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; with 1000 hours, produce 400 tons. b) Tons/hour; additional hour increases output by 2 tons. c) Real wage is \( \frac{w}{p} \) tons/hour. d) Profit: \( \pi(L) = p f(L) - w L \). e) At max profit, \( f'(L) = \frac{w}{p} \).

Step 1: Understanding the Units of f(L)

The function \( f(L) \) represents the number of tons of product produced with \( L \) hours of labor. Therefore, its units are tons. For \( f(1000) = 400 \), it means that with 1000 hours of labor, the company can produce 400 tons of product.

Step 2: Calculating the Units of f'(L)

The derivative \( f^{\prime}(L) \) is the rate of change of production with respect to labor, indicating the additional output (in tons) produced by an additional hour of labor. Thus, the units of \( f^{\prime}(L) \) are tons per hour. For \( f^{\prime}(1000) = 2 \), this implies that an additional hour of labor increases production by 2 tons.

Step 3: Deriving the Real Wage Formula

The real wage is the amount of product purchasable with one hour's wage. The wage per hour is \( w \) dollars, and the price per ton is \( p \) dollars, so each hour's wage buys \( \frac{w}{p} \) tons, expressing the real wage as \( \frac{w}{p} \) tons per hour.

Step 4: Expressing Company Profit

The profit function \( \pi(L) \) is revenue minus cost. Revenue is \( p \times f(L) \) (price per ton times number of tons) and cost is \( w \times L \) (wages per hour times hours). Therefore, profit is \( \pi(L) = p f(L) - w L \).

Step 5: Equality Condition for Maximum Profit

At maximum profit, the additional revenue from an extra hour of labor (\( p \times f^{\prime}(L) \)) equals the cost of the labor (\( w \)). Thus, setting these equal gives \( p f^{\prime}(L) = w \). Dividing both sides by \( p \) gives \( f^{\prime}(L) = \frac{w}{p} \), showing the company's marginal product of labor equals the real wage.

Key Concepts

Marginal Product of Labor

The marginal product of labor is a very important concept in understanding how labor inputs affect production output. Essentially, it measures the additional amount of output produced by using an additional unit of labor.
In mathematical terms, it's given by the derivative of the production function, represented as \( f'(L) \). So, if a company has a production function \( f(L) \), the marginal product of labor, \( f'(L) \), tells us how much more product will be produced by adding one more hour of labor.
Practically, if \( f'(L) = 2 \), this means that each extra hour of labor contributes 2 additional tons to production. This insight helps businesses decide whether employing more labor will be beneficial in terms of increased output.

Real Wage

The real wage concept helps us understand the true purchasing power of workers. It's not just about how much money workers make per hour (which would be their nominal wage), but rather about how much product they can buy with those wages.
This concept is crucial for assessing how well workers can actually utilize their earnings to purchase the goods they produce. In our exercise, the real wage is calculated by dividing the nominal wage \( w \) by the price per ton \( p \). Thus, the real wage is \( \frac{w}{p} \) tons per hour.
Understanding real wages is important for both employees and employers when considering pay rates and cost of living expenses.

Profit Function

The profit function is a cornerstone in business analyses, describing the relationship between costs, revenues, and profit. For a company selling products, profit is basically the revenue minus the cost.
Using the production and pricing information, one can derive the profit function as \( \pi(L) = p \times f(L) - w \times L \). This equation shows that a company's profit \( \pi(L) \) depends on the revenue from their product \( p \times f(L) \) and their costs in terms of labor, \( w \times L \).
By understanding the profit function, businesses can determine how changes in production levels and labor costs affect their profitability.

Maximum Profit Condition

The maximum profit condition helps businesses optimize their production in order to achieve the highest possible profit. It involves analyzing the point at which the additional revenue generated by an extra unit of labor equals the cost of that labor.
This principle is expressed through the equality \( p \cdot f'(L) = w \), which implies that the marginal product of labor (\( f'(L) \)) times the price per product (\( p \)) equals the wage (\( w \)).
Rearranging this equation gives the condition for maximum profit as \( f'(L) = \frac{w}{p} \). This means the marginal product of labor should equal the real wage, ensuring that the cost of hiring additional labor is justified by the increase in production, allowing the company to maximize profit.