-

pected profits. To solve its plant location decision, the firm must then

choose among the 227 possible sets of production locations, resulting

in a high dimensional discrete choice problem. Given the interactions

among plants described above, this maximization problem cannot be

solved analytically in general. Instead, it will be solved numerically

using the method introduced in Chapter 1 during the quantitative


exercises. A full exposition on applying the method in this context is

deferred to Section 2.4.

2.3. General equilibrium

This section characterizes the general equilibrium of the model. Opti-

mal firm behavior is summarized in Section 2.3.1, followed by aggre-

gate conditions in Section 2.3.2. Drawing on these conditions, general

equilibrium is defined and characterized.

2.3.1. Optimal firm behavior

Because the location set J is chosen before plant-specific productivi-

ties are realized, all firms with the same birthplace and core produc-

tivity z will choose the same optimal set. For this reason, there is a

well-defined policy function J ∗i (·) for each birthplace i, which maps

a firm’s core productivity to its optimal set of plant locations.

Consider the firm’s incentives when making its plant location de-

cision. A plant j’s marginal value in J given by (2.5) ultimately com-

pares the change in profit potential created by j’s inclusion against


the fixed cost of its establishment. Hence, there are both motives for

FDI and for concentrating production. These are assessed as a whole

to determine the optimal set of locations.

For this choice, taxes have both a level and dispersion effect on the

firm’s decision. To isolate the level effect, suppose there is a uniform

tax rate τ. It would still appear in the marginal profit function (2.5),

but identically for all plants j and sets J . In particular, it could be fac-

tored out of the profit potential terms so that it is simply multiplying

the firm’s core productivity z. Then, if J nti (·) represents the firm’s

optimal behavior if there were no taxes, the policy function with uni-

form taxes τ could simply be derived as J τi (z) = J nt

i (z(1− τ)). In

other words, a firm with productivity z in the uniform tax environ-

ment behaves identically to a firm with productivity z(1 − τ) in a

tax-free setting. The level effect of tax rates is therefore to simply shift

the policy function rightward.

Variation among tax rates has an additional effect. As discussed

when describing the firm’s sourcing problem, country-level tax rate

variation distorts the firm’s choice of which plant will serve market k

when it makes its sourcing decisions. This consideration is internal-


ized during the plant location stage, so countries with comparatively

lower taxes have higher marginal values of inclusion to any set com-

pared to an otherwise identical location with a high tax rate. Firms

will therefore tend towards opening plants in countries with lower

tax rates, all else equal, resulting in a distortion of the optimal plant lo-

cation strategy that cannot be described by a simple shift in the policy

function.

Note that, though firms with the same origin i and core productiv-

ity z choose the same set of locations, they will have different produc-

tion patterns since sourcing decisions are made based on the realized

draws of plant-specific productivities. However, with a law of large

numbers argument, the ex-ante probability pijk(J ) that a single firm

with plants in J chooses the plant in j to serve market k is also the

share of firms with set J that will make this choice.

2.3.2. Aggregate conditions

This section completes the model by developing the equilibrium con-

ditions which pin down the aggregate variables. In particular, ag-


gregation in the traded sector is characterized, including aggregation

over firm behavior, equilibrium sales shares, and total tax revenues. In

addition, labor market clearing across both sectors is specified. These

conditions are combined to define general equilibrium.

Price index and sales shares

Since the home plant is free, all entering firms participate in the econ-

omy. Thus, the continuous mass of entrants Mi will also be the mass

of active firms headquartered in each country. Integrating over indi-

vidual firm pricing behavior, the price index of the traded basket can

be written as

(PT

k

)1−ρ= ρA ∑

i,jMi

∫z

zΘik (J ∗i (z))ρ−1

θpijk(J ∗i (z)

)1− τj

dGi(z) . (2.6)

Since there is a continuum of firms, a law of large numbers argument

ensures that the distribution of plant productivity draws is also the re-

alized distribution of draws over the mass firms. Additionally, prices

are proportional to plant productivities, so they are also distributed

according to a Fréchet distribution. The expression for the price in-


dex uses the properties of this distribution to integrate accordingly

over the distribution of prices in market k. It includes the profit po-

tential terms Θik(J ∗i (·)) and shares pijk(J ∗i (·)) of firms with plants

in J that select the plant in j to serve market k. Both terms incorpo-

rate the firm’s location choice decision J ∗i (·), precluding an analytic

integration.

Given the expression for the price index, sales shares can be con-

structed using the CES nature of demand. Specifically, the share of

tradables sales in k paid to plants in j owned by firms from i can be

written as

λijk =Mi∫

z zΘik(J ∗i (z)

) ρ−1θ

pijk(J ∗i (z))1−τj

dGi(z)

∑i′,j′ Mi′∫

z zΘi′k′(J ∗i′ (z)

) ρ−1θ

pi′ j′k(J ∗i′ (z))1−τj

dGi′(z). (2.7)

Because these shares are the key intensive margin share of the model,

appearing in many later equilibrium conditions, the economic intu-

ition of each of the terms are now discussed. To begin, both Mi and

pijk(J ) broadly summarize the mass of varieties sold in k produced

in j by a firm based in i. Countries of origin with a higher mass of


firms Mi receive a larger share of expenditure from any market k be-

cause each firm corresponds to one variety. Likewise, the number of

varieties attributable to plants in j and firms in i is higher if plants

in j are more likely to be selected for supplying market k compared

to any other production location j′. This effect is represented by the

presence of pijk(J ), which is high if a plant in i with set J is likely to

select the plant in j to serve k.

The other terms broadly capture, for each variety, the sales volume

based on the price competitiveness of the good. For example, firms

with high core productivity z have higher productivity plants in gen-

eral, resulting in their varieties having lower prices. A production

location j will also capture a larger share of total expenditure in k if

it has a higher tax burden, corresponding to lower (1− τj). While

intuition may expect high tax countries to have a lower share because

they are less likely to be selected to produce for any market k, this

effect is already captured by the probability pijk(J ). Conditional on

being chosen, it must be that high tax locations generate high pre-tax

variable profits to compensate for their high tax rates. In turn, high

pre-tax variable profits are generated by low marginal costs, which


translate to low prices and higher sales shares. Similarly, if a plant

in j is selected to serve k from a set with high profit potential Θik(J ),

it must be profitable enough to be preferable to the other high-profit

options in J . In general, highly profitable locations are productive

and have low prices, capturing larger market shares.

Because the shares λijk apportion the sales in a market k to the origin-

production pair (i, j), observe that ∑i,j λijk = 1 for all k. Trade shares

can also be constructed as ∑i λijk. These represent the share of expen-

diture on tradables in k captured by goods produced by facilities in j,

no matter the origin of the operating firm. Likewise, total trade flows

from j to k can be written as ∑i λijkXTk . Examining the shares, it is easy

to verify that these flows depart from the standard gravity form.

FDI participation shares

While the λijk shares in (2.7) describe intensive margin multinational

activity, the second set of key shares summarize the extensive margin.

In particular, aggregating over firm behavior,

µij =∫

z1 [j ∈ J ∗i (z)]dGi(z) . (2.8)


obtains the share of firms born in i with a production facility in j. Then,

given a mass of Mi firms from i, a total mass Miµij of firms will open

plants in j while Mi(1− µij) will not. This share therefore captures

the extensive margin FDI participation of firms from i in country j.

Free entry

Because there is free entry and firms do not observe their core pro-

ductivity z until after the entry cost is sunk, firms will enter in every

country i until expected profits equal the cost of entry wi f ei . With a

convex continuum of firms in each country, the law of large numbers

implies that firm-level expectations and probabilities are exactly re-

alized over the measure of entering firms. Therefore, the expected

firm profits before entry coincide with the average operating firm’s

profits in each country i. The free entry condition can then be written

as setting total net profits to zero, as follows.

Πi =1ρ ∑

j,k

(1− τj

)λijkXT

k −∑j

Miµijwj fij −Miwi f ei = 0 (2.9)


The first term summarizes total variable profits after tax. With the CES

structure of demand, gross variable profits are simply a fraction 1/ρ

of total sales, which are expressed using the sales shares λijk described

above. These variable profits are taxed in the country of production j

then added over j and k to account for all destination markets and pro-

duction locations. The next term represents total fixed cost payments

for plant establishment, which are calculated using the FDI participa-

tion shares µij. A total mass Miµij of firms from i optimally open a

plant in j, each paying the fixed cost wj fij. Finally, fixed costs of entry

are deducted to arrive at total profits. The mass of active firms Mi is

then endogenously determined in general equilibrium.

Tax revenues

Total tax revenues raised in country j are aggregated

Rj =τj

ρ ∑i,k

λijkXTk . (2.10)

Note that taxes in a country j are levied on variable profits attributable

to production carried out by plants within its borders. Because de-


mand is CES and firms compete monopolistically, variable profits are

a constant share 1/ρ of sales. Total sales by j plants can in turn be

written using the sales shares λijk, similarly to those in the free entry

condition (2.9). Finally, taxes levied on variable profits are summed

over i and k, to account for profits made by selling to any market k

and produced by firms based in any country i.

Labor market clearing

The final aggregate condition is labor market clearing, as follows.

wjLj =ρ− 1

ρ ∑i,k

λijkXTk + ∑

iMiµijwj fij + Mjwj f e

j +1− β

βXT

j (2.11)

This condition equates total labor supply in j to labor demand. The

first three terms summarize the amount of labor demanded by the

traded sector for variable production, plant setup fixed costs, entry

fixed costs. Because labor flows freely between the traded and non-

traded sectors, the final term in the expression captures total payments

to labor from the nontraded sector.

In the traded sector, CES demand fixes the total payments to the


variable factors of production as a constant share (ρ− 1)/ρ of sales.

These are determined using the sales shares λijk similarly to the ex-

pression for total tax revenues (2.10). Likewise, the term capturing

total payments for foreign affiliate establishment incorporates Miµij,

the mass of firms from i opening a plant in j. The sum for this term is

over i, totaling the fixed costs paid by firms from all origin countries

i. The third term captures fixed costs of entry paid by firms entering

in country j. Finally, the last term reflects the fact that there is perfect

competition in the nontraded sector, so payments to labor are simply

equal to total sales in the sector.

General equilibrium definition

Let boldface letters represent vectors or matrices of aggregates. For

example, let L be the vector of populations Ljj, where each entry

is the population of country j. Similarly, let µ be the matrix of FDI

participation shares, collecting shares µij for all i and j, and λ be the

collection of sales shares. Then, general equilibrium can be defined

formally by gathering the aggregates and conditions described above

as follows.


Equilibrium. The aggregate vector υ = L, XT, PN, PT, R, M, w, λ, µ

and functions J ∗i (·)i constitute an equilibrium if:

(i) Given J ∗i (·)i, aggregates υ satisfy equations (2.1) - (2.11).

(ii) Given υ, functions J ∗i (·)i describe optimal choices for (2.4).

To summarize, the key country aggregates are populations, tradables

expenditures, price indices, tax revenues, masses of firms, and wages.

They define equilibrium together with sales shares, FDI participation

shares, and policy functions J ∗i (·) for each i which map firm produc-

tivity z to a set of plant locations for each country of origin i. For a

collection of these objects to be an equilibrium, the policy functions

J ∗i (·) must describe optimal firm behavior conditional on the aggre-

gate variables. Likewise, given firm behavior given by the policy

functions, aggregates must satisfy the conditions developed in this

section. These ensure utility equalization from free mobility, balance

of payments, CES-consistent aggregation of prices indices, zero ag-

gregate profits in line with free entry, and labor market clearing.


Trade deficits and surpluses

As discussed, trade is not balanced in general. By combining the

aggregate conditions developed in this section, the expression for net

imports can be written as follows.

XTj −∑

i,kλijkXT

k =1ρ

[∑j′,k(1− τj′)λjj′kXT

k −∑i,k(1− τj)λijkXT

k

]

+ ∑i

Miµijwj fij −∑j′

Mjµjj′wj′ f jj′

+ bj ∑k(1− η)

(wkLk + Rk − wjLj − Rj

)(2.12)

This expression combines balance of payments, the free entry condi-

tion, the expression for total tax revenues, and labor market clearing.

There are three differences in the expression for the net imports

of country j. The first results because domestic firms can earn prof-

its by operating in foreign countries, while foreign firms earn can

earn profits by operating in country j. The next difference similarly

arises because domestic firms hire labor in foreign countries to estab-

lish plants, while foreign firms hire domestic labor when establishing


plants in j. The final difference appears because each country contains

capitalists that collect a portion bj of the global portfolio. It captures

the gap between country j’s withdrawal from the portfolio and its

contribution into the portfolio.

It is clear from this expression that there would still be trade im-

balances if capitalists in each country j were simply paid the local

rents from housing, (1 − η)(wjLj + Rj). In this case, only the last

difference in expression (2.12) would disappear, while the first two

would remain. The wedge in trade balance arising from capitalists

is incorporated in the model so that empirical trade imbalances can

be matched using the portfolio shares bj. A detailed discussion of

how these shares are recovered is included in Section 2.4.1, when the

estimation procedure is described.

2.4. Computation and estimation

The structural model is now ready to be estimated. Section 2.4.1 de-

scribes the calibration and estimation strategy. The main challenge in

computing the model is solving for the distribution of optimal firm be-


havior, especially the firm solutions to plant location problems. To ad-

dress this challenge, the policy function strategy introduced in Chap-

ter 1 is applied. Section 2.4.2 discusses the algorithm in this context. To

highlight the numerical benefits of this method, the estimated model

is used to compare the algorithm’s quantitative performance to the

current approach taken by the literature.

2.4.1. Calibration and estimation

The model is calibrated and estimated in anticipation of the policy

counterfactual. First, a functional form must be chosen for the distri-

bution of core productivity Gi(·). The chapter proceeds by assuming

that Gi(·) follows a Pareto distribution with shape ξ and minimum

value zmini . The tail index θ is constant across origin countries i, but

origin-specific zmini allows the distribution’s minimum value to vary.

Then, countries with high zmini will have relatively more productive

firms. Firm productivity directly affects the set of countries selected

for production, and will indirectly shift the firm’s plant productivity

upwards in expectation. Since the vector of minimum productivities


zmin will be estimated in what follows, this section denotes the firm

productivity distribution as Gi(·; zmini ) to make explicit the parame-

terization. The notation omits the shape parameter ξ because it is

calibrated from the literature.

The model contains several parameters and location fundamentals.

Values for these are determined with three separate steps. First, sev-

eral parameters and location fundamentals are calibrated to values

from the literature and data. These are listed in Table 2.1. Next, a small

set are estimated using the general method of moments (GMM), using

moments listed in Table 2.2. Finally, the remaining parameters and

location fundamentals are recovered by inverting a set of equilibrium

conditions, conditional on the previously calibrated and estimated

values. Table 2.3 summarizes this step. Implicitly, these strategies

assume the state of the EU represented by the aggregate data is in the

general equilibrium defined in the previous section. A more detailed

description of each of these steps follows.


Calibration

The parameters in Table 2.1 are set to values in the literature or calcu-

lated directly in the data. First, the Fréchet shape parameter governing

the dispersion of plant productivity draws θ is set to 7, following es-

timations from the gravity literature. The elasticity of substitution

between traded varieties ρ is set at 6, implying a markup of 20% over

marginal costs. As discussed in Section 2.1, tax rates are set at the

statutory tax rate in each country.

Next, many values are directly determined using aggregate EU

member state data from 2014, collected by EUROSTAT. The Cobb-

Douglas shares in consumer preferences are set according aggregate

expenditure data across all EU countries. Since housing empirically

constitutes 25% of total expenditure, the weight on housing 1− η is

taken to be 0.25, implying a weight η = 0.75 on consumption goods.

Within consumption goods, β is the Cobb-Douglas share on the trad-

ables sector. The model’s tradables sector is interpreted as the man-

ufacturing sector in the data, which empirically accounts for 70% of

consumption expenditure. Accordingly, β is set at 0.7. EUROSTAT


Parameter Description Calibration strategyθ = 7 Fréchet shape Eaton and Kortum [2002]ρ = 6 CES 20% markupτ corporate taxes statutory tax rates

from EUROSTAT data

η = 0.75 C-D share, non-housing household expenditureβ = 0.7 C-D share, tradables household expenditureL workers labor forceM mass of firms manufacturing firms countXT tradables expenditure manufacturing trade flows

from Tintelnot [2016]

ξ = 6.436 Pareto shaped trade costs with CEPII distanceγ arms-length costs with CEPII distance

Table 2.1.: A summary of parameter calibration from literature anddata.

data is also used to set the value of labor forces L and masses of firms

M. In particular, each country’s labor force is set to its number of work-

ers in the data while the mass of active tradables firms is similarly set

to the number of manufacturing firms in the data.5

5While a country’s labor force Lk is an equilibrium outcome in the model, it is


The total expenditure on tradables XT is also set to match data on ag-

gregate manufacturing data. To construct these measures, trade flow

data is combined with aggregate production data for each country.

Since the model does not contain any non-member countries, the em-

pirical object corresponding to the model’s XTj would be a European

country j’s total expenditure on traded goods produced within the EU.

This expenditure is the sum of absorption and total manufacturing im-

ports from other EU countries. Total imports from other EU countries

can be derived using aggregate manufacturing trade flow data. To

calculate absorption, total manufacturing exports from the trade flow

data are subtracted from each country’s total manufacturing sales.

Next, some parameters are set following Tintelnot [2016], which

estimates similar parameters using microdata on German firms. The

parameter values are calibrated using the values from Tintelnot [2016]

because the study’s model shares many elements with the one in this

chapter. The most important similarity is that heterogeneous firms

pinned down by utility equalization (2.1). This expression contains the country util-ity shifters uk, which can be recovered to exactly rationalize the data. Similarly, fixedcosts of entry f e

i in the free entry condition (2.9) can be set to exactly rationalize theempirical number of firms Mi in each country. The estimation of these parametersis described in depth at the end of this section.


also solve a plant location problem with export platform FDI and

fixed costs of entry. Although Tintelnot [2016] confines its analysis to

German firms, the full of countries available for hosting affiliates in

the paper’s model contains several large EU members. The overlap

suggests it is reasonable to use these estimated parameters in the con-

text of the EU, which is the focus of this chapter. To begin, the shape

parameter governing the Pareto distribution of firm core productivity

draws is set at ξ = 6.436. It is identified in Tintelnot [2016] through a

maximum likelihood strategy using data on the affiliate location sets

of each multinational as well as the output of each affiliate.

Next, the bilateral iceberg trade costs d and bilateral iceberg costs

of arms-length production γ are also set according to Tintelnot [2016]

estimates. Following the gravity literature, the bilateral costs are pa-

rameterized as functions of distance, shared language dummies, and

shared border dummies between each pair of countries as follows.

djk = βd0 ×

(distjk

)βddist ×

(βd

lang

)langjk ×(

βdborder

)borderjk

γjk = βγ0 ×

(distjk

)βγdist ×

(β

γlang

)langjk ×(

βγborder

)borderjk


The coefficients βd, βγ are set to Tintelnot [2016] estimates, which

are then combined with CEPII data on bilateral proximity measures

to arrive at the bilateral costs of trade and foreign affiliate production.

Own-costs djj and γii are normalized to one. Given the presence of Tj

and zmini , these are true normalizations.6

Estimation

Following calibration, a collection of model parameters are estimated

using the general method of moments. Table 2.2 summarizes these

parameters and corresponding informative moments.

In particular, the key set of targeted moments is the share of firms

from i with an affiliate in j. These are the shares µij in the model,

defined in (2.8) by integrating over optimal plant location decisions.6To see why, observe that the iceberg costs of trade djk are always accompanied

multiplicatively by Tj. Intuitively, since Tj is a measure of the average j-plant’s pro-ductivity, it could incorporate the “average shipping effectiveness” of all plants inj. For example, under the interpretation that Tj describes the local factors availableto all plants in j, access to shipping networks could affect this average shippingeffectiveness. By normalizing djj = 1, the common component of shipping fromj to any destination market will be accounted for by Tj. Then, djk/djj = djk repre-sents the cost of shipping from the country j to the market k, relative to the cost ofdelivering the good locally. A similar argument can be made for the bilateral costsof arms-length production γ and the minimum productivity of firms born in eachorigin zmin.


Parameter Description Informative momentf plant fixed cost µzmin Pareto scale ∑j 1[j 6= i]µijT country productivity δ

Table 2.2.: A summary of parameter estimation using GMM

EUROSTAT’s Foreign Affiliates Statistics (FATS) are leveraged to con-

struct their empirical counterpart. These statistics provide the number

of foreign establishments operating in each member country where

the establishments are crucially divided by the operating firm’s coun-

try of ownership. Thus, data equivalents of µij can easily be con-

structed as

µdataij ≡ # (plants in j with i ownership)

# (firms owned in i)

for all pairs i 6= j. Since the µdataij shares describe extensive-margin

moments, they are particularly informative for the fixed costs of plant

establishment f j in the model. In general, when the fixed costs of

establishing a plant in country j increases, it will be optimal to do so


for fewer firms.

Because the FATS data set describes foreign affiliate activity, it does

not contain any information about domestically owned plants. To

pin down the country-specific component of the firm productivity

distribution zmin, the GMM strategy also targets a combined measure

∑j 1[i 6= j]µdataij . This moment counts the total number of foreign

plants in the EU owned by firms based in i, then divides by the number

of firms headquartered in i. Since a single firm could have plants in

multiple EU countries, it is possible for this number to exceed one. In

this case, there are more foreign plants operated by firms based in i

than there are firms based in i. The moment therefore serves as an

overall statistic for the foreign activity of i-based firms. In the model,

foreign activity is positively related to the average productivity of

firms born in i, which is controlled by zmini . When a country i has

high zmini , its firms tend to be productive and engage in multinational

production, implying higher overall FDI participation captured by

∑j 1[i 6= j]µij.

Finally, a second set of shares describing the distribution of manu-


facturing production within the EU

δdataj ≡ manufacturing production in j

manufacturing production in the EU

is constructed using EUROSTAT data on manufacturing. This share

describes intensive-margin production in each country carried out by

both domestic and foreign affiliates. They are likewise matched to the

counterparts calculated using the model

δj ≡∑i,k λijkXT

k

∑k XTk≡

YTj

YT

where YTj denotes total tradables production occurring in country

j and YT is total EU tradables production. Because δdataj describes

intensive margin production in a country, it mainly disciplines Tj,

which governs the country-level average of plant productivity. With

higher Tj, production in a country j increases for two reasons. First,

once plants have been placed, firms are more likely to produce and

source from plants in j for any market k because productivity draws

there tend to be higher. Then, during the plant location stage, firms


will value potential plants in j highly and will accordingly be more

likely to place plants there.

Having discussed the moments used, the GMM minimization rou-

tine can now be defined. Computationally, it is cast as a mathemat-

ical program with equilibrium constraints (MPEC). The strategy of

estimating structural general equilibrium models as MPECs was pro-

posed in Su and Judd [2012] as an alternative to the nested fixed point

routine. Further, the MPEC method was found to be faster and more

accurate in performance tests evaluated by Dubé et al. [2012]. Given

the insights from these studies, the GMM estimation is operational-

ized as an MPEC as follows.

minPT

j ,wj/Tj,wj f j,zminj j

∑j

(µij − µdata

ij

)2+ ∑

j

(δj − δdata

j

)2

s.t.(

PTk

)1−ρ= ρA ∑

i,jMi

∫z

zΘik (J ∗i (·))ρ−1

θpijk(J ∗i (·)

)1− τj

dGi(z)

∑j

1[j 6= i]µdataij = ∑

j1[j 6= i]µij

The first constraint is the equilibrium condition, which ensures that


the price indices for the traded basket are consistent with aggregation

over individual firm prices.

