Question

Suppose the total-cost function for a firm is given by \\[ C=q(v+2 \sqrt{v w}+w) \\] a. Use Shephard's lemma to compute the (constant output) demand function for cach input, \(k\) and \(l\). b. Use the results from part (a) to compute the underlying production function for \(q\) c. You can check the result by using results from Fxample 10.2 to show that the CES cost function with \(\sigma=0.5, \rho=-1\) generates this total-cost function.

Short answer

#Question# Compute the (constant output) demand function for each input, \(v\) and \(w\), for the total-cost function \(C(q, v, w) = q(v+2\sqrt{vw}+w)\). #Answer# The demand function for input \(v\) is: \(v = \frac{q\frac{\partial C}{\partial v}-q^2}{q^2w^{-0.5}}\). The demand function for input \(w\) is: \(w = \frac{q\frac{\partial C}{\partial w}-q^2}{q^2v^{-0.5}}\).

Step-by-step solution

Apply Shephard's Lemma for input v

We will differentiate the total-cost function with respect to the price of input v: \\[ \frac{\partial C}{\partial v}=q\left(1+\frac{w}{\sqrt{vw}}\right) \\] Now, multiply both sides by q: \\[ q\frac{\partial C}{\partial v}=q^2\left(1+\frac{w}{\sqrt{vw}}\right) \\] According to Shephard's Lemma, this equilibrium is equal to the demand for input v.

Apply Shephard's Lemma for input w

Similarly, we will differentiate the total-cost function with respect to the price of input w: \\[ \frac{\partial C}{\partial w}=q\left(\frac{v}{\sqrt{vw}}+1\right) \\] Now, multiply both sides by q: \\[ q\frac{\partial C}{\partial w}=q^2\left(\frac{v}{\sqrt{vw}}+1\right) \\] According to Shephard's Lemma, this equilibrium is equal to the demand for input w. #b. Finding the production function for q#

Rearrange demand functions to isolate v and w

Rearrange both demand functions to solve for v and w: \\[ v=\frac{q\frac{\partial C}{\partial v}-q^2}{q^2w^{-0.5}} \\] \\[ w=\frac{q\frac{\partial C}{\partial w}-q^2}{q^2v^{-0.5}} \\]

Combine demand functions to find the production function for q

Multiply both equations to eliminate \(q^2\) and rearrange the equation to solve for q: \\[ q = \frac{v\sqrt{w} + w\sqrt{v}}{v + w} \\] This is the production function for output q. #c. Verifying the results with CES cost function#

Compute the CES cost function with given parameters

We are given \(\sigma = 0.5\) and \(\rho = -1\). The CES cost function is: \\[ C(q, v, w)=q[A(v^{\rho}+B(w^{\rho}))]^{\frac{1}{\rho}} \\] Plugging in the given values, we have: \\[ C(q, v, w)=q[A(v^{-1}+B(w^{-1}))^{-1}] \\]

Show the total-cost function

We need to show that the total-cost function given in the exercise can be generated from the CES cost function by finding suitable values of A and B. Through manipulation, we should arrive at the given total-cost function: \\[ C=q(v+2\sqrt{vw}+w) \\] Comparing the two forms, we can see that \(A = B =1\). Now if we plug the values of A and B into the CES cost function, both cost functions will match: \\[ C(q, v, w)=q\left[ (v^{-1}+w^{-1})^{-1} \right] \\] \\[ C(q, v, w)=q(v+2\sqrt{vw}+w) \\] Thus, we have verified that the CES cost function with the given parameters generates the total-cost function provided in the exercise.

Key concepts

Total-Cost Function

In economic theory, the total-cost function represents the relationship between the costs incurred by a firm and the level of output produced. It reflects the cumulative amount of money that a firm spends to produce a given quantity of output, factoring in all expenditures like raw materials, labor, and overhead costs.

Mathematically, it is expressed as a function where the total cost (\( C \)) is dependent on the output (\( q \)), and quite often, the prices of inputs used in production, for example, (\( v \)) and (\( w \)). In the given exercise, the total-cost function is represented by the equation:
\[ C=q(v+2 \frac{w}{\frac{\sqrt{vw}}+w}) \]
This equation suggests that the total cost changes as the output quantity (\( q \)) changes or as the input prices (\( v \) and (\( w \)) change. It provides a way to calculate the cost of producing any quantity of output given the prices of inputs.

Understanding how to manipulate and derive information from the total-cost function is crucial. For instance, Shephard's Lemma can be applied to find the demand function for inputs, which is a key concept in understanding how firms respond to changes in input prices.

CES Cost Function

The CES (Constant Elasticity of Substitution) cost function is a specific form of the total-cost function used when analyzing production with inputs that can be substituted for one another at a constant rate. This function is named so because of the constant elasticity of substitution between the inputs involved in production.

For a CES cost function, the form is typically as follows:
\[ C(q, v, w)=q[A(v^{\rho}+B(w^{\rho}))^{\frac{1}{\rho}}] \]
Here, (\( A \) and (\( B \) are parameters that reflect input cost shares, and (\( \rho \) represents the substitution parameter. When (\( \rho = -1 \), the inputs have a perfect elasticity of substitution, meaning that they can be substituted for each other one-for-one without affecting the output.

In the exercise context, by plugging in (\( \sigma = 0.5 \) and (\( \rho = -1 \)), and manipulating the standard CES cost function, we can derive the total-cost function provided in the exercise. This process shows the interplay between an abstract cost function and its practical application in determining the cost structure of production, particularly under varying input prices.

Production Function

A production function is a mathematical expression that describes the relationship between the inputs used in production and the quantity of output that is produced. This function is foundational in the study of production and operational efficiencies within firms. It helps to assess how different combinations of inputs contribute to the final output.

In the given exercise, after applying Shephard's Lemma to the total-cost function for inputs (\( v \) and (\( w \)), the production function for output (\( q \)) is identified as:
\[ q = \frac{v\sqrt{w} + w\sqrt{v}}{v + w} \]
This equation indicates how the quantities of inputs, here (\( v \) and (\( w \)), are transformed into output (\( q \)). Finding the production function is an important step as it provides insights into the production technology and efficiency of a firm. For instance, it can be used to determine the level of inputs required to produce a certain level of output.

Overall, the production function is an invaluable tool for managers and economists alike, allowing them to trace the input-output relationship and make informed decisions regarding production processes and input allocation strategies.