Producer Heterogeneity, Value-Added, andInternational Trade

Patrick D. Alexander∗

June 2017

Abstract

Standard new trade models depict producers as heterogeneous in total factor productivity. In thispaper, I adapt the Eaton and Kortum (2002) model to incorporate intermediate inputs and pro-ducer heterogeneity in value-added productivity. In equilibrium, this yields a positive relationshipbetween the international trade elasticity and the share of inputs in production. This relationshipprovides a potential explanation for the rising import content of exports over time and the “distancepuzzle” in international trade. The model yields similar conclusions to the standard model regard-ing static gains from trade, but different conclusions regarding the evolution of gains from tradeover time.

JEL classification codes: F11, F12, F15Keywords: Trade elasticity; Intermediate inputs; Gains from trade: Distance puzzle

∗Senior Economist, Bank of Canada, International Economic Analysis Department, 234 Wellington StreetWest, Ottawa, Ontario, Canada, K1A 0G9. Email: alep@bank-banque-canada.ca. I would like to thank CostasArkolakis, Pinelopi Goldberg, Ian Keay, Oleksiy Kryvtsov, Beverly Lapham, Trevor Tombe, Dan Trefler and MattWebb for valuable comments. I would also like to thank participants in seminars at the Bank of Canada, Universityof Calgary, Universite Laval, UQAM, Queen’s University, SUNY Albany and the 2014 CESifo Conference onGravity Models in Munich for their input. Resources and support provided by the Bank of Canada and Queen’sUniversity are also gratefully acknowledged. All mistakes are my own, and the views expressed in this paper aremy own and do not necessarily reflect those of the Bank of Canada.

1 Introduction

The growing international fragmentation of production is a well-documented phenomenon.For example, Hummels, Ishii and Yi (2001) provide evidence that vertical specialization –measured by the share of imported inputs in products that are exported – has grown significantlyover time. In theory, this rising trend could be explained by either of two factors: the overallshare of inputs has increased, or the import content of inputs has increased, in exported productsover time.

Figure 1 breaks down vertical specialization from 2005 into these two factors for severalmajor economies. On average, intermediate inputs accounted for roughly 61% of gross exports,and 24% of these inputs were imported, hence the average vertical specialization share wasequal to the product of these, 15%. Figure 2 reports how these two factors have grown overtime. Remarkably, growth in the input share of exports has been, on average, slightly larger

(in level terms) than growth in the import content of inputs across these economies. Hence,inputs are an increasingly important component of exports, and this helps to explain the rise ofvertical specialization trade over time.

Meanwhile, exports from sectors that use intermediate inputs also tend to be traded overshorter distances and, hence, are more “regionalized,” than other types of trade. This is shownin Figure 3, which depicts a negative relationship between the sector-level average distance oftrade (ADOT) and the average input share of output across countries.1

Together, these patterns suggest that exports of products that use intermediate inputs areparticularly sensitive to variations in trade costs. With the gradual decline in trade costs overtime, export growth has been particularly high for products that use inputs. Meanwhile, thetrade costs imposed by bilateral distance are particularly burdensome for sectors that use inputs,leading to a localized pattern of trade for these products. Notably, this extra sensitivity isnot reflected in standard micro-founded general equilibrium trade models with intermediateinputs.2

In this paper, I develop an adapted Eaton and Kortum (EK) model that generates this extrasensitivity. As in many versions of this framework, industries combine intermediate inputs andvalue-added to produce output. However, where standard versions of the EK model featureheterogeneity in total factor productivity (TFP) across products, this model features cross-product heterogeneity in value-added productivity (VAP). In equilibrium, this adjustment gen-erates a positive relationship between the sensitivity of exports to variations in trade costs (alsoknown as the “trade elasticity”) and the share of intermediate inputs used in production. Impor-tantly, this relationship is driven by the extensive margin of trade and does not emerge under a

1See Johnson and Noguera (2012c) for evidence of this pattern for imported inputs specifically.2A notable recent example that demonstrates this point is the model from Caliendo and Parro (2015), where

export sensitivity to trade costs is unrelated to intermediate inputs.

2

representative-producer model with a similar production structure.In an application of the model, I show that growth in the elasticity of international trade

with respect to bilateral distance observed since the 1970s (known as the “distance puzzle”)can be potentially explained by the rising share in intermediate inputs in exports observedsince this period. Hence, this higher elasticity can be reconciled with a stable (or weakening)relationship between trade costs and distance, and is not so puzzling after all.

I also quantify the gains from trade (GFT) according to this model for over thirty countriesand compare these with the GFT suggested by the standard EK setting with intermediate in-puts. The static GFT under this model are, for the average country, quantitatively similar tothe gains derived from the standard model. Crucially, however, while the standard model fea-tures a positive relationship between the GFT and the share of inputs used in production, thismodel features no such relationship, hence the evolution of GFT is different across these twoenvironments. I show that, as the share of inputs in production has grown over time for mostcountries, the standard model suggests significant growth in GFT where the VAP model doesnot. As a result, between 1995 and 2010, the average GFT grew roughly 39% more accordingto the standard model relative to the VAP model.

Overall, my findings contribute to the literature in several ways. Numerous others haveaimed to distinguish between intermediate inputs and value-added in exports using input-outputdata. For examples, see Antras et al. (2012), Hummels, Ishii and Yi (2001), Johnson andNoguera (2012a, 2012b, 2012c), Koopman, Wang and Wei (2012, 2014) and Timmer et al.(2014). These papers have generally drawn a particular country-level distinction between im-ported inputs and domestic value-added in exports. In contrast, the present analysis emphasizesthe distinction between intermediate inputs (domestic- or foreign-produced) and value-addedat the producer level. My results broaden the findings of Johnson and Noguera (2012c), whichstresses the localized pattern of exports for industries that use imported intermediate products,to suggest that this pattern emerges for industries that use any intermediate products, whethersourced domestically or imported.

My theoretical framework is based on the EK international trade model, combined witha production setting similar to Yi (2003, 2010). Yi (2003) aims to explain growth in verticalspecialization trade with a two-country model that features industry heterogeneity and endoge-nous growth in both the trade elasticity and vertical specialization trade. In contrast, my modelcan accommodate many countries, and yields a gravity-type equation in equilibrium.3 More-over, the trade elasticity is exogenous, but varies across sectors and is higher in sectors thatuse intermediate inputs. As a result, falling trade costs lead to endogenous growth in verticalspecialization trade at the country level.

3My model adds to a long list of trade models that feature gravity model properties. See Tinbergen (1962) forthe original exposition of a gravity model.

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This paper also adds to an extensive empirical literature that explores the distance puzzlein international trade. For several examples, see Berthelon and Freund (2008), Bhavnani et al.(2002), and Disdier and Head (2008).

The welfare analysis in this paper adds to the literature on quantifying the GFT based ontrade models with producer heterogeneity. For other contributions, see Arkolakis, Costinot andRodrıguez-Clare (2012), Caliendo and Parro (2015), Costinot and Rodrıguez-Clare (2014),and Levchenko and Zhang (2014). As with these papers, I show that the GFT in this modelcan be calculated using only a few parameters from the data. Moreover, for a common set ofcertain exogenous parameters (including trade elasticities), this model adds to the set of modelsdescribed by Arkolakis, Costinot and Rodrıguez-Clare (2012) that deliver the “same old gains”as representative firm models.

