MANAGERIAL ECONOMICS:
THEORY, APPLICATIONS, AND CASES
W. Bruce Allen | Keith Weigelt | Neil Doherty | Edwin Mansfield
CHAPTER 4
Production Theory
OBJECTIVES
• Explain how managers should
determine the optimal method of
production by applying an
understanding of production processes
• Understand the linkages between
production processes and costs
PRODUCTION PROCESSES
• Production processes include all
activities associated with providing
goods and services, including
• Employment practices
• Acquisition of capital resources
• Product distribution
• Managing intellectual resources
PRODUCTION PROCESSES
• Production processes define the
relationships between resources used
and goods and services produced per
time period.
• Managers exert control over production
costs by understanding and managing
production technology.
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• A production function shows the
maximum amount that can be
produced per time period with the best
available technology from any given
combination of inputs.
• Table
• Graph
• Equation
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• Production Function Example
• Q = f(X1, X2)
• Q = Output rate
• X1 = Input 1 usage rate
• X2 = Input 2 usage rate
• Q = 30L + 20L2 – L3
• Q = Hundreds of parts produced per year
• L = Number of machinists hired
• Fixed Capital = Five machine tools
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• Unit Functions
• Average Product of Labor = APL = Q/L
• Common measuring device for estimating the
units of output, on average, per worker
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• Unit Functions (Continued)
• Marginal Product of Labor = MPL = Q/L
• Metric for estimating the efficiency of each
input in which the input’s MP is equal to the
incremental change in output created by a
small increase in the input
• Using calculus (assumes that labor can be
varied continuously): MP = dQ/dL
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• Unit Functions (Continued)
• Unit function examples from Q = 30L +
20L2 – L3
• Table 4.2 and Figure 4.2
• APL = 30 + 20L – L2
• Using calculus: MPL = 30 + 40L – 3L2
• APL is at a maximum, and MPL = APL, at L =
10 and MPL = APL = 130
• MPL is at a maximum at L = 6.67 and MPL =
163.33
PRODUCTION FUNCTION
WITH ONE VARIABLE INPUT
• Unit Functions (Continued)
• Why does MPL = APL when APL is at a maximum?
• If MPL > APL, then APL must be increasing
• If MPL < APL, then APL must be decreasing
THE LAW OF DIMINISHING
MARGINAL RETURNS
• Law of diminishing returns
• When managers add equal increments of
an input while holding other input levels
constant, the incremental gains to output
eventually get smaller
THE PRODUCTIONN FUNCTION
WITH TWO VARIABLE INPUTS
• Q = f(X1, X2)
• Q = Output rate
• X1 = Input 1 usage rate
• X2 = Input 2 usage rate
• AP1 = Q/X1 and MP1 = Q/X1 or dQ/dX1
• AP2 = Q/X2 and MP2 = Q/X2 or dQ/dX2
• Example
• Table 4.3 and Figure 4.3
ISOQUANTS
• Isoquant: Curve showing all possible
(efficient) input bundles capable of
producing a given output level.
• Graphically constructed by cutting
horizontally through the production
surface at a given output level
• Isoquants representing different output
levels are shown in Figure 4.4.
ISOQUANTS
• Properties
• Isoquants farther from the origin
represent higher input and output levels.
• Given a continuous production function,
every possible input bundle is on an
isoquant and there is an infinite number
of possible input combinations.
• Isoquants slope downward to the left and
are convex to the origin.
MARGINAL RATE OF
TECHNICAL SUBSTITUTION
• Marginal rate of technical substitution
(MRTS): Shows the rate at which one input
is substituted for another (with output
remaining constant)
• Q = f(X1, X2)
• MRTS = –X2/X1 with Q held constant and X2 on
the vertical axis
• MRTS = MP1/MP2
• MRTS = Absolute value of the slope of an
isoquant
MARGINAL RATE OF
TECHNICAL SUBSTITUTION
• MRTS and isoquants (with X2 on the
vertical axis)
• If the MRTS is large, it takes a lot of X2 to
substitute for one unit of X1, and
isoquants will be steep.
• If the MRTS is small, it takes little X2 to
substitute for one unit of X1, and
isoquants will be flat.
MARGINAL RATE OF
TECHNICAL SUBSTITUTION
• MRTS and isoquants (with X2 on the vertical
axis) (Continued)
• If X1 and X2 are perfect substitutes, MRTS is
constant, and isoquants will be straight lines.
