In this chapter you work with production functions, relating output of a
ﬁrm to the inputs it uses. This theory will look familiar to you, because
it closely parallels the theory of utility functions. In utility theory, an
indiﬀerence curve is a locus of commodity bundles, all of which give a
consumer the same utility. In production theory, an isoquant is a locus
of input combinations, all of which give the same output. In consumer
theory, you found that the slope of an indiﬀerence curve at the bundle
(x1, x2) is the ratio of marginal utilities, MU1(x1, x2)/MU2(x1, x2). In
production theory, the slope of an isoquant at the input combination
(x1, x2) is the ratio of the marginal products, MP1(x1, x2)/MP2(x1, x2).
Most of the functions that we gave as examples of utility functions can
also be used as examples of production functions.
There is one important diﬀerence between production functions and
utility functions. Remember that utility functions were only “unique up to
monotonic transformations.” In contrast, two diﬀerent production functions
that are monotonic transformations of each other describe diﬀerent
technologies.
If the utility function U(x1, x2) = x1 + x2 represents a person’s preferences,
then so would the utility function U∗
(x1, x2) = (x1+x2)2
. A person
who had the utility function U∗
(x1, x2) would have the same indiﬀerence
curves as a person with the utility function U(x1, x2) and would make
the same choices from every budget. But suppose that one ﬁrm has the
production function f(x1, x2) = x1 + x2, and another has the production
function f∗
(x1, x2) = (x1 + x2)2
. It is true that the two ﬁrms will have
the same isoquants, but they certainly do not have the same technology.
If both ﬁrms have the input combination (x1, x2) = (1, 1), then the ﬁrst
ﬁrm will have an output of 2 and the second ﬁrm will have an output of
4.
Now we investigate “returns to scale.” Here we are concerned with
the change in output if the amount of every input is multiplied by a
number t > 1. If multiplying inputs by t multiplies output by more than
t, then there are increasing returns to scale. If output is multiplied by
exactly t, there are constant returns to scale. If output is multiplied by
less than t, then there are decreasing returns to scale.
Consider the production function f(x1, x2) = x
1/2
1 x
3/4
2 . If we multiply
the amount of each input by t, then output will be f(tx1, tx2) =
(tx1)1/2
(tx2)3/4
. To compare f(tx1, tx2) to f(x1, x2), factor out the
expressions involving t from the last equation. You get f(tx1, tx2) =
t5/4
x
1/2
1 x
3/4
2 = t5/4
f(x1, x2). Therefore when you multiply the amounts
of all inputs by t, you multiply the amount of output by t5/4
. This means
there are increasing returns to scale.
Let the production function be f(x1, x2) = min{x1, x2}. Then
f(tx1, tx2) = min{tx1, tx2} = min t{x1, x2} = t min{x1, x2} = tf(x1, x2).
Therefore when all inputs are multiplied by t, output is also multiplied by
t. It follows that this production function has constant returns to scale.
You will also be asked to determine whether the marginal product
of each single factor of production increases or decreases as you increase
the amount of that factor without changing the amount of other factors.
Those of you who know calculus will recognize that the marginal product
of a factor is the ﬁrst derivative of output with respect to the amount
of that factor. Therefore the marginal product of a factor will decrease,
increase, or stay constant as the amount of the factor increases depending
on whether the second derivative of the production function with respect
to the amount of that factor is negative, positive, or zero.
Consider the production function f(x1, x2) = x
1/2
1 x
3/4
2 . The marginal
product of factor 1 is 1
2 x
−1/2
1 x
3/4
2 . This is a decreasing function of x1,
as you can verify by taking the derivative of the marginal product with
respect to x1. Similarly, you can show that the marginal product of x2
decreases as x2 increases.
18.0 Warm Up Exercise. The ﬁrst part of this exercise is to calculate
marginal products and technical rates of substitution for several
frequently encountered production functions. As an example, consider
the production function f(x1, x2) = 2x1 +
√
x2. The marginal product of
x1 is the derivative of f(x1, x2) with respect to x1, holding x2 ﬁxed. This
is just 2. The marginal product of x2 is the derivative of f(x1, x2) with
respect to x2, holding x1 ﬁxed, which in this case is 1
2
√
x2
. The TRS is
−MP1/MP2 = −4
√
x2. Those of you who do not know calculus should
ﬁll in this table from the answers in the back. The table will be a useful
reference for later problems.
