A ppendix to
Chapter
10
Producing at Least Cost
This appendix describes a set of useful tools for
studying a firm’s long-run production and costs. The
tools are isoquants and isocost lines.
I soquants
F .  ’ 
function. The figure highlights that Swanky can use
three different combinations of labor and capital to
produce 15 sweaters a day and two different combinations to produce 10 and 21 sweaters a day. These
combinations, and many others not shown in the figure, can be illustrated by using an isoquant map.
An Isoquant Map
An isoquant is a curve that shows the different
combinations of labor and capital required to pro-
duce a given quantity of output. The word isoquant
means “equal quantity”—iso meaning equal and
quant meaning quantity. There is an isoquant for
each output level. A series of isoquants is called an
isoquant map. Figure A10.2 shows an isoquant
map with three isoquants: one for 10 sweaters, one
for 15 sweaters, and one for 21 sweaters. Each isoquant shown is based on the production function in
Fig. A10.1.
Although all goods and services can be produced
by using a variety of alternative methods of production or techniques, the ease with which capital and
labor can be substituted for each other varies from
industry to industry. The shape of the production
function reflects the ease with which inputs can be
substituted for each other. Therefore the production
function can be used to calculate the degree of substitutability between inputs. Such a calculation involves
a new concept—the marginal rate of substitution of
labor for capital.

FIGURE
TO
C H A P T E R 1 0 P RO D U C I N G
AT
LEAST COST
A10.1
FIGURE
Capital (machines per day)
Swanky’s Production Function
Output (sweaters per day)
5
4
3
16
15
13
22
21
18
25
24
22
27
26
24
28
27
25
5
4
10
15
18
20
b
3
2
2
A10.2
An Isoquant Map
Capital (machines per day)
APPENDIX
a
c
21
1
1
4
10
13
15
1
2
3
15 sweaters
16
10 sweaters
0
0
21 sweaters
4
5
Labor (workers per day)
The figure shows how many sweaters can be produced per
day by various combinations of labor and capital inputs. For
example, by using 1 worker and 2 machines, Swanky can produce 10 sweaters a day; and by using 4 workers and 2 knitting
machines, Swanky can produce 20 sweaters.
Marginal Rate of Substitution
The marginal rate of substitution of labor
for capital is the decrease in capital needed per unit
increase in labor so that output remains constant.
The marginal rate of substitution is the magnitude of
the slope of the isoquant. Figure A10.3 illustrates this
relationship. The figure shows the isoquant for 13
sweaters a day. Pick any point on this isoquant and
imagine increasing labor by the smallest conceivable
amount and decreasing capital by the amount necessary to keep output constant at 13 sweaters. As we
increase the labor input and decrease the capital
input so as to keep output constant at 13 sweaters a
day, we travel down along the isoquant.
The marginal rate of substitution of labor for
capital at point a is the magnitude of the slope of the
straight red line that is tangent to the isoquant at
point a. The slope of the isoquant at point a equals
the slope of the red line. To calculate that slope, let’s
1
2
3
4
5
Labor (workers per day)
The figure illustrates an isoquant map, but one that shows only
3 isoquants—those for 10, 15, and 21 sweaters a day. These
curves correspond to the production function shown in Fig.
A10.1. If Swanky uses 2 machines and 1 worker (point a), it
produces 10 sweaters. If it uses 4 machines and 1 worker
(point b), it produces 15 sweaters. And if it uses 2 machines
and 5 workers (point c), it produces 21 sweaters.
move along the red line from 5 knitting machines
and no workers to 2.5 workers and no knitting
machines. Capital decreases by 5 knitting machines,
and labor increases by 2.5 workers. The magnitude of
the slope is 5 divided by 2.5, which equals 2. Thus
when Swanky uses technique a to produce 13
sweaters a day, the marginal rate of substitution of
labor for capital is 2.
