International Trade Melitz (2003) - Fudan Universityhomepage.fudan.edu.cn/yiqingxie/files/2014/12/Melitz.pdf · 2014. 12. 31. · Melitz (2003) develops a model featuring facts 1

International TradeMelitz (2003)

Yiqing Xie

School of EconomicsFudan University

Nov. 29, 2013

Yiqing Xie (Fudan University) Int’l Trade - Melitz Eca 2003 Nov. 29, 2013 1 / 37

Outline

Motivation

Introduction to "New New" Trade Theory

Closed Economy

Open economy with variable and fixed trade costs

Summary


Motivation

Internationalized firms are rare.

Internationalized firms only export a small amount of their totalproduction.

Internationalized firms are larger, pay higher wages, and are moreproductive.

Lowering of trade costs lead to more firms exporting (extensivemargin).



The 1990s saw a boom in the availability of micro-level datacovering firms/plants. This changed many fields and Trade was noexception.

Problem: both neoclassical and new trade theory are at odds with(or cannot account for) many micro-level facts that becameapparent in this data:

1 Within a given industry, there is firm-level heterogeneity (eg inproductivity levels).

2 Fixed costs matter in export related decisions.3 Within a given industry, more productive firms are more likely to

export.4 Trade liberalization leads to intra-industry reallocation across firms.5 These reallocations are correlated with productivity and export

status (so industry productivity can rise purely due to trade-drivenselection).



Melitz (2003) develops a model featuring facts 1 and 2 that canexplain facts 3, 4, and 5.

This has been an extremely influential paper and modelingapproach in both Trade and other fields.

Two building blocks:I Krugman (AER, 1980): CES, IRS technology, monopolistic

competition.I Hopenhayn (Ecta, 1992): equilibrium model of entry and exit.

From a normative point of view, Melitz (2003) may also providenew source of gains from trade if trade induces reallocation oflabor from least to most productive firms (more on that later).


Closed Economy: DemandLike in Krugman (1980), representative agent has CESpreferences:

U =

[∫ω∈Ω

q(ω)σ−1σ dω

] σσ−1

where σ > 1 is the elasticity of substitution (and ρ ≡ (σ − 1)/σ).

Consumption and expenditures for each variety ω are given by

q(ω) = Q[p(ω)P

]−σr(ω) = R

[p(ω)P

]1−σwhere:

P ≡[∫

ω∈Ωp(ω)1−σdω

] 11−σ

, R ≡∫ω∈Ω

r(ω)dω, and Q ≡ R/P.


Closed Economy: ProductionLike in Krugman (1980), labor is the only factor of production.

I L ≡ total employment, w = 1 ≡ wage

Like in Krugman (1980), there are IRTS in production:

l = f + q/ϕ

IMPORTANT: firms have different levels of ϕ.

Like in Krugman (1980), monopolistic competition implies price ismark-up over marginal cost.

(Normalize wage level, w = 1.)

p(ϕ) =w

ρϕ=

σ

σ − 11

ϕ

More productive firms sell at lower prices.


Closed Economy: Production

Profit of a firm:

π(ϕ) = r(ϕ)− l(ϕ) = r(ϕ)σ− f

Revenue of a firm:r(ϕ) = R (Pρϕ)σ−1

Output of a firm:q(ϕ) = Q (Pρϕ)σ

Any two firms’ outputs and revenues only depend on the ratio oftheir productivity levels:

q(ϕ1)

q(ϕ2)=

(ϕ1ϕ2

)σ,

r(ϕ1)

r(ϕ2)=

(ϕ1ϕ2

)σ−1


Closed Economy: Production

Comments:

I In (extremely common) cases where statistical agencies can’tcollect firm-specific price deflators, higher productivity ϕ in themodel also implies higher measured productivity:

r(ϕ)

l(ϕ)=

1

ρ

[1− f

l(ϕ)

]

I A more productive firm produces more, charges a lower price,earns higher revenues and profits.

I ϕ can also be interpreted in terms of quality.


Closed Economy: AggregationIn equilibrium there will be a mass of M firms and a distributionµ(ϕ) over a subset of (0,+∞).

