Question

In the perfect competitive market,

market demand for shoes : Q^D = 1,000 - 10*P

In this market, the long term average cost of the firms which produce shoes is all same and the formula is : AC = (q_i-10)² / 2 + 20

(1) Price of the shoes at the long term equilibrium?

(2) How many firms participate in this market?

Answer 1

Market demand for shoes :

Qd = 1,000 - 10P

The long term average cost of the firms which produce shoes is all same and the formula is :

AC = (q -10)² / 2 + 20

(a) In long run equilibrium, average cost is equal to marginal cost and profits are 0.;

The total cost will be derived as;

TC = AC*q
TC = q(q -10)²/2 + 20q

Marginal cost = d/dq ( q(q -10)²/2 + 20q)
MC = 20 + q(q-10) + (q -10)²/2

At equilibrium;

MC = AC
20 + q(q-10) + (q -10)²/2 = (q -10)² / 2 + 20
q² - 10q = 0
q(q-10) = 0
q = 0 or q = 10

q = 0 cannot be possible

so, q = 10

However, to find long run price, we know firm produces where;

P = MC = MR

So, we can find marginal cost;

MC = 20 + q(q-10) + (q -10)²/2
= 20 + 10(10-10) + (10-10)²/2
= 20 + 0 + 0
MC = 20

Therefore, equilibrium price will be;

P = 20

(b) Now, given Qd = 1,000 - 10P, we will determine the quantity Qd ;

Qd = 1000 - 10P
= 1000 - 10(20)
= 1000 - 200
Qd = 800

As we know, each firm is producing q = 10 units and the market quantity demanded is Qd = 800, thus there will be;

No. of frims = 800/10
No. of frims = 80