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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 2 / 37
Introduction
Figure: Bernard, Eaton, Jensen and Kortum (2003) AER
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 3 / 37
Introduction
Figure: Bernard, Jenson, Redding and Shot (2007) JEP
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 4 / 37
Introduction
Figure: Bernard, Jenson, Redding and Shot (2007) JEP
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 5 / 37
Introduction
Figure: Eaton, Kortum and Kramarz (2004) AER P&P
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 6 / 37
Introduction
Melitz Model 2003
Extension of Krugman model 1980 and Hopenhayn 1992
Chaney 2006 (NBER) & 2008 (AER)
Intesnive and Extensive margins of trade
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 7 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 8 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 9 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 10 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 11 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 12 / 37
Household
Demand for good j :
x (j) =(p (j)P
)−(σ−1)X
where
P =(∫
p (j)−(σ−1) dj)−1/(σ−1)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 13 / 37
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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 14 / 37
Producer
Profit maximization of a firm with productivity z ⇒Constant Markupm̄ = σσ−1 over price:
p (z) = m̄wzin local market
p (z) = m̄wdzin export market
is the price of a producer with effi ciency z
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 15 / 37
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
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 16 / 37
Firm’s Profit
Profit of a local producer
πo (z) =r (z)
σ− wfo
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 17 / 37
Closed Economy
Let’s consider the close economy case first.
So, there is not export.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 18 / 37
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)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 19 / 37
Local Entry
Who enters the local market?
πo (z) ≥ 0r (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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 20 / 37
Local Entry
There is an endogenous distribution of productivities: µ (z) dz
µ (z) =
{g (z )
1−G (z̄o ) z ≥ z̄o0 O.W
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 21 / 37
"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)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 22 / 37
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)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 23 / 37
Labor Market Equilibrium
Total Revenue = Total income
R = wL (LMC)
Mσ (π̃ + wfo ) = wL (1)
M =wL
σ (π̃ + wfo )(2)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 24 / 37
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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 25 / 37
Survival threshold
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 26 / 37
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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 27 / 37
Trade
Two similar economies
Iceberg trade cost: d
Fixed export cost: fX
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 28 / 37
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 )
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 29 / 37
Export Entry
Who enters the export market?
πx (z) ≥ 0r (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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 30 / 37
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.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 31 / 37
Trade Equilibrium
Profits derived from domestic and export sales:
π̃ = πD (z̃ (z̄o )) + probXπX (z̄x )
with probX =1−G (z̄X )1−G (z̄o )
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 32 / 37
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 )
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 33 / 37
Selection under trade
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 34 / 37
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 35 / 37
Trade and Aggregate productivity
Opening up to trade induces reallocation towards more productivefirms.
Aggregate productivity goes up.
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 36 / 37
References
Melitz 2003
Chaney 2006(NBER), 2008(AER)
Eaton Kortum Kramarz (2012)
Seyed Ali Madanizadeh (Sharif U. of Tech.) Melitz Model: Heterogenous Firm Model of Trade May 7, 2014 37 / 37