Solution to Homework 3
Monopolistic Competition and Increasing Returns
ECO-13101 Economia Internacional I (International Trade Theory)∗
∗
Instructor - Rahul Giri. Contact Address: Centro de Investigacion Economica, Instituto Tecnologico Autonomo de Mexico
(ITAM). E-mail: [email protected]
Question 1: Increasing Returns and Pro-Competitive Gains
1. The profit maximization problem of the monopolist is
max (a − bQ).Q − F − cQ .
{Q}
The first-order condition with-respect-to Q is
a − 2bQ − c = 0 ,
Q=
a−c
,
2b
(1)
Therefore, the monopoly price is given by:
P =a−b
(a − c)
a+c
=
2b
2
(2)
Thus,
P −c=
a−c
a+c
−c=
.
2
2
This is positive since a > c. This shows that the price is greater than the marginal
cost, which is because the monopolist has market power and charges a markup which
results in a price higher than the marginal cost.
2. The Foreign oil producer will have an incentive to sell oil in the Home market. This is
because by doing so, the producer exploit economies of scale, i.e. spread the fixed cost
over a larger output, thereby reducing the average cost of production.
3. The profit maximization problem of the Home firm is
max (a − b(Q + Q∗ ))Q − F − cQ .
{Q}
The first-order condition with-respect-to Q is
a − b(2Q + Q∗ ) − c = 0 ,
⇒Q=
a − c − bQ∗
.
2b
(3)
2
4. The profit maximization problem of the Foreign firm is
max
(a − b(Q + Q∗ ))Q∗ − F − cQ∗ .
∗
{Q }
The first-order condition with-respect-to Q∗ is
a − b(Q + 2Q∗ ) − c = 0 ,
⇒ Q∗ =
a − c − bQ
.
2b
(4)
5. Use Eq. (3) and Eq. (4) to solve for Q and Q∗ .
Q∗ = Q =
a−c
.
3b
(5)
Then, the equilibrium price is given by:
P = a − b(Q + Q∗ ) =
a + 2c
3
(6)
6. The consumers benefit from free trade. The total output consumed is higher and the
price is lower.
Qtrade + Q∗trade =
2(a − c)
(a − c)
a−c
(a − c)
= 0.66
>
= 0.5
= Qautarky ,
3b
b
2b
b
and
Ptrade =
a+c
a + 2c
<
= Pautarky
3
2
(because a > c).
Question 2: Monopolistic Competition and Gains from Product Diversity
1. The consumer’s utility maximization exercise is the following:
n
X
max
n
{Xi }i=1
s.t.
n
X
Xiα ,
i=1
pxi Xi = wL .
i=1
3
The first-order condition with-respect-to a good i is
αXiα−1 − λpxi = 0 , i = {1, . . . , n} ,
where λ is the lagrange multiplier for the consumer. The first-order condition withrespect-to the lagrange multiplier just gives the budget constraint. The first-order
condition above implies that
λpxi
Xi =
α
1/(α−1)
.
Substituting this for Xi in the budget constraint, and noting that labor is the numeraire, i.e. w = 1, gives:
1/(α−1) X
n
λ
α/(α−1)
pxi
=L ,
α
i=1
"
L
⇒ λ = α Pn
α/(α−1)
i=1
pxi
#(α−1)
.
Substitute this into the first-order condition obtained with-respect-to Xi .
"
L
αXiα−1 − α Pn
α/(α−1)
i=1 pxi
#(α−1)
1/(α−1)
p
⇒ Xi = Pnxi
L
α/(α−1)
i=1 pxi
⇒ Xi =
Let σ = 1/(1 − α) and P =
Pn
j=1
Xi =
1/(1−α)
pxi
L
Pn
pxi = 0 ,
,
−α/(1−α)
i=1 pxi
p−ασ
xj .Then
L
pσxi P
, i = {1, . . . , n} .
This expression gives the demand for good i.
2.
L −(σ+1)
dXi
.
= −σ pxi
dpxi
P
4
.
Substituting this and the expression obtained for Xi (in the previous part) in the
formula for elasticity of demand gives:
L −(σ+1)
pxi
,
ei = − −σ pxi
.
P
L/ (pσxi P )
#
"
(σ+1)
−(σ+1)
pxi P
Lpxi
,
.
ei = σ
P
L
⇒ ei = σ =
1
, i = {1, . . . , n} .
