Question

Suppose the total-cost function for a firm is given by \\[ C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma to compute the (constant output) demand functions for inputs \(l\) and \(k\). b. Use your results from part (a) to calculate the underlying production function for \(q\)

Short answer

Question: Based on Shephard's lemma, find the demand functions for inputs \(l\) and \(k\), and the underlying production function for \(q\) given the total cost function: \(C = qw^{2/3}v^{1/3}\). Answer: Using Shephard's lemma, the demand functions for inputs \(l\) and \(k\) were found to be \(l = \frac{2}{3} q w^{-1/3} v^{1/3}\) and \(k = \frac{1}{3} q w^{2/3} v^{-2/3}\), respectively. The underlying production function for \(q\) was found to be \(q = \sqrt[3]{9lk}\).

Step-by-step solution

Computing the partial derivatives of the total cost function

Compute the partial derivatives of the total cost function with respect to input prices \(w\) and \(v\): \(\frac{\partial C}{\partial w} = \frac{2}{3} q w^{-1/3} v^{1/3}\) \(\frac{\partial C}{\partial v} = \frac{1}{3} q w^{2/3} v^{-2/3} \)

Applying Shephard's lemma

According to Shephard's lemma, these derivatives will give us the demand functions for inputs \(l\) and \(k\). Therefore: Demand function for input \(l\) can be obtained from the derivative \(\frac{\partial C}{\partial w}\): \(l = \frac{2}{3} q w^{-1/3} v^{1/3}\) Demand function for input \(k\) can be obtained from the derivative \(\frac{\partial C}{\partial v}\): \(k = \frac{1}{3} q w^{2/3} v^{-2/3}\) #b. Finding the underlying production function for \(q\)#

Expressing the output \(q\) in terms of the demand functions for inputs \(l\) and \(k\)

. Rewrite both demand functions for inputs \(l\) and \(k\) in terms of \(q\): \(q = \frac{3}{2}l w^{1/3} v^{-1/3} \) \(q = 3k w^{-2/3} v^{2/3}\)

Eliminating input prices \(w\) and \(v\) from the equations to find the production function for \(q\)

. We need to eliminate input prices \(w\) and \(v\) from the equations. To do this, square the first equation, and multiply the second equation by \(3\): \(\left( \frac{3}{2}l w^{1/3} v^{-1/3} \right)^2 = \left( 3k w^{-2/3} v^{2/3} \right) \left( 3 \right)\) Solve for \(q\): \(q = \sqrt[3]{9lk}\) So, the underlying production function for \(q\) is: \\[ q = \sqrt[3]{9lk} \\]

Key concepts

Production Function

In economics, a production function represents the relationship between input resources, commonly denoted as factors of production, and the output of goods or services that a firm can produce. These inputs may include labor, capital, land, and raw materials. Mathematically, a production function can be expressed as \( q = f(l, k) \), where \( q \) is the quantity of output, \( l \) is the quantity of labor, and \( k \) is the quantity of capital.

The form and complexity of the production function can vary depending on the nature of the production process. For instance, in the textbook exercise where we are given the production function \( q = \sqrt[3]{9lk} \), the output, represented by \( q \), is a function of labor \( l \) and capital \( k \), raised to the power that represents the increasing, constant, or decreasing returns to scale. Such a function might suggest that there are increasing returns to scale if the output increases by a greater proportion than the increase in inputs.

Understanding the production function allows economists and businesses to determine the most efficient combination of inputs for producing a desired level of output, as well as to analyze the effects of scaling production up or down. It is critical in making decisions about production, considering costs, and optimizing efficiency.

Total-Cost Function

A total-cost function illustrates how total cost of production depends on the quantity of outputs produced, given the input prices. In economic models, this function can be depicted in the form \( C = g(w, v, q) \), where \( C \) is the total cost, \( w \) and \( v \) are the prices of the inputs, such as wages for labor and rent for capital, and \( q \) stands for the quantity of output.

In the given exercise, the total-cost function is represented by \( C = q w^{2/3} v^{1/3} \). This function hints at the particular relationship between the cost and the inputs, showing how these inputs are transformed into outputs. By finding the partial derivatives of the total cost function with respect to input prices, we have the foundation for deriving demand functions for inputs using Shephard's lemma. This lemma stipulates that the derivative of the cost function with respect to input prices gives the cost-minimizing input demand functions, which are critical for understanding how input demands change with prices.

Simply put, the total-cost function is used to analyze how total production cost changes as the firm's level of production changes, considering the costs of all inputs required for production. This function is central to the firm's decision on the quantity of production that minimizes costs and maximizes profits.

Demand Functions for Inputs

The demand functions for inputs signify the relationship between the quantities of inputs demanded by a firm and the prices of those inputs. Each input has an associated demand function that indicates the quantity of that input the firm needs to minimize costs for a given level of output and input prices. These functions are derived from the firm's total-cost function using the principles of Shephard's lemma.

In the exercise presented, we can ascertain the demand functions for inputs \( l \) (labor) and \( k \) (capital) through the calculus of Shephard's lemma. As we've calculated, the demand function for input labor \( l \) is: \( l = \frac{2}{3} q w^{-1/3} v^{1/3} \), and for capital \( k \), it is: \( k = \frac{1}{3} q w^{2/3} v^{-2/3} \). These functions are pivotal for understanding how the firm reacts to changes in the prices of inputs.

For instance, if the wage rate \( w \) increases, we might expect a decrease in the quantity of labor demanded, as indicated by the negative exponent on \( w \) in the demand function for \( l \). Similarly, if the rental rate of capital \( v \) decreases, the demand for capital may increase, as seen by the negative exponent on \( v \) in the demand function for \( k \). Ultimately, these demand functions help firms optimize input utilization, economize on costs, and make informed decisions about production in response to market price changes.