Question

Suppose the total cost function for a firm is given by \\[ T C-q w^{23} v^{13} \\] 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

Question: Using Shephard's lemma and given the total cost function \(TC = q w^{23} v^{13}\), find the constant output demand functions for inputs \(L\) and \(K\), and calculate the underlying production function for output \(q\). Answer: The constant output demand functions for inputs \(L\) and \(K\) are: \(L(w,v,q) = 23w^{22} v^{13}\) \(K(w,v,q) = w^{23} 13v^{12}\) The underlying production function for output \(q\) is: \(q(L,K) = \frac{13L}{23} + \frac{23K}{13}\)

Step-by-step solution

Take Partial Derivatives of Total Cost Function

First, we have to find the partial derivatives of the total cost function with respect to \(w\) and \(v\). Given the total cost function, \\[ TC = q w^{23} v^{13} \\], let's take the partial derivatives: For the partial derivative with respect to \(w\): \\[ \frac{\partial TC}{\partial w} = q \cdot 23w^{22} v^{13} \\] For the partial derivative with respect to \(v\): \\[ \frac{\partial TC}{\partial v} = q \cdot w^{23} 13v^{12} \\]

Compute Constant Output Demand Functions for Inputs \(L\) and \(K\)

According to Shephard's lemma, the partial derivatives with respect to input prices represent the constant output demand functions for the corresponding inputs. Therefore, the constant output demand functions for inputs \(L\) and \(K\) are given by: For input \(L\) (input prices correspond to \(w\)): \\[ L(w,v,q) = 23w^{22} v^{13} \\] For input \(K\) (input prices correspond to \(v\)): \\[ K(w,v,q) = w^{23} 13v^{12} \\]

Calculate the Underlying Production Function for \(q\)

Now that we have the constant output demand functions for inputs \(L\) and \(K\), we can compute the underlying production function for \(q\). To do this, set \(L(w,v,q)\) and \(K(w,v,q)\) equal to \(L\) and \(K\), respectively: For input \(L\): \\[ L = 23w^{22} v^{13} \\] For input \(K\): \\[ K = w^{23} 13v^{12} \\] Now, eliminate \(w\) and \(v\) from the above equations to get the production function. Divide the first equation by the second equation to get: \\[ \frac{L}{K} = \frac{23}{13} \] Now, solve for \(q\): \\[ q = \frac{13L}{23} + \frac{23K}{13} \\] The underlying production function for output \(q\) is given by: \\[ q(L,K) = \frac{13L}{23} + \frac{23K}{13} \\]

Key concepts

Total Cost Function

Understanding the total cost function is crucial for economics students as it represents the expenses a firm incurs from all sources to produce a given level of output. It encapsulates various costs like fixed costs, variable costs, and at times, semi-variable costs. Specifically, the total cost function in the form of \( TC = q w^{23} v^{13} \) indicates an algebraic relationship between the total cost, the quantity produced \( q \), and the prices of inputs denoted by \( w \) and \( v \).

Shephard's lemma plays a significant role here. It tells us that partial derivatives of the total cost function with respect to input prices yield the demand functions for those inputs, assuming output level stays constant. These derivatives reflect how sensitive the quantity of an input is with respect to its price, holding the output fixed. By understanding this relationship, businesses can optimize their input use to minimize costs or adjust to price changes in the input markets.

Constant Output Demand Functions

The constant output demand functions are a direct application of Shephard's lemma. They are derived from the total cost function assuming the output level remains unchanged. In the context of our exercise, the demand functions for inputs \( L \) and \( K \) denote the quantity of labor and capital required to produce a certain quantity of output at constant prices.

More formally, these functions are expressed as \( L(w,v,q) \) and \( K(w,v,q) \), representing the amount of labor and capital demanded for any combination of their respective prices \( w \) and \( v \), and the level of output \( q \). These functions allow firms to make predictions about their required inputs and help economists understand the firm's production behavior under constant output conditions. They are a fundamental concept in understanding how production inputs are allocated in the short run, where output levels are often inflexible.

Production Function

The production function represents the relationship between the quantities of inputs used and the level of output achieved. It is expressed in the form \( q(L,K) \), signifying that the output \( q \) is a function of labor \( L \) and capital \( K \). In our exercise, we have derived the production function from the constant output demand functions and Shephard's lemma. By equating the constant demand functions for labor and capital to \( L \) and \( K \) and then eliminating \( w \) and \( v \), we obtained the relationship between labor and capital which can be used to calculate output.

The production function is central to the study of producer's behavior because it embodies the technological constraints facing the firm. It indicates the highest level of output attainable with a given set of inputs, and it's essential for understanding economies of scale, marginal rates of technical substitution, and the law of diminishing returns - all key concepts in microeconomic production theory.