Lecture # 10 Inputs and Production Functions (cont.) Lecturer: Martin Paredes.
26
Lecture # 10 Lecture # 10 Inputs and Production Functions Inputs and Production Functions (cont.) (cont.) Lecturer: Martin Paredes Lecturer: Martin Paredes
-
Upload
lorin-morris -
Category
Documents
-
view
220 -
download
4
Transcript of Lecture # 10 Inputs and Production Functions (cont.) Lecturer: Martin Paredes.
Lecture # 10Lecture # 10
Inputs and Production Inputs and Production FunctionsFunctions
(cont.)(cont.)
Lecturer: Martin ParedesLecturer: Martin Paredes
2
1. The Production Function (conclusion) Elasticity of Substitution
2. Some Special Functional Forms3. Returns to Scale4. Technological Progress
3
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.
4
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.
5
Example: Elasticity of Substitution
• Suppose that…
At point A: MRTSAL,K = 4 KA/LA =
4At point B: MRTSB
L,K = 1 KB/LB = 1
• What is the elasticity of substitution?
6
Example: The Elasticity of Substitution
L
K
0
KA /LA = 4
Q
MRTSA = 4
•A
7L
K
0
KA /LA
Q
MRTSA
•A
MRTSB = 1
KB/LB = 1
•B
Example: The Elasticity of Substitution
8
Example: Elasticity of Substitution
% (K/L) = -3 / 4 = - 75%% MRTSL,K = -3 / 4 = - 75%
= % (K/L) = - 75% = 1 % MRTSL,K - 75%
9
1. Linear Production FunctionQ = aL + bK
where a,b are positive constants
Properties: MRTSL,K = MPL = a (constant)
MPK b Constant returns to scale =
10
Example: Linear Production Function
L
K
Q0
0
11
Example: Linear Production Function
L
K
Q1
0
Q0
Slope = -a/b
12
2. Fixed Proportions Production FunctionQ = 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
13Tires
Frames
2
Q = 1 (bicycles)
0
1
Example: Fixed Proportion Production Function
14Tires
Frames
2 4
Q = 1 (bicycles)
Q = 2 (bicycles)
0
1
2
Example: Fixed Proportion Production Function
15
3. Cobb-Douglas Production FunctionQ = ALK
where A, , are all positive constants
Properties: MRTSL,K = MPL = AL-1K = K
MPK ALK-1 L
= 1
16
Example: Cobb-Douglas Production Function
L
K
0
Q = Q0
17
Example: Cobb-Douglas Production Function
L
K
0
Q = Q1
Q = Q0
18
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
19
Example: The Elasticity of Substitution
L
K
0
= 0
20
Example: The Elasticity of Substitution
L
K
0
= 0
=
21
Example: The Elasticity of Substitution
L
K
0
= 0
= 1
=
22
Example: The Elasticity of Substitution
L
K
0
= 0
= 1
=
= 0.5
23
Example: The Elasticity of Substitution
L
K
0
= 0
= 1 = 5
=
= 0.5
24
"The shape of the isoquant indicates the degree of substitutability of the inputs…"
Example: The Elasticity of Substitution
L
K
0
= 0
= 1 = 5
=
= 0.5
25
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
26
Suppose we increase ALL inputs by a factor
Suppose that, as a result, output increases by a factor .
Then:1. If > ==>Increasing returns to
scale2. If = ==>Constant returns to
scale3. If < ==>Decreasing returns to
scale.
