Melitz(2003)– FirmheterogeneityintheKrugman-model · Melitz(2003)– FirmheterogeneityintheKrugman-model InternationalTrade–DICE/RGS JensSuedekum March/April2014 InternationalTrade–DICE/RGS

Setup of the ModelClosed EconomyOpen Economy

Melitz (2003) –Firm heterogeneity in the Krugman-model

International Trade – DICE/RGSJens Suedekum

March/April 2014

International Trade – DICE/RGS Jens Suedekum Melitz (2003) – Firm heterogeneity in the Krugman-model


DemandProductionFirm Entry and Exit

Demand

One monopolistically competitive sector. CES preferences.

U =[∫

ω∈Ωq (ω)ρ dω

] 1ρ

.

CES price index. Minimum expenditure per aggregate unit Q

P =[∫

ω∈Ωp (ω)1−σ dω

] 11−σ

.

Consumption and expenditure per variety(with E = R = P × Q), see eq. (13) in LN1.

q (ω) =RP

[p (ω)

P

]−σand r (ω) = R

[p (ω)

P

]1−σ.




Production

Continuum of firms; each firm produces different variety ω.Labor requirement for output q of variety ω

l (ϕ) = f +qϕ.

Isoleastic demand ⇒ constant markup [1/ρ = σ/(σ − 1)]

p (ω) = p (ϕ) =σ

σ − 1wϕ

=1ρϕ.

Firm revenue and profits

r (ϕ) = R [ρϕP]σ−1 = R[

Pp (ϕ)

]σ−1and π (ϕ) =

r (ϕ)σ− f .

More productive firms charge lower price, sell higher quantity,earn higher revenue and profits. BUT: Same markup 1/ρ.




Aggregation

Rewrite CES price index in terms of ϕ instead of ω

P =[∫

ω∈Ωp (ω)1−σ dω

] 11−σ

.

=

[∫ ∞ϕ=0

p (ϕ)1−σ Mµ(ϕ)dϕ] 1

1−σ.

= M1

1−σ · 1ρ·[∫ ∞

ϕ=0ϕσ−1µ(ϕ)dϕ

] 11−σ

︸︷︷︸=1/ϕ̃

.

= M1

1−σ · (1/ (ρ ϕ̃)) = M1

1−σ · p(ϕ̃)

Compare to eq. (15) in LN1: P = M1

1−σ · p.Krugman-model embedded in Melitz (2003) as a special case!




Entry and firm selection

Huge mass of ex ante identical entrepreneurs Me .Entry requires a sunk cost fe > 0.Entrants draw productivity ϕ randomly from distribution g (ϕ)with support over (0,∞) and cumulative distribution G (ϕ).After learning ϕ, firms decide whether to exit immediately orto remain in the market.Recall: π (ϕ) = r(ϕ)σ − f , with r (ϕ) = R [ρϕP]

σ−1 > 0.→ Firm must be productive enough to cover fixed cost f .




Cutoff productivity

Value of a firm with productivity draw ϕ is determined by:

v (ϕ) = max

(0,∞∑

t=0

(1− δ)t π (ϕ)

)= max

(0,π (ϕ)

δ

).

Constant profit stream over time.At each time instant, probability δ of facing a terminal shock(δ independent of ϕ, for δ(ϕ) see Hopenhayn, ECTA 1992)The lowest productivity level for survival (“cutoff level”) isϕ∗ = inf {ϕ : v(ϕ) > 0}.Mass of surviving firms: M = (1− G (ϕ∗)) Me .Me : Mass of entrants, 1− G (ϕ∗): survival probability(both endogenous!)




Average productivity

Productivity distribution among surviving firms:µ(ϕ) = g(ϕ)1−G(ϕ∗) for ϕ ≥ ϕ

∗ and µ(ϕ) = 0 otherwise.

Note: g(ϕ) is exogenous, µ(ϕ) is endogenous.Av.productivity in the market (conditional on firm survival):

ϕ̃ =

[∫ ∞ϕ=0

ϕσ−1µ(ϕ)dϕ] 1

σ−1

=

[1

1− G (ϕ∗)·∫ ∞ϕ∗ϕσ−1g(ϕ)dϕ

] 1σ−1

Given the propoerties of the ex ante distribution g(ϕ), thisaverage productivity is solely determined by the cutoff ϕ∗



EquilibriumProperties of equilibriumExample: The Pareto distribution

Equilibrium Conditions

Objective: solve for the cutoff and the mass of entrants.Together, ϕ∗ and ME completely characterize equilibrium

Two equilibrium conditions:1 zero cutoff profit condition (ZCPC)2 free entry condition (FEC)

Solved for ϕ∗ and π̄ = π(ϕ̃), i.e.,cutoff and average profit conditional on survivalWith this, mass of entrants Me and consumption variety M arepinned down via aggregate resource constraint




1. Zero Cutoff Profit Condition (ZCPC)

The ratio of any two firms’ revenues is:

r (ϕ1)r (ϕ2)

=R[

p(ϕ1)P

]1−σR[

p(ϕ2)P

]1−σ = R [ρϕ1P]σ−1R [ρϕ2P]σ−1 =[ϕ1ϕ2

]σ−1.

