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

a) Demand functions: \( l = \frac{2}{3} q w^{-1/3} v^{1/3} \), \( k = \frac{1}{3} q w^{2/3} v^{-2/3} \). b) Production function: \( q = l^{3/2} k^{3} \).

Step-by-step solution

Understanding Shephard's Lemma

Shephard's Lemma states that the derivative of the cost function with respect to input prices yields the input demand functions. In this context, the cost function is given, and our task is to find the demand functions for inputs \( l \) and \( k \) by differentiating the cost function with respect to their respective prices \( w \) and \( v \).

Differentiate Cost Function with respect to Price \(w\)

To find the demand function for input \( l \) (associated with price \( w \)), differentiate the cost function \( C = q w^{2/3} v^{1/3} \) with respect to \( w \). This gives: \[ \frac{\partial C}{\partial w} = \frac{2}{3} q w^{-1/3} v^{1/3}. \]

Differentiate Cost Function with respect to Price \(v\)

For input \( k \) (associated with price \( v \)), differentiate the cost function with respect to \( v \). The partial derivative is: \[ \frac{\partial C}{\partial v} = \frac{1}{3} q w^{2/3} v^{-2/3}. \]

Calculate the Production Function

Having found the input demand functions, use them to derive the production function. Since the cost function is derived from a production function of the form \( q = f(l, k) \), rearrange the demand functions from steps 2 and 3 to express \( q \) as a function of \( l \) and \( k \): - Substitute \( l = \frac{2}{3} q w^{-1/3} v^{1/3} \) in the Cobb-Douglas form \( q = l^{3/2} k^{3} \) and - \( k = \frac{1}{3} q w^{2/3} v^{-2/3} \).

Form Final Expression of Production Function

The production function is derived by relating the input demand functions and the structure of the cost function. By solving the equations:- Express \( q = \left(\frac{2}{3} q w^{-1/3} v^{1/3}\right)^{3/2} \left(\frac{1}{3} q w^{2/3} v^{-2/3}\right)^{1/3} \),- Simplifying gives the production function \( q = l^{3/2} k^{3} \).

Key concepts

Cost Function

A cost function represents the total cost incurred by a firm to produce a certain level of output. It captures the relationship between the firm’s total cost and the level of production, as well as the prices of inputs used. In our given problem, the cost function is given as:
\[ C = q w^{2/3} v^{1/3} \]
Here, \( C \) is the total cost, \( q \) represents output level, \( w \) is the price of input \( l \), and \( v \) is the price of input \( k \). This function indicates how costs change as the prices of inputs or the level of output alters.

• Costs vary directly with changes in input prices.
• The exponents \( \frac{2}{3} \) and \( \frac{1}{3} \) show how costs are sensitive to \( w \) and \( v \).

Shephard's Lemma

Shephard's Lemma is a useful tool in microeconomic theory that helps to derive input demand functions from a given cost function. This lemma states that the partial derivative of the cost function with respect to an input price gives the input demand function for that particular input.

• It connects cost functions to input demand, bridging how costs respond to price changes.
• To apply this lemma, differentiate the cost function \( C = q w^{2/3} v^{1/3} \) with respect to each input price.

For example, differentiating with respect to \( w \), the price of input \( l \), we get:
\[ \frac{\partial C}{\partial w} = \frac{2}{3} q w^{-1/3} v^{1/3} \]
This represents how demand for \( l \) changes with its price.

Input Demand Functions

Input demand functions are derived from the cost function using Shephard's Lemma. They indicate the quantity of each input that a firm demands based on the prices of inputs and the level of output required.

• They allow firms to optimize input use according to price changes.
• The demand functions inform on minimum cost combinations to achieve a specified output.

From the differentiation performed using Shephard's Lemma, we obtain:

• Demand for input \( l \):
  \[ l = \frac{2}{3} q w^{-1/3} v^{1/3} \]
• Demand for input \( k \):
  \[ k = \frac{1}{3} q w^{2/3} v^{-2/3} \]

Production Function

A production function expresses the relationship between inputs and the output they produce. In this exercise, having determined the input demand functions, we can express the production function by rearranging these functions.

• The form of the production function here is derived from the Cobb-Douglas structure.
• It shows how combining different amounts of inputs \( l \) and \( k \) results in varying levels of output \( q \).

Using the expressions derived from the input demand functions, substitute back to derive:
\[ q = l^{3/2} k^{3} \]
This suggests that the firm’s output is a function of a specific combination of these inputs.

Partial Derivatives

Partial derivatives play a crucial role in the analysis of functions of multiple variables, like cost or production functions. They measure the rate at which one variable changes as another variable changes, while keeping others constant.

• Here, they help find how costs respond to changes in input prices.
• They provide insights into marginal costs related to each input.

For instance, taking the partial derivative of our cost function
\[ C = q w^{2/3} v^{1/3} \]
with respect to \( w \) results in:
\[ \frac{\partial C}{\partial w} = \frac{2}{3} q w^{-1/3} v^{1/3} \]
This illustrates the sensitivity of the cost concerning changes in \( w \), aiding in decisions regarding input usage optimization.