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

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 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.

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

TC(y) =	0	if y = 0
	100y + F	if y > 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

  	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.

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

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

• 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.