The marginal rate of substitution of labor for capital at point b is the magnitude of the slope of the
straight red line that is tangent to the isoquant at
point b. Along this red line, if capital decreases by 2.5
knitting machines, labor increases by 5 workers. The
magnitude of the slope is 2.5 knitting machines divided by 5, which equals 1/2. Thus when Swanky uses
technique b to produce 13 sweaters a day, the marginal rate of substitution of labor for capital is 1/2.
The marginal rates of substitution we’ve just calculated obey the law of diminishing marginal
rate of substitution, which states that

ISOCOST LINES
FIGURE
A10.3
FIGURE
Swanky’s Input Possibilities
Capital (machines per day)
Capital (machines per day)
The Marginal Rate
of Substitution
5
4
MRS = 2
a
3
2
A10.4
5
4
3
a
Total cost = $100
b
c
2
1
MRS = —
2
b
1
d
1
13 sweaters
0
1
2
3
4
5
Labor (workers per day)
The marginal rate of substitution is measured by the magnitude
of the slope of the isoquant. To calculate the marginal rate of
substitution at point a, use the red line that is tangential to the
isoquant at point a. Calculate the slope of that line to find the
slope of the isoquant at point a. The magnitude of the slope at
point a is 2. Thus at a point a, the marginal rate of substitution
of labor for capital is 2. The marginal rate of substitution at
point b is found from the slope of the red line tangential to the
isoquant at that point. That slope is 1/2. Thus the marginal rate
of substitution of labor for capital at point b is 1/2.
The marginal rate of substitution of labor for
capital diminishes as the amount of labor increases
and the amount of capital decreases.
You can now see that the law of diminishing
marginal rate of substitution determines the shape of
the isoquant. When the capital input is large and the
labor input is small, the isoquant is steep and the
marginal rate of substitution of labor for capital is
large. As the capital input decreases and the labor
input increases, the isoquant becomes flatter and the
marginal rate of substitution of labor for capital
diminishes. Only curves that are bowed toward the
origin have this feature; hence isoquants are always
bowed toward the origin.
Isoquants are very nice, but what do we do with
them? The answer is that we use them to work out a
Isocost line
e
0
1
2
3
4
5
Labor (workers per day)
For a given total cost, Swanky’s input possibilities depend on
input prices. If labor and capital cost $25 a day each, for a total
cost of $100 Swanky can employ the combinations of capital
and labor shown by the points a through e. The line passing
through these points is an isocost line. It shows all possible
combinations of capital and labor that Swanky can hire for a
total cost of $100 when capital and labor cost $25 a day each.
firm’s least-cost technique of production. But to do
so, we need to illustrate the firm’s costs in the same
sort of figure that contains the isoquants.
I socost Lines
A ’      
lines. An isocost line shows all the combinations of
capital and labor that can be bought for a given total
cost. For example, suppose Swanky is going to spend
$100 a day producing sweaters. Knitting-machine
operators can be hired for $25 a day, and knitting
machines can be rented for $25 a day. The points a, b,
c, d, and e in Fig. A10.4 show five possible combinations of labor and capital that Swanky can employ for
$100. For example, point b shows that Swanky can
use 3 machines (costing $75) and 1 worker (costing

TO
C H A P T E R 1 0 P RO D U C I N G
AT
LEAST COST
$25). If Swanky can employ workers and machines
for fractions of a day, then any combination along the
line ae will cost Swanky $100 a day. This line is
Swanky’s isocost line for a total cost of $100.
The Isocost Equation
The isocost line can be described by an equation.
We’ll work out the isocost equation by using symbols
that apply to any firm and with numbers that
describe Swanky’s situation.
