MANAGERIAL ECONOMICS: THEORY, APPLICATIONS, AND CASESW. 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: Q = aLbKc

• 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

© 2009 W. W. Norton & Company, Inc.

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