Examples and exercises on a profit-maximizing monopolist that sets a single price



A monopolist has the cost function TC(y) = 200y + 15y2 and faces the demand function given by p = 1200  10y.

What output maximizes its profit? What is the profit-maximizing price? What is its maximal profit?


(A more complicated example to show the possibility of two outputs at which MR is equal to MC.)

A monopolist's cost function is TC(y) = (y/2500)(y  100)2 + y, so that MC(y) = 3y2/2500  4y/25 + 5. It faces the inverse demand function P(y) = 4  4y/100. Find its output, the associated price, and its profit.

We conclude that the outputs of 0 and 50 are both optimal for the firm.

Example: profit maximization and the effect of a lump sum tax

A monopolist's cost function is
TC(y) =0 if y = 0
100y + F  if y > 0.
It faces the demand function p = 300  5y. How much does the monopolist produce (as a function of F )? What is the price? What is the monopolist's profit?

Notice that the optimal output is independent of F  if F  > 0.

Now suppose the firm has to pay a lump sum tax of T. Then its cost is

TC(y) = 100y + F  + T if y > 0
so that its marginal cost is exactly the same as before: MC(y) = 100. Thus the the output it chooses is not affected so long as the tax T isn't so large that the firm is better off shutting down---in this case, so long as 2000  F   T > 0, or T < 2000  F .

Now suppose that the firm has to pay a fixed percentage tax on profit. Then it maximizes (1  t)(y) instead of (y), where t is the tax rate. Since t is a constant, the solution of this problem is exactly the same as the solution of the original problem of maximizing (y). Thus this tax has no effect on the monopolist's behavior.

Example: the effect of an excise tax on monopoly behavior

Suppose the monopolist in the previous example has to pay a tax of $t for every unit it sells, rather than a lump sum tax. Notice that in this case the optimal output now depends on t: the higher is t, the higher is the price and the lower is the output.
Copyright © 1997 by Martin J. Osborne