The variables that affect the firm’s total cost (TC )
are the prices of the inputs—the price of labor (PL)
and the price of capital (PK)—and the quantities of
the inputs employed—the quantity of labor (L) and
the quantity of capital (K ). In Swanky’s case, we’re
going to look at the amount of labor and capital that
can be employed when each input costs $25 a day
and the total cost is $100. The cost of the labor
employed (PLL) plus the cost of the capital employed
(PK K ) is the firm’s total cost (TC ). That is,
PL L + PK K = TC
and in Swanky’s case,
25 L + 25 K = 100.
To calculate the isocost equation, divide the
firm’s total cost by the price of capital and then subtract (PL/PK)L from both sides of the resulting equation. The isocost equation is
K = TC /PK − ( PL /PK ) L
It tells us how the firm can vary its capital input as it
varies its labor input, holding total cost constant.
Swanky’s isocost equation is
K = 4 − L.
This equation corresponds to the isocost line in Fig.
A10.4.
The Effect of Input Prices
Along the isocost line that we have just calculated,
capital and labor cost $25 a day each. Therefore, to
decrease its capital input by 1 unit and keep its total
cost at $100 a day, the firm must increase the labor
input by 1 unit. The magnitude of the slope of the
isocost line shown in Fig. A10.4 is 1. The slope tells
us that 1 unit of labor costs 1 unit of capital.
If factor prices change, the slope of the isocost
line changes. If the wage rate rises to $50 a day and
the rental rate of a machine remains at $25 a day,
then 1 worker costs 2 machines and the isocost line
becomes steeper—line B in Fig. A10.5. If the wage
rate remains at $25 a day and the rental rate of a
machine rises to $50 a day, then 1 machine costs 2
workers and the isocost line becomes less steep—line
C in Fig. A10.5.
The higher the relative price of labor, the steeper
is the isocost line. The magnitude of the slope of the
isocost line measures the relative price of labor in
terms of capital—that is, the price of labor divided by
the price of capital. As the price of either capital or
labor changes, so too does the relative price of labor
and the slope of the isocost line.
FIGURE
A10.5
Input Prices and the
Isocost Line
Capital (machines per day)
APPENDIX
4
Labor $50
Capital $25
3
Labor $25
Capital $25
2
Labor $25
Capital $50
B
1
0
1
C
2
A
3
4
Labor (workers per day)
The slope of the isocost line depends on the relative input
prices. Three cases are shown (each for a total cost of $100).
If the prices of labor and capital are $25 a day each, the isocost line is line A. If the price of labor rises to $50 but the
price of capital remains $25, the isocost line becomes steeper
and is line B. If the price of capital rises to $50 and the price of
labor remains constant at $25, the isocost line becomes flatter
and is line C.

ISOCOST LINES
The Isocost Map
An isocost map shows a series of isocost lines, each
for a different total cost when the price of each input
is constant. With a larger total cost, larger quantities
of all the inputs can be employed. Figure A10.6 illustrates an isocost map. In that figure, the middle isocost line is the one in Fig. A10.4. It is the isocost line
for a total cost of $100 when capital and labor cost
$25 a day each. The other two isocost lines in Fig.
A10.6 are for a total cost of $125 and $75, when the
input prices are constant at $25 each.
The answer can be seen in Fig. A10.7. The isoquant for
15 sweaters is shown, and the three points on that
isoquant (marked a, b, and c) illustrate the three techniques of producing 15 sweaters that are shown in
Fig. A10.1. The figure also contains two isocost
lines—each drawn for a price of capital and a price of
labor of $25. One isocost line is for a total cost of
$125, and the other is for a total cost of $100.
First, consider point a. Swanky can produce 15
sweaters at point a by using 1 worker and 4
machines. The total cost when Swanky uses this technique of production is $125. Point c, which uses 4
workers and 1 machine, is another technique by
T h e L e a s t - C o s t Te c h n i q u e
FIGURE
A10.6
An Isocost Map
5
A10.7
The Least-Cost Technique
of Production
5
a
4
Least-cost
technique
3
TC
b
2
=
25
$1
4
$1
0
5
$1
2
$1
=
00
=
TC
00
TC
2
15 sweaters
=
3
c
TC
1
TC
Capital (machines per day)
FIGURE
Capital (machines per day)
The least-cost technique is the combination of
inputs that minimizes total cost of producing a given
output. Let’s suppose that Swanky wants to produce 15
sweaters a day. What is the least-cost way of doing this?
