Chapter 5
Production
© 2004 Thomson Learning/South-Western
Production Functions


2
The purpose of a firm is to turn inputs into
outputs.
An abstract model of production is the
production function, a mathematical
relationship between inputs and outputs.
Production Functions

Letting q represent the output of a particular
good during a period, K represent capital use,
L represent labor input, and M represent raw
materials, the following equation represents a
production function.
q  f ( K , L, M )
3
Two-Input Production Function


An important question is how firms choose
their levels of output and inputs.
While the choices of inputs will obviously vary
with the type of firm, a simplifying assumption
is often made that the firm uses two inputs,
labor and capital.
q  f ( K , L)
4
Application 5.1: Everyone is a Firm
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5
Looking at people as “firms can yield some
interesting insights
Economists have tried to estimate the amount
of production that people do for themselves.
Time-use studies suggest that the time people
spend in home production is only slightly less
than the time spent working.
Application 5.1: Everyone is a Firm
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6
Some of the more straightforward things
produced at home are what might be called
“housing services”.
The production function concept is widely used
in thinking about health issues.
A somewhat more far-fetched application of
the home-production concept is to view
families as “producers of children.”
Marginal Product

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7
Marginal physical productivity, or more
simply, the marginal product of an input is the
additional output that can be produced by
adding one more unit of a particular input while
holding all other inputs constant.
The marginal product of labor (MPL) is the
extra output obtained by employing one more
unit of labor while holding the level of capital
equipment constant.
Marginal Product

8
The marginal product of capital (MPK) is the
extra output obtained by using one more
machine while holding the number of workers
constant.
Diminishing Marginal Product


9
It is expected that the marginal product of an
input will depend upon the level of the input
used.
Since, holding capital constant, production of
more output is likely to eventually decline with
adding more labor, it is expected that marginal
product will eventually diminish as shown in
Figure 5.1.
FIGURE 5.1: Relationship between Output and
Labor Input Holding Other Inputs Constant
Output
per week
Total
Output
(a) Total output
L*
MP
L
10
L*
(b) Marginal product
Labor input
per week
Labor input
per week
Diminishing Marginal Product


11
The top panel of Figure 5.1 shows the
relationship between output per week and
labor input during the week as capital is held
fixed.
Initially, output increases rapidly as new
workers are added, but eventually it diminishes
as the fixed capital becomes overutilized.
Marginal Product Curve

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12
The marginal product curve is simply the slope
of the total product curve.
The declining slope, as shown in panel b,
shows diminishing marginal productivity.
Average Product


13
Average product is simply “output per worker”
calculated by dividing total output by the
number of workers used to produce the output.
This corresponds to what many people mean
when they discuss productivity, but economists
emphasize the change in output reflected in
the marginal product.
Appraising the Marginal Product
Concept
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14
Marginal product requires the ceteris paribus
assumption that other things, such as the level
of other inputs and the firm’s technical
knowledge, are held constant.
An alternative way, that is more realistic, is to
study the entire production function for a good.
APPLICATION 5.2: Why Do the Japanese
Have a Cost Advantage in Making Cars?


In 1979 Japan overtook the United States as
the world’s largest producer of automobiles.
Japan appears to have a productivity
advantage.
–
15
For example, workers at Honda or Toyota take
about 30 hours to produce a car while workers in
General Motors or Chrysler take about 45 hours.
APPLICATION 5.2: Why Do the Japanese
Have a Cost Advantage in Making Cars?
–

U.S. producers tend to use more assembly
lines which allows greater variability in vehicle
size than in Japan.
–
16
Reasons for this are not known although it does not
appear to be explained by simple substitution of
capital for labor.
This makes it easier to use automating production,
such as robots, in Japan.
APPLICATION 5.2: Why Do the Japanese
Have a Cost Advantage in Making Cars?
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17
Also, although both U.S. and Japanese
producers tend to buy many components of
cars from independent suppliers, these
suppliers are better integrated with the
assembly firms in Japan.
In addition, there appears to be more of an
adversarial relationship between labor and
management in the U.S. than in Japan.
Isoquant Maps


An isoquant is a curve that shows the various
combinations of inputs that will produce the
same (a particular) amount of output.
An isoquant map is a contour map of a firm’s
production function.
–
18
All of the isoquants from a production function are
part of this isoquant map.
Isoquant Map

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19
In Figure 5.2, the firm is assumed to use the
production function, q = f(K,L) to produce a
single good.
The curve labeled q = 10 is an isoquant that
shows various combinations of labor and
capital, such as points A and B, that produce
exactly 10 units of output per period.
FIGURE 5.2: Isoquant Map
Capital
per week
KA
A
KB
B
0
20
LA
q = 10
LB
Labor
per week
Isoquant Map


