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

### Procedure

• Find the output(s) for which MC(y*) = MR(y*).
• For each output you find, check to see whether the condition MC'(y*)  MR'(y*) is satisfied.
• For each output that satisfies the first two conditions, check to see if profit is nonnegative.
• If there are any outputs satisfing these three conditions, the one that has the highest profit is the optimal output for the monopolist. (Most probably there is at most one output that satisfies the three conditions.) If there is no output that satisfies the three conditions the the optimal output for the monopolist is 0.

### Example

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?

• We have TR(y) = (1200  10y)y = 1200y  10y2, so
MR(y) = 1200  20y.
Also
MC(y) = 200 + 30y.
Thus any output at which MR is equal to MC satisfies
1200  20y = 200 + 30y,
or
50y = 1000, or y = 20.
• We have MR'(y) = 20 and MC'(y) = 30, so MC'(20)  MR'(20).
• The price associated with y = 20 is p = 1200  (10)(20) = 1000, so the firm's profit is
(1000)(20)  200(20)  15(20)2 = 20000  4000  6000 = 10000.
Since this profit is positive, the optimal output for the monopolist is the output we have found, namely y* = 20. The price is 1000 and the monopolist's profit is 10000.

### Example

(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.

• For MR = MC we need
3y2/2500  4y/25 + 5 = 4  8y/100,
or
3y2/2500  8y/100 + 1 = 0,
or
3y2  200y + 2500 = 0,
or
y = [200 ± (40,000  30,000)]/6 = [200 ± 100]/6 = 50 or 100/6.
Thus there are two outputs at which MR is equal to MC: 50 and 100/6.
• We have
MR'(y) = 8/100 and MC'(y) = 6y/2500  4/25.
We have MR'(50) = 8/100 = 0.08 and MC'(50) = 0.04, so that MC'(50)  MR'(50). Also we have MR'(100/6) = 8/100 = 0.08 and MC'(100/6) = 0.12, so that MC'(100/6) < MR'(100/6). Hence the slope of MC is greater than the slope of MR only at y = 50.
• For y = 50 the price is P(50) = 4  200/100 = 2, so the firm's profit is (2)(50)  TC(50) = 100  100 = 0.
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?

• We have TR(y) = (300  5y)y, so MR(y) = 300  10y; MC(y) = 100. Thus for MC(y) = MR(y) we need y = 20.
• We have MR'(y) = 10 and MC'(y) = 0, so the condition MC'(y MR'(y) is satisfied.
• For y = 20 the price is 300  5y = 200, so the profit is
TR(20)  TC(20) = (200)(20)  2000  F  = 2000  F .
Thus the optimal output is
 0 if F  > 2000 y* = both 0 and 20 if F  = 2000 20 if F  < 2000.
If the firm is in business then the price is p* = 200.
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.
• We now have TC(y) = 100y + F  + ty, so MC(y) = 100 + t. (That is, MC shifts up by t.) Thus MR(y) = MC(y) when
300  10y = 100 + t,
or y = 20  t/10.
• As before, MR'(y) = 10 and MC'(y) = 0, so the condition MC'(y MR'(y) is satisfied.
• The price is 300  5y = 300  100  t/2 = 200 + t/2 and the profit is (200 + t/2)(20  t/10)  [(100 + t)(20  t/10) + F ] = t2/20  20t + 2000  F . If t is such that this profit is nonnegative then the output y = 2  t/10 is optimal for the firm; if t is such that the profit is negative then the output 0 is optimal.
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