Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
ECO2704 Lecture Notes: Melitz Model
Xiaodong Zhu
University of Toronto
October 15, 2010
1 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
• Dynamic Industry Model with heterogeneous firms where
opening to trade leads to reallocations of resources within an
industry
• Opening to trade leads to
• Reallocations of resources across firms
• Low productivity firms exit
• High productivity firms expand so there is a change in industry
composition
• High productivity firms enter export markets
• Improvements in aggregate industry productivity
• No change in firm productivity
• Consistent with empirical evidence from trade liberalizations?
2 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
The theoretical model is consistent with a variety of other stylized
facts about industries
• Heterogeneous firm productivity
• Ongoing entry and exit
• Co-movement in (gross) entry and exit due to sunk entry costs
• Exiting firms are low productivity (selection effect)
• Explains why some firms export within industries and others do
not
• Contrast with traditional theories of comparative advantage
• Exporting firms are high productivity (selection effect)
• No feedback from exporting to productivity
3 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
• Single factor: labor, numeraire, wage normalized to 1
• Firms enter market by paying sunk entry cost (fe )
• Firms observe their productivity (ϕ) from distribution G (ϕ)
• Productivity is fixed thereafter
• Once productivity is observed, firms decide whether to produce
or exit
• Firms produce horizontally-dierentiated varieties, with a fixed
production cost (fd ) and a variable cost that depends on their
productivity
• Firms face an exogenous probability of death (δ) per period
due to force majeure events
4 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Preferences and demand
Q=
ˆ
q(k)
k∈Ω
σ−1
σ
dk
σ
σ−1
,σ > 1
Let R be the total expenditures of the representative consumer.
Then,
p(k) −σ
q(k) =
Q
P
p(k) 1−σ
R
r (k) = p(k)q(k) =
P
Here p(k) is the price of variety k and P is the price index faced by
the consumer,
1
ˆ
1−σ
1−σ
p(k)
dk
P=
k∈Ω
5 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Technology
The amount of labour needed for a firm to produce q(k) units of
variety k is
fd + q(k)/ϕ(k)
So the total cost of production is
fd + q(k)/ϕ(k)
and the marginal cost is
1/ϕ(k)
All firms with the same productivity behave in exactly the same
way. So we will use productivity ϕ rather than variety k to identify
a firm.
6 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Profit maximization
Each firm produces a differentiated variety (monopolistic
competition):
π(ϕ) = maxq(ϕ) {p(ϕ)q(ϕ) − fd + q(ϕ)/ϕ}
Constant markup pricing:
p(ϕ) =
σ
ϕ−1 = (ρϕ)−1
σ−1
(1)
which implies the following:
r (ϕ) = (̺ϕ)σ−1 P σ−1 R, q(ϕ) = [̺ϕ]σ P σ−1 R
π(ϕ) =
r (ϕ)
− fd
σ
(2)
(3)
7 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Entry and exit
Prior to entry, a firm incurs a sunk cost of entry, fe .
Upon entry, it draws a productivity ϕ randomly from a distribution
G (.). The firm will stay in the market only if
π(ϕ) ≥ 0 or
r (ϕ) ≥ σfd
Current value of firm ϕ is
∞
X
π(ϕ)
t
,0
(1 − δ) max {π(ϕ), 0} = max
v (ϕ) =
δ
t=0
Let ϕ∗ be the cutoff productivity such that π(ϕ∗ ) = 0 and
pin (ϕ∗ ) = 1 − G (ϕ∗ ) the probability that a firm will stay. Then,
productivity distribution of producing firms is given by the following
density function:
( g (ϕ)
∗
pin (ϕ∗ ) if ϕ ≥ ϕ
(4)
µ(ϕ) =
0
otherwise
8 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Equilibrium in a closed economy
A stationary equilibrium is a vector (ϕ∗ , M, Me , P, R) of cutoff
productivity, mass of producing firms, mass of entrants, price level and
aggregate expenditures such that price, revenue, quantity and profit for
each firm ϕ are given by equation (1) to (3), the distribution of producing
firms is given by µ(ϕ) in equation (4), and the following conditions hold:
Zero profit condition:
π(ϕ∗ ) = 0
Free entry condition:
pin (ϕ∗ )
ˆ
π(ϕ)
µ(ϕ)dϕ = fe
δ
Stationarity condition:
pin (ϕ∗ ) Me = δM
Market clearing condition:
R=L
9 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Price index
1
1−σ
µ(ϕ)Md ϕ
P=
ˆ
p(ϕ)
P = M 1/(1−σ) ρ−1
ˆ
ϕσ−1 µ(ϕ)d ϕ = M 1/(1−σ) (ρϕ)−1
1−σ
Here
ϕ=
ˆ
σ−1
ϕ
(5)
1
σ−1
µ(ϕ)d ϕ
is a weighted average of firm productivities.
