Question

Suppose a firm operating in a perfectly competitive industry has costs in the short run given by:

SRTC = 8 + 1/2Q^2 and therefore MC = q.

if the minimum point of the short-run ATC curve for all firms(existing and potential)is also the minimum point of the long run average cost curve (LRAC), calculate the long-run equilibrium price, market quantity, and firm quantity. What is the long-run equilibrium number of firms in the industry?

Answer 1

Solution:

SRTC = 8 + 1/2Q²                         MC = q

Derive SRATC:

SRATC = $\frac{TC}{Q} = \frac{8 + 0.5Q^{2} }{Q} = \frac{8}{Q} + 0.5Q$

Derive the derivative of the average total cost:

$\frac{\partial SRATC} {\partial q} = \frac{-8} {q^{2} } + 0.5$

Set the derivative equal to zero and solve for q:

$\frac{-8} {q^{2} } + 0.5 = 0$

0.5q² = 8

q² = 16

q = 4

Long run equilibrium price: AC = $\frac{8} {q } + 0.5$

AC = $\frac{8} {4 } + 0.5(4)$ = 2 + 2 = 4

Long run equilibrium price = 4

SRTC = 8 + 1/2Q² = 8 + 0.5(4)² = 8 + 8 = 16

The long-run equilibrium number of firms in the industry = $\frac{16} {4}$ = 4 Firms