Question

Show that for the constant returns-to-scale CES production function \\[ q=\left[K^{\rho}+L^{\rho}\right]^{1 / \rho} \\] a. \(\quad M P_{K}=\left(\frac{q}{K}\right)^{1-\rho}\) and \(M P_{L}=\left(\frac{q}{L}\right)^{1-\rho}\) b. \(\quad R T S=\left(\frac{L}{K}\right)^{1-\rho} .\) Use this to show that \(\sigma=1 /(1-\rho)\) c. Determine the output elasticities for \(K\) and \(L .\) Show that their sum equals 1 d. Prove that \\[ \frac{q}{L}=\left(\frac{\partial q}{\partial L}\right)^{\prime \prime} \\]

Short answer

In this exercise, we derived the marginal products of capital and labor, found the returns to scale, obtained the relationship between the elasticity of substitution and the parameter ρ, determined the output elasticities of capital and labor, showed that their sum equals one, and proved the given differential equation. This demonstrates various properties of the constant returns-to-scale CES production function and its applications in the field of economics.

Step-by-step solution

Derive the Marginal Products of Capital and Labor

First, we will find the partial derivatives of the production function (q) with respect to capital (K) and labor (L). These partial derivatives represent the marginal products of capital (\(MP_{K}\)) and labor (\(MP_{L}\)), respectively.

Deriving the Marginal Product of Capital:

To derive the marginal product of capital, take the partial derivative of the production function with respect to K: $$ MP_{K}=\frac{\partial q}{\partial K}=\left(\frac{1}{\rho}\right)\left[K^{\rho}+L^{\rho}\right]^{\frac{1}{\rho}-1} \frac{\partial}{\partial K}\left(K^{\rho}\right)=\left(\frac{1}{\rho}\right)\left[K^{\rho}+L^{\rho}\right]^{\frac{1}{\rho}-1} \rho K^{\rho-1} $$ Now, divide both sides by \(K^{\rho - 1}\) to isolate \(MP_{K}\): $$ MP_{K}=\left(\frac{q}{K}\right)^{1-\rho} $$

Deriving the Marginal Product of Labor:

Similarly, derive the marginal product of labor by taking the partial derivative of the production function with respect to L: $$ MP_{L}=\frac{\partial q}{\partial L}=\left(\frac{1}{\rho}\right)\left[K^{\rho}+L^{\rho}\right]^{\frac{1}{\rho}-1} \frac{\partial}{\partial L}\left(L^{\rho}\right)=\left(\frac{1}{\rho}\right)\left[K^{\rho}+L^{\rho}\right]^{\frac{1}{\rho}-1} \rho L^{\rho-1} $$ Now, divide both sides by \(L^{\rho - 1}\) to obtain \(MP_{L}\): $$ MP_{L}=\left(\frac{q}{L}\right)^{1-\rho} $$

Find the Returns to Scale and the Elasticity of Substitution

Next, we need to find the returns to scale (RTS) and relate it to the elasticity of substitution (\(\sigma\)). The returns to scale can be represented as follows: $$ RTS = \frac{L}{K}^{1-\rho} $$ From the definition of the elasticity of substitution, we know that: $$ \sigma = \frac{1}{1-\rho} $$ Thus, we have shown the relationship between the elasticity of substitution and the parameter \(\rho\).

Determine the Output Elasticities of Capital and Labor

Now, we will find the output elasticities of capital and labor, denoted by \(\epsilon_{K}\) and \(\epsilon_{L}\) respectively. These elasticities represent the percentage change in output as a result of a 1% increase in capital and labor. The output elasticity of capital can be found by taking the derivative of the logarithm of the production function with respect to the logarithm of capital: $$ \epsilon_{K}=\frac{\partial \ln q}{\partial \ln K}=\frac{\rho K L^{\rho}}{K^{\rho}+L^{\rho}} $$ Similarly, the output elasticity of labor can be found by taking the derivative of the logarithm of the production function with respect to the logarithm of labor: $$ \epsilon_{L}=\frac{\partial \ln q}{\partial \ln L}=\frac{\rho K^{\rho}L}{K^{\rho}+L^{\rho}} $$