Marginal Products and Technical Rates of Substitution
f(x1, x2) MP1(x1, x2) MP2(x1, x2) TRS(x1, x2)
x1 + 2x2
ax1 + bx2
50x1x2
x
1/4
1 x
3/4
2
1
4 x
−3/4
1 x
3/4
2
Cxa
1xb
2 Caxa−1
1 xb
2
(x1 + 2)(x2 + 1) x2 + 1
(x1 + a)(x2 + b)
ax1 + b
√
x2
xa
1 + xa
2
(xa
1 + xa
2)
b
baxa−1
1 (xa
1 + xa
2)
b−1
baxa−1
2 (xa
1 + xa
2)
b−1
Returns to Scale and Changes in Marginal Products
For each production function in the table below, put an I, C, or D in
the ﬁrst column if the production function has increasing, constant, or
decreasing returns to scale. Put an I, C, or D in the second (third)
column, depending on whether the marginal product of factor 1 (factor
2) is increasing, constant, or decreasing, as the amount of that factor
alone is varied.
f(x1, x2) Scale MP1 MP2
x1 + 2x2
√
x1 + 2x2
.2x1x2
2
x
1/4
1 x
3/4
2
x1 +
√
x2
(x1 + 1).5
(x2).5
x
1/3
1 + x
1/3
2
3
18.1 (0) Prunella raises peaches. Where L is the number of units of
labor she uses and T is the number of units of land she uses, her output
is f(L, T) = L
1
2 T
1
2 bushels of peaches.
(a) On the graph below, plot some input combinations that give her an
output of 4 bushels. Sketch a production isoquant that runs through these
points. The points on the isoquant that gives her an output of 4 bushels
all satisfy the equation T = .
T
8
6
4
2
0 4 8 12 16
L
(b) This production function exhibits (constant, increasing, decreasing)
returns to scale. .
(c) In the short run, Prunella cannot vary the amount of land she uses.
On the graph below, use blue ink to draw a curve showing Prunella’s
output as a function of labor input if she has 1 unit of land. Locate the
points on your graph at which the amount of labor is 0, 1, 4, 9, and 16 and
label them. The slope of this curve is known as the marginal
of Is this curve getting steeper or ﬂatter as the amount
of labor increase? .
Output
8
6
4
2
0 4 8 12 16
Labor
(d) Assuming she has 1 unit of land, how much extra output does she
get from adding an extra unit of labor when she previously used 1 unit
of labor? 4 units of labor? If
you know calculus, compute the marginal product of labor at the input
combination (1, 1) and compare it with the result from the unit increase
in labor output found above.
.
(e) In the long run, Prunella can change her input of land as well as
of labor. Suppose that she increases the size of her orchard to 4 units
of land. Use red ink to draw a new curve on the graph above showing
output as a function of labor input. Also use red ink to draw a curve
showing marginal product of labor as a function of labor input when the
amount of land is ﬁxed at 4.
18.2 (0) Suppose x1 and x2 are used in ﬁxed proportions and f(x1, x2) =
min{x1, x2}.
(a) Suppose that x1 < x2. The marginal product for x1 is and
(increases, remains constant, decreases) for small
increases in x1. For x2 the marginal product is , and (increases,
remains constant, decreases) for small increases in
x2. The technical rate of substitution between x2 and x1 is
This technology demonstrates (increasing, constant, decreasing)
returns to scale.
(b) Suppose that f(x1, x2) = min{x1, x2} and x1 = x2 = 20. What is
the marginal product of a small increase in x1? What is the
marginal product of a small increase in x2? The marginal
product of x1 will (increase, decrease, stay constant) if
the amount of x2 is increased by a little bit.
18.3 (0) Suppose the production function is Cobb-Douglas and
f(x1, x2) = x
1/2
1 x
3/2
2 .
(a) Write an expression for the marginal product of x1 at the point
(x1, x2). .
(b) The marginal product of x1 (increases, decreases, remains constant)
for small increases in x1, holding x2 ﬁxed.
(c) The marginal product of factor 2 is , and it (increases,
remains constant, decreases) for small increases in x2.
(d) An increase in the amount of x2 (increases, leaves unchanged, decreases)
the marginal product of x1.
(e) The technical rate of substitution between x2 and x1 is .
(f) Does this technology have diminishing technical rate of substitution?
