Inputs and Production Functions

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Lecture 10 Inputs and Production
Functions (cont.) Lecturer Martin Paredes
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Outline

• The Production Function (conclusion)
• Elasticity of Substitution
• Some Special Functional Forms
• Returns to Scale
• Technological Progress

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Elasticity Of Substitution

• 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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Elasticity Of Substitution

• 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 MRTSAL,K 4 KA/LA 4
• At point B MRTSBL,K 1 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
L
0
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K
Example The Elasticity of Substitution
MRTSA
KA /LA

A
KB/LB 1

B
Q
MRTSB 1
L
0
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Example Elasticity of Substitution ? (K/L)
-3 / 4 - 75 ? MRTSL,K -3 / 4 - 75
? ? (K/L) - 75 1 ?
MRTSL,K - 75
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Special Functional Forms

• 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
L
0
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K
Example Linear Production Function
Slope -a/b
Q1
Q0
L
0
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Special Functional Forms

• 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
Tires
2
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Frames
Example Fixed Proportion Production Function
Q 2 (bicycles)
2
Q 1 (bicycles)
1
0
Tires
2 4
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Special Functional Forms

• 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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Special Functional Forms

• Constant Elasticity of Substitution Production
  Function
• Q (aL? bK?)1/?
• where ?, ?, ? are all positive constants
• In particular, ? (?-1)/?
• Properties
• If ? 0 gt Leontieff case
• If ? 1 gt Cobb-Douglas case
• If ? ? gt Linear case

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K
Example The Elasticity of Substitution
? 0
L
0
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K
Example The Elasticity of Substitution
? 0
? ?
L
0
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K
Example The Elasticity of Substitution
? 0
? 1
? ?
L
0
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K
Example The Elasticity of Substitution
? 0
?? 0.5
? 1
? ?
L
0
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K
Example The Elasticity of Substitution
? 0
?? 0.5
? 1
?? 5
? ?
L
0
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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
? ?
L
0
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Returns to Scale
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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Returns to Scale

• Suppose we increase ALL inputs by a factor ?
• Suppose that, as a result, output increases by a
  factor ?.
• Then
• If ? gt ? gt Increasing returns to scale
• If ? ? gt Constant returns to scale
• If ? lt ? gt Decreasing returns to scale.