Advanced International Trade
Problem Set
Question 1:
Consider an economy which is inhabited by a representative consumer who offers inelastically L
units of labor and derives utility from two consumption goods c 1 and c2. The preferences can be
represented by the utility function
U (c 1 , c 2 )=ln (c1 )+ln (c 2)
Consumption expenditures are financed by labor income where the wage rate is denoted by w. The
prices of commodities are p1 and p2. Let us normalize the price of good one to be 1. Outputs of
commodities one and two, Q1 and Q2, are produced by a Ricardian technology of the following form
Q1 (L 1)=L 1 and Q 2 ( L2 )=
L2
a2
where L1 and L2 are the amounts of labor used in production of commodity one and two,
respectively, and a2>0 is a parameter.
a) Derive the autarky equilibrium for this economy, i.e., determine the value for all endogenous
variables of the model. Determine also the autarky welfare level.
Assume in the following that there is a second country. The production function for commodity 1
and preferences are identical to the first country. However, the second country has labor endowment
L* and the input coefficient in the production function of commodity two differs. Let 0<a *2<a2
(* denotes second country variables).
b) Determine the trading equilibrium for this two-country model under the assumption that each
country fully specializes in the production of the good in which it has a comparative advantage.
c) Derive conditions in terms of the exogenous parameters of the model which support the
equilibrium that you derived in b). Interpret your conditions economically.
d) Show that the utility of each country’s representative consumer in the trading equilibrium is
higher than under autarky.
Question 2:
Consider the standard two-good, two-factor model (i.e., goods 1 and 2 produced from labor and
capital in a perfectly competitive economy under constant returns to scale):
a) State the equilibrium conditions.
b) Use a unit-value-isoquant diagram to illustrate the equilibrium conditions, assuming that industry
1 is labor-intensive, industry 2 capital-intensive.
c) Explain and use a new diagram to illustrate why factor-price equalization may not occur when
there are factor-intensity reversals.
Question 3:
Consider a small open economy in which two goods (1 and 2) are produced from two factors
(capital and labor). Good 1 is labor intensive, good 2 is capital intensive. The production technology
exhibits constant returns to scale, and markets are perfectly competitive. There is free trade in
goods, and the country takes the world prices of goods 1 and 2 as given. Suppose this economy
experiences an inflow of capital.
a) Assuming that the country continues to produce both goods, explain how this capital inflow will
affect the outputs of goods 1 and 2, the wage rate, and the rental rate of capital.
b) How would your answer change, if the capital inflow led to specialization in the production of
one good.
Question 4:
Consider the standard 2x2x2 version of the Heckscher-Ohlin-Samuelson Model. Within this
framework one can prove the following important theorems: Stolper-Samuelson, Rybcynski,
Heckscher-Ohlin and Factor-Price Equalization.
a) State these theorems (no proof).
b) International trade economists often make use of the following assumptions
•
no factor intensity reversal
•
identical technologies in both countries
•
identical homothetic preferences in both countries
•
no (or incomplete) specialization
Which of these assumptions, if any, are necessary to prove each of the four theorems? Explain for
each theorem why a certain assumption is necessary. Do not provide formal proofs of the theorems.
Question 5:
Two core propositions in the theory of international trade under perfect competition and constant
returns to scale are the Heckscher-Ohlin Theorem and the so-called Heckscher-Ohlin-Vanek
Theorem.
a) State the Heckscher-Ohlin Theorem and its main assumptions. For a two-sector and two-factor
model, describe the main steps in proving the theorem and explain where the assumptions enter.
b) Suppose that there are two countries which are endowed with m factors, v 1,...,vm (country 1) and
v1*,...,vm* (country 2). Suppose that factors are indexed such that
v1
v
*
1
≥
v2
v
*
2
≥...
vm
v *m
State (no proof) and explain the Heckscher-Ohlin-Vanek Theorem (‘Factor content of trade with
factor price equalization’). What is the relationship between the Heckscher-Ohlin-Theorem and the
Heckscher-Ohlin-Vanek Theorem?
c) There exists a large literature that has tested the two Theorems. Does the data confirm the
theories? Describe briefly and discuss at least two different studies that have empirically tested the
Heckscher-Ohlin-Vanek Theorem.
