Setup of the Model
Closed Economy
Open Economy
Melitz (2003) –
Firm heterogeneity in the Krugman-model
International Trade – DICE/RGS
Jens Suedekum
March/April 2014
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Demand
Production
Firm Entry and Exit
Demand
One monopolistically competitive sector. CES preferences.
Z
ρ
U=
1
ρ
q (ω) dω
.
ω∈Ω
CES price index. Minimum expenditure per aggregate unit Q
Z
P=
1−σ
p (ω)
dω
1
1−σ
.
ω∈Ω
Consumption and expenditure per variety
(with E = R = P × Q), see eq. (13) in LN1.
R p (ω) −σ
p (ω) 1−σ
q (ω) =
and r (ω) = R
.
P
P
P
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Demand
Production
Firm Entry and Exit
Production
Continuum of firms; each firm produces different variety ω.
Labor requirement for output q of variety ω
q
l (ϕ) = f + .
ϕ
Isoleastic demand ⇒ constant markup [1/ρ = σ/(σ − 1)]
p (ω) = p (ϕ) =
σ w
1
=
.
σ−1ϕ
ρϕ
Firm revenue and profits
P σ−1
r (ϕ)
σ−1
r (ϕ) = R [ρϕP]
=R
and π (ϕ) =
−f.
p (ϕ)
σ
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 Model
Closed Economy
Open Economy
Demand
Production
Firm Entry and Exit
Aggregation
Rewrite CES price index in terms of ϕ instead of ω
Z
1
1−σ
1−σ
P =
p (ω)
dω
.
ω∈Ω
Z
∞
1−σ
p (ϕ)
=
1
1−σ
Mµ(ϕ)dϕ
.
ϕ=0
= M
1
1−σ
1
· ·
ρ
Z
ϕ
ϕ=0
|
= M
1
1−σ
∞
1
1−σ
µ(ϕ)dϕ
.
{z
}
σ−1
=1/ϕ
e
1
· (1/ (ρ ϕ))
e = M 1−σ · p(ϕ)
e
1
Compare to eq. (15) in LN1: P = M 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 Model
Closed Economy
Open Economy
Demand
Production
Firm 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 or
to remain in the market.
σ−1
Recall: π (ϕ) = r (ϕ)
> 0.
σ − f , with r (ϕ) = R [ρϕP]
→ 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 Model
Closed Economy
Open Economy
Demand
Production
Firm Entry and Exit
Cutoff productivity
Value of a firm with productivity draw ϕ is determined by:
!
∞
X
π (ϕ)
t
.
v (ϕ) = max 0,
(1 − δ) π (ϕ) = max 0,
δ
t=0
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”) is
ϕ∗ = 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 Model
Closed Economy
Open Economy
Demand
Production
Firm Entry and Exit
Average productivity
Productivity distribution among surviving firms:
g (ϕ)
∗
µ(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):
Z
ϕ
e =
σ−1
ϕ
1
σ−1
µ(ϕ)dϕ
ϕ=0
=
∞
1
·
1 − G (ϕ∗ )
Z
∞
ϕ
σ−1
1
σ−1
g (ϕ)dϕ
ϕ∗
Given the propoerties of the ex ante distribution g (ϕ), this
average 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 Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: 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
2
zero cutoff profit condition (ZCPC)
free entry condition (FEC)
Solved for ϕ∗ and π̄ = π(ϕ̃), i.e.,
cutoff and average profit conditional on survival
With this, mass of entrants Me and consumption variety M are
pinned down via aggregate resource constraint
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
1. Zero Cutoff Profit Condition (ZCPC)
The ratio of any two firms’ revenues is:
i1−σ
h
p(ϕ1 )
σ−1
R
P
r (ϕ1 )
R [ρϕ1 P]σ−1
ϕ1
=
.
= h
=
i1−σ
σ−1
r (ϕ2 )
ϕ2
p(ϕ2 )
R
[ρϕ
P]
2
R
P
Thus, for cutoff and average surviving firm:
σ−1
σ−1
r (ϕ̃)
ϕ̃
ϕ̃
=
⇔ r̄ = r (ϕ̃) =
r (ϕ∗ ) .
∗
∗
r (ϕ )
ϕ
ϕ∗
Hence, average firm makes profits
σ−1
ϕ̃
r (ϕ∗ )
r (ϕ̃)
π = π (ϕ̃) =
−f =
−f.
σ
ϕ∗
σ
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
1.The Zero Cutoff Profit Condition (ZCPC)
The profits of the cutoff firms are equal to 0:
π (ϕ∗ ) =
r (ϕ∗ )
− f = 0.
σ
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
ϕ̃ (ϕ∗ )
−f = k (ϕ )·f with k (ϕ ) =
ϕ
International Trade – DICE/RGS Jens Suedekum
∗
∗
σ−1
−1
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
2.The Free Entry Condition (FEC)
Conditional on survival, the expected net present value of
profits is
π
v=
δ
The net value of entry is thus
ve = Prin · v − fe =
1 − G (ϕ∗ )
π − fe
δ
with Prin = 1 − G (ϕ∗ ), the survival probability
Entry occurs until this net value is equal to 0:
π=
International Trade – DICE/RGS Jens Suedekum
δfe
1 − G (ϕ∗ )
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
Equilibrium
FEC upward-sloping in {ϕ∗ , π̄}-space
