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Notes on Graduate International Trade1
Costas Arkolakis2
Yale University
May 2012 [New version: preliminary]
1This set of notes and the homeworks accomodating them is a
collection of material designed for an in-ternational trade course
at the graduate level. I am grateful to my teachers at the
University of Minnesota,Cristina Arellano, Jonathan Eaton, Timothy
Kehoe and Samuel Kortum. I am also grateful to Steve Reddingand
Peter Schott for sharing their teaching material and to Federico
Esposito, Ananth Ramananarayan, OlgaTimoshenko, Philipp Stiel, and
Alex Torgovitsky for valuable input and for various comments and
suggestions.All remaining errors are purely mine.
[email protected]
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Abstract
These notes are prepared for the first part of the part of (720)
Graduate International Trade at Yale.
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Contents
1 An introduction into deductive reasoning 1
1.1 Inductive and deductive reasoning . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 1
1.2 Employing and testing a model . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 2
1.3 International Trade: The Macro Facts . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 3
2 An introduction to modeling 4
2.1 The Heckscher-Ohlin model . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 4
2.1.1 Autarky equilibrium . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 5
2.1.2 Free Trade Equilibrium . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 7
2.1.3 No specialization . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 8
2.1.4 Specialization . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 12
2.1.5 The 4 big theorems. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 12
2.2 Armington model . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 12
2.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 13
2.2.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 15
2.2.3 Welfare . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 16
2.3 Monopolistic Competition with Homogeneous Firms and CES
demand . . . . . . . . . 16
2.3.1 Consumers problem . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 17
2.3.2 Firms problem . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 17
2.3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 18
2.3.4 Gravity . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 19
2.3.5 Welfare . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 20
2.4 Monopolistic Competition with Separable Utility Function
(Krugman 79) . . . . . . . 20
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2.4.1 Consumers problem . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 21
2.4.2 Firms Problem . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 21
2.5 Ricardian model . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 23
2.5.1 The two goods case . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 23
3 Modeling production heterogeneity 26
3.1 Introduction to heterogeneity: The Ricardian model with a
continuum of goods . . . . 26
3.2 A theory of technology starting from first principles . . .
. . . . . . . . . . . . . . . . . 29
3.3 Application I: Perfect competition (Eaton-Kortum) . . . . .
. . . . . . . . . . . . . . . . 34
3.3.1 Key contributions of EK . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 37
3.4 Application II: Bertrand competition (Bernand, Jensen &
EK) . . . . . . . . . . . . . . . 37
3.4.1 Key contributions of BEJK . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 41
3.5 Application III: Monopolistic competition with CES
(Chaney-Melitz) . . . . . . . . . . 42
3.5.1 Consumer . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42
3.5.2 Firms Problem . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 42
3.5.3 Gravity-Aggregation . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 43
3.5.4 Welfare . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 44
3.5.5 Key contributions of Melitz . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 44
4 Closing the model 46
4.1 Equilibrium in the labor market . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 46
4.2 The free entry condition . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 49
4.3 Solving for the Equilibrium . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 50
4.4 Computing the Equilibrium (Alvarez-Lucas) . . . . . . . . .
. . . . . . . . . . . . . . . 51
5 Extensions: Modeling the Demand Side 53
5.1 Extension I: The Nested CES demand structure . . . . . . . .
. . . . . . . . . . . . . . . 53
5.2 Extension II: Market penetration costs . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 54
5.3 Extension III: Multiproduct firms . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 56
5.3.1 Gravity and Welfare . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 61
5.4 Extension IV: Separable Utility Functions
(Arkolakis-Costinot-Donaldson-Rodriguez
Clare) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 62
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5.4.1 Firm Problem . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 62
5.4.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 63
5.4.3 Welfare . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 63
6 Modeling Vertical Production Linkages 67
6.1 Each good is both final and intermediate . . . . . . . . . .
. . . . . . . . . . . . . . . . . 67
6.2 Each good has a single specialized intermediate input . . .
. . . . . . . . . . . . . . . . 68
6.3 Each good uses a continuum of inputs . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 68
7 Some facts on disaggregated trade flows 71
7.1 Firm heterogeneity . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 71
7.2 Trade liberalization . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 72
7.3 Trade dynamics . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 73
8 Trade Liberalization 74
8.1 Trade Liberalization and Firm Heterogeneity . . . . . . . .
. . . . . . . . . . . . . . . . 74
8.1.1 Trade Liberalization and Welfare gains
(Arkolakis-Costinot-Rodriguez-Clare) . 74
8.1.2 The Dekle-Eaton-Kortum Procedure for Counterfactual
Experiments . . . . . . 78
9 Estimating Models of Trade 80
9.1 The Anderson and van Wincoop procedure . . . . . . . . . . .
. . . . . . . . . . . . . . 80
9.2 The Head and Ries procedure . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 82
9.3 The Eaton and Kortum procedure . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 82
9.4 Calibration of a firm-level model of trade . . . . . . . . .
. . . . . . . . . . . . . . . . . 83
9.5 Estimation of a firm-level model . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 85
9.5.1 The model . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 85
9.5.2 Estimation, simulated method of moments . . . . . . . . .
. . . . . . . . . . . . 88
10 Appendix 90
10.1 Distributions . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 90
10.1.1 The Frchet Distribution . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 90
11 Figures and Tables 93
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List of Figures
11.1 Sales in France from firms grouped in terms of the minimum
number of destinations
they sell to. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 94
11.2 Market Size and French firm entry in 1986. . . . . . . . .
. . . . . . . . . . . . . . . . . 94
11.3 Distributions of average sales per good and average number
of goods sold. . . . . . . 95
11.4 Distribution of sales for Portugal and means of other
destinations group in terciles
depending on total sales of French firms there. . . . . . . . .
. . . . . . . . . . . . . . . 95
11.5 Product Rank, Product Entry and Product Sales for Brazilian
Exporters . . . . . . . . . 96
11.6 Increases in trade and initial trade. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 96
11.7 Frechet Distribution . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 97
11.8 Pareto Distribution . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 97
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Chapter 1
An introduction into deductive reasoning
1.1 Inductive and deductive reasoning
Inductive or empirical reasoning is the type of reasoning that
moves from specific observations to
broader generalizations and theories. Deductive reasoning is the
type of reasoning that moves from
axioms to theorems and then applies the predictions of the
theory to the specific observations.
Inductive reasoning has failed in several occasions in
economics. According to Prescott (see
Prescott (1998)) The reason that these inductive attempts have
failed ... is that the existence of policy
invariant laws governing the evolution of an economic system is
inconsistent with dynamic economic
theory. This point is made forcefully in Lucas famous critique
of econometric policy evaluation.
Theories developed using deductive reasoning must give
assertions that can be falsified by an
observation or a physical experiment. The consensus is that if
one cannot potentially find an obser-
vation that can falsify a theory then that theory is not
scientific (Popper).
A general methodology of approaching a question using deductive
reasoning is the following:
1) Observe a set of empirical (stylized) facts that your theory
has to address and/or are relevant
to the questions that you want to tackle,
2) Build a theory,
3) Test the theory with the data and then use your theory to
answer the relevant questions,
4) Refine the theory, going through step 1
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1.2 Employing and testing a model
A vague definition of two methodologies: calibration and
estimation
Calibration is the process of picking the parameters of the
model to obtain a match between the
observed distributions of independent variables of the model and
some key dimensions of the data.
More formally, calibration is the process of establishing the
relationship between a measuring device
and the units of measure. In other words, if you think about the
model as a measuring device
calibrating it means to parameterize it to deliver sensible
quantitative predictions.
Estimation is the process of picking the parameters of the model
to minimize a function of the
errors of the predictions of the model compared to some
pre-specified targets. It is the approximate
determination of the parameters of the model according to some
pre-specified metric of differences
between the model and the data to be explained.
It is generally considered a good practice to stick to the
following principles (see Prescott (1998)
and the discussion in Kydland and Prescott (1994)) when
constructing quantitative models:
1. Whenmodifying a standard model to address a question, the
modification continues to display
the key facts that the standard model was capturing.
2. The introduction of additional features in the model is
supported by other evidence for these
particular additional features.
3. The model is essentially a measurement instrument. Thus,
simply estimating the magnitude of
that instrument rather than calibrating the model can influence
the ability of the model to be used as
a measuring instrument. In addition the models selection (or in
particular, parametric specification)
has to depend on the specific question to be addressed, rather
than the answer we would like to
derive. For example, if the question is of the type, how much of
fact X can be accounted for by Y,
then choosing the parameter values in such a way as to make the
amount accounted for as large as
possible according to some metric makes no sense.
4. Researchers can challenge existing results by introducing new
quantitatively relevant features
in the model, that alter the predictions of the model in key
dimensions.
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1.3 International Trade: The Macro Facts
Chapter 2 of Eaton and Kortum (2011) manuscript
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Chapter 2
An introduction to modeling
2.1 The Heckscher-Ohlin model
The Heckscher-Ohlin (H-O) model of international trade is a
general equilibrium model that pre-
dicts that patterns of trade and production are based on the
relative factor endowments of trading
partners. It is a perfect competition model. In its benchmark
version it assumes two countries with
identical homothetic preferences and constant return to scale
technologies (identical across countries)
for two goods but different endowments for the two factors of
production. The models main pre-
diction is that countries will export the good that uses
intensively their relatively abundant factor
and import the good that does not. We will present a very simple
version of this model. Country is
representative consumers problem is
max a1 log ci1 + a2 log ci2
s.t. p1ci1 + p2ci2 ri ki + wi li
The production technologies of good in the two countries are
identical and given by
yi = zkib
li1b
, i, = 1, 2
and where 0 < b2 < b1 < 1. This implies that good 1 is
more capital intensive than good 2. Assume
for simplicity that k1/l1 > k2/l2. This implies that country
1 is capital abundant relative to country
2. Finally, goods, labor, and capital markets clear. One of the
common assumptions for the H-O
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model is that there is no factor intensity reversal which in our
example is always the case given the
Cobb-Douglas production function (one good is always more
capital intensive than the other, with
the capital intensity given by b).
