melitz model: heterogenous firm model of...

Melitz Model: Heterogenous Firm Model of Trade

Seyed Ali Madanizadeh

Sharif U. of Tech.

May 7, 2014

Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 1 / 37

Introduction

Challenges to Krugman model:

There is a lot of heterogeneity across firms, within any sector.Very few firms export (import, engage in FDI)Exporters are very different from non exporters.There is a lot of reallocation between firms within sectors.


Introduction

Figure: Bernard, Eaton, Jensen and Kortum (2003) AER


Introduction

Figure: Bernard, Jenson, Redding and Shot (2007) JEP


Introduction

Figure: Eaton, Kortum and Kramarz (2004) AER P&P


Introduction

Melitz Model 2003

Extension of Krugman model 1980 and Hopenhayn 1992

Chaney 2006 (NBER) & 2008 (AER)

Intesnive and Extensive margins of trade


Model Assumptions

Heterogeneity assumptions

Firms differ in labor productivityProductivity is random, unobserved before firm starts.

Trade barrier assumptions

Firms face iceberg (variable) trade costs.Firms face fixed export costs.

Simplifying assumption

Symmetric countries


Model

No differences in factor intensities

One factor of production: Call it labor L with wage w

Labor moves across producers but not across countries


Model

Effi ciencies are different across producers

Firms productivities’distribution: g (z)

(An example) Pareto distribution

G (z) = Pr [Z < z ] =

{1−

(zz0

)−θfor z > z0

0 for z < z0

The range of goods to be produced or export are endogenous

It is determined by the free entry conditions

Each producer makes a different good: so it sells it monopolistically.

The space of goods is modeled as a continuum

Index the goods by j

Consider 2 similar countries with trade costs = d


Model

Firms pay an entry sunk cost fE to draw a random productivity. Thenthey decide to enter or not

Measure M of firms are potential entrants.

Production setup requires an operational fixed cost (additional labor)fo

Measure Mo of firms actually produce.

Export also requires a fixed cost (additional labor) fxMeasure Mx of firms export.

The range of goods to be produced or exported are endogenous.

It is determined by the free entry conditions


Household

Preferences: Spence-Dixit-Stiglitz (SDS)

A CES over the continuum of goods:

U =(∫

y (j)(σ−1)/σ dj) σ

σ−1

σ > 1

Budget Constraint: ∫p (j) y (j) dj = X = wL+Π

X : Total Spending


Household

Demand for good j :

x (j) =(p (j)P

)−(σ−1)X

where

P =(∫

p (j)−(σ−1) dj)−1/(σ−1)


Producer

The market structure is monopolistic competition.

Each good is produced by a separate monopolist who takes totalspending X and the price index P in each market as given.

Markets are segmented so that producers can set a different price ineach national market.

Producers draw their random productivity from a cumulatativedistribution function F (z).

For analytical tractabality, We assume G (z) = Pr (Z > z) is paretodistributed.


Producer

Profit maximization of a firm with productivity z ⇒Constant Markupm̄ = σ

σ−1 over price:

p (z) = m̄wzin local market

p (z) = m̄wdzin export market

is the price of a producer with effi ciency z


Firm’s Revenue

Consider 2 similar countries and trade costs dHF = dFH = d

Revenue of a local producer:

r (z) = zσ−1 (m̄w)−(σ−1) Pσ−1X


Firm’s Profit

Profit of a local producer

πo (z) =r (z)

σ− wfo


Closed Economy

Let’s consider the close economy case first.

So, there is not export.


Aggregation

Define convenient aggregate productivity:

z̃ =(∫

zσ−1µ (z) dz) 1

σ−1

Aggregate variables:

R = Mr (z̃)

Π = Mπ (z̃)

P = M−1

σ−1 p (z̃)


Local Entry

Who enters the local market?