.
(g) This technology demonstrates (increasing, constant, decreasing)
returns to scale.
18.4 (0) The production function for fragles is f(K, L) = L/2 +
√
K,
where L is the amount of labor used and K the amount of capital used.
(a) There are (constant, increasing, decreasing) returns
to scale. The marginal product of labor is (constant,
increasing, decreasing).
(b) In the short run, capital is ﬁxed at 4 units. Labor is variable. On the
graph below, use blue ink to draw output as a function of labor input in
the short run. Use red ink to draw the marginal product of labor as a
function of labor input in the short run. The average product of labor is
deﬁned as total output divided by the amount of labor input. Use black
ink to draw the average product of labor as a function of labor input in
the short run.
Fragles
8
6
4
2
0 4 8 12 16
Labor
18.5 (0) General Monsters Corporation has two plants for producing
juggernauts, one in Flint and one in Inkster. The Flint plant produces
according to fF (x1, x2) = min{x1, 2x2} and the Inkster plant produces
according to fI(x1, x2) = min{2x1, x2}, where x1 and x2 are the inputs.
(a) On the graph below, use blue ink to draw the isoquant for 40 juggernauts
at the Flint plant. Use red ink to draw the isoquant for producing
40 juggernauts at the Inkster plant.
x2
80
60
40
20
0 20 40 60 80
x1
(b) Suppose that the ﬁrm wishes to produce 20 juggernauts at each plant.
How much of each input will the ﬁrm need to produce 20 juggernauts at
the Flint plant? How much of each input will the ﬁrm
need to produce 20 juggernauts at the Inkster plant?
Label with an a on the graph, the point representing the total amount
of each of the two inputs that the ﬁrm needs to produce a total of 40
juggernauts, 20 at the Flint plant and 20 at the Inkster plant.
(c) Label with a b on your graph the point that shows how much of each
of the two inputs is needed in toto if the ﬁrm is to produce 10 juggernauts
in the Flint plant and 30 juggernauts in the Inkster plant. Label with a
c the point that shows how much of each of the two inputs that the ﬁrm
needs in toto if it is to produce 30 juggernauts in the Flint plant and
10 juggernauts in the Inkster plant. Use a black pen to draw the ﬁrm’s
isoquant for producing 40 units of output if it can split production in any
manner between the two plants. Is the technology available to this ﬁrm
convex? .
18.6 (0) You manage a crew of 160 workers who could be assigned to
make either of two products. Product A requires 2 workers per unit of
output. Product B requires 4 workers per unit of output.
(a) Write an equation to express the combinations of products A and B
that could be produced using exactly 160 workers. On
the diagram below, use blue ink to shade in the area depicting the combinations
of A and B that could be produced with 160 workers. (Assume
that it is also possible for some workers to do nothing at all.)
B
80
60
40
20
0 20 40 60 80
A
(b) Suppose now that every unit of product A that is produced requires
the use of 4 shovels as well as 2 workers and that every unit of product B
produced requires 2 shovels and 4 workers. On the graph you have just
drawn, use red ink to shade in the area depicting combinations of A and
B that could be produced with 180 shovels if there were no worries about
the labor supply. Write down an equation for the set of combinations of
A and B that require exactly 180 shovels. .
(c) On the same diagram, use black ink to shade the area that represents
possible output combinations when one takes into account both the
limited supply of labor and the limited supply of shovels.
(d) On your diagram locate the feasible combination of inputs that use
up all of the labor and all of the shovels. If you didn’t have the graph,
what equations would you solve to determine this point?
.
(e) If you have 160 workers and 180 shovels, what is the largest amount of
product A that you could produce? If you produce this
amount, you will not use your entire supply of one of the inputs. Which
one? How many will be left unused? .
18.7 (0) A ﬁrm has the production function f(x, y) = min{2x, x + y}.
On the graph below, use red ink to sketch a couple of production isoquants
for this ﬁrm. A second ﬁrm has the production function f(x, y) = x +
min{x, y}. Do either or both of these ﬁrms have constant returns to scale?
On the same graph, use black ink to draw a couple of
isoquants for the second ﬁrm.
y
40
30
20
10
0 10 20 30 40
x
18.8 (0) Suppose the production function has the form
f(x1, x2, x3) = Axa
1xb
2xc
3,
where a + b + c > 1. Prove that there are increasing returns to scale.
