Question

1. A representative firm with long-run total cost given by TC = 2,000 + 20q + 5q² operates in a competitive industry where the market demand is given by Q_D = 10,000 – 40P.
   1. Find the long-run equilibrium output of the individual firm.
2. 2. Find the long-run equilibrium price.
3. 3. Find the long-run equilibrium output of the industry.

Answer 1

a.

The firm is in the competitive market and in the long-run; therefore, equilibrium output be at the minimum of average total cost (min ATC).

Given, total cost (TC) as below:

TC = 2,000 + 20q + 5q^2

ATC = TC/q = 2,000/q + 20q/q + 5q^2/q

                        = 2,000/q + 20 + 5q

ATC would be the minimum if derivative of ATC with respect to q is 0.

ATC = 2,000/q + 20 + 5q

Derivative of ATC = - 2,000/q^2 + 0 + 5

0 = - 2,000/q^2 + 5

Now, by solving,

2,000/q^2 = 5

2,000 = 5q^2

2,000/5 = q^2

400 = q^2

Square root of 400 = q

q = 20

Answer: required equilibrium output is 20 units.

b.

Equilibrium price (P) in the long-run for a competitive firm is the min ATC. ATC would be the minimum if (q = 20).

P = min ATC

   = 2,000/q + 20 + 5q

   = 2,000/20 + 20 + 5 × 20

= 100 + 20 + 100

   = 220

Answer: required equilibrium price is 220.

c.

The long-run price (220) should be placed in the demand equation in order to get equilibrium industry output (Qd).

Qd = 10,000 – 40P

      = 10,000 – 40 × 220

      = 10,000 – 8,000

      = 2,000 (Answer)