Chapter 21 NAME
Cost Minimization
Introduction. In the chapter on consumer choice, you studied a con-
sumer 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,x
2)=(
√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 out-
The amounts x1and x2that we solved for in the previous para-
graph 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 ex-
press this by saying x1(1,1,16) = 16/100 and x2(1,1,16) = 144/100.
In consumer theory, you also dealt with cases where the consumer’s
indifference “curves” were straight lines and with cases where there were
268 COST MINIMIZATION (Ch. 21)
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
21.1 (0) Nadine sells user-friendly software. Her firm’s production func-
tion is f(x1,x
2)=x1+2x2,wherex1is the amount of unskilled labor
and x2is 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.
10
20
30
40
x2
40 units
(c) If Nadine uses only unskilled labor, how much unskilled labor would
(d) If Nadine uses only skilled labor to produce output, how much skilled
2.
(e) If Nadine faces factor prices (1,1), what is the cheapest way for her
NAME 269
(f) If Nadine faces factor prices (1,3), what is the cheapest way for her
(g) If Nadine faces factor prices (w1,w
2), what will be the minimal cost
(h) If Nadine faces factor prices (w1,w
2), what will be the mini-
21.2 (0) The Ontario Brassworks produces brazen effronteries. As you
knowbrassisanalloyofcopperandzinc,usedinfixedproportions. The
production function is given by: f(x1,x
2)=min{x1,2x2},wherex1is
the amount of copper it uses and x2is the amount of zinc that it uses in
production.
(a) Illustrate a typical isoquant for this production function in the graph
below.
0102030
40
10
20
30
40
x2
x1
x2=1
_
2x
1
(b) Does this production function exhibit increasing, decreasing, or con-
(c) If the firm wanted to produce 10 effronteries, how much copper would
270 COST MINIMIZATION (Ch. 21)
(d) If the firm faces factor prices (1,1), what is the cheapest way for it
(e) If the firm faces factor prices (w1,w
2), what is the cheapest cost to
(f) If the firm faces factor prices (w1,w
2), what will be the minimal cost
Calculus 21.3 (0) A firm uses labor and machines to produce output according to
the production function f(L, M)=4L1/2M1/2,whereLis the number of
units of labor used and Mis 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 com-
binations of machines and labor that cost $400 and another isocost line
(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
(c) On the graph, sketch the production isoquant corresponding to an
output of 40. Calculate the amount of labor 5 units and the
number of machines 20 that are used to produce 40 units of output
in the cheapest possible way, given the above factor prices. Calculate the
(d) How many units of labor y/8 and how many machines y/2
would the firm use to produce yunits in the cheapest possible way? How
returns to scale.)
NAME 271
0102030
40
10
20
30
40
Machines
Labour
$400 isocost line
$200 isocost line
21.4 (0) Earl sells lemonade in a competitive market on a busy street
corner in Philadelphia. His production function is f(x1,x
2)=x1/3
1x1/3
2,
where output is measured in gallons, x1is the number of pounds of lemons
he uses, and x2is the number of labor-hours spent squeezing them.
(a) Does Earl have constant returns to scale, decreasing returns to scale,
(b) Where w1is the cost of a pound of lemons and w2is the wage rate
for lemon-squeezers, the cheapest way for Earl to produce lemonade is to
of his isoquant equal to the slope of his isocost line.)
(c) If he is going to produce yunits in the cheapest way possible,
then the number of pounds of lemons he will use is x1(w1,w
2,y)=
w1/2
2y3/2/w1/2
1and the number of hours of labor that he will use
is x2(w1,w
(d) The cost to Earl of producing yunits at factor prices w1and w2is
272 COST MINIMIZATION (Ch. 21)
(a) If the production function is given by f(x1,x
2)=min{x1,x
2}, what
(b) If the production function is given by f(x3,x
4)=x3+x4, what is the
(c) If the production function is given by f(x1,x
2,x
3,x
4)=min{x1+
x2,x
3+x4}, what is the minimum cost of producing one unit of output?
(d) If the production function is given by f(x1,x
2)=min{x1,x
2}+
min{x3,x
4}, what is the minimum cost of producing one unit of output?
21.6 (0) Joe Grow, an avid indoor gardener, has found that the number
of happy plants, h, depends on the amount of light, l,andwater,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
(b) If Suppose Joe wants to produce 4 happy plants, what are the mini-
(c) Joe’s conditional factor demand function for light is l(w1,w
2,h)=
hand his conditional factor demand function for water is
(d) If each unit of light costs w1and each unit of water costs w2,Joe’s
2h.
21.7 (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 fis the number of bags of fertilizer used
and tis the number of hours she talks to her plants, the number of happy
plants produced is exactly h=t+2f. Suppose fertilizer costs wfper bag
and talk costs wtper hour.