2.1.1 Autarky equilibrium
Wefirst solve for the autarky equilibrium for country i. This is
easy especially if we consider the social
planner problem, but we will compute the competitive equilibrium
instead. The Inada conditions for
the consumers utility function imply that both goods will be
produced in equilibrium. Thus, we just
have to take FOC for the consumer and look at cost minimization
for the firm. For the consumer we
have
max a1 log ci1 + a2 log ci2
s.t. pi1ci1 + p
i2c
i2 ri ki + wi li
which implies
a1 = ipi1ci1 (2.1)
a2 = ipi2ci2 (2.2)
pi1ci1 + p
i2c
i2 = r
i ki + wi li (2.3)
This gives
pi2ci2 =
a2a1
pi1ci1 . (2.4)
The firms cost minimization problem
min riki + wili
s.t. yi zkib
li1b
implies the following equation, under the assumption that both
countries produce both goods,
b(1 b) l
iw
i = riki . (2.5)
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We can also use the goods market clearing to obtain
ci = zkib
li1b
=)
ci = zli
b
1 bwi
ri
b. (2.6)
Zero profits in equilibrium, pizkib li1b = riki + wili, combined
with (2.5), give us
pi =riki
bzrikiwi
(1b)b
b
(1b)wiri
b=
wi1b rib
z (1 b)1b (b)b
We can also derive the labor used in each sector. From the
consumers FOCs, together with the
expressions for pi and ci derived above, we obtain:
a = ipici =)
(1 b) ai= wili
this implies that
li1(1 b2) a2(1 b1) a1 = l
i2
We can use the labor market clearing condition and get
li2 + li1 = l
i =)
li1(1 b2) a2(1 b1) a1 + l
i1 = l
i =)
li1 =(1 b1) a1
(1 b2) a2 + (1 b1) a1 li . (2.7)
The results are similar for capital and thus,
ki1 =b1a1
b1a1 + b2a2ki ,
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This impliesli
ki=
ri
wi (1 b) a ba
(2.8)
Thus, in a labor abundant country capital is relatively more
expensive as we would expect. We can
finally use the goods market clearing conditions combined with
the optimal choices for li, ki to get
the values for cis as a function solely of parameters and
endowments,
ci = z
ba
0 b0a0kib (1 b) a
0 (1 b0) a0li1b
. (2.9)
.
2.1.2 Free Trade Equilibrium
In the two country example, free trade implies that the price of
each good is the same in both coun-
tries. Therefore, we will denote free trade prices without a
country superscript. In the two country
case it is important to distinguish among three conceptually
different cases: in the first case both
countries produce both goods, in the second case one country
produces both goods and the other
produces only one good, and in the last case each country
produces only one good.
We first define the free trade equilibrium. A free trade
equilibrium is a vector of allocations for
consumersci, i, = 1, 2
, allocations for the firm
ki, li, i, = 1, 2
, and prices
wi, r, p, i, = 1, 2
such that
1. Given prices consumers allocation maximizes her utility for i
= 1, 2
2. Given prices the allocations of the firms solve the cost
minimization problem in i = 1, 2,
bpzkib1
li1b ri , with equality if yi > 0
(1 b) pzkib
lib wi , with equality if yi > 0
3. Markets clear
ici =
iyi, = 1, 2
ki = ki for each i = 1, 2
li = li for each i = 1, 2 .
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2.1.3 No specialization
We analyze the three cases separately. First, lets think of the
case in which both countries produce
both goods.
max a1 log ci1 + a2 log ci2
s.t. p1ci1 + p2ci2 ri ki + wi li
a1 = ip1ci1 (2.10)
a2 = ip2ci2 (2.11)
p1ci1 + p2ci2 = r
i ki + wi li (2.12)
This implies again that
p2ci2 =a2a1
p1ci1 (2.13)
When both countries produce both goods the firms cost
minimization problem implies the fol-
lowing two equalities,
bpzkib1
li1b
= ri ,
(1 b) pzkib
lib
= wi ,
which in turn implybliwi
(1 b) ri = ki . (2.14)
Additionally, from zero profits,
p =rib wi1b
z (b)b (1 b)1b
(2.15)
and, of course, technologies (by assumption) and prices (due to
free trade) are the same in the two
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countries. Notice that the equality (2.15) is true for i = 1, 2
this implies that
r1b
w11b
=r2b w21b = 1, 2
r1
r2
b=
w2
w1
1b = 1, 2
Noticing that the above expression holds for = 1, 2 and
replacing these two equations in one
another we have
w2
w1
(1b2)b1b2
1+b1= 1 =)
w2 = w1
and of course
r2 = r1 .
This shows that we have factor price equalization (FPE) in the
free trade equilibrium.
From the cost minimization of the firm we have
bpizkib
li1b
= riki =)
bpyi = riki =)
pyi =rikib
.
Summing up over i and using FPE we have
p
iyi
!=
rb
iki
!. (2.16)
The equations (2.10) and (2.11) imply
a1+
a2= p
c1 + c
2
(2.17)
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Using goods market clearing, i ci = i yi, we have
i
ai
= pc1 + c
2
= p
iyi =
rbiki =)
bai
1i
= riki =)
i
1i
!
ba = riki =)
i
1i
=rk1 + k2
ba
(2.18)
and in a similar manner
i
1i=
wl1 + l2
(1 b) a
. (2.19)
Using (2.18) and (2.19) we can determine the w/r ratio
l1 + l2
k1 + k2=
rw (1 b) a ba
. (2.20)
Assuming that one country is more capital abundant than the
other (say k1/l1 > k2/l2), the equi-
librium factor price ratio r/w under free trade lies in between
the autarky factor prices of the two
countries (determined in equation 2.8).
Using the relationships for the capital labor ratio (2.14)
together with the above expression and
factor market clearing conditions we can derive the equilibrium
labor used from each country in each
sector. Using the capital labor ratios for the second good and
for both countries we get:
wr
li li2
b1(1 b1) +
li2 b2(1 b2)
= ki
li2li
=(1 b2) (1 b1)
b2 b1
rwki
li b1(1 b1)
li2li
=(1 b2) (1 b1)
b1 b2
b1(1 b1)
ba (1 b) a
l1 + l2
k1 + k2ki
li
You may notice two things in this expression. First, if initial
endowments of the two countries are
inside a relative range, there is diversification since lij >
0. If the endowments of a country for a given
good are not in this range, then a country specializes in the
other good (this range of endowments
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that implies diversification in production is commonly referred
to as the cone of diversification).
Second, conditional on diversification labor abundant countries
use relatively more labor in the labor
intensive sector.
What is the share of consumption for each country? We can use
the FOC from the consumers
problem to obtain
p1ci1
1+
a2a1
= rki + wli =)
ci1 =rki + wli
p11+ a2a1
=)ci1 =
1(1 b)
wli
p11+ a2a1
(2.21)where in the last equivalence we used equation (2.14).
Obtaining the rest of the allocations and prices
is straightforward. In fact, you can show that if the production
function exhibits CRS and the capital-
labor ratio for both countries is fixed (in a given sector),
total production can be represented by1
y = z
iki
!b ili
!1b.
We can determine i ki, i li by combining expression (2.18) with
(2.16), (2.17) and using the market
clearing condition. This givesba
ba=i kii ki
, (2.22)
1Assume that k1/l1 = k2/l2. We only have to prove that given
this assumption
Ak1 + k2
b l1 + l2
1b= A
k1b
l11b
+ Ak2b
l21b
.
Using repeatedly the condition we have that
A
k1 + k2
l1 + l2
!b= A
k1
l1
!bl1
l1 + l2+ A
k2
l2
b l2l1 + l2
() k1 + k2
l1 + l2
!b=
k1
l1
!b()
k2l1/l2 + k2
l1 + l2
!=
k1
l1
!()
k2
l2=
k1
l1,
which holds by assumption completing the proof.
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and similarly for labor.
2.1.4 Specialization
[HW]
2.1.5 The 4 big theorems.
In this final section for the H-Omodel we will state main
theorems that hold in the benchmark model
with two countries and two goods. Variants of these theorems
hold under less or more restrictive
assumptions. Our approach will still be as parsimonious as
possible.2
Theorem 1 (Factor Price Equalization) Assume countries engage in
free trade, there is no specialization
(thus there is diversification) in equilibrium and there is no
factor intensity-reversal, then factor prices equalize
across countries.
Proof. See main text
Theorem 2 (Rybczynski Theorem) Assume that the economies remain
always incompletely specialized.
An increase in the relative endowment of a factor will increase
the ratio of production of the good that uses
the factor intensively.3
Theorem 3 (Stolper-Samuelson) Assume that the economies remain
always incompletely specialized. An
increase in the relative price of a good increases the real
return to the factor used intensively in the production
of that good and reduces the real return to the other
factor.
