Question

Consider a firm's profit function, II(x) = R(2) -C(2), where R(x) is total revenue as a function of output (2), and C(x) is total cost as a function of output

(a) Under perfect competition, each firm is a price taker. Assuming a competitive market price, p* = 10 and a cost function, C(x) = (x – 5), express the firm's profit as a function of x.

(b) Find the competitive firm's profit maximizing level of output, r* (Hint: maximize the firm's profit by taking the first derivative of the profit function, setting it equal to zero, and solving for the level of output, r*).

(c) If the firm were only interested in minimizing costs, what level of output would it choose?

Answer 1

1)

$Revenue = px = 10x\\ Profit = Revenue - cost\\ p(x) = 10x - (x-5)^2\\ p(x) = 10x - x^2 -25 + 10x\\ = 20x - x2 -25$

2)

$P(x) = 20-2x =0\\ x = 10$

Profit Maximizing level of output = 10

3)

$C(x) = (x-5)^2\\ C(x) = 2(x-5) = 0\\ x -5 = 0\\ x = 5$

Output that minimize cost is 5