Question

1. Suppose the demand function of an industry is P=100-0.5X and there are two firms A and B in it. Futher assume that the firm A has a constant cost function of CA = 5xA and firm B has an increasing cost function CB =0.5xB. (X= xA +xB ).

a. What would the reaction function of each firm?

b. What would be the profit of each firm? c. What would the out put of each firm? d. What would be the equilibrium price?

I'm stuck with this question. Can somebody help me? thank you so much.

Answer 1

P = 100 - 0.5xA - 0.5xB [since X = xA + xB]

MCA = dCA/dxA = 5

MCB = dCB/dxB = 0.5

(a)

For firm A,

Total revenue (TRA) = P.xA = 100xA - 0.5xA² - 0.5xA.xB

Marginal revenue (MRA) = TRA/xA = 100 - xA - 0.5xB

Setting MRA = MCA,

100 - xA - 0.5xB = 5

xA + 0.5xB = 95.............(1) [Reaction function, firm A]

For firm B,

Total revenue (TRB) = P.xB = 100xB - 0.5xA.xB - 0.5xB²

Marginal revenue (MRB) = TRB/xB = 100 - 0.5xA - xB

Setting MRB = MCB,

100 - 0.5xA - xB = 0.5

0.5xA + xB = 99.5

Multiplying by 2,

xA + 2xB = 199.............(2) [Reaction function, firm B]

(b), (c) and (d)

Subtracting (1) from (2),

1.5xB = 104

xB = 69.33

xA = 199 - 2xB [from (2)] = 199 - (2 x 69.33) = 199 - 138.66 = 60.34

X = 69.33 + 60.34 = 129.67

P = 100 - 0.5 x 129.67 = 100 - 63.84 = 36.16

So,

Profit, Firm A = xA.(P - MCA) = 60.34 x (36.16 - 5) = 60.34 x 31.16 = 1880.19

Profit, Firm B = xB.(P - MCB) = 69.33 x (36.16 - 0.5) = 69.33 x 35.66 = 2,472.31