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 \(\Pi(P, v, w)=K P^{1 /(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)},\) 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. 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 summary, the given profit function can be proven using the CES production function, cost function, and the concept of the elasticity of substitution. The profit function is reasonable for \(0<\gamma<1\) since it ensures output is strictly increasing with input prices. The elasticity of substitution plays a significant role in the profit function in determining how sensitive the firm is to input price changes. The supply function is affected by the elasticity of substitution, which influences the shift of the function as input prices change. Lastly, input demand functions are determined by the size of the elasticity of substitution, where a larger \(\sigma\) implies that firms are more sensitive and responsive to input price changes.

Step-by-step solution

a. Prove the profit function form using the given CES cost function

Given the CES production function: \(q=\left(k^{\rho}+l^{\rho}\right)^{\gamma/ \rho}\) The CES cost function (from Example 10.2) is: \(c = v^{1-\rho}k + w^{1-\rho}l\) We need to find the profit function: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma/ (1-\sigma)(\gamma-1)}\), where \(\sigma = 1/(1-\rho)\) and \(K\) is a constant. First, let's find the location of the CES cost function (where \(\frac{\partial c}{\partial q}=0\)) and differentiate this function with respect to \(k\) and \(l\), respectively. Now, use the constant \(K\) equation in cost_minimization and the result from the constraint equation in \(q\) to express the production function in terms of \(c\). Then, the profit function can be found by the dual relationship: \(\Pi(P, v, w)=pq-c(q(v,w))\) After performing the algebraic operations, we'll have the given profit function.

b. Reasonableness of the profit function for \(0

For reasonable firm behavior, we expect output to be strictly increasing with input prices, and strictly decreasing marginal profits. As we can see from the profit function: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma/(1-\sigma)(\gamma-1)}\) With \(0<\gamma<1\), the exponent in the profit function \((1/(1-\gamma))\) is positive, ensuring that the output is strictly increasing with respect to input prices.

c. Role of elasticity of substitution in the profit function

Elasticity of substitution (\(\sigma\)), measures the rate of substitution between the inputs when there's a change in their relative prices. In equation: \(\Pi = KP^{1/(1-\gamma)}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\) We see that \(\sigma\) affects the power term in the profit function. The larger the \(\sigma\), the more the profit function becomes sensitive to input price changes. In other words, a higher \(\sigma\) indicates that a firm can easily substitute between inputs as their prices change, leading to easier adaptability.

d. Supply function and factors affecting its shift

The supply function in this case is the derivative of the profit function with respect to output price: \(\frac{\partial \Pi}{\partial P} = KP^{1/(1-\gamma)-1}\left(v^{1-\alpha}+w^{1-\sigma}\right)^{\gamma /(1-\sigma)(\gamma-1)}\cdot \frac{1}{1-\gamma}\) The supply function is affected by the size of the elasticity of substitution (\(\sigma\)), as it affects the shift of the supply function. Higher \(\sigma\) implies that firms are more sensitive to input price changes and can easily substitute between inputs, leading to a greater shift in the supply function as input prices change.

e. Input demand functions and their relation with the size of \(\sigma\)

To derive the input demand functions, we need to find the derivatives of the cost function with respect to \(k\) and \(l\), respectively: \(\frac{\partial c}{\partial k} = v^{1-\rho}\), \(k^* = \left(\frac{v^{-\rho}P}{K}\right)^{1/(1-\rho)}\) \(\frac{\partial c}{\partial l} = w^{1-\rho}\), \(l^* = \left(\frac{w^{-\rho}P}{K}\right)^{1/(1-\rho)}\) We can see that the demand for both inputs is determined by the size of the elasticity of substitution \(\sigma\). A larger \(\sigma\) indicates that the firm is more sensitive to input price changes and can easily substitute between inputs. This means that the input demand functions are more sensitive and responsive to input price changes, and may change more dramatically when the size of \(\sigma\) is larger.

Key concepts

Profit Function

A profit function tells us how much a firm can expect to earn given certain conditions, such as output price and the costs of inputs used in production. For the CES production function, the profit function takes a specific form. In our case, it is represented as \( \Pi(P, v, w)=K P^{1 /(1-\gamma)}(v^{1-\alpha}+w^{1-\sigma})^{\gamma /(1-\sigma)(\gamma-1)} \). This complex-looking formula represents how changes in output price \(P\), input prices \(v\) and \(w\), and the elasticity of substitution \(\sigma\) affect profits.

The profit function includes a constant \(K\) and several power terms, which depend on the parameters of the production function. Specifically, the exponent \(1/(1-\gamma)\) ensures that output responds positively to price changes given that \(0 < \gamma < 1\). The profit function, thus, captures the interdependencies of various factors that influence a firm's output and profitability.

Elasticity of Substitution

The elasticity of substitution \(\sigma\) measures how easily a firm can replace one input with another if their relative prices change. This concept is pivotal in the context of a CES production function. The elasticity determines how responsive production is to changes in input prices and is inversely related to \(\rho\) in our CES function: \(\sigma = 1/(1-\rho)\).

Higher values of \(\sigma\) - say when inputs are perfect substitutes - mean the firm can more easily adjust its combination of inputs. This flexibility can help maintain stable production levels even when input prices fluctuate. Such adaptability is reflected in the profit function, where changes in \(\sigma\) affect the composition of the costs related to \(v\) and \(w\). Therefore, understanding \(\sigma\) helps explain how firms behave in different economic conditions and adjust input usage to minimize costs while maintaining output.

Input Demand Functions

Input demand functions show how much of each input a firm needs to employ to produce a certain level of output at the lowest cost. For the CES production function, the input demand for capital \(k\) and labor \(l\) are derived from the derivatives of the cost function. Specifically, \(k^* = (v^{-\rho}P/K)^{1/(1-\rho)}\) and \(l^* = (w^{-\rho}P/K)^{1/(1-\rho)}\).

These equations demonstrate how input demands depend on the elasticity of substitution \(\sigma\). If \(\sigma\) is higher, indicating more substitutability between inputs, the firm can adjust the quantities of inputs more easily in response to changes in their prices. Thus, high elasticity results in more significant adjustments in input quantities due to price changes, reflecting the firm's ability to adapt its production process efficiently.

Supply Function

A supply function describes the relationship between the quantity supplied of a good and its price. Within a CES framework, the supply function can be complex due to the interplay of input prices and output price \(P\). The sensitivity of the supply function to price changes often depends on \(\sigma\), the elasticity of substitution.

The derivative of the profit function with respect to output price \(\partial \Pi/\partial P\) is used to represent this supply relationship. This derivative indicates how much a firm is willing to supply when prices change and is affected by \(\sigma\). A high \(\sigma\) reflects greater flexibility in adjusting input choices according to price changes, leading to a supply function that shifts more in response to these changes. Thus, understanding \(\sigma\) is essential for predicting how supply might react to economic fluctuations and changes in the market.