-
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
-
Setup of the ModelClosed EconomyOpen Economy
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
DemandProductionFirm Entry and Exit
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
DemandProductionFirm Entry and Exit
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!
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
DemandProductionFirm Entry and Exit
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 .
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
DemandProductionFirm Entry and Exit
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!)
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
DemandProductionFirm Entry and Exit
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∗
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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 .
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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∗)
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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)!
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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!
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
EquilibriumProperties of equilibriumExample: The Pareto
distribution
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
OPEN ECONOMY
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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?
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
Equilibrium Conditions
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
Equilibrium Conditions
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 .
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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!
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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 .
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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.
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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
-
Setup of the ModelClosed EconomyOpen Economy
Basic assumptionsOpen economy equilibriumThe impacts of
trade
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).
International Trade – DICE/RGS Jens Suedekum Melitz (2003) –
Firm heterogeneity in the Krugman-model
Setup of the ModelDemandProductionFirm Entry and Exit
Closed EconomyEquilibriumProperties of equilibriumExample: The
Pareto distribution
Open EconomyBasic assumptionsOpen economy equilibriumThe impacts
of trade