Transcript Production Economics Chapter 7

Production Economics
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.
2005 South-Western Publishing
Slide 1
The Production Function
• A Production Function is the maximum 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
• Q = a • K b1• L b2
is a Cobb-Douglas
Production Function
• The number of inputs is often large. But economists
simplify by suggesting some, like materials or labor, is
variable, whereas plant and equipment is fairly fixed in
the short run.
Slide 2
The Short Run
Production Function
• Short Run Production Functions:
» Max output, from a n y set of inputs
» Q = f ( X1, X2, X3, X4, X5 ... )
FIXED IN SR
VARIABLE IN SR
_
Q = f ( K, L) for two input case, where K as Fixed
• A Production Function is has only one variable
input, labor, is easily analyzed. The one variable
input is labor, L.
Slide 3
• Average Product = Q / L
» output per labor
• Marginal Product =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, we’re at the peak of the
AP curve
Slide 4
Elasticities of Production
• The production elasticity of labor,
» EL = MPL / APL = (DQ/DL) / (Q/L) = (DQ/DL)·(L/Q)
» The production elasticity of capital has the identical in
form, except K appears in place of L.
• When MPL > APL, then the labor elasticity, EL > 1.
» A 1 percent increase in labor will increase output by more than 1
percent.
• When MPL < APL, then the labor elasticity, EL < 1.
» A 1 percent increase in labor will increase output by less than 1
percent.
Slide 5
Short Run Production Function
Numerical Example
Marginal Product
L
Q
MP
0
1
2
3
4
5
0
20
46
70
92
110
--20
26
24
22
18
AP
--20
23
23.33
23
22
Labor Elasticity is greater then one,
for labor use up through L = 3 units
Average
Product
1
2
3
4
5
Slide 6
L
• 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
Slide 7
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
Slide 8
Three stages of production
Total Output
• Stage 1: average
product rising.
Stage 1
• Stage 2: average
product declining (but
marginal product
positive).
• Stage 3: marginal
product is negative, or
total product is
declining.
Stage 2
Stage 3
L
Figure 7.4 on Page 306
Slide 9
Optimal Use of the Variable Input
• HIRE, IF GET MORE
MRP L  MP L • P Q = W
REVENUE THAN
COST
wage
• HIRE if
DTR/DL > DTC/DL
• HIRE if the marginal
revenue product >
W  MFC
W
•
marginal factor cost:
MRPL
MRP L > MFC L
MPL
• AT OPTIMUM,
L
MRP L = W  MFC
optimal labor
Slide 10
MRP L is the Demand for Labor
• If Labor is MORE
productive, demand for
labor increases
• If Labor is LESS
productive, demand for
labor decreases
• Suppose an
EARTHQUAKE destroys
capital 
• MP L declines with less
capital, wages and
labor are HURT
SL
W
DL
D’ L
L’ L
Slide 11
Long Run Production Functions
• All 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 = DQ / DL
• MP of labor declines as more labor is
applied. Also the MP of capital declines as
more capital is applied.
Slide 12
Isoquants & LR Production Functions
• In the LONG RUN, ALL
factors are variable
• Q = f ( K, L )
• ISOQUANTS -- locus of
input combinations which
produces the same output (A
& B or on the same isoquant)
• SLOPE of ISOQUANT is
ratio of Marginal Products,
called the MRTS, the
marginal rate of technical
substitution
ISOQUANT
MAP
K
Q3
C
B
Q2
A
Q1
L
Slide 13
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
Equimarginal Criterion:
Produce where
MPL/CL = MPK/CK
where marginal products per
dollar are equal
Figure 7.9 on page 316
• C0 = CL·L + CK·K
• K = C0/CK - (CL/CK)·L
• Optimal where:
» MPL/MPK = CL/CK·
» Rearranged, this becomes
the equimarginal criterion
K
D
at D, slope of
isocost = slope
of isoquant
C(1)
L
Q(1)
Slide 14
Use of the Equimarginal Criterion
• Q: Is the following firm
EFFICIENT?
• A dollar spent on labor
produces 3, and a dollar
spent on capital produces 2.
• Suppose that:
USE RELATIVELY
» MP L = 30
MORE LABOR!
» MPK = 50
• If spend $1 less in capital,
» W = 10 (cost of labor)
output falls 2 units, but rises
3 units when spent on labor
» R = 25 (cost of
• Shift to more labor until the
capital)
equimarginal condition
• Labor: 30/10 = 3
holds.
• Capital: 50/25 = 2
• That is peak efficiency.
• A: No!
Slide 15
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, they firm is not
producing as much as is
technically possible.
D
T
(1)
Q
Q(0)
Slide 16
Returns to Scale
• A function is homogeneous of degree-n
» if multiplying all inputs by  (lambda) increases
the dependent variable byn
» Q = f ( K, L)
» So, f(K,  L) = n • Q
• Constant Returns to Scale is homogeneous of
degree 1.
» 10% more all inputs leads to 10% more output.
• Cobb-Douglas Production Functions are
homogeneous of degree a + b
Slide 17
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
• Coefficients are elasticities
a is the capital elasticity of output, often about .67
b is the labor elasticity of output, often about .33
which are EK and E L
Most firms have some slight increasing returns to scale
Slide 18
Problem
Suppose: Q = 1.4 L .70 K .35
1. Is this function homogeneous?
2. Is the production function constant
returns to scale?
3. What is the production elasticity of
labor?
4. What is the production elasticity of
capital?
5. What happens to Q, if L increases 3%
and capital is cut 10%?
Slide 19
Answers
1. Yes. Increasing all inputs by , increases output
by 1.05. It is homogeneous of degree 1.05.
2. No, it is not constant returns to scale. It is
increasing Returns to Scale, since 1.05 > 1.
3. .70 is the production elasticity of labor
4. .35 is the production elasticity of capital
5. %DQ = EL• %DL+ EK • %DK =
.7(+3%) + .35(-10%) = 2.1% -3.5% =
-1.4%
Slide 20