.
18.9 (0) Suppose that the production function is f(x1, x2) = Cxa
1xb
2,
where a, b, and C are positive constants.
(a) For what positive values of a, b, and C are there decreasing returns
to scale? constant returns to scale?
increasing returns to scale? .
(b) For what positive values of a, b, and C is there decreasing marginal
product for factor 1? .
(c) For what positive values of a, b, and C is there diminishing technical
rate of substitution? .
18.10 (0) Suppose that the production function is f(x1, x2) =
(xa
1 + xa
2)
b
, where a and b are positive constants.
(a) For what positive values of a and b are there decreasing returns to
scale? Constant returns to scale? Increasing
returns to scale? .
18.11 (0) Suppose that a ﬁrm has the production function f(x1, x2) =
√
x1 + x2
2.
(a) The marginal product of factor 1 (increases, decreases, stays constant)
as the amount of factor 1 increases. The marginal product
of factor 2 (increases, decreases, stays constant) as the
amount of factor 2 increases.
(b) This production function does not satisfy the deﬁnition of increasing
returns to scale, constant returns to scale, or decreasing returns to scale.
How can this be?
Find a combination of inputs such that
doubling the amount of both inputs will more than double the amount
of output. Find a combination of inputs
such that doubling the amount of both inputs will less than double output.
.
A ﬁrm in a competitive industry cannot charge more than the market price
for its output. If it also must compete for its inputs, then it has to pay
the market price for inputs as well. Suppose that a proﬁt-maximizing
competitive ﬁrm can vary the amount of only one factor and that the
marginal product of this factor decreases as its quantity increases. Then
the ﬁrm will maximize its proﬁts by hiring enough of the variable factor
so that the value of its marginal product is equal to the wage. Even if a
ﬁrm uses several factors, only some of them may be variable in the short
run.
A ﬁrm has the production function f(x1, x2) = x
1/2
1 x
1/2
2 . Suppose that
this ﬁrm is using 16 units of factor 2 and is unable to vary this quantity
in the short run. In the short run, the only thing that is left for the ﬁrm
to choose is the amount of factor 1. Let the price of the ﬁrm’s output
be p, and let the price it pays per unit of factor 1 be w1. We want to
ﬁnd the amount of x1 that the ﬁrm will use and the amount of output
it will produce. Since the amount of factor 2 used in the short run must
be 16, we have output equal to f(x1, 16) = 4x
1/2
1 . The marginal product
of x1 is calculated by taking the derivative of output with respect to
x1. This marginal product is equal to 2x
−1/2
1 . Setting the value of the
marginal product of factor 1 equal to its wage, we have p2x
−1/2
1 = w1.
Now we can solve this for x1. We ﬁnd x1 = (2p/w1)2
. Plugging this
into the production function, we see that the ﬁrm will choose to produce
4x
1/2
1 = 8p/w1 units of output.
In the long run, a ﬁrm is able to vary all of its inputs. Consider
the case of a competitive ﬁrm that uses two inputs. Then if the ﬁrm is
maximizing its proﬁts, it must be that the value of the marginal product
of each of the two factors is equal to its wage. This gives two equations in
the two unknown factor quantities. If there are decreasing returns to scale,
these two equations are enough to determine the two factor quantities. If
there are constant returns to scale, it turns out that these two equations
are only suﬃcient to determine the ratio in which the factors are used.
In the problems on the weak axiom of proﬁt maximization, you are
asked to determine whether the observed behavior of ﬁrms is consistent
with proﬁt-maximizing behavior. To do this you will need to plot some of
the ﬁrm’s isoproﬁt lines. An isoproﬁt line relates all of the input-output
combinations that yield the same amount of proﬁt for some given input
and output prices. To get the equation for an isoproﬁt line, just write
down an equation for the ﬁrm’s proﬁts at the given input and output
prices. Then solve it for the amount of output produced as a function
of the amount of the input chosen. Graphically, you know that a ﬁrm’s
behavior is consistent with proﬁt maximization if its input-output choice
in each period lies below the isoproﬁt lines of the other periods.
19.1 (0) The short-run production function of a competitive ﬁrm is
given by f(L) = 6L2/3
, where L is the amount of labor it uses. (For
those who do not know calculus—if total output is aLb
, where a and b
are constants, and where L is the amount of some factor of production,
then the marginal product of L is given by the formula abLb−1
.) The cost
per unit of labor is w = 6 and the price per unit of output is p = 3.