In practice, the minimization searches over composites of equilib-

rium aggregates and location fundamentals PTj , wj/Tj, wj f jj. The

estimation routine proceeds as follows. Given values for these com-

posites and parameters calibrated in Table 2.1, the policy functions

J ∗i (·) capturing optimal firm behavior can be solved. Details of the

solution method are deferred until the next section.

Given the firm policy functions, values of minimum core produc-

tivities zmin are inferred by satisfying the MPEC’s second constraint.

This constraint amounts to exactly matching the overall measure of

FDI participation ∑j 1[i 6= j]µdataij described above. Since fii = 0, the

domestic plant carries no fixed cost of establishment. It is therefore

optimal for all firms with non-negative productivity to open a plant in

their countries of birth. However, firm productivities in each country

i are drawn from the distribution Gi(·; zmini ) with positive minimum

zmini . This minimum is therefore binding, so it pins down the produc-

tivity of the lowest participating firm in equilibrium. To infer its value,

the Pareto form of Gi(·; zmini ) is used to recover the values of zmin

i that


perfectly match the model’s overall FDI participation measure with

the data.

The distribution of firm behavior in the model is jointly described

by policy functions J ∗i (·) and productivity distributions Gi(·; zmini ).

Aggregating over the firm’s plant location decisions delivers FDI par-

ticipation shares µ, the first set of model-implied moments. Firm be-

havior can also be aggregated to calculate sales shares λ, which can

be combined with the values for aggregate expenditure on tradables

XT to arrive at the second set of model-implied moments δ.

To summarize, values for the composites PTj , wj/Tj, wj f jj deliver

a set of model-implied values for the moments µ and δ. The GMM

routine minimizes the sum of squared residuals between the model

moments and their empirical counterparts, conditional on the equi-

librium constraint ensuring the values for price indices are consistent

with optimal firm-level pricing behavior. Each moment is weighted

equally. Note that the GMM routine does not ultimately return pure

estimates for the location fundamentals, but instead for the compos-

ites wj f j, wj/Tjj. Separating out the location fundamentals f j and

Tj is discussed in the next section.


Parameter Description Estimation strategyfe entry fixed cost free entry (2.9)uα(1−fi)H−(1−η) utility composite utility equalization (2.1)b portfolio share balance of payments (2.2)

Table 2.3.: A summary of parameter estimation from inverting equi-librium conditions using GMM estimates

Inverting equilibrium conditions

The remaining location fundamentals are determined by inverting

equilibrium conditions from the model. These are summarized in

Table 2.3. They are the fixed cost of firm entry f ej , a utility composite

ujαη(1−β)j H−(1−η)

j , and the capitalist claim on the global portfolio bj.7

These fundamentals each appear in a single general equilibrium

condition. The fixed costs of firm entry are associated with the free

entry conditions, the utility composites with the free mobility condi-

tions, and the global portfolio shares with the balance of payments

7The composite term contains the utility shifter in each country uj, labor produc-tivity in the nontraded sector αj, and the country’s stock of housing Hj. The indi-vidual elements in the composite term are not separately identified because theyenter indirect utility (2.1) in an identical way. However, the composite term is suf-ficient to compute equilibrium outcomes of interest in the counterfactual, namelythe workforce Lj in each country j.


conditions. The location fundamentals are therefore pinned down

by their respective aggregate conditions, since data for the mass of

firms, workforce in each country, and tradables expenditure are used

as the equilibrium values of model counterparts. More specifically,

the aggregate conditions can each be inverted to recover the location

fundamentals that exactly rationalize the data. Consider the free entry

condition as an example, which is replicated below for convenience.

1ρ ∑

j,k

(1− τj

)λijkXT

k −∑j

Miµijwj fij −Miwi f ei = 0

In this expression, the elasticity of substitution ρ, tax rates τ, aggregate

expenditures on tradables XT, and mass of active firms M are all de-

termined by the calibration step. Sales shares λ and FDI participation

shares µ are constructed during the estimation step. The estimation

step also returns estimates for the composites wj fijj. Then, only the

cost of entry wi f ei remains, so ensuring that the free entry condition

holds determines its value. A similar logic can be applied to the free

mobility and balance of payments conditions conditions.

Finally, inverting the labor market clearing condition (2.11) in a


similar way solves for nominal wages wj. The fundamentals Tj, f jj

can then be separated from the composite estimates wj f j, wj/Tj re-

turned by the GMM estimation. This step completes the calibration

and estimation procedure.

2.4.2. Computing the firm’s problem

The main challenge in the calibration and estimation routine above is

determining each firm’s optimal plant location strategy. Not only is a

single firm’s problem a complex discrete choice problem over 227 pos-

sible choices, which could be computationally burdensome to solve

once, the entire distribution of firm behavior must be solved to verify

aggregate conditions. Additionally, the distribution of firm behavior

must be solved repeatedly as the estimation routine searches for a mini-

mum. The need to solve the plant location problem at all levels of firm

productivity many times has the potential to render the estimation

routine prohibitively slow. This section describes the approach taken

to maintain the computational feasibility of the estimation routine.


Solving for policy functions

To address the challenge of solving plant location problems for the

entire continuum of firms in each country, the method from Chap-

ter 1 is applied. It finds the policy functions J ∗i (·) for each origin

country i which map firm core productivity z to their optimal set of

plants. The key advantage to using the policy function method is that

it determines the entire range of firm behavior quickly, which allows

for exact aggregation while conserving computational time. A more

in-depth discussion of these benefits are described in the next section.

As outlined in Chapter 1, the policy function method can be applied

when the firm’s plant location problem is a combinatorial discrete

choice problem satisfying monotonicity in set and monotonicity in

type. In the context of the plant location problem, monotonicity in

set disciplines the way one plant can interact with others while mono-

tonicity in type characterizes how the firm’s core productivity affects

its profits. Since bilateral costs of arms-length production γ result in

different profit functions for the origin countries i, the policy function

approach must be applied separately for each. These conditions are


therefore verified for every country of origin i.

The key function needed to verify both conditions is Dij(J ; z, υ),

which defines the marginal value of adding a plant in j for a firm of

type z based in country i, conditional on opening plants in J . For the

plant location problem, it is the change in profits when the plant in j

is added to the set J , given by expression (2.5). It is replicated below

for convenience.

Dij(J ; z, υ) = Az ∑k

XTk(

PTk

)1−ρ

(κijk +

(∑

j 6=j′∈Jκij′k

)) ρ−1θ

−(

∑j 6=j′∈J

κij′k

) ρ−1θ

− wj fij

κijk =(1− τj

) ρ−1θ(wjdjkγij/Tj

)−θ

As discussed when introducing the firm’s plant location problem in

Section 2.2.3, the marginal value function encapsulates the trade-off

between the expected increase in variable profits from adding the

plant in j with the fixed cost associated with its establishment. In ac-

cordance with the notation from Chapter 1, the marginal value func-


tion is written to explicitly include υ as an argument, the vector of

aggregates relevant for the problem. In this context, it contains trad-

ables expenditure, tradables prices, and wages in each country so that

υ = XTk (PT

k )ρ−1, wkk. The composite aggregates are needed to cap-

ture the relative size and importance of each market k, while wages

are needed to summarize costs faced by the firm.

Start with monotonicity in set and let J1 ⊂ J2. To satisfy the con-

dition, it is sufficient if Dij(J1; z, υ) ≤ Dij(J2; zυ) for any i, j, z, and

υ. In other words, it is enough for the marginal value of opening a

plant in j to be decreasing as the set J grows. Recall that, since the

exponent (ρ− 1)/θ on profit potentials is below one, expected prof-

its are concave in profit potentials and exhibit decreasing differences

as the set J grows. The intuitive interpretation of this property of

the marginal value function is that the plants are substitutes for each

other. It is therefore easy to see that the plant location problem satisfies

monotonicity in set.

Now turn to monotonicity in type. This condition requires that the

marginal value function increases in z for all plants j, sets J , and ag-

gregate vectors υ. It is immediately clear that this condition holds for


every origin i. Holding all else constant, a more productive firm will

derive larger gains from adding a plant compared to a less productive

firm. The larger gains stem from the fact that the firm’s core produc-

tivity increases average plant efficiency holding all else constant. For

example, return to the interpretation that the firm’s productivity re-

flects its management style while plant efficiency reflects a plant’s

access to local factors. An additional plant is valuable to firms be-

cause it could have very good access to local factors, ultimately being

selected to serve a market k. Firms with better management styles can

extract relatively more value from the well connected plant than firms

with worse management styles. The plant location problem therefore

satisfies monotonicity in type.

Since the plant location problem exhibits both types of monotonicity,

the algorithm developed in Chapter 1 can be readily applied to solve

for the policy function J ∗i (·; υ). In fact, observe that the marginal ben-

efit of any j’s inclusion into J is linear in core productivity z. This

functional form is particularly convenient for applying Chapter 1’s

algorithm, which relies on finding productivity cutoffs z for various

sets J such that the marginal value Dij(J ; z, υ) is zeroed. With linear-


ity in z, identifying these cutoff values z is especially straightforward

given any set J and vector of aggregates υ.

Gains from policy function method

Because this chapter is the first study to apply the policy function

method developed in Chapter 1, the section concludes with a quan-

titative exploration of the algorithm’s advantages compared to the

existing method used by the literature. Crucially, the policy function

method determines the optimal plant location strategy for the entire

continuum of plant productivity z. As a result, the integrals arising in

the model’s aggregate conditions can be calculated exactly. For exam-

ple, verifying the free entry condition requires the average firm’s total

profits, which integrates over the distribution of firm choices at each

firm productivity. Because the model is both estimated by matching

moments from aggregate data then leveraged to make aggregate pre-

dictions in a policy counterfactual, precise aggregation is necessary.

In contrast, the current method employed by the literature would

solve the plant location problem separately for a discrete number of

firm productivities. More specifically, a solution method such as the


ones proposed by Jia [2008] or Arkolakis and Eckert [2017] would

be applied independently at each point. When integrating over the

distribution of productivity for aggregation, an approximation must

be calculated from these discrete grid points. The type-by-type dis-

cretization alternative therefore introduces a trade-off between solu-

tion speed and accuracy. Covering the range of productivity with a

denser grid implies more points at which to solve the plant location

problem, but likely results in better approximations when calculating

aggregates. A sparser grid entails fewer points at which to solve the

plant location problem, but could result in more inaccuracy for aggre-

gates. On the other hand, the policy function method does not require

the researcher to select a grid over productivity.

To concretely illustrate this point, consider an equilibrium variable

from the model like the sales in market k of an average firm from i

produced by its plant in j. It is aggregated as

xijk ≡ AXT

k(PT

k

)1−ρ

∫z

zΘik (J ∗i (z))ρ−1

θpijk(J ∗i (z)

)1− τj

dGi(z) . (2.13)

The expression inside the integral includes the firm’s choice of plant


locations J ∗i (z) for every possible level of productivity z. If the policy

function is approximated by discretizing the range of productivity and

finding the optimal set at each grid point, then it must be interpolated

between grid points. Aggregation error will be introduced when cal-

culating the integral. The approximation becomes more precise with

a denser discretization, but then a single-point solution method must

be evaluated at more points.

On the other hand, the integral can be exactly calculated if the full

policy function is found. Moreover, it can be greatly simplified. In

particular, the solution method specifies the policy function by parti-

tioning the range of z into subintervals. The subintervals are such that

firms with productivity levels within each subinterval all share the

same optimal set, which are also computed by the algorithm. Then,

indexing the subintervals by e so that zei is the interval’s left endpoint,

the policy function takes the form

J ∗i (z) = J ∗ei z ∈[ze

i , ze+1i

).

In other words, the policy function has a simple form where it is con-


stant within each subinterval. Observe that Θik(·) and pijk(·) only

take the optimal set J ∗i (z) as an argument for each z, so they are also

constant within the subintervals. For notational simplicity, let these

values be Θeik and pe

ijk respectively. Then, (2.13) can be rewritten as

A ∑j,k

XTk(

PTk

)1−ρ ∑e(Θe

ik)ρ−1

θpe

ijk

1− τj

∫ ze+1i

zei

zdGi(z) .

Since the core productivity distribution Gi(·) is Pareto, it can be ana-

lytically integrated over z. Then, the production of an average firm is

not only exactly calculated, but also in a simple manner.

Consider the following experiment to compare the policy function

approach with the discretization method. Fix parameters, location

fundamentals, and aggregates at the levels determined by the esti-

mation process. Conditional on these values, time the policy func-

tion method as it solves for the optimal behavior of German firms

J ∗G(·; υ).8 The outcome of this approach can be captured by a single

point in speed-accuracy space. Of course, because the policy func-

8Germany is chosen as a representative origin country, but the arguments madewill hold for any origin country.


tion method does not include any approximation, it will have perfect

aggregation accuracy. Next, discretize the range of German firm pro-

ductivity with a grid of values. Solve the plant location problem at

each point, then use the output to approximate equilibrium aggre-

gates. By repeating this process with grids of various densities, trace

out the discretization method’s speed-accuracy frontier. Finally, com-

pare the position of the point representing the policy function method

in speed-accuracy space with this frontier.

This section completes precisely this experiment, where simple

computation time is the measure of speed. Total production of an

average German firm ∑ijk xDE,j,k is used in constructing the measure

for accuracy. This aggregate is selected because it is easily interpreted

while also appearing in a majority of the equilibrium conditions. In

particular, observe that total production by i-based firms in j plants

for market k is simply the average firm’s production multiplied by

the mass of active firms Mixijk. The total production of i-based firms

in j plants for market k is in turn used in the free entry, balance of pay-

ments, and labor market clearing conditions. Computing the average

firm’s production is therefore needed to find general equilibrium and


estimate the model accurately.

Recall that the goal of the quantitative model is to capture the activ-

ity of multinational firms, since their behavior is central to ultimately

understanding the counterfactual effects of a tax harmonization pol-

icy. The model’s key feature is therefore allowing firms to choose any

set of countries in which to locate production, crucially allowing it to

occur outside of the country of birth. Then, define

xspi ≡ A ∑

j,k

XTk(

PTk

)1−ρ

∫z

zΘik(i)ρ−1

θpijk(i)

1− τjdGi(z)

as the total production of an average firm based in i when restricted

to a single (domestic) plant restriction, holding fixed all equilibrium

aggregates. The model’s key feature is therefore that this average

differs from the average firm’s production xi ≡ ∑j,k xijk if this restric-

tion were lifted, so the difference xi − xspi encapsulates the effect of

allowing multinational production. Moreover, since tax rate disper-

sion primarily acts through distorting the decisions of multinational

firms, it is important to capture these “multinational forces” well. The


Figure 2.2.: Comparing the speed and error of the policy functionmethod (red) with the discretization method (blue)

The grid density is printed next to each blue point. Sparser ones computequickly but with more aggregation error. Solving for the policy function withChapter 1’s method is both quick and exact.

accuracy measure is therefore defined as

1−∣∣∣∣∣x

gi − xsp

i

xi − xspi

∣∣∣∣∣where xg

i is the approximation of xi derived from a g-point discretized

grid. In words, the accuracy measure represents the extent of “multi-

national forces” captured by the g-point discretization.


Figure 2.2 graphs the results from this experiment. To begin, the

large red point in the lower left corner represents the results from us-

ing Chapter 1’s method. It computes the full policy function in under

0.05 seconds.9 As discussed above, it also exactly calculates the aver-

age firm’s production, so it has zero aggregation error. The smaller

points in blue plot the results from the discretized grids, ranging in

density from 13 grid points to 769. As expected, there is a negative

relationship between accuracy and speed. In particular, sparse grids

take a relatively short time to compute, but return worse approxima-

tions. The opposite is true for denser grids.

Most importantly, the policy function method outperforms the dis-

cretization method even on the sparsest grid, in terms of both compu-

tation time and aggregation error. While it completes in 0.05 seconds

and entails no aggregation error, the 13-point grid takes around the

same time to compute but has almost 20% error. Overall, it takes

considerably more computation time for the discretization method

to approach the accuracy of the full solution method. The 769-point

grid is close, but takes over 3 seconds. The main reason is that a per-9Calculations were done in Julia on a 2018 Intel i7-8550U CPU.


fect discretization would place grid points precisely according to the

interval endpoints returned by the policy function method, so there

is no interpolation error between points. However, it is difficult to

predict these values with no prior knowledge, so a dense discretiza-

tion is necessary to cover as many subintervals as possible. The policy

function approach conversely delivers numerical precision without

compromising on solution speed.10 Its performance therefore renders

feasible the estimation procedure outlined in the previous section. It

is also necessary to compute the counterfactual general equilibrium

presented in Section 2.5.

Finally, note that the numerical exercise only considered the av-

erage German firm’s production for simplicity, which sums the in-

tegrals over all possible plant locations j and markets k. However,

some equilibrium conditions like labor market clearing require each j

term separately. Therefore, aggregation accuracy may be worse than

what is represented in this simple exercise. Moreover, aggregation

error measures in this exercise are derived from computing the aver-

10For a further discussion on how the computation method is affected by thestrength of substitutability among plants, see Appendix B.


age firm’s production while holding aggregates and parameters fixed.

However, once the firm block is embedded in a general equilibrium

routine, deviations in this integral will affect many general equilib-

rium conditions. Computed general equilibrium could potentially be

very inaccurate as a result. A similar argument can be made for pa-

rameter estimates recovered from the GMM routine, which minimizes

aggregate moment subject to aggregate equilibrium constraints.

2.5. Counterfactual experiment

The estimated quantitative model can now be used to evaluate the

effects of a tax harmonization, a proposal that has been the subject of

policy discussion among EU member states in recent years. As high-

lighted by the structural model, tax rate differentials distort incentives

in the firm’s production along both the sourcing and plant location

margins. Imposing a common corporate tax rate across the member

countries would eliminate these distortions, changing the production

patterns of multinational firms. Because multinationals account for a

disproportionate share of aggregate production, there could be gen-


eral equilibrium effects on aggregates like wages, prices, and the level

of industry in each country. This section investigates these questions

with a counterfactual experiment where taxes are equalized in the

estimated quantitative model.

2.5.1. Results

The optimal tax rate is beyond the scope of this paper. Instead, the

counterfactual searches for an equalized tax rate τ generating the same

amount of EU-wide tax revenue ∑j Rj as the calibrated model. As

discussed in Section 2.2, taxes on firm profits have both a level and

dispersion effect. The rate is chosen in this way to determine the effect

of tax rate dispersion only, while holding constant the level effect as

much as possible.

The harmonized rate generating the same EU-wide tax revenue

turns out to be around 27.26%. Figure 2.3 summarizes the changes

to tax rates implied by adopting this rate across the union. The coun-

terfactual policy implies tax cuts for only five countries, which are

referred to as the “cutters”. In order from the largest to smallest cuts,


these countries are Belgium, France, Italy, Germany, and Spain which

are emphasized in the figure with a black border. Cuts range from

over 5 percentage points in Belgium and France to under one percent-

age point in Spain. The remaining countries see tax rate increases,

ranging from 1.26 percentage points in Greece to 17.26 percentage

points in Bulgaria. A complete summary of these tax rate changes are

listed by Table B.1.

The spatial distribution of production

When this counterfactual rate is implemented in the model, aggre-

gate EU production in the tradables sector increases around 0.33%.

However, this figure belies considerable redistributive effects across

countries. Once tax rate variation is eliminated, a larger portion of

real tradables production is carried out in countries with previously

high tax rates. This share is defined as

δQj ≡

QTj

QT ≡∑i,k λijk

(XT

k /PTk)

∑i,j,k λijk(XT

k /PTk

)


which captures the quantity of tradables produced in j divided by the

total quantity of tradables produced in the EU.11 Before harmoniza-

tion, the cutters combined accounted for 64.65% of total EU tradables

production. This share increases by almost one percentage point to

65.61% following the policy change. The main drivers behind this

shift are France and the United Kingdom.

Figure 2.3b depicts the percentage point change in these shares for

each country following tax rate harmonization. Countries in blue see

their share of total EU production increase while countries in red see

their share decrease. Comparing the two panels of Figure 2.3, there is

a strong link between the tax rate change in a country and the share of

EU tradables production carried out there. France and the UK stand

out the most.

The largest increase is the French share, which gains half a percent-

age point from 12.09% to 12.59%. This result is likely because France

had the second highest tax rate before harmonization, while also being

11These shares differ from the shares δj used in the estimation, because they rep-resent the real quantity of tradables produced by each country. The shares usedin the estimation instead represent the total share of nominal EU tradables salesproduced by each country.


(a) p.p. change: tax rates (b) p.p. change: share of total EU trad-ables production δ

Figure 2.3.: The 27.26% harmonization rate’s implications for tax ratesand location of industry. Countries with lowered tax rates, outlinedin black, account for a larger share of EU production.

relatively productive. In particular, it both hosts relatively productive

plants and is home to relatively productive firms as measured by the

estimated values for Tj and zminj , respectively.12 In addition, France

constitutes a large market as well as being close to other large markets.

12French plants are estimated to be around 68.35% as productive as German plants,while French firms are around 85.41% as productive.


Once its tax disadvantage relative to other countries is eliminated, it

therefore becomes very profitable to situate production there.

On the other hand, the United Kingdom loses the largest share of

EU production, dropping almost half a percentage point from 8.54%

to 8.10%. What explains this large drop? While the UK increases tax

rates as a consequence of the harmonization, it does not do so by a

relatively large amount. Before the counterfactual policy, it levied a

20% tax rate on profits, placing it squarely in the middle of the tax rates

distribution. Referring to Table B.1, which lists the full set of member

countries and their tax rates, thirteen countries have the same rate as

the UK or lower. However, upon closer inspection, these countries are

all small and located in the Eastern half of Europe. Compared to these

countries, the UK is more productive in terms of both plants and firms.

It is also well situated in relation to large European markets like France

and Germany, while itself being a large market. In terms of market

access and productivity, it is the only low tax country comparable to

the large high tax countries. Prior to tax harmonization, the UK is

therefore an attractive country in which to locate production. Once

the policy is implemented, however, its tax rate advantage disappears.


Domestic and multinational activity

Under the harmonized corporate tax regime, tradables production

relocates towards cutters. To investigate this relocation more deeply,

let x′ represent the value of a variable x under the counterfactual pol-

icy. Further, let x = x′/x be the proportional change between its

value in the estimated equilibrium and the counterfactual. Note that

100(x − 1) ≡ ∆%x is the corresponding percentage change in the

variable x.

The percentage point change in a country j’s share of EU production,

which was examined to uncover the patterns of industry relocation

as a consequence of the counterfactual policy, can now be expressed

as

100(

δQj′− δQ

j

)=

δQj

QT

[∆%QT

j − ∆%QT]

.