21
The isoquants labeled q = 20 and q = 30
represent two more of the infinite curves that
represent different levels of output.
Isoquants record successively higher levels of
output the farther away from the origin they are
in a northeasterly direction.
FIGURE 5.2: Isoquant Map
Capital
per week
KA
A
q = 30
q = 20
KB
B
0
22
LA
q = 10
LB
Labor
per week
Isoquant Map


23
Unlike indifference curves, the labeling of the
isoquants represents something measurable,
the quantity of output per period.
In addition to the location of the isoquants,
economists are also interested in their shape.
Rate of Technical Substitution
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
24
The negative of the slope of an isoquant is
called the marginal rate of technical
substitution (RTS), the amount by which one
input can be reduced when one more unit of
another input is added while holding output
constant.
It is the rate that capital can be reduced,
holding output constant, while using one more
unit of labor.
Rate of Technical Substitution
Rate of technical substituti on (of labor for capital)
 RTS (of L for K)
 - (slope of isoquant)
Change in capital input

Change in labor input
25
Rate of Technical Substitution
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
26
The particular value of this trade-off depends
upon the level of output and the quantities of
capital and labor being used.
At A in Figure 5.2, relatively large amounts of
capital can be given up if one more unit of
labor is added (large RTS), but at B only a little
capital can be sacrificed when adding one
more unit of labor (small RTS).
The RTS and Marginal Products
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27
It is likely that the RTS is positive (the isoquant
has a negative slope) because the firm can
decrease its use of capital if one more unit of
labor is employed.
If increasing labor meant the having to hire
more capital the marginal product of labor or
capital would be negative and the firm would
be unwilling to hire more of either.
Diminishing RTS
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28
Along any isoquant the (negative) slope
become flatter and the RTS diminishes.
When a relatively large amount of capital is
used (as at A in Figure 5.2) a large amount can
be replaced by a unit of labor, but when only a
small amount of capital is used (as at point B),
one more unit of labor replaces very little
capital.
APPLICATION 5.3: Engineering and
Economics
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29
Engineering studies can be used to provide
information about a production function.
Engineers have developed three different
methods (A, B, and C) to produce output.
These methods are shown in Figure 1, when
method A uses a greater capital labor ratio than
B, and the capital labor ratio at B exceeds that
at C.
FIGURE 1: Construction of an Isoquant
from Engineering Data
K per
period
A
B
a
C
b
c
q0
L per period
30
APPLICATION 5.3: Engineering and
Economics
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31
The points a, b, and c represent three different
methods to produce q0 units of output, so these
points are one the same isoquant.
This method was used by economists to
examine the production of domestic hot water
by rooftop solar collectors.
APPLICATION 5.3: Engineering and
Economics
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32
This method has been used to examine the
extent to which energy and capital can be
substituted in industrial equipment design.
Economists have also found that energy and
capital are sometimes complements in
production, which may have caused the poor
productivity of the 1970s due to high energy
costs.
Returns to Scale
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33
Returns to scale is the rate at which output
increases in response to proportional increases
in all inputs.
In the eighteenth century Adam Smith became
aware of this concept when he studied the
production of pins.
Returns to Scale

Adam Smith identified two forces that come
into play when all inputs are increased.
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–
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34
A doubling of inputs permits a greater “division of
labor” allowing persons to specialize in the
production of specific pin parts.
This specialization may increase efficiency
enough to more than double output.
However these benefits might be reversed if
firms become too large to manage.
Constant Returns to Scale

A production function is said to exhibit constant
returns to scale if a doubling of all inputs
results in a precise doubling of output.
–
35
This situation is shown in Panel (a) of Figure 5.3.
FIGURE 5.3: Isoquant Maps showing Constant,
Decreasing, and Increasing Returns to Scale
A
Capital
per week
4
q = 40
3
q = 30
2
q = 20
1
0
1
2
q = 10
Labor
3 4 per
week
(a) Constant Returns to Scale
36
Decreasing Returns to Scale

If doubling all inputs yields less than a doubling
of output, the production function is said to
exhibit decreasing returns to scale.
–
37
This is shown in Panel (b) of Figure 5.3.
FIGURE 5.3: Isoquant Maps showing Constant,
Decreasing, and Increasing Returns to Scale
A Capital
per week
Capital
per week
4
3
2
1
A
4
q = 40
q = 30
q = 20
3
2
q = 30
q = 20
1
q = 10
q = 10
Labor
Labor
0
1 2 3 4 per
0 1 2 3 4
week
per week
(a) Constant Returns to Scale (b) Decreasing Returns to Scale
38
Increasing Returns to Scale