In contrast to these other papers, however, I demonstrate that under the VAP heterogeneitymodel, the trade elasticity is an increasing function of the intermediate inputs share, and hencethe GFT are not higher in sectors that use intermediate products. In other words, even strongerthan the message from Arkolakis, Costinot and Rodrıguez-Clare (2012), this model featuresthe “same old gains” as the classic models in addition to the “same old gains” for sectors thatuse inputs in production. This finding is in particularly stark contrast to results from Melitz andRedding (2014), which emphasize that trade in intermediate inputs generates welfare gains thatare significantly above the gains generated from trading final goods alone.

This paper also contributes to an existing literature that aims to correct for biases in the linkbetween empirical estimates of trade elasticities and model-based structural parameters. Otherexamples include di Giovanni, Levchenko and Ranciere (2011), Imbs and Mejean (2015),Melitz and Redding (2015), Ruhl (2008), and Simonovska and Waugh (2014). As in these pa-pers, I use theory to relate empirical trade elasticity estimates to the parameters of my model. Ialso show, as these other papers do, that failure to do this leads to highly distorted conclusionsfrom the model.

Finally, my findings emphasize the importance of the extensive margin of trade, which isalso emphasized by other recent papers. These include Chaney (2008), Crozet and Koenig(2010), Helpman, Melitz and Rubenstein (2008), and Hillberry and Hummels (2008).

The remainder of the paper is organized as follows. Section 2 describes the theoreticalframework. Section 3 provides a short application of the model to the distance puzzle in in-ternational trade. Section 4 provides quantitative results for the gains from trade. Section 5concludes. An Appendix follows.

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2 Theory

2.1 The model

The following is a static multi-sectoral EK model of trade with intermediate inputs similarto the model derived in Caliendo and Parro (2015).4

2.1.1 Environment

Consider a world with N countries and J sectors. Labor is freely mobile across sectorsin a given country, and immobile across countries. All markets are perfectly competitive.Country n has a measure Ln of representative households. Households in n purchase Cj

n unitsof composite final goods from each sector j to maximize the following Cobb-Douglas utilityfunction:

Un =J∏j=1

Cjn

αjn , (1)

where∑J

j=1 αjn = 1. Labor is the only factor of production and the sole source of household

income. As a result, the budget constraint for consumers in n is given by:

J∑j=1

P jnC

jn = wnLn, (2)

wherewn denotes the wage rate and P jn denotes the composite price index in sector j of country

n (described in detail below).Each sector consists of a continuum of tradable products indexed by ω ∈ [0, 1]. Potential

producers of product ω in sector j of country n receive a random productivity draw zjn(ω) froma Frechet distribution of the following form:

F jn(zjn) = exp

{−T jnzjn

−θj}. (3)

This distribution varies across both countries and sectors. A higher T jn implies higher averageproductivity for the country-sector pair, while a higher θj implies lower dispersion of produc-tivity draws across countries. All producers in n have access to this technology for a givenproduct ω.5

4The basic EK model does not incorporate intermediate products, although the authors provide an extensionwith intermediates in the second half of their original paper. Other multi-sectoral versions of the EK model includeCostinot, Donalson and Komunjer (2012), Shikher (2012), Levchenko and Zhang (2014), and Caliendo and Parro(2015).

5The original Eaton and Kortum (2002) model has a single sector, so Tn depicts a parameter of country-level average productivity, while θ provides dispersion across productivity draws and, hence, a basis for GFT.

5

Producers use two types of inputs in production: labor and composite intermediate goodsfrom each of the J sectors. The corresponding production function for product ω is:

qjn(ω) =[zjn(ω)ljn(ω)

]1−βj [ J∏k=1

Mk,jn (ω)γ

k,jn

]βj, (4)

where zjn, ljn and Mk,jn denote labor productivity, labor input and intermediate input of the

composite intermediate good from sector k, respectively. The parameter γk,jn denotes theshare of the composite goods from sector k used by producers in sector j of country n, with∑J

k=1 γk,jn = 1. Equation (4) includes an important departure from standard versions of the

EK model. The parameter zjn(ω) does not enter here as total factor productivity (TFP) but asvalue-added productivity (VAP). As shown below, this difference is not trivial: it generates arelationship between intermediate inputs and the trade elasticity.

Non-traded composite goods Qjn are produced using tradable products as inputs. Producers

of composite goods minimize costs by sourcing these products from the lowest cost suppliers,whether they are located at home or abroad. These products are then assembled according tothe following CES production function:

Qjn =

(∫ 1

0

qjn(ω)σ−1σ dω

) σσ−1

, (5)

where σ > 1 denotes the elasticity of substitution across products. The composite goods fromj are demanded by both consumers as final goods Cj

n and by producers as intermediate inputsM j,k

n across all k sectors.Given the CES production function in (5), the composite goods producers in sector j of n

have the following demand for expenditures on product ω exported from i:

xjni(ω) =

[pkni(ω)

P jn

]1−σXjn, (6)

where Xjn denotes total expenditures in n on goods from sector j, and

P jn =

[∫ 1

0

pjn(ω)1−σdω

] 11−σ

(7)

denotes the composite price index for sector j in country n.As mentioned, total expenditures Xj

n consists of spending by both consumers and produc-

In the present model, variance in T jn across sectors provides an additional basis for GFT owing to across-sectorcomparative advantage.

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ers. Given (1) and (4), this can be expressed as the following:

Xjn = αjnwnLn +

J∑k=1

γj,kn βkY kn , (8)

where Y kn denotes the value of total production in sector k of country n. To clear the product

market for this sector, total production value Y kn in country nmust be equal to total expenditures

by all other countries on products produced by n. That is:

Y jn =

n∑i=1

Xjin. (9)

Substituting this into total expenditures yields the following:

Xjn = αjnwnLn +

J∑k=1

γj,kn βk

(N∑i=1

Xkin

). (10)

2.1.2 Prices

Composite goods producers in n buy tradable products from the lowest-cost producer. Pro-ducers are perfectly competitive, setting prices equal to marginal cost. Exports from i to n aresubject to an additional iceberg trade cost of the form κni > κii = 1, where κni units of agiven product need to be exported from i for each unit that arrives in n. As a result, the priceof product ω exported from i to n takes the following form:

pjni(zji (ω))V AP =

cjiκjni

(zji (ω))1−βj, (11)

where

cji = Ψjiw

1−βji

[J∏k=1

P ki

γk,jn

]βj(12)

denotes the unit cost of production and Ψji is a constant.6

Note that (11) is different here than in the standard EK model with TFP heterogeneity. Inthat setting, the analogous expression is the following:

pjni(zji (ω))TFP =

cjiκjni

zji (ω). (13)

Expression (7) can be simplified, given our choice of the Frechet distribution, to yield the

6Specifically, Ψji =

∏Jk=1(γk,jn )−γ

k,jn (1−βj)(βj)−β

j

(1− βj)βj−1.

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following expression:

P jn = Aj

[N∑i=1

T ji[cjiκ

jni

] −θj1−βj

] 1−βj

−θj

= Aj[φjn] 1−βj

−θj , (14)

where Aj is a constant.7 See the Appendix for a proof of (14).