This expression rewrites the percentage point change in a country’s

share δQj of EU production in terms of the percentage change in its

tradables production ∆%QTj in comparison to the percentage change

in total EU production ∆%QT. If a country’s production grows by

more than total EU production so that ∆%QTj − ∆%QT > 0, then


its share of total EU production δQj consequently grows. However,

large percentage changes can still correspond to relatively small level

changes since percentage changes scale a variable’s change by its ini-

tial size. Therefore, these percentage changes are scaled by the coun-

try’s initial share of total production. If the country initially accounts

for larger proportion of total EU production, it produces relatively

more. A large percentage change in its production would then trans-

late to larger level change, in turn corresponding to a larger change

in its share of total EU production.

To understand the change in a country’s share of total EU produc-

tion, it is therefore important to characterize how its own production

QTj changes. To that end, note that total production in a country j can

be decomposed into domestic and foreign activity as follows

QTj = ∑

kλjjk

(XT

k /PTk

)+ ∑

i 6=j,kλijk

(XT

k /PTk

)≡ Qd

j + Q fj .

The first term represents total production carried out by domestically-

based firms (d) while the second term captures production attributable

to foreign-operated affiliates ( f ). The change in a country’s production


can be similarly decomposed as

∆%QTj =

(Qd

j

QTj

)∆%Qd

j +

Q fj

QTj

∆%Q fj .

This expression expresses the percentage change in the country’s to-

tal production as a weighted average of the percentage change in

domestic and foreign production. The weights are simply the share

accounted for by each type of production in the country’s initial pro-

duction. Given this decomposition, the policy’s impact on the distri-

bution of domestic and multinational activity are now both examined.

Domestic activity

Begin with the patterns of domestic production. Figure 2.4a shows

the percentage change in domestic production Ydj for each country

once tax rates are equalized. It increases in all cutters except Spain

and decreases in all other countries. Notably, the increase in domestic

production is the highest in France at 4.67%, in line with France ac-

counting for a larger share δj of EU production. The UK sees domestic


(a) % change: domestic production (b) % change: firm mass M

Figure 2.4.: After harmonization, most cutters (outlined in black) seean increase in domestic production. All cutters have a larger massof active tradables firms.

production drop by 5.23%, also consistent with the change in its EU

production share.

Since there is no fixed cost to establish a domestic plant, all entering

firms remain active in their country of birth. As a result, the mass of

entrants in each country has implications for the patterns of domestic

production. Figure 2.4b summarizes the changes in the mass of en-


trants, revealing a clear relationship between the tax rate change and

the change in mass of firms. The mass of entering firms increases in

all cutters while decreasing in the other countries. The magnitudes

of these changes can be quite large. For example, the total mass of

firms entering in France increases by 12.26%. On the other hand, there

are large declines in many Eastern European countries. The mass of

firms entering in Cyprus, Latvia, and Bulgaria fall the most, by 26.93%,

19.63%, and 18.72% respectively. The magnitude of these declines may

be surprising given the small change in EU production shares of these

countries. However, while the level changes in production for these

countries are small relative to total EU production, they are large in

proportion to their small economies. consequently, the percentage

changes are large, reflecting a significant effect on their economies.

The UK, which has the largest losses in the share of EU production,

has a nontrivial drop of 12.19% in the mass of entrants.

To summarize, there is an intuitive relationship between a country’s

tax rate change, its domestic production volumes, and its mass of

entering firms. These patterns are in line with the broader patterns of

industry relocation described in the previous section.


Multinational activity

Now turn to production carried out by the establishments of foreign-

based firms. Figure 2.5a shows the percentage change in foreign affil-

iate production. In contrast to the previous variables, multinational

production expands most notably in Germany, increasing by 31.97%.

This change is qualitatively reflective of both the tax policy change

and industry relocation patterns. After harmonization, German tax

rates decrease around 3 percentage points, while the German share of

EU production increases by 0.29 percentage points. German domes-

tic production rises, as shown in Figure 2.4a, and its foreign affiliate

activity increases even more relatively.

On the other hand, Belgium sees the largest declines, with a drop

of 58.52% in foreign affiliate production. Surprisingly, this drop coin-

cides with a 6.7 percentage point drop in its tax rate. If the Belgian

tax rate decreased, why did its foreign affiliate production decrease

as well? It is likely because many firms that previously chose to open

plants in Belgium did so given its proximity to the large nearby mar-

kets of France, Germany, and the UK. At the same time, the parame-


(a) % change: hosted foreign production (b) % change: hosted foreign plants

Figure 2.5.: After harmonization, production carried out by foreignaffiliates increases considerably in Germany and decreases consid-erably in Belgium.

ter governing the average Belgian plant’s productivity Tj is relatively

high while the fixed cost of plant establishment f j is relatively low.13

Taken together, it is comparatively more attractive to locate produc-

tion in Belgium than the other cutters before harmonization. However,

13Choosing the German productivity TG = 1 as a normalization, the Belgian pa-rameter is estimated to be around 0.85. This normalization amounts to choosingthe units for the traded good. The fixed cost of establishing a Belgian plant is esti-mated to be 25% lower than the fixed cost of establishing a German plant.


once the tax rates are equalized, firms may instead open affiliates in

the large countries directly. This pattern highlights the role of allow-

ing plants to serve as export platforms in the model. If plants could

only serve the domestic market, then Belgium would likely not be as

attractive for opening establishments and completing production.

The changes in the number of foreign affiliates hosted by each coun-

try is consistent with this interpretation. As shown in Figure 2.5b, the

number of plants in the other four cutters increases. In the case of Ger-

many, there is a dramatic increase of 42.18%. Conversely, the number

of foreign affiliates owned by other countries decreases by 59.08% in

Belgium. Most of the non-cutter countries also see decreases, though

smaller.

Having examined the “inward” flows of multinational production,

Figure 2.6a now summarizes the “outward” patterns of multinational

production. For each country i, this aggregate is the total arms-length

multinational production completed by the foreign affiliates of firms

originating within the country ∑j,k λijk(XTk /PT

k ). The standout coun-

try is France, where total outward multinational production increases

by 46.09%.


(a) % change: arms-length production (b) % change: arms-length plants

Figure 2.6.: After harmonization, firms originating in most cuttersconduct more arms-length production and open more foreign affil-iates.

The changes in arms-length production are qualitatively very simi-

lar to the pattern of change in arms-length plants, which are shown

in Figure 2.6b. Because French tax rates were very high before har-

monization, it becomes significantly more profitable to operate as a

French firm following the policy. French firms are therefore able to

open more foreign plants, which pushes up arms-length production.


A similar effect occurs in the other cutters but to a lesser degree, while

the opposite is true for other countries. As they raise tax rates, it be-

comes less profitable for firms born there, resulting in firms opening

fewer foreign plants and depressed arms-length production.

Finally, the changes in the location of domestic and foreign pro-

duction taken together can explain why a selection of Eastern Euro-

pean countries see an increase in hosted affiliate foreign production,

despite their higher tax levels. This pattern is visible in Figure 2.5.

These countries experience a decrease in the mass of entering firms

and arms-length production as a result of the higher tax rates. What

makes foreign affiliate production different? The reason inward multi-

national production increases in Eastern Europe is that the workers

and capitalists based there still constitute consumer demand. With

domestic production activity decreasing in light of the increased tax

rate pressure, these consumers are increasingly served by foreign af-

filiates, especially those owned by firms based in the cutters. Because

multinational firms with foreign affiliates in these Eastern European

countries tend to be more productive on average than the domestic

firms there, they are less negatively impacted by the higher tax rates.


Labor market effects

The expansion of industry in cutters creates a labor demand effect in

these countries, as firms and plants are opened there. Labor demand

correspondingly drops in other countries. The EU labor market ad-

justs in reaction to this effect through both the spatial distribution of

workers and local wages. Figure 2.7a displays the percentage change

in worker population Lj in each country while Figure 2.7b shows the

percentage change in the local real wages, which are calculated by de-

flating nominal wages with the local price index in each country. The

adjustment accounts for the price of consumable goods, both traded

and nontraded, as well as housing.

As displayed in Figure 2.7a and Figure 2.7b, relocation of industries

to cutters causes a labor demand effect which increases both workforce

and real wages in most cutters. As more firms and affiliates open there,

real wages are bid up, attracting more workers as they freely move

within the EU. For example, real wages in France increase by 1.60%

while workforce population also grows by 1.72%. Worker population

also grows in three non-cutter countries, which are the Netherlands,


(a) % change: workforce (b) % change: wage

Figure 2.7.: As labor demand rises in cutters, workers migrate there.In tandem, local wages drop in the other countries, especially inEastern Europe.

Hungary, and the Czech Republic. Production by foreign affiliates

located in these countries also increased, possibly accounting for a

portion of the workforce increase.

An opposite effect largely occurs in the other countries. For exam-

ple, production leaves the UK as established previously, which is the

largest non-cutter country. As a reflection of this relocation, workers


also migrate out and real wages drop. Overall, the workforce pop-

ulation in the UK declines by 2.57% while real wages fall by 1.56%.

A similar effect occurs in Bulgaria, the country with the previously

lowest tax rate. Its share of total real EU production in the estimated

model is 0.37%, which decreases to 0.36% with the counterfactual pol-

icy. While it is a small percentage point change, it is relatively large for

the Bulgarian economy. In response, the Bulgarian worker population

declines by 0.67% with real wages decreasing by 4.26%.

Trade flows

As the spatial distribution of production is rearranged in response

to the counterfactual policy, trade flows adjust. Figure 2.8a summa-

rizes these changes for gross exports while Figure 2.8b shows these

changes for gross imports, which are defined using the model as

∑i,k 6=j λijk(XTk /PT

k ) and ∑i,j 6=k λijk(XTk /PT

k ) respectively. Overall, ex-

port flows from cutters increase, while import flows decrease in East-

ern European countries. On the other hand, the countries with the

largest increase in gross imports are the UK, Belgium, and the Nether-

lands.


(a) % change: exports (b) % change: imports

Figure 2.8.: Once tax rates are harmonized, trade outflows from cut-ters increase. Imports fall in many Eastern European countries butincrease for the UK, Belgium, and the Netherlands.

Of the cutters, the increases in export flows are largest for France

at 4.51%, followed by Italy at 1.92%. Since production relocates to

cutters, especially to France, they increasingly serve the remaining

countries in the union through exporting. In particular, the countries

with the largest increase in quantity imported are the UK, Belgium,

and the Netherlands, whose imports increase by 0.57%, 1.14%, and


2.03% respectively. Importing into these countries likely increases

because they maintain a relatively level worker population while also

accounting for a smaller share of total production. Given these two

changes, consumers there increasingly rely on imported tradables

over domestically-produced tradables. In the case of Belgium, real

wages also increase, as shown in Figure 2.7b, further driving possible

expenditure on imports.

On the other hand, many countries in Eastern Europe experience

a decline in total real traded imports. The largest decreases occur

for Cyprus, Latvia, and Bulgaria, where quantity imported falls by

2.29%, 1.99%, and 1.89% respectively. Though production has de-

creased in these countries, possibly increasing the reliance on imports,

their worker populations also decline. The overall effect is a decrease

in total quantity of goods imported.

Worker consumption

As production patterns relocate across space and workers respond

to spatial local demand, what happens to the worker consumption

in each country? With free mobility and utility equalization in all


countries, the real consumption per worker must remain equalized

across countries. In other words, when nominal worker expenditure

is adjusted for the price of consumer goods and housing, it must be

the same in every country. Otherwise, if one country had higher real

consumption, workers would move there, increasing the labor supply

and housing demand. Real wages and tax revenue rebated per capita

would decrease as rents increase, equalizing real consumption. With

equalized real consumption across all countries both before and after

the new tax policy is implemented, the change in real consumption

per worker is the same in each country.

However, while the real consumption per worker is equalized in

all countries, its composition can change differentially. Figure 2.9a

shows the percentage change in real traded consumption per worker

while Figure 2.9b shows the same information for real nontraded

consumption. Using the structure of the model, these quantities are

∆%(XTk /PT

k ) for tradables consumption and ∆%(XNk /PN

k ) for non-

tradables consumption in each country k.

As tradables production is relocated into the cutters, workers living

there consume more tradable goods. On the other hand, the quantity


(a) % change: real traded consumptionper worker

(b) % change real nontraded consump-tionper worker

Figure 2.9.: As traded production increases in cutters, consumptionof tradable goods increases in these countries. Consumption shiftstoward nontraded goods in other countries.

consumed of nontraded goods increase for the representative worker

living in the other countries. At 1.74%, the largest increase in quan-

tity consumed per worker of the traded basket occurs in France. On

the other hand, the largest drops occur in Cyprus at 2.62%, Latvia at

2.06%, and Romania at 2.03%. Given the Cobb-Douglas structure of


demand, workers spend a constant share of nominal income on each

sector. However, as the price of the good changes, the real amount

consumed changes accordingly. For workers in Eastern European

countries, the price of the traded basket increases as production re-

locates to cutters because prices are set at a constant markup over

marginal costs. Since a larger fraction of traded goods are now pro-

duced farther away, the price index incorporates the higher bilateral

costs of trade. In addition, the price of a traded variety reflects the

local wage of where it is produced. As nominal wages are pushed

up in cutters but fall in Eastern Europe, the price of the traded good

increases relative to Eastern European incomes. An opposite logic

holds for the cutters, which feature increased tradables consumption

per worker. As a larger proportion of the traded basket is produced lo-

cally or in nearby countries, prices do not incorporate as high a trade

cost as before. While the nominal wage increases in cutters, feeding

into prices, it also implies an increase in the nominal incomes of their

workers.

The final component of worker consumption is housing, which is

simply Hk/Lk, the total stock of housing endowed in a country di-


vided by the number of workers living there. Because Hk is fixed

and does not change between the estimated equilibrium and the equi-

librium under the counterfactual tax policy, the amount of housing

consumed per worker in each country moves perfectly against the

change in worker population. Referring to Figure 2.7a, it is therefore

easy to conclude that workers in cutters consume less housing un-

der the new policy as these countries become more congested while

workers in most Eastern European countries consume more.

Lastly, observe that these changes in consumption do not imply

that, for example, all workers originally living in Cyprus experience a

2.62% drop in real consumption of traded goods. Individual workers

originally living in Cyprus may move to a different country during the

adjustment to the new corporate tax policy. Rather, these changes re-

flect a cross sectional difference, comparing the representative worker

in Cyprus before the policy’s implementation to the representative

worker there afterwards.


2.5.2. Discussion

In the context of the political discussions regarding EU tax harmoniza-

tion, the policy has generally been supported by officials from higher

tax countries like France and Germany. Policymakers there argue that

low tax countries my have higher levels of industry and more jobs

than they otherwise would if tax incentives were neutralized. Un-

surprisingly, tax harmonization has been opposed by countries that

have set lower taxes, such as Ireland and Hungary, who argue for tax

competition.

The counterfactual experiment of the estimated structural model

predicts redistributional consequences in line with the concerns of

these policymakers. Overall, previously low tax countries lose some

of their ability to attract firms and affiliates, therefore accounting for

a lower share of total EU production. In turn, their gross exports, em-

ployment, and real wages drop as production is reallocated to those

previously high tax countries that lower their tax rates during the har-

monization. Quantitatively, the predictions are nontrivial, with up to

12.67% more firms in the cutters and up to 1.72% more workers, while


the other countries see up to a 26.93% decrease in firms as well as up

to a 2.57% decrease in workers. These redistributional consequences

could be unattractive to the low tax countries, whose co-operation

would be necessary in enacting a harmonization policy.

However, it is also possible that these redistributional consequences

receive resistance in the current high tax countries. In particular, the

counterfactual expansion of industry in these countries will be sup-

ported by population inflow from the other countries. This inflow

could cause difficulties beyond the scope of this chapter. For exam-

ple, tax revenue is lump-sum rebated to workers in the current model

to maintain simplicity. With a more complex government sector, an

increasingly diverse population could increase the per capita cost of

providing public goods and services.

2.6. Conclusion

Through free trade, a monetary union, and policy to encourage the

free movement of labor, the EU takes steps towards integrating its

member states into one economic unit. However, they still remain


considerably differentiated along other dimensions, such as fiscal pol-

icy and taxation. In particular, countries independently set their cor-

porate tax rates. This tax rate dispersion has been the topic of recent

policy debate, with some political leaders raising the possibility of tax

reform that would unify tax policy across the countries.

In the presence of taxation, firms will optimally choose production

locations taking tax burden into account. Since profits are effectively

taxed in the location of production within the EU, firms will account

for differential tax incentives across countries when they make lo-

cation decisions. Moreover, multinational firms in the EU account

for a disproportionate share of total production and sales. Therefore,

factors affecting their production locations such as tax rate disper-

sion could have general equilibrium effects on industrial location, real

wages, and other aggregate variables.

This chapter builds a quantitative general equilibrium model to bet-

ter understand the effects of tax dispersion on the spatial allocation

of economic activity and general equilibrium outcomes. The model

crucially features endogenous multinational production, where firms

optimally choose a set of countries in which to set up foreign affiliates.


To firms, each country represents a bundle of relevant characteris-

tics, including productivity factors and tax incentives. Endogenously

arising multinational firms therefore trade off a rich set of incentives

when making FDI decisions, including a potential plant’s expected

production capability, market access, tax burden, and fixed cost of

establishment. Then, the model is calibrated to aggregate EU data

before it is used to complete a counterfactual experiment to assess the

potential effects of the proposed tax reform. To compute and estimate

the general equilibrium model, the method described in Chapter 1 is

operationalized to solve the firm’s complex plant location optimiza-

tion.

The precise counterfactual exercise undertaken is one where taxes

are equalized across the EU at the common rate that would generate

the same total tax revenue as in the estimated equilibrium. Harmo-

nization leads to a reallocation of production to the countries that

lower their tax rates (“cutters”) from those that raise it. There are five

cutters, corresponding to large EU countries with currently high taxes.

They are Belgium, France, Italy, Germany, and Spain. In particular,

both domestic and foreign affiliate production increase in these coun-


tries, as both the mass of domestic entrants as well as the number of

foreign affiliates present increase. An opposite effect occurs in the

others. Overall, cutters account for a larger share of EU production

once the reform is implemented.

This redistributional effect on the spatial patterns of production

leads to aggregate effects on trade flows and labor markets. Because

more relatively production occurs in cutters, trade outflows from

these countries increase while dropping in others. In other words,

as production is shifted into cutters, these countries begin supplying

the others. In labor markets, this industry relocation creates a labor

demand effect, as more firms and establishments open. Accordingly,

real wages are bid up in cutters, which experience a corresponding

inflow of workers as they are freely mobile within the EU. On the

other hand, non-cutters see a drop in real wages and labor workforce.

Overall, the results suggest nontrivial redistributional effects of har-

monization in the long run. Currently high tax countries stand to

gain both domestic and foreign industrial activity, a higher volume

of gross exports, raised real wages, and an inflow of workers. Con-

versely, current low tax countries can expect lowered domestic and


foreign production, leading to lower gross exports, real wages, and

jobs.

Chapter 3.

Reallocation among nontraded producers:

Brazil’s import liberalization

In recent years, a large literature in international trade has investi-

gated the local labor market’s role in the adjustment to trade shocks.

Studies from this literature have emphasized that when industries are

differentially exposed to trade, it can translate to significant variation

in outcomes at the regional level. The key insight is that industrial ac-

tivity is unevenly spread across space while workers are imperfectly

mobile across space. In the face of an industry-level trade shock, this

observation implies that a region’s exposure is determined by the in-

dustrial composition of its economic activity at the time of the shock.

For example, a seminal analysis in this body of work, Autor et al.

3. Reallocation among nontraded producers 204

[2013], finds that the dramatic increase in Chinese manufacturing ex-

ports (the “China shock”) disproportionately affected workers living

in American commuting zones that were specialized in manufactur-

ing compared to workers living elsewhere, regardless of the worker’s

own industry of employment.

This chapter uses Brazil’s import liberalization episode to investi-

gate labor reallocation within the local labor market in response to

a trade shock. Specifically, the chapter characterizes the flow of jobs

between producers in the same industry. Intra-industry reallocation

of labor between heterogeneous producers has been emphasized by

a large trade literature, beginning with Melitz [2003], because it can

potentially have implications for industry-level productivity and con-

sumer welfare. Inspired by the insights of these studies, the first por-

tion of this chapter focuses on uncovering the empirical patterns of

within-industry labor reallocation within local labor markets.

By comparing Brazilian local labor markets that were more exposed

to the import liberalization to those that were less exposed, this chap-

ter first establishes sizable within-industry job reallocation that would

not be captured by only focusing on net job flow between industries.


In particular, local labor markets more exposed to the import liber-

alization saw more job destruction due to increased establishment

contraction but less job destruction due to establishment exit. In ad-

dition, job creation was lower in these regions due to lowered plant

entry. These effects were quantitatively relevant in both the traded

and nontraded sectors, with over one million jobs reallocated within

the traded sector and nearly three million jobs reallocated within the

nontraded sector.

Digging deeper, the chapter then shows that the plant-level effects

on incumbents differed by the initial size of the establishment, with the

smallest establishments in high-exposure areas being more likely to

exit than their larger counterparts. Contingent on survival, the same

pattern holds for intensive margin employment. Small survivors in

highly exposed regions were less likely to grow (or more likely to con-

tract) compared to large survivors. Again, these qualitative patterns

appeared in both traded and nontraded industries. Taken together,

these results suggest that the import liberalization caused labor re-

allocation from small to large establishments. The theory from the

heterogeneous producer literature of international trade predicts this


pattern within the traded sector, but does not directly speak to within-

industry reallocation for the nontraded sector.

Furthermore, the reallocation patterns among producers in non-

traded industries uncovers a novel adjustment margin in the local

labor market’s response to a trade shock. To introduce some formal

structure, this chapter establishes a baseline model based on Melitz

[2003] but augmented with a heterogeneous-producer nontraded sec-

tor. It is nearly identical to the traded sector, with the only difference

being that nontraded producers cannot export and consumers cannot

import nontraded varieties. As this chapter then shows, the baseline

model does not predict any reallocation among producers in the non-

traded sector. This model is taken to be the baseline model. Without

global activity in the nontraded sector, the distribution of producers

is unaffected by the import liberalization.

To take steps to reconcile the data with the theory, the chapter finally

proposes a model incorporating intermediate input and final demand

in the nontraded sector. The key innovation of this model over the

baseline model is that nontraded varieties are used both in household

consumption and as an input for traded variety production. There


are now two markets for nontraded varieties, the consumer and input

markets. Producers of nontraded varieties self select into each, so that

there is a minimum productivity threshold required in each to partic-

ipate. As long as the threshold is higher for the input market, that is,

the input market is more competitive than the consumer market, the

input model is able to deliver job reallocation patterns once tariffs are

lowered that are qualitatively consistent with the empirical results.

The central intuition underlying the input model is that producers

of nontraded goods are differentially exposed to shocks to the traded

sector. As the price of importing falls, so that households shift con-

sumption toward traded varieties, the nontraded consumer market

shrinks. For producers of nontraded goods that only serve this mar-

ket, this effect decreases profits, forcing the least productive producers

to exit. Continuing producers that only serve the consumer market

shrink. On the other hand, the growth of the traded sector also implies

an expansion of the nontraded input market. Producers supplying

the market therefore receive a positive shock to this portion of their

business. As long as it outweighs the negative effect of the shrinking

consumer market, then these producers will expand.