Question 6:
Paul Krugman considers a model of international trade in which the number of goods that are
produced and traded is endogenous (‘Increasing Returns, Monopolistic Competition, and
International Trade’, JIE 1979). Consider the following simplified version.
n
A representative consumer’s preferences are given by the utility function
U =∑ v (ci )
i =1
, where ci is
the consumption level of good i and v(c i) is a strictly concave function. Each firm produces one
good using the identical cost function l i=a+b⋅x i where a, b > 0 are parameters, l i is labor
employed, and xi is output of good i. Total labor supply L is fixed.
Since the cost structure is symmetric in all industries and preferences are symmetric over all goods,
an equilibrium for a closed economy of this type can be derived from the following set of
conditions:
(1)
p ϵb
=
w ϵ−1
(2)
p
a
=b+
w
L⋅c
(3)
n=
(4)
x= L c
where ε is the elasticity of demand ϵ=
−v '
v' ' c
L
(a+bx)
where p is the price of a typical good and w is the wage.
a) Explain the economic meaning of each of the four conditions.
d
b) Suppose the subutility function v takes the form v (ci )=(1−d )−1 c 1−
, 0 <d<1. For a closed
i
economy derive the equilibrium values of c, x, and n as function of the exogenous variables.
c) Assume now that there is a second country that is identical in all respects. Does free trade lead to
a different allocation than the one you derived in b)? If so, which are the equilibrium values of c, x
and n?
d) Are there gains from trade for the two countries? If so, identify them in the model.
Question 7:
In his 1980 AER paper on “Scale Economies, Product Differentiation, and the Pattern of Trade”
Paul Krugman develops a monopolistic competition model of international trade. Consider the
following simplified version.
θ
A representative consumer’s preferences are given by the utility function U =∑i ci , where ci is
the consumption level of good i=1,…,n and 0<θ<1. Maximizing utility subject to a budget
constraint yields the following inverse demand curve for good i:
p i= θ c θ−1
λ i
where pi is the price of good i and λ is the Lagrange multiplier.
There is one factor of production, labor, and all goods are produced with the same
cost-function: l i=a+b⋅x i , where a, b > 0 are parameters, li is labor employed, and xi is output of
good i. Total labor supply, L, is fixed, and output of good i must equal consumption of a
representative worker times the labor force: xi=Lci. It is also assumed that there is full employment,
so that:
n
L=∑ (a+b x i )
i=1
(a) For a single country in autarky derive the equilibrium price and output of each good as well as
the number of goods produced.
(b) Now suppose that two identical countries open trade with each other at zero transportation cost.
What will happen to price, output and the number of goods? Explain where the gains from trade
come from.
Question 8:
The gravity equation allows for a simple characterization of bilateral trade flows between countries.
a) State the gravity equation in its simplest form.
b) Explain the assumptions necessary to derive the gravity equation.
c) The monopolistic competition model of international trade provides a possible theoretical
foundation for the use of the gravity equation. Explain which feature of the monopolistic
competition model accounts for this. Is the gravity equation consistent with the predictions of trade
models other than the monopolistic competition model? Explain.
Question 9:
Consider two firms, h and f, that produce homogeneous goods for sale on the world market; there is
no local consumption of the good. Firm h is located in the home country, firm f in the foreign
country. Each firm i produces a quantity q i at a cost of C i (qi )=c qi . World market demand is
given by P=a−b Q , where Q denotes total sales in the country. The firms are Cournot
competitors.
a) Suppose that the home country subsidizes the exports of firm h at a rate s per unit.
Compute the export sales of each firm in Nash equilibrium. Also compute the equilibrium profits.
b) Suppose that firm h is owned by citizens of country h. Compute the social welfare of country h as
a function of s. What is the optimal subsidy?
c) How would your answer to b) change if both firms were owned by citizens of country h?
Question 10:
Consider a world with two countries, H and F. Each country i=H,F has one firm producing a
quantity qi of a homogeneous good at a cost of C i (qi )=c qi . Market demand in each country is
given by P=a−b Q , where Q denotes total sales in the country. Markets are segmented, i.e.,
there is no arbitrage between them. The per-unit transportation costs on goods shipped between the
countries is t. The firms are Cournot competitors.
a) Compute the domestic sales and export sales of each firm in Nash equilibrium. Also compute the
equilibrium profits.
b) Compute the consumer surplus and social welfare of country H as a function of t.
c) What is the minimum level of t at which trade seizes? At which level of t does social welfare
reach its minimum? Its maximum?
d) Now suppose that t is a tariff, and that this tariff is only imposed by country H. Compute the
optimal tariff of country H.
