A monopoly faces market demand Q=30-P and has a cost ...



Econ 101 - Spring 2003

PS #8 - XtraQ

ANSWERS

a. The monopoly produces at the point where mr=mc. In this question mr = 30-2X and mc=X.

Equating mr and mc gives us Xsm=10. From the demand equation we can find Psm=30-Xsm=$20. Profit = (20*10)-50 = $150.

b. nDx,p= |d XD / dP| * (P/XD) = (1)*(20/10) = 2.

c. CS=area PsmAB = (1/2)*(10)*(30-20)=50 and PS area OPsmBC = [(1/2)*(10)*(10)] + [(10)*(10)] = 150. Note PS=total revenue - variable costs or PS=profit + fixed costs. Since there are no fixed costs, PS=profit.

d. Now the monopolist can’t charge its profit-maximizing price of $20. The appropriate demand curve is now odd: it's the solid bold flat and then downward sloping demand relationship. (See diagram on right.) From this odd demand curve we get an even odder marginal revenue curve: It is the horizontal portion of the demand curve and the bold dashed segments. Notice that when the demand price is constrained by the price ceiling, demand is horizontal and demand=mr for those output levels. Once demand starts to be its old self again, marginal revenue plunges vertically and catches up to its old self, too. So now equate mr and mc and go up to the demand curve to get the price. From examination of the graph you can see that P=$18 and, therefore, X=12 (use the demand curve to get this value by plugging in P=18 so X=30-18). Profit = ( = (18*12)-[(12*12)/2]=144. CS=(1/2)*(12)*(12)=72, PS = profit + fixed costs = 144.

e. The introduction of business fee will not affect profit maximizing price and quantity (as long as monopoly decides to produce) because it doesn’t change either mr or mc. So P=$20 and X=10 as in part (a). Profit=150 - 130=20>0. Because the monopoly earns positive economic profit by selling X=10 for P=$20 and paying the business fee, it will choose to do so.

f. To get allocative efficieny, set MB=MC. Use the demand curve to get MB. That is, MB=PD=30-X. And we know that MC=X. So XAE is where: 30- XAE = XAE so that XAE = 15 units.

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30

MC

D

MR

10

20=Psm

X

$

MC

30

MR

D

A

B

C

20=Psm

10

O

A

B

C

O

10

10

18

12

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