Transcript Chapter 8

CHAPTER 8
THE DISCOVERY OF
PRODUCTION
AND ITS TECHNOLOGY
DISCOVERING PRODUCTION
• Primitive society
• Fruit and land
• Accidental discovery: jam
• Opportunity cost
• Cost of engaging in any activity
• Opportunity forgone - particular activity
• Normal profit
• Just sufficient to recover opportunity cost
• Extra-normal profit
• Return above normal profit
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PRODUCTION FUNCTION AND
TECHNOLOGY
• Technology
• Set of technological constraints
• On production
• Combine inputs into outputs
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PRODUCTION FUNCTION AND
TECHNOLOGY
• No free lunch assumption
• Production process
• Need inputs to produce outputs
• Non reversibility assumption
• Cannot run a production process in reverse
• Free disposability assumption
• Combination of inputs
• Certain output
• Or strictly less output
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PRODUCTION FUNCTION AND
TECHNOLOGY
• Additivity assumption
• Produce output x
• One combination of inputs
• Produce output y
• Another combination of inputs
• Feasible: produce x+y
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PRODUCTION FUNCTION AND
TECHNOLOGY
• Divisibility assumption
• Feasible input combination y
• Then, λy – feasible input combination
• 0≤ λ ≤ 1
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PRODUCTION FUNCTION AND
TECHNOLOGY
• Convexity assumption
• Production activity: y
• Output: z
• Particular amounts of inputs
• Production activity: w
• Output: z
• Different amounts of inputs
• Produce: at least z
• Mix activities y (λ time) and w(1- λ time)
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PRODUCTION FUNCTION AND
TECHNOLOGY
• Production function
• Maximum amount of output
• Given a certain level of inputs
• Output=f (input1, input2)
• Marginal product of input1
• the increase in output as a result of a marginal increase in
input1 holding input2 constant
• diminishing
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ISOQUANT
• Isoquant
• Set of bundles
• Given production function
• Produce same output
• Most efficiently
ISOQUANT
Capital
III
II200
I100
0
Labor
All combinations of inputs along the same isoquant yield the same
output.
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ISOQUANT
• Isoquants
• Never cross each other
• Farther from the origin greater outputs
• Slope
• Marginal rate of technical substitution
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MARGINAL RATE OF TECHNICAL
SUBSTITUTION
Capital (x2)
α
3
2
7
β
4
0
9
11
Labor (x1)
The absolute value of the isoquant’s slope measures the rate at which one
input can be substituted for the other while keeping the output level constant.
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MARGINAL RATE OF TECHNICAL
SUBSTITUTION
• Marginal rate of technical substitution (MRTS)
• Rate of substitution
• One input for another
• Constant output
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THE PRODUCTION FUNCTION
Output (y)
The level of output
is a function of the
levels of capital and
labor used.
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Labor (x1)
Capital (x2)
W
y1
W
y2
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MARGINAL RATE OF TECHNICAL
SUBSTITUTION
• Marginal product of input x2 at point α

(change
( change in output)
in the use of input x 2 given x 1 )

y
 x2
• MRTS of x2 for x1 at point α
y

Marginal product of x 1
Marginal product of x 2
 x1

y
 x2
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DESCRIBING TECHNOLOGIES
• Returns to scale – ratio of
• Change in output
• Proportionate change in all inputs
• Constant returns to scale
• All inputs - increase by λ
• Output - increases by λ
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DESCRIBING TECHNOLOGIES
• Increasing returns to scale
• All inputs - increase by λ
• Output - increases by more than λ
• Decreasing returns to scale
• All inputs - increase by λ
• Output - increases by less than λ
• Elasticity of substitution
• Substitute one input for another
• Given level of output
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RETURNS TO SCALE
Capital (x2)
Capital (x2)
(a)
Capital (x2)
(b)
(c)
p2
D
4
C
2
1
0
A
3
B p1
8
4
6
12
Labor (x1)
2
1
0
A
B p1
10
4
12
6
Labor (x1)
Constant returns to scale. Increasing returns to scale.
Doubling the levels of
Doubling the levels of both
labor (from 3 to 6) and
inputs more than doubles
capital (from 2 to 4) also the output level
doubles the level of output
(from 4 to 8)
2
1
0
A
B p1
6
4
6
12
Labor (x1)
Decreasing returns to
scale. Doubling the levels
of both inputs less than
doubles the output level
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TIME CONSTRAINTS
• Immediate run
• Period of time
• Cannot vary inputs
• Fixed factor of production
• Cannot be adjusted
• Given period of time
• Variable factor of production
• Can be adjusted
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TIME CONSTRAINTS
• Short run
• Time period
• At least one factor of production – fixed
• Long run
• Time period
• All factors of production – variable
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TIME CONSTRAINTS
• Long-run production function
• All inputs – variable
• Short-run production function
• Some inputs – variable
• Capital – fixed
• Labor – variable
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FIGURE 8.5
C
With the level of
capital fixed at x2,
the output level is a
function solely of
the level of labor.
Short-run production function
B
Labor (x1)
Capital (x2)
x2
0
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TIME CONSTRAINTS
• Total product curve
• Amount of output
• Add more and more units of variable input
• Hold one input constant
• Output – as we add more variable input
• First: increase at increasing rate
• After a point: Increase at decreasing rate
• Later: decrease
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FIGURE 8.6
Output
Short-run production function inDlabor-output space
8 14
G
8
E
1
12
1
2
0
A
1
+1
+1
2
10
15 16
30
Labor
The level of the fixed input, capital, is suppressed.
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TIME CONSTRAINTS
• Decreasing returns to factor
• Rate of output growth: decreasing
• Increase one input
• Other inputs – constant
• Marginal product curve
• Marginal product
• Factor of production
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FIGURE 8.7
Marginal
product
Marginal product
e
d
1
2
0
1
10
30
Labor (x1)
The slope of the short-run production function measures the change
in the output level resulting from the introduction of 1 additional unit
of the variable input - labor.
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THE PRODUCTION FUNCTION
• Cobb-Douglas production function
Q=AKαLβ
•
•
•
•
•
A – positive constant
0<α<1; 0<β<1
K – amount of capital
L – amount of labor
Q – output
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THE PRODUCTION FUNCTION
• Returns to scale = (α+β)
• For λ K and λL:
Q’= A(λK)α(λL)β =λ α+β Q
• If α+β=1
• Linearly homogeneous
• Constant returns to scale Q=AKαL1-α
• If α+β>1
• Increasing returns to scale
• If α+β<1
• Decreasing returns to scale
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THE PRODUCTION FUNCTION
• MRTS: dQ=0
dK
1    K


dL
   L
• Elasticity of substitution
 KL 
d ln( K / L )
1
d ln( MRTS )
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THE PRODUCTION FUNCTION
• Q=AKαLβ; α+β=1
Define : q 
Q
;k 
L
K
L
Average products : APL 
Q

 Ak ; APK 
L
Marginal products : MPL 
MPK 
Q
K
  Ak
Q
L
Q
 Ak
 1
K

 (1   ) Ak ;
 1
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THE PRODUCTION FUNCTION
• Q=AKαLβ; α+β=1
• Share of capital in output: K∙MPK/Q=α
• Share of labor in output: L∙MPL/Q=1-α
• Elasticity of output
 QK 
 QL 
Q
K
Q
L

Q

K

Q
1 
L
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