Question

The market for sandwiches consists of two sandwich shops operating in the local market that produce near identical products and that the inverse demand for sandwiches over the lunch hour is given by P = 280 − 2(Q1 + Q2). In this market each firm independently produces a quantity of output, and these quantities are then sold in the market at a price that is determined by the total amount produced by the two firms. The cost function for Shop 1 is C1(Q1) = 3Q1, and costs for Shop 2 is C2(Q2) = 2Q2. Given this information answer the following:

(a) Calculate the marginal revenue for each shop.

(b) Calculate the reaction function for each shop.

(c) Calculate the level of output each shop will produce if the market is in equilibrium.

(d) Calculate the equilibrium profits for each shop.

Answer 1

P = 280 - 2Q1 - 2Q2

MC1 = dC1/dQ1 = 3

MC2 = dC2/dQ2 = 2

(a)

For Shop 1,

TR1 = P x Q1 = 280Q1 - 2Q12 - 2Q1Q2

MR1 = TR1/Q1 = 280 - 4Q1 - 2Q2

For Shop 2,

TR2 = P x Q2 = 280Q2 - 2Q1Q2 - 2Q22

MR2 = TR2/Q2 = 280 - 2Q1 - 4Q2

(b)

Setting MR1 = MC1,

280 - 4Q1 - 2Q2 = 3

4Q1 + 2Q2 = 277..........(1) [Reaction function, Shop 1]

Setting MR2 = MC2,

280 - 2Q1 - 4Q2 = 2

2Q1 + 4Q2 = 278..........(2) [Reaction function, Shop 2]

(c)

(2) x 2 yields:

4Q1 + 8Q2 = 556.........(3)

4Q1 + 2Q2 = 277..........(1)

(3) - (1) yields:

6Q2 = 279

Q2 = 46.5

Q1 = (278 - 4Q2)/2 [From (2)] = [278 - (4 x 46.5)]/2 = (278 - 186)/2 = 92/2 = 46

(d)

Q = Q1 + Q2 = 46 + 46.5 = 92.5

P = 280 - (2 x 92.5) = 280 - 185 = 95

Profit, Shop 1 = Q1 x (P - MC1) = 46 x (95 - 3) = 46 x 92 = 4,232

Profit, Shop 2 = Q2 x (P - MC2) = 46.5 x (95 - 2) = 46.5 x 93 = 4,324.5