Suppose that inverse demand is given by D(Q) = 56 2Q, Q = q1 + q2 and the cost f

Question

Suppose that inverse demand is given by D(Q) = 56 2Q, Q = q1 + q2 and the cost function is TC(qi ) = 20qi + f Find the Stackelberg equilibrium and compare it to the Cournot equilibrium. 6. Demand and costs are as given in the preceding question. (a) Find the limit output for fixed costs ( f ) equal to 50, 32, 18, and 2. (b) What is the SPNE for the entry game with the following timing: in the first-stage firm 1 can commit to its output; in the second stage firm 2 can enter and choose its output for fixed costs equal to 50, 32, 18, and 2?

Explanation / Answer

Q = Q1 + Q2

So, D(Q) = P = 56 – 2Q1 – 2Q2

TR1 = P x Q1 = 56Q1 – 2Q12 – 2Q1Q2

MR1 = dTR1 / dQ1 = 56 – 4Q1 – 2Q2

TR2 = P x Q2 = 56Q2 – 2Q1Q2 – 2Q22

MR2 = dTR2 / dQ2 = 56 – 2Q1 – 4Q2

TC = 20Q + F

MC = dTC / dQ = 20

(a) Cournot equilibrium:

Firm 1 will equate MR1 with MC:

56 – 4Q1 – 2Q2 = 20

Or,

2Q1 + Q2 = 18    (1) [Firm 1’s reaction function]

Firm 2 will equate its MR with MC:

56 – 2Q1 – 4Q2 = 20

Q1 + 2Q2 = 18   (2) [Firm 2’s reaction function]

Solving (1) & (2):

3Q2 = 18, or Q2 = 6

Q1 = 18 – 2Q2 = 6

Q = 12

P = 56 – 2Q = 32

(b) Stackelberg

Firm 1 sets output first & firm 2 takes firm 1's output as fixed.

Firm 2's response function is the same as in Cournot's model:

Q1 + 2Q2 = 18   Or, Q2 = 9 – 0.5Q1

Substituting this in Firm 1’s total revenue, TR1:

TR1 = 56Q1 – 2Q12 – 2Q1Q2

= 56Q1 – 2Q12 – 2Q1 x (9 – 0.5Q1)

= 38Q1 - 2Q12

MR1 = 38 – 4Q1

Equating MR1 = MC,

38 – 4Q1 = 20

4Q1 = 18, Or Q1 = 4.5

So, Q2 = 9 – 0.5Q1 = 6.75

Q = 11.25

P = 56 – 2Q = 33.5

NOTE: There are 3 questions in total, the 1st one has been answered.

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