Theorem 4 (Heckscher-Ohlin) Each country will produce the good
which uses its abundant factor of pro-
duction more intensively.
2.2 Armington model
The Armington (1969) model is based on the (ad-hoc) assumption
that consumers have a certaindesire for foreign goods.
2For a detailed treament you can look at the books of Feenstra
(2003) and Bhagwati, Panagariya, and Srinivasan (1998).3If prices
were fixed a stronger version of the theorem can be proved.
12
-
Though primitive, it can generate relationships that are key to
understanding the next genera-tion of models. The main
relationships that arise in the model appear in many of the models
of
the next generation. This turns out to be great given that
empirically the gravity approach had
some success (gravity is a good statistical representation of
trade flows).
The gravity equation that we will get will serve as a great
benchmark for thinking about bilat-eral trade flows.
2.2.1 The model
Consider N countries denoted as i when a country is importing
and j when a country is ex-porting. We assume an endowment economy
where each consumer is endowed with 1 unit of
endowment and there is a measure of Li consumers in country
i.
Supply: Markets are competitive and the price for each unit of
the good produced iswi. In orderto ship the good to a different
country an iceberg transportation cost ij has to be incurred -
ij
units of the good have to be produced in country i so that 1
unit arrives at the final destination
j. Therefore, the final at-the-dock price of the good in
destination j is pij = wiij and total
output is Li = Nk=1 ikxik., where xik is the total quantity
demanded by country j, which we
characterize next.
Demand: The representative consumer in country j is assumed to
have the symmetric utilityfunction
Uj =
"N
k=1
1/kj x(1)/kj
#/(1)where kj is an exogenous preference parameter and 2
(0,+).
To derive consumers demand we look at the first order conditions
of the consumer in countryj for goods originating from countries k,
i, which imply
1/ij x(1)/1ij
1/kj x(1)/1kj
=pijpkj
.
Raising the last expression to the power of , rearranging kj,
ijs and summing over all
13
-
origins k = 1, ..,N.
pijpkj
xijxkj
=ij
kj
pijpkj
1=)
kkj
p1kj
=
1pijxij
ijpij1
kpkjxkj .
Rearranging we obtain the Marshallian demand for the
consumer
xij =
pij
Pj1 aij
kpkjxkj
where we define the Dixit-Stiglitz price index
Pj =
"N
k=1
kjpkj1#1/(1) . (2.23)
Given the optimal choices for xij, given prices, we can also
define the consumption aggregator
Cj =
"N
k=1
1/kj x(1)/kj
#/(1).
For the case of the CES utility function we can easily show that
CjPj equals total spending
(simply replace the Marshallian demand function in the
expression k pkjxkj) and thus CjPj =
k pkjxkj = Xj. Using this relationship we can write bilateral
sales in country j are given by
pijxij = ij
pijPj
1Xj. (2.24)
Thus, a change in the price of a good sold in country j changes
total sales in j with an elasticity
of 1 (ignoring the General Equilibrium effect in Pj).
14
-
2.2.2 Gravity
Now we will derive an expression that is closer to the gravity
equation. First, notice that the trade
share of country i in country j is given by
ij =xijpij
k xkjpkj
=aijw1i
1ij
k akjw1k
1kj
(2.25)
where we have replaced for equation (2.24).
From the budget constraint of the consumer and trade balance we
also have
Xi = wiLi = wijijxij =
jpijxij.
Let pij = piiij and sum across j0s.
Xi =jpijxij = w1i
kik
ikPk
1Xk =) (2.26)
w1i =Xi
k ikikPk
1Xk
(2.27)
Replacing in 2.24 we get
Xij = pijxij = ij
pijPj
1Xj =)
Xij = ijw1i
ij
Pj
1Xj
= XiXjij
k ikikPk
1Xk
ij
Pj
1(2.28)
which shows that the bilateral trade spending is related to the
product of the GDPs of the two coun-
tries (gravity!!), the distance/tradecost and a GE
component.
The price index Pj is a measure of bilateral trade resistance.
If a country is very isolated from the
rest of the world, the domestic prices will be high on average
and, thus, the domestic price index will
15
-
also be high. Given that large countries trade a lot with
themselves the changes in trade costs affect
their price index only by a little. The opposite is true for
small countries. Additionally, jijPj
1Xj
is a measure of how many selling opportunities i has
overall.
2.2.3 Welfare
We will now show that welfare in relationship to trade is given
by a simple relationship involving
the trade to GDP ratio and parameters of the model (but no other
equilibrium variables). We will be
revisiting this relationship multiple times in this course.
Using expression (2.25) we have
ij =aijp1ijP1j
.
Thus
Pj =
ajjw1jjj
!1/(1)Since pij = ijwi then welfare for the representative
consumer can be written as
Wj =wjPj=
wj
a1/(1)jj wj1/(1)jj = a
1/(1)jj
1/(1)jj .
Notice that the welfare depends only on changes in the trade to
GDP ratio, jj, with an elasticity of
1/ ( 1)which is the inverse of the trade elasticity (see
expression 2.25 the coefficient on the tradecosts)
2.3 Monopolistic Competition with Homogeneous Firms and CES
de-
mand
The formulation that we will develop in this section is based on
the monopolistic competition frame-
work of Krugman (1980) and the subsequent analysis of Arkolakis,
Demidova, Klenow, and Rodrguez-
Clare (2008). We assume that there are i = 1, ...,N countries
and there is a measure Li of consumers
in each country i. Let 2 be a potentially differentiated
variety, where is the set of all poten-tial varieties. The
representative consumer in each country has symmetric CES
preferences over all
16
-
goods 2 .4 In order for a firm from country i to produce a
differentiated variety it has to incura fixed cost of entry in
terms of domestic labor, f ei . We define as Xj the total spending
of country i.
Since labor is the only factor of production and in equilibrium
all profits would be accrued to labor
the total spending equals labor income, Xj = wjLj, where wj is
the wage and Lj the labor force.
2.3.1 Consumers problem
The problem of the representative consumer from country j is
max
N
i=1
Zi
xj ()1 d
! 1
,
s.t.N
i=1
Zi
pj () xj () d = wj,
where xij () is the quantity demanded of good , pij () is the
price of that good, and > 1 is the
elasticity of substitution. The consumer inelastically supplies
his unit of labor endowment which is
an assumption we maintain throughout the rest of this note
The above implies the following CES demand for the consumers in
country j5
xij () =pij ()Pj
wjLjPj
,
Pj =
N
i=1
Zi
pj ()1 d
! 11
.
2.3.2 Firms problem
Each firm is a monopolist of a variety. All firms in country i
have a common productivity, zi, and
produce one unit of the good using 1zi units of labor. Firms
have to pay a fixed cost of production
in terms of domestic labor, fi. They also incur an iceberg
transportation cost, ij, to ship the good
from country i to country j.6 Profit maximization implies that
optimal pricing for a firm selling from
4We make the standard assumption that if a good is not consumed
pj () = + so that xj () = 0.5The steps for derivation of the demand
function are the same as in the Armington case in section 2.2. The
only differ-
ence in the derivation is the use of integration instead of
summation.6Notice that here we havent introduced fixed costs of
exporting. Introducing these costs will change the analysis in
that we may have countries for which all the firms chose not to
export depending on values of the fixed costs and othervariables.
More extreme predictions can be delivered if the production cost fi
is only a cost to produce domestically andindependent of the
exporting cost. However, in order to create a true extensive margin
of firms (i.e. more firms exporting
17
-
country i to country j is
pij (zi) =
1ijwizi
.
We will make the notation a bit cumbersome by carrying around
the zis in order to allow for direct
comparison of our results with the heterogeneous firms example
that will be studied later on. Given
the Dixit-Stiglitz demand, the variable profit of the firm from
operating in country j (revenues in
country j minus related labor cost of production) is simply
revenues divided by the elasticity of
substitution ,
piVij (zi) =pij (zi)
1
P1j
wjLj
.
Based on variable profits, the firmwill have to decide whether
paying the fixed entry cost is profitable
to which we turn next.
2.3.3 Equilibrium
In order to determine the equilibrium of the model, we have to
consider the free entry condition and
the labor market clearing condition. The free entry condition
implies that in a given country entry
occurs until the point where expected profits are equal to zero
for the firms of that country: the sum
of the revenue in each destination minus its total labor cost of
production minus the fixed cost of
entry equals to zero for each firm. This implies that
j
pij (zi)1
P1j
wjLj
wi f ei = 0 =)
j
1ijwizi
1P1j
wjLj
= wi f ei =)
1 1
wizixi = wi f ei =)
xi = f ei zi ( 1) ,
where by slightly abusing the notation we define xi jij
1ijwizi
P1j
wjLj.
The labor market clearing condition implies that total labor
used for production and the entry
cost must equal to the total labor endowment. Using the above
definition for xi we have that
when trade costs decrease) requires heterogeneity either in the
productivities of firms (as we will do later on in the notes)or in
the fixed costs of selling to a market (see Romer (1994)).
18
-
Ni
264 Nk=1
1ikwizi
P1k
wkLkikzi+ f ei
375 = Li =)Ni
xizi+ f ei
= Li =)
Ni [ f ei ( 1) + f ei ] = Li =)
Ni =Li f ei
, (2.29)
where Ni is the measure of entrants, i.e. the firms that incur
the fixed entry cost. Notice that in this
case the equilibrium measure of entrants is independent of
variable trade costs.
