Examples and exercises on a profit-maximizing monopolist that sets a single price
- 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.
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.
MC(y) = 200 + 30y.
Thus any output at which MR is equal to MC satisfies
1200 20y = 200 + 30y,
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.
(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.
- For MR = MC we need
3y2/2500 4y/25 + 5 = 4 8y/100,
3y2/2500 8y/100 + 1 = 0,
3y2 200y + 2500 = 0,
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.
Example: profit maximization and the effect of a lump sum tax
A monopolist's cost function is
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?
|TC(y) =||0|| if y = 0|
|100y + F || if y > 0.|
Notice that the optimal output is independent of F if F > 0.
- 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
If the firm is in business then the price is p* = 200.
|0|| if F > 2000|
|y* = ||both 0 and 20|| if F = 2000|
|20|| if F < 2000.|
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.
- 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.
Copyright © 1997 by Martin J. Osborne