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

Inverse demand function: P = 100 - Q

Private Total cost function: TC = 50 + 80Q + 0.5Q²

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

Total social cost (SC) = TC + External total cost = 50 + 80Q + 0.5Q² + Q² = 50 + 80Q + 1.5Q²

Marginal social cost (MSC) = dSC/dQ = 80 + 3Q

(Task 1)

Free market competitive equilibrium is obtained when price equals PMC.

100 - Q = 80 + Q

2Q = 20

Q = 20/2 = 10 (Market quantity)

(Task 2)

Socially optimal (efficienct) outcome is derived by equating price with MSC.

100 - Q = 80 + 3Q

4Q = 20

Q = 20/4 = 5

In following graph, D, PMC and SMC are demand, private marginal cost and social marginal cost curves respectively. Free market competitive equilibrium is at point A where D intersects PMC curve, with free market equilibrium price P0 and output Q0. Socially efficient outcome is at point B where D intersects SMC curve with optimal price P1 and output Q1. Since Q1 < Q0 and P1 > P0, efficient output is less than market output, and optimal price is higher than market price.

(Task 3)

When Q = 5 (socially efficient output),

PMC = 80 + 5 = 85

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

Pigouvian tax per unit = Difference between MSC and PMC curves at socially efficient output = 95 - 85 = $10

(Task 4)

As a monopolist, WheatsRUs will maximize profit by at intersection of Marginal revenue (MR) and PMC curves.

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

MR = dTR/dQ = 100 - 2Q

Equating MR and PMC,

100 - 2Q = 80 + Q

3Q = 20

Q = 6.67

P = 100 - 6.67 = 93.33