ZCPC (weakly) decreasing in {ϕ∗ , π̄}-space for a wide class of
distributions g (ϕ), see footnote 15 in Melitz (2003)
Equilibrium {ϕ∗ , π̄} uniquely determined irrespective of the
precise functional form of g (ϕ)!
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
Some aggregate accounting
Aggregate resource constraint: L = Lp + Le
Labor endowment equals production and investment workers
Aggregate payment to production workers is the difference
between aggregate revenues and profits: Lp = R − Π
Aggregate payment to investment workers: Le = Me fe
Aggregate firm revenue must equal aggregate consumption
spending, R = L, hence: Π = Me fe
Representative portfolio across all firms in the economy yields
zero profits!
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
Mass of entrants and surviving firms, welfare
From L = R = M · r we get
M=
R
L
=
r
σ · (π + f )
In stationary equilibrium, condition Prin Me = δM must hold.
Hence,
δM
Me =
1 − G (ϕ∗ )
Welfare determined solely by CES price index:
1
W = P −1 = M σ−1 ρϕ.
e
An increase of the country size L raises the mass of firms in
equilibrium and, hence, welfare.
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
Example: The Pareto distribution
Let G (ϕ) = 1 −
ϕmin
ϕ
k
, so that g (ϕ) = k · ϕkmin · ϕ−k−1
k > 1 – shape parameter, ϕkmin – lower bound for ϕ-draw
g HjL
jMIN
low
jMIN
high
0
j
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 Model
Closed Economy
Open Economy
Equilibrium
Properties of equilibrium
Example: The Pareto distribution
Example: The Pareto distribution
Average productivity of surviving firms (assuming k > (σ − 1)):
1
1
Z ∞
σ−1
σ−1
1
k
σ−1
ϕ
g (ϕ)dϕ
ϕ̃ =
·
=
ϕ∗
∗
1 − G (ϕ ) ϕ∗
k +1−σ
Average ϕ̃ proportional to cutoff ϕ∗ → Flat ZCPC!
ZCPC: π̄ =
(σ−1)f
k+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) L
,
σ fe k
International Trade – DICE/RGS Jens Suedekum
and
M=
(k + 1 − σ)L
σf k
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
OPEN ECONOMY
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Basic assumptions
Two types of trade costs:
per-unit iceberg trade costs τ > 1
per-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 outside
good ("agriculture") to ensure wage equalization.
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Prices, revenue, profits in the open economy
Isoelastic demands → constant markups ("mill pricing")
pd (ϕ) =
1
τ
and px (ϕ) =
= τ pd (ϕ) .
ρϕ
ρϕ
Revenue on different markets:
σ−1
Domestic revenue: rd (ϕ) = R (ρϕP)
Export revenue (foreign aggregate vars with *):
n · rx (ϕ) = n · R ∗
ρϕ
P∗
σ−1
= n · τ 1−σ · rd (ϕ),
τ
since P ∗ = P and R ∗ = R due to symmetry
Domestic and export profits
πd (ϕ) = rd (ϕ)/σ − f
International Trade – DICE/RGS Jens Suedekum
and
πx (ϕ) = rx (ϕ)/σ − fx
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Domestic and export cutoff
The value of a firm
v (ϕ) = max
π (ϕ)
0,
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The 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∗x
P
τ
σ−1
We also know that: rd (ϕ∗ ) = σf and rx (ϕ∗x ) = σfx
Hence, we have
∗ σ−1
ϕx
rx (ϕ∗x )
fx
1−σ
=τ
·
=
∗
∗
rd (ϕ )
ϕ
f
⇒
ϕ∗x
ϕ∗
σ−1
= τ σ−1 ·
fx
f
With τ σ−1 fx > 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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The 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:
k/(σ−1)
1−G (ϕ∗x )
f
∗ /ϕ∗ )k =
=
(ϕ
· τ −k ≡ Prx
∗
x
fx
1−G (ϕ )
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Equilibrium Conditions
Deriving ex ante expected profits
πd (ϕ∗ ) = 0 ⇔ rd (ϕ∗ ) = σf ⇔ πd (ϕ)
e =f
"
πx (ϕ∗x ) = 0 ⇔ rx (ϕ∗x ) = σfx ⇔ πx (f
ϕx ) = fx
ϕ
e
ϕ∗
"
σ−1
ϕ
fx
ϕ∗x
#
− 1 =k (ϕ∗ ) f
σ−1 #
− 1 =k (ϕ∗x ) fx .
where ϕ
e is the average productivity among all domestic firms, and
ϕ
fx > ϕ
e is the average productivity among all domestic exporters.
Using the Pareto distribution:
πd (ϕ)
e = k (ϕ∗ ) f =
(σ − 1)f
,
k +1−σ
International Trade – DICE/RGS Jens Suedekum
πx (f
ϕx ) = k (ϕ∗x ) fx =
(σ − 1)fx
k +1−σ
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Equilibrium Conditions
The new ZCPC (general and with Pareto)
k (ϕ∗ ) f + Prx · n · k (ϕ∗x ) fx

