Question

Suppose there are two firms operating in a market. The firms produce identical products, and the total cost for each firm is given by C = 10qi, i = 1,2, where qi is the quantity of output produced by firm i. Therefore the marginal cost for each firm is constant at MC = 10. Also, the market demand is given by P = 106 –2Q, where Q= q1 + q2 is the total industry output.

The following formulas will be useful:

If market demand is given by P = a –bQ, then

• MR1 = a – 2bq1 – bq2

• MR2 = a – bq1 – 2bq2

For parts a-c, assume the firms choose their quantities simultaneously.

a) What is firm 1’s reaction function? (Write the equation.)

b) Determine the Nash equilibrium in quantities; that is, how much output will each firm

produce in equilibrium?

c) What will be the market price?

Answer 1

For firm 1, MC1 = dC/dq1 = 10

For firm 2, MC2 = dC/dq2 = 10

P = 106 - 2Q1 - 2Q2

(a)

For firm 1,

MR1 = 106 - 4Q1 - 2Q2

Equating with MC1,

106 - 4Q1 - 2Q2 = 10

4Q1 + 2Q2 = 96

2Q1 + Q2 = 48.........(1) [Reaction function, firm 1]

(b)

For firm 2,

MR2 = 106 - 2Q1 - 4Q2

Equating with MC2.

106 - 2Q1 - 4Q2 = 10

2Q1 + 4Q2 = 96........(2) [Reaction function, firm 2]

Nash equilibrium is obtained by solving (1) and (2). Subtracting (1) from (2),

3Q2 = 48

Q2 = 16

Q1 = (48 - Q2)/2 [From (1)] = (48 - 16)/2 = 32/2 = 16

(c)

Q = Q1 + Q2 = 16 + 16 = 32

P = 106 - (2 x 32) = 106 - 64 = 42