Question

The demand curve for luminous socks is given by

Q=50 - 0.5P

and the total cost function for any curve in the industry is C=4Q

(d) Suppose two Cournot firms operate in the market.

1. (i) Derive reaction functions

2. (ii) Equilibrium outputs

3. (iii) Price

4. (iv) Profit levels for the two firms

5. (e) Draw the reaction curves and indicate the Cournot equilibrium on the diagram.

6. (f) In your own words, what does the reaction curve represent?

(g) If the two firms decide to collude, what is industry output and the market price? What is the total profit?

Answer 1

Q = 50 - 0.5P

0.5P = 50 - Q

P = 100 - 2Q = 100 - 2Q1 - 2Q2

TC1 = 4Q1, therefore MC1 = dTC1/dQ1 = 4

TC2 = 4Q2, therefore MC2 = dTC2/dQ2 = 4

(d)

(i)

For firm 1,

Total revenue (TR1) = P x Q1 = 100Q1 - 2Q1² - 2Q1Q2

Marginal revenue (MR1) = TR1/Q1 = 100 - 4Q1 - 2Q2

Equating MR1 and MC1,

100 - 4Q1 - 2Q2 = 4

4Q1 + 2Q2 = 96

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

For firm 2,

Total revenue (TR2) = P x Q2 = 100Q2 - 2Q1Q2 - 2Q2²

Marginal revenue (MR2) = TR2/Q2 = 100 - 2Q1 - 4Q2

Equating MR2 and MC2,

100 - 2Q1 - 4Q2 = 4

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

(ii)

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

3Q2 = 48

Q2 = 16

Q1 = (96 - 4Q2)/2 [From (2)] = [96 - (4 x 16)]/2 = (96 - 64)/2 = 32/2 = 16

(iii)

Q = Q1 + Q2 = 16 + 16 = 32

P = 100 - (2 x 32) = 100 - 64 = 36

(iv)

Profit, firm 1 = Q1 x (P - MC1) = 16 x (36 - 4) = 16 x 32 = 512

Profit, firm 2 = Q2 x (P - MC2) = 16 x (36 - 4) = 16 x 32 = 512

(v)

From (1), When Q1 = 0, Q2 = 48 (Vertical intercept) and when Q2 = 0, Q1 = 48/2 = 24 (Horizontal intercept).

From (2), When Q1 = 0, Q2 = 48/2 = 24 (Vertical intercept) and when Q2 = 0, Q1 = 48 (Horizontal intercept).

In following graph, BR1 and BR2 are reaction functions for firm 1 and firm 2 respectively, intersecting at point A with equilibrium output being Q1* for firm 1 (= 16) and Q2* for firm 2 (= 16).

NOTE: As per Answering Policy, first 5 parts of first sub-parts are answered.