Calculus Worked-Out Problem 20.1:
The Problem Two hundred paper mills compete in the paper market. The total cost of
production (in dollars) for each mill is given by the formula
(
)
where
indicates the mill’s annual production in thousands of tons. The external cost of a
mill’s production (in dollars) is given by the formula
(
)
where A is a constant. (We’ll supply values for A later.) Finally, annual market demand (in
thousands of tons) is given by the formula
where P is the price of paper per ton. Assume that A = 100. Find the competitive equilibrium
price and quantity, as well as the efficient quantity. Calculate the magnitude of the deadweight
loss resulting from the externality.
The Solution First we’ll find the competitive equilibrium. Taking the derivative of any mill’s
cost curve, we see that
Each mill produces the quantity for which price equals marginal cost:
It follows that at any price, P, each mill supplies the quantity
quantity by 200 (the number of mills), we obtain the market supply curve:
.
. Multiplying that
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:
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:
.
It follows that at any price, each mill would supply the quantity
. Multiplying
that quantity by 200 (the number of mills), we obtain a new formula for market supply:
̂
In equilibrium, the quantities supplied and demanded must be equal:
. Solving that equation, we find the competitive price that would prevail
without externalities: P = 1,200. Substituting that price into either the new market supply
function or the demand function, we obtain the efficient quantity: Qeffi cient = 30,000. (Another
way to find the socially efficient outcome is to solve for the level of production that equates
marginal social benefit and marginal social cost. Read More Online 20.1 shows how to solve this
part of the problem that way.)
Notice that the formulas for each mill’s marginal cost, marginal external cost, and marginal
social cost correspond to the curves drawn in Figure 20.1(a). Likewise, the formulas for Qs and
Qd correspond to the market supply and demand curves drawn in Figure 20.1(b), while the
formula for ̂ (the market supply curve without externalities) corresponds to the curve labeled
MSCmarket. Therefore, Figure 20.1 illustrates the competitive equilibrium and efficient allocation
for which we solved.
The deadweight loss created by the externality equals the area of the red triangle in Figure 20.1.
Looked at sideways, the base of that triangle—that is, the vertical distance between the red and
brown lines—equals the magnitude of the marginal external cost of production when the market
output is 50 million tons. With 200 mills, production per mill is 250,000 tons. Substituting that
value into our formula for marginal external cost (and remembering that Qmill measures
thousands of tons), we find that MEC(250) = $600 per ton. The height of the triangle, 20 million
tons, equals the difference between the competitive quantity (50 million tons) and the efficient
quantity (30 million tons). To find the deadweight loss, we compute the area of the triangle:
($600 per ton × 20,000,000 tons)/2 = $6 billion per year.