2.3.4 Gravity
Now we compute the fraction of total income in country j spent
on goods from country i, ij,
ij =
Ni
1
ijwizi
1P1j
wjLj
Nk=1
Nk
1
kjwkzk
1P1j
wjLj
=Niijwizi
1Ni=1
Nkkjwkzk
1and using equation (2.29), we have7
ij =
Lif ei
ijwi
1(zi)
1
Nk=1
Lkf ek
kjwk
1(zk)
1. (2.30)
In order to compute wages across countries notice that trade
balance implies that total income equals
to sales in all destinations so that
wiLi =N
k=1
ikLkwk.
7Given that there is free entry of firms and payments for entry
costs are accrued to labor, total labor income wiLi is thetotal
income in each country i.
19
-
2.3.5 Welfare
Denote bilateral sales of country i to country j as Xij. Related
to the above, we also have
ij =XijXj=
=
Ni
1
ijwizi
1P1j
wjLj
wjLj,
which implies that
Pj = (Ni)1
1
1ijwizi
ij 1
1 .
Looking at the domestic market share of j, we have
Pj =Nj 1
1
1wjzj
jj 1
1 . (2.31)
Finally, welfare is given by
wjPj=
1
zj
Nj 1
1jj 1
1,
with Nj given by equation (2.29).
If we interpret the effect of trade liberalization as a change
in trade to GDP ratio, jj, then the
change in welfare before and after a trade liberalization is
given by
w0jP0jwjPj
=
0jjjj
! 11. (2.32)
2.4 Monopolistic Competitionwith Separable Utility Function
(Krugman
79)
We now retain the monopolistic competition structure and all the
notation from the previous section
but consider a general symmetric separable utility function as
in Krugman (1979).
20
-
2.4.1 Consumers problem
The problem of the representative consumer from country j is
max
N
i=1
Zi
uxj ()
d
!,
s.t.N
i=1
Zi
pj () xj () d = wj,
with u0 > 0 and u00 < 0. We assume a particular regularity
condition on the utility function that will
allow us to focus on the empirically relevant cases, and in
particular
ln u0 (x)
ln x= xj () u
00 xj ()u0xj ()
> 0 . (2.33)This condition implies
u0xj ()
= jpj ()
where j is the Lagrange multiplier of the consumer in country j.
The demand function implied by
this solution is given by
xjpj ()
= u01
jpj ()
(2.34)
We focus on demand functions that have the choke price property,
i.e. there exists a pj so that given
j xjpj= 0 for all p pj . It is straightforward to show that this
property requires u0 (0) < +
and that pj = u0 (0) /j and thus xj
pj, pj
= xj
pj()pj
= u01
u0 (0) pj()pj
.
2.4.2 Firms Problem
We can now incorporate this demand as a constraint to the firms
problem. In particular, a firm from
country i with productivity z = zi choses price in country j to
maximize
piij (z) =p () ijwiz
u01
jp ()
(2.35)
21
-
The first order condition of this problem is given by (making
use of the inverse function theorem)
u01jpj ()
+p () ijwiz
u01
jp ()
0j = 0 =)
p =1
1+xj()u00(xj())
u0(xj())
ijwiz
,
and the markup can be written as
pj ()pj
!=
1
1+x
pj()
pj
u00xj
pj()
pj
u0xj
pj()
pj
Notice that assumption (2.33) implies that firm markup is
increasing with firm sales. Now define
zij wiij/pj then we can finally express
xj = x
zzij
!
and
j =
zzij
!
In this environment we can write bilateral trade shares as
ij =
Ni
zizij
ijwizi
q
zizij
k Nk
zkzkj
kjwkzk
q
zkzkj
=
Niijwizi
zizij
q
zizij
k Nk
kjwkzk
zkzkj
q
zkzkj
Define vij as z/zij
ij =Niwiij e
(vij)zij
qvij vij
k Nkwkkj
e(vkj)zkj
qvkj vkj
Welfare
22
-
TBD
2.5 Ricardian model
The Ricardianmodel is amodel of perfect competitionwhere
countries produce the same goods using
different technologies. The Ricardian model predicts that
countries may specialize in the production
of certain ranges of goods.
2.5.1 The two goods case
We consider the simple version of the model with two countries
and two goods. In order to get
as much intuition as possible we will first consider the case
where both countries specialize in the
production of one good.
The production technologies in the two countries i = 1, 2 are
different for the two goods = 1, 2
and given by
yi () = zi () li () , i, = 1, 2 .
Assume that country 1 has absolute advantage in the production
of both goods
z2 (1) < z1 (1) ,
z2 (2) < z1 (2) .
Assume that country 1 has comparative advantage in the
production of good 1 and country 2 in good
2z1 (2)z2 (2)
, but if HF = FH then = . A simple
condition that determines which are the goods that will be
produced by country F dictates that the
price of these goods in country F has to be lower than the price
of imported goods, i.e.
wFzF ()
1 governs variation in productivity
across different goods (comparative advantage) (higher less
dispersed).
Now we will split the [0, 1] interval by thinking of as the
probability that the relative produc-
tivity of F to H is less than A, where A can either be defined
by (3.2). Therefore, in order to determine
which is defined as the share of goods that the domestic country
produces we simply compute
the probability that the domestic country is the cheapest
provider of the good across all the range of
2See the appendix for the properties of the Frechet distribution
and the next chapter for a derivation fromfirst principles.
28
-
productivities. For example using (3.2) for the definition of A
we can derive
= HH
= PrzFzH A
= Pr
zF AzH
=
Z +0
exphAF
Ai| {z }
Pr(z()AzH())
dFH (z)| {z }density of zH()
=Z +0
exphAF
Azi
AH (z)1 exp
hAH (z)
idz
=AH
AH + AF A
Country H is spending (1 )wHLH on imports (given Cobb-Douglas)
which implies
XFH =AF (wFHF)
AH + AF (wFHF)wHLH
Notice that this relationship is similar to the relationship
(2.25) derived with the assumption of
the Armington aggregator but with an exponent . A lower value of
generates more heterogene-ity. This means that the comparative
advantage exerts a stronger force for trade against resistance
impose by the geographic barrier in. In other words with low
there are many outliers that over-
come differences in geographic barriers (and prices overall) so
that changes in ws and s are not so
important for determining trade.
3.2 A theory of technology starting from first principles
We start with a very general framework under the following
assumptions
Time is continuous and there is a continuum of goods with
measure ().
Ideas for good (ways to produce the same good with different
efficiency) arrive at location iat date t at a Poisson rate with
intensity
aRi (, t)
29
-
where we think of a as research productivity and R as research
effort.
The quality of ideas is a realization from a random variable Q
drawn independently from aPareto distribution with > 1, so
that
Pr [Q > q] =q/q
, q q
where qis a lower bound of productivities. Note that the
probability of an idea being bigger
than q conditional on ideas being bigger than a threshold, is
also Pareto (see appendix for the
properties of the Pareto distribution).
The above assumptions together imply that the arrival rate of an
idea of efficiency Q q is
aRi (, t)q/q
.
(normalize this with q! 0, a! + such that aq
! 1 in order to consider all the ideas in (0,+)).
Important assumption: There is no forgetting of ideas. Thus, we
can summarize the history of
ideas for good by
Ai (, t) =Z t
Ri (, ) d.
The number of ideaswith efficiencyQ > q0 is therefore
distributed Poissonwith a parameter A (, t) (q0)
(using the previous normalization).
The unit cost for a location i of producing good with an
efficiency of q is c = wi/q. Given all
the above, the expected number of techniques providing unit cost
less than c is distributed Poisson
with parameter
i (, t) c
where
i (, t) = Ai (, t)wi .
30
-
But notice that this delivers back unit costs that are
conditionally Pareto distributed
PrC c0jC c = Pr hQ w
c0= q0jQ w
c= q
i=
A (, t) (q0)
A (, t) (q)=
qq0
=
c0
c
.
In what follows set
= i (, t) .
Definition 6 C(k) is the kth lowest unit cost technology for
producing a particular good. Given this definition
we have the main theorem for the joint distribution of the order
statistics C(k)
Theorem 7 The joint density C(k),C(k+1) is
gC(k) = ck,C(k+1) = ck+1
gk,k+1 (ck, ck+1)
=2
(k 1)!k+1ck1k c
1k+1 exp
ck+1
for 0 < ck ck+1 < while the marginal density of C(k)
is:
gk (ck) =
(k 1)!kck1k exp
ck
for 0 < ck < +
Proof. We start by looking at costs C c. The distribution of a
cost C conditional on C c is:
F (cjc) = cc
c c
F (cjc) = 1 c > c
The probability that a cost is less than ck is F (ckjc). Thus,
if we have n techniques with unit cost lessthan c, where ck ck+1 c,
the probability that k are less than ck while the remaining are
greater
31
-
than ck+1 is given by the multinomial:
PrhC(k) ck,C(k+1) ck+1jn
i=
0B@ nk
1CA F (ckjc)k (1 F (ck+1jc))nk
Taking the negative of the cross derivative of this expression
wrt to ck, ck+1 gives
gk,k+1 (ck, ck+1jc, n) = n!F (ckjc)k1 [1 F (ck+1jc)]nk1 F0
(ckjc) F0 (ck+1jc)
(k 1)! (n k 1)!
for ck+1 ck and n k+ 1. For n < k+ 1 we can define gk,k+1
(ck, ck+1jc, n) = 0. We also know thatn is drawn from a Poisson
distribution with parameter c , the expectation of this joint
distribution
unconditional on n is:
gk,k+1 (ck, ck+1jc) =
n=0
expc cn
n!| {z }prob n ideas arrived for
a particular good
gk,k+1 (ck, ck+1jc, n)| {z }conditional on n prob C(k)=ck ,
C(k+1)=ck+1
=
=
n=k+1
expc cn
n!n!F (ckjc)k1 [1 F (ck+1jc)]nk1 F0 (ckjc) F0 (ck+1jc)
(k 1)! (n k 1)!
