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Chapter 23 NAME
Firm Supply
Introduction. The short-run supply curve of a competitive firm is the
portion of its short-run marginal cost curve that is upward sloping and
lies above its average variable cost curve. The long-run supply curve of a
competitive firm is the portion of its short-run marginal cost curve that
is upward-sloping and lies above its long-run average cost curve.
Example: A firm has the long-run cost function c(y)=2y2+ 200 for
y>0andc(0) = 0. Let us find its long-run supply curve. The firm’s
marginal cost when its output is yis MC(y)=4y. If we graph output on
the horizontal axis and dollars on the vertical axis, then we find that the
long-run marginal cost curve is an upward-sloping straight line through
the origin with slope 4. The long-run supply curve is the portion of this
curve that lies above the long-run average cost curve. When output is y,
long-run average costs of this firm are AC(y)=2y+ 200/y.ThisisaU-
shaped curve. As ygets close to zero, AC(y) becomes very large because
200/y becomes very large. When yis very large, AC(y) becomes very
large because 2yis very large. When is it true that AC(y)<MC(y)?
This happens when 2y+ 200/y < 4y. Simplify this inequality to find that
AC(y)<MC(y) when y>10. Therefore the long-run supply curve is
the piece of the long-run marginal cost curve for which y>10. So the
long-run supply curve has the equation p=4yfor y>10. If we want to
find quantity supplied as a function of price, we just solve this expression
for yas a function of p.Thenwehavey=p/4 whenever p>40.
Suppose that p<40. For example, what if p= 20, how much will
the firm supply? At a price of 20, if the firm produces where price equals
long-run marginal cost, it will produce 5 = 20/4 units of output. When
the firm produces only 5 units, its average costs are 2 ×5 + 200/5 = 50.
Therefore when the price is 20, the best the firm can do if it produces a
positive amount is to produce 5 units. But then it will have total costs of
5×50 = 250 and total revenue of 5×20 = 100. It will be losing money. It
would be better off producing nothing at all. In fact, for any price p<40,
the firm will choose to produce zero output.
23.1 (0) Remember Otto’s brother Dent Carr, who is in the auto repair
business? Dent found that the total cost of repairing scars is c(s)=
2s2+ 100.
(a) This implies that Dent’s average cost is equal to 2s+ 100/s ,
his average variable cost is equal to 2s, and his marginal cost is
286 FIRM SUPPLY (Ch. 23)
0 5 10 15 20
20
40
60
Output
Dollars
80
Supply
mc
ac
avc
Revenue
Costs
Profit
(b) If the market price is $20, how many cars will Dent be willing to
(c) Suppose the market price is $40 and Dent maximizes his profits. On
Calculus 23.2 (0) A competitive firm has the following short-run cost function:
c(y)=y3−8y2+30y+5.
(c) On the axes below, sketch and label a graph of the marginal cost
(d) Average variable cost is falling as output rises if output is less than
(e) Marginal cost equals average variable cost when output is 4.
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(g) The smallest positive amount that the firm will ever supply at any
0246
8
10
20
30
y
Costs
40
mc
avc
Calculus 23.3 (0) Mr. McGregor owns a 5-acre cabbage patch. He forces his
wife, Flopsy, and his son, Peter, to work in the cabbage patch without
wages. Assume for the time being that the land can be used for nothing
other than cabbages and that Flopsy and Peter can find no alternative
employment. The only input that Mr. McGregor pays for is fertilizer. If
he uses xsacks of fertilizer, the amount of cabbages that he gets is 10√x.
Fertilizer costs $1 per sack.
(a) What is the total cost of the fertilizer needed to produce 100 cabbages?
(b) If the only way that Mr. McGregor can vary his output is by varying
the amount of fertilizer applied to his cabbage patch, write an expression
(c) If the price of cabbages is $2 each, how many cabbages will Mr. Mc-
Gregor produce? 100. How many sacks of fertilizer will he buy?
288 FIRM SUPPLY (Ch. 23)
(d) The price of fertilizer and of cabbages remain as before, but Mr. Mc-
Gregor learns that he could find summer jobs for Flopsy and Peter in
a local sweatshop. Flopsy and Peter would together earn $300 for the
summer, which Mr. McGregor could pocket, but they would have no time
to work in the cabbage patch. Without their labor, he would get no cab-
bages. Now what is Mr. McGregor’s total cost of producing ycabbages?
(e) Should he continue to grow cabbages or should he put Flopsy and
23.4 (0) Severin, the herbalist, is famous for his hepatica. His total cost
function is c(y)=y2+10 for y>0andc(0) = 0. (That is, his cost of
producing zero units of output is zero.)
(a) What is his marginal cost function? 2y.What is his average cost
(b) At what quantity is his marginal cost equal to his average cost?
(c) In a competitive market, what is the lowest price at which he will
23.5 (1) Stanley Ford makes mountains out of molehills. He can do this
with almost no effort, so for the purposes of this problem, let us assume
that molehills are the only input used in the production of mountains.
Suppose mountains are produced at constant returns to scale and that
it takes 100 molehills to make 1 mountain. The current market price of
molehills is $20 each. A few years ago, Stan bought an “option” that
permits him to buy up to 2,000 molehills at $10 each. His option contract
explicitly says that he can buy fewer than 2,000 molehills if he wishes, but
he can not resell the molehills that he buys under this contract. In or-
der to get governmental permission to produce mountains from molehills,
Stanley would have to pay $10,000 for a molehill-masher’s license.
(a) The marginal cost of producing a mountain for Stanley is $1,000
if he produces fewer than 20 mountains. The marginal cost of producing
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(b) On the graph below, show Stanley Ford’s marginal cost curve (in blue
ink) and his average cost curve (in red ink).
