ECOS3005 Mid-semester Test Answers
Industrial Organisation Practice
Section A: Answer all short answer questions in the booklet provided. Please be sure to show your working
and explain your reasoning where appropriate. (Total 40 marks)
1. The demand for good A is given by Q(P) = 75? 0.5P, where Q is the market quantity, and P is the
market price. Production of good A involves costs of C(q) = 400+30q, where q is firm output.
(a) Suppose a single firm operates in the market. Find the profit-maximising price and quantity of
the monopolist. [4 marks]
ANS: Rewrite the demand curve as P= 150?2Q. Monopoly profits are then given by
pim = Q(150?2Q)?30Q?400.
First order conditions yield:
150?4Q?30 = 0
4Q= 120? Q= 30.
Substituting into the demand curve, we have P= 90.
(b) Suppose two firms operate in the market. The firms engage in simultaneous quantity competition
in a single period.
i. Find the reaction function for each firm. [4 marks]
ANS: Firm i chooses output to maximise profits given by
pii = qi(150?2Q?30)?400.
This leads to first order conditions
150?4qi?2q j?30 = 0
qi =
120?2q j
4
qi = 30? q j2 .
This is the reaction function for each firm.
ii. Find the Nash equilibrium outputs of both firms. [3 marks]
ANS: Substituting firm 2’s reaction function into firm 1’s, we obtain
q1 = 30? 30?q1/22
3
4
q1 = 15
q1 = 20.
Similarly, q2 = 20. Each firm produces an output of 20 in equilibrium.
1
(c) Suppose that two firms operate in the market. The firms engage in Stackelberg competition. Firm
1 chooses its output first, then Firm 2 chooses it’s output. Find the output of each firm.
[4 marks]
ANS: The Stackelberg leader (Firm 1) earns profits
pi1 = q1
(
150?2q1?2
(
30? q1
2
)
?30
)
?400
= q1(60?q1)?400
Firm 1 then has FOCs
60?2q1 = 0
q1 = 30.
Firm 1 produces an output of 6 and Firm 2 produces q2 = 30?q1/2 = 15 units.
2. Consider the following game. Firm 1 and Firm 2 have three strategies available, A, B, and C. The first
entry in each cell contains the payoffs for Firm 1 and the second entry contains the payoffs for Firm 2.
Both firms have the common discount factor, δ, where 0 < δ< 1.
Firm 1
Firm 2
A B C
A 0,0 6,?1 2,?2
B ?1,6 4,4 2,2
C ?2,2 2,2 3,3
(a) Suppose the game above is played once. Identify any Nash equilibria. Explain briefly.
[4 marks]
ANS: There are two Nash equilibria: (A,A) and (C,C). In each case, both players are choosing an
optimal strategy given the strategy of their rival. That is, if Firm 1 chooses A, the best response
of Firm 2 is also to play A. Similarly, if Firm 2 chooses A, the best response of Firm 1 is also to
play A. The same reasoning applies to the Nash equilibrium at (C,C).
(b) Suppose the game above is played twice. Consider the following strategy:
? in period 1: play B;
? in period 2: play C if both played B in period 1; otherwise play A.
For what value of δ (if any) is there a subgame perfect Nash equilibria in which both players play
the above strategy? [6 marks]
ANS: To identify a subgame perfect Nash equilibrium, we must ensure that the strategies consti-
tute a Nash equilibrium to each subgame. We can sole this by backward induction. First, note
that in stage 2, the strategies call on either both players to play A or both players to playC. From
part (a), we know that (A,A) and (C,C) are Nash equilibria. Therefore, there is no incentive to
deviate in stage 2 for either player.
2
Next, consider stage 1. Consider the perspective of Firm 1 (everything is symmetric for Firm 2).
By playing B in stage 1, Firm 1 anticipates a total payoff of 4+δ×3. If instead, Firm 1 were to
deviate, the optimal deviation involves playing A instead of B. This yields a payoff of 6+δ×0.
There is no incentive for Firm 1 to deviate if
4+3δ≥ 6
δ≥ 2/3.
Therefore, there is a subgame perfect Nash equilibrium with these strategies if δ≥ 2/3.
3. Two firms compete in the market for a homogeneous product. Each firm has a capacity of 200 units.
Market demand is given by
Q(p) = 500?10p,
where Q= q1 +q2. Each firm has a constant marginal cost and no fixed costs:
C(q) = 10q.
The firms compete by choosing price. The lowest priced firm captures the whole market (up to their
capacity constraints), while the higher priced firm serves any residual demand. The firms interact
sequentially in a single period. First, Firm 1 chooses a price p1. Then, after observing p1, Firm 2
chooses a price p2.
(a) Solve for the reaction function for Firm 2. [8 marks]
ANS: Firm 2 has an incentive to undercut and steal market share if p1 is high, while Firm 2 has
an incentive to relent if p1 is close to marginal cost. To identify Firm 2’s reaction function, we
must find the point of indifference.
If relenting, Firm 2 will choose p2 to maximise profits based on residual demand. Relenting
profits are given by
piR = (500?200?10p2)(p2?10).
Maximising profits leads to the FOCs
300?10p2?10(p2?10) = 0
p2 = 20
Relenting profits are then
piR = (300?10p2)(p2?10) = 100.10 = 1000.
The optimal way to undercut for Firm 2 is to charge a price just below p1. Undercutting profits
will then be given by
piU ≈ 200(p1?10).
3
Firm 2 is indifferent between undercutting and relenting when
piU = piR
200(p1?10) = 1000
p1 = 15.
In fact, when p1 is exactly equal to 15, Firm 2 just prefers relenting to undercutting. This is
because relenting gives a profit of 1000, while undercutting will give a profit just below 1000.
Finally, for completeness, let us calculate the monopoly price. Monopoly profits are pi= (500?
10p)(p?10). Solving first order conditions yields:
500?20p+100 = 0
pm = 30.
Notice that the monopolist’s output at this price is Q= 200, the same as the firm’s capacity.
Summarising, Firm 2’s reaction function is:
p2 = R2(p1) =
?????????
30 if p1 > 30
p1? ε if 15 < p1 ≤ 30
20 if p1 ≤ 15.
(1)
(b) Solve for a subgame perfect Nash equilibrium to this game. Explain. [7 marks]
ANS: Firm 1 will understand Firm 2’s reaction function when setting price. Suppose Firm 1 sets
p1 > 15, but below the monopoly price. Then, Firm 2 will undercut, giving Firm 1 profits of
pi1 = (500?200?10p1)(p1?10).
Firm 1 could choose p1 to maximise these profits. Notice that this problem is exactly the same
as Firm 2’s relenting problem above. Hence, p1 = 20 and pi1 = 1000.
Suppose instead that Firm 1 sets p1 ≤ 15. Then, Firm 2 will relent to p2 = 20. Firm 1 would
then earn profits of
pi1 = 200(p1?10).
This would be maximised (subject to the constraint that p1 ≤ 15) by setting p1 = 15. This leads
to profits of pi1 = 200×5 = 1000.
Hence, Firm 1 is indifferent between p1 = 20 and p1 = 15. There are 2 subgame perfect Nash
equilibria. In the first, Firm 1 sets p1 = 15 and Firm 2 follows the reaction function from equation
(1) above. In the second, Firm 2 sets p1 = 20 and Firm 2 follows the reaction function from
equation (1) above.
4