Question

Suppose a perfectly competitive market consists of identical firms with the same cost function given by

C(q)=2q³ - 3q² + 70q

The market demand is

Q_D= 2200 - 10p

What will be the long-run equilibrium price in this market?

Round your answer to the nearest cent (0.01)

Answer 1

The long run equilibrium price in perfectly competetive market is equal to,

Average cost of production = marginal cost of production

So that the economic profit in perfectly competitive market is always zero in the long run.

So let's first calculate the average cost of production,

C(q) = 2q^3 - 3q^2 + 70q

To calculate the average cost of production we need to divide the cost function by quantity q,

Average cost = C(q)/q

Average cost = (2q^3 - 3q^2 + 70q)/q

Average cost = 2q^2 - 3q + 70

Now let's calculate the marginal cost of production.

MC = dC(q)/dq

MC = d(2q^3 - 3q^2 + 70q)/dq

MC = 6q^2 - 6q + 70

Now putting MC = AC

6q^2 - 6q + 70 = 2q^2 - 3q + 70

Now let's solve for q,

6q^2 - 2q^2 = -3q + 6q

4q^2 = 3q

q = 3/4

Now let's calculate either MC or AC since they both going to be equal at q = 3/4.

MC = 6q^2 - 6q + 70

MC = 6(3/4)^2 - 6(3/4) + 70

MC = 6×9/16 - 6×3/4 + 70

MC = 54/16 - 18/4 + 70

MC = (54 - 18×4 + 70×16)/16

MC = (54 - 72 + 1120)/16

MC = (1120 - 18)/16

MC = 1102/16

MC = 68.875.

So the long run equilibrium price in the perfectly competetive market will be equal to, $68.87.