Production Theory
and Estimation
Chapter 7
• Managers must decide not only what to produce for the
market, but also how to produce it in the most efficient or least
cost manner.
• Economics offers widely accepted tools for judging whether
the production choices are least cost.
• A production function relates the most that can be produced
from a given set of inputs.
» Production functions allow measures of the marginal
product of each input.
Green Power Initiatives
 California permitted and encouraged buying
cheap power from other states.
 So, PG&E and Southern Cal Edison scaled
back their expansion of production facilities.
 Off-Peak and Peak costs per MWh ranged
from $25 to over $65, but regulators tried to
keep prices low.

Resulting in PG&E bankruptcy
Green Power Initiatives
 Carbon dioxide emission trading schemes
in Europe encouraged construction of
greener nuclear & wind generation.
Q: If you were asked to pay 3 times more for
electricity in the day than night, would you
change your usage?
The Organization of Production
 Inputs

Labor, Capital, Land
 Fixed Inputs
 Variable Inputs
 Short Run

At least one input is fixed
 Long Run

All inputs are variable
Production Function With Two Inputs
Q = f(L, K)
K
6
5
4
3
2
1
Q
10
12
12
10
7
3
1
24
28
28
23
18
8
2
31
36
36
33
28
12
3
36
40
40
36
30
14
4
40
42
40
36
30
14
5
39
40
36
33
28
12
6 L
Production Function With One Variable Input
Total Product
Marginal Product
Average Product
Production or
Output Elasticity
TP = Q = f(L)
MPL =
TP
L
APL =
TP
L
EL =
MPL
APL
Production Function With One Variable Input
Total, Marginal, and Average Product of Labor, and Output Elasticity
L
0
1
2
3
4
5
6
Q
0
3
8
12
14
14
12
MPL
3
5
4
2
0
-2
APL
3
4
4
3.5
2.8
2
EL
1
1.25
1
0.57
0
-1
Production Function With One Variable Input
Production Function With One Variable Input
Optimal Use of the Variable Input
Marginal Revenue
Product of Labor
Marginal Resource
Cost of Labor
Optimal Use of Labor
MRPL = (MPL)(MR)
MRCL =
TC
L
MRPL = MRCL
Optimal Use of the Variable Input
Use of Labor is Optimal When L = 3.50
L
2.50
3.00
3.50
4.00
4.50
MPL
4
3
2
1
0
MR = P
$10
10
10
10
10
MRPL
$40
30
20
10
0
MRCL
$20
20
20
20
20
Optimal Use of the Variable Input
Production With Two Variable Inputs
Isoquants show combinations of two inputs that can produce
the same level of output.
Firms will only use combinations of two inputs that are in the
economic region of production, which is defined by the portion
of each isoquant that is negatively sloped.
Production With Two Variable Inputs
Isoquants
Production With Two Variable Inputs
Economic
Region of
Production
Production With Two Variable Inputs
Marginal Rate of Technical Substitution
MRTS = -K/L = MPL/MPK
Production With Two Variable Inputs
MRTS = -(-2.5/1) = 2.5
Production With Two Variable Inputs
Perfect Substitutes
Perfect Complements
Optimal Combination of Inputs
Isocost lines represent all combinations of two inputs that a
firm can purchase with the same total cost.
C  wL  rK
C  Total Cost
w  Wage Rate of Labor ( L)
C w
K  L
r r
r  Cost of Capital ( K )
Optimal Combination of Inputs
Isocost Lines
Prepared by Robert F. Brooker,
Ph.D. Copyright ©2004 by
South-Western, a division of
AB
C = $100, w = r = $10
A’B’
C = $140, w = r = $10
A’’B’’
C = $80, w = r = $10
AB*
C = $100, w = $5, r = $10
Optimal Combination of Inputs
MRTS = w/r
Prepared by Robert F. Brooker,
Ph.D. Copyright ©2004 by
South-Western, a division of
Optimal Combination of Inputs
Effect of a Change in Input Prices
Returns to Scale
Production Function Q = f(L, K)
Q = f(hL, hK)
If  = h, then f has constant returns to scale.
If  > h, then f has increasing returns to scale.
If  < h, the f has decreasing returns to scale.
Returns to Scale
Constant Returns
to Scale
Prepared by Robert F. Brooker,
Ph.D. Copyright ©2004 by
South-Western, a division of
Increasing
Returns to Scale
Decreasing
Returns to Scale
Empirical Production Functions
Cobb-Douglas Production Function
Q = AKaLb
Estimated using Natural Logarithms
ln Q = ln A + a ln K + b ln L
The Production Function
 A Production Function is the maximum feasible
quantity from any amounts of inputs
 If L is labor and K is capital, one popular
functional form is known as the Cobb-Douglas
Production Function
The Production Function (con’t)
Q = a • K
b1•
L b2
is a Cobb-
Douglas
Production Function
 The number of inputs is typically greater than
just K & L. But economists simplify by
suggesting some, like materials or labor, is
variable, whereas plant and equipment is
fairly fixed in the short run.
The Short Run
Production Function
 Short Run Production Functions:
 Max
Q
output, from a n y set of inputs
= f ( X1, X2, X3, X4, X5 ... )
FIXED IN SR
VARIABLE IN SR
_
_
Q = f ( K, L) for two input case, where K is Fixed
© 2011 Cengage Learning. All Rights Reserved. May
not be scanned, copied or duplicated, or posted to a
publicly accessible website, in whole or in part.
The Short Run
Production Function (con’t)
 A Production Function with only
one variable input is easily
analyzed. The one variable input is
labor, L. Q = f( L )
Average Product = Q / L
 output
per labor
Marginal Product = Q/L =Q/L =
dQ/dL
 output
attributable to last unit of labor
applied
 Similar to profit functions, the Peak of
MP occurs before the Peak of average
product
 When
MP = AP, this is the peak of the
AP curve
Law of Diminishing Returns
INCREASES IN ONE FACTOR OF PRODUCTION,
HOLDING ONE OR OTHER FACTORS FIXED,
AFTER SOME POINT,
MARGINAL PRODUCT DIMINISHES.
MP
A SHORT
RUN LAW
point of
diminishing
returns
Variable input
Bottlenecks in Production
Plants
 Boeing found diminishing returns in ramping up
production.
 It sought ways to adopt lean production
techniques, cut order sizes, and outsourced
work at bottlenecked plants.
Increasing Returns and Network
Effects
 There are exceptions to the law of
diminishing returns.
 When the installed base of a network product
makes efforts to acquire new customers
increasing more productive, we have
network effects

