Question

A perfectly competitive market exists for wheat. The inverse demand is P = 200 − Q where P is the price of wheat and Q is the total quantity of wheat. The private total cost for the unregulated market is C = 50 + 80Q + 0.5Q2 . The production of wheat creates an externality where the total external cost is E = 0.5Q2 .

(a) Solve for the unregulated competitive equilibrium of wheat and the socially optimal level of wheat.

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

(c) One big company, WheatsRUs, buys out all the farmers of wheat and becomes a monopolist. Using the same functional forms, solve for the unregulated monopoly equilibrium.

(d) Given the socially optimal level of wheat in (a), what is the optimal tax that should be placed on the monopolist?

Answer 1

A. For a perfectly competitive unregulated market, the market equilibrium is where Price = Marginal Cost (Private Marginal cost).

We are given private total cost = 50 + 80Q + 0.5Q²

Thus, PMC = dPTC/dq = 80+ 2*0.5Q = 80+Q

Thus, equilibrium condition, P= MC => 200-Q= 80+Q => 280=2Q => Q*= 140

And,P* = 200-140 => P* = 60

Socially optimal level of price and quantity occurs where, Social marginal cost equals to Price(inverse demand equation).

Social total cost = Private total cost+total external cost => Social total cost = 50+80Q+0.5Q²+0.5Q²=> STC= 50+80Q+Q²

Using differentiation, Social marginal cost=dSTC/dQ= 80+2Q

For optimal level, P(Marginal Social benefit/inverse demand) =SMC

P=SMC => 200-Q= 80+2Q=> 280=3Q=> Q= 93.33

And P= 200-93.33 => P= 106.66

B. Pigovian tax should be equal to the difference between the social optimal price and private optimal price.

Here t(Pigovian tax) = P-P*=> 106.66-60 =>t= 40.66

This tax would bring up price to 106.66 from price in unregulated market, 60 and thus socially optimal quantity will be produced.

C. For a monopolist optimality condition is MR= MC

We are given, P= 200-Q

TR= P*Q=>TR= (200-Q)*Q=> 200Q-Q²

MR= dTR/dQ= 200-2Q

Unregulated MC, PMC= 80+Q

Optimal: 200-2Q= 80+Q=>280=3Q=>Q=93.33

Amd P= 106.66

D. Since the private allocation by monopolist produces the social optimal level of output. We do not need to use the Pigovian tax.