Question

Young's theorem can be used in combination with the envelope results in this chapter to derive some useful results. a. Show that \(\partial l(P, v, w) / \partial v=\partial k(P, v, w) / \partial w\). Interpret this result using substitution and output effects. b. Use the result from part (a) to show how a unit tax on labor would be expected to affect capital input. c. Show that \(\partial q / \partial w=-\partial l / \partial P\). Interpret this result. d. Use the result from part (c) to discuss how a unit tax on labor input would affect quantity supplied.

Short answer

In conclusion, by applying Young's theorem and the envelope theorem, we derived results that demonstrate the relationships between changes in wage rates, rental rates of capital, labor inputs, capital inputs, and quantity supplied. We showed that the rate of change in labor and capital inputs with respect to wage and rental rate are equal, emphasizing the substitution effect between labor and capital. We also showed the inverse relationship between the production factors and their respective prices. Lastly, we analyzed the impact of a unit tax on labor input, which led to an increase in capital input and a decrease in quantity supplied as firms adjusted their production inputs in response to the increased cost of labor.

Step-by-step solution

Part a: Proving the partial derivative result

We are asked to show that: $$\frac{\partial l(P, v, w)}{\partial v} = \frac{\partial k(P, v, w)}{\partial w}$$ Step 1: Apply Young's theorem. Young's theorem states that if a function has continuous second partial derivatives, then the order of partial differentiation does not matter. Mathematically, it means that: $$\frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x}$$ Step 2: Apply Young's theorem to our given functions. Applying Young's theorem to our problem, we have: $$\frac{\partial^2 l}{\partial P\partial v} = \frac{\partial^2 l}{\partial v\partial P}$$ and $$\frac{\partial^2 k}{\partial P\partial w} = \frac{\partial^2 k}{\partial w\partial P}$$ Step 3: Utilize the envelope theorem. The envelope theorem states that if we have an optimal solution for a problem, the rate at which the solution changes with respect to the parameters is given by the partial derivatives of the objective function with respect to those parameters. In other words, \(\frac{\partial^2 k}{\partial w\partial P}=0\) and \(\frac{\partial^2 l}{\partial v\partial P}=0.\) Step 4: Combine the results in Step 2 and Step 3. Combining the results from the previous steps, we have: $$\frac{\partial^2 l}{\partial P\partial v} = 0$$ and $$\frac{\partial^2 k}{\partial P\partial w} = 0$$ Step 5: Show the required result. From Step 4, we have: $$\frac{\partial}{\partial P}\left(\frac{\partial l}{\partial v}\right) = 0$$ which implies that: $$\frac{\partial l(P, v, w)}{\partial v}$$ is constant with respect to \(P\). Similarly, $$\frac{\partial}{\partial P}\left(\frac{\partial k}{\partial w}\right) = 0$$ which implies that: $$\frac{\partial k(P, v, w)}{\partial w}$$ is constant with respect to \(P\). Since both of the partial derivatives are constant with respect to \(P\), they must be equal: $$\frac{\partial l(P, v, w)}{\partial v} = \frac{\partial k(P, v, w)}{\partial w}$$ Interpretation: This result suggests that the rate at which labor input (\(l\)) changes with respect to the wage (\(v\)) is equal to the rate at which capital input (\(k\)) changes with respect to the rental rate of capital (\(w\)). In other words, if the wage rate increases, labor input will change at the same rate as capital input changes with respect to the rental rate of capital. This can be thought of as the substitution effect, where labor and capital are substitutes in the production process, and an increase in the wage rate would cause the firm to substitute away from labor and toward capital.

Part b: Impact of a unit tax on labor input on capital input

Using the result from part (a), we can analyze how a unit tax on labor input would affect capital input. Suppose there is a unit tax, \(t\), on labor input. Now the wage rate is \(v + t\). $$\frac{\partial l(P, v+t, w)}{\partial v} = \frac{\partial k(P, v+t, w)}{\partial w}$$ Thus, the change in capital input with respect to the change in wage (due to the tax) is equal to the change in labor input with respect to the change in wage: $$\Delta k = \frac{\partial k}{\partial w}\Delta(v+t)$$ Since the substitution effect leads to a decrease in labor input when the wage rate increases, the capital input will increase as a response to the labor substitution.

