Anonymous
2 days ago
Illustrate that the constant elasticity of substitution of production below will collapse into any of the specific forms of production functions: when e = 0, e = -1, e = 2.
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ChatGPT
2 days ago
The constant elasticity of substitution (CES) production function is given by:
Y = [αK^ρ + (1-α)L^ρ]^(1/ρ)
where Y is output, K is capital input, L is labor input, α is the share parameter, and ρ is the elasticity of substitution between capital and labor.
When ρ = 0, the CES production function collapses into the Leontief production function:
Y = min{αK, (1-α)L}
This means that capital and labor inputs are perfect complements and must be used in fixed proportions to produce output.
When ρ = -1, the CES production function collapses into the Cobb-Douglas production function:
Y = K^α * L^(1-α)
This means that capital and labor inputs are used in fixed proportions, with the elasticity of substitution being equal to 1.
When ρ = 2, the CES production function collapses into the linear production function:
Y = αK + (1-α)L
This means that capital and labor inputs are perfect substitutes and can be used in any proportion to produce output.
Therefore, the CES production function can collapse into the Leontief, Cobb-Douglas, or linear production functions depending on the value of the elasticity of substitution parameter ρ.
Y = [αK^ρ + (1-α)L^ρ]^(1/ρ)
where Y is output, K is capital input, L is labor input, α is the share parameter, and ρ is the elasticity of substitution between capital and labor.
When ρ = 0, the CES production function collapses into the Leontief production function:
Y = min{αK, (1-α)L}
This means that capital and labor inputs are perfect complements and must be used in fixed proportions to produce output.
When ρ = -1, the CES production function collapses into the Cobb-Douglas production function:
Y = K^α * L^(1-α)
This means that capital and labor inputs are used in fixed proportions, with the elasticity of substitution being equal to 1.
When ρ = 2, the CES production function collapses into the linear production function:
Y = αK + (1-α)L
This means that capital and labor inputs are perfect substitutes and can be used in any proportion to produce output.
Therefore, the CES production function can collapse into the Leontief, Cobb-Douglas, or linear production functions depending on the value of the elasticity of substitution parameter ρ.
0
Google Gemini AI
2 days ago
Constant Elasticity of Substitution (CES) Production Function:
```
Q = A - [α - K^e + (1-α) - L^e]^(1/e)
```
where:
- Q is output
- K is capital
- L is labor
- A is a constant
- α is the elasticity of substitution
Collapse to Specific Production Functions:
1. When e = 0:
The CES production function collapses to the Cobb-Douglas production function:
```
Q = A - K^α - L^(1-α)
```
This implies that the elasticity of substitution is equal to 1.
2. When e = -1:
The CES production function collapses to the Leontief production function:
```
Q = min(A - K, (1-A) - L)
```
This implies that the elasticity of substitution is zero, indicating that capital and labor are perfect complements.
3. When e = 2:
The CES production function collapses to the quadratic production function:
```
Q = A - (α - K^2 + (1-α) - L^2)
```
This implies that the elasticity of substitution is 2, indicating that capital and labor are perfect substitutes.
```
Q = A - [α - K^e + (1-α) - L^e]^(1/e)
```
where:
- Q is output
- K is capital
- L is labor
- A is a constant
- α is the elasticity of substitution
Collapse to Specific Production Functions:
1. When e = 0:
The CES production function collapses to the Cobb-Douglas production function:
```
Q = A - K^α - L^(1-α)
```
This implies that the elasticity of substitution is equal to 1.
2. When e = -1:
The CES production function collapses to the Leontief production function:
```
Q = min(A - K, (1-A) - L)
```
This implies that the elasticity of substitution is zero, indicating that capital and labor are perfect complements.
3. When e = 2:
The CES production function collapses to the quadratic production function:
```
Q = A - (α - K^2 + (1-α) - L^2)
```
This implies that the elasticity of substitution is 2, indicating that capital and labor are perfect substitutes.
0
