ECON6006 Market structure and strategic behaviour Mid-semester Test Answers 2022
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ECON6006
Market structure and strategic behaviour
Mid-semester Test Answers
2022
1. Take a look at your student ID number (SID). Define x as the sum of the digits of your SID. For example, if your SID is 123456789, then x = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.
Consider the following market. There are n firms in the market. Demand is given by P(Q) = 12(x + 1) - Q,
and each firm has the cost function
C(q) = 100 + 12q,
where q is firm quantity, Q is market quantity, and P is the market price.
(a) What is the value of x?
ANS: We will use the example SID from above: SID = 123456789.
(b) Suppose n = 1. Find the monopolist’s output and price.
ANS: The monopolist earns profits of
[1 mark ]
[3 marks]
π = Q(552 - Q - 12) - 100
= Q(540 - Q) - 100.
Taking FOCs gives
0 = 540 - 2Q
Q = 270.
The market price is given by P = 552 - Q = 282.
(c) Suppose n = 2 firms engage in simultaneous quantity competition in a single period.
i. Find the reaction function for each firm. [3 marks] ANS: For Firm 1,
π 1 = q1 (552 - Q - 12) - 100
= q1 (540 - Q) - 100.
Taking FOCs gives
0 = 540 - 2q1 - q2
q1 = 270 - q2 /2.
This is the reaction function for Firm 1. Similarly, the reaction function for Firm 2 is q2 = 270 - q1 /2.
ii. Find the Nash equilibrium outputs of both firms. [3 marks] ANS: In the Nash equilibrium, both firms operate on their reaction function. Let q1 = q2 = q in a symmetric equilibrium. Then,
q = 270 - q/2
q = = 180.
(d) Suppose n = 2 firms engage in sequential quantity competition. Firm 1 chooses their output
q1 . Then, Firm 2 observes q1 before choosing their output, q2 . Find the subgame perfect Nash equilibrium outputs of both firms. [5 marks] ANS: We will solve by backward induction. First, observe that Firm 2 will play according to the reaction function that we derived above. Firm 1 will anticipate this reaction function when choosing output. Rewrite Firm 1’s profit function as follows:
π 1 = q1 (540 - Q) - 100
= q1 (540 - q1 - 270 + q1 /2) - 100
= q1 (270 - q1 /2) - 100.
Take FOCs for Firm 1:
0 = 270 - q1
q1 = 270.
Substituting into Firm 2’s reaction function gives q2 = 270 - 270/2 = 135.
(e) Suppose there are n = 2 firms. Bob makes output decisions for Firm 1 and Jane makes output decisions for Firm 2. Bob and Jane simultaneously choose output. Bob owns 100% of Firm 1 and 50% of Firm 2. Jane owns the remaining 50% of Firm 2. Find the Nash equilibrium outputs set by Bob and Jane. Explain why your answer differs from part 1(c)ii. [5 marks]
ANS: Just as before, Jane wants to maximise the profits earned by Firm 2. Jane therefore has the same reaction function as before:
q2 = 270 - q1 /2.
Bob has payoffs:
πB = π 1 + 0.5π2
= q1 (552 - Q - 12) - 100 + 0.5(q2 (552 - Q - 12) - 100) = q1 (540 - Q) + q2 (540 - Q)/2 - 150.
Take FOCs for Bob:
0 = 540 - 2q1 - q2 - q2 /2
q1 = 270 - 3q2
Combining the reaction functions gives
q1 = 270 - (270 - q1 /2)
q1 =
q1 = 108,
q2 = 216.
In the Nash equilibrium, Firm 1 produces a lot less than before. This is because Bob internalises the effect of Firm 1’s output on the profits of Firm 2. Producing less for Firm 1 increases the profits of Firm 2. This improves Bob’s payoff because he has a 50% stake in Firm 2.
2. Two firms compete in the market for a homogeneous product. Each firm has a capacity of K units. Market demand is given by
Q(p) = 6(x +10) - p,
where Q = q1 + q2 is aggregate output, and p is the price paid by consumers. For each firm, the cost of producing q units of output is given by:
C(q) = 60q.