0102030
40
1000
2000
3000
Output
Dollars
4000
Red ac
curve
Pencil
mc
curve
Blue mc curve
(c) If the price of mountains is $1,600, how many mountains will Stanley
(d) The government is considering raising the price of a molehill-masher’s
license to $11,000. Stanley claims that if it does so he will have to go out
of business. Is Stanley telling the truth? No. What is the highest
fee for a license that the government could charge without driving him
(e) Stanley’s lawyer, Eliot Sleaze, has discovered a clause in Stanley’s
option contract that allows him to resell the molehills that he purchased
under the option contract at the market price. On the graph above,
use a pencil to draw Stanley’s new marginal cost curve. If the price of
mountains remains $1,600, how many mountains will Stanley produce
23.6 (1) Lady Wellesleigh makes silk purses out of sows’ ears. She is
the only person in the world who knows how to do so. It takes one sow’s
ear and 1 hour of her labor to make a silk purse. She can buy as many
290 FIRM SUPPLY (Ch. 23)
sows’ ears as she likes for $1 each. Lady Wellesleigh has no other source
of income than her labor. Her utility function is a Cobb-Douglas function
U(c, r)=c1/3r2/3,wherecis the amount of money per day that she has
to spend on consumption goods and ris the amount of leisure that she
has. Lady Wellesleigh has 24 hours a day that she can devote either to
leisure or to working.
(a) Lady Wellesleigh can either make silk purses or she can earn $5 an
hour as a seamstress in a sweatshop. If she worked in the sweat shop, how
many hours would she work? 8. (Hint: To solve for this amount,
write down Lady Wellesleigh’s budget constraint and recall how to find
the demand function for someone with a Cobb-Douglas utility function.)
(b) If she could earn a wage of $wan hour as a seamstress, how much
(c) If the price of silk purses is $p, how much money will Lady Wellesleigh
earn per purse after she pays for the sows’ ears that she uses? p−1.
(d) If she can earn $5 an hour as a seamstress, what is the lowest price
(e) What is the supply function for silk purses? (Hint: The price of silk
purses determines the “wage rate” that Lady W. can earn by making silk
purses. This determines the number of hours she will choose to work and
Calculus 23.7 (0) Remember Earl, who sells lemonade in Philadelphia? You
met him in the chapter on cost functions. Earl’s production function is
f(x1,x
2)=x1/3
1x1/3
2,wherex1is the number of pounds of lemons he
uses and x2is the number of hours he spends squeezing them. As you
found out, his cost function is c(w1,w
2,y)=2w1/2
1w1/2
2y3/2,whereyis
the number of units of lemonade produced.
(a) If lemons cost $1 per pound, the wage rate is $1 per hour, and the
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(b) In general, Earl’s marginal cost depends on the price of lemons and
the wage rate. At prices w1for lemons and w2for labor, his mar-
ginal cost when he is producing yunits of lemonade is MC(w1,w
2,y)=
3w1/2
1w1/2
2y1/2.The amount that Earl will supply depends on the
three variables, p,w1,w2. As a function of these three variables, Earl’s
Calculus 23.8 (0) As you may recall from the chapter on cost functions, Irma’s
handicrafts has the production function f(x1,x
2)=(min{x1,2x2})1/2,
where x1is the amount of plastic used, x2is the amount of labor used,
and f(x1,x
2) is the number of lawn ornaments produced. Let w1be the
price per unit of plastic and w2be the wage per unit of labor.
(b) If w1=w2= 1, then Irma’s marginal cost of producing yunits of
output is MC(y)= 3y.The number of units of output that she would
supply at price pis S(p)= p/3.At these factor prices, her average
(c) If the competitive price of the lawn ornaments she sells is p= 48, and
(d) More generally, at factor prices w1and w2, her marginal cost is a
function MC(w1,w
2,y)= (2w1+w2)y.At these factor prices and
an output price of p, the number of units she will choose to supply is
23.9 (0) Jack Benny can get blood from a stone. If he has xstones, the
number of pints of blood he can extract from them is f(x)=2x1
3.Stones
cost Jack $weach. Jack can sell each pint of blood for $p.
(a) How many stones does Jack need to extract ypints of blood?
292 FIRM SUPPLY (Ch. 23)
(c) What is Jack’s supply function when stones cost $8 each? y=
(d) If Jack has 19 relatives who can also get blood from a stone in the
same way, what is the aggregate supply function for blood when stones
23.10 (1) The Miss Manners Refinery in Dry Rock, Oklahoma, converts
crude oil into gasoline. It takes 1 barrel of crude oil to produce 1 barrel of
gasoline. In addition to the cost of oil there are some other costs involved
in refining gasoline. Total costs of producing ybarrels of gasoline are
described by the cost function c(y)=y2/2+poy,wherepois the price of
a barrel of crude oil.
(a) Express the marginal cost of producing gasoline as a function of po
(b) Suppose that the refinery can buy 50 barrels of crude oil for $5 a
barrel but must pay $15 a barrel for any more that it buys beyond 50
(c) Plot Miss Manners’ supply curve in the diagram below using blue ink.
0255075
100
20
40
60
Barrels of gasoline
Price of gasoline
80
30
Red line
Black line Blue lines
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(d) Suppose that Miss Manners faces a horizontal demand curve for gaso-
line at a price of $30 per barrel. Plot this demand curve on the graph
(e) If Miss Manners could no longer get the first 50 barrels of crude for
$5, but had to pay $15 a barrel for all crude oil, how would her output
(f) Now suppose that an entitlement program is introduced that permits
refineries to buy one barrel of oil at $5 for each barrel of oil that they
buy for $15. What will Miss Manners’ supply curve be now? S(p)=
manner. Plot this supply curve on the graph above using black ink. If
the demand curve is horizontal at $30 a barrel, how much gasoline will
