Transcript MANAGERIAL ECONOMICS 11th Edition

Production Analysis and
Compensation Policy
Chapter 7
Chapter 7
OVERVIEW
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Production Functions
Total, Marginal, and Average Product
Law of Diminishing Returns to a Factor
Input Combination Choice
Marginal Revenue Product and Optimal
Employment
Optimal Combination of Multiple Inputs
Optimal Levels of Multiple Inputs
Returns to Scale
Productivity Measurement
Production Functions
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Properties of Production Functions
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Determined by technology, equipment, and labor
Discrete functions are lumpy.
Continuous functions employ inputs in small
increments.
Returns to Scale and Returns to a Factor
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Returns to scale measure output effect of increasing
all inputs.
Returns to a factor measure output effect of
increasing one input.
Production Function
𝑄 = 𝑓 𝑋, 𝑌
𝑄 = 𝑓 𝐿𝑎𝑏𝑜𝑟, 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
𝑄 = 𝑓 𝐿𝑎𝑏𝑜𝑟, 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
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Total Product (TP) is whole output.
Marginal Product (MP) is the change in output caused by
increasing any input X.
Δ𝑄 𝜕𝑄
𝑀𝑃𝑋 =
=
Δ𝑋 𝜕𝑋
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Average product
𝑄
𝐴𝑃𝑋 =
𝑋
Law of Diminishing Returns
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Returns to a Factor
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Shows what happens to MPX as X usage grows.
• MPX > 0 is common.
• MPX < 0 implies irrational input use (rare).
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Diminishing Returns to a Factor Concept
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MPX shrinks as X usage grows, ∂2Q/∂X2 < 0.
If MPX grew with use of X, there would be no limit to
input usage.
Total, Marginal, and Avg Product
∆𝑇𝑃𝑋 𝑑𝑇𝑃𝑋
𝑀𝑃 =
=
∆𝑋
𝑑𝑋
Q
80
70
𝑇𝑃
60
𝑇𝑃𝑋
𝐴𝑃 =
𝑋
50
40
30
𝑋
𝑇𝑃
𝑀𝑃
𝐴𝑃
1
15
15
15.0
20
10
0
0
2
31
16
15.5
3
48
17
16.0
4
59
11
14.8
5
6
68
72
9
4
13.6
12.0
7
73
1
10.4
10
8
72
-1
9.0
5
9
70
-2
7.8
10
67
-3
6.7
𝑄
𝑋
1
2
3
4
5
6
7
8
9
10
𝑋
Marginal Product
20
15
𝐴𝑃
𝑋
0
0
-5
1
2
3
4
5
6
7
8
9
10
𝑀𝑃
Marginal Revenue Product (one input)
- A fixed resource
- Production efficiency
Tractor and Wagon
Implies Maximum Output per Worker
- Perfect Competition
Optimal level of a single input
Labor
Hay per
Hour
0
1
0
10
2
3
𝑀𝑃𝐿
Hay Price
𝑀𝑅𝑃𝐿 = 𝑃𝐿
𝑀𝑅𝑃𝐿
Wage
10
$1
$10
$8
25
50
15
25
$1
$1
$15
$25
$8
$8
4
5
65
75
15
$1
$15
$8
10
$1
$10
$8
6
7
80
80
5
0
$1
$5
$8
Multiple Input Choice
𝑄 = 𝑓 𝐿𝑎𝑏𝑜𝑟, 𝐶𝑎𝑝𝑖𝑡𝑎𝑙
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Production Isoquants
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All inputs are variable
Show efficient input combinations.
Technical efficiency is least-cost production.
Isoquant shape shows input
substitutability.
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Straight line isoquants depict perfect substitutes.
C-shaped isoquants depict imperfect substitutes.
L-shaped isoquants imply no substitutability.
Optimal Combination of Multiple Inputs
Slope of the isoquant is
𝑀𝑅𝑇𝑆𝑋𝑌
𝜕𝑌
𝑀𝑃𝑋
=
=−
𝜕𝑋
𝑀𝑃𝑌
Marginal Rate of Technical
Substitution which shows
amount of one input that
must be substituted for
another to maintain constant
output
Y
Q3
Q2
Q1
X
Input Combination Choice
Perfect Compliments
Imperfect Substitutes
Capital
Frames
Gas
Perfect Substitutes
C1
Q 1 Q2
Q3=3
C2
Q2=2
C3
Q3
Q2
Q1=1
Q3
Q1
Diesel
Where: Q1 < Q2 < Q3
Wheels
L1 L2 L3
Labor
Marginal Rate of Technical
Substitution
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Marginal Rate of Technical Substitution
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Shows amount of one input that must be
substituted for another to maintain constant
output.