The firms compete by simultaneously choosing price in a single period. The lowest priced firm cap- tures the whole market (up to their capacity constraints), while the higher priced firm serves any residual demand.
(a) Suppose that K = 300.
i. Calculate the monopoly price. [2 marks] ANS: The monopolist earns profits
π = (6 x 55 - p)(p - 60)
= (330 - p)(p - 60).
Solving the FOCs for a maximum gives
0 = 330 - 2p +60
p = 195.
At this price, the quantity demanded is Q = 135. Notice that Q < K = 300, so the monopolist is able to serve market demand at this price.
ii. Solve for the reaction function of each firm. Is there a Nash equilibrium in pure strategies? Explain why or why not. [5 marks] ANS: Consider the perspective of Firm 1, and suppose that Firm 2 were to set a price at or above the marginal cost of 60. At this price, market demand is equal to Q(p2 ) = 330 - p2 <
270. But this is less than the capacity of each firm, K = 300. Therefore, it is not possible for firms to obtain positive residual demand if they were to relent. As a result, the model is reduced to the Bertrand model.
Each firm has the following reaction function. Undercut to the monopoly price if your rival sets a price above the monopoly price; marginally undercut if your rival sets a price above marginal cost (but below the monopoly price); and set a price above your rival if your rival sets a price below marginal cost. In the unique Nash equilibrium, both firms set a price equal to marginal cost. This is a Nash equilibrium because neither firm has an incentive to change price.
(b) Suppose that K = 100.
i. Calculate the monopoly price. [2 marks] ANS: In the previous question, we derived a monopoly price of p = 195 and an output of Q = 135. Notice that, with a capacity of K = 100, the monopolist is unable to satisfy market demand. She is better off producing at her capacity, and setting the price
= 330 - K = 230.
ii. Solve for the reaction function of each firm. Is there a Nash equilibrium in pure strategies? Explain why or why not. [5 marks] ANS: Consider the perspective of Firm 1 and suppose that Firm 2 sets the price p2. If Firm
1 relents, they obtain profits:
πr = (330 - p - K)(p - 60)
= (230 - p)(p - 60).
The FOCs for optimisation are
0 = 230 - 2p + 60
pr = 145.
At this price, residual demand is
Qr = 230 - pr
= 85 < K.
Firm 1 is indifferent between relenting and undercutting when
πr = πu
852 = K(p2 - 60)
p2(*) = 60 + = 132.25.
Firm 1’s reaction function is: undercut to the monopoly price of 230 if p2 > 230; marginally undercut p2 when p2 > p2(*); and set the price pr when p2 < p2(*). Firm 1 has an equivalent reaction function. There is no Nash equilibrium in pure strategies because the reaction functions do not intersect.
(c) Suppose that K = 30.
i. Calculate the monopoly price. [1 mark ] ANS: In the first part, we derived a monopoly price of p = 195 and an output of Q = 135. Notice that, with a capacity of K = 30, the monopolist is unable to satisfy market demand. She is better off producing at her capacity, and setting the price
= 330 - K = 300.
ii. Is there a Nash equilibrium in pure strategies? Explain why or why not. [5 marks] ANS: Consider the perspective of Firm 1 and suppose that Firm 2 sets the price p2. If Firm 1 relents, they obtain profits:
πr = (330 - p - K)(p - 60)
= (300 - p)(p - 60)
The FOCs for optimisation are
0 = 300 - 2p + 60
pr = 180.
At this price, residual demand is
Qr = 330 - K - pr
= 300 - 180 = 120.
Notice that Firm 1 is unable to satisfy the residual demand with their capacity of K = 30. In this case, Firm 1 and Firm 2 will both use all of their capacity. Market output is Q = 2K = 60 and the market price is p = 330 - Q = 270. In the Nash equilibrium, both firms set a price of p = 270 and use all of their capacity.
2022-05-31