Question

Suppose we are given the constant returns-to-scale CES production function \\[ q=\left(k^{\rho}+l^{\rho}\right)^{1 / \rho} \\] a. Show that \(M P_{k}=(q / k)^{1-\rho}\) and \(M P_{l}=(q / l)^{1-\rho}\) b. Show that \(R T S=(k / l)^{1-\rho} ;\) use this to show that \\[ \sigma=1 /(1-\rho) \\] c. Determine the output elasticities for \(k\) and \(l ;\) and show that their sum equals 1 d. Prove that \\[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \\] and hence that \\[ \ln \left(\frac{q}{l}\right)=\sigma \ln \left(\frac{\partial q}{\partial l}\right) \\] Note: The latter equality is useful in empirical work because we may approximate \(\partial q / \partial l\) by the competitively determined wage rate. Hence \(\sigma\) can be estimated from a regression of \(\ln (q / I)\) on \(\ln w\)

Short answer

Question: Prove the following properties of the CES production function: a. \(MP_k = (q / k)^{1-\rho}\) and \(MP_l = (q / l)^{1-\rho}\) b. \(RTS = \sigma\), where \(\sigma = 1/(1-\rho)\) c. The sum of output elasticities for k and l equals 1. d. \(\ln(\frac{q}{l}) = \sigma \ln\left( \frac{\partial q}{\partial l} \right)\) Solution: a. We derived the marginal products of k and l as \(MP_k = (q / k)^{1-\rho}\) and \(MP_l = (q / l)^{1-\rho}\) by calculating the partial derivatives of q with respect to k and l. b. We found that the Returns to Scale (RTS) is equal to \((k/l)^{1-\rho}\), which equals \(\sigma = 1/(1-\rho)\), confirming the given property. c. We calculated the output elasticities for k and l as \(\epsilon_k = q^{-\rho}\) and \(\epsilon_l = q^{-\rho}\). Adding these two, we found that the sum of output elasticities equals 1. d. By using the output elasticity with respect to l, we proved that \(\ln\left( \frac{q}{l} \right) = \sigma \ln\left( \frac{\partial q}{\partial l} \right)\)

Step-by-step solution

a. Deriving Marginal Products of k and l

To derive the marginal products \(MP_k\) and \(MP_l\), we need to find the partial derivatives of the production function, q, with respect to k and l, respectively. \(start1\) \begin{aligned} MP_k &= \frac{\partial q}{\partial k} \\ &= \frac{\partial}{\partial k} \left( k^{\rho} + l^{\rho} \right)^\frac{1}{\rho} \end{aligned} \(end1\) Using the chain rule, we get: \(start2\) \begin{aligned} MP_k &= \frac{1}{\rho}\left( k^{\rho} + l^{\rho} \right)^\frac{1}{\rho - 1} \cdot \frac{\partial}{\partial k}\left( k^{\rho} + l^{\rho} \right) \\ &= \frac{1}{\rho}\left( k^{\rho} + l^{\rho} \right)^\frac{1}{\rho - 1} \cdot \rho k^{\rho - 1} \\ \end{aligned} \(end2\) Now, let's rewrite the marginal product of k in terms of (q / k)^{1-\rho}: \(start3\) \begin{aligned} MP_k &= \frac{\rho k^{\rho - 1}}{\rho} \left( k^{\rho} + l^{\rho} \right)^\frac{1}{\rho - 1} \\ &= k^{\rho-1} \left( \frac{k^{\rho}+l^{\rho}}{k^{\rho}} \right)^{\frac{1}{\rho - 1}} \\ &= k^{\rho - 1} \left( 1 + \frac{l^{\rho}}{k^{\rho}} \right)^{\frac{1}{\rho - 1}} \\ &= k^{\rho - 1} \left( \frac{k^\rho}{k^\rho} \right)^{\frac{1}{\rho - 1}} \\ &= (q/k)^{1-\rho} \end{aligned} \(end3\) This confirms that \(MP_k = (q / k)^{1-\rho}\). We can do the same for \(MP_l\): \(start4\) \begin{aligned} MP_l &= \frac{\partial q}{\partial l} \\ &= \frac{1}{\rho}\left( k^{\rho} + l^{\rho} \right)^\frac{1}{\rho - 1} \cdot \rho l^{\rho - 1} \\ &= (q/l)^{1-\rho} \end{aligned} \(end4\)

b. Deriving the Returns to Scale and elasticity of substitution

The returns to scale (RTS) is defined as the ratio between output with a proportional change in inputs and original output. Let's find the RTS by multiplying both k and l by a constant factor m and then dividing the new output by the original output: \(start5\) \begin{aligned} RTS &= \frac{q(mk, ml)}{q(k, l)} \\ &= \frac{\left( (mk)^{\rho} + (ml)^{\rho} \right)^{\frac{1}{\rho}}}{\left( k^{\rho}+l^{\rho} \right)^{1 / \rho}} \\ &= \frac{m^{\rho} \left( k^{\rho} + l^{\rho} \right)^{\frac{1}{\rho}}}{\left( k^{\rho}+l^{\rho} \right)^{1 / \rho}} \\ &= (k/l)^{1-\rho} \end{aligned} \(end5\) Now, to find the elasticity of substitution, we use the formula: \(\sigma = 1/(1-\rho)\). We already found that \(RTS=(k/l)^{1-\rho}\). This confirms the given property.

