In the chapter on consumer choice, you studied a consumer who tries to
maximize his utility subject to the constraint that he has a ﬁxed amount
of money to spend. In this chapter you study the behavior of a ﬁrm that
is trying to produce a ﬁxed amount of output in the cheapest possible
way. In both theories, you look for a point of tangency between a curved
line and a straight line. In consumer theory, there is an “indiﬀerence
curve” and a “budget line.” In producer theory, there is a “production
isoquant” and an “isocost line.” As you recall, in consumer theory, ﬁnding
a tangency gives you only one of the two equations you need to locate the
consumer’s chosen point. The second equation you used was the budget
equation. In cost-minimization theory, again the tangency condition gives
you one equation. This time you don’t know in advance how much the
producer is spending; instead you are told how much output he wants to
produce and must ﬁnd the cheapest way to produce it. So your second
equation is the equation that tells you that the desired amount is being
produced.
Example. A ﬁrm has the production function f(x1, x2) = (
√
x1 +
3
√
x2)2
. The price of factor 1 is w1 = 1 and the price of factor 2
is w2 = 1. Let us ﬁnd the cheapest way to produce 16 units of output.
We will be looking for a point where the technical rate of substitution
equals −w1/w2. If you calculate the technical rate of substitution
(or look it up from the warm up exercise in Chapter 18),
you ﬁnd TRS(x1, x2) = −(1/3)(x2/x1)1/2
. Therefore we must have
−(1/3)(x2/x1)1/2
= −w1/w2 = −1. This equation can be simpliﬁed
to x2 = 9x1. So we know that the combination of inputs chosen has to
lie somewhere on the line x2 = 9x1. We are looking for the cheapest way
to produce 16 units of output. So the point we are looking for must satisfy
the equation (
√
x1 + 3
√
x2)2
= 16, or equivalently
√
x1 + 3
√
x2 = 4.
Since x2 = 9x1, we can substitute for x2 in the previous equation to get
√
x1 +3
√
9x1 = 4. This equation simpliﬁes further to 10
√
x1 = 4. Solving
this for x1, we have x1 = 16/100. Then x2 = 9x1 = 144/100.
The amounts x1 and x2 that we solved for in the previous paragraph
are known as the conditional factor demands for factors 1 and 2,
conditional on the wages w1 = 1, w2 = 1, and output y = 16. We express
this by saying x1(1, 1, 16) = 16/100 and x2(1, 1, 16) = 144/100.
Since we know the amount of each factor that will be used to produce
16 units of output and since we know the price of each factor,
we can now calculate the cost of producing 16 units. This cost is
c(w1, w2, 16) = w1x1(w1, w2, 16)+w2x2(w1, w2, 16). In this instance since
w1 = w2 = 1, we have c(1, 1, 16) = x1(1, 1, 16) + x2(1, 1, 16) = 160/100.
In consumer theory, you also dealt with cases where the consumer’s
indiﬀerence “curves” were straight lines and with cases where there were
kinks in the indiﬀerence curves. Then you found that the consumer’s
choice might occur at a boundary or at a kink. Usually a careful look
at the diagram would tell you what is going on. The story with kinks
and boundary solutions is almost exactly the same in the case of costminimizing
ﬁrms. You will ﬁnd some exercises that show how this works.
20.1 (0) Nadine sells user-friendly software. Her ﬁrm’s production function
is f(x1, x2) = x1 + 2x2, where x1 is the amount of unskilled labor
and x2 is the amount of skilled labor that she employs.
(a) In the graph below, draw a production isoquant representing input
combinations that will produce 20 units of output. Draw another isoquant
representing input combinations that will produce 40 units of output.
x2
40
30
20
10
0 10 20 30 40
x1
(b) Does this production function exhibit increasing, decreasing, or constant
returns to scale? .
(c) If Nadine uses only unskilled labor, how much unskilled labor would
she need in order to produce y units of output? .
(d) If Nadine uses only skilled labor to produce output, how much skilled
labor would she need in order to produce y units of output? .
(e) If Nadine faces factor prices (1, 1), what is the cheapest way for her
to produce 20 units of output? x1 = , x2 = .
(f) If Nadine faces factor prices (1, 3), what is the cheapest way for her
to produce 20 units of output? x1 = , x2 = .
(g) If Nadine faces factor prices (w1, w2), what will be the minimal cost
of producing 20 units of output? .
(h) If Nadine faces factor prices (w1, w2), what will be the minimal cost
of producing y units of output? .
