Daniel Trefler
ECO2304: Advanced Topics in International Trade
Problem Set: CES Model
This assignment will take you about 2 hours. Feel free to do it with friends – I suggest
classmates since doing it with other friends may end the relationship. Consider the
monopolistic competition model with CES utility developed by Dixit and Stiglitz (AER,
1977). There is a continuum of varieties indexed by ω. Let N be the measure or ‘number’
of varieties.1 Let q(ω ) denote a quantity of ω and let p(ω ) be the price of ω. Utility can be
written as:
U=
N
Z
[q(ω )]
0
(σ−1)/σ
σ/(σ−1)
dω
(1)
where σ > 1. Letting R denote income, the budget constraint can be written as:2
R=
Z N
0
p(ω )q(ω )dω .
The Consumer Problem
1. Let L be the Lagrangean for the consumer problem of maximizing utility subject
to the budget constraint. Use the first-order conditions ∂U/∂q(ω ) = λp(ω ) and
∂U/∂q(ω 0 ) = λp(ω 0 ) to derive
q(ω )
=
q(ω 0 )
p(ω )
p(ω 0 )
−σ
.
(2)
2. Use this expression to show that σ is the elasticity of substitution. That is,
q(ω )
∂ ln q(ω 0 )
∂ ln
p(ω )
p(ω 0 )
= −σ.
1 Our
set of varieties is the interval [0,N ]. Melitz considers a more general set of varieties, which he
denotes by Ω.
2 If it is easier for you, work with
!σ/(σ−1)
N
U=
N
(σ−1)/σ
∑ qi
and R =
i =1
∑ pi qi .
i =1
1
3. Derive the demand for variety ω, denoted q(ω ), by substituting equation (2) into the
budget contraint:
q (ω ) = p (ω )−σ
R
P 1− σ
.
where the price index P is given by:3
Z N
1/(1−σ)
1− σ
P=
dω
.
[ p(ω )]
0
(3)
(4)
4. An indirect utility function is utility as a function of price and income. To find it,
plug equation (3) into equation (1). The answer is:
U=
R
.
P
(5)
Interpret this.
5. Suppose that price is the same for all varieties ω and let p be the common price.
Then P in equation (4) can be written as a simple function of N and p. Show that,
holding p constant,
P = N 1/(1−σ) p
is decreasing in N. Interpret this. Show that
U = N 1/(σ−1)
R
p
(6)
is increasing in N. Why?
6. With a continuum of firms, dP/dp(ω ) = 0. Use this and your expression for q(ω ) to
show that
e(ω ) ≡ −
∂ ln q(ω )
=σ
∂ ln p(ω )
i.e., the elasticity of demand equals the elasticity of substitution. This establishes
a very important property of monopolistic competition with CES demands: the
elasticity of demand is a constant i.e., it does not change as a firm moves down
its demand function and it does not depend on the prices charged by other firms.
This property is what makes CES easy to work with.
3 The
finite equivalent is (ΣiN=1 p1i −σ )1/(1−σ) .
2
7. A minor point: Let λ be the Lagrange multiplier (marginal utility of income). Show
that λ = 1/P. The easiest way is to prove this is to remember that λ is the marginal
utility of income so that λ = dU/dR.
The Producer Problem
There is a single ‘representative’ consumer. Let c(ω ) be the marginal cost of production.
Let f be the fixed cost of the plant — it is the same for all plants. The variable profit
function for the producer of variety ω is
π (ω ) = [ p(ω ) − c(ω )] q(ω ).
(7)
The (total) profit function is π (ω ) − f . Firms are price setters.
1. Before doing heavy lifting, let’s approach the firm’s problem using ECO101 i.e., MC
1
= MR. Marginal revenue is, as always, p(ω ) 1 − e(1ω ) = p(ω ) σ−
σ . Equating this
to marginal cost yields
p(ω ) =
σ
c(ω ) .
σ−1
(8)
This is sometimes stated as a markup rule. The markup is usually defined as the
ratio of price to marginal cost. Thus, σ/(σ − 1) is the markup. Note also that a
monopolist always sets price in the elastic region of the demand function i.e., where
σ > 1. Since σ is a constant, we must assume σ > 1.
2. As you now know, q(ω ) = p(ω )−σ P1R−σ is the demand function. Since each firm is
tiny, a change in p(ω ) has no effect on P. (A consumer’s income R is exogenous to
the firm.) This makes it convenient to write demand as q(ω ) = p(ω )−σ A where A is
a constant from the firm’s perspective. What is A? (The answer is obvious.) In what
follows, use A to simplify expressions.
3. Plug in your equation (3) expression for demand into the profit function. Derive the
profit-maximizing price. In doing so note that since each firm is tiny, a change in
p(ω ) has no effect on P: ∂P/∂p(ω ) = 0. (A consumer’s income R is also exogenous
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to the firm i.e., ∂R/∂p(ω ) = 0.) Interpret the optimal price as a mark-up of price
over marginal cost.
4. Assume that all plants have the same marginal costs c = c(ω ). Does the optimal
price vary across plants (easy answer)? Does the optimal output vary across plants?
5. Let r (ω ) ≡ p(ω )q(ω ) be a firm’s revenue function. Show that
π (ω ) =
r (ω )
.
σ
(9)
This is important because it shows that profits, revenues and hence marginal costs
are all proportional to one another. This will be very helpful in solving the math of
complex problems.
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Harder General Equilibrium Results
You do not have to do this section now. It will prove helpful later in the course. Note
that I have not checked my math for this section. Please let me know if there are mistakes.
1. Derive the consumer’s demand by plugging the optimal price into the consumer
demand c(ω ):
d
q (ω ) = c(ω )
−σ
R
P 1− σ
σ
σ−1
−σ
.
2. Suppose that the firm producing variety ω earns zero profits. Use equations (7) and
(8) to show that the optimal supply is:
q π =0 ( ω ) = ( σ − 1 )
f
.
c(ω )
3. The variety ω 0 that just breaks even satisfies qd (ω 0 ) = qπ −0 (ω 0 ). Using this equation, solve for the break-even marginal cost:
c ( ω 0 ) σ −1 =
( σ − 1 ) σ −1 σ − σ R
.
f
P 1− σ
(10)
Note that all firms with c < c(ω 0 ) enter and all firms with c > c(ω 0 ) exit.
4. Suppose that all firms are identical i.e., have the same marginal costs c. Solve for
the general equilibrium measure of firms N when there is free entry so that all firms
earn zero profits. Start with equation (10). In this equation set c(ω 0 ) equal to c. Next,
σ
set P = N 1/(1−σ) p (from above) and use equation (8) to set P = N 1/(1−σ) c σ−
1 . Next,
we must know what revenue is. Assume that all costs (c and f ) are wage costs and
choose the wage as the numeraire (w = 1). Also assume that it takes c units of labour
to produce 1 unit of output and that fixed costs require f units of labour i.e., c and
f are now technological constants. Finally, assume that there are L workers in the
economy so that total income is wL = L. But all income is spent on goods so that
R = wL = L. Plugging all of this back into equation (10) yields:
c σ −1 =
( σ − 1 ) σ −1 σ − σ
f
5
L
N 1/(1−σ) c
σ
( σ −1)
1− σ .
Simplifying yields the general equilibrium number of (identical) firms with free
entry:
N∗ =
L
.
σf
(11)
By the way, this is identical to equation (10) in Krugman (AER, 1980). The derivation
is different in order to be consistent with Melitz (Econometrica, 2003).
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