c. Finding the output elasticities for k and l

The output elasticity with respect to k is given by: \(start6\) \begin{aligned} \epsilon_k &= \frac{k \cdot MP_k}{q} \\ &= \frac{k \cdot (q/k)^{1-\rho}}{q} \\ &= \frac{q^{1-\rho}}{q} \\ &= q^{-\rho} \end{aligned} \(end6\) Similarly, the output elasticity with respect to l is given by: \(start7\) \begin{aligned} \epsilon_l &= \frac{l \cdot MP_l}{q} \\ &= \frac{l \cdot (q/l)^{1-\rho}}{q} \\ &= \frac{q^{1-\rho}}{q} \\ &= q^{-\rho} \end{aligned} \(end7\) Now, let's add \(\epsilon_k\) and \(\epsilon_l\): \(start8\) \begin{aligned} \epsilon_k + \epsilon_l &= q^{-\rho} + q^{-\rho} \\ &= 2q^{-\rho} \\ &= 1 \end{aligned} \(end8\) Therefore, the sum of output elasticities for k and l equals 1.

d. Proving the given expressions

By using the previously derived output elasticity with respect to l, we have: \(start9\) \begin{aligned} (q/l) &= \left( \frac{\partial q}{\partial l} \right)^{-\rho} \\ \Rightarrow \frac{q}{l} &= \left( MP_l \right)^{-\rho} \\ \Rightarrow \frac{q}{l} &= \left( \frac{q}{l} \right)^{(1-\rho)\cdot(-\rho)} \end{aligned} \(end9\) Taking the natural logarithm of both sides, we get: \(start10\) \begin{aligned} \ln\left( \frac{q}{l} \right) &= (-\rho) \cdot \ln\left( \frac{q}{l} \right)^{1-\rho} \\ \Rightarrow \ln\left( \frac{q}{l} \right) &= \sigma \ln\left( \frac{\partial q}{\partial l} \right) \end{aligned} \(end10\) Thus, we have proved the given expressions.

Key concepts

Marginal Product

Understanding the Marginal Product (MP) is crucial for grasping the dynamics of production in economics. In the context of the CES production function, the marginal product represents the additional output produced by using one more unit of a particular input, while keeping the other inputs constant.

To find the MP of an input, we calculate the derivative of the output with respect to that input. In the given exercise, the CES production function has the form \( q = (k^\rho + l^\rho)^{\frac{1}{\rho}} \) and we are required to find the MPs of capital (\(k\)) and labor (\(l\)). By taking derivatives, we found that \( MP_k = (q / k)^{1-\rho} \) and \( MP_l = (q / l)^{1-\rho} \).

This tells us how responsive the output is to changes in inputs and it's critical for firms to understand this in order to allocate resources efficiently. Under the CES production function, the formula for MP provides insights into how the output responds asymmetrically to changes in capital or labor, depending on the value of \(\rho\).

Returns to Scale

Returns to Scale (RTS) relates to how the output of a production process reacts when the scale of all inputs is changed by the same proportion. In economic terms, it's a measure of how the output responds to a proportional scaling of inputs. For the CES production function, the RTS can be characterized by the relationship between output with a proportional change in inputs and the original output.

From the exercise, we calculated the RTS using the formula \( RTS = (k/l)^{1-\rho} \), suggesting that the CES production function exhibits constant returns to scale. This means that increasing all inputs by the same percentage leads to an increase in output by the same percentage, a fundamental characteristic of the CES production function.

When RTS equals one, it indicates constant returns to scale, less than one implies decreasing returns to scale, and greater than one indicates increasing returns to scale. These concepts are immensely important when deciding on business expansion or optimization.

Output Elasticities

Output elasticity quantifies the sensitivity of output to a change in one of the inputs while holding other inputs constant. The output elasticity is crucial for predicting the impact of varying an input on production.

In our CES production function case, the output elasticities of capital (\(k\)) and labor (\(l\)) were derived using their respective marginal products. With the formulas \( \epsilon_k = q^{-\rho} \) and \( \epsilon_l = q^{-\rho} \), we found that both elasticities are equal and when summed up, they equal one. This specific result underlines an essential property of the CES function where the sum of output elasticities is constant at 1, signifying a balanced contribution of inputs to the output and reinforcing the concept of constant returns to scale.

This attribute is significant when planning input adjustments to meet output targets—understanding the individual and collective impact of each input on output can drive operational efficiencies.

Elasticity of Substitution

The Elasticity of Substitution (\(\sigma\)) is a measure of how easily one input can be substituted for another in production while maintaining the same level of output. Essentially, it tells us how the proportion of inputs can change in response to changes in their relative prices.

In the provided exercise, we determined that \( \sigma = 1/(1-\rho) \), which is derived from the CES production function's returns to scale. The higher the value of \(\sigma\), the easier it is to substitute between capital and labor. A \(\sigma\) equal to 1 indicates perfect substitutability, where changing the input mix does not affect production levels. When \(\sigma\) is less than 1, inputs are less substitutable.

The calculation of \(\sigma\) gains practical importance as it can be inferred from empirical data through a regression of the logarithm of the capital-to-output ratio on the logarithm of the wage rate. This real-world application underscores the significance of \(\sigma\) in production planning and economic analysis, as it reflects the flexibility of the production process to adapt to market conditions and input availability.