=
c
k+1 exp cF (ck+1jc) F (ckjc)k1 F0 (ckjc) F0 (ck+1jc)(k 1)!
m=0
expc
c
m exp cF (ck+1jc) [1 F (ck+jc)]mm!
=
c
k+1 exp cF (ck+1jc) F (ckjc)k1 F0 (ckjc) F0 (ck+1jc)(k 1)! 1
Substituting using the expression F (cjc) we have that
gk,k+1 (ck, ck+1jc) = 2
(k 1)!k+1ck1k c
1k+1 exp
ck+1
Now by letting c! we can integrate for the entire range of c ck
and derive the marginal densityby making use of the above
expression. We have that
gk(ck) =Z ck
gk,k+1(ck, ck+1)dck+1
=2
(k 1)!k+1ck1k
Z ck
c1k+1eck+1dck+1.
32
-
Now by making the substitution u = ck+1,
Z ck
c1k+1eck+1dck+1 = 1
Z ck
eudu
= 11eck .
Therefore
gk(ck) =2
(k 1)!k+1ck1k
11ec
k
=
(k 1)!kck1k e
ck , (3.3)
as asserted.
This result will be the base for a series of lemmas to be
discussed later on. First, by noticing that
F0k (ck) = gk(ck) we can directly compute the probability
PrhC(k) ck
i:
Lemma 8 The distribution of the k0th lowest cost C(k) is:
PrhC(k) ck
i= Fk (ck) = 1
k1=0
ck
!
ec (3.4)
This Lemma implies that the distribution of the lowest cost (k =
1) is the Frechet distribution
F1 (c1) = 1 expc1
Now in this context we will assume that ideas are randomly
assigned to goods across the contin-
uum. Given that there is a large number of goods (say of measure
()) in the continuum we can
drop the notation by simply denoting Ai (, t) = Ai (t) / () to
be the average number of ideas
available for a good, in location i at time t. Given the above,
the measure of goods with cost less than
c is i (t) / () c and the distribution of the lowest cost C(1)
(the frontier idea) is
F1 (c1) = 1 exp (i (t) / ()) c1
Thus a set of () F1 (c1) ideas can be produced at a cost less
than c1. We will proceed under this
convention in the rest of this chapter.
33
-
Using the general technology framework we developed above and
different assumptions on the
competition structure we will be able to derive main
quantitative models that are widely used in the
recent international trade literature.
3.3 Application I: Perfect competition (Eaton-Kortum)
Assume perfect competition
Assume that there is a unit continuum of available goods and CES
preferences. The utilityfunction of the representative consumer
is
Uj =Z 1
0xj ()
(1)//(1)
where the elasticity of substitution is > 0. If the spending
in country j is Xj and the demand
function for good is
xj () =
pj ()
Pj1 Xj.
Assume iceberg transportation costs of trade
In perfect competition only the lowest cost producer of a good
will supply that particular good.
Thus, we want to derive the distribution of the minimum price
over a set of prices offered by
producers in different countries
pj = minp1j, ..., pNj
In order to find this distribution we take advantage of the
properties of extreme value distributions.
Everything turns out to work beautifully! Let ci be the unit
cost and labor be the only factor of
production. The probability that country i can sell in j with a
price less than p is simply given by
Gij (p) = 1 expijp
34
-
where ij = Aiciij
.3 Taking in account all the source countries (including the
home country)we have that the distribution of realized prices in
country j is given by
Grj (p) = PrPj p
= 1 Pr Pj p= 1
N
i=1
PrPij p
= 1
N
i=1
1 Gij (p)
= 1 ejp
where j Ni=1ij.Given the above technology framework and the
assumption of perfect competition we have that
the market share of country i to country j (which is the same as
the probability of supplying a good
given that the distribution of prices is source independent)
is:4
ij =Z +0
s 6=i1 Gsj (q)
dGij (q) =
Aiciij
N=1 A
cj
= ijj (3.5)Using the following proposition and the assumption of
the CES demand we can directly derive
the price index
3Starting from the Frechet disribution of productivities we can
derive the distribution of prices offered by country i ton :
Gij (p) = Prpij p
= Pr
ciZi
ij p
= Prcipij Zi
= 1 Fi
cipij
= 1 exp
Ti
cipij
!= 1 exp
ijp
where ij = iciij
.
4We can show that the distribution of realized prices of goods
that country j actually buys from i is independent of i.This result
can be obtained by showing that
Grij (p) =
R p0
k 6=i
h1 Gkj (q)
idGij (q)
ij.
This also implies that the fraction of goods for coutnry i to
country n is the same as the fraction of sales of country i
tocountry n.
35
-
Proposition 9 For each order k, the bth moment (b > k) is
E
C(k)b
=1/
b [(k+ b) /](k 1)! ,
where () =R +0 y
1eydy.
Proof. First consider k = 1, where suppressing notation we
denote by the marginal density of C(k),
gk (c) =
(k 1)!kck1k exp
hck
i
E
C(1)b
=Z +0
cbg1 (c) dc
=Z +0
c+b1 exphc
idc
changing the variable of integration to = c and applying the
definition of the gamma function,
we get
E
C(1)b
=Z +0
(/)b/ exp [] d
= ()b/ + b
well defined for + b > 0. For general k we have
E
C(k)b
=Z +0
cbgk (c) dc
=Z +0
cb
(k 1)!kck1 exp
hc
idc
=k1
(k 1)!Z +0
cb+kc1 exphc
idc
=k1
(k 1)!E
C(1)b+(k1)
(3.6)
36
-
The above proposition shows that the price index for a country i
under perfect competition is
Pi =E
C(k)11/(1)
= PC1/i
where
PC = + 1
1/(1)and is the gamma function and > 1.
Finally, using a methodology similar to the one outlined above
for the Krugman model we can
compute the welfare index which is given by
wjPj=
Aj1/
PCjj 1 . (3.7)
3.3.1 Key contributions of EK
Models heterogeneous sectors in a Ricardian GE model of
trade.
Estimates of using alternative empirical specifications.
Estimation of gravity models will be atopic of section 9.
Implement a model for counterfactual experiments.
Most importantly: Laid the foundation for the quantitative
analysis of international trade whenheterogeneity plays an
important role. It was followed by models such as those of
Bernard,
Eaton, Jensen, and Kortum (2003) and Melitz (2003) that model
other forms of competition
(Bertrand and monopolistic, respectively) and also work that
brings models of trade closer to
the trade data in many dimensions.
3.4 Application II: Bertrand competition (Bernand, Jensen &
EK)
Consider the case where different producers have access to
different technologies. If we assumeBertrand competition, the cost
distributionwill be given by the frontier producer (k = 1 in the
expression 3.4)
but prices are related to the distribution of the second lowest
cost (k = 2). Since the lowest cost
37
-
supplier is the one that will sell the good, the probability
that a good is supplied from i to j is
Aiwiij
Nk=1 Ak
wkkj
. (3.8)The price of a good in market j is:
pj () = minC2j () , mC1j ()
where we will define Cij () to be the cost of the ith minimum
cost producer of good in
country j, and m = / ( 1) is the optimal markup that a
monopolist firm would charge (as-suming CES preferences with a
demand elasticity ). Given heterogeneity among technological
costs for firms we will derive the distribution of costs and
markups in each given country.
Efficiency, markups, and measured productivity
Define again C(k) as the kth lowest unit cost technology for
producing a particular good. We have
the following Lemma
Lemma 10 The distribution of C(k+1) conditional on C(k) = ck
is:
PrhC(k+1) ck+1jC(k) = ck
i= 1 exp
h
ck+1 ck
i, ck+1 ck 0
Proof. Using Bayes rule we have
PrhC(k+1) ck+1jC(k) = ck
i=
Z ck+1ck
gk,k+1 (ck, c)gk (ck)
dc
=Z ck+1ck
c1 exphc +ck
idc
= 1 exph
ck+1 ck
i.
Thus, the distribution of C(2) is stochastically increasing in
c1 and hence decreasing in z1 = w/c1.
Thus, high productivity producers are more likely to charge a
lower price. The above relationship
38
-
also implies that (define m0 = ck+1/ck)
Pr
"C(k+1)
ck ck+1
ckjC(k) = ck
#= Pr
"C(k+1)
C(k) m0jC(k) = ck
#= 1 exp
hck
m0 1i
The distribution of the ratio M0 = C(2)/C(1) given C(1) = c1
is:
PrhM0 m0jC(1) = c1
i= 1 exp
hc1
m0 1i .
We have that the lower c1, the more likely a high markup. Thus,
in this context low-cost producers
are more likely to charge a high markup and their measured
(revenue) productivity is more likely to
appear as higher. In this model, revenue productivity is
associated one-to-one with the markup that
the firm charges, since it equals
M ()wiijq () /z ()
| {z }revenue
/ l ()| {z }labor used
=M ()wiijq () /z ()
/ [q () /z ()] = M ()wiij.
Notice that from this expression is straightforward that market
structures/demand functions which
imply costant markup imply no revenue productivity variation
across firms.
