Question

Pineapples grow in Hawaii all year round and there are many farms and gardens that produce pineapples for the local market. The demand curve for pineapples in Hawaii is given by Q = 240−2P, where Q is in tons of pineapples. The marginal cost of producing an additional ton of pineapples is constant and equal to $50 for all producers in the market.

(a) Given that there are many producers of pineapples in Hawaii and that pineapples are a homogeneous good, let’s assume that the market for pineapples is perfectly competitive. Calculate and show on a graph the equilibrium quantity sold in the market, total sur- plus, and deadweight loss. (Hint: recall the relationship between the supply curve and marginal cost curves of firms.)

(b) Now suppose one of the pineapple producers, Noa, inherits a large sum of money and decides to buy out all of the island’s land on which pineapples can be grown (including all the farms and gardens of his competitors). That is, Noa becomes a monopolist in the market for pineapples in Hawaii. Calculate and show on a graph the new equilibrium quantity and price; consumer, producer, and total surplus; and deadweight loss. [For this problem it will help to recall that when the demand curve is linear, the monopolist’s marginal revenue curve is a line with the same “y-intercept” as the demand curve, but a slope that’s twice as steep as the demand curve.]

(c) If Noa produces Q tons of pineapples, his total cost is 250 + 50Q. Calculate Noa’s profit if he operates at his profit-maximizing quantity and price.

(d) Sketch Noa’s average total cost function on a new graph. Does Noa experience increasing, decreasing, or constant returns to scale?

Answer 1

(a) Case of perfect competition

Given the demand curve as Q = 240 - 2P, the inverse demand function can be derived as: 2P = 240 - Q

=> P = 120 - 0.5Q

At competitive equilibrium, price = AR = MR = MC = 50 (since MC = 50 is given)

Hence, Q = 240 - 2(50) = 140

Therefore, equilibrium price = 50 and equilibrium quantity = 140 under perfect competition

In the below diagram, AD is the inverse demand curve. The equilibrium is at point E, where equilibrium price (P*) = 50 and equilibrium output = 140 (Q*).

Consumer surplus (CS) = area of the traingle AP*E = 0.5 (120 - 50) (140) = 0.5(70)(140) = 4900

Producer surplus (PS) = 0

Hence, total welfare = CS + PS = 4900

There is no deadweight loss (DWL), i.e., DWL = 0 since the equilibrium is socially efficient.

(b) Case of monopolist

The inverse demand as derived above is P = 120 - 0.5Q. Hence, total revenue (TR) = 120 Q - 0.5Q2.

=> Marginal revenue (MR) = d (TR)/d(Q) = 120 - Q

At equilibrium, MR = MC => 120 - Q = 50 => Q = 120 - 50 = 70

and P = 120 - 0.5Q = 120 - 0.5(70) = 120 - 35 = 85

The case of monopilist is shown in figure below, where AD is the demand curve and MR is the marginal revenue curve. The equilibrium E is determined corresponding to MR = MC. Hence, equilibrium output (Q*) = 70 and equilibrium price P = 120-0.5(70) = 85

CS = area of the triangle AP*E = 0.5(120-85)(70) = 1225

PS = (P-MC)xequilibrium output = (85-50)70 = 35*70 = 2450

Hence, total welfare = CS+PS = 1225 + 2450 = 3675

Deadweight loss = efficient level of welfare - current welfare = 4900 - 3675 (since efficient level of welfare = 4900 from (a))

=> deadweight loss = 1225

(c) If total cost = 250+50Q, then average cost (AC) = (250/Q) + 50 and MC = 50. The AC curve is drawn as follows in the diagram with the existing demand, MR and MC curves.

At profit maximizing output, AC = (250/70) + 50 = 53.57 (up to 2 decimal places)

At profit maximizing quantity and price (i.e., Q* = 70 and P* = 85), profit will be (price-AC)*equilibrium output

i.e., profit = (85-53.57)*70 = 2200

(d) The average cost curve is shown in the digram below and the equilibrium output is mapped to it.

As it can be observed, the average cost curve is declining at the equilibrium level of output. Hence, it exhibits decresing returns to scale or economies of scale at this level of output.