(a) Plot a few points on the graph of this ﬁrm’s production function and
sketch the graph of the production function, using blue ink. Use black
ink to draw the isoproﬁt line that passes through the point (0, 12), the
isoproﬁt line that passes through (0, 8), and the isoproﬁt line that passes
through the point (0, 4). What is the slope of each of the isoproﬁt lines?
How many points on the isoproﬁt line through
(0, 12) consist of input-output points that are actually possible?
Make a squiggly line over the part of the isoproﬁt line through
(0, 4) that consists of outputs that are actually possible.
(b) How many units of labor will the ﬁrm hire? How much
output will it produce? If the ﬁrm has no other costs, how much
will its total proﬁts be? .
Output
48
36
24
12
0 4 8 12 16 20 24
Labor input
(c) Suppose that the wage of labor falls to 4, and the price of output
remains at p. On the graph, use red ink to draw the new isoproﬁt line
for the ﬁrm that passes through its old choice of input and output. Will
the ﬁrm increase its output at the new price? Explain why,
referring to your diagram.
.
19.2 (0) A Los Angeles ﬁrm uses a single input to produce a recreational
commodity according to a production function f(x) = 4
√
x, where x is
the number of units of input. The commodity sells for $100 per unit. The
input costs $50 per unit.
(a) Write down a function that states the ﬁrm’s proﬁt as a function of
the amount of input. .
(b) What is the proﬁt-maximizing amount of input? of output?
How much proﬁts does it make when it maximizes proﬁts?
.
(c) Suppose that the ﬁrm is taxed $20 per unit of its output and the
price of its input is subsidized by $10. What is its new input level?
What is its new output level? How much proﬁt does
it make now? (Hint: A good way to solve this is to write an
expression for the ﬁrm’s proﬁt as a function of its input and solve for the
proﬁt-maximizing amount of input.)
(d) Suppose that instead of these taxes and subsidies, the ﬁrm is taxed
at 50% of its proﬁts. Write down its after-tax proﬁts as a function
of the amount of input. What is the proﬁtmaximizing
amount of output? How much proﬁt does it make
after taxes? .
19.3 (0) Brother Jed takes heathens and reforms them into righteous
individuals. There are two inputs needed in this process: heathens (who
are widely available) and preaching. The production function has the
following form: rp = min{h, p}, where rp is the number of righteous
persons produced, h is the number of heathens who attend Jed’s sermons,
and p is the number of hours of preaching. For every person converted,
Jed receives a payment of s from the grateful convert. Sad to say, heathens
do not ﬂock to Jed’s sermons of their own accord. Jed must oﬀer heathens
a payment of w to attract them to his sermons. Suppose the amount of
preaching is ﬁxed at ¯p and that Jed is a proﬁt-maximizing prophet.
(a) If h < ¯p, what is the marginal product of heathens? What
is the value of the marginal product of an additional heathen? .
(b) If h > ¯p, what is the marginal product of heathens? What
is the value of the marginal product of an additional heathen in this case?
.
(c) Sketch the shape of this production function in the graph below. Label
the axes, and indicate the amount of the input where h = ¯p.
(d) If w < s, how many heathens will be converted? If w > s,
how many heathens will be converted? .
19.4 (0) Allie’s Apples, Inc. purchases apples in bulk and sells two products,
boxes of apples and jugs of cider. Allie’s has capacity limitations of
three kinds: warehouse space, crating facilities, and pressing facilities. A
box of apples requires 6 units of warehouse space, 2 units of crating facilities,
and no pressing facilities. A jug of cider requires 3 units of warehouse
space, 2 units of crating facilities, and 1 unit of pressing facilities. The
total amounts available each day are: 1,200 units of warehouse space, 600
units of crating facilities, and 250 units of pressing facilities.
(a) If the only capacity limitations were on warehouse facilities, and if
all warehouse space were used for the production of apples, how many
boxes of apples could be produced in one day? How many
jugs of cider could be produced each day if, instead, all warehouse space
were used in the production of cider and there were no other capacity
constraints? Draw a blue line in the following graph to
represent the warehouse space constraint on production combinations.
(b) Following the same reasoning, draw a red line to represent the constraints
on output to limitations on crating capacity. How many boxes of
apples could Allie produce if he only had to worry about crating capacity?