Re-write the aggregate price P as:

P =

[∫ω∈Ω

p(ω)1−σdω

] 11−σ

=

[∫ +∞0

p(ϕ)1−σMµ(ϕ)dϕ

] 11−σ

Use the pricing rule to re-write the previous expression:

P = M1

1−σ1

ρ

[∫ +∞0

ϕσ−1µ(ϕ)dϕ

] 11−σ

Denote some kind of weighted average of the productivities offirms:

ϕ̃ =

[∫ +∞0

ϕσ−1µ(ϕ)dϕ

] 11−σ


Closed Economy: AggregationThe price of a firm with productivity ϕ̃ is

p(ϕ̃) =1

ρϕ̃=

1

ρ

[∫ +∞0

ϕσ−1µ(ϕ)dϕ

] 11−σ

Thus,P = M

11−σ p(ϕ̃)

Using a similar methodology, we can find

R = Mr(ϕ̃), Π = Mπ(ϕ̃), Q = Mσ

1−σ q(ϕ̃)

Comments:I These are the same aggregate variables we would get in a

Krugman (1980) model with a mass M of identical firms withproductivity ϕ̃.

I But productivity ϕ̃ now is an endogenous variable which mayrespond to changes in trade cost, leading to aggregate productivitychanges.


Closed Economy: Entry and Exit

In order to determine how µ(ϕ) and ϕ̃ get determined inequilibrium, one needs to specify the entry and exit of firms.

Timing is similar to Hopenhayn (1992):

1 There is a large pool of identical potential entrants deciding whetherto become active or not.

2 To enter, firms to pay a fixed cost (in terms of labor), fe.

3 After paying fe, firms get a productivity draw ϕ from a cdf G(ϕ).

4 After drawing, firms can either immediately exit or produce.

5 Firms deciding to remain active exit with a constant probability δ.


Closed Economy: Entry and ExitAside: Pareto Distribution

In variations and extensions of Melitz (2003), most commonassumption on the productivity distribution G(.) is Pareto:

G(ϕ) ≡ 1−(ϕminϕ

)θfor ϕ ≥ ϕmin

g(ϕ) ≡ θϕθminϕ−θ−1 for ϕ ≥ ϕmin

Pareto distributions have two advantages:I Combined with CES, it delivers closed form solutions.I Distribution of firm sizes remains Pareto, which is not a bad

approximation empirically (at least for the upper tail).

But like CES, Pareto distributions will have very strong implicationsfor equilibrium properties (more on this on ‘gravity models’).



In a stationary equilibrium, a firm either exits immediately orproduces and earns the same profits π(ϕ) in each period.

A firm’s value function is

v(ϕ) = max{

0,∑+∞

t=0(1− δ)tπ(ϕ)

}= max

{0,π(ϕ)

δ

}Exit a unique productivity level ϕ∗ ≡ inf

{ϕ ≥ 0 : π(ϕ)δ > 0

}.

Productivity cutoff ϕ∗ can also be written as:

π(ϕ∗) = 0



Once we know ϕ∗, we can compute the pdf of firm-productivitylevels among the active firms:

µ(ϕ) =

{g(ϕ)

1−G(ϕ∗) if ϕ ≥ ϕ∗

0 if ϕ ≥ ϕ∗

where pin = 1−G(ϕ∗) is the ex ante probability of successfulentry.

Accordingly, the measure of aggregate productivity is given by:

ϕ̃(ϕ∗) =

[1

1−G(ϕ∗)

∫ +∞ϕ∗

ϕσ−1g(ϕ)dϕ

] 1σ−1


Closed Economy: Entry and ExitFree Entry Condition

Let π̄ = π(ϕ̃) ≡ Π/M denote average profits per period forsurviving firms.

Free entry requires the total expected value of profits to be equalto the fixed cost of entry:

0×G(ϕ∗) + π̄δ× [1−G(ϕ∗)] = fe

Free Entry Condition (FE):

π̄ =δfe

1−G(ϕ∗)

Holding constant the fixed costs of entry, if firms are less likely tosurvive, they need to be compensated by higher average profits.


Closed Economy: Entry and ExitZero Cutoff Profit Condition

Definition of ϕ∗ can be rearranged to obtain a second relationshipbetween ϕ∗ and π̄.