1−α
3. The profit function of the firm producing good i is given by πxi = T Rxi − T Cxi , where
T Rxi is the total revenue of the firm and T Cxi is the total costs of the firm. The
total cost function has been given to us. What we need is to derive the total revenue
function. Well, total revenue is nothing but the value of the sales, i.e. quantity sold
(what is demanded is sold) times the price at which it sold - T Rxi = pxi Xi . Therefore,
after substituting for Xi , the profit maximization problem is given by:
1−σ
max πxi = pxi
pxi
L
L
− F − M Cxi . σ
.
P
pxi P
Note that choosing a price is equivalent to choosing output since the two are uniquely
related through the demand function (the expression for Xi derived in (1)). The firsorder condition with-respect-to price is
(1 −
σ)p−σ
xi
L
−(σ+1) L
− M Cxi . −σpxi
.
=0 ,
P
P
⇒ −(σ − 1)pxi + σM Cxi = 0 ,
⇒ pxi =
1
σ
M Cxi = M Cxi , i = {1, . . . , n} .
σ−1
α
Since 0 < α < 1, it implies that the price is greater than marginal cost. So 1/α
represents the markup. Notice that
ei = σ =
1
1
ei
1
⇒α=1− ⇒ =
.
1−α
ei
α
ei − 1
Therefore, the price of good i could also be written as:
pxi =
ei
M Cxi , i = {1, . . . , n} .
ei − 1
5
4. Due to free entry of firms (to produce each good i), the profit of each firm is driven to
zero, i.e. pxi = ACxi , where ACxi is the average cost of producing good i. The average
costs is nothing but T Cxi /Xi . Using the expression for price obtained in the previous
part, we get
T Cxi
F
1
M Cxi =
=
+ M Cxi ,
α
Xi
Xi
F
1−α
⇒
M Cxi ,
=
Xi
α
αF
, i = {1, . . . , n} .
⇒ Xi =
M Cxi (1 − α)
5. Substituting the expression for Xi obtained in the previous part in the expression for
total cost gives
T Cxi = F + M Cxi .
αF
,
M Cxi (1 − α)
1
F , i = {1, . . . , n} .
1−α
This expression shows that the total cost is the same for every good i = {1, . . . , n}.
⇒ T Cxi =
Since labor is the only factor, it must be that the total cost of producing the n goods
P
is equal to the total labor cost, i.e. ni=1 T Cxi = wL.
n
X
i=1
1
F = wL ,
1−α
1
F =L ,
1−α
L
.
⇒ n = (1 − α)
F
Thus, the numbers of good produced in the economy is going to depend on the total
⇒ n.
labor income relative to the fixed cost (per good).
6. Thus, if the endowment of labor doubled, it would increase the number of goods produced in the economy, i.e. increase product diversity, which in turn will cause the
utility to increase. This tells us that when two such (identical) economies open up
to trade, then the combined economy will have a larger labor endowment (size), 2L,
and therefore this will increase the product diversity. Increase in product diversity will
increase welfare. Thus, trade allows the two countries to exploit scale economies and
therefore increase product diversity.
6
Question 3: Mexico and its Competitors in the US Market
0
Share in US Imports
.05
.1
.15
.2
Share in US Imports
1993
1999
2007
2013
Year
Canada
Germany
Great Britain
Japan
Mexico
China
France
Italy
South Korea
Change in Share in US Imports
0
1
2
3
4
Share in US Imports Relative to 1993
1993
1999
2007
Year
Canada
Germany
Great Britain
Japan
Mexico
7
China
France
Italy
South Korea
2013
0
Share in US Imports
.05
.1
.15
.2
Share in US Imports of Manufactures
1993
1999
2007
2013
Year
Canada
Germany
Great Britain
Japan
Mexico
China
France
Italy
South Korea
Change in Share in US Imports
0
1
2
3
4
Share in US Imports of Manufactures Relative to 1993
1993
1999
2007
Year
Canada
Germany
Great Britain
Japan
Mexico
8
China
France
Italy
South Korea
2013
Mexico versus China − Number of Products Exported
1993
1999
2007
2013
0
1,000
2,000
3,000
Number of products
China higher share
Mexico exports, China not
Mexico & China do not export
9
4,000
5,000
Mexico higher share
China exports, Mexico not