1
2
3
4
5
Labor (workers per day)
=
5
$7
1
0
1
2
3
5
4
Labor (workers per day)
This isocost map shows three isocost lines, one for a total
cost of $75, one for $100, and one for $125. For each isocost
line, the prices of capital and labor are $25 a day each. The
slope of each isocost line is equal to the price of labor divided
by the price of capital—a constant. The larger the total cost,
the farther is the isocost line from the origin.
The least-cost technique of producing 15 sweaters is 2
machines and 2 workers (point b) and the total cost is $100.
An output of 15 sweaters can be produced by using 4
machines and 1 worker (point a) or 1 machine and 4 workers
(point c). But with either of these techniques, the total cost is
greater, at $125. At b, the isoquant for 15 sweaters is tangential to the isocost line for $100. The isocost line and the isoquant have the same slope. If the isoquant intersects the isocost line—for example, at a and c—the least-cost technique
has not been found. With the least-cost technique, the marginal rate of substitution (slope of isoquant) equals the relative
price of the inputs (slope of isocost line).

APPENDIX
TO
C H A P T E R 1 0 P RO D U C I N G
AT
LEAST COST
which the firm can produce 15 sweaters for a cost of
$125.
Next look at point b. At this point, Swanky uses
2 machines and 2 workers to produce 15 sweaters at a
total cost of $100. Point b is the least-cost technique or
the economically efficient technique for producing 15
sweaters when knitting machines and workers cost
$25 a day each.
Notice that although there is only one way in
which Swanky can produce 15 sweaters for $100,
there are several ways of producing 15 sweaters for
more than $100. Techniques shown by points a and c
are two examples. All the points between a and b and
all the points between b and c are also ways of producing 15 sweaters for a cost that exceeds $100 but is
less than $125. That is, there are isocost lines
between those shown, for total costs between $100
and $125. Those isocost lines cut the isoquant for 15
sweaters at the points between a and b and between b
and c. Swanky can also produce 15 sweaters for a cost
that exceeds $125. That is, the firm can change its
technique of production by moving along the isoquant to a point above point a or to a point to the
right of point c. All of these ways of producing 15
sweaters are economically inefficient.
You can see that Swanky cannot produce 15
sweaters for less than $100 by imagining the isocost
line for $99. That isocost line will not touch the isoquant for 15 sweaters. That is, the firm cannot produce 15 sweaters for $99. At $25 for a unit of each
input, $99 will not buy the inputs required to produce 15 sweaters.
Marginal Rate of Substitution
E q u a l s R e l a t i ve I n p u t P r i c e
At the least-cost technique point b, the slope of the
isoquant is equal to the slope of the isocost line.
Equivalently, when a firm is using the least-cost technique of production, the marginal rate of substitution
between the inputs equals their relative price. Recall
that the marginal rate of substitution is the magnitude of the slope of an isoquant. Relative input prices
are measured by the magnitude of the slope of the
isocost line. We’ve just seen that producing at least
cost means producing at a point where the isocost
line is tangential to the isoquant. Because the two
curves are tangential, their slopes are equal. Hence
the marginal rate of substitution (the magnitude of
the slope of isoquant) equals the relative input price
(the magnitude of the slope of isocost line).
M
a r g i n a l P ro d u c t a n d
Marginal Cost in the
Long Run
W -   ,  ginal cost of increasing output by using one more
unit of capital is equal to the marginal cost of increasing output by using one more unit of labor. To see
why, we’re first going to learn about the relationship
between the marginal rate of substitution and marginal product.