20.2 (0) The Ontario Brassworks produces brazen eﬀronteries. As you
know brass is an alloy of copper and zinc, used in ﬁxed proportions. The
production function is given by: f(x1, x2) = min{x1, 2x2}, where x1 is
the amount of copper it uses and x2 is the amount of zinc that it uses in
production.
(a) Illustrate a typical isoquant for this production function in the graph
below.
x2
40
30
20
10
0 10 20 30 40
x1
(b) Does this production function exhibit increasing, decreasing, or constant
returns to scale? .
(c) If the ﬁrm wanted to produce 10 eﬀronteries, how much copper would
it need? How much zinc would it need? .
(d) If the ﬁrm faces factor prices (1, 1), what is the cheapest way for it to
produce 10 eﬀronteries? How much will this cost?
.
(e) If the ﬁrm faces factor prices (w1, w2), what is the cheapest cost to
produce 10 eﬀronteries? .
(f) If the ﬁrm faces factor prices (w1, w2), what will be the minimal cost
of producing y eﬀronteries? .
20.3 (0) A ﬁrm uses labor and machines to produce output according to
the production function f(L, M) = 4L1/2
M1/2
, where L is the number of
units of labor used and M is the number of machines. The cost of labor
is $40 per unit and the cost of using a machine is $10.
(a) On the graph below, draw an isocost line for this ﬁrm, showing combinations
of machines and labor that cost $400 and another isocost line
showing combinations that cost $200. What is the slope of these isocost
lines? .
(b) Suppose that the ﬁrm wants to produce its output in the cheapest
possible way. Find the number of machines it would use per worker.
(Hint: The ﬁrm will produce at a point where the slope of the production
isoquant equals the slope of the isocost line.) .
(c) On the graph, sketch the production isoquant corresponding to an
output of 40. Calculate the amount of labor and the number
of machines that are used to produce 40 units of output in the
cheapest possible way, given the above factor prices. Calculate the cost
of producing 40 units at these factor prices: c(40, 10, 40) = .
(d) How many units of labor and how many machines
would the ﬁrm use to produce y units in the cheapest possible way? How
much would this cost? (Hint: Notice that there are constant
returns to scale.)
Machines
40
30
20
10
0 10 20 30 40
Labor
20.4 (0) Earl sells lemonade in a competitive market on a busy street
corner in Philadelphia. His production function is f(x1, x2) = x
1/3
1 x
1/3
2 ,
where output is measured in gallons, x1 is the number of pounds of lemons
he uses, and x2 is the number of labor-hours spent squeezing them.
(a) Does Earl have constant returns to scale, decreasing returns to scale,
or increasing returns to scale? .
(b) Where w1 is the cost of a pound of lemons and w2 is the wage rate
for lemon-squeezers, the cheapest way for Earl to produce lemonade is to
use hours of labor per pound of lemons. (Hint: Set the slope
of his isoquant equal to the slope of his isocost line.)
(c) If he is going to produce y units in the cheapest way possible, then the
number of pounds of lemons he will use is x1(w1, w2, y) =
and the number of hours of labor that he will use is x2(w1, w2, y) =
(Hint: Use the production function and the equation you
found in the last part of the answer to solve for the input quantities.)
(d) The cost to Earl of producing y units at factor prices w1 and w2 is
c(w1, w2, y) = w1x1(w1, w2, y) + w2x2(w1, w2, y) = .
20.5 (0) The prices of inputs (x1, x2, x3, x4) are (4, 1, 3, 2).
(a) If the production function is given by f(x1, x2) = min{x1, x2}, what
is the minimum cost of producing one unit of output? .
(b) If the production function is given by f(x3, x4) = x3 +x4, what is the
minimum cost of producing one unit of output? .
(c) If the production function is given by f(x1, x2, x3, x4) = min{x1 +
x2, x3 + x4}, what is the minimum cost of producing one unit of output?
.
(d) If the production function is given by f(x1, x2) = min{x1, x2} +
min{x3, x4}, what is the minimum cost of producing one unit of output?
.
20.6 (0) Joe Grow, an avid indoor gardener, has found that the number
of happy plants, h, depends on the amount of light, l, and water, w. In
fact, Joe noticed that plants require twice as much light as water, and any
more or less is wasted. Thus, Joe’s production function is h = min{l, 2w}.
(a) Suppose Joe is using 1 unit of light, what is the least amount of water
he can use and still produce a happy plant? .
(b) If Suppose Joe wants to produce 4 happy plants, what are the minimum
amounts of light and water required? .
(c) Joe’s conditional factor demand function for light is l(w1, w2, h) =
and his conditional factor demand function for water is
w(w1, w2, h) = .
