Question

Consider an industry composed by two firms -- Argyle (A) and Blantyre (B) -- that sell a standardized product. They maximize their profits by choosing how much to produce. The total output of this industry (X) is the sum of the output of the two firms (X = x_A+x_B )

Both firms have no fixed cost, and a constant marginal cost equal to c1=10. So the cost function is the same for the two firms, and equal to c(x)=10x

The demand function of consumers is as follows:

X=(210-p)/(2)

Where p is the price of the product.

(b) What is the profit function for Argyle?

(C) what is the profit function for Blantyre?

(d) -URGENT PLEASE ANSWER - Use the principle of profit-maximization to find the best response function (BRF) of each firm.

(e) What is the profit-maximizing level of output for firm Argyle?

(f) What is the profit-maximizing level of output for firm Blantyre?

Answer 1

Demand function is X = (210 - p)/2 or 2X = 210 - p. This implies inverse demand function is P = 210 - 2X.

a) Profit function for Argyle = ?A = revenue - cost = price * quantity - cost

= (210 - 2X)xA - 10xA

= (210 - 2xA - 2xB)xA - 10xA

= 210xA - 2xA^2 - 2xBxA - 10xA

= 200xA - 2xA^2 - 2xBxA

c) Since cost function is same, the game is symmetric and so profit function for Blantyre is ?B = 200xB - 2xB^2 - 2xBxA

d) Maximize profits by placing the marginal profit equal to zero for both firms

200 - 4xA - 2xB = 0 200 - 4xB - 2xA = 0

xA = 50 - 0.5xB and xB = 50 - 0.5xA

These are the best response functions

e) Solve the two BRFs

xA = 50 - 0.5*(50 - 0.5xA)

xA = 50 - 25 + 0.25xA

xA = 25/0.75 = 100/3 = 33.33. The profit-maximizing level of output for firm Argyle is 33.33 units

f) The profit-maximizing level of output for Blantyre xB = 50 - 0.5*33.33 = 33.33units.