Two hundred paper mills compete in the paper market. The total cost of productio

Question

Two hundred paper mills compete in the paper market. The total cost of production (in dollars) for each mill is given by the formula TC=500Q^mill + (Q^mill)^2 where Q^mill indicates the mills annual production in thousands of tons. The marginal cost of production is MC=500 + 2Q^mill. The external cost of a mill's production (in dollars) is given by the formula EC=40Q^mill + (Q^mill)^2 and the marginal external cost of production is MEC=40+2Q^mill. Finally, annual market demand (in thousands of tons) is given by the formula Qd=150,000 - 100P where P is the price of paper per ton. Using algebra, find the competitive equilibrium price and quantity, as well as the efficient quantity. Calculate the magnitude of the deadweight loss resulting from the externality. Illustrate your solution with graphs.

Explanation / Answer

Answer:

First we’ll find the competitive equilibrium. Taking the derivative of any mill’s cost curve, we see that:                  MC = 500 + 2Qmill

Each mill produces the quantity for which price equals marginal cost: P = MC = 500 + 2Qmill. It follows that at any price, P, each mill supplies the quantity is:

                                Qmill = P/2 -250

Multiplying that quantity by 200 (the number of mills), we obtain the market supply curve:

                                QS = 100P – 150,000

In equilibrium, the quantities supplied and demanded must be equal: 100P 50,000 = 150,000 100P. Solving that equation, we obtain the competitive price: P = 1,000. Substituting that price into either the market supply or demand function, we obtain the market quantity:

                                Qmarket = 50,000.

Next we’ll find the efficient level of production. Taking the derivative of any mill’s external cost curve, we see that:

                                MEC = 100 + 2 Qmill

If the external costs of production were borne directly by the mills, each mill would produce the quantity for which price equals marginal social cost:

                                P = MSC = MC + MEC

                                                = 600 + 4Qmill

It follows that at any price, each mill would supply the quantity Qmill = P/4 - 150.

Multiplying that quantity by 200 (the number of mills), we obtain a new formula for market supply:

                                QS = 50P – 30,000

In equilibrium, the quantities supplied and demanded must be equal:

                                50P – 30,000 = 100P – 150,000

By solving that equation, we find the competitive price that would prevail without externalities: P = 1,200.

Now, substituting that price into either the new market supply function or the demand function, we obtain the efficient quantity: Qefficient = 30,000

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