This chapter builds on the local labor market literature. The most

closely related works are Dix-Carneiro and Kovak [2017] and Dix-

Carneiro and Kovak [2019], which study the Brazilian import liber-

alization taking the local labor market as the unit of analysis. Dix-

Carneiro and Kovak [2017] traces out the dynamic response of pro-

ducers to the liberalization, with the analysis ending around ten years

after the initial policy shock. It establishes that the tariff cuts had ef-

fects that were long lasting. For example, exposed regions experi-

enced a persistent decline in the number of establishments as well as

the average size of an establishment. However, the study focuses its

analysis on establishments in the traded sector. This chapter instead

is concerned with long differences in both sectors rather than the dy-

namic adjustment process. In addition, the main goal is to describe the

differential establishment-level effects according to initial plant size.

On the other hand, Dix-Carneiro and Kovak [2019] takes the per-

spective of the worker, following them over time to understand the

impact of the liberalization on labor market outcomes. Reemploy-

ment of displaced workers from nontraded industries was largely

in low-paying service industries and the informal sector. Addition-


ally, there negative earnings effects on workers in nontraded indus-

tries that were not directly import competing. These are attributed

to falling nontraded sector labor demand in import-exposed regions.

This result is in a similar spirit to the empirical analysis of this chapter,

since it uncovers indirect effects in the nontraded sectors of local labor

markets that were highly exposed to the tariff policy. In contrast to the

study, this chapter focuses on the differential employment outcomes

of formal establishments over time.

In related work, Pavcnik [2002] studies the effect of Chile’s dramatic

import liberalization on the outcomes of plants in import-competing

sectors. In particular, the study finds empirical support for import ex-

posure increasing plant level productivity of surviving plants. At the

same time, exiting establishments are estimated to be lower produc-

tivity than survivors. While this chapter does not directly estimate

plant-level productivity, it finds evidence that plants of initially larger

employment were less likely to shrink (more likely to grow) and less

likely to exit than plants of initially smaller employment. Moreover,

the chapter completes this analysis for both traded and nontraded

industries, while Pavcnik [2002] focuses on import-competing indus-


tries.

Finally, Asquith et al. [2019] investigate the effect of the China shock

on job flow patterns within the local labor markets of the United

States. The analysis discovers that establishment death was an im-

portant channel through which the shock affected employment, and

also that jobs were reallocated from import-competing industries to in-

dustries that were not import-competing. Similarly to the results from

this chapter, the paper finds between-establishment job reallocation

within import-competing industries. However, it does not conduct

plant-level analysis to investigate potential differential effects by plant

characteristics.

Section 3.1 describes the policy environment of Brazil’s import lib-

eralization in more detail. The empirical strategy and results are then

discussed in Section 3.2 and Section 3.3 respectively. Section 3.4 moves

towards theory by describing and characterizing the baseline model,

crucially discussing the implications of an import liberalization un-

der this model. Section 3.5 then presents the input model. Finally,

Section 3.6 concludes.


3.1. Import liberalization in Brazil

The Brazilian import liberalization has been used as a setting to in-

vestigate the effects of trade policy in several existing studies, and

has been described most recently in Dix-Carneiro [2019]. Its salient

aspects are therefore briefly summarized here.

The policy event crucially involved large unexpected one-sided tar-

iff declines across the board between 1990–1995. In March 1990, a

newly elected government suddenly announced a tarificaçao (“tariffi-

cation”), in which nearly all non-tariff import barriers were abolished

and tariffs were adjusted to maintain the same level of import protec-

tion.1 The primary effect of this process was to convert the main trade

policy instrument from non-tariff barriers to tariffs. These were then

steadily lowered in the next five years. Given these events, the trade

policy has been argued to be plausibly exogenous in timing.

Moreover, the import liberalization featured sizable variation indus-

try by industry. The reason is that the pre-liberalization protection

1More precisely, tariffs were set to maintain the same gap between internal andexternal prices to Brazil.


level was the main determinant for how much a tariff was lowered.2

These were set initially in 1957, a significant time before the liberal-

ization.

Given the policy environment, it is unlikely that contemporaneous

political economy factors influenced either the timing of the liberal-

ization or the variation in tariff cuts at the industry level. For this

reason, the tariff declines are taken as exogenous in the analysis of

this chapter, lending a causal interpretation to the empirical results.

3.2. Empirical approach

This section describes the empirical approach taken by the main anal-

ysis of the chapter. Section 3.2.1 describes the data sources used. The

two main measures of interest, local labor market exposure to the tar-

iff policy and gross job flows, are then described by Section 3.2.2 and

Section 3.2.3 respectively.

2Goldberg and Pavcnik [2003] show that the correlation was a near perfect−0.90.


3.2.1. Data

The main data source is detailed matched employer-employee panel

data from Brazil’s Relação Anual de Informações Socials (RAIS). The data

set is collected by the Instituto Brasileiro de Geografia e Estatística (IBGE)

for administrative purposes such as verifying worker eligibility for

government benefits. It consequently contains accurate information

covering the entire universe of formal Brazilian employees.3 Because

it is both high quality and describes all formal economic activity in

the country, the RAIS has been used by many papers.4

The key variable at the worker level is the identity of the establish-

ment where they hold employment. Additionally, information is also

collected about educational attainment and wage. Establishments

themselves are each associated with a firm, an industry, and a geo-

graphic location.5 Since the RAIS is a panel data set, every worker and

3A limitation of the data is that it omits self-employed and informally employedworkers. Some work that has studied Brazil’s informal sector in the context of tradeliberalization have been Dix-Carneiro and Kovak [2019] and Dix-Carneiro et al.[2019].

4A recent selection which study it in a trade context include Aquino Menezes-Filho and Muendler [2011], Helpman et al. [2016], Dix-Carneiro and Kovak [2017],and Dix-Carneiro and Kovak [2019].

5The empirical analysis excludes establishments in the mining, utilities, and pub-


establishment has a unique identifier that allows them to be followed

over time.

For the analysis in this chapter, the critical feature of the data is that

every observation constitutes a worker-establishment pair, where an

establishment is a physical location belonging to a firm. As such, ev-

ery job observed can be assigned to a geographic region. Regional

job creation and destruction can then be measured by tracking the

number of employees at each establishment across time. For example,

if an establishment employs fewer people in one period compared to

the next, jobs are interpreted as having been created in the establish-

ment’s region. Crucially, job flows at the regional level can be inferred

because the data collects information by establishment, which has a

well-defined geographic location, rather than by firm, which could

potentially have a presence in many physical locations. Because the

RAIS includes information on the universe of Brazilian formal em-

ployment, data sparsity is not a concern even with analysis at very

fine geographic levels.

Two other data sets are used to supplement the RAIS. The first

lic industries.


is the 1991 Censo Demográfico (Demographic Census). It is used to

characterize the initial distribution of industrial activity across space,

specifically the shares of workers employed in each industry by re-

gion.6 The second is a collection of tariff levels over time compiled

by Kume et al. [2003]. The changes in these tariffs across time are the

primary measure of import liberalization used in the analysis of this

chapter.

3.2.2. Liberalization exposure in local labor markets

The chapter’s empirical analysis follows a difference in difference

strategy inspired by the insights from the local labor market litera-

ture, where outcomes in local labor markets relatively more exposed

to the liberalization are compared across time to outcomes in those

that are less exposed to the policy. Local labor markets are defined in

the data as the geographic microregion units developed by the IBGE,

of which there are 404 in the years analyzed. These are groupings

of economically related municipalities that border each other while6Although the liberalization was announced in 1990, the 1991 census is used be-

cause the census is only conducted every ten years. The 1991 collection is the closestin time to the liberalization announcement.


also being similar geographically and in terms of productivity. The

grouped municipalities are also economically integrated, allowing

the microregions to be interpreted as local labor markets. In what

follows, the terms “local labor market” and “microregion” are used

interchangeably.

The main variable of interest is a microregion’s effective exposure

to the industrial tariff cuts, which is constructed for the microregion

r as follows.

libr ≡ −∑k

λrk/αk

∑` λr`/α`d ln(1 + τk) . (3.1)

In this expression, λrk is the share of employees working in microre-

gion r that are employed in industry k in the initial period, while αk

is the share of industry k’s inputs that are non-labor. Because tariff

changes vary at the industry level, the liberalization measure weights

them by how much the microregion’s workers are initially concen-

trated in the corresponding industry, adjusted for that industry’s ex-

penditure on labor input.

This measure is a modified Bartik instrument first proposed in Ko-


vak [2013], which develops a multi-sector model with sector-specific

capital and perfect labor mobility across sectors. When tariffs are ex-

ogenously modified in this model, the liberalization measure defined

above is exactly proportional to the regional change in labor demand.

The model features one liberalizing home region, so each Brazilian lo-

cal labor market can be interpreted as an independent instance of one

of the model’s home regions as long as there is no interaction among

the microregions. In particular, the movement of labor or goods across

microregions would violate this assumption. Though Dix-Carneiro

and Kovak [2017] provide supporting evidence that there was little

movement of workers across microregions in response to the liberal-

ization, lack of data availability precludes investigating whether there

was trade between the microregions. Consequently, this measure is

used for the empirical analysis in the rest of the chapter but interpreted

as a reduced form approximation of liberalization exposure at the lo-

cal labor market level rather than a strict structural measurement of

exposure.7

7Alternate structurally-motivated Bartik measures appear in Autor et al. [2013]and Kim and Vogel [2020].


The employment shares λrk for each microregion r and industry k

are calculated using 1991 census data. The employment shares λrk con-

sequently provide a snapshot of the composition of industrial activity

for each microregion shortly after the trade policies were announced

in 1990. By comparing the outcomes of two microregions with differ-

ent values of the liberalization measure, the effect of initially being

more or less exposed to the shock can be inferred. This effect there-

fore incorporates the different patterns of adjustment that may occur

based on the local labor market’s initial level of exposure.

3.2.3. Gross job flows

Having defined the liberalization measure, this section now turns to

specifying how gross job flows are measured. They are defined at the

establishment level as the difference in employment between the two

periods. For example, if an establishment employed more people in

the initial period, then job destruction has occurred. The opposite is

true for job creation.8

8Jobs are defined as employment positions at a plant, rather than a worker-plantpair as in some studies. Since the empirical analysis takes long differences across a


Previous studies have largely focused on net job flows across indus-

tries or sectors in response to trade. This chapter instead examines the

reaction of gross job flows, that is, the reallocation of labor across es-

tablishments.9 To highlight the conceptual difference between the two

types of flows, consider a simplified scenario where a worker moves

from one establishment to another and no other adjustments occur in

the economy. If the two establishments were in different industries,

then the industry of the first would have one less job while the in-

dustry of the second would have one more job. In other words, the

first industry experiences a net job flow of −1 while the second expe-

riences a net job flow of +1. On the other hand, if the plants are in the

same industry, there is no net job flow across the industries. However,

the gross job flow measure would register one job created and one job

destroyed in either case, since it records changes in employment at

the establishment level.

This simplified scenario highlights the key difference between mea-

ten year period to uncover medium- to long-run results, jobs are not defined as aplant-worker pair to avoid introducing life cycle effects.

9Analyses of gross job flows in the context of trade include Pavcnik [2002] andAsquith et al. [2019].


suring gross job flows to net job flows. While the job that switches

between industries is captured by net job flows and gross job flows

alike, the job that moves from one plant to another within the same

industry will have no effect on net job flows while registering both a

gross job creation and a gross job destruction. In light of this example,

it is easy to see that net job flows characterize the patterns of labor real-

location between different industries, but gross job flows also include

the patterns of labor reallocation within industries among different es-

tablishments.10 As has been highlighted by the literature on hetero-

geneous producers and trade, within-industry reallocation can have

industry-level implications. For example, Melitz [2003] develops a

theoretical framework where the compositional change induced by

labor reallocation from small unproductive producers to large produc-

tive producers within the same industry results in increased aggregate

productivity.11

In the empirical exercises that follow, all differences will be long10Davis et al. [1998] provides a more in depth discussion of gross job flows, jobcreation, and job destruction.11Recent work by Holmes and Stevens [2014] complicates this notion. This studyproposes that large producers fabricate generic goods and small producers artisanvarieties.


differences taken across the years 1990 and 2000.

3.3. Empirical results

This section presents the main empirical results. First, Section 3.3.1

establishes that job flows at the local labor market responded in both

traded and nontraded industries. These are quantitatively compara-

ble in both. Then, Section 3.3.2 further investigates the job flows at the

plant level, showing that they uncover labor reallocation away from

small establishments toward large establishments in both groups of

industries.

3.3.1. Effects at the local labor market level

To characterize the overall job flows at the local labor market level,

plants are first categorized into four sets based on whether they exit,

contract, expand, or enter from 1991 to 2000. The sets are correspond-

ingly labeled Ωxrk, Ωc

rk, Ωerk, and Ωn

rk for every microregion r and in-

dustry k. In particular, note that job destruction occurs when an es-

tablishment exits or contracts while job creation occurs when an es-


tablishment expands or enters. Then, indexing plants by ω, four mea-

surements of gross job flow can be defined

Jxrk = ∑

ω∈Ωxrk

Lt0(ω) Jcrk = ∑

ω∈Ωcrk

[Lt0(ω)− Lt1(ω)]

Jnrk = ∑

ω∈Ωnrk

Lt1(ω) Jerk = ∑

ω∈Ωerk

[Lt1(ω)− Lt0(ω)]

where Lt(ω) is plant ω’s employment in time t ∈ 1991, 2000. The

first line captures total job destruction in a microregion r and indus-

try k, either through extensive margin plant exit or intensive margin

plant contraction. Likewise, the second line measure total job creation,

through plant entry or expansion. The four measures together sum-

marize the overall reallocation patterns within an industry in each

microregion, where net flows into the microregion’s industry k can

be derived by simply subtracting total job destruction from total job

creation Jnrk + Je

rk − Jxrk + Jc

rk.

For the empirical specification, job destruction flows J frk for f ∈

x, c are scaled by the initial employment of industry k and microre-

gion r. Job creation flows are similarly scaled by the ending employ-


ment of industry k and microregion r, so that

g frk =

J frk

∑ω∈Ω f

rk

Lt0(ω)≡

J frk

Lrk,t0

for f ∈ x, c

g frk =

J frk

∑ω∈Ω f

rk

Lt1(ω)≡

J frk

Lrk,t1

for f ∈ n, e

will be the four dependent variables studied in this section. With

the rescaling, the job destruction variables g frk for f ∈ x, c can be

thought of as capturing the share of initial jobs that were destroyed

through either plant exit or contraction. Likewise, the job creation

variables g frk for f ∈ n, e can be interpreted as the share of final jobs

that were created either through plant entry or expansion.

Having defined the dependent variables, the empirical specification

can now be introduced as follows.

g frk = α

f0 + α

fs(r) + α

fk + β f Xrk + θ

fK(k)libr + ε

frk (3.2)

Each observation is a region-industry pair, with the dependent vari-


able being a job flow measure across 1990–2000 as described above.

The key variable in specification (3.2) is libr, which is the measure

of region r’s exposure to the liberalization policy developed in the

previous section. This regression therefore captures the effect of a

microregion’s liberalization exposure on its job flow patterns. In par-

ticular, the coefficient on the liberalization measure is θfK(k), where

K(k) ∈ T, NT indicates whether industry k is tradable or not. Since

the coefficient θfK(k) crucially varies by industry tradability, it is possi-

ble for liberalization to have a different effect on job flows within the

traded sector compared to those within the nontraded sector.

The right hand side of the specification also includes a constant, a

fixed effect for the Brazilian state in which microregion r is located,

an industry k fixed effect, and a vector of controls. These controls

include the industry k’s share of microregion r’s total workers in 1991,

the share of total payroll in microregion r attributable to industry k

in 1991, the log mean wage in industry k in 1991, and the pre-trend

in these variables. The pre-trend controls are calculated over 1986

to 1990 using the RAIS.12 For example, if wrk denotes the log mean12The period 1986–1990 is selected for the pre-trend calculations since 1986 is the


Exit Contraction Expansion Entrylib × T -0.397∗∗∗ 0.576∗∗∗ 0.076 -0.492∗∗∗

(0.139) (0.073) (0.049) (0.111)lib × NT -0.526∗∗∗ 0.400∗∗∗ 0.186∗∗∗ -0.277∗∗∗

(0.115) (0.061) (0.036) (0.090)Observations 6,829 6,829 6,680 6,680

∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

Table 3.1.: The effect of liberalization exposure on gross job flows byindustry tradability, as in specification (3.2).

Liberalization exposure has the same qualitative effect on gross job flowpatterns in traded and nontraded industries. Increased contraction and de-creased entry imply fewer net jobs, but lowered exit implies more net jobs.Standard errors are clustered at the local labor market level.

wage of industry k workers in region r, then the pre-trend control

would be wrk,1990 − wrk,1986. Finally, the pre-trend value of g frk over

the 1986–1990 period is also included within the controls.

This specification is run for each type of flow f ∈ x, c, n, e sepa-

rately. Results are reported in Table 3.1. The key observation is that

exposure to liberalization had qualitatively similar effects on gross

job flow patterns in both the traded and nontraded industries. In par-

earliest year of RAIS data accessible.


ticular, regions that were relatively more exposed to the tariff cuts

experienced more job destruction due to contraction and less job cre-

ation due to entry compared to regions that were not as exposed to

the tariff cuts. Both these effects imply fewer net jobs in these region-

industries.

On the other hand, regions with higher exposure to the liberaliza-

tion also saw decreased job destruction due to plant exit. This patterns

works in the opposite direction as the other two effects, implying more

jobs on net in the affected region-industries. Finally, in nontraded in-

dustries, there is also higher job creation due to plant expansion in

more exposed areas when compared to less exposed areas. This effect

also works in direction of increasing implied net jobs.

These results take a step toward illuminating the liberalization-

induced churn within industries that would not be captured by calcu-

lating net flows only. To further understand the quantitative impor-

tance of these estimates, they are used to in a calculate the implied

effect on jobs attributable to variation liberalization exposure. Con-

sider job destruction via plant contraction as an example. Rearranging

the empirical specification in (3.2) to arrive at jobs destroyed due to


plant contraction,

Jcrk = ∑

ω∈Ωcrk

[Lt0(ω)− Lt1(ω)]

=[αc

0 + αcs(r) + αc

k + βcXrk + θcK(k)libr + εc

rk

]∑

ω∈Ωcrk

Lt0(ω)

so the implied number of jobs destroyed through plant contraction at-

tributable to variation in liberalization exposure is θcK(k)librLrk,t0 . This

expression multiplies the realized liberalization exposure in region r

with its marginal effect on the share of initial jobs destroyed due to

plant contraction, then rescales by the initial total number of jobs. A

similar argument can be made to derive the implied number of jobs

destroyed due to plant exit attributable to liberalization, θxK(k)librLrk,t0 .

Conversely, since the job creation dependent variables for plant entry

and expansion are in terms of the total number of jobs in the final

period, rescale by Lrk,t1 to calculate the number of created jobs that

can be attributed to the liberalization policy. Finally, summing each

of these measures over all industries k and microregions r results in

the total number of Brazilian jobs created or destroyed that can be


attributed to differential exposure to the liberalization policy.

Note that these reduced-form calculations quantify the effect on

jobs from variation in liberalization exposure. The reason is that the

coefficients θfK(k) for each type of flow f ∈ x, c, n, e are identified

with a difference in difference strategy, capturing the relative effect of

high liberalization exposure to low liberalization exposure. The over-

all level effect to jobs from the liberalization event will be eliminated

as a result. In other words, the quantification from this exercise can

be interpreted as the total effect of the liberalization on job flow under

the assumption that an unexposed microregion would see no change

to its pattern of job flows following liberalization.13

Table 3.2 presents the results of the quantification. The first row lists

the gross job flows, obtained by adding the absolute value of all quan-

tified job flows. The remaining rows report the quantified job flows

for each of the four plant events. Beginning with the traded sector, al-

most 500,000 jobs were destroyed due to plant contraction but around

13As a more easily interpreted example, suppose all workers in a microregion areemployed in a nontraded industry. Then, this microregion’s liberalization exposureis zero. The assumption would therefore correspond to job flows in this microregionremaining unchanged due to the liberalization.


Traded NontradedGross job flow 1,322,616 2,754,888

jobs destroyed:

Exit -340,458 -1,043,708Contraction 494,232 792,670

jobs created:

Expansion 65,267 368,354Entry -422,659 -550,157

Table 3.2.: Total implied jobs destroyed or created attributable to dif-ferential liberalization exposure.

Though the estimates were mostly smaller for the nontraded sector, the im-plied quantitative effect is larger because the nontraded sector is larger thanthe traded sector. Overall, around twice as many jobs turned over in thenontraded sector than the traded sector.

340,000 fewer were destroyed because of decreased plant exit. Addi-

tionally, over 400,000 fewer jobs were created due to decreased plant

entry. These effects taken together, more than one million jobs were

reallocated among establishments in the traded industries, especially

from entrants to incumbents.

The second column in Table 3.2 presents the quantification for the


nontraded sector. The first immediate observation is that the figures

for the nontraded sector are much larger than those for the traded sec-

tor. Though some estimated coefficients were smaller in the nontraded

sector compared to their traded sector counterparts, all quantified job

flows are larger because the nontraded sector is simply much bigger

than the traded sector, accounting for almost double the amount of

total employment. The qualitative patterns remain comparable across

the sectors, in line with the qualitative similarity of the estimated co-

efficients. Overall, almost three million jobs were reallocated among

establishments in the nontraded sector, with labor being redirected

from entrants to incumbents. Among incumbents, there was both

increased job creation due to plant expansion as well as increased

job destruction due to plant contraction. Import exposure therefore

caused nontrivial reallocation between plants within the nontraded

industries as well as the traded industries.


3.3.2. Effects at the establishment level

The results from the previous section reveal quantitatively relevant

liberalization-induced job flows among establishments within a re-

gion and industry. The goal of this section is to better understand the

characteristics of the incumbents that exited, expanded, or contracted.

In particular, the exercises will reveal that these establishment-level

events were related to the establishment’s initial size.

In particular, there are two plant-level outcomes of interest. The

first is whether or not the plant survives. Then, conditional on sur-

vival, the second is the change in employment at the plant. These are

extensive and intensive margin measures, with the first correspond-

ing to the exit flows from the previous section and the second to the

expansion and contraction flows. Formally, the plant-level variables

will be defined as

vkr(ω) = 1[ω survives]

gkr(ω) =[Lkr,t1(ω)− Lkr,t0(ω)

]/Lkr,t0(ω) .


The first variable v is a simple dummy, taking on the value of one

if plant ω survives into 2000 and zero if does not. The second is the

plant-level analogy to the regional job flow variables in the previous

section. It represents the change in the amount of jobs at plant ω,

scaled by the plant’s original employment.14

The plant level analysis proceeds in two steps. First, a probit model

is used for the survival variable, so that

vkr(ω) = Φ[αv

0 + αvs(r) + αv

k + αvb

+ βvXrk(ω) + θvbK(k)libr + εv

rk(ω)]

(3.3)

where Φ(·) is the CDF of the normal distribution. In this specification,

every incumbent in 1990 establishment is one observation. They are

associated with an industry k, microregion r, and size bin b. Specifi-

cally, plants are categorized into six bins according to their employ-

ment in 1991, where b indexes the bins. These correspond to plant

employment ranging from 0–5, 6–20, 21–100, 101–250, 251–500, and

over 500.14The notation denotes v for survival and g for growth.