Notice that the markup with Bertrand competition that BEJK
consider is
M () = min
(C(2) ()C(1) ()
, m
)(3.9)
We start by characterizing the distribution of the ratio M0 =
C(2)/C(1). Conditional on C(2)i = c2, we
have
PrhM0 m0jC(2)i = c2
i= Pr
hc2/m0 C(1)i c2jC(2)i = c2
i=
R c2c2/m0 gi (c1, c2) dc1R c20 gi (c1, c2) dc1
=c2 (c2/m0)
c2= 1 m0 .
39
-
This derivation implies that the distribution of this M0 does
not depend on c2 and is also Pareto.5
Thus, the unconditional distribution is also Pareto.6 Given the
markup function for the case of Be-
trand competition, equation (3.9), we have proved the following
proposition:
Proposition 11 Under Bertrand competition the distribution of
the markup M is:
Pr [M m] = FM (m) =
8>: 1m if m < m
1 if m m.
With probability m the markup is m. The distribution of the
markup is independent of C(2).
Gravity and Welfare
Lengthy derivations can be used (see the online appendix of
BEJK) to show that the joint distribu-
tion of the lowest and the second lowest cost of supplying a
country, conditional on a certain country
being the supplier, is the same, and independent of the country
of origin. Therefore, the market share
of country i in j equals to the probability that country i is
the supplier and thus
ij =Aiwiij
Nk=1 Ak
wkkj
= ijj5Using again the results of the theorem we can derive the
distribution of m for each k. Notice that
PrhC(k) ckjC(k+1) = ck+1
i=
Z ck0
gk,k+1 (c, ck+1) /gk+1 (ck+1) dc
=Z ck0
2
(k1)!k+1ck1c1k+1 exp
ck+1
(k)!
k+1c(k+1)1k+1 eck+1
dc
=
ckck+1
k(3.10)
and simply replacing for ck =ck+1m in expression (3.10) (and
given that C
(k+1) = ck+1) we can get
Pr
"C(k+1)
C(k) mjC(k+1) = ck+1
#= 1 Pr
hC(k) ck+1
mjC(k+1) = ck+1
i= 1mk
6An alternative derivation of the distribution of the markups
can be obtained for m < m. To compute the
unconditionaldistribution of productivities for m < m we have
that
Pr [M m] =Z +0
1 exp
hc1
m 1
ic11 exp
c1
dc1
= expc1
exphc1m
im
+
0
= 1 1/m
40
-
Using the derivations for the distribution of markups, we can
also derive a price index for this case.
We have
P1i =Z 1
Ehp1i jM = m
im1dm
=Z m1
E
C(2)i1
m(+1)dm+Z +m
E
mC(2)i /m1
m(+1)dm
= E
C(2)i1
1 m+ m
1+
where in the second equality we used the fact that the
distribution of markups is independent of the
second lowest cost. We have already calculated E
C(2)i1
in equation (3.6). Thus, the price index
under Bertrand competition is
Pi = BC1/i
BC =
1 m
+ m
1+ 1/(1)
2 + 1
1/(1)and given the gravity expression the welfare as a function
of trade is the same as in the case of Perfect
competition
3.4.1 Key contributions of BEJK
Develop a firm level model and explicitly tests its predictions
with firm-level data.
Model a framework where firms markups are variable and depending
on competition. Alter-native models of variable markups can be
developed in monopolistic competition by allowing
for a preference structure that departs from the CES aggregator
(see section 5.4).
Develop a methodology of simulating an artificial economywith
heterogeneous firms and find-ing the parameters of this economy
that brings the predictions of the model closer to the data.
Acknowledges the fact that measured productivity when measured
as nominal output overemployment is constant inmodels with constant
markups. It developes amodel that can deliver
variable markups, thus measured productivity differentials.
41
-
3.5 Application III: Monopolistic competition with CES
(Chaney-Melitz)
3.5.1 Consumer
Utility functionUj =
Zxj ()
(1)/ d/(1)
where the measure of is () and xj() is a quantity of a variety
available to consumers in
j.
Consumers derive income from labor the ownership of domestic
firms.
() [0,+) is the set of available varieties. Let Ii the measure
of ideas that fall randomlyinto goods. In some sense Ii/ () ideas
correspond to each good.
In the probabilistic context we described above, the
monopolistic competition model arises in avery natural way. Let the
distribution of the lowest cost for a good to be Frechet such
that
F1 (c1) = 1 exp Ii ()
c1
.
The measure of firms with unit cost less than C(1) c1, is () F1
(c1) . Taking the limit ofthis expression for the number of
potential varieties () ! + we can show that the dis-tribution of
the best producers cost of a variety is Pareto. The Pareto
distribution in terms of
productivities is defined as Pr (Z z) = 1 Aiz , where z 2 [A1/i
,+). More details are givenin appendix (10.1).
In particular, given the fact that the CES preferences always
imply an interior solution, we needto introduce a marketing cost of
selling to a market that is sizable compared to the marginal
profit from entering. Melitz uses a simple uniform fixed
marketing cost.
3.5.2 Firms Problem
Firms decide to enter each market by maximizing their profit
piij (z) = maxpp1
P1jXj
ij
zwi
p
P1jXj wj fij
42
-
where Xj is the expenditure inmarket j and fixedmarketing costs
are paid in terms of labor in country
j and cij = ijw/z. Prices are
pij (z) =
1ij
zwi
which implies that variable profits are
1ijz wi
1P1j
Xj
and thus only the firms that have
z zij
where zij1
=wj fijP1j
1ijwi1 Xj (3.11)
will enter. Replacing this definition, sales can be rewritten in
the simple form:
pij (z) qij (z) = wj fij
zzij
!1.
3.5.3 Gravity-Aggregation
Given the Pareto distribution of productivities, we can compute
trade flows from a country to another
by performing the simple integration
Xij = Ni PrZ zij
| {z }# exporters from i to j
ZzijwjFj
zzij
!1 (
Ai)
z+1
PrZ zij
dz| {z }
av. sales per exporter
= Ni
Aizij
!
+ 1wj fij ,
with > 1. Derivations reveal that the share of profits in
this model is a fixed proportion of totalincome = ( 1) / (). Using
that result, the measure of entrants can be written as a function
oftrade flows,
Mij =Xijwj fij11/
,
43
-
where
= / ( 1) .
Finally, trade shares are given by
ij =Aiij Niwi f 1/(1)ij
k Akkj Nk (wk) f 1/(1)kj (3.12)
3.5.4 Welfare
Using equation (3.12) and the price index
P1j =kMij
Zzkj
pkj (z)1 kj (z) dz
where kj (z) is again the conditional productivity density of
firms from i that sell to j as well as
(3.11) we can write real wage as7
wjPj=jj1/0@ cj
L1/(1)j
+ 1
1A1/ (3.13)where
cj =f jj1/(1) Aj1/(1)
1
(1 )1/(1) (3.14)
3.5.5 Key contributions of Melitz
Established a macroeconomic framework where the concept of the
firm had a meaning whilethe model was tractable and amenable to a
variety of exercises. In this framework it is possible
to think about trade liberalization and firms in GE.
In addition the number of varieties offered to the consumer is
potentially changing when thefundamentals change (e.g. trade
liberalization).
Explicitly modeled the importance of reallocation of production
through the death of the leastproductive firms.
7The original Melitz setup assumed a free entry condition and
that firms learn their productivity after incurring a fixedcost of
entry f e. The implications of such an assumption will be studied
later on.
44
-
Extensions by Helpman, Melitz, and Yeaple (2004), Chaney (2008),
Helpman, Melitz, and Ru-binstein (2008) and Eaton, Kortum, and
Kramarz (2011) (henceforth EKK11) made the frame-
work applicable to actual quantitative exercises. In particular,
the specification of a Pareto dis-
tribution of productivities is key in aggregating the model but
also in capturing many of the
empirical regularities trade economists have documented. (see
EKK11 and Arkolakis (2010))
45
-
Chapter 4
Closing the model
In order to determine the solution of the endogenous variables
of the models we constructed above
in the general equilibrium. The individual goods markets are
already assumed to clear since we
replaced the consumer demand directly in the sales of the firm
for each good.
4.1 Equilibrium in the labor market
We now show how we can determine the wages, wi, and spending, Xi
that solve for the models
general equilibrium (income, yi, can be written always as a
straightforward function of spending
in the cases we will analyze). It turns out that under the
technological distributions and demand
structures that we introduced above, we can first solve for
wages and spending and all the rest of the
variables can be written as simple functions of these two
variables. To create a formal mapping to the
data, where trade deficits are a commonplace, we can also allow
for exogenous transfer payments
to countries, Di, (following Dekle, Eaton, and Kortum (2008))
which in a static model will imply an
equal amount of trade deficit. Of course, these trade deficits
have to sum up to zero across countries,
i Di = 0.
In principle, to solve for wi, Xi we need to consider two sets
of equations
i) the budget constraint of the representative consumer
kXki Xi = wiLi + pii + Di , (4.1)
where pii is total profits earned by firms from country i net of
fixed marketing costs (if any).
46
-
ii) and the current account balance (there are no capital flows)
that consists of exports, imports and re-
lated payment to labor for fixed marketing costs but can be
equivalently written as total expenditure
equals total income and transfers
0 = k 6=i
Xik| {z }exports
k 6=i
Xki| {z }imports
+k 6=i
kiXki k 6=i
ikXik| {z }net foreign income
+ Di =)
0 = kXik
kXki +
kkiXki
kikXik + Di =)
Xi =kXki =
kXik +
kkiXki
kikXik + Di , (4.2)
where ij is the share of bilateral sales from i to j that
accrues to labor for payments of fixedmarketing
costs, and trade flows Xij can also be written as ijXj, ij being
the bilateral market shares. If there
are no fixed marketing costs the third and the fourth term in
the right hand side of equation (4.2) are
simply zero.