How many jugs of cider? .
(c) Finally draw a black line to represent constraints on output combinations
due to limitations on pressing facilities. How many boxes of apples
could Allie produce if he only had to worry about the pressing capacity
and no other constraints? How many jugs of
cider? .
(d) Now shade the area that represents feasible combinations of daily
production of apples and cider for Allie’s Apples.
Cider
600
500
400
300
200
100
0 100 200 300 400 500 600
Apples
(e) Allie’s can sell apples for $5 per box of apples and cider for $2 per
jug. Draw a black line to show the combinations of sales of apples and
cider that would generate a revenue of $1,000 per day. At the proﬁtmaximizing
production plan, Allie’s is producing boxes of apples
and jugs of cider. Total revenues are .
19.5 (0) A proﬁt-maximizing ﬁrm produces one output, y, and uses one
input, x, to produce it. The price per unit of the factor is denoted by
w and the price of the output is denoted by p. You observe the ﬁrm’s
behavior over three periods and ﬁnd the following:
Period y x w p
1 1 1 1 1
2 2.5 3 .5 1
3 4 8 .25 1
(a) Write an equation that gives the ﬁrm’s proﬁts, π, as a function of
the amount of input x it uses, the amount of output y it produces, the
per-unit cost of the input w, and the price of output p. .
(b) In the diagram below, draw an isoproﬁt line for each of the three
periods, showing combinations of input and output that would yield the
same proﬁts that period as the combination actually chosen. What are the
equations for these three lines? Using
the theory of revealed proﬁtability, shade in the region on the graph that
represents input-output combinations that could be feasible as far as one
can tell from the evidence that is available. How would you describe this
region in words? .
Output
12
10
8
6
4
2
0 2 4 6 8 10 12
Input
19.6 (0) T-bone Pickens is a corporate raider. This means that he looks
for companies that are not maximizing proﬁts, buys them, and then tries
to operate them at higher proﬁts. T-bone is examining the ﬁnancial
records of two reﬁneries that he might buy, the Shill Oil Company and
the Golf Oil Company. Each of these companies buys oil and produces
gasoline. During the time period covered by these records, the price of
gasoline ﬂuctuated signiﬁcantly, while the cost of oil remained constant
at $10 a barrel. For simplicity, we assume that oil is the only input to
gasoline production.
Shill Oil produced 1 million barrels of gasoline using 1 million barrels
of oil when the price of gasoline was $10 a barrel. When the price of
gasoline was $20 a barrel, Shill produced 3 million barrels of gasoline
using 4 million barrels of oil. Finally, when the price of gasoline was $40
a barrel, Shill used 10 million barrels of oil to produce 5 million barrels
of gasoline.
Golf Oil (which is managed by Martin E. Lunch III) did exactly the
same when the price of gasoline was $10 and $20, but when the price
of gasoline hit $40, Golf produced 3.5 million barrels of gasoline using 8
million barrels of oil.
(a) Using black ink, plot Shill Oil’s isoproﬁt lines and choices for the three
diﬀerent periods. Label them 10, 20, and 40. Using red ink draw Golf
Oil’s isoproﬁt line and production choice. Label it with a 40 in red ink.
Million barrels of gasoline
12
10
8
6
4
2
0 2 4 6 8 10 12
Million barrels of oil
(b) How much proﬁts could Golf Oil have made when the price of gasoline
was $40 a barrel if it had chosen to produce the same amount that it did
when the price was $20 a barrel? What proﬁts did
Golf actually make when the price of gasoline was $40? .
(c) Is there any evidence that Shill Oil is not maximizing proﬁts? Explain.
.
(d) Is there any evidence that Golf Oil is not maximizing proﬁts? Explain.
.
19.7 (0) After carefully studying Shill Oil, T-bone Pickens decides that
it has probably been maximizing its proﬁts. But he still is very interested
in buying Shill Oil. He wants to use the gasoline they produce to fuel his
delivery ﬂeet for his chicken farms, Capon Truckin’. In order to do this
Shill Oil would have to be able to produce 5 million barrels of gasoline
from 8 million barrels of oil. Mark this point on your graph. Assuming
that Shill always maximizes proﬁts, would it be technologically feasible
for it to produce this input-output combination? Why or why not?
.
