- 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.

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

- We have TR(
*y*) = (1200 10*y*)*y*= 1200*y*10*y*^{2}, soMR(

Also*y*) = 1200 20*y*.MC(

Thus any output at which MR is equal to MC satisfies*y*) = 200 + 30*y*.1200 20

or*y*= 200 + 30*y*,50

*y*= 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)

Since this profit is positive, the optimal output for the monopolist is the output we have found, namely^{2}= 20000 4000 6000 = 10000.*y** = 20. The price is 1000 and the monopolist's profit is 10000.

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

- For MR = MC we need
3

or*y*^{2}/2500 4*y*/25 + 5 = 4 8*y*/100,3

or*y*^{2}/2500 8*y*/100 + 1 = 0,3

or*y*^{2}200*y*+ 2500 = 0,

Thus there are two outputs at which MR is equal to MC: 50 and 100/6.*y*= [200 ± (40,000 30,000)]/6 = [200 ± 100]/6 = 50 or 100/6. - We have
MR'(

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*) = 8/100 and MC'(*y*) = 6*y*/2500 4/25.*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.

It faces the demand function

TC( y) =0 if y= 0100 y+Fif y> 0.

- We have TR(
*y*) = (300 5*y*)*y*, so MR(*y*) = 300 10*y*; 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 5*y*= 200, so the profit isTR(20) TC(20) = (200)(20) 2000

Thus the optimal output is*F*= 2000*F*.

If the firm is in business then the price is0 if *F*> 2000*y** =both 0 and 20 if *F*= 200020 if *F*< 2000.*p** = 200.

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

TC(so that its marginal cost is exactly the same as before: MC(y) = 100y+F+Tify> 0

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.

- We now have TC(
*y*) = 100*y*+*F*+*t**y*, so MC(*y*) = 100 +*t*. (That is, MC shifts up by*t*.) Thus MR(*y*) = MC(*y*) when300 10

or*y*= 100 +*t*,*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 5
*y*= 300 100*t*/2 = 200 +*t*/2 and the profit is (200 +*t*/2)(20*t*/10) [(100 +*t*)(20*t*/10) +*F*] =*t*^{2}/20 20*t*+ 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