Thus, for cutoff and average surviving firm:

r (ϕ̃)r (ϕ∗)

=

[ϕ̃

ϕ∗

]σ−1⇔ r̄ = r (ϕ̃) =

[ϕ̃

ϕ∗

]σ−1r (ϕ∗) .

Hence, average firm makes profits

π = π (ϕ̃) =r (ϕ̃)σ− f =

[ϕ̃

ϕ∗

]σ−1 r (ϕ∗)σ− f .




1.The Zero Cutoff Profit Condition (ZCPC)

The profits of the cutoff firms are equal to 0:

π (ϕ∗) =r (ϕ∗)σ− f = 0.

Hence, the revenues of the cutoff firm are

r (ϕ∗) = σf .

For the profits of the average firm, this implies

π = f[ϕ̃ (ϕ∗)

ϕ

]σ−1−f = k (ϕ∗)·f with k (ϕ∗) =

[ϕ̃ (ϕ∗)

ϕ

]σ−1−1




2.The Free Entry Condition (FEC)

Conditional on survival, the expected net present value ofprofits is

v =π

δ

The net value of entry is thus

ve = Prin · v − fe =1− G (ϕ∗)

δπ − fe

with Prin = 1− G (ϕ∗), the survival probabilityEntry occurs until this net value is equal to 0:

π =δfe

1− G (ϕ∗)




Equilibrium

FEC upward-sloping in {ϕ∗, π̄}-spaceZCPC (weakly) decreasing in {ϕ∗, π̄}-space for a wide class ofdistributions g(ϕ), see footnote 15 in Melitz (2003)Equilibrium {ϕ∗, π̄} uniquely determined irrespective of theprecise functional form of g(ϕ)!




Some aggregate accounting

Aggregate resource constraint: L = Lp + LeLabor endowment equals production and investment workersAggregate payment to production workers is the differencebetween aggregate revenues and profits: Lp = R − ΠAggregate payment to investment workers: Le = Me feAggregate firm revenue must equal aggregate consumptionspending, R = L, hence: Π = Me feRepresentative portfolio across all firms in the economy yieldszero profits!




Mass of entrants and surviving firms, welfare

From L = R = M · r we get

M =Rr

=L

σ · (π + f )

In stationary equilibrium, condition PrinMe = δM must hold.Hence,

Me =δM

1− G (ϕ∗)Welfare determined solely by CES price index:W = P−1 = M

1σ−1 ρϕ̃.

An increase of the country size L raises the mass of firms inequilibrium and, hence, welfare.




Example: The Pareto distribution

Let G (ϕ) = 1−(ϕminϕ

)k, so that g(ϕ) = k · ϕkmin · ϕ−k−1

k > 1 – shape parameter, ϕkmin – lower bound for ϕ-draw

00j

g HjL

jhighMIN

jlowMIN

Matches empirical firm-size distributions well (Axtell, 2001)Left-truncated Pareto is still a Pareto!Easy to handle analytically, widely used in the literature




Example: The Pareto distribution

Average productivity of surviving firms (assuming k > (σ−1)):

ϕ̃ =

[1

1− G (ϕ∗)·∫ ∞ϕ∗ϕσ−1g(ϕ)dϕ

] 1σ−1

=

(k

k + 1− σ

) 1σ−1

ϕ∗

Average ϕ̃ proportional to cutoff ϕ∗ → Flat ZCPC!ZCPC: π̄ = (σ−1)fk+1−σ , FEC: π̄ =

δfe(ϕmin)k

(ϕ∗)k

Equilibrium cutoff:

ϕ∗ =

((σ − 1) f

δ fe (k + 1− σ)

)1/k· ϕmin

Mass of entrants and consumption variety:

Me =(σ − 1) Lσ fe k

, and M =(k + 1− σ)L

σ f k



Basic assumptionsOpen economy equilibriumThe impacts of trade

OPEN ECONOMY




Basic assumptions

Two types of trade costs:per-unit iceberg trade costs τ > 1per-period fixed costs of exporting fx

For simplicity: World consists of n identical countries→ Same aggregate variables, wage equalization (w = 1)across countries.

Demidova (IER 2008), Pflueger/Suedekum (JPubE 2013):Asymmetric countries, existence of freely tradable outsidegood ("agriculture") to ensure wage equalization.