19.8 (0) Suppose that ﬁrms operate in a competitive market, attempt to
maximize proﬁts, and only use one factor of production. Then we know
that for any changes in the input and output price, the input choice and
the output choice must obey the Weak Axiom of Proﬁt Maximization,
∆p∆y − ∆w∆x ≥ 0.
Which of the following propositions can be proven by the Weak Axiom
of Proﬁt Maximizing Behavior (WAPM)? Respond yes or no, and
give a short argument.
(a) If the price of the input does not change, then a decrease in the price
of the output will imply that the ﬁrm will produce the same amount
or less output.
.
(b) If the price of the output remains constant, then a decrease in
the input price will imply that the ﬁrm will use the same amount or
more of the input.
.
(c) If both the price of the output and the input increase and the ﬁrm
produces less output, then the ﬁrm will use more of the input.
.
19.9 (1) Farmer Hoglund has discovered that on his farm, he can get
30 bushels of corn per acre if he applies no fertilizer. When he applies N
pounds of fertilizer to an acre of land, the marginal product of fertilizer is
1 − N/200 bushels of corn per pound of fertilizer.
(a) If the price of corn is $3 a bushel and the price of fertilizer is $p per
pound (where p < 3), how many pounds of fertilizer should he use per
acre in order to maximize proﬁts? .
(b) (Only for those who remember a bit of easy integral calculus.) Write
down a function that states Farmer Hoglund’s yield per acre as a function
of the amount of fertilizer he uses. .
(c) Hoglund’s neighbor, Skoglund, has better land than Hoglund. In fact,
for any amount of fertilizer that he applies, he gets exactly twice as much
corn per acre as Hoglund would get with the same amount of fertilizer.
How much fertilizer will Skoglund use per acre when the price of corn
is $3 a bushel and the price of fertilizer is $p a pound?
(Hint: Start by writing down Skoglund’s marginal product of fertilizer as
a function of N.)
(d) When Hoglund and Skoglund are both maximizing proﬁts, will
Skoglund’s output be more than twice as much, less than twice as much
or exactly twice as much as Hoglund’s? Explain.
.
(e) Explain how someone who looked at Hoglund’s and Skoglund’s
corn yields and their fertilizer inputs but couldn’t observe the quality
of their land, would get a misleading idea of the productivity
of fertilizer.
.
19.10 (0) A ﬁrm has two variable factors and a production function,
f(x1, x2) = x
1/2
1 x
1/4
2 . The price of its output is 4. Factor 1 receives a
wage of w1 and factor 2 receives a wage of w2.
(a) Write an equation that says that the value of the marginal product
of factor 1 is equal to the wage of factor 1 and an
equation that says that the value of the marginal product of factor 2 is
equal to the wage of factor 2. Solve two equations in
the two unknowns, x1 and x2, to give the amounts of factors 1 and 2
that maximize the ﬁrm’s proﬁts as a function of w1 and w2. This gives
x1 = and x2 = (Hint: You could use the
ﬁrst equation to solve for x1 as a function of x2 and of the factor wages.
Then substitute the answer into the second equation and solve for x2 as
a function of the two wage rates. Finally use your solution for x2 to ﬁnd
the solution for x1.)
(b) If the wage of factor 1 is 2, and the wage of factor 2 is 1, how many
units of factor 1 will the ﬁrm demand? How many units of
factor 2 will it demand? How much output will it produce?
How much proﬁt will it make? .
19.11 (0) A ﬁrm has two variable factors and a production function
f(x1, x2) = x
1/2
1 x
1/2
2 . The price of its output is 4, the price of factor 1 is
w1, and the price of factor 2 is w2.
(a) Write the two equations that say that the value of the marginal product
of each factor is equal to its wage.
If w1 = 2w2, these two equations imply that x1/x2 = .
(b) For this production function, is it possible to solve the two marginal
productivity equations uniquely for x1 and x2? .
19.12 (1) A ﬁrm has two variable factors and a production function
f(x1, x2) =
√
2x1 + 4x2. On the graph below, draw production isoquants
corresponding to an ouput of 3 and to an output of 4.
(a) If the price of the output good is 4, the price of factor 1 is 2, and
the price of factor 2 is 3, ﬁnd the proﬁt-maximizing amount of factor 1
, the proﬁt-maximizing amount of factor 2 , and the
proﬁt-maximizing output .