πo (z) ≥ 0

r (z) ≥ σwfo

So there is a threshold z̄o such that firms enter if and only if:

z ≥ z̄o =m̄wP

(σwfoX

)1/(σ−1)

It means that only firms more productive than z̃o would produce inthe local market.


Local Entry

There is an endogenous distribution of productivities: µ (z) dz

µ (z) =

{g (z )

1−G (z̄o ) z ≥ z̄o0 O.W


"Zero Cutoff Profits" condition

Aggregate productivity:

z̃ (z̄o ) ==(

11− G (z̄o )

∫ ∞

z̄ozσ−1g (z) dz

) 1σ−1

Average profits are profits of the "average" firm π̃ = π (z̃ (z̄))

Profits of the marginal firm directly related to average profits:

π (z̄o ) = 0⇐⇒ π̃ = wfo

[(z̃z̄o

)σ−1− 1]

(ZCP)


Endogenous Entry

Value of entering is the expected profits minus sunk entry cost:

VE = E [π (z)− wfE ] = (1− G (φ̄)) π̄ − wfE

Free entry drives down the expected profits to zero:

V E ≤ 0

⇒π̃ =

wfE1− G (z) (FE)


Labor Market Equilibrium

Total Revenue = Total income

R = wL (LMC)

Mσ (π̃ + wfo ) = wL (1)

M =wL

σ (π̃ + wfo )(2)


General Equilibrium

Firms below z̄o exit (ZCP).

Free entry drives down (expected) profits to zero (FE).

Labor markets clear (LMC)

π̃ = wfo

[(z̃z̄o

)σ−1− 1]

π̄ =wfE

1− G (z)

M =wL

σ (π̃ + wfo )

(ZCP) and (FE) solves for z̄o and π̃.

(LMC) and π̃ solves for M.


Survival threshold


Ownership structure

Stationary equilibrium: exit mass = entry mass:

ME =M

1− G (z̄o )

Expected profits of firm owners cover the financing of entrants:

Π = MEwf E

All firms owned by a continuum of competitive mutual funds thathold diversified portfolios of firms.

Note: no financing frictions.


Trade

Two similar economies

Iceberg trade cost: d

Fixed export cost: fX


Exporters’s Revenue and Profit

Revenue of an exporter:

rx (z) =

(m̄wzP

)−(σ−1)X +

(m̄wdzP

)−(σ−1)X

=[zσ−1 (m̄w)−(σ−1)

] [Pσ−1X

] [1+ d−(σ−1)

]Profit of an exporter

πx (z) =x (z)

σ− (wfo + wfx )


Export Entry

Who enters the export market?

πx (z) ≥ 0

r (z) ≥ σ (wfo + wfx )

z ≥ z̄x =m̄wP

(σw (fo + fx )

X

)1/(σ−1)d̃

where d̃ =(1+ d−(σ−1)

)−1/(σ−1)

So, only firms with productivity greater than z̃x would export.


Selecting into Export

πx (z̄x ) = 0⇐⇒ z̄x = d(fxfo

)1/(σ−1)z̄o (ZCPX )

Empirically motivated restriction:

d(fxfo

)1/(σ−1)> 1⇒ z̄x > z̄o

So not all firms export.


Trade Equilibrium

Profits derived from domestic and export sales:

π̃ = πD (z̃ (z̄o )) + probXπX (z̄x )

with probX =1−G (z̄X )1−G (z̄o )


Equilibrium Conditions

z̄x = d(fxfo

)1/(σ−1)z̄o

π̃ = wfo

[(z̃ (z̄o )z̄o

)σ−1− 1]+ probxwfx

[(z̃ (z̄x )z̄o

)σ−1− 1]

π̄ =wfE

1− G (z)

M =wL

σ (π̃ + wfo + probxwfx )


Selection under trade


Trade and Aggregate productivity

Opening up to trade induces reallocation towards more productivefirms.

Aggregate productivity goes up.


References

Melitz 2003

Chaney 2006(NBER), 2008(AER)

Eaton Kortum Kramarz (2012)


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