# COMM 295 Study Guide - Quiz Guide: Nash Equilibrium, Cournot Competition, Inverse Demand Function

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Commerce 295/FRE 295

Fall 2015

Assignment 2

Assigned: Nov. 9/10.

Due: Nov. 25/26/27.

There are 10 questions. Each question is worth 10 pts. Show your working. Unless otherwise

stated each part of each question is worth 5 pts.

1. Oligopoly and Cartels

a) Recall the Cournot duopoly model of quantity competition between two firms (pp. 360-365).

Suppose Firm 1 has a constant marginal cost equal to $10 and Firm 2 has a constant marginal

cost equal to $20. Fixed costs are zero for both firms. Market demand is Q = 300 – 4P.

Calculate each firm’s best response function as determine the Nash equilibrium quantities.

Show the best response functions and the Nash equilibrium quantities on your graph.

For Firm 1, we calculate the residual marginal revenue as

follows.

Residual demand is given by q1 = 300 – q2 – 4P and inverse

residual demand is given by P =75 -0.25q2 – 0.25q1. Revenue

is q1*P = (75-0.25q2)q1-0.25q1^2. Marginal

revenue is (75-0.25q2)-0.5q1. Similarly, marginal l

revenue for Firm 2 is (75-0.25q1) – 0.5q^2. To find each

Firm’s best response we set the marginal revenue equal

to the marginal cost. Hence we have q1 = (65-0.25q2)/0.5 and

q2 = (55-0.25q1)/0.5. Solving simultaneously we get q1 = 100

and q2=60 as the Nash equilibrium quantities.

Marking Guide: 1 mark for each best response functions. 1 mark for correct Nash equilibrium

quantities. 2 marks for correct diagram with best response functions plotted and Nash equilibrium

labeled.

b) Suppose Firm 2 proposes to Firm 1 that they should form a cartel. According to their

agreement the entire quantity of the product will be produced by the firm with the lower

marginal cost (Firm 1) and they will split the profits evenly. Would Firm 1 benefit from such

an agreement? Explain. (Hint: Calculate the profits for Firm 1 from Cournot competition and

from the cartel.)

The cartel will act like a monopoly hence MR = 75- 0.5Q. Setting this equal to the marginal cost

for Firm 1, we get Q=130. The market price will be 42.5 and total profit will be (42.5 –

10)*130=4225 and the profit for Firm 1 will be 2,112.5. In part (a), Firm 1’s profit is (35-10)*100

= 2,500. Hence profit will be lower with the cartel than without so Firm 1 will not benefit from the

cartel.

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2. Monopolistic Competition and Bertrand Oligopoly

a) Consider a market with many identical firms selling identical products and free entry

(Monopolistic Competition – pp. 378-380). Illustrate the long-run equilibrium outcome for a

representative firm including cost curves. What can you say about each firm’s profit? Is this

outcome efficient (surplus maximizing)? Explain.

In the long-run, with free entry, firms will have zero profit,

i.e. the price will be equal to average cost. On the other hand,

since firms have market power, i.e. they face a downward

sloping demand curve, when they set MR=MC this implies

P > MC and hence the outcome does not maximize total

surplus, i.e. it is not efficient.

Marking Guide: 1 mark for zero profits. 2 marks for

inefficient and indicating how either with P>MC or

showing deadweight loss on diagram (not shown). 2

marks for correct diagram with MC, AC, D, and MR

curves.

b) Consider a market with a small number of identical firms selling identical products without

free entry where firms compete by setting prices (Bertrand Oligopoly – pp. 374-377). There

are 200 customers, each of whom will purchase one unit of output at the lowest available price.

(Thus demand is perfectly inelastic at Q = 200.) Also assume that each firm has a constant

marginal cost of 8 and no fixed costs. What is the Nash equilibrium price (i.e. the Bertrand

equilibrium price) charged by each firm? What can say you about firms’ profits? What

important aspects of the Bertrand equilibrium would change if products were differentiated?

The Nash equilibrium of this model is for both firms to charge P=8 and to split the demand

equally. In this case each firm will earn zero profits. With differentiated products three things

will change: 1) The prices will be above the marginal cost, 2) Firms will earn positive profits,

and 3) the outcome will be sensitive to changes in demand.

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3. Static Games. Two competing firms are each planning to introduce a new product. Each will

decide whether to produce Product A, Product B, or Product C. They will make their choices at the

same time. The resulting payoffs are shown below.

a) Are there any Nash equilibria in pure strategies? If so, what are they? (State the strategies

chosen and the payoffs.) Would the Pareto Criterion help firms coordinate? Explain. (See pp.

391-402.)

There are two Nash equilibria in pure strategies. Each one involves one firm introducing

Product A and the other firm introducing Product C. We can write these two strategy pairs as

(A, C) and (C, A), where the first strategy is for Firm 1. The payoffs for these two strategies

are, respectively, (10, 20) and (20, 10). The Pareto Criterion would not help firms coordinate

here, since there is not a Nash Equilibrium that is mutually beneficial to players.

b) If the managers of both firms are conservative and each follows a maximin strategy, what will

be the outcome? If Firm 1 uses a maximin strategy and Firm 2 knows this, what will Firm 2 do

to maximize profit? Justify your reasoning. (See pp. 408-411)

Recall that maximin strategies maximize the minimum payoff for both players. If Firm 1

chooses A, the worst payoff is -10, with B the worst payoff is -20, and with C the worst is -30.

So Firm 1 would choose A because -10 is better than the other two payoff amounts. The same

reasoning applies for Firm 2. Thus (A, A) will result, and payoffs will be (-10, -10). Each

player is much worse off than at either of the pure-strategy Nash equilibria. If Firm 1 plays its

maximin strategy of A, and Firm 2 knows this, then Firm 2 would get the highest payoff by

playing C. Notice that when Firm 1 plays conservatively, the outcome that results gives Firm 2

the higher payoff of the two Nash equilibria.