Outlook and Microsoft Office
Table 7.2:
Total, Marginal & Average Products
Total, Marginal & Average Products
Marginal Product
Average
Product
3
4
5
6
7
8
The maximum MP occurs before the maximum AP
 When MP > AP, then AP is RISING

IF YOUR MARGINAL GRADE IN THIS CLASS IS HIGHER
THAN YOUR GRADE POINT AVERAGE, THEN YOUR G.P.A.
IS RISING
 When MP < AP, then AP is
FALLING

IF YOUR MARGINAL BATTING AVERAGE IS LESS THAN
THAT OF THE NEW YORK YANKEES, YOUR ADDITION TO
THE TEAM WOULD LOWER THE YANKEE’S TEAM BATTING
AVERAGE
 When MP = AP, then AP is at its
MAX

IF THE NEW HIRE IS JUST AS EFFICIENT AS THE AVERAGE
EMPLOYEE, THEN AVERAGE PRODUCTIVITY DOESN’T
CHANGE
Three stages of production
Three stages of production
 Stage 1: average product rising.

Increasing returns
 Stage 2: average product declining (but marginal
product positive).

Decreasing returns
 Stage 3: marginal product is negative, or total
product is declining.

Negative returns
Determining the Optimal Use of
the Variable Input
 HIRE, IF GET MORE
REVENUE THAN
COST
 HIRE if
TR/L > TC/L
 HIRE if the marginal
revenue product >
marginal factor cost:
MRP L > MFC L
 AT OPTIMUM,
MRP L = W  MFC
MRP L  MP L • P Q = W
wage
W
•
W  MFC
MRPL
optimal labor
MPL
L
Optimal Input Use at L = 6
Table 7.3
© 2011 Cengage Learning. All Rights Reserved. May
not be scanned, copied or duplicated, or posted to a
publicly accessible website, in whole or in part.
Production Functions with multiple
variable inputs
 Suppose several inputs are variable

greatest output from any set of inputs
 Q = f( K, L ) is two input example
 MP of capital and MP of labor are the derivatives of
the production function

