Question

In: Economics

In this market, price is given by P= 24 - Q/2. Firm 1 moves first, the...

In this market, price is given by P= 24 - Q/2. Firm 1 moves first, the firm 2. The firms have the cost functions C(q)= q^2.

1. Find the marginal revenue for firm 2.

2. What is the reaciton function for firm 2?

3. Find the marginal revenue for firm 1.

4. What is the equilibrium price and quantity?

Expert Solution

P = 24 - 0.5Q = 24 - 0.5q1 - 0.5q²

TC1 = q1², so MC1 = dTC1/dq1 = 2q1

TC2 = q2², so MC2 = dTC2/dq2 = 2q2

(1)

For Firm 2 (Follower),

Total revenue (TR2) = P x q2 = 24q2 - 0.5q1q2 - 0.5q2²

Marginal revenue (MR2) = TR2/q2 = 24 - 0.5q1 - q2

(2)

Equating MR2 and MC2,

24 - 0.5q1 - q2 = 2q2

0.5q1 + 3q2 = 24

Multiplying by 2,

q1 + 6q2 = 48 ............(1) (Reaction function, firm 2)

(3)

Firm 1 considers this as known. For Firm 1,

q2 = (48 - q1)/6

TR1 = P x q1 = 24q1 - 0.5q1² - 0.5q1q2

= 24q1 - 0.5q1² - 0.5q1 x [(48 - q1)/6]

= 24q1 - 0.5q1² - q1 x (24 - 0.5q1)/6]

= 24q1 - 0.5q1² - 4q1 + (q1²/12)

= 20q1 - (5q1²/12)

MR1 = dTR1/dq1 = 20 - (10q1/12)

MR1 = 20 - (5q1/6)

(4)

Equating MR1 and MC1,

20 - (5q1/6) = 2q1

120 - 5q1 = 12q1

17q1 = 120

q1 = 7.06 (considering integer value only for output)

q2 = (48 - 7.06)/6 = 40.94/6 = 6.82

Q = q1 + q2 = 7.06 + 6.82 = 13.88

P = 24 - (0.5 x 13.88) = 24 - 6.94 = 17.06

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