Problem Set 2

Problem 1 (20 points)
In Krugman’s monopolistic competition model, consider trade between two countries (as in the
class). Consumer preferences are identical (no home market e¤ect). Assume that there are
iceberg trade costs represented by > 1.
1) Derive the indirect utility function in this case. Interpret the derived expression.
2) Show that higher trade costs lead to welfare losses (reduce the consumer utility).
Problem 2 (20 points)
Consider an economy with a continuum of sectors, ! 2 [0; 1], and a …nite number N of …rms
within each sector, with preferences de…ned as
with
where 1 < < .
Z 1
U =
y ! =
0
NX
n=1
y 1 1
! d!
!
(q ! (n)) 1 1
1) Discuss the assumption 1 < < and try to give the intuition for it.
2) De…ne the price index for the aggregate good of each sector !, P ! , and the price index
for all consumption goods, P .
3) Derive the demand for the aggregate good !, y ! , as a function of the price of the aggregate
good !, P ! , and the overall price index P , and aggregate expenditures Y .
4) Derive the demand for a single good n in sector !, q ! (n), as a function of the price of this
good, p ! (n), the price of the aggregate good, P ! , and the aggregate demand for good !, y ! .
Problem 3 (60 points)
Take a simple Krugman/CES model with one factor of production, labor, and a homogenous
good and a di¤erentiated good (there are many varieties produced by identical …rms under the
increasing returns to scale technology). Preferences are Cobb-Douglas between the 2 goods
(with share for the homogeneous good and 1 for the di¤erentiated good), with a constant
elasticity of substitution between di¤erentiated varieties. Speci…cally, the preferences are given
by
nP
(1 )=
U = X q i
;
i=1
where X is the consumption of the homogenous product, q i is the consumption of each variety,
and n is the number of available varieties.
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The homogenous good is produced one to one with labor: to produce one unit of the product
requires one unit of the labor (no …xed costs). Take the homogenous good as the numeraire (the
homogenous good is produced under perfect competition). Consider a case where there are two
countries that are identical, except perhaps for di¤erences in sizes: L and L .
1) Imagine that there are no transportation costs of any type. Show that there is only
intra-industry trade (trade in di¤erentiated varieties) in the equilibrium.
2) Imagine now that there is an iceberg-type transportation cost applied only for trade in
di¤erentiated varieties (there is free trade in a homogenous good). Show that if the countries
are identical, then there is only intra-industry trade, while if they di¤er in size, then there is
both inter-industry and intra-industry trade (note, you can assume that the equilibrium entails
all countries producing some amount of the homogenous good). What can you say about the
direction of inter-industry trade?
Hint: If the homogenous good is produced in all countries and freely traded, then its price
is the same in all countries. Because of perfect competition, wage levels in all countries are equal
to the price of the homogenous good (price is to marginal cost). As a result, despite the presence
of transport costs, there is FPE! That is, in your model, the presence of the homogenous good
leads to factor price equalization.
Problem 4 (15 points)
How does an increase in transport costs change the equilibrium outcomes and the gains from
trade in Melitz’s (2003) model? Explain. Are the e¤ects of an increase in transport costs
qualitatively di¤erent from the e¤ects of an increase in the …xed cost of becoming an exporter?
Explain.
Problem 5 (20 points)
Consider the Melitz (2003) model with …rm heterogeneity in the closed economy case. Assume
that the …xed cost of production f is now equal to zero.
(a) Derive the equilibrium conditions in this case. (Do not forget to explain the intuition
behind them.)
(b) What happens in the equilibrium if the entry cost f e rises? (Provide the mathematical
proof for your answer.)
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Problem 6 (15 points)
Show that in the Melitz model, in the case of trade between n+1 identical countries, the average
pro…ts are given by
= d (~'(' )) + np x x (~'(' x)):
Problem 7 (50 points)
Derive the analogue of the gravity equation in the Melitz model.
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