Consider the Cournot duopoly game with demand p = 100 - (q_1 + q_2) and variable

Question

Consider the Cournot duopoly game with demand p = 100 - (q_1 + q_2) and variable costs c_i(q_i) = 0 for i element {1, 2}. The twist is that there is now a fixed cost of production k > 0 that is the same for both firms. Assume first that both firms choose their quantities simultaneously. Model this as a normal-form game. Write down the firm's best-response function for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique Now assume that firm 1 is a "Stackelberg leader" in the sense that it moves first and chooses q_1. Then after observing q_1 firm 2 chooses q_2. Also assume that if firm 2 cannot make strictly positive profits then it will not produce at all. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium of this game for k = 25. Is it unique How does your answer in (c) change for k = 225

Explanation / Answer

Firm 1’s profit is

            1         = revenue of firm 1 – cost of firm 1

                        = PQ1 – k

                        = (100 – Q1 – Q2)Q1 – k

                        = 100Q1 – Q12 – Q1Q2 – k

To maximize profit, differentiate the above function with respect to Q1 and equate to 0.

            1/Q1 = 0

            100 – 2Q1 – Q2 = 0

            Q1 = 50 – Q2 /2                                                …… (1)

which is the best response function of firm 1.

Similarly, firm 2’s best response function is

            Q2 = 50 – Q1/2                                     …… (2)

The strategy profile is (50 – Q2 /2, 50 – Q1 /2).

(b)

Substitute (2) in (1).

            Q1 = 50 – (50 – Q1/2)/ 2

            Q1 = 50 – (25 – Q1/4)

            Q1 = 25 + Q1/4

            3Q1/4 = 25

            Q1 = 100/3

And

            Q2 = 50 – (100/3)/2 = 100/3

Therefore, under Cournot equilibrium, each firm will produce approximately 33 units, regardless of the value of k (provided that the profit should be positive).

Firm 1’s profit is

            1         = 100(33.33) – (33.33)2 – (33.33) (33.33) – 1000

                        = $111.22

c.

Given firm 1 produces Q1 units of output, firm 2’s best response function is

            Q2 = 50 – Q1/2

Firm 1’s profit function is

Firm 1’s profit is

            1         = revenue of firm 1 – cost of firm 1

                        = PQ1 – k

                        = (100 – Q1 – Q2)Q1 – k

                        = 100Q1 – Q12 – Q1Q2 – k

Substitute Firm 2’s best response function in firm 1’s profit function.

            1         = 100Q1 – Q12 – Q1(50 – Q1/2) – k

                        = 100Q1 – Q12 – 50Q1 + Q12/2 – k

                        = 50Q1 – Q12/2 –k

To maximize profit, differentiate the above function with respect to Q1 and equate to 0.

            1/Q1 = 0

            50 – Q1 = 0

            Q1 = 50

Therefore, Firm 1 will produce 50 units.

Firm 2’s best response is

            Q2 = 50 – 50/2

            Q2 = 25

Firm 2’s profit is

            2         = revenue of firm 2 – cost of firm 2

                        = PQ2 – k

                        = (100 – 50 – Q2)Q2 – k

                        = (50 – 25)25 – k

                        = 625 – k

Firm 2’s profit is positive when

2         >          0

625 – k            >          0

            k          < 625

For k = 25, firm 2’s profit is positive. So the Nash solution is that firm 1 will produce 50 units and firm 2 twenty five.

d. When k is 225, it is still below 625. So the answer does not change.

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