Prices, revenue, profits in the open economy

Isoelastic demands → constant markups ("mill pricing")

pd (ϕ) =1ρϕ

and px (ϕ) =τ

ρϕ= τpd (ϕ) .

Revenue on different markets:Domestic revenue: rd(ϕ) = R (ρϕP)

σ−1

Export revenue (foreign aggregate vars with *):

n · rx(ϕ) = n · R∗(ρϕτ

P∗)σ−1

= n · τ1−σ · rd(ϕ),

since P∗ = P and R∗ = R due to symmetry

Domestic and export profits

πd (ϕ) = rd (ϕ)/σ − f and πx(ϕ) = rx(ϕ)/σ − fx




Domestic and export cutoff

The value of a firm

v (ϕ) = max{0,π (ϕ)

δ

}.

Domestic cutoff:

ϕ∗ = inf {ϕ : v (ϕ) > 0}

Is firm productive enough to cover the domestic fixed costs f ?Export cutoff:

ϕ∗x = inf {ϕ : ϕ > ϕ∗ and πx (ϕ) > 0} .

Can the firm also cover the additional fixed costs fx?




Domestic and export cutoff

Revenue of domestic and export cutoff firm

rd (ϕ∗) = R (ρϕ∗P)σ−1 and rx(ϕ∗x) = R

(ρϕ∗xτ

P)σ−1

We also know that: rd (ϕ∗) = σf and rx(ϕ∗x) = σfxHence, we have

rx(ϕ∗x)rd (ϕ∗)

= τ1−σ ·(ϕ∗xϕ∗

)σ−1=

fxf

⇒(ϕ∗xϕ∗

)σ−1= τσ−1 · fx

f

With τσ−1fx > f : ϕ∗x > ϕ∗ ("partitioning")

→ Self-selection of more productive firms into exporting




Exporting probability (general and Pareto)

Survival probability among all entrants:1− G (ϕ∗) = (ϕmin/ϕ∗)k

Probability of exporting among all entrants:1− G (ϕ∗x) = (ϕmin/ϕ∗x)k

Probability of exporting conditional on survival:1−G(ϕ∗x )1−G(ϕ∗) = (ϕ

∗/ϕ∗x)k =

(ffx

)k/(σ−1)· τ−k ≡ Prx

Note: Prx is then also the share of exporters in each country!This share is decreasing in both trade costs, τ and fx .

Consumption variety: Mt = M + nMx , where Mx = Prx ·M.





Deriving ex ante expected profits

πd (ϕ∗) = 0⇔ rd (ϕ∗) = σf ⇔ πd (ϕ̃) = f

[(ϕ̃

ϕ∗

)σ−1− 1

]=k (ϕ∗) f

πx (ϕ∗x) = 0⇔ rx (ϕ∗x) = σfx ⇔ πx (ϕ̃x) = fx

[(ϕ̃xϕ∗x

)σ−1− 1

]=k (ϕ∗x) fx .

where ϕ̃ is the average productivity among all domestic firms, andϕ̃x > ϕ̃ is the average productivity among all domestic exporters.

Using the Pareto distribution:

πd (ϕ̃) = k (ϕ∗) f =(σ − 1)fk + 1− σ

, πx (ϕ̃x) = k (ϕ∗x) fx =(σ − 1)fxk + 1− σ





The new ZCPC (general and with Pareto)

π = πd (ϕ̃) + Prx · n · πx (ϕ̃) = k (ϕ∗) f + Prx · n · k (ϕ∗x) fx

=(σ − 1)fk + 1− σ

·

1 + n τ−k(

ffx

) k+1−σσ−1

︸︷︷︸≡φ

= (σ − 1)fk + 1− σ · [1 + φ]with φ > 0 the measure of trade freeness (decreasing in τ and fx).

The (old and new) FEC (general and with Pareto)Net present value of average profit stream: v = πδ .Zero expected profits:Prin · v − fe = 0⇔ π = δfe1−G(ϕ∗) = δfe · (ϕ

∗/ϕmin)k .




Open economy equilibrium

Effects of moving from autarky to trade:

ϕ∗ > ϕ∗a π > πa

Rising trade freeness increases the domestic cutoff→ trade leads to tougher domestic firm selection!Pareto: ZCPC is flat in {ϕ∗, π}-space; shifts upwards.Open economy cutoff: ϕ∗ = (1 + φ)1/k · ϕ∗a > ϕ∗a




Mass of firms in the open economy

Aggregate resource constraint

L = R = M · r(ϕ̃) = M [rd (ϕ̃) + Prx · n · rx(ϕ̃x)]

= M σ

(πd (ϕ̃) + Prx · n · πx(ϕ̃x))︸︷︷︸=π>πa

+f + Prx · n · fx

The mass of surviving firms in the domestic economy is thus

M =L

σ (π + f + Prx · n · fx)< Ma

Under the Pareto (verify for yourself!):

