Problem 32 - Applied Calculus

Question

A company can produce and sell \(f(L)\) tons of a product per month using \(L\) hours of labor per month. The wage of the workers is \(w\) dollars per hour, and the finished product sells for \(p\) dollars per ton. (a) The function \(f(L)\) is the company's production function. Give the units of \(f(L) .\) What is the practical significance of \(f(1000)=400 ?\) (b) The derivative \(f^{\prime}(L)\) is the company's marginal product of labor. Give the units of \(f^{\prime}(L) .\) What is the practical significance of \(f^{\prime}(1000)=2 ?\) (c) The real wage of the workers is the quantity of product that can be bought with one hour's wages. Show that the real wage is \(w / p\) tons per hour. (d) Show that the monthly profit of the company is $$ \pi(L)=p f(L)-w L $$ (e) Show that when operating at maximum profit, the company's marginal product of labor equals the real wage: $$ f^{\prime}(L)=\frac{w}{p} $$

Step-by-Step Solution

Answer

(a) Tons; 400 tons with 1000 hours. (b) Tons/hour; 2 extra tons per hour. (c) \(\frac{w}{p}\) tons/hour. (d) \(\pi(L)=pf(L)-wL\). (e) \(f^{\prime}(L)=\frac{w}{p}\).

Step 1: Units of Production Function

The production function \(f(L)\) gives the quantity of product (tons) produced using \(L\) hours of labor. Therefore, the units of \(f(L)\) are tons. The expression \(f(1000)=400\) means that with 1000 hours of labor per month, the company can produce 400 tons of the product. This signifies the productivity of labor in terms of output.

Step 2: Units of Marginal Product of Labor

The derivative \(f^{\prime}(L)\) represents the marginal product of labor, which is the additional output (in tons) resulting from one additional hour of labor. Therefore, the units of \(f^{\prime}(L)\) are tons per hour. The statement \(f^{\prime}(1000)=2\) means that if you add one more hour of labor when already using 1000 hours, you will produce an additional 2 tons of product.

Step 3: Calculation of Real Wage

The real wage is the amount of product that can be bought with the wages earned from one hour of work. With a wage rate of \(w\) dollars per hour and the product selling at \(p\) dollars per ton, the real wage is calculated as \(\frac{w}{p}\) tons per hour. This means for every dollar earned, the worker can buy \(\frac{w}{p}\) tons of the product.

Step 4: Derivation of Profit Function

Profit \(\pi(L)\) is calculated as total revenue minus total cost. Total revenue is obtained by multiplying the price per ton \(p\) by the production function \(f(L)\), which gives \(pf(L)\). Total labor cost is the wage \(w\) per hour times the number of hours \(L\), resulting in \(wL\). Therefore, the profit function is \(\pi(L) = pf(L) - wL\).

Step 5: Marginal Product and Real Wage at Maximum Profit

At maximum profit, the cost of hiring one more hour of labor should equal the revenue generated from the additional output produced by that hour. Mathematically, this condition is expressed as: \(pf^{\prime}(L)=w\). Solving for \(f^{\prime}(L)\), we divide both sides by \(p\) to get \(f^{\prime}(L) = \frac{w}{p}\), showing that at maximum profit, the company's marginal product of labor equals the real wage.

Key Concepts

Marginal Product of Labor

When we talk about the Marginal Product of Labor (MPL), we're essentially examining how additional hours of labor can increase production. Specifically, MPL refers to the extra output produced from employing one more unit of labor, holding everything else constant.
This concept is vital in understanding production efficiency. In this context, if the MPL is high, it means that each additional worker or hour significantly boosts production. Conversely, a low MPL implies additional labor contributes less to total output.
- **Units**: MPL is expressed in terms of output per additional unit of labor, or tons per hour in this example.- **Practical Example**: The expression \(f'(1000) = 2\) suggests that at 1000 hours of labor, each additional hour increases production by 2 tons. This is a direct measure of the efficiency of labor at that level.Understand that MPL is crucial for businesses to decide the optimal level of labor to employ for maximum productivity. Increasing labor may lead to higher output, but the cost-effectiveness of this increase depends on MPL.

Real Wage

Real Wage is a way of understanding how much a worker can actually buy with their wage, in terms of the goods they produce. It contrasts the nominal wage, which is simply the amount of money earned per hour.
In simple terms, Real Wage is the purchasing power of a worker's wage with respect to the commodity they help produce. If a worker earns \(w\) dollars per hour and the product is sold at \(p\) dollars per ton, the Real Wage in terms of the product is given by the formula: \(\frac{w}{p}\) tons per hour.
- **Importance**: This concept helps workers understand their earnings' true value. More importantly, it allows businesses to gauge the effective cost of labor in terms of production.- **Contextual Example**: If a worker's hourly wage and the selling price of the product align such that \(\frac{w}{p}\) equals some value, it specifies how much of the product can be bought with their earnings.
Understanding Real Wage helps in negotiating fair wages and making informed decisions about labor pricing in relation to market conditions.

Profit Function

The Profit Function is a fundamental concept that helps businesses evaluate how productive and cost-effective their operations are. It is calculated by subtracting total costs from total revenue.
In the given equation, the company's monthly profit \(\pi(L)\) is calculated as: - **Total Revenue**: \(pf(L)\), which is the product of the price per ton \(p\) and the quantity produced \(f(L)\).- **Total Cost**: \(wL\), which represents the wage rate \(w\) times the number of labor hours \(L\).Thus, the profit function can be written as \(\pi(L) = pf(L) - wL\). This equation shows that profit is derived from maximizing revenue and minimizing costs - a central aim for any successful business.
- **Maximizing Profit**: To reach maximum profit, a company must ensure that the marginal profit from additional labor equals the extra real wage paid. Hence, maximum profit occurs when the condition \(f'(L) = \frac{w}{p}\) holds true.Grasping the concept of the profit function can lead to better decision-making, ensuring that operational inefficiencies are minimized for sustainable business growth.