Question

1. Suppose an industry has a duopoly structure. Duopolist 1 has a cost function given by: c1 (y1) = (y1)2 for y1 ≥ 0 . Duopolist 2 has a cost function given by: c2(y2 ) = 12y2 for y2 ≥ 0. Denoting total output produced in the industry by y = (y1 + y2), the inverse demand function for the good produced in the industry is given by: p = 100 – y

   1. Find the reaction function of each duopolist.

   2. Using your answer to part (a), obtain the output levels that will be produced in a Cournot-Nash equilibrium, and the price level in such an equilibrium.

   3. Illustrate your solution in (b) above in a suitable diagram.  

   4. Calculate profits for each firm.

   5. Suppose the two firms merged so that they have cost structure 12y + y2

   6. Find the monopoly price, output and profit. Compare it to the competitive price, output, and profit.

Answer 1

MC1 = dc1/dy1 = 2y1

MC2 = dc2/dy2 = 24y2

p = 100 - y = 100 - y1 - y2

(Part a)

For firm 1,

Total revenue (TR1) = p x y1 = 100y1 - y1² - y1y2

Marginal revenue (MR1) = TR1/y1 = 100 - 2y1 - y2

Equating MR1 and MC1,

100 - 2y1 - y2 = 2y1

4y1 + y2 = 100...........(1) [Reaction function, firm 1]

For firm 2,

Total revenue (TR2) = p x y2 = 100y2 - y1y2 - y2²

Marginal revenue (MR2) = TR2/y2 = 100 - y1 - 2y2

Equating MR2 and MC2,

100 - y1 - 2y2 = 24y2

y1 + 26y2 = 100...........(2) [Reaction function, firm 2]

(Part b)

Cournot equilibrium is obtained by solving (1) and (2). Multiplying (2) by 4,

4y1 + 104y2 = 400......(3)

4y1 + y2 = 100......(1)

(3) - (1) yields:

103y2 = 300

y2 = 2.91

y1 = 100 - 26y2 [From (2) = 100 - (26 x 2.91) = 100 - 75.66 = 24.34

y = 24.34 + 2.91 = 27.25

p = 100 - 27.25 = 72.75

(Part c)

From (1): When y1 = 0, y2 = 100 (Vertical intercept) & when y2 = 0, y1 = 100/4 = 25 (Horizontal intercept).

From (2): When y1 = 0, y2 = 100/26 = 3.85 (Vertical intercept) & when y2 = 0, y1 = 100 (Horizontal intercept).

In following graph, VR1 and BR2 are reaction functions of firms 1 & 2 respectively. Cournot equilibrium is at intersection of BR1 & BR2 at point E with optimal quantities being y1* (= 24.34) and y2* (= 2.91).

(Part d)

MC1 = 2 x 24.34 = 48.68

MC2 = 24 x 2.91 = 69.84

Profit, firm 1 = y1 x (p - MC1) = 24.34 x (72.75 - 48.68) = 24.34 x 24.07 = 585.86

Profit, firm 2 = y2 x (p - MC2) = 2.91 x (72.75 - 69.84) = 2.91 x 2.91 = 8.45

NOTE: As per Chegg Answering Policy, 1st 4 parts are answered.