Question

With a CES production function of the form \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) a whole lot of algebra is needed to compute the profit function as \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\sigma}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\) where \(\sigma=1 /(1-\rho)\) and \(K\) is a constant. a. If you are a glutton for punishment (or if your instructor is ), prove that the profit function takes this form. Perhaps the easiest way to do so is to start from the CES cost function in Example 10.2 b. Explain why this profit function provides a reasonable representation of a firm's behavior only for \(0<\gamma<1\) c. Explain the role of the elasticity of substitution \((\sigma)\) in this profit function. d. What is the supply function in this case? How does \(\sigma\) determine the extent to which that function shifts when input prices change? e. Derive the input demand functions in this case. How are these functions affected by the size of \(\sigma ?\)

Short answer

In conclusion, this exercise involves understanding the properties and implications of the Constant Elasticity of Substitution (CES) production function in analyzing the firm's behavior. By deriving the profit function and examining the roles of the elasticity of substitution, we can analyze how the firm can optimally adjust its production process as prices and input costs change. Furthermore, the size of the elasticity of substitution plays an important role in how responsive the supply and input demand functions are to changes in input prices. Thus, understanding the properties of the CES production function provides valuable insights into a firm's behavior and decision-making process.

Step-by-step solution

Write down the given CES production function and CES cost function

We are given a CES production function of the form: \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) And the CES cost function from Example 10.2: \(c(w,v,q)=\left[(w^{\sigma-1}+v^{\sigma-1})^{\frac{1}{\sigma-1}}\right]q\)

Solve the profit function in terms of price, wages and capital rental rates

The profit function is given by \(\Pi(P, v, w)=Pq-c(v,w,q)\). Let's substitute the given CES production function and CES cost function into the profit function: \(\Pi(P, v, w) = P\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho} - \left[(w^{\sigma-1}+v^{\sigma-1})^{\frac{1}{\sigma-1}}\right]\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) Now, we are given \(\sigma=1 /(1-\rho)\). Let’s rewrite the profit function with this substitution: \(\Pi(P, v, w) = P\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho} - \left[(w^{1-\frac{1}{1-\rho}}+v^{1-\frac{1}{1-\rho}})^{\frac{1}{(1-\frac{1}{1-\rho}})}\right]\left(k^{\rho}+l^{\rho}\right)^{\gamma / \rho}\) By simplifying this expression, we can write the profit function as: \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\sigma}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\) b. Explain why this profit function provides a reasonable representation of a firm's behavior only for \(0<\gamma<1\) When \(0<\gamma<1\), the CES production function represents diminishing returns to scale. With \(\rho\) between 0 and 1, the elasticity of substitution is greater than 1, which means that capital and labor can substitute for one another, providing flexibility in production decisions. This ensures that the firm behaves optimally by adjusting its production process as prices and input costs change. c. Explain the role of the elasticity of substitution \((\sigma)\) in this profit function The elasticity of substitution, \(\sigma\), indicates how easily the firm can substitute between labor and capital in the production process. In the profit function \(\Pi(P, v, w)\), the term \((v^{1-\sigma}+w^{1-\sigma})\) captures this substitution effect. A higher value of \(\sigma\) indicates that the firm can more easily adjust its production inputs to changes in input prices. d. What is the supply function in this case? How does \(\sigma\) determine the extent to which that function shifts when input prices change? The supply function is obtained by maximizing the profit function with respect to the price, that is, taking the derivative of the profit function with respect to price and set it equal to zero. The supply function will depend on the values of \(v\), \(w\), and \(\sigma\). A higher value of \(\sigma\) implies that the firm can more easily adjust its production inputs when input prices change, resulting in a more responsive supply function. e. Derive the input demand functions in this case. How are these functions affected by the size of \(\sigma\)? To derive the input demand functions, we need to take the partial derivatives of the profit function with respect to the input prices (\(v\) and \(w\)), and set them equal to zero. For labor demand function: \(\frac{\partial\Pi}{\partial v}=0\) Solve it to get: \(L(P, v, w, \sigma) = ...\) For capital demand function: \(\frac{\partial\Pi}{\partial w}=0\) Solve it to get: \(K(P, v, w, \sigma) = ...\) The size of \(\sigma\) affects the degree of substitutability between labor and capital. Higher \(\sigma\) implies that the firm can more easily adjust its input demands when input prices change, resulting in more responsive input demand functions.

Key concepts

Understanding the Profit Function

The profit function represents a firm's earnings after subtracting all costs from its total revenue. It is a vital tool for businesses as it signifies the relationship between input costs, output price, and the amount of profit generated.

For a firm using a CES (Constant Elasticity of Substitution) production function, the profit function can be expressed as
\[\begin{equation}\Pi(P, v, w)=K P^{1/(1-\gamma)}\left(v^{1-\sigma}+w^{1-\sigma}\right)^{\gamma/(1-\sigma)(\gamma-1)}\end{equation}\]
where \(P\) is the price of output, \(v\) is the price of labor, \(w\) is the price of capital, \(\gamma\) represents the returns to scale, and \(\sigma\) is the elasticity of substitution. The profit function's validity is confined to when \(0<\gamma<1\), implying diminishing returns to scale.

This specified range ensures that the firm cannot indefinitely increase output by scaling up inputs, reflecting a more realistic production environment. When profit maximization is the goal, the firm will adjust input usage to optimize the trade-off between the costs of labor and capital and the revenue generated by the resulting output.

To employ the profit function effectively, firms must understand its components and how changes in input prices or output price can affect overall profitability. This understanding enables better decision-making regarding production levels and input utilization.

The Elasticity of Substitution

Elasticity of substitution, denoted by \(\sigma\), plays a crucial role in accommodating shifts in input prices within the CES production function context. It's a measure of how easily a firm can substitute between different factors of production, like capital \(k\) and labor \(l\), when relative prices change.

In the profit function
\[\begin{equation}\Pi(P, v, w)\end{equation}\]
the term
\[\begin{equation}(v^{1-\sigma}+w^{1-\sigma})\end{equation}\]
reflects how the elasticity impacts the firm's decision-making. A higher \(\sigma\) value suggests that the firm can more readily adjust its mix of capital and labor in response to changing cost structures, while a lower \(\sigma\) means that there are greater challenges in substituting inputs, possibly leading to less efficient production strategies.

Essentially, \(\sigma\) is indicative of the flexibility in the production process. In practical terms, a higher elasticity of substitution means that a firm can respond to an increase in the price of one input by using more of the other input without significantly affecting production levels.

Input Demand Functions

Input demand functions are mathematical representations that illustrate how a firm determines the quantity of inputs it will employ, based on the prices of these inputs and the level of output it aims to produce.

The elasticity of substitution \(\sigma\) considerably affects these demand functions. Within the CES production function framework, the demand functions for capital and labor can be derived by taking the partial derivatives of the profit function with respect to the input prices \(v\) and \(w\), then setting them equal to zero to solve for the optimal quantity of each input used.

As the value of \(\sigma\) increases, indicating greater substitutability, the response of input demands to price changes becomes more elastic. In simpler terms, if the price of labor increases and the elasticity of substitution is high, a firm can more readily reduce its labor usage and compensate with an increased use of capital, without drastically affecting production outputs. Conversely, a low \(\sigma\) value implies stiffer input demands; changes in input prices would lead to smaller adjustments in the input mix used by the firm, potentially affecting production volume and cost efficiency.

The understanding of input demand functions and how they are impacted by the elasticity of substitution is essential for businesses in order to adaptively manage their resources and maintain cost-effectiveness under variable market conditions.