For inputs X and Y, MRTSXY = -MPX / MPY
Rational Limits of Input Substitution
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Ridge lines show rational limits of input
substitution.
MPX < 0 or MPY < 0 are never observed.
Marginal Rate of Technical
Substitution
Y
Q3
Ridge Line Y or X
Q2
Q1
Ridge Line X or Y
𝑋
Marginal Revenue Product
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Marginal Revenue Product of labor is the net
revenue gain after all variable costs except labor costs.
𝜕𝑇𝑅
𝑀𝑅𝑃𝐿 = 𝑀𝑃𝐿 ∙ 𝑀𝑅𝑄 =
𝜕𝐿
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MRPL is the maximum amount that could be paid to
increase employment.
Optimal Level of a Single Input
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Set MRPL=PL to get optimal employment.
If MRPL=PL, then input marginal revenue equals input
marginal cost.
Optimal Combination of Multiple Inputs
Budget Lines show how many inputs can be bought.
𝐵 = 𝑃𝑋 ∙ 𝑋 + 𝑃𝑌 ∙ 𝑌
𝐵 𝑃𝑋
𝑌=
− 𝑋
𝑃𝑌 𝑃𝑌
𝑃𝑌 = $250
Y
𝐵1 = $1,000
𝐵2 = $2,000
𝐵3 = $3,000
𝑃𝑋 = $500
Then slope of
$500
the budget
−
=2
$250
line is
X
Optimal Combination of Multiple Inputs
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Budget Lines
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Show how many inputs can be bought.
Least-cost production occurs when
𝑃𝑋 𝑀𝑃𝑋
=
𝑃𝑌 𝑀𝑃𝑌
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or
𝑀𝑃𝑋 𝑀𝑃𝑌
=
𝑃𝑋
𝑃𝑌
Expansion Path shows efficient input combinations as
output grows.
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Illustration of Optimal Input Proportions
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Input proportions are optimal when no additional output
could be produce for the same cost.
Optimal input proportions is a necessary but not
sufficient condition for profit maximization.
Optimal Combination of Multiple Inputs
Optimal when
𝑀𝑃𝑋 𝑀𝑃𝑌
=
𝑃𝑋
𝑃𝑌
Y
expansion
path
Equilibrium (minimum cost output) when
the slope of the budget line is equal to the
isoquant slope
Y3
Q3
Y2
Y1
Q2
Q1
X1
X2
X3
𝑃𝑋 𝑀𝑃𝑋
=
𝑃𝑌 𝑀𝑃𝑌
X
When
𝑀𝑃𝑋 𝑀𝑃𝑌
>
𝑃𝑋
𝑃𝑌
𝑀𝑃𝑋 𝑀𝑃𝑌
>
𝑃𝑋
𝑃𝑌
Optimal Combination of Multiple Inputs
The Tax Advisors, Inc. currently has three CPAs and four bookkeepers.
Bookkeeper wages are $30 per hour and have MP=0.3. CPAs currently
receive hourly pay of $70 per hour with MP=1.4.
𝑀𝑃𝐵𝐾 𝑀𝑃𝐶𝑃𝐴
<
𝑃𝐵𝐾
𝑃𝐶𝑃𝐴
0.3
1.4
<
$30 $70
0.01 < 0.02
The Tax Advisors, Inc. should increase use of CPAs which will decrease their MP
and also reduce use of bookkeepers thus increasing their MP
𝑀𝑃𝐵𝐾 𝑀𝑃𝐶𝑃𝐴
<
𝑃𝐵𝐾
𝑃𝐶𝑃𝐴
Optimal Combination of Multiple Inputs
The numbers may also be used to calculate MC of production using
alternative labor sources (bookkeepers vs. CPAs).
𝑀𝐶𝐵𝐾
𝑃𝐵𝐾
$30
=
=
= $100
𝑀𝑃𝐵𝐾
.3
𝑀𝐶𝐶𝑃𝐴
𝑃𝐶𝑃𝐴
$70
=
=
= $50
𝑀𝑃𝐶𝑃𝐴
1.4
The CPAs have lowest MC as the moment thus hire a CPA next.
Education Example
MPta
=
Pta
MPct
=
MPsp
Pct
Pta = $15,000
Pct = $30,000
Psp = $35,000
Pad = $50,000
MPad
=
Psp
Pad
ta – teacher aid
ct – certified teacher
sp – Specialist
ad – Administrator
MPta = 30,000
MPct = 70,000
MPsp = 70,000
MPad = 90,000
Aid
Certified
Specialist
Administrator
30,000
70,000
70,000
90,000
$15,000
$30,000
$35,000
$50,000
2
2.33
2
1.8
The next person hired should be a:
Certified Teacher
Optimal Levels of Multiple Inputs
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Optimal Employment and Profit
Maximization
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Profits are maximized when MRPX = PX for all
inputs.