If doubling all inputs results in more than a
doubling of output, the production function
exhibits increasing returns to scale.
–
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39
This is demonstrated in Panel (c) of Figure 5.3.
In the real world, more complicated possibilities
may exist such as a production function that
changes from increasing to constant to
decreasing returns to scale.
FIGURE 5.3: Isoquant Maps showing Constant,
Decreasing, and Increasing Returns to Scale
A Capital
per week
Capital
per week
A
4
4
q = 40
3
q = 30
2
q = 20
1
3
2
q = 30
q = 20
1
q = 10
q = 10
Labor
Labor
0
1 2 3 4 per
0 1 2 3 4
week
per week
(a) Constant Returns to Scale (b) Decreasing Returns to Scale
A
Capital
per week
4
3
40
q = 40
2
q = 30
q = 20
1
q = 10
Labor
0 1 2 3 4
per week
(c) Increasing Returns to Scale
APPLICATION 5.4: Returns to Scale in
Beer Brewing
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41
Because beer is produced in volume (barrels
per year) but capital has costs that are
proportional to surface area, larger breweries
were able to achieve increasing returns to
scale.
Economies to scale were also achieved
through automated control systems in filling
beer cans.
APPLICATION 5.4: Returns to Scale in
Beer Brewing
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42
National markets may also foster economies of
scale in distribution, advertising (especially
television), and marketing.
These factors became especially important
after World War II and the number of U.S.
brewing firms fell by over 90 percent between
1945 and the mid-1980s.
APPLICATION 5.4: Returns to Scale in
Beer Brewing
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43
The output of the industry became
consolidated in a few large firms which
operated very large breweries in multiple
locations to reduce shipping costs.
Beginning the the mid-1980s, however, smaller
firms offering premium brands, provided and
opening for local microbreweries.
APPLICATION 5.4: Returns to Scale in
Beer Brewing
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44
The 1990s have seen a virtual explosion of
new brands with odd names or local appeal.
A similar event occurred in great Britain during
the 1980s with the “real ale” movements, but it
was followed by small firms being absorbed by
national brands.
A similar absorption may be starting to take
place in the United States.
Input Substitution
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45
Another important characteristic of a
production function is how easily inputs can be
substituted for each other.
This characteristic depends upon the slope of a
given isoquant, rather than the whole isoquant
map.
Fixed-Proportions Production
Function



It may be the case that absolutely no
substitution between inputs is possible.
This case is shown in Figure 5.4.
If K1 units of capital are used, exactly L1 units
of labor are required to produce q1 units of
output.
–
46
If K1 units of capital are used and less than L1 units
of labor are used, q1 can not be produced.
Fixed-Proportions Production
Function
–
–
–
–
47
If K1 units of capital are used and more than L1 units
of labor are used, no more than q1 units of output
are produced.
With K = K1, the marginal physical product of labor
is zero beyond L1 units of labor.
The q1 isoquant is horizontal beyond L1.
Similarly, with L1 units of labor, the marginal physical
product of capital is zero beyond K1 resulting in the
vertical portion of the isoquant.
FIGURE 5.4: Isoquant Map with Fixed
Proportions
Capital
per week
A
K2
q2
K1
q1
q0
K0
0
48
L0
L1
L2
Labor
per week
Fixed-proportions Production
Function
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49
This type of production function is called a
fixed-proportion production function
because the inputs must be used in a fixed
ratio to one another.
Many machines require a fixed complement of
workers so this type of production function may
be relevant in the real world.
The Relevance of Input
Substitutability
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50
Over the past century the U.S. economy has
shifted away from agricultural production and
towards manufacturing and service industries.
Economists are interested in the degree to
which certain factors of production (notable
labor) can be moved from agriculture into the
growing industries.
Changes in Technology
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51
Technical progress is a shift in the production
function that allows a given output level to be
produced using fewer inputs.
Isoquant q0 in Figure 5.5, summarized the
initial state of technical knowledge.
K0 and L0 units of capital and labor respectively
can produce this level of output.
Changes in Technology
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52
After a technology improvement, the same
level of output can be produced with the same
level of capital and reduced labor, L1.
The improvement in technology is represented
in Figure 5.5 by the shift of the q0 isoquant to
q’0.
Technical progress represents a real savings in
inputs.
FIGURE 5.5: Technical Change
Capital
per week
K1
K0
A
q0
q’0
53
0
L1
L0
Labor
per week
Technical Progress versus Input
Substitution
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54
In studying productivity data, especially data on
output per worker, it is important to make the
distinction between technical improvements
and capital substitution.
In Figure 5.5, the first is shown by the
movement from L0, K0 to L1, K0, while the latter
is L0, K0 to L1, K1.
Technical Progress versus Input
Substitution
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55
In both cases, output per worker would rise
(q0/L0 to q0/L1)
With technical progress there is a real
improvement in the way things are produced.
With substitution, no real improvement in the
production of the good takes place.
APPLICATION 5.5: Multifactor
Productivity
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56
Table 1 shows the rates of change in
productivity for three countries measured as
output per hour.
While the data show declines during the 1974
to 1991 period, they still averaged over 2
percent per year.
However, this measure may simply reflect
simple capital-labor substitution.
APPLICATION 5.5: Multifactor
Productivity
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57
A measure that attempts to control for such
substitution is called mutifactor productivity.
As table 2 shows, the 1974 - 91 period looks
much worse using the multifactor productivity
measure.
APPLICATION 5.5: Multifactor
Productivity
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58
Reasons for the decline in post 1973
productivity include rising energy prices, high
rates of inflation, increasing environmental
regulations, deteriorating education systems,
or a general decline in the work ethic.
Clearly after 1991, productivity has improved
greatly.
TABLE 1: Annual Average Change in Output
per Hour in Manufacturing
59
1956-73
1974-91
1992-00
United States
2.84
2.36
4.03
Germany
6.29
2.80
3.08
France
6.22
2.80
4.29
TABLE 2: Annual Average Change in
Multifactor Productivity Manufacturing
60
1956-73
1974-91
1992-00
United States
1.57
0.74
1.96
Germany
3.40
0.94
1.66
France
4.42
1.23
3.05
A Numerical Example

