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: Using 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: The demand functions for inputs \(l\) and \(k\) are: $$l(q,w,v)=q\frac{2}{3}w^{-1/3}v^{1/3}$$ $$k(q,w,v)=q\frac{1}{3}w^{2/3}v^{-2/3}$$ The underlying production function for \(q\) is: $$q=\frac{l}{2k}$$

Step-by-step solution

Calculate partial derivative of the cost function with respect to \(w\)

In order to find the partial derivative of the cost function with respect to \(w\), we perform the following operation: $$\frac{\partial C}{\partial w}$$ Applying the power rule for partial derivatives: $$\frac{\partial C}{\partial w}=q\frac{2}{3}w^{-1/3}v^{1/3}$$

Calculate partial derivative of the cost function with respect to \(v\)

In order to find the partial derivative of the cost function with respect to \(v\), we perform the following operation: $$\frac{\partial C}{\partial v}$$ Applying the power rule for partial derivatives: $$\frac{\partial C}{\partial v}=q\frac{1}{3}w^{2/3}v^{-2/3}$$

Find the demand functions for inputs \(l\) and \(k\)

According to Shephard's lemma, we have the constant-output demand functions for inputs \(l\) and \(k\) as: $$l(q,w,v)=\frac{\partial C}{\partial w}=q\frac{2}{3}w^{-1/3}v^{1/3}$$ $$k(q,w,v)=\frac{\partial C}{\partial v}=q\frac{1}{3}w^{2/3}v^{-2/3}$$

Finding the underlying production function for \(q\)

We can now use the demand functions for inputs \(l\) and \(k\) to derive the underlying production function for \(q\). First, we need to express \(l\) and \(k\) in terms of \(q\). From the demand functions, we can do this by isolating \(q\): $$q=\frac{3}{2}l(w^{1/3}v^{-1/3})$$ $$q=\frac{3}{1}k(w^{-2/3}v^{2/3})$$ In order to find the production function, we need to eliminate one of the variables (\(w\) or \(v\)). Let's eliminate variable \(w\). To do this, divide the first equation by the second equation: $$\frac{\frac{3}{2}l(w^{1/3}v^{-1/3})}{\frac{3}{1}k(w^{-2/3}v^{2/3})} = \frac{l(w^{1/3}v^{-1/3})}{2k(w^{-2/3}v^{2/3})}$$ The right-hand side simplifies to: $$\frac{lw^{3/3}}{2kv^{3/3}} = \frac{l}{2k}$$ Therefore, the production function is: $$q=\frac{l}{2k}$$

Key concepts

Partial Derivative

In economics and mathematics, the concept of the partial derivative is crucial when dealing with functions that depend on more than one variable. These different variables each can affect the outcome of the function.

Imagine you're cooking and your recipe (the function) depends on various ingredients (variables) like salt, pepper, and garlic. The partial derivative tells you how much changing the amount of just one ingredient, say salt, tweaks the flavor of your dish, while keeping the rest constant.

The process involves differentiating the function with respect to one variable at a time, holding others fixed, much like adjusting one seasoning in your recipe without altering others. You might see an expression like \( \frac{\partial C}{\partial w} \), which seeks to calculate how changes in the wage rate (\(w\)) affect the firm's total cost (\(C\)). The value obtained through this derivative provides insight into the rate at which costs are changing with respect to wages.

Demand Functions for Inputs

Moving to the economics of production, the demand functions for inputs are the relationships that show how much of each input a firm requires to produce a certain level of output at given input prices.

These functions often stem from the firm's desire to minimize costs while producing a certain output level. They can be visualized as recipes for producing a certain dish. Just as a recipe dictates the amount of each ingredient for a dish, demand functions show the quantities of labor, capital, and other inputs needed to produce goods efficiently.

In our exercise, Shephard's Lemma assists in deriving these functions directly from the cost function. By taking the partial derivatives of the cost function with respect to input prices, we obtain the firm's input demand functions, symbolizing how many units of each input are ordered given their prices and the firm's production scale.

Production Function

The production function, a cornerstone in the study of economics, represents the technological relationship between the quantities of inputs used and the quantity of output produced.

Think of it like a blueprint that describes how to convert raw materials into finished goods using a set of instructions. For example, the production function might tell you how many loaves of bread (output) you can bake from flour, yeast, and water (inputs).

In an exercise context such as ours, the production function could be a bit like reverse-engineering the recipe from the finished dishes and the quantities used. By analyzing the demand functions and input costs, we can determine the original 'recipe' for producing a certain quantity of goods. In the given solution, by eliminating one variable, we simplify the input demand functions to articulate a production function, expressing the output in terms of input quantities alone.