MPL = Q/L = Q/L
 MP of labor declines as more labor is applied. Also
the MP of capital declines as more capital is
applied.
Isoquants & LR Production Functions
ISOQUANT
MAP
 In the LONG RUN, ALL factors are
variable
 Q = f ( K, L )
 ISOQUANTS -- locus of input
combinations which produces the
same output
 Points A & B are on the same
isoquant
 SLOPE of ISOQUANT from A to
B is ratio of Marginal Products,
called the MRTS, the marginal rate
of technical substitution = -K /L
K
Q3
C
B
Q2
A
Q1
L
Optimal Combination of Inputs
• The objective is to minimize cost for a given output
• ISOCOST lines are the combination of inputs for a given cost,
C0
• C0 = CL·L + CK·K
• K = C0/CK - (CL/CK)·L
• Optimal where:
» MPL/MPK = CL/CK·
» Rearranged, this becomes the equimarginal criterion
Optimal Combination of Inputs
Equimarginal Criterion:
Produce where
MPL/CL = MPK/CK where
marginal products per dollar
are equal
at D, slope of
isocost = slope
of isoquant
Use of the Equimarginal Criterion
• Q: Is the following firm EFFICIENT?
• Suppose that:
» MP L = 30
» MPK = 50
» W = 10 (cost of labor)
» R = 25 (cost of capital)
• Labor: 30/10 = 3
• Capital: 50/25 = 2
• A: No!
Use of the Equimarginal Criterion
 A dollar spent on labor produces 3, and a dollar
spent on capital produces 2.
USE RELATIVELY
MORE LABOR!
 If spend $1 less in capital, output falls 2 units, but
rises 3 units when spent on labor
 Shift to more labor until the equimarginal condition
holds.
 That is peak efficiency.
Production Processes and Process Rays
under Fixed Proportions
 If a firm has five computers and
just one person, typically only one
computer is used at a time. You
really need five people to work on
the five computers.
 The isoquants for processes with
fixed proportions are L-shaped.
Small changes in the prices of
input may lead to no change in the
process.
 M is the process ray of one
worker and one machine
people
5
4
M
3
2
1
1 2 3 4 5 6 7 8 9
computers
Allocative & Technical Efficiency
 Allocative Efficiency – asks if the firm using the
least cost combination of input

It satisfies: MPL/CL = MPK/CK
 Technical Efficiency – asks if the firm is maximizing
potential output from a given set of inputs

When a firm produces at point T
rather than point D on a lower
isoquant, that firm is not
producing as much as is
technically possible.
D
T
Q(1)
Q(0)
Overall Production Efficiency
 Suppose a plant produces 93% of what the
technical efficient plant (the benchmark)
would produce.
 Suppose a plant produces 85.7% of what an
allocatively efficient plant would produce,
due to a misaligning the input mix.
 Overall Production Efficiency = (technical
efficiency)*(allocative efficiency)
 In this case: overall production efficiency =
(.93)(.857) = 0.79701 or about 79.7%.
Returns to Scale
 If multiplying all inputs by  (lambda) increases the
dependent variable by,the firm has constant returns
to scale (CRS).



Q = f ( K, L)
So, f(K,  L) =  • Q is Constant Returns to Scale
So if 10% more all inputs leads to 10% more output the
firm is constant returns to scale.
 Cobb-Douglas Production Functions are constant
returns if a + b 1
Cobb-Douglas Production Functions
Q = A • Ka • Lb is a Cobb-Douglas Production Function
IMPLIES:
Can be CRS, DRS, or IRS
if a + b 1, then constant returns to scale
if a + b< 1, then decreasing returns to scale
if a + b> 1, then increasing returns to scale
Suppose: Q = 1.4 K .35 L .70
Is this production function constant returns to scale?
No, it is Increasing Returns to Scale, because 1.05 > 1.
Reasons for Increasing & Decreasing
Returns to Scale
Some Reasons for IRS
Some Reasons for DRS
 The advantage of
 Problems with coordination
specialization in capital and
labor – become more adept at
a task
 Engineering size and volume
effects – doubling the size of
motor more than doubles its
power
 Network effects
 Pecuniary advantages of
buying in bulk
and control – as a organization
gets larger, harder to get
everyone to work together
 Shirking increases
 Bottlenecks appear – a form of
the law of diminishing returns
appears
 CEO can’t oversee a
gigantically complex operation
Interpreting the Exponents of the
Cobb-Douglas Production Functions
 The exponents a and b are elasticities
a is the capital elasticity of output

The a is [% change in Q / % change in K]
b is the labor elasticity of output

The bis a [% change in Q / % change in L]
These elasticities can be written as EK and E L
Most firms have some slight increasing returns to scale.
Empirical Production Elasticities
Table 7.4: Most are statistically close to CRS or have IRS
˟ such as management or other staff personnel.