π = πd (ϕ)
e + Prx · n · πx (ϕ)
e =

=
(σ − 1)f
k +1−σ
k+1−σ 

σ−1
f
(σ − 1)f


−k
· 1 + n τ
· [1 + φ]
=
fx

 k +1−σ
|
{z
}
≡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:
∗
k
e
Prin · v − fe = 0 ⇔ π = 1−Gδf(ϕ
∗ ) = δfe · (ϕ /ϕmin ) .
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The 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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Mass of firms in the open economy
Aggregate resource constraint
L = R = M · r (ϕ)
e = M [rd (ϕ)
e + Prx · n · rx (f
ϕx )]




= M σ (πd (ϕ)
e + Prx · n · πx (f
ϕx )) +f + Prx · n · fx 
|
{z
}
=π>π a
The mass of surviving firms in the domestic economy is thus
M=
L
< Ma
σ (π + f + Prx · n · fx )
Under the Pareto (verify for yourself!):
M=
k +1−σ
k +1−σ
· L < Ma =
·L
σ f k (1 + φ)
σf k
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Gradual trade liberalization
So far, move from autarky to (imperfect) trade
Measure φ also allows to consider gradual liberalization
Three mechanisms:
1
2
3
an increase in the number of available trading partners n
a decrease in the variable trade costs τ
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The 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 stay
active, 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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Reallocation
Change of firm-level profits after opening up to trade:
∆π (ϕ) = π (ϕ) − πa (ϕ) =
International Trade – DICE/RGS Jens Suedekum
1
(rd (ϕ) + nrx (ϕ) − ra (ϕ)) − nfx .
σ
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The 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 productive
firms?
Two channels:
1
2
an increase in product market competition and
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Mass of exporters and consumption variety
Recall: mass of surviving firms
1
k +1−σ
·L =
·Ma ,
M=
σ f k (1 + φ)
1+φ
with φ = n τ
−k
k+1−σ
σ−1
f
fx
Mass of domestic exporters:
k
f
f σ−1 −k
τ ·M =
Mx = Prx · M =
·φ·M
fx
n · fx
Consumption variety:
1 + ffx · φ
f
Ma
Mt = M + n · Mx = M 1 + · φ =
fx
1+φ
Trade raises consumption variety if τ 1−σ f < fx < f
Trade replaces domestic varieties from low productive firms by
imported varieties from high productive foreign firms.
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Welfare
Welfare comparison: Autarky versus free trade
1/(σ−1)
Wa = (1/Pa ) = Ma
·ϕ
ea ·ρ,
1/(σ−1)
Wt = (1/Pt ) = Mt
·ϕ
et ·ρ
where ϕ
et is average productivity among all (domestic+foreign)
firms active in the domestic market.
Clearly, ϕ
et > ϕ
ea . 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 proportional
to the domestic cutoff:
Wa = ρ · (L/σf )1/(σ−1) · ϕ∗a
Hence,
Wt = ρ · (L/σf )1/(σ−1) · ϕ∗
Wt
ϕ∗
= ∗ = (1 + φ)1/k
Wa
ϕa
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model
Setup of the Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
Total export sales – A preview of gravity
Total export sales of the domestic country in any foreign
market:
X = Mx · r x (f
ϕx ) = Mx · σ (π x (f
ϕx ) + fx ) = Mx ·
Using Mx =
X =
f
nfx
σ fx k
k +1−σ
· φ · M, we thus have
f
k +1−σ
σ fx k
1
φ
·φ·
·
·L= ·
·L
nfx
σ f k (1 + φ) k + 1 − σ
n 1+φ
Share of domestic spending in total expenditure E = L is thus
L − nX
1
=
≡λ
L
1+φ
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 Model
Closed Economy
Open Economy
Basic assumptions
Open economy equilibrium
The impacts of trade
New new trade theory - same old gains?
Recall the welfare gains of moving from autarky to trade
∆W =
Wt
ϕ∗
= ∗ = (1 + φ)1/k
Wa
ϕa
Using the domestic expenditure share, we get: ∆W = λ−1/k
Recall that k is the Pareto shape-parameter. At the same
time, 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 and
Rodriguez-Clare (AER 2011).
They show that the formula ∆W = λ1/ can be used to assess
the welfare gains from trade in a wide class of CES- and
similar models (with and without firm heterogeneity).
International Trade – DICE/RGS Jens Suedekum
Melitz (2003) – Firm heterogeneity in the Krugman-model