M =k + 1− σ

σ f k (1 + φ)· L < Ma =

k + 1− σσ f k

· L

Trade causes exit of less productive domestic firms! (M < Ma)International Trade – DICE/RGS Jens Suedekum Melitz (2003) – Firm heterogeneity in the Krugman-model



Gradual trade liberalization

So far, move from autarky to (imperfect) tradeMeasure φ also allows to consider gradual liberalization

Three mechanisms:1 an increase in the number of available trading partners n2 a decrease in the variable trade costs τ3 a decrease in fixed trade costs fx

All of these increase φ, which intensifies selection even further




Reallocation

Consider a firm with productivity ϕ > ϕ∗a:In autarky: Positive revenue ra (ϕ) and profits πa (ϕ).Opening up to trade: reallocation of resources across firms!

rd (ϕ) < ra (ϕ) < rd (ϕ) + nrx (ϕ) .

Least productive firms exit, medium ones shrink but stayactive, most productive ones turn to exporters and gain!




Reallocation

Change of firm-level profits after opening up to trade:

∆π (ϕ) = π (ϕ)−πa (ϕ) =1σ

(rd (ϕ) + nrx (ϕ)− ra (ϕ))−nfx .




Explaining selection and reallocation

Why does trade force the least productive firms to exit?Why does it lead to a reallocation towards more productivefirms?

Two channels:1 an increase in product market competition and2 an increase in competition in the domestic factor/labor market




Mass of exporters and consumption variety

Recall: mass of surviving firms

M =k + 1− σ

σ f k (1 + φ)·L = 1

1 + φ·Ma, with φ = n τ−k

(ffx

) k+1−σσ−1

Mass of domestic exporters:

Mx = Prx ·M =(

ffx

) kσ−1

τ−k ·M = fn · fx

· φ ·M

Consumption variety:

Mt = M + n ·Mx = M(1 +

ffx· φ)

=1 + ffx · φ1 + φ

Ma

Trade raises consumption variety if τ1−σf < fx < fTrade replaces domestic varieties from low productive firms byimported varieties from high productive foreign firms.




Welfare

Welfare comparison: Autarky versus free trade

Wa = (1/Pa) = M1/(σ−1)a ·ϕ̃a·ρ, Wt = (1/Pt) = M1/(σ−1)t ·ϕ̃t ·ρ

where ϕ̃t is average productivity among all (domestic+foreign)firms active in the domestic market.Clearly, ϕ̃t > ϕ̃a. Yet, we may have Mt < Ma. But even then,there are welfare gains from trade! See problem set...In fact, both in autarky and with trade, welfare is proportionalto the domestic cutoff:

Wa = ρ · (L/σf )1/(σ−1) · ϕ∗a Wt = ρ · (L/σf )1/(σ−1) · ϕ∗

Hence,WtWa

=ϕ∗

ϕ∗a= (1 + φ)1/k




Total export sales – A preview of gravity

Total export sales of the domestic country in any foreignmarket:

X = Mx · r x(ϕ̃x) = Mx · σ (πx(ϕ̃x) + fx) = Mx ·σ fx k

k + 1− σ

Using Mx = fnfx · φ ·M, we thus have

X =f

nfx· φ · k + 1− σ

σ f k (1 + φ)· σ fx kk + 1− σ

· L = 1n· φ1 + φ

· L

Share of domestic spending in total expenditure E = L is thus

L− nXL

=1

1 + φ≡ λ

Autarky (φ = 0): λ = 1; free trade (φ→ n): λ→ 1/(n + 1)International Trade – DICE/RGS Jens Suedekum Melitz (2003) – Firm heterogeneity in the Krugman-model



New new trade theory - same old gains?

Recall the welfare gains of moving from autarky to trade

∆W =WtWa

=ϕ∗

ϕ∗a= (1 + φ)1/k

Using the domestic expenditure share, we get: ∆W = λ−1/k

Recall that k is the Pareto shape-parameter. At the sametime, k is the elasticity of trade flows with respect to variable(iceberg) trade costs (k = −�), the "trade elasticity".This verifies the results by Arkolakis, Costinot andRodriguez-Clare (AER 2011).They show that the formula ∆W = λ1/� can be used to assessthe welfare gains from trade in a wide class of CES- andsimilar models (with and without firm heterogeneity).


Setup of the ModelDemandProductionFirm Entry and Exit

Closed EconomyEquilibriumProperties of equilibriumExample: The Pareto distribution

Open EconomyBasic assumptionsOpen economy equilibriumThe impacts of trade

Documents

Melitz(2003)– FirmheterogeneityintheKrugman-model · Melitz(2003)– FirmheterogeneityintheKrugman-model InternationalTrade–DICE/RGS JensSuedekum March/April2014 InternationalTrade–DICE/RGS