Question

A perfectly competitive industry has a large number of potential entrants. All firms have identical cost structure and minimize the unit cost at the same point. (a.) If $STC=0.5{}^{}+50,$derive the MC and AC functions. (b.) Find the quantity and the cost at the point where the unit cost is minimized. (c.) If total market demand is P = 30 - 1/30Q, what is the price and the number of firms needed to satisfy the total market demand. (d.) Derive the short run market supply. (e.) If the demand shifts to P = 60 - 1/30Q, what will the new market equilibrium price and quantity be? (f.) What will output be per firm be in the short-run?

Answer 1

Answer :

(1)

MC = dSTC / dq = q

AC = STC / q = 0.5q + (50 / q)

(2)

AC is minimized when dAC / dq = 0

0.5 - (50 / q2) = 0

(50 / q2) = 0.5

q2 = 100

q = 10

STC ($) = 0.5 x 10 x 10 + 50 = 50 + 50 = 100

(3) Since all firms minimize unit cost at same point, industry is in long run equilibrium where

AC = MC = Price

When AC = MC,

0.5q + (50 / q) = q

(50 / q) = 0.5q

q = 10 (From part (2))

MC = q = 10

Equating with price,

30 - (Q / 30) = 10

Q / 30 = 20

Q = 600

Number of firms = Q / q = 600 / 10 = 60

(4)

Short run supply curve for each firm is its MC. So,

Supply function: P = q

Market supply is horizontal summation of individual supply curves.

Q / q = 60

q = Q / 60

Market supply curve: P = Q / 60