Question

The CES production function can be generalized to permit weighting of the inputs. In the two-input case, this function is \\[ q=f(k, l)=\left[(a k)^{\rho}+(b l)^{\rho}\right]^{\gamma / \rho} \\] a. What is the total-cost function for a firm with this production function? Hint: You can, of course, work this out from scratch; easier perhaps is to use the results from Example 10.2 and reason that the price for a unit of capital input in this production function is \(v / a\) and for a unit of labor input is \(w / b\) b. If \(\gamma=1\) and \(a+b=1,\) it can be shown that this production function converges to the CobbDouglas form \(q=k^{a} l^{b}\) as \(\rho \rightarrow 0 .\) What is the total cost function for this particular version of the CES function? c. The relative labor cost share for a two-input production function is given by \(w l / v k\). Show that this share is constant for the Cobb-Douglas function in part (b). How is the relative labor share affected by the parameters \(a\) and \(b\) ? d. Calculate the relative labor cost share for the general CES function introduced above. How is that share affected by changes in \(w / v\) ? How is the direction of this effect determined by the elasticity of substitution, \(\sigma\) ? How is it affected by the sizes of the parameters \(a\) and \(b\) ?

Short answer

The total cost function assumes form \( C = vk + wl \), which simplifies to \( C = q \) for Cobb-Douglas. Labor cost share is consistent for Cobb-Douglas but varies with price and \( \rho\) in CES.

Step-by-step solution

Identify Total Cost Function Expression

The production function is provided as \( q = \left[ (ak)^\rho + (bl)^\rho \right]^{\gamma/\rho} \). From the hint, substitute the input prices: capital has a price of \( \frac{v}{a} \) per unit and labor \( \frac{w}{b} \) per unit. Total cost \( C \) is \( vk + wl \) based on unit prices.

Derive Total Cost Function for Standard CES

For a general CES function, express the inputs in terms of their marginal products set against their respective prices: \( v = \frac{d}{dk} \left[ (ak)^\rho+(bl)^\rho \right]^{\gamma/\rho} \) and \( w = \frac{d}{dl} \left[ (ak)^\rho+(bl)^\rho \right]^{\gamma/\rho} \). Solve these to find expressions relating \( k \) and \( l \) to \( q \) and substitute into \( C = vk + wl \).

Simplify to Cobb-Douglas Form

When \( \gamma = 1 \) and \( a + b = 1 \), the CES production simplifies to Cobb-Douglas form as \( q = k^a l^b \) when \( \rho \rightarrow 0 \). The cost function simplifies as substitution elasticity equals 1, simplifying marginal rate terms to \( vk = aq \) and \( wl = bq \), thereby giving \( C = q \).

Evaluate Relative Labor Cost Share for Cobb-Douglas

In the Cobb-Douglas form, relative cost share \( \frac{wl}{vk} = \frac{b}{a} \). Show that \( abla q = a + b = 1 \) ensures constant cost shares regardless of \( q \) scale. Parameters \( a \) and \( b \) dictate their respective proportions in cost.

Analyze CES Function Relative Labor Share

General CES has share \( \frac{wl}{v k} = \left( \frac{b l}{a k} \right)^{\rho-1} \). Effects of changing \( \frac{w}{v} \): greater price effect on labor share. With \( \sigma = \frac{1}{1-\rho} \), if \( \rho > 0 \): \( \sigma < 1 \), cost share increases with labor price increase, opposite if \( \rho < 0 \), \( \sigma > 1 \). Parameters \( a, b \) maintain their influence on proportionality.

Key concepts

Cobb-Douglas Production Function

The Cobb-Douglas production function is a specific form of a production function that represents the output as a product of two inputs raised to constant powers. It can be expressed as \( q = k^a l^b \), where \( q \) is the quantity of output, \( k \) and \( l \) are the inputs of capital and labor, and \( a \) and \( b \) are the output elasticities of these inputs. This form is significant because it assumes constant returns to scale if \( a + b = 1 \), meaning doubling both inputs will double the output.

In economic analysis, the Cobb-Douglas function is often used due to its simplicity and the realistic representation of production processes. When transformed from the CES (Constant Elasticity of Substitution) form, it indicates a case where the elasticity of substitution equals one. This means inputs can be substituted for one another at a constant rate without changing the production function's shape.

In the context of the CES function with parameters \( \gamma = 1 \) and \( a + b = 1 \), as \( \rho \to 0 \), the CES function converges to the Cobb-Douglas form. This convergence implies that under such conditions, the structure of substitutability between inputs aligns with the assumptions of the Cobb-Douglas function.

Elasticity of Substitution

Elasticity of substitution, denoted as \( \sigma \), is a concept that measures how easily one input can be substituted for another in response to changes in relative input prices. Mathematically, in the CES production function context, it is defined as \( \sigma = \frac{1}{1-\rho} \), where \( \rho \) is the parameter in the CES function controlling the degree of substitutability.

When \( \rho = 0 \), the elasticity of substitution becomes equal to one, which signifies that inputs are perfectly substitutable at a constant rate, consistent with the Cobb-Douglas scenario. If \( \rho > 0 \), \( \sigma < 1 \) indicates inelastic substitution, meaning inputs are not easily substituted with each other. Conversely, if \( \rho < 0 \), \( \sigma > 1 \) suggests elastic substitution, making it easier to swap one input for another.

This elasticity plays a critical role in understanding how firms adjust input usage as input prices fluctuate. It dictates the flexibility a firm has in response to changing economic conditions, influencing both production decisions and cost structures.

Total Cost Function

The total cost function in production economics represents the total cost incurred by a firm in producing a given level of output. For a production function, knowing the input prices and the quantities used, a firm can compute total cost as \( C = vk + wl \), where \( v \) and \( w \) are the unit prices of capital and labor, and \( k \) and \( l \) are the quantities of capital and labor used.

For the CES production function, this involves substituting derived input quantities from marginal product conditions into the cost equation. In the specific case of the Cobb-Douglas form, the cost function simplifies due to constant returns to scale and elasticity of substitution being one. Here, the cost relates directly to output levels, indicating that the firm's cost can be calculated proportionally through marginal contributions of capital and labor, expressed as \( C = q \).

• This simplification allows easier computation and comparison of costs across different levels of production.
• Understanding the structural form of the total cost helps firms to optimize their production efficiency.

Relative Labor Cost Share

The relative labor cost share represents the proportion of total cost attributed to labor in relation to capital in a production process. It is calculated as the ratio \( \frac{w l}{v k} \), where \( w \) and \( v \) are the unit prices of labor and capital, and \( l \) and \( k \) are the quantities used.

For the Cobb-Douglas production function, this share remains constant, as the ratio \( \frac{b}{a} \), meaning that labor's share of costs does not change with levels of output or input usage. The parameters \( a \) and \( b \) determine the proportion of each input's contribution to total output.

In the general CES function, however, the relative labor cost share can vary with changes in the relative prices \( \frac{w}{v} \). The degree and direction of this effect also depend on the elasticity of substitution, \( \sigma \). A higher elasticity indicates greater sensitivity of input substitution to price changes, impacting how cost shares adjust in response to price fluctuations.

• This understanding helps firms strategize resource allocation efficiently under changing economic conditions.
• Assessing cost share dynamics is key for forecasting and managing profitability under different market scenarios.