Question

Given an aggregate demand function Q( p) = 54 − 2 p and a cost function for each firm of C(q) = 3q^ 3+ 29. (Hint: we must have q ≥ 0, so when you look at the roots, pick the non- negative one.)

(a) Suppose there are 36 firms. Setup and solve the firm profit maximization problem. Then solve for the price, quantity, and profits for each individual firm and aggregate equilibrium quantity, given that the number of firms is fixed.

(b) Does the firm shutdown in the short run? Explain. Do you expect firms to exit or enter in the long run? Explain.

(c) Suppose now that firms can enter and exit. Solve for the equilibrium quantity for each firm, aggregate quantity, price, and number of firms

Answer 1

For a single firm, cost is 3q^3 + 29. This implies MC = 9q^2. Supply function of a firm is rising MC. Here supply function is P = 9q^2 or q = P^0.5/3. For 36 firms, supply function is Qs = 36q = 12P^0.5

Since at the market equilibrium, Qd = Qs, we have

12P^0.5 = 54 – 2P

Let P^0.5 = x. then we have

12x = 54 – 2x^2

x^2 + 6x – 27 = 0

(x – 3)(x + 9) = 0

This gives x = 3 and so market price is P = x^2 = 9. At P= 9, Qs = 12*3 = 36 units and each firm produces 1 unit.

At q = 1 , ATC = (3q^3 + 29)/q = 32 and minimum AVC = 0. Hence there is loss but firm will leave in the long run and will continue operating in short run.

Long run price is ATC = MC. (3q^3 + 29)/q = 9q^2 or 3q^3 + 29 = 9q^3. This gives 6q^3 = 29 or q = 1.69. Each Firm produces 1.69. Hence long run price is 9*(1.69^2) = 25.72. Market quantity = 54 – 2*25.72= 2.54 and so there will only be 1 firms in the long run.