2.1.3 Expenditures and trade balance

We denote the share of total expenditures in n on products exported from i in sector j asπjni = Xj

ni/Xjn. Again, using some useful properties of the Frechet distribution, this share can

be represented by the following:

πjni =Xjni

Xjn

=T ji[cki κ

jni

] −θj1−βj

φjn. (15)

See the Appendix for a proof of equation (15).Finally, the total trade surplus for country n can be defined as Dn =

∑Jj=1D

jn, where

Djn =

∑Ni=1X

jni −

∑Ni=1X

jin denotes the total trade surplus in sector j. Trade is balanced for

all countries when Dn = 0 for all n, which is equivalent to the following:

Dn =J∑j=1

(N∑i=1

Xjni −

N∑i=1

Xjin

)= 0 (16)

for all n.

2.1.4 Equilibrium

Following Alvarez and Lucas (2007) and Caliendo and Parro (2015), I define an equilib-rium as a set of wages and prices that satisfy (10), (12) (14), (15), and (16) for all n and j.8

Total bilateral exports: A gravity equationRearranging (15) in terms of Xj

ni and substituting this into the market-clearing equation in(9) yields:

Y ji = T ji

(cji) −θj

1−βjN∑n=1

(κjniX

jn

) −θj1−βj

φjn. (17)

7Specifically, Aj = Γ

(θj+(1−σ)(1−βj)

θj

) 1(1−σ)

and Γ is the Gamma function.8Other versions of the EK model often allow for trade deficits at the country level. This element could easily

be included in this model as well. However, for simplicity, I assume that trade is balanced for each country.

8

Solving this expression for T ji(cki) −θj

1−βj and substituting into (15) yields the following gravityequation:

Xjni

V AP= Xj

nYji

(κjni/P

jn

) −θj1−βj∑N

n=1

(κjni/P

jn

) −θj1−βj

. (18)

Equation (18) is different from the standard multi-sectoral EK gravity equation (e.g., Caliendoand Parro, 2015). In the standard setting with TFP heterogeneity, the gravity equation is thefollowing:

Xjni

TFP= Xj

nYji

(κjni/P

jn

)−θj∑Nn=1

(κjni/P

jn

)−θj . (19)

Clearly, the main difference between these expressions relates to the 1− βj in the exponent ofthe (κjni/P

jn) terms in (18). Denoting the trade elasticity with respect to variable trade costs as

ηjX,κ, we can derive the following simple expression (controlling for Xjn, Y j

n and P jn):

ηjX,κV AP

=−θj

1− βj. (20)

In contrast, the trade elasticity according to (19) is:

ηjX,κTFP

= −θj. (21)

In the model with heterogeneity in VAP, sectors that use a higher share of intermediateinputs have a higher elasticity of trade with respect to trade costs. In the model with TFP, thisrelationship is absent.

2.2 Extensive and intensive margins

To illustrate the role of the extensive and intensive margins separately, I reproduce thefollowing product-level bilateral exports equation from (6):

xjni(ω) =

[cjiκ

jni(z

ji )βj−1

P jn

]1−σXjn. (22)

Note that I have substituted in the price equation from (11). Clearly, the elasticity of trade withrespect to trade costs for a given product (i.e., the intensive margin) is 1 − σ, and hence therelationship between the trade elasticity and the intermediate inputs share does not appear atthe producer level.

As discussed in Eaton and Kortum (2002), expression (15) represents not only total bilateralexpenditure share, but also the total share of bilaterally traded products, and thus captures the

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extensive margin of trade in this model. Accordingly, expression (20) represents not only thetotal trade elasticity, but extensive margin trade elasticity. The extensive margin describes theentire trade elasticity, as well as the entire equilibrium relationship between the trade elasticityand the intermediate inputs share under the VAP heterogeneity setting.9

This result is interesting since it reveals the important role of producer heterogeneity inthis model. A representative producer model with VAP as opposed to TFP in the productionfunction would not generate any relationship between the trade elasticity and the intermediateinputs share.

2.3 Discussion

To summarize, the model described here yields a positive relationship between the inter-national trade elasticity and the intermediate inputs share in production. As discussed, thismechanism is not present in standard versions of the EK model with intermediate products.

The explanation for this mechanism is fairly intuitive. In the heterogeneous producer envi-ronment, the international trade elasticity is governed by product-level productivity dispersionacross countries. In sectors with high productivity dispersion (low θj), production tends tobe more concentrated internationally (assuming countries trade), the extensive margin reactslittle when trade costs change, and the international trade elasticity is low. When intermediateinputs are used in production, all countries have access to these products from the same sup-pliers at the same price (excluding freight), and the impact, in terms of productivity, is neutralacross countries. Thus, as the share of value-added in production is replaced with intermediateproducts, the distribution of market shares across countries becomes more even, the extensivemargin becomes more responsive to changes in trade costs, and the international trade elasticityrises.

In models where productivity enhances all factors equally, such as in most versions of theEK model, more productive countries are equally more productive in all inputs of production,so including intermediates does not flatten the global productivity distribution in any way.Under the value-added productivity setting, however, including intermediates does flatten thisdistribution.

The Frechet and Pareto distributions are popular choices in this literature mainly becausethey deliver clean analytical solutions. Another significant factor, however, is evidence that thesize distribution of firms (and especially exporting firms) in advanced economies often closely

9This result can be derived from other models with producer heterogeneity as well. For example, Chaney(2008) develops a gravity model from a Melitz (2003) framework with Pareto-distributed firm heterogeneity thatyields separate closed-form expressions for the intensive, extensive and compositional margins. In adapting theChaney (2008) model to incorporate VAP heterogeneity, one can recover an extensive margin elasticity that is alsoequivalent to the total trade elasticity expression found in (20).

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resembles a type II extreme value distribution, such as Frechet or Pareto. For evidence, seeAxtell (2001), Luttmer (2007), and di Giovanni, Levchenko and Ranciere (2011).

Importantly, this feature remains true under the value-added productivity setting describedhere. In fact, for both the producer size distribution and calculations of the static GFT (seeSection 2.4), the economic role of the international trade elasticity is essentially unchanged.What does change is the definition of the trade elasticity which, under the value-added setting,becomes positively related to the intermediate inputs share.

2.4 Gains from trade

To illustrate the welfare impact of international trade in this framework, I consider a sim-plified model where γj,j = 1 and γj,kn = 0 for all k 6= j. That is, sector j uses only intermediateinputs from its own sector in production.10 Welfare per capita in country n for this case is equalto that country’s real wage, depicted as the following:

Wn =wnP cn

, (23)

where P cn = χn

∏Jj=1 P

jnαjn denotes the composite price index for consumers in n, and χn is a

constant.11

We can rearrange (15) to find the following expression for P jn:

P jn =

(T jnπjnn

) 1−βj

−θj

cjn, (24)

where κjnn is assumed to be 1. Note that, given the simplified input-output assumption, theunit cost from (12) reduces to cjn = Ψj

nw1−βjn P j

nβj . Substituting this expression into (24) and

solving for the price index P jn yields the following:

P jn =

(T jnπjnn

) 1

−θj

Ψjnwn. (25)

Finally, substituting this expression into (23) yields the following expression for welfare per

10I consider this version strictly to simplify the analysis, since when γj,j = 1, the GFT reduce to a simplersolution under both the VAP and TFP frameworks. Intermediate input linkages affect the GFT in a similar wayacross the VAP and TFP models, so similar conclusions to those reached this section would be reached if we wereto focus on the versions of the models with linkages.

