Question

A perfectly competitive market exists for wheat. The inverse demand is P = 100?Q where P is the price of wheat and Q is the total quantity of wheat. The private total cost for the unregulated market to produce a quantity of Q is 50+80Q +0.5Q 2 . The production of wheat creates some pollution where the total externality cost is EC = Q 2 .

Task 1: Solve for the free market competitive equilibrium of wheat.

Task 2: Solve for the socially optimal level of wheat. Illustrate it in the graph.

Task 3: Derive the Pigouvian tax (per unit of output of wheat) that results in the social optimum.

Task 4: One big company, WheatsRUs, buys out all the farmers of wheat and becomes a monopolist. Using the same demand and cost information, solve for the quantity and price under the unregulated monopolist.

Answer 1

P = 100 - Q

TC = 50 + 80Q + 0.5Q², therefore

Private Marginal cost (PMC) = dTC/dQ = 80 + Q

Total social cost (TSC) = TC + EC = 50 + 80Q + 0.5Q² + Q² = 50 + 80Q + 1.5Q², therefore

Social marginal cost (SMC) = dTSC/dQ = 80 + 3Q

(Task 1) In competitive equilibrium, Demand = PMC

100 - Q = 80 + Q

2Q = 20

Q = 10

(Task 1) In socially optimal outcome, Demand = SMC

100 - Q = 80 + 3Q

4Q = 20

Q = 5

In following graph, D, PMC and SMC are relevant curves. Market equilibrium is at point A where Demand intersects PMC with price P0 and output Q0. Socially efficient is at point B where Demand intersects SMC with price P1 and output Q1, where P1 > P0 and Q1 < Q0.

(Task 3) When Q = 5,

PMC = 80 + 5 = 85

SMC = 80 + (3 x 5) = 80 + 15 = 95

Pigouvian tax per unit = (SMC - PC) at socially optimal output = 95 - 85 = 10

(Task 4) Monopolist will maximize profit by equating Marginal revenue (MR) with PMC.

Total revenue (TR) = P x Q = 100Q - Q²

MR = dTR/dQ = 100 - 2Q

Equating with PMC,

100 - 2Q = 80 + Q

3Q = 20

Q = 6.67

P = 100 - 6.67 = 93.33