Marginal Rate of Substitution and
M a r g i n a l P ro d u c t s
The marginal rate of substitution and the marginal
products are linked together in a formula:
The marginal rate of substitution of labor for
capital equals the marginal product of labor divided
by the marginal product of capital.
The change in output resulting from a change in
inputs is determined by the marginal products of the
inputs. That is,
Change in output = ( MPL × ∆L ) + ( MPK × ∆K ).
That is, the change in output equals the marginal
product of labor, MPL multiplied by the change in
labor, ∆L, plus the marginal product of capital, MPK,
multiplied by the change in the capital, ∆K.
Suppose that Swanky wants to change its inputs
but remain on an isoquant—that is, it wants to
change its inputs of labor and capital but produce the
same number of sweaters. To remain on an isoquant,
the change in output must be zero. We can make the
change of output zero in the above equation; doing
so yields the equation
MPL × ∆L = − MPK × ∆K .
Divide both sides of this equation by the increase
in labor (∆L) and also divide both sides by the marginal product of capital (MPK) to give
MPL /MPK = − ∆K /∆L
This equation tells us that, when Swanky
remains on an isoquant, the decrease in its capital
input (∆K) divided by the increase in its labor input
(∆L) is equal to the marginal product of labor (MPL)

M A R G I N A L P RO D U C T
divided by the marginal product of capital (MPK).
The decrease in capital divided by the increase in
labor when we remain on a given isoquant is the marginal rate of substitution of labor for capital. What
we have discovered, then, is that the marginal rate of
substitution of labor for capital equals the ratio of the
marginal product of labor to the marginal product of
capital.
Marginal Cost
When the least-cost technique is employed, the slope
of the isoquant and the isocost line are the same.
That is,
MPL /MPK = PL /PK
Rearrange the above equation in the following way.
First, multiply both sides by the marginal product of
capital and then divide both sides by the price of
labor. We then get
MPL /PL = MPK /PK
This equation says that the marginal product of labor
per dollar spent on labor is equal to the marginal
product of capital per dollar spent on capital. In other words, the extra output from the last dollar spent
on labor equals the extra output from the last dollar
spent on capital. This makes sense. If the extra output from the last dollar spent on labor exceeds the
extra output from the last dollar spent on capital, it
will pay the firm to use less capital and use more
labor. By doing so, it can produce the same output at
a lower total cost. Conversely, if the extra output
from the last dollar spent on capital exceeds the extra
output from the last dollar spent on labor, the firm
can lower its cost of producing a given output by
AND
MARGINAL COST
IN THE
LONG RUN
using less labor and more capital. A firm achieves the
least-cost technique of production only when the
extra output from the last dollar spent on all the
inputs is the same.
Marginal cost with fixed capital and variable
labor equals marginal cost with fixed labor and variable capital. To see this proposition, simply flip the
last equation over and write it as
PL /MPL = PK /MPK .
Expressed in words, this equation says that the price
of labor divided by its marginal product must equal
the price of capital divided by its marginal product.
But what is the price of an input divided by its marginal product? The price of labor divided by the marginal product of labor is marginal cost when the capital input is held constant. To see why this is so, first
recall the definition of marginal cost: Marginal cost is
the change in total cost resulting from a unit increase
in output. If output increases because one more unit
of labor is employed, total cost increases by the cost
of the extra labor, and output increases by the marginal product of the labor. So marginal cost is the
price of labor divided by the marginal product of
labor. For example, if labor costs $25 a day and the
marginal product of labor is 2 sweaters, then the marginal cost of a sweater is $12.50 ($25 divided by 2).
The price of capital divided by the marginal
product of capital has a similar interpretation. The
price of capital divided by the marginal product of
capital is marginal cost when the labor input is constant. As you can see from the above equation, with
the least-cost technique of production, marginal cost
is the same regardless of whether the capital input is
constant and more labor is used or the labor input is
constant and more capital is used.