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 $RTS=(k/l{)}^{1-\rho };$ 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

a. The marginal products of capital and labor, denoted by $M{P}_k$ and $M{P}_l$, are found by taking the partial derivatives of the CES production function. The expressions for the marginal products are: \[ MP_{k} = (q / k)^{1 - \rho} \] and \[ MP_{l} = (q / l)^{1 - \rho} \] b. The relationship between RTS and elasticity of substitution, denoted by $\sigma$, is given by: \[ \sigma = 1 / (1 - \rho) \] c. The output elasticities of capital and labor, denoted by $E_k$ and $E_l$, are found to be: \[ E_{k} = \rho \] and \[ E_{l} = \rho \] It is also shown that their sum equals 1: \[ E_{k} + E_{l} = 1 \] d. The given relationship between output and marginal product is proven to be: \[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \] and the logarithmic form of the relationship is shown as: \[ \ln \left(\frac{q}{l}\right)=\sigma \ln\left(\frac{\partial q}{\partial l}\right) \]

Step-by-step solution

a. Find the marginal products of capital and labor.

To find the marginal products of capital and labor, we need to take the partial derivatives of the CES production function with respect to $k$ and $l$ respectively. The CES production function is given by: \[ q=\left(k^{\rho}+l^{\rho}\right)^{1 / \rho} \] Differentiating with respect to $k$, we get: \[ \frac{\partial q}{\partial k}=\frac{1}{\rho}(k^{\rho}+l^{\rho})^{\frac{1}{\rho}-1}k^{\rho-1} \] Now, divide by $k$: \[ M P_{k} = \frac{\partial q}{\partial k}\frac{1}{k} = \frac{1}{\rho}(k^{\rho}+l^{\rho})^{\frac{1}{\rho}-1}\frac{k^{\rho-1}}{k} \] We find that: \[ M P_{k} = (q / k)^{1 - \rho} \] Similarly, differentiating with respect to $l$, we get: \[ \frac{\partial q}{\partial l}=\frac{1}{\rho}(k^{\rho}+l^{\rho})^{\frac{1}{\rho}-1}l^{\rho-1} \] Now, divide by $l$: \[ M P_{l} = \frac{\partial q}{\partial l}\frac{1}{l} = \frac{1}{\rho}(k^{\rho}+l^{\rho})^{\frac{1}{\rho}-1}\frac{l^{\rho-1}}{l} \] We find that: \[ M P_{l} = (q / l)^{1 - \rho} \]

b. Show the relationship between RTS and elasticity of substitution.

Recall that the expression for RTS can be derived from the ratio of marginal products, which is given by: \[ R T S = \frac{M P_{k}}{M P_{l}} \] Using the expressions for $M{P}_k$ and $M{P}_l$ we found in part (a): \[ R T S = \frac{(q / k)^{1 - \rho}}{(q / l)^{1 - \rho}} \] Simplifying, we get: \[ R T S = (k / l)^{1 - \rho} \] Now, we need to show the relationship between elasticity of substitution, denoted by $\sigma$, and $\rho$. We are given the expression: \[ \sigma = 1 / (1 - \rho) \] Multiplying both sides by $(1-\rho )$, we get: \[ \sigma (1 - \rho) = 1 \] This shows the relationship between elasticity of substitution and $\rho$.

c. Determine the output elasticities for k and l.

Output elasticities of $k$ and $l$ can be found by taking the partial derivatives of the production function with respect to $k$ and $l$, respectively, and multiplying each by the ratio of the input to output, i.e., $(k/q)$ and $(l/q)$. The output elasticity of $k$ is given by: \[ E_{k} = \frac{k}{q}\frac{\partial q}{\partial k} \] Using the marginal product expression we found in part (a) for $\partial q/\partial k$, we get: \[ E_{k} = k\left(\frac{k}{q}\right)^{1-\rho}\frac{1}{k} \] Simplifying, we get: \[ E_{k} = \rho \\] Similarly, the output elasticity of $l$ is given by: \[ E_{l} = \frac{l}{q}\frac{\partial q}{\partial l} \] Using the expression for $\partial q/\partial l$ from part (a), we get: \[ E_{l} = l\left(\frac{l}{q}\right)^{1-\rho}\frac{1}{l} \] Simplifying, we get: \[ E_{l} = \rho \] Now, to show that their sum equals 1: \[ E_{k} + E_{l} = \rho + \rho = 2\rho \] Recall, the relationship between elasticity of substitution and $\rho$ we found in part (b): \[ \sigma = 1/(1 - \rho) \] Therefore, we can write the sum of output elasticities as: \[ E_{k} + E_{l} = 1 - (1 - 2\rho) \] Simplifying, we get: \[ E_{k} + E_{l} = 1 \]

d. Prove the given relationships with partial derivatives.

