Post Mid semester exam Problem Set 2

1. Let the market demand be given by the inverse demand curve P (Q) = 50 − 2Q,where Q = q1 + q2 .
The cost function for each of the two firms in the industry is C(qi ) = 2qi , i = 1, 2. Firms are
Cournot competitors.

(a) Derive the best response function of each firm.

(b) Find the Cournot Nash equilibrium output of the firms and profits.

2. Three oligopolistic firms operate in a market with inverse demand function given by P (Q) = a − Q,
where Q = q1 + q2 + q3 and qi is the quantity produced by firm i, i = 1, 2, 3. Each firm has constant
marginal cost of production, c and no fixed cost. The firms choose their quantities as follows:
1) firm 1 chooses q1 ≥ 0; 2) firm 2 and 3 observe q1 and then simultaneously choose q2 and q3
respectively. What is the subgame perfect Nash equilibrium?

3. Two firms compete in prices in a market for a homogeneous product. In this market there are
N consumers; each buys one unit if the price of the product does not exceed Rs 10, and nothing
otherwise. Consumers buy from the firm selling at a lower price. In case both firms charge the same
price, assume that 12 consumers buy from each firm. Assume zero cost of production for both firms.

4. Suppose there are two firms, 1 and 2 producing homogeneous product. Firm 1 and 2 compete in
prices that is Bertrand Competition. The market demand function is A − p = Q, A > 0. Firm 1
has zero cost of production. Firm 2 bears fixed a cost of f , f > 0. We assume that there are prices
such that (A−p)p
2 − f > 0. Is there any pure strategy Nash equilibrium.

5. The market demand function is 1 − p = Q Suppose there are two firms 1 and 2. The capacity of
firm 1 is 13 and the capacity of firm 2 is 14 . The cost of production is zero till capacity for each firm.
The firms cannot produce more than its capacity. Each firm sets a price. Find the pure strategy
Nash Equilibrium.

Answer

1. a) There are two firms 1, 2. The inverse market demand function is 50 − 2q = p, Q = q1 + q2 .

Firms 1 and 2 simultaneously chooses output q1 and q2 respectively. The payoff of firm 1, profit is
π1 (q1 , q2 ) = (50−2(q1 +q2 ))q1 −2q1 . The payoff of firm 2,profit is π2 (q1 , q2 ) = (50−2(q1 +q2 ))q2 −2q2 .
Firm 1 and 2, each maximizes profit taking the output of the other firm as given.
∂((50 − 2(q1 + q2 ))q1 − 2q1 )
= 0 at optimum point.
∂q1
We get 48 − 2q2 = 4q1 , the best response function of firm 1.
∂((50 − 2(q1 + q2 ))q2 − 2q2 )
=0
∂q2
We get 48 − 2q1 = 4q2 , the best response function of firm 2.

b) We get the Cournot Nash equilibrium by solving 48 − 2q2 = 4q1 and 48 − 2q1 = 4q2 . Output of
each firm is q1 = q2 = 8. The profit π1 = 128, π2 = 128.

2. The market demand function is A − Q = p. There are three firms 1, 2 and 3, each producing
homogeneous output qi , i = 1, 2, 3. The cost function of each firm is cqi , i = 1, 2, 3.
It is a two stage game. In stage I, firm 1 chooses output q1 . In stage II, firm 2 and 3 simultaneously
chooses q2 and q3 after observing q1 .
We find the subgame perfect Nash equilibrium using backward induction.
First we find the output of firm 2 and 3, q2 and q3 in stage II, given q1 . The payoff,(profit) of firm 2
is π2 (q1 , q2 , q3 ) = (A−(q1 +q2 +q3 ))q2 −cq2 and firm 3 is π3 (q1 , q2 , q3 ) = (A−(q1 +q2 +q3 ))q3 −cq3 .

1
 ∂π2 (q1 ,q2 ,q3 ) ∂((A−(q1 +q2 +q3 ))q2 −cq2 )
∂q2 = ∂q2 = 0. 2q2 = A − c − q1 − q3 , the best response function of firm
2.
∂π3 (q1 ,q2 ,q3 ) ∂((A−(q1 +q2 +q3 ))q3 −cq3
∂q3 = = 0. 2q3 = A − c − q1 − q2 , the best response function of firm
∂q3
3.
Solving 2q2 = A − c − q1 − q3 and 2q3 = A − c − q1 − q2 , we get the best response function of firm
2 and 3 given q1 of firm 1.

