Problem 31 - Applied Calculus

Question

A company manufactures only one product. The quantity, \(q,\) of this product produced per month depends on the amount of capital, \(K,\) invested (i.e., the number of machines the company owns, the size of its building, and so on) and the amount of labor, \(L,\) available each month. We assume that \(q\) can be expressed as a Cobb-Douglas production function: $$ q=c K^{\alpha} L^{\beta} $$ where \(c, \alpha, \beta\) are positive constants, with \(0<\alpha<1\) and \(0<\beta<1 .\) In this problem we will see how the Russian government could use a Cobb-Douglas function to estimate how many people a newly privatized industry might employ. A company in such an industry has only a small amount of capital available to it and needs to use all of it, so \(K\) is fixed. Suppose \(L\) is measured in man-hours per month, and that each man-hour costs the company \(w\) rubles (a ruble is the unit of Russian currency). Suppose the company has no other costs besides labor, and that each unit of the good can be sold for a fixed price of \(p\) rubles. How many man-hours of labor per month should the company use in order to maximize its profit?

Step-by-Step Solution

Answer

Use \(L = \left(\frac{w}{p \times c K^{\alpha} \beta}\right)^{\frac{1}{\beta - 1}}\) man-hours for maximum profit.

Step 1: Understand the Problem

To optimize the number of man-hours of labor (\(L\)) the company should use to maximize its profit, we must consider a Cobb-Douglas production function \(q = c K^{\alpha} L^{\beta}\) where \(K\) is fixed. Profits \(\Pi\) depend on cost and revenue: \(\Pi = p \times q - w \times L\) where \(p\) is the price per unit and \(w\) is the wage per man-hour.

Step 2: Express Revenue and Cost

The revenue generated by producing \(q\) units is \(R = p \times q = p \times c K^{\alpha} L^{\beta}\). The labor cost is \(C = w \times L\). Therefore, the profit function becomes \(\Pi = p \times c K^{\alpha} L^{\beta} - w \times L\).

Step 3: Find the Derivative of Profit Function

Differentiate the profit function \(\Pi = p \times c K^{\alpha} L^{\beta} - w \times L\) with respect to \(L\): \(\frac{d\Pi}{dL} = \frac{d}{dL}(p \times c K^{\alpha} L^{\beta} - w \times L) = p \times c K^{\alpha} \beta L^{\beta - 1} - w\).

Step 4: Set Derivative to Zero

For profit maximization, set the derivative \(\frac{d\Pi}{dL} = 0\). Solve for \(L\) to find: \(p \times c K^{\alpha} \beta L^{\beta - 1} = w\).

Step 5: Solve for Optimal Labor

Rearrange the equation to solve for \(L\): \(L^{\beta - 1} = \frac{w}{p \times c K^{\alpha} \beta}\). Raise both sides to the power of \(\frac{1}{\beta - 1}\) to find \(L\): \(L = \left(\frac{w}{p \times c K^{\alpha} \beta}\right)^{\frac{1}{\beta - 1}}\).

Key Concepts

Cobb-Douglas Production Function

The Cobb-Douglas production function is a well-known mathematical model used in economics to describe the output of a company as a function of two inputs: capital (\(K\)) and labor (\(L\)). This function is commonly expressed as \[q = c K^{\alpha} L^{\beta}\] where \(q\) stands for the quantity of output, and \(c\), \(\alpha\), and \(\beta\) are positive constants. These constants define the nature of returns to scale.
The exponents \(\alpha\) and \(\beta\) indicate the output elasticity with respect to capital and labor, respectively. This means they show how sensitive the output is to changes in either capital or labor. In the given problem, \(K\) is fixed, pointing out that the capital input remains constant, and the company adjusts only labor to maximize profit.
Understanding this function is crucial for analyzing how a company can adjust labor to achieve maximum productivity when capital is a limiting factor.

Revenue

Revenue represents the income a company generates from selling its goods or services. In this exercise, the company's revenue is tied to its output \(q\) and the fixed price per unit \(p\). Mathematically, the revenue \(R\) is calculated as the product of the price and the output: \[R = p \times q\]
Substituting the Cobb-Douglas function for \(q\), the revenue function becomes \[R = p \times c K^{\alpha} L^{\beta}\], which expresses revenue in terms of labor, when capital is constant.
This relationship highlights how changes in labor demand affect revenue, given the fixed price and capital level. Knowing how revenue interacts with cost is key to understanding profit maximization, as it serves as the top-line figure in the profit equation.

Cost Optimization

Cost optimization involves minimizing expenses while maintaining necessary levels of production. In this context, the main focus is on labor costs, as labor is the only variable expense mentioned.
The cost of using labor \(L\) is calculated by multiplying the number of labor hours by the wage per hour \(w\): \[C = w \times L\].
For profit maximization, the company seeks an optimal level of labor that results in the greatest difference between revenue and cost. This involves analyzing the profit function \(\Pi = p \times c K^{\alpha} L^{\beta} - w \times L\).
To find the optimal labor quantity, one differentiates this function with respect to \(L\) and solves for when the derivative equals zero. This optimization helps determine the labor level that maximizes profits, given the fixed capital.

Labor Economics

Labor economics studies how labor is used within the marketplace. In this exercise, labor is a crucial variable due to its direct effect on cost and productivity potential.
Labor \(L\) is treated in terms of hours worked. The cost associated with labor greatly affects the overall cost structure of production, making it vital for profit calculations.
Labor economics within this framework implies understanding the balance between employing sufficient labor to maximize output and retaining cost efficiency.
The problem showcases how analyzing labor input intricately ties into optimizing production processes under budget constraints and what labor should be inputted for maximum output without unnecessary expenditure. It underlines the essence of strategic labor deployment in achieving operating efficiency and company goals.