11χn is equal to∏Jj=1

(αjn)−αjn .

11

capita in n:

W V APn =

J∏j=1

(λjnWπjnn

)αjnθj

, (26)

where λjnW = (T jn)1

θj αjn−αjnΨj

n is a constant.To represent the gains from international trade, I follow the “exact hat algebra” approach

from Dekle, Eaton and Kortum (2007), denoting x = x/x as the relative change betweensome initial value x and counterfactual value x of a variable due to a counterfactual change ininternational trade costs. Using this notation, we can produce the following expressions fromfor the change in equation (26) going from the the status quo, where πjnn ≤ 1 for all j, to acounterfactual of autarky where πjnn = 1 for all j.

GFT can be denoted as:

GFT V APn =1

W V APn

=J∏j=1

(πjnn)αjnθj =

J∏j=1

(πjnn)−αjn

θj − 1, (27)

where πjnn = (πjnn)−1 since πjnn = 1 for all n and j under autarky. In words, to calculate

GFT in n, all that one needs is data on three variables: sectoral consumption shares (αjn) forall j in n, the sectoral domestic exepnditure shares (πjnn) for all j in n, and sectoral dispersionparameters (θj).

Equation (26) is different here than it would be for the case with heterogeneity in TFP. Inthat environment, welfare simplifies to the following:

W TFPn =

J∏j=1

(λjnTπjnn

) αjn

θj(1−βj)

, (28)

where λjnT = (T jn)1

θj αjn−αjnΨj

nT and ΨjnT = (βj)−β

j .GFT with TFP heterogeneity are the following:

GFT TFPn =1

W TFPn

=J∏j=1

(πjnn) α

jn

θj(1−βj) =J∏j=1

(πjnn) −αjnθj(1−βj) − 1. (29)

Based on equations (27) and (29), we can make the following proposition:

Proposition 1: For a common set of parameters θjθjθj , αjnαjnαjn, and βjβjβj , and a common set of ob-

served domestic expenditure shares πjnnπjnnπjnn, the GFT are higher under the TFP heterogeneitymodel than the VAP heterogeneity model for all nnn.

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Since βj∈(0, 1) for all j, it is clear that GFT are higher in (29) than (27) for a common setof πjnn, αji , β

j and θj across the two models. In the TFP model, intermediate inputs generatean input-output loop that amplifies the GFT. As a result, the larger the share of intermediateproducts, the higher the GFT. In contrast, when productivity enhances value-added, as in mymodel, this amplification effect disappears. This reveals that the amplification in the TFPmodel is not only due to the use of intermediate inputs, but also depends on the form of theproductivity parameter in the production function. In the standard TFP heterogeneity model,productivity enhances both value-added and intermediate products by the same factor, creatinga compounding effect for productivity through the input-output loop. This mechanism is absentfrom the VAP heterogeneity framework.12

These points are not to say, however, that estimates of GFT will necessarily be higher us-ing the TFP model in cases where direct estimates of θj , αjn, βj and πjnn are unavailable. Infact, most studies that calculate the GFT do so using estimates of the trade elasticity, ηjX,κ, asopposed to using direct values for the primitives that govern the trade elasticity (θj or, in thecase of the VAP model, both θj and βj). In cases where the GFT are calculated using estimatesof ηjX,κ instead of direct information about θj , then the following proposition holds:

Proposition 2: For a common set of parameters ηjX,κηjX,κηjX,κ, αjnαjnαjn, and βjβjβj , and a common set of

observed domestic expenditure shares πjnnπjnnπjnn, the GFT under the TFP heterogeneity model andVAP heterogeneity model are identical.

This statement can be shown to be true by substituting the values for ηjX,κ from equations(20) and (21) into equations (27) and (29) respectively. This result reaffirms findings fromArkolakis, Costinot and Rodrıguez-Clare (2012) in that, for a common set of ηjX,κ, αjn, βj andπjnn, the VAP model delivers the same GFT as the TFP model and, in turn, the “same old gains”as the entire set of models described in that paper.

Proposition 2 implies that θj must be different across the two models in order to permitcommon values of ηjX,κ and βj parameters. For the VAP model, GFT are higher in sectors withintermediate inputs not because of their direct effect, but because higher intermediate inputsimply a lower value of θj for a given value of ηjX,κ.

Despite the fact that GFT are equivalent across these models for a common set of ηjX,κparameters, the VAP model suggests that ηjX,κ varies over time if there is time variation in βj ,whereas the TFP model does not. Many studies that attempt to quantify the GFT do so usingan exogenous set of estimates for ηjX,κ that are used to back out fixed structural parameters

12Melitz and Redding (2014) reveal that GFT can become arbitrarily large in a framework with sequentialproduction. This point can be equally demonstrated by setting βj close to one in expression (29). Notably, settingβj close to one has no impact on expression (27), hence the insights emphasized by Melitz and Redding (2014)do not carry over to the VAP model.

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such as θj . For cases like this, the VAP model implies different conclusions than the TFPmodel for how the GFT evolve over time. Particularly, the TFP model implies that, as shares ofintermediate inputs in production have generally increased over time in most sectors, the GFThave become larger. In contrast, the VAP model implies that the GFT do not rise as values ofβj increase, since higher values of βj lead to higher trade elasticities that counter the positiveeffect of intermediate inputs on the GFT.

As I demonstrate in Section 4, while country-level estimates for the static GFT are, on av-erage, only slightly different across these models, growth in the GFT over time is significantlydifferent across these models and generally larger under the TFP model than under the VAPmodel.

3 Application I: The Distance Puzzle

According to numerous studies, the elasticity of international trade with respect to bilateraldistance has remained fairly stable over time, and in fact rose from the 1970s to the 2000s.13

Since bilateral distance is often viewed as a proxy variable for transportation and communica-tion costs, and these costs have declined over time due to technological change, it is consideredsurprising that distance has in fact increased as a hindrance to international trade in recentyears. This surprising result has been coined the “distance puzzle” in international trade.

This puzzle, however, can be explained according to the VAP heterogeneity model de-scribed in this paper. Table 1 describes how the average share of intermediate inputs in exportshas evolved over time across the same set of OECD economies depicted in Figures 1 and 2.The average share of intermediate inputs in exports across these countries rose from 0.535 inthe pre-1985 period to 0.59 in 2005, which represents a 0.055 increase.

Now consider the definition of the trade elasticity under the VAP heterogeneity model inequation (20). Suppose we define trade costs as being related to distance in such a way thatκni = dρni, where dni denotes the distance between countries n and i and ρ represents theelasticity of trade costs with respect to distance. Next, suppose that we calibrate ρ = 0.2,θ = 2.09 and βt = 0.535 so that a value of ηX,κ = 0.9 represents the average elasticity oftrade with respect to distance in the 1970s.14 Then, suppose that ρ and θ remained unchangedbetween the 1970s and 2000s, so that the only factor that changes over time and influencesηX,κ is βt. Under these parameters, we would see the observed distance elasticity of trade rise

13For examples of studies, see Berthelon and Freund (2008), Bhavnani et al. (2002), and Disdier and Head(2008).