• If X1 and X2 are perfect complements, no
substitution is possible, MRTS is undefined, and
isoquants will be right angles.
MARGINAL RATE OF
TECHNICAL SUBSTITUTION
• Ridge Lines
• Ridge lines: The lines that profit-maximizing firms
operate within, because outside of them, marginal
products of inputs are negative
• Economic region of production is located within the
ridge lines.
THE OPTIMAL COMBINATION
OF INPUTS
• Isocost curve: Curve showing all the
input bundles that can be purchased
at a specified cost
• PLL + PKK = M
• L = Labor use rate
• PL = Price of labor
• K = Capital use rate
• PK = Price of capital
• M = Total outlay
THE OPTIMAL COMBINATION
OF INPUTS
• Isocost curve (Continued)
• K = M/PK – (PL/PK)L
• Vertical intercept = M/PK
• Horizontal intercept = M/PL
• Slope = – PL/PK
THE OPTIMAL COMBINATION
OF INPUTS
• Optimal Combination of Inputs
• Tangency between isocost and isoquant
• MRTS = MPL/MPK = PL/PK
• MPL/PL = MPK/PK
• Marginal product per dollar spent should be the same
for all inputs.
• MPa/Pa = MPb/Pb =  = MPn/Pn
• Maximize output for given cost: Figure 4.8
• Minimize cost for a given output: Figure 4.9
CORNER SOLUTIONS
• Optimal input combination does not
occur at a point of tangency between
isocost and isoquant curves.
• In a two-input case, one of the inputs will
not be used at all in production.
• Example: Figure 4.10
CORNER SOLUTIONS
• If two inputs are perfect complements
(isoquants are right angles), then both
inputs will be used, but the optimal
combination will not occur at a point of
tangency between isocost and isoquant
curves.
RETURNS TO SCALE
• Long-run effect of an equal proportional
increase in all inputs
• Increasing returns to scale: When output
increases by a larger proportion than inputs
• Decreasing returns to scale: When output
increases by a smaller proportion than inputs
• Constant returns to scale: When output
increases by the same proportion as inputs
RETURNS TO SCALE
• Sources of increasing returns to scale
• Indivisibilities: Some technologies can
only be implemented at a large scale of
production.
• Subdivision of tasks: Larger scale allows
increased division of tasks and increases
specialization.
RETURNS TO SCALE
• Sources of increasing returns to scale
(Continued)
• Probabilistic efficiencies: Law of large numbers
may reduce risk as scale increases.
• Geometric relationships: Doubling the size of a
box from 1 X 1 X 1 to 2 X 2 X 2 multiplies the
surface area by four times (from 3 to 12) but
increases the volume by eight times (from 1 to
8). This applies to storage devices,
transportation devices, etc.
RETURNS TO SCALE
• Sources of decreasing returns to scale
• Coordination inefficiencies: Larger
organizations are more difficult to
manage.
• Incentive problems: Designing efficient
compensation systems in large
organizations is difficult.
THE OUTPUT ELASTICITY
• Output elasticity: The percentage change in
output resulting from a 1 percent increase
in all inputs.
• Note: A more common definition of output
elasticity is the percentage change in output
resulting from a 1 percent increase in a single
input. Accordingly, the coefficients 0.3 and 0.8 in
the Cobb-Douglas function below would be
referred to as the output elasticities of labor and
capital, respectively.
THE OUTPUT ELASTICITY
• Cobb-Douglas production function example:
Q = 0.8L0.3K0.8
• Q = Parts produced by the Lone Star Company
per year
• L = Number of workers
• K = Amount of capital
• Output elasticity = 1.1 for infinitesimal changes
in inputs
• Example calculation for 1 percent increase in
both inputs
• Q' = 0.8(1.01L)0.3(1.01K)0.8 = 1.011005484Q
ESTIMATIONS OF
PRODUCTION FUNCTIONS
• Cobb-Douglas Mathematical form:
= aLbKc
Q
• MPL = Q/L = b(Q/L) = b(APL)
• Linear estimation: log Q = log a + b log L
+ c log K
• Returns to scale
• b + c > 1 => increasing returns
• b + c = 1 => constant returns
• b + c < 1 => decreasing returns
This concludes the
Lecture PowerPoint
presentation for Chapter 4
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