978-0393123982 Chapter 27 Solution Manual

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Chapter 27 NAME

Factor Markets

Introduction. In this chapter you will examine the factor demand de-

cision of a monopolist. If a firm is a monopolist in some industry, it

will produce less output than if the industry were competitively orga-

nized. Therefore it will in general want to use less inputs than does a

competitive firm. The value marginal product is just the value of the ex-

tra output produced by hiring an extra unit of the factor. The ordinary

logic of competitive profit maximization implies that a competitive firm

will hire a factor up until the point where the value marginal product

equals the price of the factor.

The marginal revenue product is the extra revenue produced by

hiring an extra unit of a factor. For a competitive firm, the marginal

revenue product is the same as the value of the marginal product, but

they differ for monopolist. A monopolist has to take account of the fact

that increasing its production will force the price down, so the marginal

revenue product of an extra unit of a factor will be less than the value

marginal product.

Another thing we study in this chapters is monopsony,whichisthe

case of a market dominated by a single buyer of some good. The case of

monopsony is very similar to the case of a monopoly: The monopsonist

hires less of a factor than a similar competitive firm because the monop-

sony recognizes that the price it has to pay for the factor depends on how

much it buys.

Finally, we consider an interesting example of factor supply, in which

a monopolist produces a good that is used by another monopolist.

Example: Suppose a monopolist faces a demand curve for output of the

form p(y) = 100 −2y. The production function takes the simple form

y=2x, and the factor costs $4 per unit. How much of the factor of

production will the monopolist want to employ? How much of the factor

would a competitive industry employ if all the firms in the industry had

the same production function?

Answer: The monopolist will employ the factor up to the point where

the marginal revenue product equals the price of the factor. Revenue as

a function of output is R(y)=p(y)y= (100 −2y)y. To find revenue as a

function of the input, we substitute y=2x:

R(x) = (100 −4x)2x= (200 −8x)x.

The marginal revenue product function will have the form MRPx= 200−

16x. Setting marginal revenue product equal to factor price gives us the

equation

200 −16x=4.

Solving this equation gives us x∗=12.25.

332 FACTOR MARKETS (Ch. 27)

If the industry were competitive, then the industry would employ the

factor up to the point where the value of the marginal product was equal

to 4. This gives us the equation

p2=4,

so p∗= 2. How much output would be demanded at this price? We plug

this into the demand function to get the equation 2 = 100 −2y,which

implies y∗= 49. Since the production function is y=2x,wecansolve

for x∗=y∗/2=24.5.

27.1 (0) Gargantuan Enterprises has a monopoly in the production of

antimacassars. Its factory is located in the town of Pantagruel. There is

no other industry in Pantagruel, and the labor supply equation there is

W=10+.1L,whereWis the daily wage and Lis the number of person-

days of work performed. Antimacassars are produced with a production

function, Q=10L,whereLis daily labor supply and Qis daily output.

The demand curve for antimacassars is P=41−Q

1,000 ,wherePis the

price and Qis the number of sales per day.

(a) Find the profit-maximizing output for Gargantuan. (Hint: Use the

production function to find the labor input requirements for any level of

output. Make substitutions so you can write the firm’s total costs as a

function of its output and then its profit as a function of output. Solve

(b) How much labor does it use? 1,000. What is the wage rate that

27.2 (0) The residents of Seltzer Springs, Michigan, consume bottles of

mineral water according to the demand function D(p)=1,000 −p.Here

D(p) is the demand per year for bottles of mineral water if the price per

bottle is p.

The sole distributor of mineral water in Seltzer Springs, Bubble Up,

purchases mineral water at cper bottle from their supplier Perry Air.

Perry Air is the only supplier of mineral water in the area and behaves

as a profit-maximizing monopolist. For simplicity we suppose that it has

zero costs of production.

(a) What is the equilibrium price charged by the distributor Bubble Up?

NAME 333

(b) What is the equilibrium quantity sold by Bubble Up? D(p∗)=

(d) What is the equilibrium quantity sold by Perry Air? D(c∗)=

(e) What are the profits of Bubble Up? πb= (500−250)(750−

(h) Suppose that this situation is expected to persist forever and that

the interest rate is expected to be constant at 10% per year. What is the

minimum lump sum payment that Perry Air would need to pay to Bubble

(i) Suppose that Perry Air does this. What will be the new price and

(k) What is the total amount of consumers’ surplus generated? How does

Calculus 27.3 (0) Upper Peninsula Underground Recordings (UPUR) has a mon-

opoly on the recordings of the famous rock group Moosecake. Moosecake’s

music is only provided on digital tape, and blank digital tapes cost them

cper tape. There are no other manufacturing or distribution costs. Let

p(x) be the inverse demand function for Moosecake’s music as a function

of x, the number of tapes sold.

334 FACTOR MARKETS (Ch. 27)

(a) What is the first-order condition for profit maximization? For future

reference, let x∗be the profit-maximizing amount produced and p∗be the

price at which it sells. (In this part, assume that tapes cannot be copied.)

p(x∗)+p(x∗)x∗=c.

Now a new kind of consumer digital tape recorder becomes widely

available that allows the user to make 1 and only 1 copy of a prerecorded

digital tape. The copies are a perfect substitute in consumption value for

the original prerecorded tape, and there are no barriers to their use or

sale. However, everyone can see the difference between the copies and the

orginals and recognizes that the copies cannot be used to make further

copies. Blank tapes cost the consumers cper tape, the same price the

monopolist pays.

(b) All Moosecake fans take advantage of the opportunity to make a single

copy of the tape and sell it on the secondary market. How is the price of an

original tape related to the price of a copy? Derive the inverse demand

curve for original tapes facing UPUR. (Hint: There are two sources of

demand for a new tape: the pleasure of listening to it, and the profits

from selling a copy.) If UPUR produces xtapes, 2x

tapes reach the market, so UPUR can sell

[p(2x)+p(2x)−c]x−cx =2p(2x)x−2cx.

(d) Let x∗∗ be the profit-maximizing level of production by UPUR. How

does it compare to the former profit-maximizing level of production?

(e) How does the price of a copy of a Moosecake tape compare to the

(f) If p∗∗ is the price of a copy of a Moosecake tape, how much will a new