Question

Consider a firm’s profit function, p(x)=R(x) - C(x), where R(x) is total revenue as a function is output (x), and C(x) is total cost as a function of output (x).

1/ Under perfect competition, each firm is a price taker. Assuming a competitive market price , p* = 10 and a cost function, C(x) = (x-5)^2 , express the firm’s profit as a function of x.

2/ Find the competitive firm’s profit maximizing level of output, x*.

3/ If the firm we’re only interested in minimizing costs, what level of output would it choose?

Answer 1

1.

$revenue=Px=10x\\pofit=revenue-cost\\=10x-(x-5)^2\\=10x-x^2-25+10x\\=20x-x^2-25$

2.

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

profit maximizing level of output =10

3.

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

output that maximizes cost is 5