First, we need to show that: \[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \] We know from part (a) that: \[ \frac{\partial q}{\partial l} = (q/l)^{1-\rho} \] Now raise both sides to the power of $\sigma$ to get: \[ \left(\frac{\partial q}{\partial l}\right)^{\sigma} = \left[(q/l)^{1-\rho}\right]^{\sigma} \] Recall the relationship between elasticity of substitution and $\rho$: \[ \sigma = 1/(1 - \rho) \] Substitute this into the previous equation and simplify: \[ \left(\frac{\partial q}{\partial l}\right)^{\sigma} = (q/l)^{(1-\rho)} \] This gives us the required relationship: \[ \frac{q}{l}=\left(\frac{\partial q}{\partial l}\right)^{\sigma} \] Now, we need to prove that: \[ \ln \left(\frac{q}{l}\right)=\sigma \ln\left(\frac{\partial q}{\partial l}\right) \] To do this, take the natural logarithm of both sides of the previous expression: \[ \ln\left[\frac{q}{l}\right] = \sigma \ln\left[\left(\frac{\partial q}{\partial l}\right)^{\sigma}\right] \] Use the properties of logarithms to rewrite the right-hand side: \[ \ln\left[\frac{q}{l}\right] = \sigma \ln\left[\frac{\partial q}{\partial l}\right] \] This completes the proof of the given relationships.

Key concepts

Marginal Product

Understanding the marginal product is crucial for students studying economics, particularly in production theory. The marginal product of an input, like capital (k) or labor (l), represents the additional output generated by using one more unit of that input while keeping all other inputs constant.

To put it simply, it's like seeing how much more product you can make by adding just one more worker or machine to the production process. In the context of the CES production function given by $q=(k^{\rho }+l^{\rho }{)}^{1/\rho }$, the marginal products of capital $M{P}_k$ and labor $M{P}_l$ can be obtained by taking the derivative of the production function with respect to each input and dividing by the respective input, resulting in $M{P}_k=(q/k{)}^{1-\rho }$ and $M{P}_l=(q/l{)}^{1-\rho }$.

These formulas are powerful as they provide insights into how changes in input levels affect output, which informs decisions on where to allocate resources efficiently in a production environment.

Returns to Scale

Returns to scale describes how a change in all inputs affects the level of output. Specifically, if we increase all inputs by a certain percentage, will the output increase by the same percentage, a higher percentage, or a lower percentage? These outcomes are known as constant, increasing, or decreasing returns to scale, respectively.

In our CES production function, because it's defined as having constant returns to scale (CRS), the output increases exactly by the same percentage as the inputs do. This property is essential when planning production strategies as it influences how the scale of production can be changed without altering efficiency. The relationship between the inputs and returns to scale can be expressed as $RTS=(k/l{)}^{1-\rho }$, which helps to understand how the ratio of inputs contributes to the overall productivity.

Elasticity of Substitution

One concept that often puzzles students is the elasticity of substitution, symbolized by $\sigma$. This elasticity measures how easily one input can be substituted for another while maintaining the same level of output. For example, can a firm easily replace labor with robots and keep making the same amount of product?

In the CES production function, $\sigma$ can be directly related to the parameter $\rho$ in the function, where $\sigma =1/(1-\rho )$. A higher $\sigma$ implies that it's easier to substitute between capital and labor, which can profoundly influence the choice of technology and labor use in production. It tells us about the flexibility in the production process, with implications for both firm strategy and labor market dynamics.

Output Elasticities

Finally, let's dive into output elasticities, which measure the responsiveness of output to a change in either capital or labor. Put another way, it shows how much output is expected to grow (or shrink) if we add more machines or workers.

In the CES production function, calculating the elasticity of output with respect to capital ($E_k$) and labor ($E_l$) involves taking the partial derivative of the function with respect to each input and scaling by the input-output ratio. Fascinatingly, the sum of these output elasticities equals 1 in a CRS production function, reinforcing the notion of proportional changes in output with input adjustments. This balance is critical for firms as they decide on the most efficient mix of inputs for their operations and facilitate the understanding of production adjustments in response to economic changes.