10 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Profit function and entrants’ expected value
From (2), (3), (5) and the market clearing condition:
σ−1
L
ϕ
− fd
π (ϕ) =
ϕ
σM
Entrants’ expected value:
ˆ
pin (ϕ∗ ) L
π (ϕ)
∗ π (ϕ)
∗
µ(ϕ)d ϕ = pin (ϕ )
=
− fd
v = pin (ϕ )
δ
δ
δ
σM
11 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Mass of producing firms
Zero profit condition=⇒
ϕ∗
ϕ
σ−1
L
= fd
σM
Free entry condition =⇒
pin (ϕ∗ ) L
− fd = fe
δ
σM
Thus,
"
ϕ∗
ϕ
1−σ
#
− 1 fd =
(6)
δfe
pin (ϕ∗ )
12 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Cutoff productivity in equilibrium
"
∗
=⇒ H(ϕ ) ≡
ϕ∗
ϕ
ˆ
1−σ
∞
ϕ∗
"
#
− 1 fd =
ϕ
ϕ∗
σ−1
δfe
pin (ϕ∗ )
#
− 1 g (ϕ)d ϕ = δfe /fd
(7)
LHS is a decreasing function, so ϕ∗ increases in fixed production
cost fd , but decreases in entry cost fe
13 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Number of firms in equilibrium
From equation (6),
M=
L
σ [fd + δfe /pin (ϕ∗ )]
14 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Size distribution in equilibrium
Firm ϕ employs z(ϕ) =q(ϕ)/ϕ number of worker for production
z(ϕ) =
p(ϕ)
P
−σ
Q/ϕ = σρfd
ϕ
ϕ∗
σ−1
Thus, the average firm size is
z = σρfd
ϕ
ϕ∗
σ−1
= σρ [fd + δfe /pin (ϕ∗ )]
15 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Properties of Closed Economy Equilibrium
• Increasing fixed production cost fd will
• raise average productivity
• raise average firm size
• but lower the number of producing firms
• Increasing sunk entry cost fe will
• lower average productivity
• lower average firm size
• and lower the number of producing firms
16 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
• n + 1 identical countries, so identical wage, price level and
income across countries
• wage is normalized to 1
• To produce, a firm will first incur a fixed production cost fd
• If the firm decides to export, it will also incur a fixed export
cost fx
• Iceberg trade cost τ
17 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Trade costs
Same markup pricing for exports:
px (ϕ) =
τ
= τ pd (ϕ)
̺ϕ
Revenue functions:
rd (ϕ) = (̺ϕ)σ−1 P σ−1 R
rx (ϕ) = τ 1−σ rd (ϕ)
Profit functions:
πd (ϕ) =
rd (ϕ)
− fd
σ
πx (ϕ) =
rx (ϕ)
− fx
σ
π(ϕ) = πd (ϕ) + nπx (ϕ)
18 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Cutoff productivities
The cutoff productivity below which a firm will now produce: ϕ∗d
πd (ϕ∗d ) = 0
Nominal income remains the same as in closed economy: R = L
Price level is lower than that in the closed economy
=⇒ πd (ϕ) < πa (ϕ)
=⇒ ϕ∗a < ϕ∗d
Trade increases competition and forces some
firms to exit
inefficient
The cutoff productivity for exporting: max ϕ∗d , ϕ∗x
πx (ϕ∗x ) = 0
If τ σ−1 fx > fd , then ϕ∗d < ϕ∗x : not all producing firms export, and
exporting firms are more productive
19 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Open economy results
• The opening of trade leads to:
• Rise in the zero profit cutoff productivity
• Rise in average firm revenue and profit
• Low productivity firms between ϕ∗a and ϕ∗d exit
• Intermediate productivity firms betwen ϕ∗d and ϕ∗x contract
• Only firms with productivities greater than ϕ∗x enter export
markets and expand
• All of the above lead to a change in industry composition that
raises aggregate industry productivity
20 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Subsequent literature
• Helpman, Melitz and Rubinstein (2004) “Export Versus FDI
with Heterogeneous Firms," American Economic Review, 94,
300-316.
• Introduces both exports and FDI as alternative means of
serving a foreign market
• Introduces an outside sector to tractably characterize
equilibrium with many asymmetric sectors
• Antras and Helpman (2004) “Global Sourcing," Journal of
Political Economy, 112(3), 552-580.
• Combines the Melitz model with the Antras (2003) model of
incomplete contracts and trade
• Bernard, Redding and Schott (2007) “Comparative Advantage
and Heterogeneous Firms," Review of Economic Studies,
73(1), 31-66.
• Incorporates the Melitz model into the framework of integrated
equilibrium of Helpman and Krugman (1985)
21 / 22
Introduction
Closed Economy Model
Open Economy Model
Subsequent literature
Subsequent literature
• Chaney, Thomas (2008) “Distorted Gravity: the Intensive and
Extensive Margins of International Trade," American
Economic Review, September.
• Provides a simplified static version of the Melitz model without
ongoing firm entry and with an outside sector
• Examines the model’s implications for the extensive and
intensive margins of international trade
• Arkolakis, Costas, Klenow, Peter, Demidova, Svetlana and
Andres Rodriguez-Clare (2009) “The Gains from Trade with
Endogenous Variety," American Economic Review, Papers and
Proceedings, 98 (4), 444-450.
• Solves the Chaney version of the model without an outside
sector
• Derives a sufficient statistic for welfare of the same form as
that in Eaton and Kortum (2002)
22 / 22