Notice first a couple of simple cases. WIth perfect competition
pii = 0 and there are no fixed
marketing costs. Thus, these two sets of equations can be
thought as one set of equations with wages
remaining to be solved. Another simple case is that of
monopolistic competition with homogeneous
firms. In that case there are again no fixed marketing costs and
all income is ultimately accrued to
labor due to free entry so that the budget constraint can be
written as Xi = wiLi + Di and given this
relationship (4.2) can be used to solve for all wages.
In the case of monopolistic competition with heterogeneous firms
things are a tad more com-
plicated. In general, replacing for the profits, we need to
solve (4.1) and (4.2) to jointly solve for
wi and Xis given that Xik = ikXk where ik is ultimately a
function of wages alone. Neverthe-
less, under the assumption of Pareto distributions and fixed
marketing costs it can be shown that
ij = ( + 1) / () so that (4.2) can be written as
Xi =kXik +
kXki
kXik + Di (4.3)
Nowwe can combine the above reformulation of the current account
balance condition and the labor
47
-
market clearing condition (equivalently the budget constraint of
the consumer), to obtain
wiLi + pii kXki = (1 )
kXik
and using again the budget constraint of the individual we can
rewrite this equation as
kXki =
kXik +
Di(1 ) (4.4)
Using (4.4) and (4.3) and (4.1) once again we have
wiLi + pii + Di = (1 )kXik +
kXik +
Di(1 )
!+ Di
wiLi + pii Di1 = kXik (4.5)
Nownote that in the case of a Pareto distributionpii is a
constant share of output = ( 1) / ()so that
pii = kXik =
kikXk . (4.6)
Combined with (4.5)
wiLi + pii Di1 =pii=)
1 wiLi Di1
= pii (4.7)
and thus we can write
wiLi + pii =1
1 wiLi
1 Di
1
and the wages can be found by solving the system of
equations
wiLi Di1 = (1 )kik
1
1 wkLk
1 Dk
1 + Dk
wiLi Di1 = kik
wkLk + Dk
1 + 1
. (4.8)
Finally, Xi can be found by utilizing the solution for wages,
equation (4.7) which expresses profits as
a function of wages, and the budget constraint of the
representative consumer, equation (4.1).
48
-
4.2 The free entry condition
Many papers assuming firm heterogeneity and monopolistic
competition, including the original pa-
per of Melitz (2003), assume that new firms can choose to enter
in the economy. These papers assume
that by paying a fixed entry cost, f e, in advance, firms can
enter the market and draw a productiv-
ity.1 If a firm gets a productivity draw that is below zii, then
it exits immediately without operat-
ing.2 It turns out that in the simple monopolistic competition
framework with Pareto distribution of
productivities of firms the free entry condition is irrelevant
(see Arkolakis, Demidova, Klenow, and
Rodrguez-Clare (2008)): the model with free entry is simply
isomorphic to one with a predetermined
number of entrants (essentially the Chaney (2008) version of
Melitz (2003)) where the number of en-
trants is proportionate to the population. The only difference
between the two models is that all the
profits are accrued to labor allocated for the production of the
fixed cost of entry.
To show the point of Arkolakis, Demidova, Klenow, and
Rodrguez-Clare (2008) first note that, in
the equilibrium, because of free entry, the expected profits of
a firm must be equal to entry costs.3
Using the free entry condition and a Pareto distribution with
shape parameter > 1, c.d.f.G (z; Ai) = 1 Aiz , and support [A1/i
,+) we have4
kpiik (z) dG (z; Ai) = wi f e =)
k
Zzik
1ikwiz
1P1k
wkLkAiz+1
dzk
Zzikwk fik
zik
z+1dz = wi f e =)
kwk fik
+ 1(zii)
zik
kwk(zii)
zik fik = wi f eAi
(zii)
=)
k
wkwi
fik(zii)
zik 1 + 1 = f eAi
(zii)
. (4.9)
1This setup, where the final realized productivity is unknown,
is the main difference with the way the free entry condi-tion is
treated in Krugman (1980). This two stages setup is postulated in
order to accomodate productivity heterogeneity.
2We assume that the parameters of the model are such that the
lower productivity threshold zij > zii > bi, 8i, j, i 6=
j.
3Essentially, we assume that there exists a perfect capital
market, which requires firms to pay a fixed entry cost
beforedrawing a productivity realization. Consequently, we multiply
the LHS by 1 G zii, bi, the probability of obtainingthe average
profit, since firms with profits below this average necessarily
exit the market. Alternatively, we could havespecified a more
general case with intertemporal discounting, . In this case the
expected profits from entry should equalthe discounted entry cost
in the equilibrium.
4An implication of free entry is that in the equilibrium all the
profits are accrued to labor for the production of the
entrycost.
49
-
4.3 Solving for the Equilibrium
We now combine the free entry condition and a reformulation of
the labor market clearing condition
to compute the equilibrium of the model. Notice that the
equilibrium number of entrants in country
i, Ni, is determined by the following labor market clearing
condition:
Ni
0BBBBB@kZzik
1ikwiz
P1k
ikzwkLk
zik
z+1Aizik dz| {z }
labor used into production
+ f e
1CCCCCA+k Nkzkk
zki fki| {z }
labor for fixed costs
= Li =)
Ni
k( 1) wk
wifik
Aizik + 1 + f e
!+
kNk
zkk
zki fki = Li. (4.10)
Substituting out equation (4.9), we obtain
Ni ( f e + f e) +kNk
zkk
zki fki = Li,
which, together with the price index, implies that
wiLi =
+ 1
kNk
zkk
zki wi fki
!,
and so
Ni = 1 f e
Li , (4.11)
which completes the derivation of the number of entrants.5
5With a slightly altered proof the same results hold under the
assumption that fixed costs are paid in terms of domesticlabor.
50
-
Notice that total export sales from country i to j are6
Xij = NiAizij
| {z }firms from i in j
wj fij
+ 1| {z } .average sales of operating firms
(4.12)
Define the fraction of total income of country j spent on goods
from country i by ij. Using the
definition of total sales from i to j and equations (3.11) and
(4.11), we have
ij =Xijk Xkj
,
which gives that
ij =LiAi
ijwi
f 1/(1)ijk LkAk
kjwk
f 1/(1)kj . (4.13)and the equilibriumwages can be simply
determined using the labor market clearing condition (4.8).
All the key expressions derived in this context are the same as
in the model of restricted firm
entry of Chaney (2008). In fact, the share of spending for fixed
costs of entry is ( 1) / (). Thisis exactly the same as the profits
share out of total income that we would get in the Chaney
(2008)
model if we assumed that domestic consumers own equal shares of
domestic firms only.
4.4 Computing the Equilibrium (Alvarez-Lucas)
Using the methodology of Alvarez and Lucas (2007) it can be
proven that the model with this gravity
structure has a unique equilibrium. To show existence Alvarez
and Lucas (2007) define the analog of
an excess demand function, which in our context and with zero
exogenous deficits is given by,
fi (w) =1wi
24j
LiAiijwi
f 1/(1)ijk LkAk
kjwk
f 1/(1)kj wjLj wiLi35 ,
6Average sales of firms from i conditional on operating in j are
the same in the model with free entry and the one witha
predetermined number of entrants.
51
-
where w is the vector of wages, and they show that it satisfies
the standard properties of an excess
demand function.7 To show uniqueness, the gross substitute
property has to be proven
fi (w)wk
> 0 for all i 6= k fi (w)wi
> 0 for all i
and uniqueness follows from Propositon 17.F.3 of Mas-Colell,
Whinston, and Green (1995).
To compute the equilibrium, notice that for some 2 (0, 1] we can
define a mapping
Ti (w) = wi [1+ fi (w) /Li] (4.14)
Now if we start with wages that satisfy i wiLi = 1, we have
iTi (w) Li =
iwiLi +
iwi f (w)
= 1+ i
24j
LiAiijwi
f 1/(1)ijk LkAk
kjwk
f 1/(1)kj wjLj wiLi35
= 1 jwjLj +
jwjLj = 1
so that Ti (w) is mapping w such that it maps i wiLi = 1 to
itself. By starting with an initial guess
of the wages, and updating according to (4.14) the system is
guaranteed to converge to the solution
Ti (w) = wi (see Alvarez and Lucas (2007)).
7These properties are continuity, homogeneity of degree zero,
Warlas law, boundness from below and infinite excessdemand if one
wage is 0. See Mas-Colell, Whinston, and Green (1995), Chapter
17.
52
-
Chapter 5
Extensions: Modeling the Demand Side
We now discuss a number of ways that the simple model can be
extended by with assumptions
that have implications for the effective demand of the firm. We
will discuss a nested CES structure,
endogenous marketing costs, multi-product firms, and other
preferences structure different than the
constant elasticity demand.
5.1 Extension I: The Nested CES demand structure
We can consider a nested CES structure
0@Z
N
k=1
xk ()1
! 1
1
d
1A
1
that delivers the demand
xi () =pi ()P ()
P ()P
X ,
with
P () =
"N
k=1
pk ()1 d
#1/(1),
P =ZP ()1 d
1/(1).
and X being the overall spending.
