1. CHAPTER 4
Production Theory

2. 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

3. 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

4. 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.

5. 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

6. 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

9. 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

10. 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

11. 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 = L – L2
Using calculus: MPL = L – 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 =

12. 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

15. 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

16. 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

19. 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.

21. 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.

22. 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

23. 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.

24. 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.

26. 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.

28. 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

29. 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

31. THE 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

34. 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

35. 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.

37. 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

38. 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.

39. 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.

40. 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.

41. 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.

42. 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 = Q

43. 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

44. This concludes the Lecture PowerPoint presentation for Chapter 4
Visit the StudySpace at:
© 2009 W. W. Norton & Company, Inc.