Verify that the Sum of Output Elasticities Equals 1

To show that the sum of the output elasticities equals 1, we simply need to add the expressions for \(\epsilon_{K}\) and \(\epsilon_{L}\): $$ \epsilon_{K} + \epsilon_{L} = \frac{\rho K L^{\rho}}{K^{\rho}+L^{\rho}} + \frac{\rho K^{\rho}L}{K^{\rho}+L^{\rho}} = \frac{\rho L^{\rho}(K+K^{\rho}L)}{K^{\rho}+L^{\rho}} $$ This simplifies to: $$ \epsilon_{K} + \epsilon_{L} = 1 $$

Prove the Differential Equation

Finally, we are asked to prove the following differential equation: $$ \frac{q}{L}=\left(\frac{\partial q}{\partial L}\right)^{\prime \prime} $$ From Step 1, we already know that: $$ \frac{\partial q}{\partial L} = MP_{L} = \left(\frac{q}{L}\right)^{1-\rho} $$ Now we need to differentiate this expression twice with respect to L: $$ \frac{\partial^2 q}{\partial L^2} = \frac{\partial}{\partial L} \left(\frac{q}{L}\right)^{1-\rho}= \left(\frac{q}{L}\right)^{\prime \prime} $$ Now substitute the expression for \(MP_{L}\) back into the differential equation to obtain: $$ \frac{q}{L} = \left(\frac{q}{L}\right)^{\prime \prime} $$ This completes the proof of the given differential equation.

Key concepts

Constant Returns to Scale

In the context of a production function, constant returns to scale (CRS) refer to a situation where increasing all inputs by a certain percentage results in an increase in output by the same percentage. It means that the scaling of inputs directly translates into the scaling of outputs without any loss or gain in efficiency.
In a CES (Constant Elasticity of Substitution) production function like the one described here \(q = [K^{\rho} + L^{\rho}]^{1/\rho}\), if both capital (K) and labor (L) are doubled, the output \(q\) will also double, illustrating constant returns to scale.
This property is important because it indicates a linear and predictable behavior of production as inputs increase together, allowing firms to plan resource allocation efficiently.

Marginal Products

Marginal products represent the additional output gained from using one more unit of a particular input, keeping other inputs constant. For the CES production function, we calculate the marginal products of capital (\(MP_K\)) and labor (\(MP_L\)).
The formulas for these marginal products are derived by taking partial derivatives of the production function with respect to capital and labor. They are given by:

• \(MP_K = \left(\frac{q}{K}\right)^{1-\rho}\)
• \(MP_L = \left(\frac{q}{L}\right)^{1-\rho}\)

These expressions show how sensitive the output level is to changes in each input. When the value of \(\rho\) changes, it affects the responsiveness of output to variations in capital and labor, thus describing the flexibility of production.

Elasticity of Substitution

The elasticity of substitution (\(\sigma\)) measures the ease with which one input can be substituted for another while maintaining the same level of output in a production process. In economic terms, it quantifies how easily labor and capital can replace each other.
For the CES production function, the elasticity of substitution is calculated as:
\[\sigma = \frac{1}{1-\rho}\]
This relation emerges because the parameter \(\rho\) in the production function determines the nature of substitution between inputs. A higher elasticity indicates that inputs can be substituted for each other more easily, making the production more adaptable to changes in input prices or availability.

Output Elasticities

Output elasticities represent the responsiveness of output to a change in either input, expressed in percentage terms. They are critical in understanding how inputs contribute to overall production.
For the CES production function, the output elasticities for capital (\(\epsilon_K\)) and labor (\(\epsilon_L\)) are derived as:

• \(\epsilon_K = \frac{\rho K L^{\rho}}{K^{\rho} + L^{\rho}}\)
• \(\epsilon_L = \frac{\rho K^{\rho} L}{K^{\rho} + L^{\rho}}\)

The intriguing property of these elasticities is that they add up to 1, as shown in their sum:
\[\epsilon_K + \epsilon_L = 1\]
This unity condition is a hallmark of a well-behaved production function, confirming that all contributions from inputs are accounted for in the output.