14According to Disdier and Head (2008), the average elasticity of trade with respect to distance across numerousstudies was approximately 0.9 in the 1970s. We suppress country and sector scripts here since this section relieson data for averages across countries and sectors, but include time scripts t for βt since this variable changes overtime in the data.

14

by roughly 13.4% on account of the higher share of βt = 0.59 in the 2000s compared to thepre-1985 period.

This growth is remarkably similar to estimates for the change in distance elasticity of tradeover this period. For example, estimates from Disdier and Head (2008) suggest that the averagedistance elasticity rose by 8% between the 1970s and the 1990s, and Berthelon and Freund(2008) find that the distance elasticity rose, on average, by 10% between the 1985 and 2005.

Hence the VAP heterogeneity model provides a mechanism that reconciles growth in theelasticity of trade with respect to distance with stability in the the elasticity of trade costs withrespect to distance, through the enhanced role that intermediate inputs have in production inrecent years.

4 Application II: Quantifying Gains from Trade

4.1 Data

To compute GFT under the specification in (27), data are needed for sectoral dispersionparameters (θj), sectoral domestic expenditure shares (πjnnt) and sectoral consumption shares(αjnt). To compare with GFT under the standard TFP heterogeneity model according to (29),data for sectoral intermediate inputs shares in production (βjit) are also needed. Since I havedata from several time periods, I add a t subscript for variables that change over time (sourcesof data are described below). Since values of intermediate inputs shares vary across countriesin the data, I also add i subscripts to these parameters.

Note that using values for βjit that vary across countries and time automatically implies thatthe trade elasticity also changes across countries and time according to the VAP heterogeneitymodel. Empirical applications of the TFP heterogeneity model typically assume that the tradeelasticity, which represents the structural parameter θj in that model, is constant across coun-tries and time. Hence, in cases where values of θjV AP and θjTFP are backed out from estimatesof ηX,κ, and then GFT are calculated for a different time period and/or set of countries fromwhich ηX,κ were derived (as is the case for the calculation of GFT in this paper), the cross-sectional estimates of GFT will be slightly different across these models. Estimates of changesin GFT over time will also differ across the models.

4.1.1 Sectoral dispersion

To find values for θj , it is common in the literature to estimate a gravity equation based onthe theoretical model. In our model with VAP heterogeneity, this equation is represented by(18). In the standard TFP heterogeneity framework, this equation is represented by (19).

15

Caliendo and Parro (2015) provide a prominent recent example of sectoral estimates forθj under the TFP specification. The authors develop a multi-sectoral EK model similar to themodel in Section 2.1. They derive the following trade-share equation for exports from i to n insector j:

πjni =Xjni

Xjn

=T ji[cki κ

jni

]−θj∑Ni=1 T

ji

[cjiκ

jni

]−θj . (30)

This equation is analogous to (15) in the VAP heterogeneity model. To estimate −θj , theyconsider the following tetradic ratio for trade between n, i and a reference country, h, in sectorj, based on (30):

XjniX

jihX

jhn

XjinX

jhiX

jnh

=

(κjniκ

jihκ

jhn

κjinκjhiκ

jnh

)−θjTFP. (31)

This ratio conveniently eliminates everything in (30) except for bilateral trade costs and thedispersion parameter to be estimated. Note that any symmetric components of trade costs alsocancel out in this expression. In fact, any country fixed effects cancel as well. To estimate(31), the authors gather asymmetric tariff data from UNCTAD-TRAINS from 1989 to 1993across 16 economies and 20 sectors (18 manufacturing and 2 non-manufacturing).15 Denotingbilateral tariffs imposed by country n on imports from i in sector j as τ jni, they specify thefollowing estimation equation based on the logarithm of (31):

ln

(XjniX

jihX

jhn

XjinX

jhiX

jnh

)= −θjTFP ln

(τ jniτ

jihτ

jhn

τ jinτjhiτ

jnh

)+ εj, (32)

where εj denotes an i.i.d. error term. Caliendo and Parro estimate (32) using OLS withheteroskedasticity-robust standard errors, dropping observations with zero flows. In the firsttwo columns of Table 2, I report the estimates and standard errors from their baseline full-sample estimation.16

According to the model with VAP heterogeneity described in Section 2, I derive the fol-lowing analog to (31) based on (15):

XjniX

jihX

jhn

XjinX

jhiX

jnh

=

(κjniκ

jihκ

jhn

κjinκjhiκ

jnh

)−θjV AP

1−βj

. (33)

Note that the right-hand side of (33) is equal to Caliendo and Parro’s expression, to theexponent of 1/(1−βj). That is, θjV AP = θjTFP×(1− βj) can be backed out from Caliendo and

15The economies included are Argentina, Australia, Brazil, Canada, Chile, China, the European Union, India,Indonesia, Japan, South Korea, New Zealand, Norway, Switzerland, Thailand and the United States.

16Caliendo and Parro estimate parameters for 20 ISIC Revision 3 industries. The values in Table 2 are convertedinto ISIC Revision 2 classification.

16

Parro’s estimates of θTFP coupled with data on βj . If we then substitute θV AP into equation(27) to find GFT under the value-added specification, it is clear that this formula becomesidentical to (29), which corresponds to GFT under the TFP specification.

In other words, for cases where values for θj and GFT are found using the same data for asingle time period, and βj is constant across countries for a given j, Proposition 2 holds andthe GFT are equivalent under the VAP and TFP specifications.

In the present case, however, and in Caliendo and Parro (2015), values for θj and GFT arefound using a different set of countries and a different time period. To quantify the impact ofthis, I adjust Caliendo and Parro’s results to be consistent with the specification in (33). Thatis, I adjust their estimates for θjTFP to the following:

θjV AP = θjTFP ×(

1− βj), (34)

where βj

denotes the observed mean intermediate inputs share across countries.17

In the third column of Table 2, I report calculated values for θjV AP based on this exercise.Not surprisingly, values for θjV AP are significantly lower than θjTFP owing to the impact ofthe intermediate inputs adjustment. This translates into a higher degree of dispersion withinsectors according to the VAP model.

The GFT are higher when the sectoral dispersion parameters are low. This is true for boththe VAP and TFP frameworks, as indicated by equations (27) and (29). Meanwhile, the TFPspecification in (29) has a (1−βj) term that equation (27) is missing. This raises the GFT underTFP heterogeneity. In the end, the lower estimates of θj from the value-added specificationcounterbalance the welfare-reducing impact of the missing (1−βj), resulting in an ambiguousoverall difference in the GFT across the two theoretical models.

In sum, when θj and GFT are calculated using the same data for a single time period, andβj is constant across countries for a given j, measured GFT are equal across these specifica-tions. In the present case, however, since θj and GFT are estimated using different data and adifferent time period, the cross-sectional GFT across specifications will differ in a theoreticallyambiguous way.