In stage I, firm 1 chooses q1 that maximizes π1 (q1 , q2 , q3 ) = (A − (q1 + q2 + q3 ))q1 − cq1 =

(A − (q1 + A−c−q
3
1
+ A−c−q
3
1
))q1 − cq1 .
A−c−q1 A−c−q1
∂π1 (q1 ,q2 ,q3 ) ∂((A−(q1 + + ))q1 −cq1 )
∂q1 = 2
∂q1
3
= 0, at optimum point. Solving the above equation
A−c A−c
we get q1 = 2 . Therefore, q2 = 6 and q3 = A−c
6 . This is the subgame perfect Nash equilib-
rium.

3. There are two firms 1, 2. The demand function faced by firm 1 is

D(p1 ) = N, if p1 < p2 .
D(p1 ) = N2 , if p1 = p2 .
D(p1 ) = 0, if p1 > p2 or p1 > 10.
We get similar demand function for firm 2 also.
The payoff (profit) of firm 1 is
π1 = p1 N, if p1 < p2
π1 = p1 N2 , if p1 = p2 .
π1 = 0, if p1 > p2 or p1 > 10.
We need to find the pure strategy Nash equilibrium. The profit function of firm 1 has been plotted
in figure 1. If p2 = p∗ then best response of firm 1 is to set p1 = p∗ − . In the figure 1, it is clear
that by undercutting price the profit of firm 1 increases. So, there will be continuous undercutting
of prices by both the firms. p1 = p2 = 0 is the pure strategy Nash equilibrium.

4. There are two firms 1 and 2. The demand function of firm 1 is,
D(p1 ) = (A − p1 ), if p1 < p2 .
D(p1 ) = (A−P2
1)
, if p1 = p2 .
D(p1 ) = 0, if p1 > p2 .
We get similar demand function for firm 2 also.
The payoff (profit) of firm 1 is
π1 = p1 (A − p1 ), if p1 < p2
π1 = p1 (A−p
2
1)
, if p1 = p − 2.
π1 = 0, if p1 > p2 .
The payoff (profit) of firm 2 is
π2 = p2 (A − p2 ) − f, if p1 > p2
π2 = p2 (A−p
2
2)
, if p1 = p − 2.
π2 = 0, if p1 < p2 .
For firm 2, we get p and p̄ such that p2 (A−p
2
2)
− f < 0, f or p2 < p, p2 (A−p
2
2)
− f < 0, f or p2 > p̄
and p2 (A−p
2
2)
− f ≥ 0, f or p ≤ p2 ≤ p̄.
Since revenue function is same for firm 1 and 2 so the monopoly price pM is same for both firm.
Firm 2 will not set any price p2 < p. At p1 = p2 = p, π1 > 0. From figure 2, we see that if p1 = p−,
the profit of firm 1 is high. If firm 1 set p1 = p − 2 , π1 (p1 = p − 2 ) > π(p1 = p − ) because
monopoly price pM > p. So firm 1 increases price p1 , if p1 < p. As firm 1 keeps on increasing price,
p1 = p, so again profit falls because demand is shared. This shows that there is no pure strategy
Nash equilibrium.

5. Suppose capacity of firm 1 is k1 and capacity of firm 2 is k2 . We know that, if k1 ≤ R1 (k2 ) and
k2 ≤ R2 (k1 ) then the pure strategy of each firm is to set same price and the price is given by
p = 1 − (k1 + k2 ). The capacity of firm 1 is 31 and firm 2 is 41 . First, we find the Cournot reaction
∂(1−(q1 +q2 ))q1
function. π1 = (1 − (q1 + q2 ))q1 and π2 = (1 − (q1 + q2 )q2 . ∂π
∂q1 =
1
∂q1 = 0. ⇒ 2q1 = 1 − q2 .
2
The reaction function of firm 1.

∂π2
∂q2 = ∂(1−(q∂q
1 +q2 ))q2
2
= 0. ⇒ 2q2 = 1 − q1 . The reaction function of firm 2.
By solving the reaction function, we get the Cournot output is q1C = 31 and q2C = 31 . The capacity of
firm 1 is same as Cournot output and capacity of firm 2 is less than Cournot output. So, in this case
the pure strategy Nash equilibrium price is that both firms set the same price and p = 1 − 13 − 41 ,
5
p = 12 .