Question

(a) The CES production function rules out \(\rho=-1 .\) If \(\rho=-1,\) however, what would be the general shape of the isoquants for positive \(K\) and \(l ?\) (b) is \(\sigma\) defined for \(\rho=-1 ?\) What is the limit of \(\sigma\) as \(\rho \rightarrow-1 ?\) (c) Interpret economically the results for parts \((a)\) and \((b)\)

Short answer

For \(\rho = -1\), isoquants are right angles, resembling perfect complements. The elasticity of substitution \(\sigma\) is undefined as inputs are used in fixed proportions without substitution.

Step-by-step solution

Understanding the CES Production Function

The Constant Elasticity of Substitution (CES) production function is a specific type of production function that has the form \[ Q = A \left( \delta K^{\rho} + (1-\delta) L^{\rho} \right)^{\frac{1}{\rho}} \], where \(A\) is a constant, \(K\) and \(L\) are input factors (usually capital and labor), and \(\rho\) determines the elasticity of substitution.

Analyze the Implication of \(\rho = -1\)

For \(\rho = -1\), the expression inside the CES function takes a specific form that isn't defined in the general CES framework. Substituting \(\rho = -1\) into the production function results in the function becoming undefined because it would lead to division by zero when trying to simplify the substitution elasticity.

Shape of Isoquants for \(\rho = -1\)

Using the mathematical identity \(\lim_{\rho \to -1} \left( \delta K^{\rho} + (1-\delta) L^{\rho} \right) = \frac{1}{\frac{1}{K} + \frac{1}{L}}\), the function resembles a Leontief production function or fixed proportions. The isoquants would form right-angle shapes, indicating perfect complements between \(K\) and \(L\).

Elasticity of Substitution \(\sigma\)

For the CES function, the elasticity of substitution \(\sigma\) is defined as \(\frac{1}{1+\rho}\). When \(\rho = -1\), \(\sigma\) would theoretically approach infinity, but is not well-defined directly since substitution in Leontief cases does not occur.

Limit Behavior of \(\sigma\) as \(\rho \rightarrow -1\)

The limit of \(\sigma\) as \(\rho \to -1\) becomes undefined mathematically as the equation turns into a division by zero issue (\(1 + \rho = 0\)). Therefore, practically speaking, \(\sigma\) represents a scenario where elastic substitution becomes irrelevant because inputs are used in fixed proportions.

Economic Interpretation

The results suggest that when \(\rho = -1\), capital and labor must be used together in fixed proportions without substitutes, like a technology where each machine requires exactly one operator. This implies no flexibility in varying input ratios without losing output, which leads to an undefined elasticity of substitution.

Key concepts

Constant Elasticity of Substitution

The Constant Elasticity of Substitution (CES) production function is a versatile model used to describe how inputs are transformed into outputs in an economic setting. It is expressed as \[ Q = A \left( \delta K^{\rho} + (1-\delta) L^{\rho} \right)^{\frac{1}{\rho}} \]where:

• \( Q \) is the quantity of output produced.
• \( A \) is a constant that represents technology level.
• \( \delta \) is a distribution parameter, representing the weight given to capital \( K \).
• \( \rho \) determines the degree of substitutability between the inputs.
• \( K \) and \( L \) represent the quantities of capital and labor, respectively.

This function uniquely incorporates the elasticity of substitution between different inputs, a key concept in understanding how easily one factor can replace another in the production process. When \(\rho\) varies, it influences the way inputs can be substituted, tailoring input combinations exactly to suit different technological conditions.

Isoquants

Isoquants play a crucial role in understanding production functions. They are curves that represent different combinations of two inputs, say capital \( K \) and labor \( L \), which yield the same level of output.When we talk about isoquants, we refer to their shape and slope, which inform us about substitution between inputs:

• Smooth, convex isoquants suggest that inputs are easily substitutable over a range.
• Right-angle isoquants imply perfect complements, where inputs are used in fixed proportions.

In the case of \(\rho = -1\), within the CES production function, the isoquants become right angles. This indicates that inputs are perfect complements, meaning substitution is impossible. So, for \(K\) and \(L\), the only way to maintain output is by using them in a specific, unchanging ratio.

Elasticity of Substitution

The elasticity of substitution, represented by \(\sigma\), is a measure of how easily one input can be substituted for another while maintaining the same level of output. For the CES production function, \(\sigma\) is calculated as \[ \sigma = \frac{1}{1+\rho} \]This elasticity informs us about the flexibility of the production process:

• A high \(\sigma\) value indicates greater ease of substitution.
• A low \(\sigma\) value means inputs are less substitutable.
• When \(\rho = -1\), \(\sigma\) approaches infinity, theoretically suggesting perfect substitution, but in reality, this becomes undefined because inputs are instead used in fixed proportions.

Thus, when inputs are required in precise ratios (like in a Leontief production function), substitution doesn't occur as each unit of one input needs an exact amount of the other.

Leontief Production Function

The Leontief production function is a special case of production functions where inputs must be used in fixed proportions to produce output. It is characterized by right-angle isoquants, indicating perfect complementarity between inputs.In a Leontief function, output depends on the smallest of the proportional input amounts, formulated as \[ Q = \min(aK, bL) \] where:

• \(a\) and \(b\) are fixed coefficients representing the necessary inputs per unit of output.
• \(K\) and \(L\) are inputs of capital and labor.

No substitution is possible here, as each unit of output requires a precise combination of inputs. In the context of \(\rho = -1\) for a CES function, the isoquants transform into those typical of a Leontief function, reinforcing the concept that inputs must be used in fixed quantities. This ensures that changes in input ratios cannot compensate for changes in output, underlining the rigidity inherent in some production technologies.