Profit maximization requires optimal input
proportions plus an optimal level of output.
Profit maximization means efficiently
producing what customers want.
Returns to Scale
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Returns to scale show the output effect of
increasing all inputs.
• Output elasticity is ε
Q = ∂Q/Q ÷ ∂Xi/Xi where
Xi is all inputs (labor, capital, etc.)
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Output Elasticity and Returns to Scale
• εQ > 1 implies increasing returns.
• εQ = 1 implies constant returns.
• εQ < 1 implies decreasing returns.
Returns to Scale
Total Product 350
(Q)
300
𝑄 = 10𝑋 .8 𝑌 .7
Increasing
Returns
250
200
𝑄=15𝑋^.5 𝑌^.5
150
Constant
Returns
100
𝑄 = 20𝑋 .4 𝑌 .2
Decreasing
Returns
50
0
0
2
4
6
8
10
12
Units of X and Y
Productivity Measurement
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Economic Productivity
• Productivity growth is the rate of change in
output per unit of input.
• Labor productivity is the change in output per
worker hour.
• Causes of Productivity Growth
• Efficiency gains reflect better input use.
• Capital deepening is growth in the amount of
capital workers have available for use.
Example (Labor Productivity)
Assume 𝑄 = 𝑓 𝐾, 𝐿 = 𝐾 .5 𝐿.5
an isoquant
Suppose K is fixed at 16 units
𝑄 = 16.5 𝐿.5 = 4𝐿.5
short run assumption
What is production when labor is 100
𝑄 = 4 ∙ 100.5 = 40
What is the marginal productivity of labor
𝜕𝑄
2
.5
−.5
𝑀𝑃𝐿 =
= 4𝐿 = .5 ∙ 4𝐿 = .5 3
𝜕𝐿
𝐿
2
1
0
0 5 10 15 20 25 30
Example (Labor Productivity)
𝜕𝑄
2
.5
−.5
𝑀𝑃𝐿 =
= 4𝐿 = .5 ∙ 4𝐿 = .5
𝜕𝐿
𝐿
When labor is 40, MP is
𝜕𝑄
2
𝑀𝑃40 =
= .5 = 0.316
𝜕𝐿 40
Suppose the product price is $40
Again if this market is
perfectly competitive
MR=P
𝑀𝑅𝑃40 = 𝑀𝑃40 ∙ 𝑃 = 0.316 ∙ $40 = $12.64
If wage rate is $10, would you use the 40th
worker?
Example (Advertising)
TV ads cost $400 each, radio ads $300 each
You have an advertising budget of $2,000
The marginal value of running additional ads are list below
Marginal Productivity
(Gross Rating Points)
𝑀𝑃𝑡𝑣 𝑀𝑃𝑟
=
𝑃𝑡𝑣
𝑃𝑟
Marginal Productivity
per dollar spend
Ads
TV
Radio
Ads
TV
Radio
Radio
$300
1
400
360
1
1
1.20
TV
$400 $700
2
300
270
2
0.75
0.90
Radio
$300 $1,000
3
280
240
3
0.70
0.80
Radio
$300 $1,300
4
260
225
4
0.65
0.75
Radio
$300 $1,600
5
240
150
5
0.60
0.50
TV
$400 $2,000
6
200
120
6
0.5
0.40
𝑀𝑃𝑡𝑣
400
=
=1
𝑃𝑡𝑣
$400
Purchase two TV spots
and four Radio Spots
Example (Appliances)
A manufacturer of home appliances faces the production function
𝑄 = 40𝐿 − 𝐿2 + 54𝐾 − 1.5𝐾 2 and input cost of PL = $10 and PK=$15.
𝜕𝑄
𝑀𝑃𝐿 =
= 40 − 2𝐿
𝜕𝐿
𝑀𝑃𝐾 =
𝜕𝑄
= 54 − 3𝐿
𝜕𝐾
𝑀𝑃𝐿 𝑀𝑃𝐾
=
𝑃𝐿
𝑃𝐾
40 − 2𝐿 54 − 3𝐾
=
$10
$15
𝐿 =𝐾+2
If K = 8 and L = 10, then
𝑄 = 40 10 − 102 + 54 8 − 1.5 8
𝑇𝐶 = $10 10 + $15 8 = $220
Minimum cost of producing 636 units
4 − .2𝐿 = 3.6 − .2𝐾
.2𝐿 = .2𝐾 + .4
.2𝐿 .2𝐾 .4
=
+
.2
.2
.2
𝐿 =𝐾+2
2
= 636