(d) If each unit of light costs w1 and each unit of water costs w2, Joe’s
cost function is c(w1, w2, h) = .
20.7 (1) Joe’s sister, Flo Grow, is a university administrator. She uses
an alternative method of gardening. Flo has found that happy plants
only need fertilizer and talk. (Warning: Frivolous observations about
university administrators’ talk being a perfect substitute for fertilizer is
in extremely poor taste.) Where f is the number of bags of fertilizer used
and t is the number of hours she talks to her plants, the number of happy
plants produced is exactly h = t+2f. Suppose fertilizer costs wf per bag
and talk costs wt per hour.
(a) If Flo uses no fertilizer, how many hours of talk must she devote if
she wants one happy plant? If she doesn’t talk to her plants
at all, how many bags of fertilizer will she need for one happy plant?
.
(b) If wt < wf /2, would it be cheaper for Flo to use fertilizer or talk to
raise one happy plant? .
(c) Flo’s cost function is c(wf , wt, h) = .
(d) Her conditional factor demand for talk is t(wf , wt, h) = if
wt < wf /2 and if wt > wf /2.
20.8 (0) Remember T-bone Pickens, the corporate raider? Now he’s concerned
about his chicken farms, Pickens’s Chickens. He feeds his chickens
on a mixture of soybeans and corn, depending on the prices of each. According
to the data submitted by his managers, when the price of soybeans
was $10 a bushel and the price of corn was $10 a bushel, they used 50
bushels of corn and 150 bushels of soybeans for each coop of chickens.
When the price of soybeans was $20 a bushel and the price of corn was
$10 a bushel, they used 300 bushels of corn and no soybeans per coop
of chickens. When the price of corn was $20 a bushel and the price of
soybeans was $10 a bushel, they used 250 bushels of soybeans and no corn
for each coop of chickens.
(a) Graph these three input combinations and isocost lines in the following
diagram.
Corn
400
300
200
100
0 100 200 300 400
Soybeans
(b) How much money did Pickens’ managers spend per coop of chickens
when the prices were (10, 10)? When the prices were (10, 20)?
When the prices were (20, 10)? .
(c) Is there any evidence that Pickens’s managers were not minimizing
costs? Why or why not?
.
(d) Pickens wonders whether there are any prices of corn and soybeans at
which his managers will use 150 bushels of corn and 50 bushels of soybeans
to produce a coop of chickens. How much would this production method
cost per coop of chickens if the prices were ps = 10 and pc = 10?
if the prices were ps = 10, pc = 20? if the prices
were ps = 20, pc = 10? .
(e) If Pickens’s managers were always minimizing costs, can it be
possible to produce a coop of chickens using 150 bushels and 50
bushels of soybeans?
.
20.9 (0) A genealogical ﬁrm called Roots produces its output using only
one input. Its production function is f(x) =
√
x.
(a) Does the ﬁrm have increasing, constant, or decreasing returns to scale?
.
(b) How many units of input does it take to produce 10 units of output?
If the input costs w per unit, what does it cost to produce
10 units of output? .
(c) How many units of input does it take to produce y units of output?
If the input costs w per unit, what does it cost to produce y
units of output? .
(d) If the input costs w per unit, what is the average cost of producing y
units? AC(w, y) = .
20.10 (0) A university cafeteria produces square meals, using only one
input and a rather remarkable production process. We are not allowed to
say what that ingredient is, but an authoritative kitchen source says that
“fungus is involved.” The cafeteria’s production function is f(x) = x2
,
where x is the amount of input and f(x) is the number of square meals
produced.
(a) Does the cafeteria have increasing, constant, or decreasing returns to
scale? .
(b) How many units of input does it take to produce 144 square meals?
If the input costs w per unit, what does it cost to produce 144
square meals? .
(c) How many units of input does it take to produce y square meals?
If the input costs w per unit, what does it cost to produce y
square meals? .
(d) If the input costs w per unit, what is the average cost of producing y
square meals? AC(w, y) = .
20.11 (0) Irma’s Handicrafts produces plastic deer for lawn ornaments.
“It’s hard work,” says Irma, “but anything to make a buck.” Her production
function is given by f(x1, x2) = (min{x1, 2x2})1/2
, where x1 is the
amount of plastic used, x2 is the amount of labor used, and f(x1, x2) is
the number of deer produced.
(a) In the graph below, draw a production isoquant representing input
combinations that will produce 4 deer. Draw another production isoquant
representing input combinations that will produce 5 deer.
x2
40
30
20
10
0 10 20 30 40
x1
(b) Does this production function exhibit increasing, decreasing, or constant
returns to scale? .