4.1.2 Intermediate products shares

For sectoral intermediate products shares (βjit), I use data from the World Input-OutputDatabase (WIOD). Because these shares vary across countries, I allow βjit to vary across i for

17To correspond with the 16 countries from Caliendo and Parro (2015), I calculate βj

using the data from theWorld Input-Output Database (WIOD) for 1995. Unfortunately, the WIOD does not perfectly overlap with the setof countries used in Caliendo and Parro’s tariff calculations. However, there is significant overlap, so I expect thatobserved values for β

jare not severely biased.

17

the remainder of this analysis.For each country, data for 14 manufacturing ISIC Revision 2 sectors are available from the

WIOD for 1995 to 2011. I use data for 1995 and 2010. I exclude one of the sectors, Leather andFootwear, which reduces the number of sectors to 13. Although the WIOD provides data for40 countries, I restrict the analysis to 33 countries.18 A list of the sectors included is providedin Table 2, and a list of the countries included is provided in Table 5.

Table 3 reports average shares of domestic-sourced (βjith) and foreign-sourced (βjitf ) inter-mediate inputs by sector and over time across these 33 countries. As we see, βjith is consistentlyhigher than βjitf across all sectors except for Coke and Refined Petroleum. However, from 1995to 2010, the domestic share generally decreased while the foreign share increased for most sec-tors. On average, the total share of intermediate inputs used in production rose a few percentagepoints over the 1995 to 2010 period across these countries.

4.1.3 Other input-output parameters

For measures of domestic expenditure shares (πjnnt) and sectoral consumption shares (αjnt)across countries and sectors, I use derivations from the WIOD. Although the WIOD reportstrade data for non-manufacturing sectors, I focus specifically on manufacturing trade for thisanalysis.

Summary statistics for data used in the GFT exercise are provided in Table 4.

4.2 Gains from Trade: Empirical Results

I calculate the gains from manufacturing trade using data for sectoral final consumptionshares (αjnt), the sectoral domestic expenditure shares (πjnnt), sectoral dispersion parameterestimates (θj), and sectoral intermediate products shares (βjnt) described in Section 4.1.

These are calculated according to the VAP heterogeneity specification based on the follow-ing equation (derived in Section 2.4):

GFT V APnt =J∏j=1

(πjnnt

) −αjntθjV AP − 1, (35)

I also calculate GFT under the TFP heterogeneity specification. This is calculated usingCaliendo and Parro’s estimates of θjTFP according to the following standard GFT equation:

GFT TFPn =J∏j=1

(πjnnt

) −αjntθjTFP (1−βjnt) − 1. (36)

18The decision to restrict the analysis to these 33 countries and 13 manufacturing sectors follows Costinot andRodrıguez-Clare (2014). The main reason for these restrictions appears to be data limitations.

18

I report the results under the VAP specification in columns 1 and 2 of Table 5 for all 33countries for 1995 and 2010. The results based on the TFP specification are reported in columns3 and 4.19

The gains from manufacturing trade are, on average, larger by 0.8 percentage points in1995 and smaller by 2.1 percentage points in 2010 under the VAP specification. In 1995, GFTare larger under the VAP specification for 20 of the 33 countries; in 2010, GFT are larger foronly 10 of the countries.

These results show that the difference in static GFT across these two specifications aremodest on average and ambiguously vary from country to country. In cases where GFT arehigher under the VAP specification, the positive effect of lower θj estimates outweighs themissing direct effect of βjit in the GFT equation. Intuitively, these cases correspond to countriesthat used intermediate shares in these years that were lower than the shares used to derive θjV AP .For example, while the average weighted share of inputs used to calculate θjV AP parameters wasroughly 0.62, Denmark’s average share in 1995 was significantly lower than this at 0.57. Thisdifference helps to explain why, for Denmark, GFT in 1995 are significantly larger accordingto the VAP model compared to the TFP model.

In cases where the GFT are higher under the TFP specification, the opposite is true. Forexample, Portugal’s average share of inputs in 1995 was 0.65, which is significant higher thanthe average share of 0.62 used to derive θjV AP parameters. As a result, for Portugal, GFT in1995 are significantly larger according to the TFP model compared to the VAP model.

Again, if θj are estimated with the same cross-section of countries and the same time periodthat the GFT are calculated for, and βj is equal across countries for a given sector, then the GFTare equivalent across these models (see Proposition 2). In contrast, in the present case valuesfor θj are estimated using a different series of countries and time period than the GFT, and βj

is not equal across countries for a given sector, so there are differences in the calculated GFTacross these models.

Figure 3 depicts growth in the GFT from 1995 to 2010 for each country. Growth is, onaverage, roughly 39% higher under the TFP specification relative to the VAP specification.This result is driven by the share of intermediate inputs in production, which grew over time formost sectors (see Table 3). This growth positively influences GFT under TFP but not under theVAP specification. Note that, for some countries, GFT actually decline over time. Moreover,for most cases, this decline is larger under the VAP specification. For example, in Canada, GFTdeclined by 20 and 14 percentage points under the VAP and TFP specifications, respectively.This decline is mostly concentrated in the Transport Equipment sector where Canadian importsdeclined significantly over the period.20 Meanwhile, the use of intermediate inputs in this

19The WIOD reports exports for non-manufacturing sectors as well. Since this project is focused on manufac-turing sectors, the figures in Table 5 are calculated with non-manufacturing trade set to zero across all countries.

20This decline is likely due to the reconfiguration of the North American automotive supply chain away from

19

sector has grown over time, so the decline in Canadian import share is counterbalanced underthe TFP specification. Under VAP heterogeneity, since intermediate inputs do not enter theGFT formula, the fall in imports had relatively more influence on the measured GFT.

Perhaps the most important difference across these models relates to trade policy. The GFTformula from the TFP model might lead one to conclude that sectors that use intermediateproducts should be targeted for trade promotion since they have a greater relative impact onwelfare. In contrast, the VAP framework invites no such policy. In addition, as emphasizedby Melitz and Redding (2014), the TFP formula suggests that future GFT might be inevitable,owing to the rising share of intermediate inputs in production over time. In contrast, accordingto the VAP specification, future GFT must come about through greater openness, which is notnecessarily expected to consistently rise over time.

5 Conclusion

In this paper, I extend the EK model of international trade to a setting with intermediateinputs and producer heterogeneity in value-added productivity. This adjustment generates apositive relationship between the international trade elasticity and the share of intermediateinputs in production, which is absent from the standard model. According to the theory, thispositive relationship is driven by the extensive margin of trade.

This relationship helps explain the rapid growth, and regional concentration, of vertical spe-cialization trade since the 1970s. Since sectors that use intermediate products are particularlysensitive to trade costs, falling trade costs have had an especially large impact on exports andimports for theses sectors. Meanwhile, distance-related trade costs also have a larger impact onthese sectors, leading to a “regionalized” pattern in this type of trade. In an application of themodel, I show that this relationship also provides a potential explanation for the “distance puz-zle” in international trade, since the rising importance of intermediate inputs over time impliesa greater elasticity of trade with respect to distance according to my model.

