Question

Suppose the total cost function for a firm is given by \\[ T C=q w^{2 / 3} v^{1 / 3} \\] a. Use Shephard's lemma (footnote 8 ) 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

Answer: The constant output demand functions for labor (L) and capital (K) are L = (2/3)q * w^(-1/3)v^(1/3) and K = (1/3)q * w^(2/3)v^(-2/3).

Step-by-step solution

Calculate the partial derivatives with respect to wages (w) and rental rates (v)

We are given the total cost function: \\[ T C = q w^{2 / 3} v^{1 / 3} \\] We need to compute the partial derivatives with respect to wages (w) and rental rates (v). Partial derivative with respect to w: \\[ \frac{\partial TC}{\partial w} = q \cdot \frac{2}{3} w^{-1 / 3} v^{1 / 3} \\] Partial derivative with respect to v: \\[ \frac{\partial TC}{\partial v} = q \cdot \frac{1}{3} w^{2 / 3} v^{-2 / 3} \\]

Use Shephard's Lemma to find the input demand functions for labor (L) and capital (K)

According to Shephard's Lemma, the partial derivatives of the cost function with respect to input prices are equal to the input demand functions. Therefore, we can write the input demand functions for labor (L) and capital (K) as follows: Constant output demand function for labor (L): \\[ L = \frac{\partial TC}{\partial w} \\] \\[ L = q \cdot \frac{2}{3} w^{-1 / 3} v^{1 / 3} \\] Constant output demand function for capital (K): \\[ K = \frac{\partial TC}{\partial v} \\] \\[ K = q \cdot \frac{1}{3} w^{2 / 3} v^{-2 / 3} \\]

Calculate the underlying production function for output (q)

Now we have the input demand functions for labor (L) and capital (K). To find the underlying production function for output (q), we need to eliminate the labor and capital terms from the demand functions. First, rearrange the demand functions to solve for w and v: From the labor demand function, \\[ w = \left(\frac{3L}{2qv}\right)^{3/2} \\] From the capital demand function, \\[ v = \left(\frac{3K}{qw}\right)^{3/2} \\] Now, substitute the expression for w from the labor demand function into the capital demand function to eliminate w: \\[ v = \left(\frac{3K}{q\left(\frac{3L}{2qv}\right)^{3/2}}\right)^{3/2} \\] Simplify the expression to find the production function, which represents the relationship between the quantity of output (q) and the inputs of labor (L) and capital (K): \\[ q = \sqrt[3]{\frac{9K^2L^3}{4}} \\] This is the underlying production function for output (q) in terms of labor (L) and capital (K).

Key concepts

Total Cost Function

In economics, the total cost function plays a fundamental role in understanding how costs vary with input prices and output level. It reflects the total expense borne by a firm to produce a given level of output. The total cost function given in this exercise is \( TC = q w^{2/3} v^{1/3} \), where \( q \) is the level of output, \( w \) is the price of labor, and \( v \) is the rental rate of capital.
Under this function, costs are dependent on both \( w \) and \( v \), indicating that changes in wages or capital rent can significantly impact the firm's overall expenses. The exponents \( 2/3 \) and \( 1/3 \) show the elasticity of the cost concerning labor and capital prices, respectively.
This cost function allows us to derive the input demand functions through Shephard's Lemma, and if needed, we can further explore the associated technology by reconstructing the production function.

Input Demand Functions

The input demand functions specify the quantities of each input a firm demands to produce a certain level of output at given input prices. Using Shephard's Lemma, these functions are derived from the total cost function by taking partial derivatives with respect to the input prices.
Here, the input demand functions are obtained as follows:

• For labor (L), the demand function is derived from \( L = \frac{\partial TC}{\partial w} = q \cdot \frac{2}{3} w^{-1/3} v^{1/3} \).
• For capital (K), it is \( K = \frac{\partial TC}{\partial v} = q \cdot \frac{1}{3} w^{2/3} v^{-2/3} \).

These functions provide valuable insights into how much labor and capital a firm needs for any given change in wages or capital costs, holding output constant. Understanding these relationships helps in budgeting and resource allocation.

Production Function

A production function describes the relationship between input quantities and the resultant level of output. It is a critical concept for firms aiming to understand how effectively they can convert inputs into outputs.
In this problem, the production function is derived by solving the input demand functions for the output \( q \), and it ultimately takes the form:
\[ q = \sqrt[3]{\frac{9K^2L^3}{4}} \]
This production function suggests a specific elasticity of input substitution, meaning that the output increases by specific proportions regarding increases or decreases in inputs of labor (L) and capital (K).
By understanding this equation, firms can optimize input use to achieve desired output levels efficiently.

Partial Derivatives

Partial derivatives are used in multivariable calculus to analyze how a function changes with respect to one variable while keeping other variables constant. In economics, they are especially useful to determine how changes in input prices affect total production costs.
In this exercise, the total cost function's partial derivatives with respect to the prices of labor (\( w \)) and capital (\( v \)) are calculated as follows:

• With respect to \( w \): \( \frac{\partial TC}{\partial w} = q \cdot \frac{2}{3} w^{-1/3} v^{1/3} \).
• With respect to \( v \): \( \frac{\partial TC}{\partial v} = q \cdot \frac{1/3} w^{2/3} v^{-2/3} \).

This computation reveals how sensitive total cost is to changes in input prices, which in turn determines the input demand functions through Shephard's Lemma.
Proficiently applying partial derivatives allows economists and businesses to decipher cost structures and streamline decision-making regarding resource use.