By definition of π̄, we know that

π̄ = π(ϕ̃) =r(ϕ̃)

σ− f = f

[r(ϕ̃)

σf− 1]

By definition of ϕ∗, we know that

π(ϕ∗) = 0⇔ r(ϕ∗) = σf

Zero Cutoff Profit Condition (ZCP):

π̄ = f

[r(ϕ̃)

r(ϕ∗)− 1]

= f

[(ϕ̃(ϕ∗)

ϕ∗

)σ−1− 1

]


Closed Economy: Equilibrium


Closed Economy: EquilibriumAside: The Shape of the ZCP Schedule

FE and ZCP determine a unique (ϕ∗, π̄) and therefore ϕ̃,independently of country size L.

However, ZCP is not necessarily downward sloping:I it depends on whether ϕ∗ or ϕ̃ increases relatively faster.I ZCP is downward sloping for most common distributions.

In the Pareto case, it is easy to check that ϕ̃/ϕ∗ is constant:I So ZCP is flat and average profits are independent of ϕ∗.


Closed Economy: EquilibriumNumber of Varieties and Welfare

Free entry and labor market clearing imply

L = R = r̄M

We can rearrange the previous expression

M =L

r̄=

L

σ(π̄ + f)

Welfare per worker is given by

W = P−1 = M1

σ−1 ρϕ̃

Since ϕ̃ and π̄ are independent of L, growth in country size (oropenness to free trade) will have the same impact as in Krugman(1980):

Increase in welfare is due ONLY to increased product variety.


Open Economy Model

In the absence of trade costs, we have seen that trade integrationdoes not lead to any intra-industry reallocation (ie ϕ̃ is fixed).

In order to move away from such (counterfactual) predictions,Melitz (2003) introduces two types of trade costs:

I Iceberg Trade Costs: in order to sell 1 unit abroad, firms need toship τ ≥ 1 units.

I Fixed Exporting Costs: in order to export abroad, firms must incuran additional fixed cost fex (information, distribution, or regulationcosts) after learning their productivity ϕ.

In addition, Melitz (2003) assumes that c = 1, ..., n countries aresymmetric so that wc = 1 in all countries.


Open Economy Model: Production

Monopoly pricing now implies:

pd(ϕ) =1

ρϕ, px(ϕ) =

τ

ρϕ

Revenues in the domestic and export markets are:

rd(ϕ) = RH (PHρϕ)σ−1 , rx(ϕ) = τ

1−σRF (PFρϕ)σ−1

Note that by symmetry, we must have:

PH = PF = P, RH = RF = R

Let fx ≡ δfex. Profits in the domestic and export markets are:

πd(ϕ) =rd(ϕ)

σ− f, πx(ϕ) =

rx(ϕ)

σ− fx


Open Economy Model: Production

Revenue of a firm is

r(ϕ) =

rd(ϕ) if the firm does not exportrd(ϕ) + nrx(ϕ) = (1 + nτ

1−σ)rd(ϕ)if the firm exports to all countries

Two comments:I Because of symmetry, if firm exports to one country, it exports to all

countries.I Although not all countries consume the same bundle of goods

(because not all firms export) the aggregate variables across allcountries are identical.


Open Economy Model: Entry, Exit and Export Status

Expected value of a firm with productivity ϕ is:

v(ϕ) = max{

0,∑+∞

t=0(1− δ)tπ(ϕ)

}= max

{0,π(ϕ)

δ

}But total profits of are now given by:

π(ϕ) = πd(ϕ) + max {0, πx(ϕ)}

Like in the closed economy, we let ϕ∗ ≡ inf{ϕ ≥ 0 : π(ϕ)δ > 0

}Similarly, we let ϕ∗x ≡ inf

{ϕ ≥ ϕ∗ : πx(ϕ)δ > 0

}be the export

cutoff.

In order to have both exporters and non-exporters in equilibrium,ϕ∗ > ϕ, we assume that τσ−1fx > f .