The linear component of the model is similar to the regional specifi-

cations (3.2) from the previous section, including a constant αv0, state-

level fixed effects αvs(r), and industry level-fixed effects αv

k . It also in-

corporates fixed effects for each size bin αvb as well as a vector of plant-

level controls Xrk(ω). Included in these controls are the plant’s initial

share of the microregion’s total industry k employment, its share of

total industry k’s wage bill payments in microregion r, its log mean

wage, and pre-trends in these variables. Additionally, the plant’s em-

ployment growth during the pre-trend period is included as a control

along with its skill intensity. Since there is no measurement of pre-

trend employment growth for plants entering after 1986 but before

1991, the variable is artificially assigned to a uniform constant for these

plants.15 The specification also includes a dummy variable indicating

whether or not the plant entered in the pre-trend period, so that this

artificial value has no effect on the coefficient for pre-trend growth.

The coefficient of interest is θvbK(k), which captures the effect of liber-

alization exposure on a plant’s probability of survival. The coefficient15Although it is possible to calculate the pre-trend growth for entrants taking thedifference in employment between the year they entered and 1991, it would resultin the pre-trend growth variable being inconsistently defined.


varies both by the tradability of the plant’s industry K(k), in addition

to the plant’s initial size b. Since the coefficient varies by the plant’s

size bin, liberalization exposure can have differential effects varying

both by the tradability of the plant’s industry as well as the plant’s size.

This formulation is effectively a triple difference in outcomes across

time, liberalization exposure, and initial plant size. The coefficients

not being equal across different size bins indicates that exposure to

liberalization causes extensive margin labor reallocation among the

size classes of plants. Note that the main effect of the establishment’s

initial size bin is captured by the size bin fixed effect αb.

Following the probit model, the employment growth for surviving

incumbents is specified as follows.

grk(ω) = αg0 + α

gs(r) + α

gk + α

gb + βgXrk(ω) + θ

gbK(k)libr + ε

grk(ω) (3.4)

Each surviving incumbent is an observation for this specification. The

dependent variable is the long difference in employment across the

1990–2000 time period as defined previously. Equation (3.4) is sim-

ilar to the linear component of the probit model (3.3), but with two


additional controls. The first is a Heckman correction, to account for

the fact that surviving incumbents are a selected group of plants.16

The second is a dummy variable indicating whether or not the firm

owning the plant becomes an importer in the years following the lib-

eralization policy.

Once again, the key feature of expression (3.4) is that the effect of

liberalization exposure is allowed to vary both by the plant industry’s

tradability as well as the plant’s initial size bin. Just as variation in the

θvK(k)b coefficients across different size bins indicates extensive margin

labor reallocation along the plant size distribution among incumbents,

variation in the θgK(k)b coefficients indicates intensive margin labor

reallocation along the plant size distribution among survivors.

Table 3.3 presents the extensive margin results while Table 3.4 the in-

tensive margin results.17 The main observation is that, in both traded

and nontraded sectors and along both extensive and intensive mar-

gins, the coefficients are not equal across initial plant size bins but16The Heckman correction is derived from the functional form of the probit exten-sive margin model and linear intensive margin model.17The same qualitative patterns broadly hold if the liberalization coefficient is al-lowed to vary by industry rather than industry tradability. Appendix C.1 presentsthese results.


Traded Nontradedlib × (0–5) −1.621∗∗∗ (0.076) −0.780∗∗∗ (0.056)lib × (6–21) −0.225∗∗ (0.096) −0.134∗ (0.071)lib × (21–100) −0.289∗∗ (0.117) −0.503∗∗∗ (0.096)lib × (101–250) 0.323 (0.243) −0.066 (0.220)lib × (251–500) 0.583 (0.404) −0.306 (0.377)lib × (500+) 2.301∗∗∗ (0.466) 0.475 (0.430)Observations 1,016,933

∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

Table 3.3.: The differential effect of liberalization exposure on survivalprobability plants of different sizes by sector, as in (3.3).

Liberalization reallocates labor towards large (in general, 500+ workers)plants in both traded and nontraded industries. Smaller plants in high ex-posure areas are more likely to exit.

broadly increasing.

Begin with the extensive margin for the traded sector. The coeffi-

cients are roughly monotonically increasing, ranging from −1.621 for

the smallest plants to 2.301 for the largest plants. The dispersion in the

coefficients implies that, for otherwise similar plants of different sizes,

the same level of liberalization exposure had a differential impact on

survival probability. In particular, the increasing pattern implies that


Traded Nontradedlib × (0–5) −13.907∗∗∗ (2.548) −7.911∗∗∗ (1.206)lib × (6–21) −12.406∗∗∗ (0.988) −1.982∗∗∗ (0.686)lib × (21–100) −12.918∗∗∗ (1.123) −6.438∗∗∗ (0.944)lib × (101–250) −6.043∗∗∗ (2.225) −0.701 (1.800)lib × (251–500) −3.153 (3.532) −2.253 (3.058)lib × (500+) 17.159∗∗∗ (4.908) 8.028∗∗ (3.544)Observations 274,379

∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

Table 3.4.: The differential effect of liberalization exposure on survivoremployment growth of different sizes by sector, as in (3.4).

Liberalization reallocates labor towards large (in general, 500+ workers) sur-vivors in both traded and nontraded industries. Conditional on survival,smaller plants in high exposure areas are more likely to shrink.

liberalization had a redistributive effect towards larger plants, since

they are more likely to survive compared to their smaller counterparts

given the same exposure to the tariff cuts.

Moreover, since the liberalization measure varies by region, their

coefficients compare outcomes for plants in more exposed regions

with otherwise similar plants in less exposed regions. For example,

the negative coefficient for plants with five or fewer employees implies


that plants of this size in regions more exposed to liberalization were

more likely to exit than plants of a similar size in regions less exposed

to liberalization. Similarly, the positive coefficient on plants with 500

or more employees implies that these very large plants were more

likely to survive if in a more exposed area than in a less exposed

area. Observe that results compare similar plants in microregions

of different liberalization exposures, so they do not incorporate the

overall level effect of the import liberalization on survival.

A similar pattern holds in the traded sector along the intensive mar-

gin, that is, the employment growth of surviving incumbents. The

coefficients are roughly increasing, indicating that labor is increas-

ingly employed at large establishments compared to small ones. The

sign of the coefficients is negative and statistically significant for the

first four bins, so plants in high exposure areas with employment 250

or below were less likely to expand (or more likely to contract) com-

pared to their counterparts in lower exposure areas. Large plants with

employment over 500 once again relatively gained from liberalization

exposure, the positive coefficient implying they were more likely to

expand (or less likely to contract) compared to similarly sized plants


in microregions less exposed to the tariff cuts.

A qualitatively similar pattern holds in the nontraded sector, espe-

cially along the intensive margin. Both margins feature coefficients

increasing in the initial size of the plant, so that the same amount of

liberalization exposure had differential effects based on initial plant

size. The coefficients are also negative and statistically significant for

the small plant size bins along both margins, so that small plants were

less likely to survive or grow compared to similar plants in less ex-

posed areas. The coefficient for the largest plants is positive along

both margins but only significant along the intensive margin. Large

plants in nontraded industries can therefore be interpreted as having

had their survival probability relatively unaffected by liberalization

exposure, but their expected growth positively impacted relative to

similar establishments in a low-exposure microregion. On the whole,

coefficient estimates for the nontraded sector are smaller in magnitude

than those for the traded sector.

To better understand the quantitative impact of these plant-level

estimates on job flows, the section concludes with an exercise similar

to the one conducted at the microregion level. Specifically, the job


flow at each plant that can be attributed to variation in liberalization

exposure is calculated for both the extensive and intensive margins.

For the extensive margin, it is computed according to (3.3) as follows.

Lrk,t0(ω)[Φ(

αv0 + αv

s(r) + αvk + αv

b + βvXrk(ω) + θvbK(k)libr

)−Φ

(αv

0 + αvs(r) + αv

k + αvb + βvXrk(ω)

)]

Since the probability of survival is assumed to follow a probit model,

the marginal effect of liberalization exposure is not constant. This

expression therefore captures the change in the predicted survival

probability of a plant as the liberalization measure is changed from

zero to its observed value. All other regressors are held at their origi-

nal levels. The probability change is then multiplied by the number

of original employees to arrive at the effect of liberalization on exten-

sive margin job flow. For example, take the extreme case where the

plant’s predicted survival probability is one with the liberalization

measure and zero without the liberalization measure. Then, liberal-

ization exposure converts the plant’s predicted fate from definitely

exiting to definitely surviving. The number of jobs that were “saved”,


or not destroyed, as a result are simply the number of jobs at the plant

initially.

Quantifying the effect of liberalization on intensive margin job flow

follows a similar argument from the quantifications of the previous

section. They are derived using equation (3.4) as θgbK(k)librLrk,t0 . It

multiplies the marginal effect of liberalization exposure with the ob-

served value of liberalization exposure, then rescales by the plant’s

original size. The rescaling step is included because liberalization ex-

posure is assumed to have a linear effect on the plant’s employment

growth rate.

Observe that the coefficients on the liberalization exposure are iden-

tified through a triple difference strategy. As was the case in the pre-

vious section, they therefore do not capture the overall level effect of

liberalization on a plant’s survival chance or growth rate. The quan-

tifications can then be interpreted as approximating the aggregate

effect of liberalization on plant-level job flow if it were the case that

establishments in an unexposed microregion would see no effect on

their survival and growth rates.

The plant-level quantification results are presented in Table 3.5. In


Extensive IntensiveBin Traded Nontraded Traded Nontraded(0–5) -9,163 -29,153 -34,819 -90,557(6–21) -5,276 -11,236 -16,266 -36,953(21–100) -17,129 -73,528 -52,211 -232,125(101–250) 14,635 -5,484 49,386 -18,330(251–500) 27,567 -16,095 84,067 -63,257(500+) 324,208 93,689 1,003,774 266,324Gross job flow 397,977 229,184 1,240,523 707,546

Table 3.5.: Total implied plant-level job flow attributable to differentialliberalization exposure.

Though estimated coefficients were smaller for the nontraded sector, impliedjob flow is similar in magnitude across both sectors. Very large plants (over500 employees) constitute a sizable portion of these flows.

line with the microregion-level results, job flow quantities are com-

parable in traded and nontraded sectors despite coefficient estimates

being smaller for the nontraded sector. This observation reiterates the

magnitude of the nontraded sector compared to the traded sector.

Next, though there is a relatively small number of large plants, they

constitute a disproportionately large share of total gross job flow. For

example, the increased survival probability attributable to liberaliza-


tion exposure of plants with 500 employees or more “saves” many

more jobs than the number destroyed from the corresponding in-

creased exit probability of small plants. Moreover, the systematic

reallocation pattern of labor from small plants to large plants could

have effects beyond a simple net change in the number of jobs, if large

plants are fundamentally different from small plants. For example, as

discussed above, if large plants are more productive than small plants,

industry-level productivity could increase.

3.4. Baseline model

This section specifies a baseline model with a traded sector modeled

in the spirit of Melitz [2003] and a similar nontraded sector with no ex-

port activity. Section 3.4.2 shows that in equilibrium, it turns out that

each sector can be characterized separately from the other, with out-

comes in the nontraded sector crucially independent of the variable

costs of importing traded goods. A critical model component deliver-

ing this separability is that household preferences over the traded and

nontraded sectors are Cobb-Douglas. Then, Section 3.4.3 analyzes a


comparative static of changing the variable cost in line with the Brazil-

ian import liberalization. Given the effectively segmented nature of

the two sectors in equilibrium, the policy only generates reallocation

among traded sector producers, contrary to the empirical evidence.

3.4.1. Theoretical framework

This section describes the stylized baseline model of the analysis,

where the domestic economy is interpreted to represent a single Brazil-

ian local labor market while the foreign economy stands in for the

rest of the world. Implicitly, this interpretation disallows labor flows

among microregions, with the main goal instead being to characterize

labor flows within the microregion.18

Given the interpretation that the model’s domestic economy repre-

sents a Brazilian local labor market, the model makes the small open

economy assumptions proposed in Demidova and Rodríguez-Clare

[2013]. In particular, foreign aggregates are taken as exogenous and

fixed, so that the foreign wage is implicitly chosen as the numeraire.18This assumption is in line with the evidence from Dix-Carneiro and Kovak [2017]that there was little cross-microregion mobility in response to the import liberaliza-tion.


A more in-depth description is provided when the foreign variables

are discussed.

Domestic households

The economy is populated by a mass L of representative households.

They have Cobb-Douglas preferences over a traded basket and a non-

traded basket, with a weight β on the traded composite. The traded

basket in turn contains a continuum of traded varieties, aggregated

with a constant elasticity of substitution σT, while the nontraded bas-

ket similarly includes a continuum of varieties but is combined with

a constant elasticity of substitution σN.

Given these preferences, consumers maximize utility subject to

their total income, the only source of which is labor earnings. House-

holds each inelastically supply one unit of labor, freely moving be-

tween the traded and the nontraded sectors. There is therefore a single

domestic wage w in equilibrium, so total earnings are wL.

The Cobb-Douglas structure of demand over the two sectors im-

plies that a fraction β of total expenditure is dedicated to the traded

basket, while the rest is spent on the nontraded basket. Moreover,


within the traded sector, demand by domestic households for a par-

ticular variety ω can be expressed as

qT(ω) = βwLPT

(pT(ω)

PT

)−σT

, PT =

∫ω∈ΩT

pT(ω)1−σTdω

11−σT

where PT is the CES aggregate price index of the tradables basket

and ΩT is the endogenous set of all varieties available for purchase. A

similar argument holds for the nontraded basket, so that total demand

for a particular nontraded variety ω is as follows.

qN(ω) = (1− β)wLPN

(pN(ω)

PN

)−σN

, PN =

∫ω∈ΩN

pN(ω)1−σNdω

11−σN

The expressions are the same as for the traded varieties, but with the

nontraded elasticity of substitution σN , set of available goods ΩN , and

consumption weight (1− β).


Traded sector

The traded sector is populated with a continuum of producers that

each hold the template for fabricating a single differentiated variety ω.

Producers also vary in their inherent productivity z, which is drawn

upon entry from a fixed distribution GT(·). This productivity influ-

ences efficiency, so that a producer with productivity z must employ

aT(z) units of labor to make one unit of output.19 The productivity

function aT(z) is assumed to be decreasing in z, so that a producer

with higher productivity z produces more with the same amount of

labor compared to one with lower productivity.20

In order to enter the domestic market, producers must pay a fixed

cost f Tc in labor. They compete monopolistically given the continuum

assumption of producers. The optimal pricing choice is therefore to

set a constant markup σT/(σT − 1) over marginal costs waT(z) since

demand features constant elasticity of substitution. Using the well-

known properties of CES demand, variable profits in the domestic

19In the model of Melitz [2003], the unit cost function takes the form c(z) = 1/z.20This assumption is without loss of generality since productivity can always berelabeled so that aT(·) is decreasing.


market are a constant fraction 1/σT of sales.

Since variable profits increase monotonically in z but fixed costs do

not depend on productivity, there is a marginal value of productivity

zTd at which variables profits derived from the domestic market are ex-

actly equal to the fixed cost of participating. Producers with this level

of productivity are indifferent between doing so or not, with lower

productivity producers optimally deciding not to. This productivity

will be referred to as the traded domestic cutoff productivity, and is

explicitly defined below.

w f Td =

β

σT wL

(σT

σT − 1waT(zT

d)

PT

)1−σT

(3.5)

Observe that variable profits in the domestic market for any producer

with productivity z can simply be written as

w f Td

(aT(z)

aT(zT

d

))1−σT

≡ w f Td hT

(z, zT

d

).

The function hT(z, z′) captures the ratio of variable profits for any two

productivities z and z′. In addition, the zero profit condition anchors


the profits of the marginal producer as exactly equal to the fixed cost

w f Td . Then, to make variable profits in excess of the fixed cost, produc-

ers must be more productive than the marginal producer. Naturally,

variable profits are decreasing in the domestic cutoff zTd , all else equal.

Rewriting the profits of any producer in terms of the threshold pro-

ductivity will be key in easily characterizing the equilibrium in what

follows.

In addition to serving the domestic market, producers can also pay

a fixed cost f Tx in domestic labor to access the export markets. In addi-

tion to the fixed cost, there are also iceberg variable costs of exporting,

so that d > 1 units of output must be shipped abroad for one unit to

arrive. The marginal cost of serving the export markets is therefore

dwaT(z). Producers choose to serve the export markets only if the

variable profits of doing so outweigh the fixed cost of access. Simi-

larly to the domestic market, there will be a marginal productivity zTx

below which producers optimally do not decide to export and above


which they do. This cutoff is defined as

w f Tx =

1σT XT∗

(σT

σT − 1dwaT(zT

x)

PT∗

)1−σT

(3.6)

where XT∗ is total foreign expenditure on traded goods and PT∗ is the

foreign price index.21 Symmetrically to the domestic market, variable

profits earned by a producer with productivity z in the export market

can be written as w f Tx hT(z, zT

x ), so that they are summarized by own

productivity, the marginal productivity, and the fixed cost of access.

There is free entry into the traded sector, but producers must pay

a sunk cost f Te in labor to develop the blueprint for a differentiated

variety ω and draw their productivity z. Since this cost is paid before

productivity is realized, there will be entry whenever the expected

profits from doing so exceed the fixed entry cost. In equilibrium, there

must therefore be exactly zero expected profits from entry once entry

costs have been taken into account. Formally,

21In line with the small open economy assumptions, these are taken to be exoge-nously fixed. A more detailed discussion of the foreign variables follows once do-mestic production has been characterized.


w f Te = w f T

d

∫zT

d

[hT(

z, zTd

)− 1]

dGT(z)

+ w f Tx

∫zT

x

[hT(

z, zTx

)− 1]

dGT(z) (3.7)

describes the free entry condition. This expression integrates over

the profits earned by all possible productivity draws, with the first

term summarizing expected profits from domestic activity while the

second expected profits from exporting. Both have incorporated the

fact that the ratio of variable profits for any pair of producers with

productivities (z, z′) is simply hT(z, z′), while the productivity of the

threshold producer is given by the zero profit conditions (3.5) and

(3.6) as the fixed costs w f Td and w f T

x of participation. Note that both

integrals start at the domestic and exporting productivity thresholds

zTd and zT

x respectively, since producers with productivity below these

thresholds optimally choose not to participate in the corresponding

activity. Let the mass of entrants in the sector be denoted with T.

Just as domestic producers can pay a fixed cost to access the export

market, foreign producers can access the domestic market through


their own exporting activity. In order to do so, foreign producers

must pay a fixed cost f T∗x in units of the numeraire. Further, there are

variable costs of trade for foreign producers so that d∗ > 1 units of

their good must be shipped to ensure one unit arrives. This variable

trade cost magnifies the original marginal costs of foreign producers,

so that a foreign producer with productivity z faces a marginal cost

of d∗cT∗(z) in units of the numeraire to supply the domestic market.

Observe that, unlike the notation for domestic production which spec-

ifies fixed and variable costs in terms of domestic labor, f T∗x and cT∗(·)

are in terms of the numeraire. Specifying foreign costs in terms of

the numeraire allows the model to avoid taking a stand on the factors

used by foreign production.

If the cost function cT∗(·) is increasing in productivity, the expres-

sion below identifies the foreign productivity level zT∗x above which

foreign producers choose to serve the domestic country.

f T∗x =

β

σT wL

(σT

σT − 1d∗cT∗(zT∗

x)

PT

)1−σT

(3.8)

The logic in deriving this expression is in parallel to the argument


for domestic producers. A mass T∗ of tradables producers entering

abroad are assumed to draw their productivities from the fixed distri-

bution GT∗(·).

Putting it together, the set of tradable varieties ΩT available to the

domestic consumers are the those originating from domestic produc-

ers with with productivity above zTd as well as those imported from

foreign producers with productivity above zT∗x . Then, the price index

in the tradables sector PT can be derived by aggregating over prices

set by these producers, as below.

(PT)1−σT

= T∫zT

d

(σT

σT − 1waT(z)

)1−σT

dGT(z)

+ T∗∫

zT∗x

(σT

σT − 1d∗cT∗(z)

)1−σT

dGT∗(z) (3.9)

This expression writes the tradables price index in terms of producer

productivity z instead of blueprint ω by observing that all producers

with the same productivity behave identically. It includes the con-

tribution of active domestic producers as well as exporting foreign


producers, starting at the relevant cutoff productivities zTd and zT∗

x .

Any producer with productivity below the relevant threshold does

not optimally participate in the corresponding activity.

Nontraded sector

The nontraded sector is broadly similar to the traded sector, but pro-

ducers can only serve the domestic market and cannot export. Each

are associated with a unique variety ω as well as a productivity z. This

productivity is drawn on entry from the fixed distribution GN(·) and

likewise affects production capability so that aN(z) units of labor are

required to produce one unit of output. Accordingly, marginal costs

for a producer with productivity z are waN(z). As with the traded

sector, productivity is assigned so that the unit cost function is de-

creasing.

Producers must pay a fixed cost f Nc in labor to sell their variety to

households.22 If they do, they compete monopolistically and there-

fore optimally set a fixed markup σN/(σN − 1) over marginal costs.

22The subscript c refers to the consumer market, in anticipation of the input modelintroduced in the next section.


Identically to the productivity cutoffs in the traded sector, there is a

threshold productivity for nontraded producers such that any pro-

ducer with a lower productivity value will not earn profits from serv-

ing households once the fixed cost is taken into account. It is identified

by setting variable profits net of the fixed entry cost to zero, as below.

w f Nc =

1− β

σN wL

(σN

σN − 1waN(zN

c)

PN

)1−σN

(3.10)

It follows that the variable profits of any producer can be written in

terms of its productivity and the fixed cost f Nc as below.

w f Nc

(aN(z)

aN(zNc )

)1−σN

≡ w f Nc hN

(z, zN

c

).

The logic used to arrive at this expression is exactly as in the traded

sector. The relative variable profits made by one producer compared

to another depends only on their productivities and is summarized

by hN(·, ·). Moreover, the variable profits earned by the marginal pro-

ducer in the nontraded sector is pinned down by the fixed cost of

market access. Combining the two observations derives the expres-


sion above.

Nontraded producers may enter freely, developing the blueprint

ω and drawing their productivity z after paying a fixed cost f Ne in

domestic labor. Just like in the traded sector, free entry into the non-

traded sector ensures that expected profits net of the entry cost are

zero. Formally, the condition ensures that

w f Ne = w f N

c

∫zN

c

[hN(

z, zNc

)− 1]

dGN(z) (3.11)

so that producers enter into the nontraded sector until it is no longer

profitable to do so. Let this mass of nontradables entrants be N. The

main difference between this free entry condition compared to the

one in the tradables sector is that there is only one potential source

of profits, which is serving household demand. Because the good is

nontraded, producers in this sector cannot export or, as a consequence,

earn profits from exporting.

