Transcript Lecture # 10 Inputs and Production Functions (cont.) Lecturer: Martin Paredes

Lecture # 10
Inputs and Production Functions
(cont.)
Lecturer: Martin Paredes
1. The Production Function (conclusion)
 Elasticity of Substitution
2. Some Special Functional Forms
3. Returns to Scale
4. Technological Progress
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Definition: The elasticity of substitution measures
how the capital-labor ratio, K/L, changes
relative to the change in the MRTSL,K.
 = % (K/L) = d (K/L) . MRTSL,K
% MRTSL,K d MRTSL,K (K/L)
 In other words, it measures how quickly the
MRTSL,K changes as we move along an
isoquant.
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Notes:
 In other words, the elasticity of substitution
measures how quickly the MRTSL,K changes as
we move along an isoquant.
 The capital-labor ratio (K/L) is the slope of any
ray from the origin to the isoquant.
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Example: Elasticity of Substitution
• Suppose that…
At point A:
At point B:
MRTSAL,K = 4
MRTSBL,K = 1
KA/LA = 4
KB/LB = 1
• What is the elasticity of substitution?
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K
Example: The Elasticity of Substitution
MRTSA = 4
KA /LA = 4
•A
Q
0
L
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K
MRTSA
Example: The Elasticity of Substitution
KA /LA
•A
KB/LB = 1
•
B
Q
MRTSB = 1
0
L
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Example: Elasticity of Substitution
% (K/L)
= -3 / 4 = - 75%
% MRTSL,K = -3 / 4 = - 75%
 = % (K/L) =
% MRTSL,K
- 75% = 1
- 75%
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1. Linear Production Function
Q = aL + bK
where a,b are positive constants
 Properties:
 MRTSL,K = MPL = a
(constant)
MPK
b
 Constant returns to scale
 =
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K
Example: Linear Production Function
Q0
0
L
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K
Example: Linear Production Function
Slope = -a/b
Q0
0
Q1
L
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2. Fixed Proportions Production Function
Q = min(aL, bK)
where a,b are positive constants
 Also called the Leontief Production Function
 L-shaped isoquants
 Properties:
 MRTSL,K = 0 or  or undefined
 =0
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Frames
Example: Fixed Proportion Production Function
Q = 1 (bicycles)
1
0
2
Tires
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Frames
Example: Fixed Proportion Production Function
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Q = 2 (bicycles)
Q = 1 (bicycles)
1
0
2
4
Tires
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3. Cobb-Douglas Production Function
Q = ALK
where A, ,  are all positive constants
 Properties:
 MRTSL,K = MPL = AL-1K = K
MPK
ALK-1
L
 =1
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K
Example: Cobb-Douglas Production Function
Q = Q0
0
L
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K
Example: Cobb-Douglas Production Function
Q = Q1
Q = Q0
0
L
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4. Constant Elasticity of Substitution Production
Function
Q = (aL + bK)1/
where , ,  are all positive constants
 In particular,  = (-1)/
 Properties:
 If  = 0 => Leontieff case
 If  = 1 => Cobb-Douglas case
 If  =  => Linear case
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K
Example: The Elasticity of Substitution
=0
0
L
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K
Example: The Elasticity of Substitution
=0
=
0
L
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K
Example: The Elasticity of Substitution
=0
=1
=
0
L
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K
Example: The Elasticity of Substitution
=0
 = 0.5
=1
=
0
L
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K
Example: The Elasticity of Substitution
=0
 = 0.5
=1
 = 5
=
0
L
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K
Example: The Elasticity of Substitution
"The shape of the
isoquant indicates the
degree of substitutability
of the inputs…"
=0
 = 0.5
=1
 = 5
=
0
L
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Definition: Returns to scale is the concept that tells
us the percentage increase in output when all
inputs are increased by a given percentage.
Returns to scale =
% Output .
% ALL Inputs
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 Suppose we increase ALL inputs by a factor 
 Suppose that, as a result, output increases by a
factor .
 Then:
1. If  > 
==> Increasing returns to scale
2. If  = 
==> Constant returns to scale
3. If  < 
==> Decreasing returns to scale.
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