Despite this heightened elasticity, the country-level static gains from international trade are,for the average country, quantitatively similar to the gains under standard EK model with in-termediate inputs. However, whereas the standard model treats the trade elasticity as constant,the value-added productivity specification proposes that the trade elasticity has increased overtime. As a result, the standard model suggests that gains from trade increased roughly 39%more between 1995 and 2010 relative to the value-added productivity model. This findinghighlights the point that researchers should be wary of treating trade elasticities as exogenousconstant parameters when calculating the gains from international trade.

Canada and towards Mexico that occurred over the 1995 to 2010 period.

20

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23

A Appendix

A.1 Proofs

A.1.1 Proof of equation (14)

Let F jni (p) denote the probability that the price at which country i can supply a given

variety in sector j to country n is lower than or equal to p. Rearranging (11) in terms of zji andusing the distribution expression in (3), we find that:21

F jni (p)

V AP = 1− F ji

(zji (ω)

)= 1− F j

i

(cjiκjnip

) 1

1−βj

= 1− exp{−T ji

(cjiκ

jni

)− θj

1−βj pθj

1−βj

}.

(37)

It follows that the probability of receiving a price in n below p for a given variety from any

country is equal to F jn (p) =

∏Ni=1 F

jni (p). Solving for this expression yields:

F jn (p) = 1−

N∏i=1

exp

{−T ji

(cjiκ

jni

)− θj

1−βj pθj

1−βj

}= 1− exp

{N∑i=1

−φjnpθj

1−βj

}, (38)

where

φjn =N∑i=1

T ji[cjiκ

jni

] −θj1−βj . (39)

Expression (7) can now be rearranged to (P jn)

1−σ=∫ 1

0pjn(ω)1−σdω =

∫∞0p1−σdF j

n(p). Ex-panding dF j

n(p) using (38) and substituting this into the price index yields the following:

(P jn

)1−σ=

∫ ∞0

p1−σφjn

(θj

1− βj

)p

θj

1−βj−1exp

{−φjnp

θj

1−βj

}dp. (40)

From here, one can employ integration by substitution. Letting x = φjnpθj

1−βj , it follows that

dx = φjn

(θj

1−βj

)p

θj

1−βj−1dp and p = (x/φjn)

1−βj

θj . Substituting these expressions into (40)

21This probability is different from the standard EK model, where the analogous expression is the following:

F jni (p)TFP

= 1− F ji(zji (ω)

)= 1− F ji

(cjiκ

jni

p

).

24

yields the following:

(P jn

)1−σ=

∫ ∞0

(x

φjn

) (1−σ)(1−βj)θj

exp {−x} dx

=(φjn) (1−σ)(1−βj)

−θj

∫ ∞0

x(1−σ)(1−βj)

θj exp {−x} dx.

(41)

The second part of this expression can be simplified as∫ ∞0

x(1−σ)(1−βj)

θj exp {−x} dx = Γ

(θj + (1− σ) (1− βj)

θj

), (42)

where Γ denotes the Gamma function (a constant).22 Substituting (42) into (41) and multi-plying by the exponent of 1/(1− σ) yields the expression (14):

P jn =

(φjn) 1−βj

−θj Γ

(θj + (1− σ) (1− βj)

θj

) 1(1−σ)

. (43)

A.1.2 Proof of equation (15)

We can represent πjni as πjni = Pr(pjni(ω

j) ≤ min{pjnk(ω

j); k 6= i})

. Suppose that pjni(ωj) =

p; then, this probability can be represented as:

∏k 6=i

Pr(pjnk(ω

j) ≥ p)

=∏k 6=i

[1− F j

nk (p)]

= exp

{∑k 6=i

−T ji(cjiκ

jni

) −θj1−βj p

θj

1−βj

}

= exp

{−φjn6=ip

θj

1−βj

},

(44)

where φjn6=i =∑

k 6=i−Tji

(cjiκ

jni

) −θj1−βj . To find πjni, we multiply (44) by the density dF j

ni(p)

and integrate this product over all possible p’s. The density dF jni(p) is equal to:

dF jni(p) = T ji

(cjiκ

jni

) −θj1−βj

θj

1− βjp

θj

1−βj−1exp

{−T ji

(cjiκ

jni

) −θj1−βj p

θj

1−βj dp

}. (45)

22The general formula for the Gamma function is Γ(a) =∫∞− xa−1e−xdx.

25

We can therefore solve for πjni as:

πjni =

∫ ∞0

exp

{−φjn6=ip

θj

1−βj dp

}dF j

ni(p)

=

T ji [cjiκjni] −θj1−βj

φjn

∫ ∞0

θj

1− βjφjnexp

{−φjnp

θj

1−βj

}p

θj

1−βj−1dp.

(46)

The portion of the expression above that is to the right of the integral is equal to dF jn(p)dp.

Since∫∞0dF j

n(p)dp = 1, (15) has been proven.

A.2 Tables and Figures

Table 1: Intermediate Inputs Shares over Time

Pre-1985 2005Country (1) (2)Australia 0.493 0.521Canada 0.606 0.574Germany 0.572 0.623Denmark 0.524 0.606France 0.547 0.657United Kingdom 0.546 0.538Japan 0.601 0.588Netherlands 0.517 0.661United States 0.415 0.550Average 0.535 0.590

Notes: Figures in columns (1) and (2) represent the average share of intermediate inputs in country-level exports,calculated using data from OECD input-output tables from the late 1960s (for Australia and the United Kingdom)or the early 1970s (for Canada, Germany, Denmark, France, Japan, the Netherlands and the United States) to 2005.Data for non-manufacturing sectors are excluded.

26

Table 2: Dispersion Parameters for ISIC Revision 2 Groups

θTFPθTFPθTFP Se. θV APθV APθV AP ObsISIC Revision 2 Group (1) (2) (3) (4)Food, Beverages and Tobacco -2.55 (0.61) -0.75 495Textiles and Products -5.56 (1.14) -1.92 437Wood and Products -10.83 (2.53) -3.72 315Pulp, Paper and Printing -9.07 (1.69) -3.57 507Coke, Ref. Petroleum -51.08 (18.05) -15.41 91Chemicals and Products -4.75 (1.77) -1.69 430Rubber and Plastics -1.66 (1.41) -0.59 376Other Non-Metallic Min. -2.76 (1.44) -1.16 342Basic Metals and Fabricated -7.99 (2.53) -2.75 388Machinery, Nec -1.52 (1.81) -0.55 397Electrical and Optical -10.60 (1.38) -3.73 343Transport Equipment -0.37 (1.08) -0.11 245Manufacturing, Nec -5.00 (0.92) -1.93 412Aggregate -4.55 (0.35) -2.91 7212

Notes: Values for θjV AP and θjTFP are derived using estimates from Caliendo and Parro (2015) and data fromthe WIOD for 1995.