Finally, the price index in the nontraded sector can be rewritten in


terms of producer productivity z instead of blueprint ω as follows.

(PN)1−σN

= N∫

zNc

(σN

σN − 1waN(z)

)1−σN

dGN(z) (3.12)

This expression integrates over the pricing behavior of all active pro-

ducers making nontraded goods. Only domestic producers are active

in this market, with nontradability precluding foreign imports.

Foreign variables

There are foreign aggregates in the model, namely XT∗, PT∗, T∗,

in addition to the foreign cutoff for exporting zT∗x . Since the model

represents a Brazilian microregion, the domestic economy is taken to

be a small open economy which cannot affect aggregates in the rest of

the world. Therefore, the model proceeds with the assumption that

the foreign expenditure on traded goods XT∗, the foreign price index

for traded goods PT∗, and the foreign mass of entering producers

into the traded sector T∗ are exogenously determined. Observe that

setting exogenous values for the foreign aggregates implicitly chooses


a numeraire.

This set of assumptions characterizing a small open economy was

proposed by Demidova and Rodríguez-Clare [2013] and discussed

by Melitz and Redding [2014]. The Demidova and Rodríguez-Clare

[2013] study shows that, with a two country setup where the relative

size L/L∗ of the countries approaches zero, the equilibrium character-

ization approaches the assumptions made here.

While the foreign aggregates are exogenously set, the foreign ex-

porting threshold zT∗x is assumed to be endogenously determined ac-

cording to the zero profit condition (3.8). Therefore, while domestic

activity cannot affect the aggregates in the rest of the world, it can still

have an impact on the equilibrium outcome of imports consumed do-

mestically. In particular, changes in d∗, which represent the iceberg

costs of trade faced by foreign exporters, can potentially have an ef-

fect on the distribution of imported goods by changing the foreign

exporting cutoff zT∗x . Tariff cuts will later be precisely modeled as an

exogenous change in these variable costs.



Labor is freely mobile across the traded and nontraded sectors, so that

there is a single equilibrium wage in the domestic economy. It is deter-

mined through labor market clearing, which ensures that total labor

demanded by both sectors is equal to total labor supplied. Formally,

the condition is as follows.

wL = T

w f Te + w f T

d

∫zT

d

[(σT − 1

)hT(z, zT

d ) + 1]

dGT(z)

+ w f Tx

∫zT

x

[(σT − 1

)hT(

z, zTx

)+ 1]

dGT(z)

+ N

w f Ne + w f N

c

∫zN

c

[(σN − 1

)hN(

z, zNc

)+ 1]

dGN(z)

(3.13)

The left hand side of the condition is the total labor supplied, while

the right hand side gathers all labor demanded. The first sum on

the right hand side collects all the sources of labor demand in the

traded sector while the second all the sources of labor demand in the


nontraded sector. In the traded sector, labor is employed to pay the

fixed costs of entry, domestic market participation, and export market

participation, as well as for variable production for both domestic and

foreign markets. In the nontraded sector, it is similarly used to pay the

fixed costs of entry, the fixed costs of market access, and variable costs

of production. The expression makes use of the fact that, with CES

demand, total expenditure on variable costs are a constant proportion

(σT − 1) of variable profits.

3.4.2. Equilibrium

Having described the model fundamentals of the model, equilibrium

is now defined.

Equilibrium. Given a vector of foreign aggregates XT∗, PT∗, T∗, a col-

lection of cutoff productivities zTd , zT

x , zT∗x , zN

c , price indices PT, PN, en-

trant masses T, N, and wage w is an equilibrium if it satisfies

(i) zero cutoff conditions for the domestic tradables market (3.5), the ex-

port market (3.6), the import market (3.8), and the domestic nontraded

market (3.10);


(ii) price aggregation in the traded (3.9) and nontraded (3.12) sectors;

(iii) free entry in the traded (3.7) and nontraded (3.11) sectors; and

(iv) labor market clearing (3.13) across both.

Equilibrium in the nontraded sector is very simple and character-

ized first. Observe that free entry (3.11) immediately determines the

marginal productivity zNc in terms of fundamentals. The result fol-

lows from the fact that expected variable profits can be expressed in

terms of the fixed cost of market access and the marginal productiv-

ity only. Free entry pins total expected profits after entry to the fixed

cost of entry, completely determining the productivity distribution of

active producers in the nontraded sector.

Moreover, the price index expression (3.12) can be combined with

the sector’s zero profit condition (3.10) to derive goods market clear-

ing for the nontraded sector as follows, which pins down the total

mass of entrants N in equilibrium.

(1− β)wL = σNw f Nc N

∫zN

c

hN(

z, zNc

)dGN(z)


What is the intuition for why N is fixed in terms of fundamentals?

The left hand side of this equation denotes total consumer expendi-

ture on nontraded goods while the right hand side is the total sales

in the nontraded sector. Because these goods are not traded, it must

be that domestic consumption is exactly equal to domestic produc-

tion. Further, free entry guarantees that there are no overall profits, so

that sales receipts are paid back to workers either as fixed costs or as

variable costs of production. In other words, total labor demand from

nontraded producers is also (1− β)wL. This result can be explicitly

seen by substituting the free entry condition (3.11) into the total labor

hired by the nontraded sector. Taken together, a constant share of

total labor (1− β) must be employed by the nontraded sector. Finally,

since the free entry condition determines the distribution of active

productivities, the mass of entrants N must adjust so that total labor

hired by the nontraded sector is perfectly equal to (1− β)L.

This logic completely characterizes the distribution of producers

in the nontraded sector. In turn, by inspection of price aggregation

(3.12), it determines the price index in the nontraded sector as pro-

portional to domestic wages w. In other words, w/PN is equal to a


known combination of fundamentals. This argument completes the

description of the nontraded sector.

Observe that the traded block of the economy is effectively inde-

pendent of the nontraded sector. In other words, all traded sector

outcomes will be exactly identical to the ones that would result from

a model where the traded sector is specified the same way, there is

no nontraded sector, and total population is βL instead of L. In this

economy, there ultimately exists a unique equilibrium determined

by two aggregate conditions in (zTx , w) space.23 Figure 3.1 depicts the

equilibrium.

The first condition is an “international competitiveness” condition

represented by the zero profit cutoff for domestic exporters (3.6). This

condition implies a positive relationship between domestic wages w

and the exporting cutoff zTx . The reason is that, as the domestic wage

w increases, the costs of production increases relative to profitability

of entering the foreign market XT∗PT∗(σT−1). Then, it must be the case

that the baseline productivity zTx for selection into exporting increases.

23Appendix C sketches the proof. The argument follows a similar intuition as inDemidova and Rodríguez-Clare [2013].


zTx

w (3.6)

(3.14)

Figure 3.1.: Equilibrium in the traded sector is determined by thecutoff for exporters (3.6) and trade balance (3.14). Decreasing d∗

shifts the trade balance condition downward.

The second condition determining equilibrium is trade balance,

which equates total sales from exports with total sales from imports. It

is obtained by combining the zero profit conditions from the tradable

sector (3.5), (3.6), and (3.8) along with the free entry condition (3.7),

and price aggregation (3.9) to arrive at

w f Tx T

∫zT

x

hT(

z, zTx

)dGT(z) = f T∗

x T∗∫

zT∗x

hT∗(

z, zT∗x

)dGT∗(z) (3.14)

where hT∗(·, ·) is defined analogously to hT(·, ·) but with the foreign

unit cost function cT∗(·). The left hand side represents exports, while

the right hand side imports. Trade balance implies a positive relation-


ship between domestic wage and the exporting cutoff. Broadly, net

exports decrease as the exporting cutoff increases, while net imports

decrease as the domestic wage decreases. The domestic wage must

therefore move in the opposite direction as the exporting cutoff in

order to maintain trade balance.

These two conditions together determine equilibrium in the traded

sector, as depicted in Figure 3.1. Lastly, the welfare of the representa-

tive household can be characterized using the zero profit condition

for domestic activity (3.5). Since the household only receives labor

income and only derives utility from consumption goods, welfare is

simply real wages. These are in turn proportional to

w

(PT)β(PN)

1−β∝( w

PT

)β∝ aT

(zT

d

)−β

where the first proportionality is because w/PN is a constant as dis-

cussed. The second derives from the zero profit condition for domes-

tic activity (3.10). Then, the domestic cutoff is a sufficient statistic for

household welfare in this model. As it increases, so does welfare.


3.4.3. Liberalization counterfactual

In this model, import tax cuts can be modeled by decreasing d∗, the

iceberg cost of trade for foreign exporters.24 This changes causes a

downward shift in the trade balance condition, as shown by the dot-

ted line in Figure 3.1.25 Intuitively, fix w and suppose d∗ falls. Without

any adjustment, imports increase since they are now less costly. Then,

exports must also increase to maintain trade balance, so the exporter

cutoff zTx must decrease. In equilibrium, the actual change in exporter

cutoff is tempered by movement along the competitiveness line de-

fined by the zero profit condition for exporting (3.6). Domestic wages

w fall along with the exporter cutoff.

In the event of a tariff decrease, the baseline model can generate

empirically consistent reallocation patterns in the traded sector but

not in the nontraded sector. To begin, consider the extensive margin

in the traded sector. As the exporter cutoff zTx falls, the free entry

24Modeling import tariffs this way implicitly assumes that tariff revenue is dis-carded. Since the domestic economy is taken to be a small open economy, there areno terms of trade effects from imposing or changing tariffs.25Appendix C sketches the proof, following a similar argument as Demidova andRodríguez-Clare [2013].


condition implies that the domestic cutoff zTd increases. The negative

relationship between these two cutoffs is explained by the fact that

total expected profits must be equal to the fixed cost of entry. If total

exports increase to accommodate the increased volume of imports,

then expected profits from exporting activity increase for a producer

that considers entering. To maintain the same level of total expected

profits, it must then be that the cutoff for serving the domestic market

zTd increases to lower the expected profits from domestic activity. The

higher domestic cutoff implies that there is a range of productivity

for which it was profitable to participate domestically before the tariff

cut policy, but is no longer once the policy is implemented. In the

empirically relevant scenario where the productivity threshold for

exporting zTx exceeds the threshold for domestic activity zT

d , this result

is consistent with the empirical result that the smallest establishments

in high-exposure microregions were more likely to exit.

Next, consider the intensive margin in the traded sector. The total

amount of labor hired by a producer after entry with productivity z is

f Td

(σT − 1

)hT(

z, zTd

)+ 1+ 1x(z) f T

x

(σT − 1

)hT(

z, zTx

)+ 1


where 1x(z) indicates whether the producer exports, that is, whether

its productivity z exceeds the exporting cutoff zTx . The first expression

summarizes total labor employed for serving the domestic market

while the second the export market. Focusing first on producers that

don’t export, it is easy to see that their employment decreases un-

ambiguously. The marginal productivity of domestic activity zTd has

increased, lowering the profits and therefore labor employed for serv-

ing this market. On the other hand, consider exporters. While the

labor they employ for domestic activity also decreases, the labor they

employ for export market activity increases as the exporting cutoff zTx

falls. It is therefore possible for exporting producers, which are larger

on average, to expand as long as the increase in export activity offsets

the decrease in domestic activity.

In contrast, the baseline model is not able to qualitatively match the

empirical patterns of reallocation among nontraded producers along

either the extensive or the intensive margin. As discussed, the thresh-

old productivity for nontraded producers zNc is fixed by fundamentals

unrelated to d∗. Consequently, the distribution of participating pro-

ducers does not change when tariffs are dropped. This result is coun-


terfactual to the empirical finding that small producers were more

likely to exit compared to large ones in microregions more exposed

to the liberalization policy. Likewise, the model is unable to match

the qualitative patterns of intensive margin labor reallocation. In the

model, a producer of nontraded goods with productivity z hires

f Nc

(σN − 1)hN

(z, zN

c

)+ 1

units of labor after entry. Therefore, there is no change in labor de-

manded at the producer level since the cutoff zNc remains unchanged.

In contrast, small plants were empirically more likely to shrink com-

pared to large plants in local labor markets more exposed to the tariff

cuts.

What explains the model’s different predictions across the two sec-

tors for labor reallocation? Consider Figure 3.2, which summarizes

the economy’s flows in equilibrium. Activity involving nontraded

varieties is represented with blue diagonal lines while activity con-

cerning traded varieties is shown in red, with domestic activity solid

and trade activity in dots. Expenditure flows to producers, either


households

βwL (1−β)wL

nontradedproducerstraded producers

exportactivity

domesticactivityforeign

==imports

exports=

foreigntraded

domestictraded

nontraded wagebill(sales)

expenditure

Figure 3.2.: Equilibrium flows in the baseline economy.Solid arrows represent expenditure flows while dashed arrows representwage bill flows (which, with no profits under free entry, are equal to salesflows). The size of the nontraded sector is fixed at (1− β)wL and representedwith blue diagonal lines. The size of the traded sector, shown in red, is corre-spondingly fixed at βwL. However, within the traded sector, the relative sizeof domestic (solid) to foreign (dots) activity is determined in equilibrium.


from domestic or foreign consumption, are shown with solid arrows.

Similarly, wage bill flows from producers to households are shown

with dashed arrows.

The diagram makes use of the fact that, with no profits under free

entry, payments to labor are equal to sales. For both sectors, total

household consumption must therefore be equal to total wage bill.

Following this logic, both sectors are fixed in size as previously argued.

The traded sector employs βL units of labor while the nontraded sector

employs (1 − β)L units. The fixed nature of the nontraded sector

was used to conclude that the productivity distribution, the mass of

entrants, and the size of each producer is unrelated to d∗.

In contrast, though the size of the traded sector is also fixed, its

counterparts of these variables change in equilibrium as d∗ changes.

Examining the schematic in Figure 3.2 reveals the reason. Despite

the fact that the size of the overall traded sector is fixed, its composi-

tion is not. In other words, the domestic and foreign components of

the traded sector adjust in response to a change in tariffs. As d∗ falls,

household consumption shifts toward imports compared to domesti-

cally produced traded varieties. Export volume increases to maintain


trade balance, as production for the domestic market falls in tandem.

The import liberalization therefore shifts labor from domestic activity

to foreign activity, both on the extensive and intensive margins. As

discussed above, this reallocation translates to differential effects at

the producer level which depend on the composition of the producer’s

economic activity.

Finally, the import liberalization represents a welfare-improving

policy. Recall that welfare in the baseline model is proportional to

aT(zTd )−β so that, as long as the threshold for traded producers to par-

ticipate domestically increases, welfare increase. Because the trade

policy reorganizes economic activity to exporters, increasing the com-

petitiveness of the domestic market, this cutoff and therefore welfare

increases.

To summarize, a reduction in tariffs constitutes a shock reallocat-

ing economic activity in the traded sector from domestic to foreign

activity, resulting in differential producer-level effects in the sector de-

pending on the producer’s composition of activity. On the other hand,

there is no reallocation in the nontraded sector, creating no variation

in shock incidence at the producer level in the sector. Welfare, which


is summarized by the domestic participation constraint, increases as

a consequence.

3.5. Model with intermediate inputs

Having established that the baseline model cannot predict with-in

sector labor flows in the nontraded sector in response to import lib-

eralization, this section now presents the input model. It can gener-

ate between-producer flows in both sectors while introducing only

two innovations on the baseline model. The interaction of these de-

partures is key in generating these reallocation patterns within the

nontraded industry. The first is introducing nontraded varieties as an

input to traded variety production. The second is CES substitutability

of household demand between the traded and nontraded baskets, so

that the relative size of each sector responds to the price of its basket.

Intuitively, the import liberalization is a shock increasing the size of

the traded sector by pushing down import prices. This shock has dif-

ferential producer-level effects in the nontraded sector according to

each’s relative involvement in serving consumers compared to the


input market.

3.5.1. Theoretical framework

This section describes the building blocks of the model. In particular,

it highlights the input model’s departures from the baseline model.

Domestic households

The only difference to domestic households from the baseline model

is that they now combine the traded and nontraded baskets with con-

stant elasticity ρ. Then, with PT still denoting the price of the traded

basket and PN the price of the nontraded basket, households will di-

rect a fraction

β =PT(1−ρ)

PT(1−ρ) + PN(1−ρ)(3.15)

of expenditure toward the traded sector. This fraction is determined

in equilibrium, and adjusts based on the relative prices in the two

sectors. It is denoted with a tilde to distinguish it from the fixed share

β in the baseline model.


Traded sector

The key difference of the traded sector in the input model compared

to the baseline model is that producers now use both labor and non-

traded inputs for variable production. In particular, labor and non-

traded inputs are combined in a Cobb-Douglas way with a weight

α on labor. Then, a producer with productivity z will face marginal

costs of wαP1−αi , where Pi represents the price of the nontraded input.

The cost Pi may differ from price of the consumer bundle PN, since

a different set of nontraded producers may choose to participate in

each market. This selection process will be characterized more fully

with the nontraded sector. For simplicity, fixed costs continue to be

paid in labor only.

Given the new formulation of marginal costs, the marginal pro-

ducer for domestic and export participation have productivities de-

fined by

w f Td =

β

σT wL

(σT

σT − 1wαP1−α

i aT(zTd)

PT

)1−σT

(3.16)


w f Tx =

1σT XT∗

(σT

σT − 1dwαP1−α

i aT(zTx)

PT∗

)1−σT

(3.17)

respectively. Observe that both wages w and the price of the input

Pi are common across all producers of tradable goods. It is therefore

still possible to write a producer’s variable profits in either market in

terms of the productivity of the marginal producer. Specifically, the

variable profits in market m ∈ d, x of a producer with productivity

z are

w f TmhT

(z, zT

m

)where h(·, ·) takes the same form as in the baseline model.

Free entry in the sector remains, ensuring that there are zero profits

in equilibrium. Since the profits of any producer can be expressed

only in terms of the fixed costs of market participation and marginal

participation productivities, the free entry condition is as follows.

f Te = ∑

m∈d,xf Tm

∫zT

m

[hT(

z, zTm

)− 1]

dGT(z) (3.18)

This condition sums over the expected profits net of fixed costs in both


markets, taking on the same form as in the baseline model. It stays

the same precisely because, in both markets, the variable profits of the

producer with productivity z can be written in terms of the marginal

participant’s productivity just as in the baseline model.

Foreign producers are once again able to pay a fixed cost f T∗x to

export to the domestic market. Since the foreign cost function cT∗(·)

is in terms of the numeraire, the model can remain agnostic on the

foreign factors of production. The foreign exporting cutoff is therefore

formulated similarly as in the baseline model, as follows.

f T∗x =

β

σT wL

(σT

σT − 1d∗cT∗(zT∗

x)

PT

)1−σT

(3.19)

The main difference between this expression and its counterpart from

the baseline model is that the share of domestic expenditure β dedi-

cated to the traded sector replaces β.

Aggregating prices across active domestic and foreign producers

delivers the price index of the tradable basket as follows.


(PT)1−σT

= T∫zT

d

(σT

σT − 1wαP1−α

i aT(z))1−σ

dGT(z)

+ T∗∫

zT∗x

(σT

σT − 1d∗cT∗(z)

)1−σT

dGT∗(z) (3.20)

Once again, the main distinction between the expression for the price

index here compared to the baseline model is the introduction of non-

traded inputs for domestic production.

Nontraded sector

Nontraded varieties still cannot be shipped abroad or imported. How-

ever, producers in this sector now can serve both domestic consumers

as well as domestic traded producers. In what follows, these will be

referred to as the consumer and input markets, respectively. The con-

sumer market remains the same as in the baseline model, aside from

the fact that consumers now spend a fraction (1− β) of expenditure

on the nontraded basket. Then, the productivity of the marginal pro-

ducer of nontraded varieties serving domestic consumers is as follows.


w f Nc =

1− β

σN wL

(σN

σN − 1waN(zN

c)

PN

)1−σT

(3.21)

In particular, nontraded production continues to only employ labor.

There is now an additional source of demand for nontraded vari-

eties, namely from producers of traded varieties to use as inputs for

production. For this purpose, nontraded varieties available on the in-

put market are aggregated at a constant elasticity σN by the producers

of traded varieties. The producer of a nontraded variety charging a

price p on the input market therefore faces the demand curve

Xi

Pi

(pPi

)−σN

, Pi =

∫ω∈ΩN

i

pNi (ω)1−σN

dω

1

1−σN

where Xi is total input demand from the traded sector, ΩNi represents

the set of available varieties on the input market, and pNi (ω) is the

price charged for variety ω on the input market. Given the CES na-

ture of input aggregation, producers will charge the constant markup

σN/(σN − 1) over marginal costs.


A fixed cost f Ni units of labor must be paid to access the input mar-

ket. It therefore defines the marginal productivity zNi for producers

of nontraded varieties to participate on the input market. Specifically,

the zero profit condition for these types is

w f Ni =

1σN Xi

(σN

σN − 1waN(zN

i)

Pi

)1−σN

. (3.22)

In general, the set of producers participating in the input market will

not be the same as the set of producers serving domestic households.

The reason is that both markets feature different total demand and

fixed costs of access, similarly to the domestic and export markets

faced by the producers of traded goods. Consumers are additionally

disallowed from purchasing nontraded varieties on the consumer

market then reselling them on the input market, ensuring that the

set of available varieties on each market will be characterized by the

threshold productivities zNc and zN

i .26 As a direct consequence, the

26One interpretation of this restriction is that only the producer is able to converttheir good between the consumer market version and input (commercial) marketversion. For example, the fixed cost f N

i could be seen as developing this infrastruc-ture or proprietary technology.


price indices PN and Pi in each are not equal.

The total expenditure on inputs Xi can be further characterized

by recognizing that traded producers spend a constant fraction (σT −

1)/σT of sales on variable inputs. Of that expenditure, a constant share

α is paid to labor while the remaining (1− α) share is paid to inputs.

Using the expression relating a traded producer’s variable profits to

the cutoff productivity of domestic and export market participation,

total expenditure on inputs is expressed as below.

Xi = (1− α)(

σT − 1)

T ∑m∈d,x

w f Tm

∫zT

m

hT(

z, zTm

)dGT(z) (3.23)

Then, the size of input market demand naturally depends on the size

of the traded sector.

Producers can freely enter into the nontraded sector upon paying

the fixed cost w f Ne , at which point they draw their productivity from

the distribution GN(·). Potential entrants decide to do so as long

as the expected profits are positive, so these are pushed to zero in

equilibrium. Once again, the free entry condition is greatly simplified

by the fact that it is possible to write the variable profits of an input


market supplier in terms of the productivity of the threshold supplier.

In this case, the variable profits earned by a producer of nontraded

varieties will be

w f Ni hN

(z, zN

i

)from participating in market m ∈ c, i. Then, the free entry condition

is written as follows.

f Ne = ∑

m∈c,if Nm

∫zN

m

[hN(

z, zNm

)− 1]

dGN(z) (3.24)

Let the mass of entrants in equilibrium be denoted with N.

Finally, the price indices in the consumer and input markets can be

rewritten in terms of productivity instead of blueprint ω. These are

(PN)1−σN

= N∫

zNc

(σN

σN − 1waN(z)

)1−σN

dGN(z) (3.25)

(Pi

)1−σN

= N∫

zNi

(σN

σN − 1waN(z)

)1−σN

dGN(z) (3.26)

where the fact that producers with the same efficiency will set prices


identically is leveraged. The nontraded nature varieties in this sector

precludes the participation of foreign producers in either market.