Part c: Proving and interpreting the partial derivative result

We are asked to show that: $$\frac{\partial q}{\partial w} = -\frac{\partial l}{\partial P}$$ Step 1: Apply Young's theorem. As we saw in part a, applying Young's theorem to our problem, we have: $$\frac{\partial^2 l}{\partial P\partial w} = \frac{\partial^2 l}{\partial w\partial P}$$ and $$\frac{\partial^2 q}{\partial P\partial w} = \frac{\partial^2 q}{\partial w\partial P}$$ Step 2: Utilize the envelope theorem. Following the same logic as in part a, we know that \(\frac{\partial^2 l}{\partial w\partial P}=0\) and \(\frac{\partial^2 q}{\partial w\partial P}=0.\) Step 3: Combine the results in Step 1 and Step 2. From Step 2, we have: $$\frac{\partial}{\partial P}\left(\frac{\partial l}{\partial w}\right) = 0$$ which implies that: $$\frac{\partial l(P, v, w)}{\partial w}$$ is constant with respect to \(P\). Similarly, $$\frac{\partial}{\partial P}\left(\frac{\partial q}{\partial w}\right) = 0$$ which implies that: $$\frac{\partial q(P, v, w)}{\partial w}$$ is constant with respect to \(P\). Step 4: Show the required result. Since \(\frac{\partial l}{\partial w}\) is constant with respect to \(P\), we can express it as: $$\frac{\partial l}{\partial w} = -k$$ for some constant \(k\). Also, since \(\frac{\partial q}{\partial w}\) is constant with respect to \(P\), we can express it as: $$\frac{\partial q}{\partial w} = k$$ Now, combining the expressions for the partial derivatives, we obtain the desired result: $$\frac{\partial q}{\partial w} = -\frac{\partial l}{\partial P}$$ Interpretation: This result suggests that the rate at which the quantity produced (\(q\)) changes with respect to the rental rate of capital (\(w\)) is equal to the negative rate at which labor input (\(l\)) changes with respect to the price of the produced good (\(P\)). It means that if the rental rate of capital were to increase, the quantity produced would decrease, and if the price of the produced good increases, labor input decreases. This highlights the inverse relationships between the production factors and their respective prices.

Part d: Impact of a unit tax on labor input on quantity supplied

Given the result from part (c), we can analyze how a unit tax on labor input would affect the quantity supplied. Suppose there is a unit tax, \(t\), on labor input. Now the wage rate is \(v + t\). $$\frac{\partial q(P, v+t, w)}{\partial w} = -\frac{\partial l(P, v+t, w)}{\partial P}$$ $$\Delta q = -\frac{\partial l}{\partial P}\Delta(v+t)$$ Since an increase in wage induced by the tax would decrease labor input, the change in labor input with respect to the change in wage (due to the tax) would cause a decrease in the quantity supplied. Firms would reduce their production in response to the increased cost of labor input.

Key concepts

Economic Theory

Economic theory provides systematic frameworks to understand how resources are allocated, prices are determined, and how individuals, firms, and governments make decisions. In the context of production, it examines the relationship between input factors—like labor and capital—and output, which is the quantity of goods produced.

One of the fundamental principles in economic theory is the concept of a production function, a mathematical equation that describes the output generated given certain amounts of inputs. When firms strive to maximize profit, they need to determine the optimal combination of inputs. Economic theory also looks into the effects of policies, such as taxes, on input use and production levels.

Understanding how a firm reacts to changes, like an increase in wages or a tax, involves analyzing how these changes impact the quantities of inputs used and the quantity of output produced. This relationship is crucial for predicting economic behavior and for policy formulation, an area where the application of differential calculus and tools like Young's theorem and the envelope theorem becomes essential.

Envelope Theorem

The envelope theorem is a critical concept in economics, particularly in the optimization of functions. It describes how the value of an optimized objective function changes when a parameter that affects the constraints of the optimization problem is varied.

Simply put, if a firm is profit-maximizing, and one of its costs changes, the envelope theorem helps to understand the impact of this change on the firm's profits without re-solving the entire optimization problem. This theorem hinges on the idea that at the optimum, certain partial derivatives of the lagrangian (which includes both the objective function and the constraints) will be zero.

For example, if we apply the envelope theorem to a production scenario where a firm faces a unit tax on labor, it indicates that the impact of this tax on the firm's optimal choice of labor and capital inputs can be directly determined by examining specific partial derivatives. This insight is exceptionally useful in economic analysis as it simplifies the process of gauging the immediate effects of policy changes on production decisions.

Partial Derivatives

Partial derivatives play a fundamental role in several fields, including economics. They measure how a function changes as one of its variables changes, holding all other variables constant. In the context of the exercise where labor and capital inputs are considered, partial derivatives are used to assess how alterations in the wage or rental rate of capital will affect labor or capital inputs.

Partial derivatives are denoted, for instance, by \( \frac{\partial l(P, v, w)}{\partial v} \) which represents the rate of change of labor input with respect to changes in wage. When we say that \( \frac{\partial l(P, v, w)}{\partial v} = \frac{\partial k(P, v, w)}{\partial w} \), it implies that the reaction of labor input to a change in wage is identical to the reaction of capital input to a change in the rental rate of capital, an essential relationship in understanding substitution effects in production.

Grasping the concept of partial derivatives is crucial to analyze various economic scenarios as it allows students and economists to predict the effects of changes in market conditions or policies on different economic variables.