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NAME 259
(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
(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
(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. No.
20.9 (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 $pper
pound (where p<3), how many pounds of fertilizer should he use per
(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
(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
(Hint: Start by writing down Skoglund’s marginal product of fertilizer as
a function of N.)
260 PROFIT MAXIMIZATION (Ch. 20)
(d) When Hoglund and Skoglund are both maximizing profits, will
Skoglund’s output be more than twice as much, less than twice as much
(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 fertil-
20.10 (0) A firm has two variable factors and a production function,
f(x1,x
2)=x1/2
1x1/4
2. The price of its output is 4. Factor 1 receives a
wage of w1and 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 2x−1/2
1x1/4
2=w1and
an equation that says that the value of the marginal product of factor
gives x1=8/(w3
1w2)and x2=4/(w2
1w2
2).(Hint: You could
use the first equation to solve for x1as a function of x2andofthefactor
(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? 1. How many units of
20.11 (0) A firm has two variable factors and a production function
f(x1,x
2)=x1/2
1x1/2
2. The price of its output is 4, the price of factor 1 is
w1, and the price of factor 2 is w2.
NAME 261
(a) Write the two equations that say that the value of the marginal prod-
uct of each factor is equal to its wage. 2x−1/2
(b) For this production function, is it possible to solve the two marginal
20.12 (1) A firm has two variable factors and a production function
f(x1,x
2)=√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
04812
16
4
8
12
Factor 1
Factor 2
16
9
_
4
20.13 (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 wand the price of the output is denoted by p. You observe the firm’s
behavior over three periods and find the following:
262 PROFIT MAXIMIZATION (Ch. 20)
Period y x w p
(a) Write an equation that gives the firm’s profits, π, as a function of the
amount of input xit uses, the amount of output yit produces, the per-unit
(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? y=x,y=1+.5x,
y=2+.25x.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.
06810
2
4
6
8
10
12
2
Output
4
Input
12
Period 3
Period 2
Period 1
20.14 (0) T-bone Pickens is a corporate raider. This means that he
looks for companies that are not maximizing profits, buys them, and then
NAME 263
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.
06810
2
4
6
8
10
12
2
Million barrels of gasoline
4
Million barrels of oil
12
10
20
40
Red 40
(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
264 PROFIT MAXIMIZATION (Ch. 20)
(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.
20.15 (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?
20.16 (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,
ΔpΔy−ΔwΔx≥0.
Which of the following propositions can be proven by the Weak Ax-
iom 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
NAME 265
(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
(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. No.
