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# 9.9.

Rons Window Washing Service is a small business that operates in the perfectly competitive residential
window washing industry in Evanston, Illinois. The short-run total cost of production is STC(Q) = 40+ 10Q +
0.1Q2, where Q is the number of windows washed per day. The corresponding short-run marginal cost
function is SMC(Q) = 10 + 0.2Q. The prevailing market price is \$20 per window.
a) How many windows should Ron wash to maximize profit?
b) What is Rons maximum daily profit?
d) What is Rons short-run supply curve, assuming that all of the \$40 per day fixed costs are sunk?
e) What is Rons short-run supply curve, assuming that if he produces zero output, he can rent or sell his fixed
assets and therefore avoid all his fixed costs?
a)

b)

## In order to maximize profit Ron should operate at the point where

20 10 0.20Q
Q 50

P MC

TR TC
Rons profit is given by
.
20(50) (40 10(50) 0.10(50) 2 )
210

d) If all fixed costs are sunk, then ANSC = AVC = (10Q + 0.1Q2)/Q = 10 + 0.1Q. So the first
step is to find the minimum of ANSC by setting ANSC = SMC, or 10 + 0.1Q = 10 + 0.2Q which
occurs when Q = 0. The minimum level of ANSC is thus 10. For prices below 10 the firm will
not produce and for prices above 10, its supply curve is found by setting P = SMC:

P 10 .2Q
Q 5 P 50

## The firms short-run supply curve is thus

0
s( P)
5P 50

if P 10
if P 10

e)If all fixed costs are non-sunk, as in this case, then ANSC = ATC = (40/Q) + 10 + 0.1Q. The

40
10 .1Q 10 .2Q
Q
Q 20
minimum point of ANSC occurs where ANSC = SMC:

The minimum level of ANSC is thus 14. For prices below 14 the firm will not produce and for prices above
14, its supply curve is found by setting P = SMC as before.

0
5P 50

s( P)

if P 14
if P 14

9.11. Newsprint (the paper used for newspapers) is produced in a perfectly competitive market. Each identical
firm has a total variable cost TVC(Q) = 40Q + 0.5Q2, with an associated marginal cost curve SMC(Q) = 40 +
Q. A firms fixed cost is entirely nonsunk and equal to 50.
a) Calculate the price below which the firm will not produce any output in the short run.
b) Assume that there are 12 identical firms in this industry. Currently, the market demand for newsprint is
D(P) = 360 2P, where D(P) is the quantity consumed in the market when the price is P. What is the shortrun equilibrium price?
a) The firm will not produce any output when the price falls below the point where SMC = ANSC, i.e. the
50 / Q 40 0.5Q 40 Q
minimum of the ANSC curve. Therefore
This implies Q = 10. The corresponding price, below which the firms will not produce, is equal to MC(10) =
ANSC(10) = 50.
P 40 Q

## b) Each firm will produce according to the relation, P = MC, or

. This means that each firms
Q P 40
12( P 40)
supply curve is
if P > 50 and zero if P < 50. Therefore market supply equals
and in
360 2 P
equilibrium this must equal market demand,
. Therefore the equilibrium price is P = 60. At this
price, each firm produces 20 units of output.
9.19. A competitive industry consists of 6 type A firms and 4 type B firms. Each firm of type A operates with
the supply curve:

## a) Suppose the market demand is

At the market equilibrium, which firms are producing, and what is the equilibrium price?
b) Suppose the market demand is
At the market equilibrium, which firms are producing, and what is the equilibrium price?
a)
When P <10, only Type B firms will operate, and the market supply will be 4(2P) = 8P.
When P >10, both types of firms will operate, and the market supply will be
4(2P) + 6(-10 + P) = -60 + 14P.
Supply

QMarket

8P60, 14P,

when P 10
when P 10

## To summarize, the market supply will be

Lets first assume the equilibrium price exceeds 10, so that all firms are producing. If this is true, setting
market supply equal to market demand: -60 + 14P = 108 10P, so that P = 7; however, the market supply we
have used is valid for P>10, but not valid for P = 7.