Factor 2
16
12
8
4
0 4 8 12 16
Factor 1
19.13 (0) A proﬁt-maximizing ﬁrm produces one output, y, and uses
one input, x, to produce it. The price per unit of the factor is denoted
by w and the price of the output is denoted by p. You observe the ﬁrm’s
behavior over three periods and ﬁnd the following:
Period y x w p
1 1 1 1 1
2 2.5 3 .5 1
3 4 8 .25 1
(a) Write an equation that gives the ﬁrm’s proﬁts, π, as a function of
the amount of input x it uses, the amount of output y it produces, the
per-unit cost of the input w, and the price of output p. .
(b) In the diagram below, draw an isoproﬁt line for each of the three
periods, showing combinations of input and output that would yield the
same proﬁts that period as the combination actually chosen. What are the
equations for these three lines? Using
the theory of revealed proﬁtability, shade in the region on the graph that
represents input-output combinations that could be feasible as far as one
can tell from the evidence that is available. How would you describe this
region in words? .
Output
12
10
8
6
4
2
0 2 4 6 8 10 12
Input
19.14 (0) T-bone Pickens is a corporate raider. This means that he
looks for companies that are not maximizing proﬁts, buys them, and then
tries to operate them at higher proﬁts. T-bone is examining the ﬁnancial
records of two reﬁneries that he might buy, the Shill Oil Company and
the Golf Oil Company. Each of these companies buys oil and produces
gasoline. During the time period covered by these records, the price of
gasoline ﬂuctuated signiﬁcantly, while the cost of oil remained constant
at $10 a barrel. For simplicity, we assume that oil is the only input to
gasoline production.
Shill Oil produced 1 million barrels of gasoline using 1 million barrels
of oil when the price of gasoline was $10 a barrel. When the price of
gasoline was $20 a barrel, Shill produced 3 million barrels of gasoline
using 4 million barrels of oil. Finally, when the price of gasoline was $40
a barrel, Shill used 10 million barrels of oil to produce 5 million barrels
of gasoline.
Golf Oil (which is managed by Martin E. Lunch III) did exactly the
same when the price of gasoline was $10 and $20, but when the price
of gasoline hit $40, Golf produced 3.5 million barrels of gasoline using 8
million barrels of oil.
(a) Using black ink, plot Shill Oil’s isoproﬁt lines and choices for the three
diﬀerent periods. Label them 10, 20, and 40. Using red ink draw Golf
Oil’s isoproﬁt line and production choice. Label it with a 40 in red ink.
Million barrels of gasoline
12
10
8
6
4
2
0 2 4 6 8 10 12
Million barrels of oil
(b) How much proﬁts could Golf Oil have made when the price of gasoline
was $40 a barrel if it had chosen to produce the same amount that it did
when the price was $20 a barrel? What proﬁts did
Golf actually make when the price of gasoline was $40? .
(c) Is there any evidence that Shill Oil is not maximizing proﬁts? Explain.
.
(d) Is there any evidence that Golf Oil is not maximizing proﬁts? Explain.
.
19.15 (0) After carefully studying Shill Oil, T-bone Pickens decides that
it has probably been maximizing its proﬁts. But he still is very interested
in buying Shill Oil. He wants to use the gasoline they produce to fuel his
delivery ﬂeet for his chicken farms, Capon Truckin’. In order to do this
Shill Oil would have to be able to produce 5 million barrels of gasoline
from 8 million barrels of oil. Mark this point on your graph. Assuming
that Shill always maximizes proﬁts, would it be technologically feasible
for it to produce this input-output combination? Why or why not?
.
19.16 (0) Suppose that ﬁrms operate in a competitive market, attempt
to maximize proﬁts, and only use one factor of production. Then we know
that for any changes in the input and output price, the input choice and
the output choice must obey the Weak Axiom of Proﬁt Maximization,
∆p∆y − ∆w∆x ≥ 0.
Which of the following propositions can be proven by the Weak Axiom
of Proﬁt Maximizing Behavior (WAPM)? Respond yes or no, and
give a short argument.
(a) If the price of the input does not change, then a decrease in the price
of the output will imply that the ﬁrm will produce the same amount
or less output.
.
(b) If the price of the output remains constant, then a decrease in
the input price will imply that the ﬁrm will use the same amount or
more of the input.
.
(c) If both the price of the output and the input increase and the ﬁrm
produces less output, then the ﬁrm will use more of the input.
.