Labor is freely mobile across both sectors, so a single labor market

clearing condition equates total labor supply with total labor demand

as follows.

L = T

f Te + ∑

m∈d,zf Tm

∫zT

m

[α(

σT − 1)

hT(

z, zTm

)+ 1]

dGT(z)

+ N

f Ne + ∑

m∈c,if Nm

∫zN

m

[(σN − 1

)hN(

z, zNm

)+ 1]

dGN(z)

(3.27)

The left hand side of this expression represents total labor supply,

while the right hand side summarizes all sources of labor demand.

The first sum contains all labor demanded from traded producers,

including for fixed costs as well as for variable production. The second

sum displays the analogous terms for the nontraded producers. The


characterization of labor market clearing in this model differs from the

expression in baseline model both because only a share α of the traded

sector variable costs are devoted to labor and also because nontraded

producers demand labor to serve the input market.

3.5.2. Equilibrium

Equilibrium in this model is defined similarly to as in the baseline

model. The main difference is the presence of one new cutoff in the

nontraded sector for input participation zNi , the corresponding price

index Pi and demand Xi in that market, as well as the share of con-

sumption β spent on traded goods.

Equilibrium. Given a vector for foreign aggregates XT∗, PT∗, T∗, the

cutoff productivities zTd , zT

x , zT∗x , zN

c , zNi , price indices PT, PN, Pi, en-

trant masses T, N, wages w, expenditure share β, and input expenditure

Xi are an equilibrium if they satisfy

(i) zero cutoff conditions for the domestic tradables market (3.16), the ex-

port market (3.16), the import market (3.19), the nontraded consumer

market (3.21), and the nontraded input market (3.22);


(ii) price aggregation in the traded (3.20), nontraded consumer (3.25),

and nontraded input (3.26) markets;

(iii) free entry in the traded (3.18) and nontraded (3.24) sectors;

(iv) labor market clearing (3.27) across both;

(v) consumption behavior (3.15) between both; and

(vi) total demand in the input market (3.23).

The general strategy to solve for equilibrium is as follows. First,

producer behavior in both sectors is characterized in terms of the ag-

gregate β, the share of consumer expenditure directed at the traded

sector. Then, the value for β is determined by combining these con-

clusions on producer behavior in sectoral equilibrium with consumer

substitution across them. Appendix C provides the parameter restric-

tions necessary for a unique equilibrium.

Beginning with the nontraded sector, observe that the free entry

condition (3.24) implies a fixed negative relationship between the two

cutoffs zNc and zN

i of the sector. This relationship results because the

expected profits of entry must be zero, tying operating profits to the


fixed cost of entry f Ne . If the participation cutoff for the consumer

market decreases, expected profits from serving this market increases.

The participation cutoff for the input market must consequently in-

crease in order to keep total expected profits fixed.

Next, combine the remaining aggregate conditions characterizing

the nontraded sector to derive a “sales ratio” condition as follows.

β

1− β

σT − 1σT (1− α) =

f Ni∫

zNi

hN(z, zNi)

dGN(z)

f Ni

∫zN

ihN(z, zN

i)

dGN(z)(3.28)

This expression governs the ratio of sales made on the input market to

those made on the consumer market. The left hand side summarizes

this ratio using aggregate variables, while the right hand side does

so with aggregation over individual producers. In particular, the left

hand side makes use of the fact that the CES nature of demand for

traded goods implies that their producers spend a constant fraction

(σT − 1)/σT of sales on variable inputs, of which a share (1− α) is

devoted to nontraded inputs. In addition, total sales in the traded

sector are simply βwL because trade is balanced as will be discussed

when equilibrium in the traded sector is described. Similarly, total


sales on the consumer market is (1− β)wL. These are divided to arrive

at the term on the left hand side.

The fraction on the right hand side of (3.28) leverages the fact that

the variable profits in market m ∈ c, i of a producer with produc-

tivity z can be written as w f Nm hN(z, zN

m). Given any value of β, this

condition imposes an increasing relationship between the two cutoffs

zNc for participating in the consumer market and zN

i the input market.

To see why, fix β. Then, the ratio of total sales in the two markets is

fixed. If the participation cutoff increases in one, decreasing total sales,

then it must be that the participation cutoff increases in the other to

hold the ratio fixed. In other words, the relative “toughness” of the

two markets is fixed by β.

Combining the free entry condition (3.24) with the sales ratio con-

dition (3.28), the thresholds zNc and zN

i can be determined for a single

value of β. Figure 3.3 depicts equilibrium in the nontraded sector. A

decline in β shifts the sales ratio curve outward, since it increases the

ratio of sales on the input market relative to the consumer market.

Then, the cutoff for participation in the input falls as the cutoff for par-

ticipation in the consumer market rises as the input market becomes


zNi

zNc

(3.28)

(3.24)

Figure 3.3.: Equilibrium in the nontraded sector for a single value ofβ is determined by the free entry (3.24) and sales ratio (3.28) condi-tions. An increase in β shifts the sales ratio condition outward, sothat zN

c falls and zNi increases.

relatively less competitive. The remaining aggregate variables in the

nontraded sector follow given the determination of the cutoffs.

Now consider the traded sector. A similar logic used to determine

equilibrium in the baseline model can be used here given a value for

β. To that end, fix β. The cutoff for export participation (3.17) can

be rewritten so that it specifies an increasing relationship between

domestic wages w and the exporting cutoff zTx as follows.

σTwσTf Tx = X∗PT∗(σT−1)

(σT

σT − 1aT(

zTx

)(Pi

w

)1−α)1−σT

(3.29)


Note that the condition determining the threshold productivity for

nontraded producers to serve the import market completely deter-

mines Pi/w, given a level of β. Then, rising domestic wages increase

costs for exporters, while the size of the export market X∗PT∗(σT−1)

remains fixed. The minimum productivity to export must therefore

increase accordingly, defining the competitiveness condition in an

analogous way as in the baseline model.

Likewise, the remaining aggregate conditions in the traded sector

arrive at the trade balance condition below.

σT f T∗x T∗

∫zT∗

x

hT∗(

z, zT∗x

)dGT∗(z) =

βwL f Tx∫

zTx

hT(z, zTx)

dGT(z)

∑m∈d,x

f Tm∫

zTm

hT(z, zTm)dGT(z)

(3.30)

This condition is the same as in the baseline model, but rewritten to

highlight the dependence on β. It equates total import volume on the

left hand side with total export volume on the right hand side. In par-

ticular, total export volume is written as a fraction of total sales in the

traded sector, given by βwL. The fraction derives from aggregating

over producers of all productivities. As argued in the baseline model,


trade balance implies a negative relationship between the domestic

wage w and the exporting cutoff zTx . Together with the competitive-

ness condition and a given value for β, trade balance determines the

equilibrium value of the domestic wage w and exporting cutoff zTx .

The remaining aggregates in the traded sector follow from the equi-

librium conditions characterizing the sector.

Finally, having characterized both sectors contingent on a value for

the share of consumer expenditure β directed toward traded goods, β

is now determined jointly with zTx . In particular, the characterization

of both sectors so far implies an increasing relationship between β

and the exporting cutoff zTx under reasonable parameter restrictions.

27 Broadly, increasing the share of household expenditure β on the

traded sector increases the right hand side of the trade balance con-

dition (3.30), implying increased exports. The exporting cutoff must

increase in response to restore trade balance. This increasing relation-

ship between β and zTx takes into account the effect of β’s increase

on all sectoral equilibrium variables characterized so far. It is conse-

27For example, σN > 2 is sufficient to ensure the relationship is increasing. SeeAppendix C.2 for the full discussion.


quently referred to as the sectoral equilibrium condition.

However, the share β is also characterized by household CES prefer-

ences over the two sectoral baskets given by (3.15). This condition dis-

ciplines the share of household consumption spent on tradable goods

by the relative price of the traded basket compared to the nontraded

basket. It implies a negative relationship between β and zTx under rea-

sonable parameter restrictions. 28 The reason is that, as the exporter

cutoff zTx increases, the minimum productivity zT

d of those supplying

the domestic market decreases according to the free entry condition.

In turn, the price of the traded basket for domestic consumers rises as

producers with lower productivity begin to participate. In response,

β falls as consumers shift expenditure toward the nontraded basket.

To summarize, equilibrium in the two sectors implies a positive

relationship between the exporter cutoff zTx and share of consumer

expenditure β directed toward the traded sector while consumer sub-

stitution behavior implies a negative relationship. These two condi-

28For example, it is enough for (ρ− 1)/(σT − 1) + (1− α)(ρ− 1)/(σN − 1) < 1.Observe that (ρ− 1)/(σT − 1) and (ρ− 1)/(σN − 1) both are less than one as longas ρ < σT and ρ < σN . See Appendix C.2 for the full discussion.


zTx

β

sectoral equilibrium

(3.15)

Figure 3.4.: Equilibrium in the input economy is determined by sec-torial equilibrium forces and consumer substitution between thetwo sectors (3.15). A decrease in d∗ shifts the sectoral equilibriumrelationship upward.

tions combine to determine the unique equilibrium in this economy.29

Figure 3.4 plots the equilibrium’s determination graphically.

Finally, welfare is equivalent to real wages, which in this model are

written as

wP

=

[(PT

w

)1−ρ

+

(PN

w

)1−ρ] 1

ρ−1

=w

PN

(1

1− β

) 1ρ−1

∝ aN(

zNc

)(1− β)

11−σ+

11−ρ

where the second line follows from the zero cutoff condition for non-

29The proof is provided in Appendix C.2.


traded producers serving the consumer market. Then, welfare is in-

creasing in both the share of tradables in consumer consumption β as

well as the consumer market cutoff zNc faced by nontraded producers.

3.5.3. Liberalization counterfactual

Falling import tariffs are once again modeled by decreasing the ice-

berg cost faced by foreign exports d∗. By the same logic as in the

baseline model, this change results in a shift downward of the trade

balance condition for any given β. Then, for the same value of β,

sectoral equilibrium delivers a lower exporter cutoff zTx just as in the

baseline model. In contrast to the baseline model, this change now

translates into a shift inward of the sectoral equilibrium relationship

shown in Figure 3.4 between the exporter cutoff zTx and β. The shift

represents the fact that, for any given value of β, the exporter cutoff

zTx implied by sectoral equilibria has been lowered. Then, as the econ-

omy adjusts to the import tariff, the share β spent on the traded basket

increases while the exporter cutoff zTx declines.

This adjustment occurs since consumer demand is elastic between


the traded and nontraded bundle, that is, ρ > 1. Decreasing the cost

of importing lowers the cost of the traded bundle, causing consumers

to consume more imports. The share of expenditure on the traded sec-

tor β increases accordingly. As discussed previously, the expanded

consumption of imports is balanced by increased exporting activity,

which is achieved through increased participation in exporting as the

exporting cutoff zTx is pushed down. In the nontraded sector, expan-

sion of traded sector activity necessarily implies contraction of the con-

sumer market for nontraded varieties. Fewer producers participate

in this market as the participation constraint zNc rises in response to

decreased demand. In parallel, as demand in the input market grows

along with traded activity, it becomes more profitable for any given

producer to participate. The minimum productivity zNi required to

make positive profits net of the fixed cost of participation drops as

more producers begin converting their varieties for use as inputs.

Adjustment to the new tariffs also implies reallocation among pro-

ducers in both sectors that are qualitatively consistent with the em-

pirical patterns observed. Labor flows among traded producers are

similar to those from the baseline model. The increase in the exporter


cutoff zTx is accompanied by a decrease in the domestic participation

cutoff zTd . Then, along the extensive margin, there will be a measure

of small producers that prefer to exit than operate once the tariff cuts

have been implemented. Similarly, the total labor employed after en-

try by a producer of traded goods is

f Td

(σT − 1

)hT(

z, zTd

)+ 1+ 1x(z) f T

x

(σT − 1

)hT(

z, zTx

)+ 1

if the producer has productivity z. This expression mirrors the one

from the baseline model, so that continuing producers serving only

the domestic market unambiguously shrink in scale as the domestic

cutoff zTd rises. Exporters will scale up their exporting activity while

shrinking their operations aimed at the domestic market. If the first

effect outweighs the second, exports will expand in employment.

Now consider the producers of nontraded goods. The participa-

tion requirement tightens for the consumer market but slackens in

the input market. As long as there is selection into serving the input

market, that is zNc < zN

i , then these changes observationally translate

into the exit of the smallest producers of nontraded varieties available.


This pattern would be consistent with the qualitative patterns from

the Brazilian data, where the smallest nontraded establishments in an

area exposed to tariff cuts were more likely to exit compared to the

largest establishments. Conditional on the assumption that more pro-

ducers serve the consumer market than the input market, the model is

able to qualitatively match the reallocation patterns among nontraded

producers in import-exposed areas along the extensive margin. This

section therefore proceeds under this assumption.

The model is also able to explain the qualitative patterns of labor

flow among nontraded producers along the intensive margin. Con-

sider the employment of a nontraded producer with productivity z

after entry.

f Nd

(σN − 1

)hN(

z, zNc

)+ 1+ 1i(z) f N

i

(σN − 1

)hi(

z, zNi

)+ 1

Producers only engaging in the consumer market unambiguously

shrink as the consumer market becomes more competitive. On the

other hand, producers also converting their varieties into inputs for

traded production will see an expansion in that portion of their busi-


ness. Their employment for serving the consumer market will shrink,

so provided that the second effect does not overwhelm the first, those

producers that serve the input market will grow in employment. The

intuition is simple. Ultimately, consumption demand increases for

tradables and decreases for nontradables since the price of imports

has dropped. Economic activity in the nontraded sector is therefore

reorganized toward serving the traded sector compared to serving

consumers.

The input model is able to generate reallocation among nontraded

producers in response to the import policy due to the interaction of

two key elements. The first is introducing the input market, which

itself has two effects. It both provides a selection mechanism for the

nontraded producers, wherein there are two types of economic ac-

tivity but only the most productive engage in both. As the relative

importance of these two activities changes, reallocation occurs among

producers because they are differentially engaged in each. The sec-

ond effect of the input market is that it links the nontraded and traded

sector, so that policy changes in the traded sector translate into the

nontraded sector.


Furthermore, the consumers in the input model have CES prefer-

ences over the baskets from the two sectors. The relative size of the

two sectors are therefore determined in equilibrium, and crucially re-

sponds to the policy change. As consumer demand for the nontraded

basket shrinks relative to the traded basket, for example, it reorga-

nizes labor into production of the traded good, which is borne out

differentially on the nontraded producers based on the composition

of their activity. These two elements together therefore generate the

reallocation among nontraded producers that was not present in the

baseline model.

To more clearly see the important interaction between the two key

elements, suppose only one is present. Begin with introducing the

input market but continuing to hold fixed the relative household con-

sumption (1− β) on nontraded goods. Then, the free entry and the

sales ratio conditions in the nontraded sector completely determine

the cutoffs for consumer and input market participation. The reason

is because the sales ratio condition fixes the relative variable profits

from each activity while the free entry condition fixes the sum of these

profits (net of fixed costs). In this model, if the trade policy is imple-


mented, there would be reallocation in the traded sector between do-

mestic and exporting activity. However, the total amount of economic

activity in the traded sector would still be fixed at βwL, so the mass

of producers serving the input market would not change. Similarly,

the total amount of consumer demand for nontraded varieties also

remains fixed, requiring no change to the mass of producers serving

the consumer market.

Now suppose there is no input market, but consumers substitute

between the two sectoral baskets according to their relative prices.

Then, the import liberalization would still trigger an increase in β as

imports become cheaper. However, a change to the ratio β would now

affect all producers of nontraded goods the same way. In other words,

total demand shrinks symmetrically regardless of the producer’s pro-

ductivity. The free entry condition for the nontraded sector would

be identical to the one from the baseline model, fixing the minimum

productivity of the participating producer. As an implication, the to-

tal labor hired by any continuing producer would remain the same.

The sector therefore adjusts by a decrease in the mass of entrants N,

uniformly scaling the sector down to match decreased consumer de-


mand. In this model, therefore, there are no differential effects across

nontraded producers.

Finally, consider the impact on household welfare of the import

liberalization in this model. Recall that it was proportional to

aN(

zNc

)(1− β)

11−σ+

11−ρ

so that it is increasing in the share β of tradables in consumption

as well as zNc the cutoff for nontraded producer participation in the

consumer market. Both increase in response to the tariff decreases,

implying an unambiguous positive effect on household welfare.

In summary, it is consequently the interaction of these two elements

that implies labor reallocation patterns among nontraded producers.

Differential producer level participation in the input market creates

variation in the exposure to the demand for traded goods among non-

traded producers, while an endogenously determined consumption

share β changes the relative demand for traded goods. The import

liberalization shock increases the relative demand for traded goods,

which translates as a positive shock to the input market. This shock is


of course then propagated differentially to the producers of nontraded

goods based on whether they serve the input market. Since the share

β of tradables in household consumption and the competitiveness of

the nontraded variety consumer market zNc are sufficient statistics for

welfare, the import liberalization unambiguously raises welfare.

3.6. Conclusion

With a bold and surprising policy change, Brazil’s import market was

unilaterally liberalized in the early 1990s in a series of steep tariff de-

clines. Given its unexpected nature, this liberalization episode has

been studied extensively to understand how an economy adjusts to

import activity. In particular, a recent growing literature has investi-

gated the role of the local labor market in this adjustment, relying on

the fact that they were exposed to the policy differently depending on

the initial industrial composition of their economic activity.

This study uncovers patterns of labor reallocation in local labor mar-

kets between producers participating in the same industry. Within

traded industries, these patterns confirm intuitions from the heteroge-


neous producer international trade literature. Specifically, in regions

more exposed to the import liberalization, large producers were more

likely to survive than smaller ones in the same industry. Moreover,

large survivors were less likely to contract (or more likely to expand)

than their smaller counterparts. These patterns have been empha-

sized by the literature because they represent potential productivity

and welfare gains as resources are reallocated toward more produc-

tive establishments.

In addition, the study uncovers a new margin of adjustment within

the local labor market, which is labor reallocation between producers

within nontraded industries. These patterns are qualitatively simi-

lar to those from traded industries, with large producers in import-

exposed areas more likely to survive and grow than their smaller

equivalents in the same industry. This reorganization suggests a novel

implication of importing for industrial productivity and concentration

of nontraded industries. However, these patterns are not consistent

with a natural extension of the Melitz [2003] model where a hetero-

geneous producer nontraded sector is added. This model predicts no

change within the nontraded sector, so that it continues to feature the


same mass and distribution of producers.

The chapter then proposes an input model where nontraded vari-

eties, in addition to being consumed by households, are also used

as inputs in the production of traded varieties. It is able to qualita-

tively generate the differential effects across producers of nontraded

varieties, with the smallest producers exiting and smallest survivors

shrinking. The key feature generating this effect is that only a por-

tion of nontraded producers select into supplying the input market.

Differential participation in the input market across producers trans-

lates into differential exposure to traded sector shocks. As importing

lowers the price of tradables and consumers shift their consumption

towards tradables in favor of nontradables, producers only serving

households see sales decline. On the other hand, the consumer shift to-

wards tradables also has a positive sales effect for those that serve the

input market. The producer size distribution therefore tilts, similarly

to in the data, with the smallest producers exiting and the smallest

survivors shrinking.

Appendix A.

Combinatorial discrete choices

During this appendix, the vector of aggregates υ ∈ Υ is assumed to be

fixed at an arbitrary value. It is omitted as an argument in all functions

for notational brevity.

A.1. Theoretical proofs

The section proves the two propositions from the chapter, Proposition

1 and Proposition 2.

Proposition 1. Suppose that (Z, L, U) are so that, for all z ∈ Z, L ⊆

J ∗(z) ⊆ U. Then, for each Ze returned by Algorithm 1, z ∈ Ze implies

L ⊆ Le ⊆ J ∗(z) ⊆ Ue ⊆ U.

A. Appendix: Combinatorial discrete choices 305

Proof. The goal is to show that if z ∈ Ze, Le ⊆ L ⊆ J ∗(z) ⊆ Ue ⊆ U

for any Ze, Le, Uee returned by Algorithm 1.

First, observe that the set of triples Ze, Le, Uee is derived from the

functions L(·) and U(·), which are defined over Z. The triples are a

partitioning of Z such that L(·) and U(·) are constant over each cell.

It is equivalent to show that L ⊆ L(z) ⊆ J ∗(z) ⊆ U(z) ⊆ U for all

z ∈ Z.

Step (ii) from Algorithm 1 ensures that L ⊆ L(z) and U(z) ⊆ U

for every z ∈ Z. What remains is to show that L(z) ⊆ J ∗(z) and

J ∗(z) ⊆ U(·).

Fix a z ∈ Z and start with L(z). Select an arbitrary element j ∈ L(z).

If j ∈ J ∗(z), then L(z) ⊆ J ∗(z) since j was an arbitrary element. Step

(ii) constructs L(z) as

L(z) =

L ∪ j ∈ U \ L | z ∈ Zj(U) if (MS-S)

L ∪ j ∈ U \ L | z ∈ Zj(L) if (MS-C).

If j ∈ L, then j ∈ L ⊆ J∗(z) by assumption on L. The only case


remaining is if

j ∈

j ∈ U \ L | z ∈ Zj(U) if (MS-S)

j ∈ U \ L | z 6∈ Zj(L) if (MS-C).

By definition on Zj(U) and Zj(L), Dj(U; z) > 0 and Dj(L; z) ≥ 0

respectively. Then,

0 < Dj(U; z) ≤ Dj(J ∗(z); z) when (MS-S)

0 ≤ Dj(L; z) ≤ Dj(J ∗(z); z) when (MS-C)

so it must be that j ∈ J ∗(z).

A similar argument can be made to show J ∗(z) ⊆ U(z) for all

z ∈ Z. Fix z ∈ Z and select an arbitrary element j ∈ J ∗(z). Observe

that it must be that Dj(J ∗(z); z) ≥ 0. Showing j ∈ U(z) is equivalent

to showing J ∗(z) ⊆ U(z). By the definition of U(z) in step (ii),

U(z) =

U \ j ∈ U \ L | z 6∈ Zj(L) if (MS-S)

U \ j ∈ U \ L | z ∈ Zj(U) if (MS-C)


so it is sufficient to show that

j 6∈

j ∈ U \ L | z 6∈ Zj(L) if (MS-S)

j ∈ U \ L | z ∈ Zj(U) if (MS-C).

For a contradiction, suppose that j is in these sets. By the definition

of Zj(L) and Zj(U), Dj(L; z) > 0 and Dj(U; z) < 0. If j ∈ U \ L |

z 6∈ Zj(L) so Dj(L; z) ≤ 0. By (MS-S), Dj(J ∗(z); z) ≤ Dj(L; z) ≤ 0,

a contradiction to the initial assertion that j ∈ J ∗(z). Likewise, if

j ∈ j ∈ U \ L | z ∈ Zj(U), then Dj(U; z) < 0. However, by (MS-C),

Dj(J ∗(z); z) ≤ Dj(J ∗(z); z) < 0, a contradiction to the assertion

that j ∈ J ∗(z).