(c) If Irma faces factor prices (1, 1), what is the cheapest way for her to
produce 4 deer? How much does this cost? .
(d) At the factor prices (1, 1), what is the cheapest way to produce 5
deer? How much does this cost? .
(e) At the factor prices (1, 1), the cost of producing y deer with this
technology is c(1, 1, y) = .
(f) At the factor prices (w1, w2), the cost of producing y deer with this
technology is c(w1, w2, y) = .
20.12 (0) Al Deardwarf also makes plastic deer for lawn ornaments.
Al has found a way to automate the production process completely. He
doesn’t use any labor–only wood and plastic. Al says he likes the business
“because I need the doe.” Al’s production function is given by f(x1, x2) =
(2x1 + x2)1/2
, where x1 is the amount of plastic used, x2 is the amount
of wood used, and f(x1, x2) is the number of deer produced.
(a) In the graph below, draw a production isoquant representing input
combinations that will produce 4 deer. Draw another production isoquant
representing input combinations that will produce 6 deer.
x2
40
30
20
10
0 10 20 30 40
x1
(b) Does this production function exhibit increasing, decreasing, or constant
returns to scale? .
(c) If Al faces factor prices (1, 1), what is the cheapest way for him to
produce 4 deer? How much does this cost? .
(d) At the factor prices (1, 1), what is the cheapest way to produce 6
deer? How much does this cost? .
(e) At the factor prices (1, 1), the cost of producing y deer with this
technology is c(1, 1, y) = .
(f) At the factor prices (3, 1), the cost of producing y deer with this
technology is c(3, 1, y) = .
20.13 (0) Suppose that Al Deardwarf from the last problem cannot vary
the amount of wood that he uses in the short run and is stuck with using
20 units of wood. Suppose that he can change the amount of plastic that
he uses, even in the short run.
(a) How much plastic would Al need in order to make 100 deer?
.
(b) If the cost of plastic is $1 per unit and the cost of wood is $1 per unit,
how much would it cost Al to make 100 deer? .
(c) Write down Al’s short-run cost function at these factor prices.
.
Here you continue to work on cost functions. Total cost can be divided
into ﬁxed cost, the part that doesn’t change as output changes, and variable
cost. To get the average (total) cost, average ﬁxed cost, and average
variable cost, just divide the appropriate cost function by y, the level of
output. The marginal cost function is the derivative of the total cost
function with respect to output—or the rate of increase in cost as output
increases, if you don’t know calculus.
Remember that the marginal cost curve intersects both the average
cost curve and the average variable cost curve at their minimum points.
So to ﬁnd the minimum point on the average cost curve, you simply set
marginal cost equal to average cost and similarly for the minimum of
average variable cost.
A ﬁrm has the total cost function C(y) = 100 + 10y. Let us ﬁnd the
equations for its various cost curves. Total ﬁxed costs are 100, so the
equation of the average ﬁxed cost curve is 100/y. Total variable costs are
10y, so average variable costs are 10y/y = 10 for all y. Marginal cost is
10 for all y. Average total costs are (100 + 10y)/y = 10 + 10/y. Notice
that for this ﬁrm, average total cost decreases as y increases. Notice also
that marginal cost is less than average total cost for all y.
21.1 (0) Mr. Otto Carr, owner of Otto’s Autos, sells cars. Otto buys
autos for $c each and has no other costs.
(a) What is his total cost if he sells 10 cars? What if he
sells 20 cars? Write down the equation for Otto’s total costs
assuming he sells y cars: TC(y) = .
(b) What is Otto’s average cost function? AC(y) = For every
additional auto Otto sells, by how much do his costs increase?
Write down Otto’s marginal cost function: MC(y) = .
(c) In the graph below draw Otto’s average and marginal cost curves if
c = 20.
AC, MC
40
30
20
10
0 10 20 30 40
Output
(d) Suppose Otto has to pay $b a year to produce obnoxious television
commercials. Otto’s total cost curve is now TC(y) = , his
average cost curve is now AC(y) = , and his marginal cost
curve is MC(y) = .
(e) If b = $100, use red ink to draw Otto’s average cost curve on the
graph above.
21.2 (0) Otto’s brother, Dent Carr, is in the auto repair business. Dent
recently had little else to do and decided to calculate his cost conditions.
He found that the total cost of repairing s cars is TC(s) = 2s2
+ 10. But
Dent’s attention was diverted to other things . . . and that’s where you
come in. Please complete the following:
Dent’s Total Variable Costs: .
Total Fixed Costs: .
Average Variable Costs: .
Average Fixed Costs: .
