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The inverse market demand in a homogeneous product Cournot duopoly is

P=100-2(Q1+Q2), and the costs are given by C(Q1) = 12Q1 and C(Q2) = 20Q2. The implied marginal costs are $12 for firm 1 and $20 for firm 2.

Determine the reaction function for firm 1.

P*Q1 = 100Q1 - 2Q1^2 - 2Q1Q2

HR= 100-4Q1-2Q2 = 12

88-2Q2/4 = Q1

Determine the reaction function for firm 2.

P*Q2 = 100Q2-2Q1Q2 - 2Q2

HR= 100 - 2Q1 - 4Q2 = 20

80 - 2Q1 / 4 = Q2

Calculate the Cournot equilibrium price and quantity.

Suppose firm 1 is a monopoly (firm 2 does not exist), what is firm 1's monopoly output and price?

How does the monopoly price and quantity comparing with Cournot equilibrium in part (c)?

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Kritika Krishnakumar
Kritika KrishnakumarLv10
28 Sep 2019
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Related questions

Suppose the inverse market demand function for a two-firm Cournot model is given by P=49-Q where Q=q1+q2 is the total output produced in the market, P is the market price, q1 is the quantity of output produced by firm 1, q2 is the quantity of output produced by firm 2. The marginal costs of the two firms are MC1=2 and MC2=3. For a linear inverse demand function P=a+bq, the marginal revenue is given by MR=a+2bq.

1. Find the marginal revenue MR1 of firm 1 as a function of q1 and q2.

2. Setting MR1=MC1, determine the reaction function of firm 1.

3. Find the marginal revenue MR2 of firm 2 as a function of q1 and q2.

4. Setting MR2=MC2, determine the reaction function of firm 2.

5. What quantity of output will each firm produce? That is, use the two reaction functions to find q1 and q2.

6. What is the market price P.

7. Determine the profits of each firm assuming there are no fixed costs.

Consider the following game. Firm 1, the leader, selects an output q1, after which firm 2, the follower, observes the choice of q1 and then selects its own output q2. The resulting price is one satisfying the industry demand curve P = 200 - q1 - q2. Both firms have zero fixed costs and a constant marginal cost of 10.

A. Derive the equation for the follower firm's best response function. Draw this equation on a graph with q2 on the vertical axis and q1 on the horizontal axis. Indicate the vertical intercept, horizontal intercept, and slope of the best response function.

B. Determine the equilibrium output of each firm in the leader-follower game. What are each firm's profits in the equilibrium?

C. Now let the two firms choose their outputs simultaneously. Compute the Cournot equilibrium outputs and industry price. Who loses and who gains when the firms play a Cournot game instead of the Stackelberg one?

Stackelberg Duopoly:

Two firms both have the same constant marginal cost of $20 and zero fixed cost; market price P = 140 - 2(q1 + q2). Both firms choose outputs to compete.

(a) Find the subgame perfect equilibrium outcome of the Stackelberg Duopoly game with firm 1 moving first. First, solve for the follower's (firm 2's) best response function. Then solve for the leader's optimal strategy.

(b) Find the Nash equilibrium of the Cournot duopoly under the same assumptions on costs and demand.

(c) Compare the two equilibria. Discuss the differences.

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