Serving a market incurs an entry cost
53
-
a) ! , F = 0 PC EK02b) ! , F = 0 Bertrand BEJKc) = , F > 0
monopolistic competition Melitz-Chaney
d) > , F 0 (with either F > 0 or ! ) and Cournot, Atkeson
and Burstein.
5.2 Extension II: Market penetration costs
The CES benchmark proved extremely useful for many applications.
Its main weakness is in pre-
dicting the behavior of small firms-goods as Eaton, Kortum, and
Kramarz (2011). These firms-goods
tend to be a very large part of trade in a trade liberalization
and as time evolves. To address this fact,
a simple extension presented in Arkolakis (2010) does the job by
modeling the fixed entry costs as
cost of reaching individual consumers into individual
destinations.
Each good is produced by at most a single firm and firms differ
ex-ante only in their productivities
z and their country of origin i = 1, ...,N. We denote the
destination country by j. The preferences for
consumer l are given by the standard symmetric constant
elasticity of substitution (CES) objective
function:
Ul =Z
2lx ()
1 d
1
,
where 2 (1,+) is the elasticity of substitution. When a good
produced with a productivity z fromcountry i is included in the
choice set of consumer l, l , the demand of this consumer is given
by,
xij (z) = yjpij (z)
P1j, (5.1)
where pij (z) is the price charged in country j, yj the income
per capita of the consumer, and Pj a
price aggregate of the goods in the choice set of the consumer.
An unrealistic assumption of the
CES framework introduced by Dixit and Stiglitz is that all the
consumers have access to the same
set of goods l . This formulation departs from the standard
formulation of trade models with CES
preferences by proposing a formulation where l can be different
for different consumers. In order
to be able to fully characterize the general equilibrium of the
model, we assume that consumers are
reached independently by different firms and that each firm pays
a cost to reach a fraction n of the
consumers. In equilibrium, all firms z from country i will reach
the same fraction of consumers in
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country j and thus their effective sales will be:1
tij(z) = nij (z) Lj| {z }consumersreached in j
yjpij (z)
1
P1j| {z }sales per-consumer
where Lj is the measure of the population of country j. In order
to give foundations to the market
penetration cost function as an explicit function of nij(z) we
depart from the standard formulation
where there is a uniform fixed marketing cost to enter the
market and sell to all the consumers there.
Instead, we consider an alternative formulation that intends to
broadly capture the marketing costs
incurred by the firm in order to increase their sales in a
particular market. The marketing costs are
modeled as increasing access costs that the firms pay in order
to access an increasing number of
customers in each given country. Due to market saturation,
reaching additional consumers becomes
increasingly difficult once a relatively large fraction of them
has already been reached. Based on a
derivation of a marketing technology from first principles the
cost function of reaching a fraction n
of a population of L consumers in Arkolakis (2010) is derived to
be
f (n) =
8>:L
1(1n)11 if 2 [0, 1) [ (1,+)
L log (1 n) if = 1
.
1/ denotes the productivity of search effort and a 2 [0, 1]
regulates returns to scale of marketingcosts with respect to the
population size of the destination country. The parameter
determines how
steeply the cost to reach additional consumers is rising.
However, for any parametrization of the
marginal cost to reach the very first consumers in a given
market j is always positive (the derivative
is always bigger than zero). Thus, only firms with productivity
above some threshold zij will have
high enough revenues from the very first consumers to find it
profitable to enter the market.2 The
case where = 1 corresponds to the benchmark random search case
of Butters (1977) and Grossman
and Shapiro (1984). If = 0 the function implies a linear cost to
reach additional consumers, which
in turn is isomorphic to the case of Melitz (2003)-Chaney (2008)
given that firms reach either all the
consumers in a market or none.1Given the existence of a
continuum of firms and consumers I am making use of the Law of
Large Numbers. This
implies that nij(z) from a probability becomes a fraction. The
application of the Law of Large Numbers also implies that Pjis now
a function of nij(z)s and has a given value for all consumers.
2With no additional heterogeneity across firms this implies a
hierarchy of exporting destinations
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The production side of the firm is standard. Labor is the only
factor of production. The firm z uses
a production function that exhibits constant returns to scale
and productivity z. It incurs an iceberg
transportation cost ij to ship a good from country i to country
j. This implies that the optimal price
of the firm is a constant markup /( 1) over the unit cost of
producing and shipping the good,wjij/z. The equilibrium of the
model retains many of the desirable properties of the benchmark
quantitative framework for considering bilateral trade flows
develop by Eaton and Kortum (2002)
and particularly the gravity structure. It also allows for
endogenous decision of exporting and non-
exporting of firms as in Melitz (2003).
How can this additional feature of endogenous market penetration
costs help the model to ad-
dress facts on exporters? The following version of the
proposition proved in Arkolakis (2010) com-
putes the responses of firms sales in a trade liberalization
episode:
Proposition 12 (Elasticity of trade flows and firm size)
The partial elasticity of a firms sales in market j with respect
to variable trade costs, ij (z) = ln tij (z) / ln ij,
is decreasing with firm productivity, z, i.e. dij (z) /dz < 0
for all z zij.
Proof. Compute the partial elasticity of trade flows tij (z)
with respect to a change in ij, namely ln tij (z) / ln ij = j (z)j
ln zij/ ln ij, where (z) = ( 1)| {z }
intensive marginof per-consumersales elasticity
+ 1
24 zzij
!(1)/ 1351
| {z }extensive margin ofconsumers elasticity
.
Notice that (z) 0 for z zij. (z) is also decreasing in z and
thus decreasing in initial exportsales. In fact, as ! 0 then (z)! (
1) for all z zij.
The proposition implies that trade liberalization benefits
relatively more the smaller exporters in
a market. The parameter governs both the heterogeneity of
exporters cross-sectional sales and also
the heterogeneity of the growth rates of sales after a trade
liberalization.
56
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5.3 Extension III: Multiproduct firms
We now turn to an extension of the basic CES setup that can
accomodate multiproduct firms. This ex-
tension is suggested byArkolakis andMuendler (2010) and
ismodeling the idea of core-competency
within the standard heterogeneous firms setup of Melitz
(2003).
A conventional variety offered by a firm from source country i
to destination j is the product
composite
xij() 0@Gij()
g=1xijg()
1
1A 1 ,where Gij() is the number of products that firm sells in
country d and xijg() is the quantity of
product g that consumers consume. The consumers utility at
destination j is a CES aggregation over
these bundles
Uj =
N
i=1
Z2ij
xij()1 d
! 1
for > 1, (5.2)
whereij is the set of firms that ship from source country i to
destination j. For simplicity we assume
that the elasticity of substitution across a firms products is
the same as the elasticity of substitution
between varieties of different firms.3
The consumers first-order conditions of utility maximization
imply a product demand
xijg() =
pijg()
P1j
Xj, (5.3)
where pijg is the price of variety product g in market j and we
denote by Xj the total spending of
consumers in country j. The corresponding price index is defined
as
Pj24 Nk=1
Z2kj
Gkj()
g=1
pkjg()(1) d
35 11 . (5.4)A firm of type z chooses the number of products
Gij(z) to sell to a given market j. The firmmakes
each product g 2 1, 2, . . . ,Gij(z) with a linear production
technology, employing local labor with3Arkolakis and Muendler
(2010) generalize the model to consumer preferences with two nests.
The inner nest contains
the products of a firm, which are substitutes with an elasticity
of . The outer nest aggregates those firm-level product linesover
firms and source countries, where the product lines are substitutes
with a different elasticity 6= . The general caseof 6= generates
similar predictions at the firm-level and at the aggregate
bilateral country level.
57
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efficiency zg. When exported, a product incurs a standard
iceberg trade cost so that ij > 1 units
must be shipped from i for one unit to arrive at destination j.
We normalize ii = 1 for domestic
sales. Note that this iceberg trade cost is common to all firms
and to all firm-products shipping from
i to j.
Without loss of generality we order each firms products in terms
of their efficiency so that z1 z2 . . . zGij . A firmwill enter
export market jwith the most efficient product first and then
expandits scope moving up the marginal-cost ladder product by
product. Under this convention we write
the efficiency of the g-th product of a firm z as
zg zh(g) with h0(g) > 0. (5.5)
We normalize h(1) = 1 so that z1 = z. We think of the function
h(g) : [0,+)! [1,+) as a contin-uous and differentiable function
but we will consider its values at discrete points g = 1, 2, . . .
,Gij as
appropriate.
Related to the marginal-cost schedule h(g) we define firm zs
product efficiency index as
H(Gij) Gijg=1
h(g)(1)! 11
. (5.6)
This efficiency index will play an important role in the firms
optimality conditions for scope choice.
As the firmwidens its exporter scope, it also faces a
product-destination specific incremental local
entry cost fij(g) that is zero at zero scope and strictly
positive otherwise:
fij(0) = 0 and fij(g) > 0 for all g = 1, 2, . . . ,Gij,
(5.7)
where fij(g) is a continuous function in [1,+).
The incremental local entry cost fij(g) accommodates fixed costs
of marketing (e.g. with 0 0. We assume that the
incremental local entry costs fij(g) are paid in terms of
importer (destination country) wages so that
Fij(Gij) is homogeneous of degree one in wj. Combined with the
preceding varying firm-product
efficiencies, this local entry cost structure allows us to
endogenize the exporter scope choice at each
destination j.
A firm with a productivity z from country i faces the following
optimization problem for selling
to destination market j
piij(z) = maxGij,pijg
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