So the equilibrium price must be less than 10, with only Type B firms producing (and Type A firms not
producing).
Setting market supply equal to market demand: 8P = 108 10P, so that P = 6.
We have found that in equilibrium, only Type B firms produce, and the equilibrium price is 6.
b)
Lets first assume the equilibrium price exceeds 10, so that all firms are producing. If this is true,
setting market supply equal to market demand: -60 + 14P = 228 10P, so that P = 12; the market supply we
have used is valid for P=12. At this equilibrium both types of firms will be producing.
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11.7. A monopolist operates with the following data on cost and demand. It has a total fixed cost of \$1,400
and a total variable cost of Q2, where Q is the number of units of output it produces. The firms demand curve
is P = \$120 - 2Q. The size of its sunk cost is \$600. The firm expects the conditions of demand and cost to
continue in the foreseeable future.
a) What is the firms profit if it operates and it maximizes profit?
b) Should the firm continue to operate in the short run, or should it shut down? Explain.
a)
The monopolist chooses Q so that MR = MC: 120 4Q = 2Q => Q = 20.
P = 120 2(20) = 80.
Profit = PQ V F = 80(20) 202 1400 = - 200.
The firm has nonsunk fixed costs: FNonsunk = F - FSunk = 1400 600 = 800.
b)
Producer surplus = PQ V FNonsunk = 80(20) 202 800 = 400. So the firm should continue to operate in
the short run. If it operates, its profit is -200. But if it shuts down, its profit = - FSunk = -600. So it can lessen its
losses by 400 if it continues to operate (and this is why producer surplus is +400 annually.)
11.11. Assume that a monopolist sells a product with a total cost function TC = 1,200 + 0.5Q2 and a
corresponding marginal cost function MC = Q. The market demand curve is given by the equation P = 300 Q.
a) Find the profit-maximizing output and price for this monopolist. Is the monopolist profitable?
b) Calculate the price elasticity of demand at the monopolists profit-maximizing price. Also calculate the
marginal cost at the monopolists profit-maximizing output. Verify that the IEPR holds.
P 300 Q

a)

If demand is given by

MR 300 2Q

then
300 2Q Q

## . To find the optimum set

MR MC

Q 100
Q 100

At
price will be
TR 200(100) 20, 000

## . At this price and quantity total revenue will be

TC 1200 .5(100) 2 6, 200
and total cost will be
. Therefore, the firm will earn a
TR TC 13,800
profit of
.
b)

Q P
Q,P
P Q
Q 300 P

## With the demand curve

1
. Therefore, at the profit-maximizing price

200

100

Q , P 1
Q , P 2

The marginal cost at the profit-maximizing output is MC = Q = 100. The inverse elasticity pricing rule states
that at the profit-maximizing price
P MC
1

P
Q,P
In this case we have

200 100
1

200
2
1 1

2 2

## Thus, the IEPR holds for this monopolist.

11.21. Imagine that Gillette has a monopoly in the market for razor blades in Mexico. The market demand
curve for blades in Mexico is P = 968 - 20Q, where P is the price of blades in cents and Q is annual demand
for blades expressed in millions. Gillette has two plants in which it can produce blades for the Mexican
market: one in Los Angeles and one in Mexico City. In its L.A. plant, Gillette can produce any quantity of
blades it wants at a marginal cost of 8 cents per blade. Letting Q1 and MC1 denote the output and marginal cost
at the L.A. plant, we have MC1(Q1) = 8. The Mexican plant has a marginal cost function given by MC2(Q2) = 1
+ 0.5Q2.
a) Find Gillettes profit-maximizing price and quantity of output for the Mexican market overall. How will
Gillette allocate production between its Mexican plant and its U.S. plant?
b) Suppose Gillettes L.A. plant had a marginal cost of 10 cents rather than 8 cents per blade. How would your
a) Profit-maximizing firms generally allocate output among plants so as to keep marginal costs equal. But
notice that MC2 < MC1 whenever 1 + 0.5Q2 < 8, or Q2 < 14. So for small levels of output, specifically Q < 14,
Gillette will only use the first plant. For Q > 14, the cost-minimizing approach will set Q2 = 14 and Q1 = Q
14.
Suppose the monopolists profit-maximizing quantity is Q > 14. Then the relevant MC = 8, and with
MR 968 40Q
we have
968 40Q 8
Q 24
Since we have found that Q > 14, we know this approach is valid. (You should verify that had we supposed
the optimal output was Q < 14 and set MR = MC2 = 1 + 0.5Q, we would have found Q > 14. So this
approach would be invalid.) The allocation between plants will be Q2 = 14 and Q1 = 10. With a total quantity
Q = 24, the firm will charge a price of P = 968 20(24) = 488. Therefore the price will be \$4.88 per blade.

b) If

MC 10

at plant 1, by the logic in part (a) Gillette will only use plant 2 if Q < 18. It will produce all
MR MC
output above Q = 18 in plant 1 at MC = 10. Assuming Q > 18, setting
implies
968 40Q 10
Q 23.95
(So again, this approach is valid. You can verify that setting MR = MC2 would again lead to Q > 18.) The
firm will allocate production so that Q2 = 18 and Q1 = 5.95. At Q = 23.95, price will be \$4.89.
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