27

Table 3: Average Sectoral Intermediate Inputs Shares (βjit) across Countries and Time

β95hβ95hβ95h β10hβ10hβ10h β95fβ95fβ95f β10fβ10fβ10f β95β95β95 β10β10β10ISIC Revision 2 Group (1) (2) (3) (4) (5) (6)Food, Beverages and Tobacco 0.64 ↓ 0.59 0.09 ↑ 0.11 0.72 ↓ 0.71Textiles and Products 0.46 ↓ 0.43 0.17 ↑ 0.20 0.63 ↑ 0.63Wood and Products 0.53 ↓ 0.52 0.12 ↑ 0.13 0.65 − 0.65Pulp, Paper and Printing 0.48 − 0.48 0.13 ↑ 0.15 0.62 ↑ 0.63Coke, Ref. Petroleum 0.39 ↓ 0.36 0.34 ↑ 0.42 0.73 ↑ 0.79Chemicals and Products 0.46 − 0.46 0.17 ↑ 0.21 0.63 ↑ 0.67Rubber and Plastics 0.46 ↓ 0.45 0.19 ↑ 0.22 0.64 ↑ 0.66Other Non-Metallic Min. 0.46 ↑ 0.48 0.11 ↑ 0.13 0.57 ↑ 0.61Basic Metals and Fabricated 0.48 ↓ 0.47 0.18 ↑ 0.23 0.66 ↑ 0.70Machinery, Nec 0.45 ↓ 0.43 0.17 ↑ 0.21 0.62 ↑ 0.65Electrical and Optical 0.42 ↓ 0.41 0.22 ↑ 0.27 0.64 ↑ 0.68Transport Equipment 0.46 ↓ 0.45 0.21 ↑ 0.27 0.68 ↑ 0.72Manufacturing, Nec 0.47 ↓ 0.45 0.14 ↑ 0.19 0.61 ↑ 0.64Average 0.47 ↓ 0.46 0.17 ↑ 0.21 0.65 ↑ 0.67

Notes: Values for βh, βf and β are derived from the WIOD for 1995 and 2010.

Table 4: Summary Statistics: Gains from Trade DataMean Std. Dev. Min. Max. N

Variable (1) (2) (3) (4) (5)αjnt 0.02 0.02 0 0.14 858βjit 0.66 0.09 0.36 0.95 858πjnnt 0.62 0.25 0 1.00 858θjV AP 3.24 3.42 0.09 12.47 13θjTFP 9.47 12.76 0.37 51.08 13

Notes: Summary statistics for αjnt, βjit, and πjnnt are derived from the WIOD for 1995 and 2010. Values for

θjV AP and θjTFP are derived using estimates from Caliendo and Parro (2015) and data from the WIOD for 1995and 2010. Data for non-manufacturing sectors are excluded.

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Table 5: Gains from Manufacturing Trade

VAP TFP1995 2010 1995 2010

Country (1) (2) (3) (4)Australia 15.2% 20.1% 12.5% 21.3%Austria 47.3% 60.7% 41.7% 59.6%Belgium 70.1% 71.8% 72.4% 78.0%Brazil 7.1% 9.6% 6.7% 9.9%Canada 49.9% 30.3% 48.0% 33.7%China 8.1% 7.8% 9.3% 11.6%Czech Republic 26.5% 38.3% 31.1% 45.8%Germany 21.4% 33.5% 18.1% 34.7%Denmark 47.6% 70.9% 37.6% 86.7%Spain 20.2% 22.6% 21.3% 26.5%Finland 22.4% 32.0% 17.8% 28.7%France 19.5% 25.8% 22.7% 33.5%Great Britain 28.4% 37.0% 24.6% 34.8%Greece 24.0% 43.0% 16.2% 29.5%Hungary 23.2% 50.9% 28.6% 55.5%Indonesia 16.2% 13.1% 14.0% 11.8%India 4.1% 9.3% 5.5% 12.8%Ireland 44.7% 45.5% 35.9% 38.5%Italy 18.2% 21.4% 19.8% 25.5%Japan 2.5% 3.3% 2.4% 3.5%Korea 12.0% 9.9% 12.8% 12.9%Mexico 17.8% 31.1% 17.1% 28.5%Netherlands 55.0% 56.5% 54.8% 56.6%Poland 12.2% 40.2% 11.3% 51.3%Portugal 43.0% 46.7% 48.8% 46.0%Romania 12.5% 29.8% 11.1% 22.0%Russia 18.9% 31.4% 16.9% 36.1%Slovakia 36.6% 44.2% 38.5% 49.3%Slovenia 58.8% 69.4% 65.1% 75.1%Sweden 22.0% 27.7% 19.6% 32.8%Turkey 18.5% 33.4% 12.9% 31.5%Taiwan 23.8% 23.4% 25.6% 32.5%United States 11.0% 13.0% 11.4% 15.6%Average 26.0% 33.4% 25.2% 35.5%

Notes: GFT are calculated according to equations (27) and (29) using data for αjnt, βjit and πjnnt constructed from

the WIOD, and values of θjTFP and θjV AP estimated using data from Caliendo and Parro (2015). Non-manufacturingsectors are excluded.

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Figure 1: Vertical Specialization and Components, 2005

Notes: The input share of exports (ISE) is calculated as uATX/Xk, where u is a 1×n vector of 1s, AT is the1×n intermediate coefficient matrix, X is an n×1 vector of exports, Xk is total country exports, and n is thenumber of sectors. The vertical specialization (VS) share is calculated as uAMX/Xk, where AM is the 1×nimport coefficient matrix. The import content of inputs (MCI) is calculated as VS divided by ISE. Data are takenfrom OECD input-output tables for 2005. Countries included are Australia, Canada, Germany, Denmark, France,the United Kingdom, Japan, the Netherlands and the United States. Data for non-manufacturing sectors areexcluded.

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Figure 2: Vertical Specialization and Components, Growth over time

Notes: The input share of exports (ISE) is calculated as uATX/Xk, where u is a 1×n vector of 1s, AT is the1×n intermediate coefficient matrix, X is an n×1 vector of exports, Xk is total country exports, and n is thenumber of sectors. The vertical specialization (VS) share is calculated as uAMX/Xk, where AM is the 1×nimport coefficient matrix. The import content of inputs (MCI) is calculated as VS divided by ISE. Data arecalculated from OECD input-output tables as growth from the late 1960s (for Australia and the United Kingdom)or the early 1970s (for Canada, Germany, Denmark, France, Japan, the Netherlands and the United States) to2005. Data for non-manufacturing sectors are excluded.

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Figure 3: Cross-sectional relationship between the ADOT and the intermediate share, 2005

3500

4000

4500

5000

5500

AD

OT

.5 .6 .7 .8Intermediate Share

Notes: The average distance of trade (ADOT) is calculated as∑n,i (Xni × dni) /

∑n,i (Xni), where Xni

denotes total exports from n to i, and dni denotes bilateral distance between n and i. Data for exports are usedfor 33 exporters and 80 importers, derived from the BACI data set provided by Centre d’Etudes Prospectives etd’Informations Internationales (CEPII) and aggregated up to two-digit ISIC Revision 2 sectors. Distance isapproximated with the distw variable provided by CEPII. Intermediate inputs shares are calculated from the2005 WIOD for 33 countries. Data for non-manufacturing sectors are excluded.

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Figure 4: % Growth in Gains from Trade, 1995 to 2010

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Figure 4: (continued) % Growth in Gains from Trade, 1995 to 2010

Notes: GFT are calculated according to equations (27) and (29) using data for αjnt, βjit and πjnnt constructed

from the WIOD, and values of θjTFP and θjV AP estimated using data from Caliendo and Parro (2015).Non-manufacturing sectors are excluded.

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