Proposition 2. With any initialization, Algorithm 2 repeats step (iii) a

maximum of |J| times.

Consider a triple (Z, L, U) with L ⊆ J ∗(z) ⊆ U for all z ∈ Z, to

which Algorithm 2 is applied.

Let each iteration of step (iii) be indexed by n. Further, let the Cn =

(Zcn, Lc

n, Ucn)c be the collection of triples still being iterated on while

Rn = (Zrn, Lr

n, Urn)r is the collection of triples for which iteration has


stopped.

Lemma. For any iteration n, Pn ≡ Zcnc ∪ Zr

nr partitions Z. That is,

for each z ∈ Z, there exists at most one c with z ∈ Zcn. If no such c exists,

there is a unique r with z ∈ Zrn.

Proof. We use induction on n. When n = 0, C0 = (Z, L, U) and

R0 = , so P0 = Z. Then, P0 partitions Z trivially.

Now suppose Pn partitions Z. Notice that step (iii) does not affect

any triples within Rn. Consider an arbitrary triple (Zcn, Lc

n, Ucn) ∈ Cn.

Step (iii) will apply the updating Algorithm 1 to this triple. If it returns

(Zcn, Lc

n, Ucn), then substep (iii)c. places this triple into Rn+1. Other-

wise, it returns Zcen , Lce

n , Ucen e, a further partitioning of Zc

n. Each of

these triples is placed into Rn+1. Then, each Zcn is (weakly) subparti-

tioned by step (iii). As a result, Pn+1 is still a partitioning of Z.

By the lemma, for every z ∈ Z, there is a unique cell in Pn to which

z belongs. At each iteration step, the functions Ln(·) and Un(·) can be


defined as follows

Ln(z) =

Lc

n, z ∈ Zcn

Lrn, z ∈ Zr

n

, Un(z) =

Uc

n, z ∈ Zcn

Urn, z ∈ Zr

n

.

Another way of interpreting step (ii) is therefore that it updates Ln(·)

and Un(·).

We now prove that the pair Ln(·) and Un(·) can only be updated a

maximum of |J| times. The argument follows the same spirit as the

parallel proof provided in Arkolakis and Eckert [2017].

Proof. Fix z ∈ Z and consider the sequence of Ln(z) and Un(z) func-

tions for each iteration n. Suppose Ln(z) = Ln+1(z) and likewise

Un(z) = Un+1(z). Then, the (n + 1)th iteration of step (iii) has not

determined any of the choices that were left undetermined from iter-

ation n. By step (iii)c., the algorithm will stop iteration for Ln(z) and

Un(z).

For continued iteration, it must be that one undetermined decision

left from the last repetition is determined. That is, it must be that

either Ln(z) ⊂ Ln+1(z) or Un+1(z) ⊂ Un(z). The algorithm can be


initialized with a maximum |J| decisions undetermined. The maxi-

mum number of continued iterations on Ln(z) and Un(z) is therefore

|J|, if each iteration determines only one decision.

A.2. Non-binary extension

Consider a CDCP where each discrete choice has multiple discrete

options. For example, suppose a firm decides whether or not to have

zero, one, or two plants in Germany. Likewise, it must decide whether

to have zero, one, or two plants in Romania. Then, the full set can be

represented as the multi-set G, G, R, R containing two copies of G

and R. The firm then optimizes over all possible subsets of this multi-

set. For example, the decision to open two plants in Germany and one

in Romania can be represented as choosing the subset G, G, R. The

method presented in the paper can then be applied to this problem,

conditional on monotonicity in set and type.

Appendix B.

European Union corporate tax equalization

This Appendix discusses the practical efficiency of the iterative step

defined by Algorithm 2. Recall that the iteration does not necessarily

fix every binary decision. Undetermined binary decisions dealt with

by the branching step Algorithm 3, which guesses a choice for one

binary decision at a time. When the iterative algorithm cannot deter-

mine many binary choices, the full Algorithm reduces to the brute

force method. It is therefore important to characterize the effective-

ness of the iterative algorithm at fixing the binary choices.

As the interactions between them become stronger, the iterative

step becomes less effective. To see why, consider the updating step. It

checks a potential plant j’s marginal value from inclusion in subsets

B. Appendix: EU corporate tax reform 312

and subsets of the optimal set to reach a decision. In the substitutes

case (MS-S), the subset represents the least amount of possible profit

cannibalization among the firm’s plants, while the superset features

the most. Then, an element is undetermined when it has positive

marginal value in the smallest set but negative marginal value in the

largest set. In this case, the plant’s fixed cost is covered by its con-

tribution to expected variable profits when the firm does not have

many other plants to choose from. However, once there are, the extra

plant’s marginal value falls below the fixed cost. This outcome is more

likely if the plants are more substitutable. A similar logic holds in the

complements case, with an element possibly adding very little on the

margin if included into a small set. However, if complementarities

are large, it could be very valuable if added into a large set. Thus,

the effectiveness of iterative Algorithm 2 depends on the strength of

substitutability between plants in this context.

As discussed, the strength of substitutability in this model decreases

in the exponent (ρ− 1)/θ from (2.4). Thus, there is likely a negative

relationship between the exponent value and the number of binary

decisions determined by the iterative step. To quantify it, the iter-


Figure B.1.: Average number of choices that are undetermined by theiterative step

More binary decisions are undetermined when substitutability among themis high. However, even when substitutability is high, the iterative step solvesa majority of firm problems without requiring the branching step.

ative step is run holding aggregates constant for various exponent

values. Then, the number of undetermined binary solutions at each

productivity level z is calculated to assess the iterative step.

The blue line in Figure B.1 graphs the result for exponent values

between 0.1 and 0.9. Since the decisions are independent, therefore

featuring no substitutability, when the exponent approaches 1, rela-

tive substitutabiliy is represented as 1− (ρ− 1)/θ. As substitutabil-

ity among plants decreases, the average number of unfixed decisions


also decreases. A uniform distribution is chosen for this average since

the metric is ultimately meant to speak to computational time, so the

density of firms gi(·) at each productivity does not play a role. The

uniform distribution’s left bound is taken to be lowest productivity

at which a firm opens any plant. Likewise, the right bound is the

productivity at which a firm operates a plant in every country. As

depicted, the average number of unfixed decisions is small, falling

below 0.5. This statement holds even with an exponent value of 0.1,

when plants are relatively substitutable. Thus, all binary decisions are

determined by the iterative algorithm for most productivity values.

The logic of the algorithm eliminates a significant number of possible

sets by using both monotonicity conditions.

To contextualize Figure B.1, the calibration described in Section 2.4.1

sets ρ = 6 and θ = 7, resulting in an exponent of around 0.71. This

point is marked in Figure B.1 with a red point, corresponding to an

average of 0.012 decisions left unfixed by the iterative procedure. As

above, all binary decisions are determined for a large majority of firm

productivities, leaving a maximum of four undetermined in rare cases.

The CDCP’s structure is thus leveraged to determines the optimal set


with no need to guess at decisions for large ranges of the productivity

distribution.


country tax rate tax changeBulgaria 10% -17.26%Cyprus 12% -14.76%Ireland 12% -14.76%Lithuania 15% -12.26%Latvia 15% -12.26%Romania 16% -11.26%Slovenia 17% -10.26%Czech Republic 19% -8.26%Hungary 19% -8.26%Poland 19% -8.26%Estonia 20% -7.26%Finland 20% -7.26%United Kingdom 20% -7.26%Croatia 20% -7.26%Portugal 21% -6.26%Luxembourg 22% -4.79%Sweden 22% -5.26%Slovakia 22% -5.26%Denmark 24% -3.66%Austria 25% -2.26%Netherlands 25% -2.26%Greece 26% -1.26%Spain 28% 0.74%Germany 30% 2.74%Italy 31% 4.14%France 33% 6.07%Belgium 34% 6.73%

Table B.1.: Statutory corporate tax levels in 2014 and their counterfac-tual changes under the equalization policy.

Appendix C.

Brazil’s import liberalization

This Appendix provides supplemental material for Chapter 3. Sec-

tion C.1 contains supplemental empirical results while Section C.2

the proofs for the theoretical models.

C.1. Liberalization exposure coefficients by industry

Coefficients on the liberalization exposure measure in the chapter are

allowed to vary by initial plant size and industry tradability for ease

of presentation. This appendix displays the coefficients derived from

repeating the specifications of the chapter, with the modification that

the liberalization coefficient can vary by industry. In particular, the

C. Appendix: Brazil’s import liberalization 318

specifications are now

vkr(ω) = Φ[αv

0 + αvs(r) + αv

k + αvb + βvXrk(ω)

+ θvbklibr + εv

rk(ω)]

(C.1)

grk(ω) = αg0 + α

gs(r) + α

gk + α

gb + βgXrk(ω)

+ θgbklibr + ε

grk(ω)

(C.2)

where (C.1) is the extensive margin specification and (C.2) the inten-

sive margin specification. The key change is that θvbk and θ

gbk now vary

by industry k rather than industry tradability K(k) as in the chapter.

Figure C.1 presents the liberalization exposure measure coefficients

for the extensive margin specification (C.1). Every industry is repre-

sented with six bars, each representing the liberalization coefficient

for a plant size bin. The bars are colored according to size bin. The

stack of bars for every industry are ordered with the largest plant

bin at the top and smallest plant bin at the bottom. Though there is

variation across industries, the main pattern discussed in the chapter

holds. Large plants were more likely to survive than small plants. In

particular, the pattern holds in both nontraded industries (the top ten


Figure C.1.: Plant-level exit probability coefficients by industry. Thesame between-plant patterns hold within industry.


industries) and traded industries.

Figure C.2 displays the analogous bar graph for intensive margin

coefficients. There is once again considerable variation in coefficient

magnitude across industries. However, the between-plant pattern

remains comparable. Small surviving plants were more likely to con-

tract (less likely to grow) compared to large surviving plants.

C.2. Theoretical proofs

This section presents the proofs for the model equilibria and compara-

tive statics. Section C.2.1 provides these for the baseline model while

Section C.2.2 does so for the input model.

C.2.1. Baseline model

This section sketches the argument establishing the unique equilib-

rium of the baseline model. It then describes how to show that drop-

ping foreign costs of exporting d∗ results in a shift upward of the trade

balance condition. The arguments are similar to those from Demidova

and Rodríguez-Clare [2013].


Figure C.2.: Plant-level growth probability coefficients by industry.The same between-plant patterns hold within industry.


Equilibrium

Begin with the competitiveness condition defined by the exporter cut-

off condition, replicated below for convenience.

w f Tx =

1σT XT∗

(σT

σT − 1dwaT(zT

x)

PT∗

)1−σT

(3.5)

By inspection, this expression defines a positive relationship between

domestic wages w and the exporter cutoff zTx .

Now consider the trade balance condition, replicated below.

σT f T∗xw

T∗∫

zT∗x

hT∗(

z, zT∗x

)dGT∗(z) = βL

f Tx∫

zTx

hT(z, zTx)

dGT(z)

∑m∈d,x

f Tm∫

zTm

hT(z, zTm)dGT(z)

(3.14)

Note that the left hand side represents import volume while the right

hand side represents export volume. Once combined with the other

equilibrium conditions, this expression implies a negative relation-

ship between the domestic wage w and exporter cutoff zTx . To begin,


note that the free entry condition

f Te = ∑

m∈d,x

f Tm

∫zT

m

hT(

z, zTm

)dGT(z) (3.7)

implies a negative relationship between the domestic cutoff zTd and

exporting cutoff zTx . In other words, the domestic cutoff can be written

as a decreasing function of the exporter cutoff zTd (z

Tx ). Then, holding

w fixed, export volume on the right hand side of the trade balance

condition is decreasing in zTx . Further, combining the foreign export-

ing (3.8) and the domestic participation (3.5) conditions imply that

the foreign exporting cutoff zT∗x is negative related to the domestic

exporting cutoff zTx . In particular, zT∗

x can be written as a function of

zTx and w accordingly.

wσTf Td

f T∗x

=

(aT(zT

d(zT

x))

d∗aT∗(zT∗x )

)1−σT

Let this function be zT∗x (zT

x , w). Then, import volume on the left hand

side of the trade balance condition is increasing in the exporter cutoff


zTx . Therefore, net exports are decreasing in the exporter cutoff zT

x .

To establish the negative relationship between domestic wage w

and exporter cutoff zTx implied by trade balance, it must be that net

exports are increasing in w when zTx is held fixed. Note that the right

hand side is unaffected by w. Then, it is sufficient to show that the left

hand side is decreasing in w when zTx . Taking the partial derivative of

f T∗xw

T∗∫

zT∗x (zT

x ,w)hT∗(

z, zT∗x

)dGT∗(z)

with respect to w establishes this result.

Liberalization counterfactual

Now this section shows that decreasing d∗ shifts the trade balance

condition downward while leaving the competitiveness condition as

before. By inspection, the relationship between domestic wages w

and exporter cutoff zTx defined by the competitiveness condition is

unchanged by changes in d∗.

Note however that zT∗x depends on d∗. Redefine the function link-

ing the home and foreign exporting cutoffs as zT∗x (zT

x , w, d∗) to empha-


size this dependence. Note that the trade balance condition will be

shifted in turn by changing d∗. Fix w and suppose d∗ falls. Then, zT∗x

decreases, increasing import volume. To maintain trade balance, it

must be that zTx falls. Decreasing zT

x both increases export volume and

decreases import volume, returning net exports to zero as required

by trade balance. Then, when d∗ is lowered, any given value of w is

associated with a lowered value of zTx . The trade balance condition

shifts downward accordingly.

C.2.2. Model with intermediate inputs

This section establishes the parameter restrictions sufficient for equi-

librium uniqueness of the input model. It then discusses the import

liberalization comparative static.

Equilibrium

Begin with the sectoral equilibrium relationship between the con-

sumption share β and exporter cutoff zTx . For ease of notation, let

the relative price of the nontraded input bundle Pi/w ≡ n.

The traded sector is similarly characterized by a competitiveness


and trade balance condition as in the baseline model. These are repli-

cated below respectively for convenience.

w f Tx =

1σT XT∗

(σT

σT − 1dwaT(zT

x)

PT∗ n1−α

)1−σT

(3.16)

σT f T∗xw

T∗∫

zT∗x

hT∗(

z, zT∗x

)dGT∗(z) = βL

f Tx∫

zTx

hT(z, zTx)

dGT(z)

∑m∈d,x

f Tm∫

zTm

hT(z, zTm)dGT(z)

(3.30)

First, let the competitiveness condition define a relationship w(zTx , n)

between the domestic wage, exporting cutoff, and relative price of

the nontraded input bundle. Additionally, let the left hand side of the

trade balance condition be denoted IM for imports and the right hand

side EX for exports.

Identically to the baseline model, the domestic participation cut-

off zTd (z

Tx ) can be written in terms of the exporting cutoff using the

free entry condition. Similarly, the foreign exporter cutoff is given by

its zero profit condition combined with the zero profit condition for


domestic participation, as follows.

wσT f Td

f T∗x

=

(aT(zT

d(zT

x))

d∗aT∗(zT∗x )

n1−α

)1−σT

Let this relationship be denoted by zT∗x (zT

x , w, n). Observe that the cen-

tral difference of equilibrium in the traded sector between the baseline

model and the input model is the presence of n. It appears in the com-

petitiveness condition delivering the relationship w(zTx , n) and the

determination of the foreign exporting cutoff delivering the relation-

ship zT∗x (zT

x , w, n).

Using the relationships established above, exports on the right hand

side of the trade balance condition can be written EX(zTx , zT

d (zTx ), β) ≡

EX(zTx , β). Similarly, imports can be written IM(w, zT∗

x (zTx , w, n)) ≡

IM(w, zTx , n). Then, given a value of β and n, the traded sector in the

input model resembles the traded sector in the baseline model. In

particular, the competitiveness condition specifies that w(zTx , n) is in-

creasing in zTx . At the same time, the trade balance condition given by

IM(w, zTx , n) = EX(zT

x , β) specifies a decreasing relationship between


w and zTx . The pair is therefore uniquely determined contingent on

values for β and n.

Observe further that the price of the input bundle n is determined by

nontraded sector equilibrium as described in the chapter and depends

only on β. In particular, this relationship is derived by using the zero

profit condition for nontraded producers participating in the input

market as below

n(β) =Pi

w=

[βL

σN f Ni

] 11−σN

aN(

zNi (β)

)

where zNi (β) is the relationship between the cutoff for participation

in the input market and β. This relationship is pinned down by the

nontraded sector free entry condition and sales ratio condition as in

the chapter. Then, fixing β determines n(β) so that a value for β is

sufficient to equilibrium in the traded sector.

The system above can be differentiated to arrive at the conditions

for which ∂zTx /∂β as implied by sectoral equilibrium is increasing. The

resulting relationship is


∂zTx

∂β

ddzT

xEX(zT

x , β)

EX(zT

x , β) − d

dzTx

IM(w(zT

x , n(β))

, zTx , n(β)

)IM(w(zT

x , n(β))

, zTx , n(β)

)

= − 1β+

σT − 1σT (1− α)

n′(β)

n(β)(SE)

resulting from differentiating the trade balance condition.1 All vari-

ables besides zTx and β in this condition have been substituted with

conditions from sectoral equilibrium. The terms in square brackets

are

ddzT

xEX(zT

x , β)

EX(zT

x , β) = sd

[(σT − 1)εT

(zT

x

)− `T

(zT

x

)]−[(σT − 1)εT

(zT

d

)− `T

(zT

d

)]zT

d′(

zTx

)< 0

ddzT

xIM(w(zT

x , n(β))

, zTx , n(β)

)IM(w(zT

x , n(β))

, zTx , n(β)

)= −εT

(zT

x

)[(σT − 1)2

σT − 1εT∗(zT∗

x )`T∗(

zT∗x

)]

+

[(σT − 1

)− 1

εT∗(zT∗x )

`T∗(

zT∗x

)]zT

d′ (

zTx

)> 0

1The notation (SE) stands for sectoral equilibrium, as it is referred to in the chapter.


where

εK(z) =aK ′(z)aK(z)

< 0 `K(

zKm

)=

1∫zK

mhK(z, zK

m)dGK(z)≥ 0

for sector K ∈ T, T∗, N and market m ∈ d, x, c, i, and

sd =

∫zT

dhT(z, zT

d )dGT(z)∫zT

dhT(z, zT

d )dGT(z) +∫

zTx

hT(z, zTx )dGT(z)

is the share of traded sector activity that is related to the domestic mar-

ket compared to the export market. Putting it together, the expression

in square brackets is negative. Intuitively, the bracketed expression

captures the effect on net exports of changing the exporting cutoff zTx

while holding β and n constant, but allowing all equilibrium variables

in the traded sector as well as wages w to adjust. Gross export flows

EX fall as zTx increases since it shifts the distribution of active produc-

ers so that a smaller share of them export. Increasing zTx also pushes

up gross imports through two channels. First, as the exporting cutoff

zTx rises, the domestic cutoff zT

d falls due to the free entry condition,

implying a decrease in zT∗x . The foreign exporting cutoff falls because


the domestic market has become less competitive. Additionally, in-

creasing zTx while holding n fixed implies that wages w rise due to

the competitiveness condition. Imports become relatively cheaper in

terms of the numeraire.

The right hand side of (SE) is the change in net imports due to chang-

ing β, and in turn n(β), but holding fixed aggregates in the nontraded

sector as well as domestic wages w. It must equal the change in net

exports from moving zTx to maintain trade balance. As long as the

right hand side is also negative, then ∂zTx /∂β is positive.

Using the expression for n(β),

n′(β)

n(β)=

11− σN

1β+ εN

(zN

i (β))

zNi′(β)

where a change in β has opposing effects on n since the first term in

the expression is negative while the second is positive. Substituting

into the right hand side of (SE) obtains

− 1β

[1 +

σT − 1σT

1− α

σN − 1

]+

σT − 1σT (1− α)εN

(zN

i (β))

zNi′(β)


where the first term is negative but the second is positive. It is there-

fore necessary to fully characterize zNi′(β) using the sales ratio and

free entry condition from the nontraded sector. Deriving both in terms

of β and combining,

zNi′(β) =

1β(1− β)

1A(β)

F(β) =(

σN − 1)

εN(

zNi

)[1 +

β

1− β

σT − 1σT (1− α)

]− `N

(zN

i (β))[

1 +(

β

1− β

σT − 1σT (1− α)

)2 fc

fi

εN(zNi (β))

εN(zNc (β))

]< 0

so the right hand side of (SE) becomes

− 1β

σT − 1σT (1− α)

[σT

σT − 11

1− α+

1σN − 1

−εN(zN

i )

(1− β)F(β)

]

where it is necessary for the expression in square brackets to exceed

0 for zTx (β) to be increasing. Then,

σT

σT − 11

1− α+

1σN − 1

−εN(zN

i )

(1− β)F(β)


>σT

σT − 11

1− α+

1σN − 1

(1− 1

(1− β) + β σT−1σT (1− α)

)

>σT

σT − 11

1− α+

1σN − 1

(1− σT

σT − 11

1− α

)=

1σN − 1

+σT

σT − 11

1− α

σN − 2σN − 1

where the second line follows from the definition of F(β) and the third

from the observation that β ∈ [0, 1]. Then, it is sufficient for σN > 2

for ∂zTx /∂β > 0 as described by sectoral equilibrium.

Now it remains to characterize the conditions under which (3.15)

delivers a decreasing relationship between zTx and β. Observe that

(3.15) implies that1− β

β=

(PN

wwPT

)1−ρ

while the price indices can be written in terms of β and zTx using the

zero profit conditions as follows.

PN

w=

[(1− β)L

σN f Nc

] 11−σN

aN(

zNc (β)

)PT

w=

[βL

σT f Td

] 11−σT

aT(

zTd

(zT

x

))n(β)1−α


Then, differentiate with respect to zTx and β to arrive at

− εT(

zTd

(zT

x

))zT

d′(

zTx

) ∂zTx

∂β

=1

β(1− β)

[1

ρ− 1− β

σN − 1−(

1− β

σT − 1+ (1− α)

1− β

σN − 1

)]− εN

(zN

c(

β))

zNc′(

β)+ (1− α)εN

(zN

i(

β))

zNi′(

β)

.

Note the coefficient on ∂zTx /∂β on the left hand side is negative, so

∂zTx /∂β < 0 if the right hand side is positive. The second line of the

right hand side is positive, since zNc increases in β while the opposite

is true for zNi . Then, it is sufficient for the term in square brackets to be

positive. As long as σK > ρ > 0 where K ∈ T, N, then it is sufficient

forρ− 1

σT − 1+ (1− α)

ρ− 1σN − 1

< 1 .

Under the conditions discussed in this section, there is a unique

equilibrium of the input model.


Liberalization counterfactual

The trade balance condition has the same forces as the one from the

baseline model, provided that β and in turn n(β) are fixed. Then, for

any value of β, reducing d∗ shifts the trade balance condition down-

ward and implies a lower zTx . This shift translates to a downward shift

of the sectoral equilibrium condition. Because d∗ does not appear in

the household substitution condition, it does not move.

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Heterogeneous Firms in International Trade · 2020. 11. 20. · sional CDCP described previously in three ways. First, if each agent has a diﬀerent optimal solution to the CDCP,