Question

Suppose a firm's total cost function in the long run cost c(y) = (y²/2048) + (1/y)

a) At what price will the firm start offering positive supply?

b) What amount will it offer at the price in (a)?

Answer 1

in the long run the firm will produce an output corresponding to the minimum of average total cost. ALso the price would be equal to the average total cost. The firm in the long run earns only normal profits.

c(y) = (y²/2048) + (1/y)

AC = y/2048 + 1/y^2

min ATC = dATC/dy = 1/2048 - 2/y^3 = 0

2/y^3 = 1/2048

y^3 = 2048*2 = 4096

y^3 = 16*16*16 or y = 16 units

so at an output y= 16 the average cost is lowest. This means that the firm will maximize output at y=16

At y=16, AC = y/2048 + 1/y^2 =16/2048+1/((16)^(2)) = P* = $0.01172 (this is the price at which ATC is minimum and also the price charged by the firm in long run. In the long run the firm will start positive supply at this price. Below this price the firm will shut down in long run.