ECON 201, Summer 2016
July 28, 2016
Problem Set 5 Solutions
1. Assume that there are two firms, firm A and firm B. They have the
same cost function T C(yi ) = 20yi for i ∈ {A, B} and the inverse market
demand is P (y) = 100 − 2y.
(a) What would be the equilibrium price and quantities if these firms
are setting quantites simultaneously.
Let’s write down the profit-maximization problem of firm A:
max π(yA ) = (100 − 2(yA + yBe ))y1 − 20yA
yA
When we take the derivative with respect to yA , we have
100yA − 4yA − 2yAe = 20
40 − yBe
yA =
2
When we do the same calculations for firm 2, we have y2 =
40 − yAe
. We know that at the equilibrium yAe = yA and yBe = yB .
2
yA∗ =
1
40 −
40 − yA∗
2
2
40
yA∗ =
3
40 − yAe
40
, we find yB∗ =
.
2
3
80
The total amount produced in the economy is yA∗ + yB∗ =
.
3
Therefore, equilibrium price is
When you substitute this into yB =
P = 100 − 2
80
= 140/3
3
(b) Find the equilibrium price and quantites if firm A is the leader
firm, i.e. chooses a level of quantity first and then firm B follows
firm A.
Let’s write down the follower’s, i.e. firm B problem first.
max(100 − 2(yA + yB ))yB − 20yB
yB
When we take the derivative with respect to yB , we have
100 − 2yA − 4yB = 20
40 − yA
yB =
2
Now let’s work out the leader’s problem:
max(100 − 2(yA + yB ))yA − 20yA
yA
40 − yA
))yA − 20yA
2
40 + yA
max(100 − 2(
)) − 2yA
yA
2
max(100 − 2(yA +
yA
Taking derivative with respect to yA
60 − 2yA = 20
yA∗ = 20
40 − yA
and when we substitute this into yB =
, we find yB∗ = 10.
2
Since the total amount produced in the economy is yA∗ + yB∗ = 30,
the equilibrium price is p∗ = 100 − 2.30 = 40.
2
2. Demand for steel is D(p) = 120 − 4p, where output is measured in tons
and prices are dollars per ton. The industry is competitive, and each
firm has constant marginal cost of 10 dollars per ton. However, each
ton of steel that is produced als causes $5 of damage to surrounding
households and firms.
(a) Find the competitive equilibrium price and quantity.
In the competitive equilibrium, we have p = M C, therefore p∗ =
10 and q ∗ = 120 − 4 ∗ 10 = 80.
(b) What is the Pareto efficient quantity of steel? What should the
per-ton tax put on steel be to make competitive equilibrium quantity efficient?
Social marginal cost is going to be SM C = 10 + 5, therefore
efficient price will be pe = 15 and the pareto-efficient quantity will
be q e = 120 − 4 ∗ 15 = 60.
3. Solve 34.2 in Workouts in Intermediate Microeconomics: Suppose that
a honey farm is located next to an apple orchard and each acts as
a competitive firm. Let the amount of apples produced be measured
by A and the amount of honey produced be measured by H. The
cost functions of the two firms are cH (H) = H 2 /100 and cA (A) =
A2 /100 − H. The price of honey is $2 and the apples is $3.
(a) If the firms operate independently, the profit maxmization prob3
lem of honey producer is as follows:
max π(H) = 2H −
H2
100
Taking derivative with respect to H gives H ∗ = 100. Apple producer will take H ∗ = 100 and will solve its profit-maximization
problem
max π(A) = 3A −
A2
+ 100
100
which is maximized when A∗ = 150. Total profit of apple and
honey producer is π(A) + π(H) = 325 + 100 = 425.
(b) Suppose that the honey and apple firms are merged. What would
be the profit-maximizing output of honey and apple for the combined firm?
Let’s write down the profit-maximization problem of the combined
firm
A2
H2
−
+H
max π(H, A) = 3A + 2H −
100 100
when we take the derivative with respect to A and H separately,
we find A∗ = 150 and H ∗ = 150. The total profit is π(H, A) = 450.
(c) What is the socially efficient output of honey? If the firms stayed
separate, how much would the honey production have to be subsized to induce an efficient supply?
The socially efficient output of honey is 150. To induce honey
producer to produce this amount, we need
H2
100
When we take the derivative with respect to H
max π(H) = (2 + s)H −
H
=0
50
150
=0
2+s−
50
s=1
2+s−
4
Therefore, we need a subsidy of $1 per unit of honey produced to
induce honey producer to produce the socially efficient amount.
5