Question 11:
World demand for goods X and Y (which are imperfect substitutes) is given by
X =1−P x +P y
Y =1−P y +P x
Good X is produced by the home firm and good Y by the foreign firm. There is no domestic
consumption of the two goods in either country, so all output is exported. The two firms compete in
prices; production costs are normalized to zero.
a) Consider the following trade policy game. The home government chooses a per unit tax T on the
exports of the home firm to maximize social welfare (the sum of the home firm’s profit and tax
revenue). The foreign government remains inactive. Both firms observe the tax and then
simultaneously and non-cooperatively choose prices. Derive the subgame-perfect equilibrium of the
game (i.e., prices and the optimal home country tax).
b) Explain the intuition for your results in a) with the help of a reaction-function diagram.
c) If the two firms were Cournot competitors, would your policy recommendation in a) change? If
yes, explain why; if not, explain why not.
Question 12:
Consider the Baldwin/Melitz model of heterogeneous firms and trade.
a) Explain how the pattern of trade differs between this model and the standard monopolistic
competition model of international trade. (Hint: do not restate the models but rather explain how the
different types of firms behave.)
b) Use a diagram to show how trade liberalization affects the cut-off conditions between the
different types of firms. Give an intuitive explanation for these changes.
c)How does trade liberalization affect industry productivity? Explain.
d) Explain the mechanism through which trade liberalization raises social welfare. How does this
mechanism differ from that in the standard monopolistic competition model with symmetric firms?
Question 13:
The Heckscher-Ohlin-Vanek theorem states that (with factor price equalization)
F i=A A T i=(V i−s i V w )
where Fi is the vector of net factor content embodied in country i’s trade, Ti is the vector of net
exports, AA is the factor content matrix for some reference country A, V is the country’s factor
endowment vector, si is the country’s share in global expenditures, and VW is the vector of world
factor endowments.
a) Discuss how one can carry out a complete test of the HOV Theorem. What are the results of
existing empirical tests?
b) Trefler (1995) argued that the HOV Theorem does a poor job in predicting trade flows, because
tests based on F i=A A T i ignore cross-country differences in technology. Discuss two ways in
which technological differences can be introduced into the HOV model and critically assess their
success in improving the predictions of the model.
Question 14:
Consider a foreign monopolist supplying the domestic market. Each domestic consumer buys either
one unit of the monopolist’s product or nothing; net surplus from buying one unit of the product of
quality q at price p is
I u (q) , where u ' >0 and u ' ' <0 ,
where I is the index of a particular buyer. I is uniformly distributed over the interval [0,1].
The foreign monopolist can produce a single quality and marginal cost c is independent of output. It
is given by
c=c(q) , where c ' >0 and c ' '>0 .
a)Derive the inverse demand function
p= p( x , q)
b)Show that a specific tariff, t, imposed by the domestic government reduces the quantity of imports
and leads to quality upgrading. (Make any assumptions you deem necessary to prove this result.)
Question 15:
The Heckscher-Ohlin-Vanek theorem states that
F i=A A T i=(V i−s i V w )
where Fi is the vector of net factor content embodied in country i’s trade, Ti is the vector of net
exports, AA is the factor content matrix for some reference country A, V is the country’s factor
endowment vector, si is the country’s share in global expenditures, and V W is the vector of world
factor endowments.
a) Which of the following assumptions, if any, are needed to prove the theorem. For each
assumption explain why it is or is not necessary.
•
no factor intensity reversal
•
identical technologies in both countries
•
identical homothetic preferences in both countries
•
no (or incomplete) specialization
b) Discuss how one can carry out a complete test of the HOV Theorem. What are the results of
existing empirical tests?
Question 16:
a) State the Heckscher-Ohlin-Vanek Theorem.
b) Which of the following assumptions, if any, are needed to prove the theorem. For each
assumption explain why it is or is not necessary.
•
no factor intensity reversal
•
identical technologies in both countries
•
identical homothetic preferences in both countries 9
•
no (or incomplete) specialization
c) (Due to Alan Deardorff via Jim Tybout) The following table contains very approximate data on
national income and population for the United States and for the world as a whole (including the
U.S.). Assume for simplicity that in all countries the available labor force equals exactly 1/3 of the
population and that U.S. trade is balanced.
Applying the HOV Theorem, how much labor should be embodied in U.S. net exports?
Population (millions)
Income (billions of $)
United States
300
8000
World
6000
32000
Question 17:
Consider a world with two identical countries, h and f. Each country has N firms, each producing a
quantity qi of a homogeneous good at a cost of C (qi )=c qi +F . Market demand in each country
is given by the linear demand curve P=a−b z , where z denotes total sales in the country.