Assume a production function for the fast-food
chain Hamburger Heaven (HH):
Hamburgers per hour  q  10 KL
–
61
where K represents the number of grills used and L
represents the number of workers employed during
an hour of production.
A Numerical Example

This function exhibits constant returns to scale
as demonstrated in Table 5.1.
–
62
As both workers and grills are increased together,
hourly hamburger output rises proportionally.
TABLE 5.1: Hamburger Production Exhibits
Constant Returns to Scale
Grills (K)
1
2
3
4
5
6
7
8
9
10
63
Workers (L)
1
2
3
4
5
6
7
8
9
10
Hamburgers per hour
10
20
30
40
50
60
70
80
90
100
Average and Marginal
Productivities

Holding capital constant (K = 4), to show labor
productivity, we have
q  10 4  L  20 L

64
Table 5.2 shows this relationship and
demonstrates that output per worker declines
as more labor is employed.
TABLE 5.2: Total Output, Average Productivity,
and Marginal Productivity with Four Grills
Grills (K)
4
4
4
4
4
4
4
4
4
4
65
Workers (L) Hamburgers per Hour (q)
1
2
3
4
5
6
7
8
9
10
20.0
28.3
34.6
40.0
44.7
49.0
52.9
56.6
60.0
63.2
q/L
20.0
14.1
11.5
10.0
8.9
8.2
7.6
7.1
6.7
6.3
MPL
8.3
6.3
5.4
4.7
4.3
3.9
3.7
3.4
3.2
Average and Marginal
Productivities
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66
Also, Table 5.2 shows that the productivity of
each additional worker hired declines.
Holding one input constant yields the expected
declining average and marginal productivities.
The Isoquant Map

Suppose HH wants to produce 40 hamburgers
per hour. Then its production function
becomes
q  40 hamburgers per hour  10 KL
4  KL
or
16  K  L
67
The Isoquant Map



68
Table 5.3 show several K, L combinations that
satisfy this equation.
All possible combinations in the “q = 40”
isoquant are shown in Figure 5.6.
All other isoquants would have the same
shape showing that HH has many substitution
possibilities.
TABLE 5.3: Construction of the q = 40
Isoquant
69
Hamburgers per Hour (q)
Grills (K)
Workers (L)
40
40
40
40
40
40
40
40
40
40
16.0
8.0
5.3
4.0
3.2
2.7
2.3
2.0
1.8
1.6
1
2
3
4
5
6
7
8
9
10
FIGURE 5.6: Technical Progress in
Hamburger Production
Grills
(K)
10
4
q = 40
1
70
4
10
Workers
(L)
Technical Progress


Technical advancement can be reflected in the
equation q  20 K  L
Comparing this to the old technology by
recalculating the q = 40 isoquant
q  40  20 KL
or
2
or
71
KL
4  KL
Technical Progress



72
In Figure 5.6 the new technology is the
isoquant labeled “q = 40 after invention.”
With 4 grills, average productivity is now 40
hamburgers per hour per worker whereas it
was 10 hamburgers per hour before the
invention.
This level of output per worker would have
required 16 grills with the old technology.
FIGURE 5.6: Technical Progress in
Hamburger Production
Grills
(K)
10
4
q = 40 after invention
q = 40
1
73
4
10
Workers
(L)