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 each input, \(k\) and \(i\) 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 Example 10.2 to show that the CES cost function with \(\sigma=0.5, \rho=-1\) generates this total-cost function.

Short answer

The input demands are \( v^* = q(1 + \frac{\sqrt{w}}{\sqrt{v}}) \) and \( w^* = q(1 + \frac{\sqrt{v}}{\sqrt{w}}) \) with the production function being \( q = v + w \).

Step-by-step solution

Apply Shephard's Lemma

Shephard's lemma states that the derivative of the total cost function with respect to the price of an input gives the conditional factor demand for that input. For input \( v \), the partial derivative is \( \frac{\partial C}{\partial v} = q(1 + \frac{\sqrt{w}}{\sqrt{v}}) \). For input \( w \), the partial derivative is \( \frac{\partial C}{\partial w} = q(1 + \frac{\sqrt{v}}{\sqrt{w}}) \). These equations give the demand functions for inputs \( v \) and \( w \).

Derive the Production Function from Input Demands

We have the input demand functions from Step 1. Assume the firm uses inputs optimally, and production is given by a function \( q = f(v, w) \). The derived input demand functions suggest a specific formulation of the production function. By rearranging, one might infer that the production involves a relation like \( q^2 = v + 2\sqrt{vw} + w \), leading to \( q = v + w \) (under choosing equal inputs).

Verify with CES Cost Function

From Example 10.2, the CES cost function with elasticity of substitution \( \sigma=0.5 \) and \( \rho=-1 \) is mathematically identical to \( C(q) = q(v + 2 \sqrt{vw} + w) \). This verifies the compatibility and consistency of the original cost function with its derived forms under the given parameters.

Key concepts

Shephard's Lemma

Shephard's Lemma is a key principle in microeconomics which links cost functions to input demand functions. It can be particularly helpful when analyzing how firms choose their mix of inputs to minimize costs while producing a given level of output.

• **Partial Derivatives**: The lemma states that if you take the derivative of a cost function concerning an input's price, you get the conditional demand function for that input.
• **Applications**: This helps in understanding how changes in input costs influence the amounts of each input a firm demands.

In the given problem, the cost function is differentiated concerning inputs \( v \) and \( w \). This yields the functions \( \frac{\partial C}{\partial v} = q(1 + \frac{\sqrt{w}}{\sqrt{v}}) \) and \( \frac{\partial C}{\partial w} = q(1 + \frac{\sqrt{v}}{\sqrt{w}}) \).
These equations represent the demand functions for the respective inputs, emphasizing Shephard's lemma's application. They reflect how the quantity demanded for inputs \( v \) and \( w \) depend on their respective prices, given the constant output \( q \).

This is crucial for firms aiming to adjust their input mix in response to price changes, ensuring efficient production.

Production Function

The production function is an equation that expresses the relationship between inputs used by a firm and the resulting output. It is foundational to understanding how firms optimize their output given constraints like cost and input availability.

• **Understanding Inputs and Outputs**: The production function shows how various combinations of inputs like labor and capital can produce output.
• **Function Form**: Companies often assume specific forms depending on their technological capabilities.

In the problem exercise, after computing the input demand functions from Shephard's lemma, the production function \( q = f(v, w) \) emerges.
The transformation suggests the relationship \( q^2 = v + 2\sqrt{vw} + w \), simplifying to \( q = v + w \) under certain symmetrical conditions.
This derivation shows how understanding the input costs and their optimal utilization leads to insights into the overall production process of a firm, guiding them towards production efficiency.

CES Cost Function

The CES, or Constant Elasticity of Substitution, cost function, is a specific form of cost function that helps to express how easily inputs in production can be substituted for one another. It provides insights into the flexibility of production processes and helps firms manage input combinations effectively.

• **Elasticity**: This parameter measures the ease of substituting one input for another while maintaining the same level of output.
• **Parameters in CES**: The function includes parameters like \( \sigma \) (the elasticity of substitution) and \( \rho \) which influences the shape of the cost function.

In the problem context, the CES cost function, with \( \sigma = 0.5 \) and \( \rho = -1 \), results in a mathematically identical form to the original total-cost function, \( C(q) = q(v + 2 \sqrt{vw} + w) \).
This alignment confirms that the initial cost function can adequately reflect the firm's cost structure when the inputs are substitutable in a specific manner depicted by the CES framework.
Such a function is particularly useful for firms in dynamic markets where input prices fluctuate, as it helps to understand how changes in input costs might impact overall expenses without sacrificing output levels. By grasping the utility of the CES cost function, firms can strategically plan for varying production conditions.