Markets are segmented, i.e., there is no arbitrage between them. The firms are Cournot
competitors, and there is free entry and exit of firms.
a) Assuming that the countries are in autarky, derive the equilibrium number of firms, the
equilibrium output per firm, and the equilibrium price.
b) Now assume that there is free trade. Compute the equilibrium number of firms, output per firm
and the equilibrium price in the integrated economy.
c) Compare your answers to a) and b) and explain the effects of trade liberalization on industry
productivity and social welfare.
More Questions
For the first two questions consider the following version of the Melitz/Ottaviano model. The utility
function of an individual consumer is given by
1
1
c
c
c 2
U =q 0+ ∫ qi di− γ ∫ (q i ) di−
2 i ∈Ω
2
i ∈Ω
2
c
q i di
∫
( )
.
i∈Ω
From this utility function we can derive the following market demand faced by firm i:
q i=
L
L
N L
− γ p i+
̄p ,
N +γ
N +γ γ
where N is the mass of firms, L is the number of consumers, γ>0 is a parameter that measures
the degree of product differentiation between varieties, p i is the price of variety i and ̄p is the
average industry price. The marginal cost of production is 0≤c≤1 . Firms entering the market
incur a one-time sunk cost fE. Firms are monopolistically competitive so that they take N and ̄p as
fixed when making their decisions.
Q1. Suppose that the marginal cost of production c is identical across firms. The country is in
autarky.
a) Derive the profit-maximizing price of firm i as a function of ̄p .
b) Note that with identical firms the profit maximizing prices have to be identical across firms, so
that for any firm i we have p i = ̄p = p . Use this fact to show that the equilibrium price of a firm is
equal to p=c +
γ(1−c)
.
N +2 γ
c) Show that the equilibrium output of a firm is then given by q=
L(1−c )
.
N +2 γ
d) Write down the zero-profit condition of a firm and solve this zero-profit condition for the
equilibrium mass of firms.
e) Show that in an autarky equilibrium with free entry and exit of firms
√
γL
N̂ =(1−c)
−2 γ
fE
p̂ =c+
√
γfE
L
√
LfE .
and q=
̂
γ
f) Now suppose that two identical countries integrate their markets. Show how this affects the
equilibrium mass of firms, price and firm output.
g) Explain how industry productivity and social welfare are affected by integration.
Q2. Suppose that the marginal cost of production c is uniformly distributed on the interval [0, 1]
with density one. (The cumulative distribution is hence equal to G(c)=c .) The country is
in autarky. The profit maximizing price, output, operating profit of a firm with marginal cost
c can be shown to equal
1
p (c)= ( c D +c)
2
q (c)=
π( c)=
where c D =
L
(c −c)
2γ D
L
(c −c)2
4γ D
γ+N ̄p
is the cost of a firm that is just indifferent between remaining in the market
N +γ
and leaving.
a) Write down the zero-profit condition of a firm and use it to show that
(
12 γ f E
c D=
L
)
1
3
b) In the autarky equilibrium the average price, average firm output, and the mass of firms are given
by:
3
̄p = c D
4
q=
̄
L cD 3 f E
=
4 γ (c D )2
N =4 γ
1−c D
cD
Suppose now that two identical countries integrate their markets. Explain how economic integration
affects the average price and average output, as well as the mass of firms.
c) Discuss how economic integration affects industry productivity and social welfare.
Q3. Consider the reciprocal dumping model of Long/Raff/Stähler. There are two countries, home
and foreign, each with an inverse demand function
P= A−Q
There are n firms in each country, each with a marginal cost of c. The trade cost between countries
is equal to t per unit.
a) For the case of homogeneous firms show that domestic output and exports in Cournot
equilibrium are, respectively, given by:
y=
A−c+nt
,
2n+1
x=
A−c−( n+1)t
2n+1
b) Now suppose that firms are heterogeneous and that c is distributed on the interval [0, c-]
according to distribution function F(c). Show that the expected domestic and foreign sales in the
Bayesian Nash equilibrium are given by
c̃y
1
̂y = ∫ F (c ) dc ,
20
c̃x
1
x̂ = ∫ F (c) dc
2 0
where
c̃y ≡ A−(n−1) ̂y −n ̂x ,
c̃x ≡ A−( n−1) ̂x −n ŷ −t .
c) Show that
d ̂y
d x̂
>0 and
<0 .
dt
dt
d) Determine the sign of
d c̃y
and interpret the result.
dt