Average Total Costs: .
Marginal Costs: .
21.3 (0) A third brother, Rex Carr, owns a junk yard. Rex can use one
of two methods to destroy cars. The ﬁrst involves purchasing a hydraulic
car smasher that costs $200 a year to own and then spending $1 for every
car smashed into oblivion; the second method involves purchasing a shovel
that will last one year and costs $10 and paying the last Carr brother,
Scoop, to bury the cars at a cost of $5 each.
(a) Write down the total cost functions for the two methods, where y is
output per year: TC1(y) = , TC2(y) = .
(b) The ﬁrst method has an average cost function and a
marginal cost function For the second method these costs are
and .
(c) If Rex wrecks 40 cars per year, which method should he use?
If Rex wrecks 50 cars per year, which method should he use?
What is the smallest number of cars per year for which
it would pay him to buy the hydraulic smasher? .
21.4 (0) Mary Magnolia wants to open a ﬂower shop, the Petal Pusher,
in a new mall. She has her choice of three diﬀerent ﬂoor sizes, 200 square
feet, 500 square feet, or 1,000 square feet. The monthly rent will be $1 a
square foot. Mary estimates that if she has F square feet of ﬂoor space
and sells y bouquets a month, her variable costs will be cv(y) = y2
/F per
month.
(a) If she has 200 square feet of ﬂoor space, write down her marginal cost
function: and her average cost function:
At what amount of output is average cost minimized?
At this level of output, how much is average cost? .
(b) If she has 500 square feet, write down her marginal cost function:
and her average cost function: At
what amount of output is average cost minimized? At this
level of output, how much is average cost? .
(c) If she has 1,000 square feet of ﬂoor space, write down her marginal
cost function: and her average cost function:
At what amount of output is average cost minimized?
At this level of output, how much is average cost? .
(d) Use red ink to show Mary’s average cost curve and her marginal cost
curves if she has 200 square feet. Use blue ink to show her average cost
curve and her marginal cost curve if she has 500 square feet. Use black
ink to show her average cost curve and her marginal cost curve if she has
1,000 square feet. Label the average cost curves AC and the marginal
cost curves MC.
Dollars
4
3
2
1
0 200 400 600 800 1000 1200
Bouquets
(e) Use yellow marker to show Mary’s long-run average cost curve and
her long-run marginal cost curve in your graph. Label them LRAC and
LRMC.
21.5 (0) Touchie MacFeelie publishes comic books. The only inputs he
needs are old jokes and cartoonists. His production function is
Q = .1J
1
2 L3/4
,
where J is the number of old jokes used, L the number of hours of cartoonists’
labor used as inputs, and Q is the number of comic books produced.
(a) Does this production process exhibit increasing, decreasing, or constant
returns to scale? Explain your answer.
.
(b) If the number of old jokes used is 100, write an expression for the
marginal product of cartoonists’ labor as a function of L.
Is the marginal product of labor decreasing or increasing as the amount
of labor increases? .
21.6 (0) Touchie MacFeelie’s irascible business manager, Gander MacGrope,
announces that old jokes can be purchased for $1 each and that
the wage rate of cartoonists’ labor is $2.
(a) Suppose that in the short run, Touchie is stuck with exactly 100 old
jokes (for which he paid $1 each) but is able to hire as much labor as he
wishes. How much labor would he have to hire in order produce Q comic
books? .
(b) Write down Touchie’s short-run total cost as a function of his output
.
(c) His short-run marginal cost function is .
(d) His short-run average cost function is .
21.7 (1) Touchie asks his brother, Sir Francis MacFeelie, to study the
long-run picture. Sir Francis, who has carefully studied the appendix to
Chapter 19 in your text, prepared the following report.
(a) If all inputs are variable, and if old jokes cost $1 each and cartoonist
labor costs $2 per hour, the cheapest way to produce exactly one comic
book is to use jokes and hours of
labor. (Fractional jokes are certainly allowable.)
(b) This would cost dollars.
(c) Given our production function, the cheapest proportions in which to
use jokes and labor are the same no matter how many comic books we
print. But when we double the amount of both inputs, the number of
comic books produced is multiplied by .
21.8 (0) Consider the cost function c(y) = 4y2
+ 16.
(a) The average cost function is .
(b) The marginal cost function is .
(c) The level of output that yields the minimum average cost of production
is .
(d) The average variable cost function is .
(e) At what level of output does average variable cost equal marginal
cost? .
21.9 (0) A competitive ﬁrm has a production function of the form
Y = 2L + 5K. If w = $2 and r = $3, what will be the minimum cost of
producing 10 units of output? .