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Production Functions
• Let q represent output, K represent capital use, L represent labor, and M represent raw materials, the following equation represents a production function.
),,( MLKfq
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Marginal Physical Productivity
• Marginal physical productivity is the additional output produced by adding one more unit of an 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 holding capital constant.
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Marginal Physical Productivity
• The marginal product of capital (MPK) is the extra output obtained by using one more machine while holding the number of workers constant.
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Diminishing Marginal Physical Product
• Marginal physical product of an input will depend upon the level of the input used.
• It is expected that marginal physical product will eventually diminish as shown in Figure 5.1.
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Outputper week
Labor inputper week
TotalOutput
L*(a) Total OutputMP
L
L*(b) Marginal Productivity
FIGURE 5.1: Relationship between Output and Labor Input Holding Other Inputs Constant
Labor inputper week
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Diminishing Marginal Physical Product
• 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.
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Marginal Physical Productivity Curve
• The marginal physical product curve is simply the slope of the total product curve.
• The declining slope, as shown in panel b, shows diminishing marginal productivity.
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Average Physical Productivity
• Average physical 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 marginal physical product.
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Appraising the Marginal Physical Productivity Concept
• Marginal physical productivity 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.
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Isoquant Maps
• An isoquant is a curve that shows the various combinations of inputs that will produce the same amount of output.
• The curve labeled q = 10 is an isoquant that shows combinations of labor and capital, such as points A and B, that produce exactly 10 units of output per period
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Capitalper week
KA
A
B q = 10 KB
Laborper week
LBLA0
FIGURE 5.2: Isoquant Map
q = 20
q = 30
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Rate of Technical Substitution
• Theslope of an isoquant is called the marginal rate of technical substitution (RTS), the amount by which one input can be reduced when one unit of another input is added.
• It is the rate that capital can be reduced, holding output constant, while using one more unit of labor.
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Rate of Technical Substitution
inputlabor in Change
input capital in Change
isoquant) of (slope-
for K) L(of RTS
capital)for labor (of onsubstituti technicalof Rate
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Diminishing RTS
• 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.
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Returns to Scale
• 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.
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Returns to Scale
• Adam Smith identified two forces that come into play when all inputs are increased.– 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.
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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.– This situation is shown in Panel (a) of Figure
5.3.
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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.– This is shown in Panel (b) of Figure 5.3.
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Increasing Returns to Scale
• If doubling all inputs results in more than a doubling of output, the production function exhibits increasing returns to scale.– 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.
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Capitalper week
4
A
q = 10
3
2
1
Laborper week1 2 3
(a) Constant Returns to Scale40
Capitalper week
4
A
3
2
1
Laborper week
1 2 3(c) Increasing Returns to Scale
40
Capitalper week
4
A
3
2
1
Laborper week1 2 3
(b) Decreasing Returns to Scale40
FIGURE 5.3: Isoquant Maps showing Constant, Decreasing, and Increasing Returns to Scale
q = 10 q = 10
q = 20
q = 20q = 20
q = 30q = 30
q = 30q = 40
q = 40
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APPLICATION 5.3: Returns to Scale in Beer Brewing
• Economies to scale were achieved through automated control systems in filling beer cans.
• National markets foster economies of scale in distribution and advertising (especially television).
• 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.
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APPLICATION 5.3: Returns to Scale in Beer Brewing
• The industry consolidated with a few firms operating large breweries in multiple locations to reduce shipping costs.
• Beginning the 1980s, smaller firms offering premium brands, provided an opening for local microbreweries.
• A similar event occurred in Britain in 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.
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Input Substitution
• 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.
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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.
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Capitalper week
K2
A
q2
q1
q0
K1
K0
Laborper week
L0 L1 L20
FIGURE 5.4: Isoquant Map with Fixed Proportions
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Fixed-proportions Production Function
• 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.
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The Relevance of Input Substitutability
• 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.
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Changes in Technology
• Technical progress is a shift in the production function that allows a given output level to be produced using fewer inputs.
• In studying productivity data, especially data on output per worker, it is important to make the distinction between technical improvements and capital substitution.
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APPLICATION 5.4: Multifactor Productivity
• 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.
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APPLICATION 5.4: Multifactor Productivity
• 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.
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APPLICATION 5.4: Multifactor Productivity
• 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.
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TABLE 1: Annual Average Change in Output per Hour in Manufacturing
1956-73 1974-91 1992-97
United States 2.84 2.36 4.03
Germany 6.29 2.80 3.94
France 6.22 2.80 4.60
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TABLE 2: Annual Average Change in Multifactor Productivity Manufacturing
1956-73 1974-91 1992-97
United States 1.57 0.74 2.12
Germany 3.40 0.94 2.91
France 4.42 1.23 2.74
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A Numerical Example
• Assume a production function for the fast-food chain Hamburger Heaven (HH):
KLq 10hourper Hamburgers
- where K represents the number of grills used and L represents the number of workers employed during an hour of production.
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TABLE 5.1: Hamburger Production Exhibits Constant Returns to Scale
Grills (K) Workers (L) Hamburgers per hour1 1 102 2 203 3 304 4 405 5 506 6 607 7 708 8 809 9 9010 10 100
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Average and Marginal Productivities
• Holding capital constant (K = 4), to show labor productivity, we have
LLq 20410
• Table 5.2 shows this relationship and demonstrates that output per worker declines as more labor is employed.
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TABLE 5.2: Total Output, Average Productivity, and Marginal Productivity with Four Grills
Grills (K) Workers (L) Hamburgers per Hour (q) q/L MPL
4 1 20.0 20.0 -4 2 28.3 14.1 8.34 3 34.6 11.5 6.34 4 40.0 10.0 5.44 5 44.7 8.9 4.74 6 49.0 8.2 4.34 7 52.9 7.6 3.94 8 56.6 7.1 3.74 9 60.0 6.7 3.44 10 63.2 6.3 3.2
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The Isoquant Map
• Suppose HH wants to produce 40 hamburgers per hour. Then its production function becomes
KLq 10hourper hamburgers 40
LK
KL
16
or
4
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TABLE 5.3: Construction of the q = 40 Isoquant
Hamburgers per Hour (q) Grills (K) Workers (L)40 16.0 140 8.0 240 5.3 340 4.0 440 3.2 540 2.7 640 2.3 740 2.0 840 1.8 940 1.6 10
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Technical Progress
• Technical advancement can be reflected in the equation LKq 20
KL
KL
KLq
4
or
2
or
2040
• Comparing this to the old technology by recalculating the q = 40 isoquant
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Grills(K)
4
10
Workers(L)
q = 40 before invention
4 101
FIGURE 5.6: Technical Progress in Hamburger Production
q = 40