Open Economy Model: Entry, Exit and Export Status


Open Economy Model: AggregationIn the open economy, aggregate productivity is now given by:

ϕ̃t =

{1

Mt

[Mϕ̃σ−1 + nMx (ϕ̃x/τ)

σ−1]} 11−σ

where:I Total number of varieties (domestic plus foreign): Mt = M + nMxI Average productivity across all domestic firms:

ϕ̃ =

[1

1−G(ϕ∗)

∫ +∞ϕ∗

ϕσ−1g(ϕ)dϕ

] 1σ−1

I Average productivity across all exporters:

ϕ̃x =

[1

1−G(ϕ∗x)

∫ +∞ϕ∗x

ϕσ−1g(ϕ)dϕ

] 1σ−1


Open Economy Model: Aggregation

Once we know ϕ̃t, we can compute all aggregate variables as:

P = M1

1−σt /ρϕ̃t

R = Mtr(ϕ̃t)

Π = Mtπ(ϕ̃t)

Q = Mσ

1−σt q(ϕ̃t)

W =R

LM

1σ−1t ρϕ̃t


Open Economy Model: Equilibrium ConditionsFree Entry Condition

The condition for free entry is unchanged.

Free Entry Condition (FE):

π̄ =δfe

1−G(ϕ∗)

The only difference is that average profits now depend on exportprofits as well:

π̄ = πd(ϕ̃) + npxπx(ϕ̃x)

where:

px =1−G(ϕ∗x)1−G(ϕ∗) is the probability of exporting conditional on

successful entry.


Open Economy Model: Equilibrium ConditionsZero Cutoff Profit Condition

By definition of the cut od’ productivity levels, we know that:

πd(ϕ∗) = 0⇔ rd(ϕ∗) = σf

πx(ϕ∗) = 0⇔ rx(ϕ∗x) = σfx

This implies:

rx(ϕ∗x)

rd(ϕ∗)=fxf⇔ ϕ∗x = ϕ∗τ

(fxf

) 1σ−1

New Zero Cutoff Profit Condition (ZCP):

π̄ = f

[(ϕ̃(ϕ∗)

ϕ∗

)σ−1− 1

]+ npxfx

[(ϕ̃x(ϕ

∗)

ϕ∗x(ϕ∗)

)σ−1− 1

]Trade shifts up ZCP.


Open Economy Model: Equilibrium ConditionsZero Cutoff Profit Condition

ZCP and FE determine π̄ and ϕ∗. Note: trade shifts up ZCP andleaves FE unchanged. Therefore, π̄ and ϕ∗ increase.

ϕ∗ determines ϕ∗x, ϕ̃ and ϕ̃x.

We can show that L = R, and r̄ = σ(π̄ + f + npxfx), so that

M =R

r̄=

L

σ(π̄ + f + npxfx)

This, then, determines the mass of varieties Mt = (1 + npx)Mavailable in each country, and the other aggregate variables.


Open Economy Model: The Impact of Trade



In line with empirical evidence, exposure to trade forces the leastproductive firms to exit: ϕ∗ > ϕ∗a.

Intuition:I For exporters, profits↗ due to export opportunities, but↘ due to

the entry of foreign firms in the domestic market (P ↘).I For non-exporters: only the negative second effect is active.I Mechanism for the least firms to exit: the increase in competition for

domestic factor (labor) due to exporting behaviors

Welfare unambiguously↗ though no. of domestic varieties↘.

M =R

r̄=

L

σ(π̄ + f + npxfx)< Ma < Mt


Open Economy ModelAside: HMY(2004) FDI versus Exports


Summary

Heterogeneous firms under monopolistic competition with IRS

Fixed entry cost – Fixed production cost – Fixed exporting cost

Aggregate productivity gains due toI Exit of least efficient firmsI Reallocation of market shares from less from more productive firms

Gains from trade also stem from the increase in total varieties.

Melitz (2003) also considers:I Increase in the number of trading partners nI Decrease in iceberg trade costs τI Decrease in fixed exporting costs fx


OutlineMotivationIntroduction to "New New" Trade TheoryClosed Economy ModelDemandProductionAggregationEntry and ExitClosed Economy Equilibrium

Open economy with variable and fixed trade costsProductionEntry, Exit and Export StatusAggregationEquilibrium ConditionsThe Impact of Trade

Summary

Documents

International Trade Melitz (2003) - Fudan Universityhomepage.fudan.edu.cn/yiqingxie/files/2014/12/Melitz.pdf · 2014. 12. 31. · Melitz (2003) develops a model featuring facts 1