Question

Marty's Frozen Yogurt has the production function per day shown in the accompanying table. The equilibrium wage rate for a worker is \(\$ 80\) per day. Each cup of frozen yogurt sells for \(\$ 2\). a. Calculate the marginal product of labor for each worker and the value of the marginal product of labor per worker. b. How many workers should Marty employ?

Short answer

#Question# Based on the given production function for frozen yogurt and the equilibrium wage rate, calculate the marginal product of labor (MPL) and value of marginal product of labor (VMPL) for each worker, and determine the optimal number of workers that Marty should employ.

Step-by-step solution

Calculate the Marginal Product of Labor (MPL)

We will find the MPL for each worker by finding the difference in output between a worker and the previous worker in the table. The MPL formula is as follows: MPL = (Output with n workers) - (Output with n-1 workers) Calculate the MPL for each worker in the table.

Calculate the Value of Marginal Product of Labor (VMPL)

Next, we calculate the VMPL for each worker, by multiplying the marginal product of labor (MPL) obtained in step 1 by the price of each cup of frozen yogurt ($2). The VMPL formula is as follows: VMPL = MPL * Price of frozen yogurt Calculate the VMPL for each worker.

Determine the optimal number of workers

Now, we'll compare the VMPL for each worker to the equilibrium wage rate ($80) to determine the optimal number of workers to employ. Marty should employ workers as long as VMPL is greater than or equal to the wage rate. Once the VMPL drops below the wage rate, Marty should stop hiring more workers. Compare the VMPL of each worker to the wage rate and determine the optimal number of workers that Marty should employ. For part a, we have calculated MPL and VMPL for each worker employing the steps above. For part b, we have determined the optimal number of workers by comparing VMPL to the wage rate. This complete the exercise.

Key concepts

Value of Marginal Product

To understand the concept of Value of Marginal Product (VMPL), it's important to first know what Marginal Product of Labor (MPL) is. MPL measures the additional output created by hiring one more worker. For example, if hiring an additional worker increases the production of frozen yogurt cups from 90 to 100, the MPL is 10 cups.

Once the MPL is calculated, we determine the VMPL by multiplying the MPL by the price of each product. For Marty’s Frozen Yogurt, if the price per cup is $2, and the MPL is 10, then the VMPL would be \( 10 \, \text{cups} \times 2 \, \text{dollars per cup} = 20 \, \text{dollars} \). This calculation gives us the dollar value added by employing an additional worker.

The VMPL is crucial in deciding how many workers a company should hire. As long as the VMPL is greater than or equal to the wage rate, it is profitable for the firm to hire more people.

Equilibrium Wage Rate

The equilibrium wage rate is the rate at which the supply of labor equals the demand for labor in the market. It represents the amount that a firm pays to hire workers under normal conditions without any economic pressure to change. For Marty’s Frozen Yogurt, this rate is $80 per day.

Understanding the equilibrium wage rate is vital because it helps businesses determine their labor costs and budgets. Companies like Marty’s compare their VMPL to the equilibrium wage to decide if hiring more workers will be profitable. If the VMPL exceeds $80, it suggests hiring more workers until the additional value they bring is less than what they cost in wages. Conversely, if the VMPL is below the equilibrium wage rate, it would signal that hiring additional workers would result in a loss.

In the world of economics, this equilibrium point helps maintain balance in the labor market, ensuring neither surplus nor shortage of labor.

Optimal Employment Decision

Making the optimal employment decision involves choosing the right number of workers to hire, ensuring maximum profitability. This decision hinges on the comparison between VMPL and the equilibrium wage rate. In simple terms, Marty's Frozen Yogurt should hire workers up to the point where the VMPL is equal to or just above the wage rate of $80.

If VMPL is greater than $80, it means each additional worker is adding more to the revenue than they are costing, which is desirable. Hiring more workers under such conditions benefits Marty by increasing total profit.

However, once VMPL falls below this rate, it signals that an additional worker would cost more than the revenue increase they bring, and hiring should be halted. This ensures that resources are used efficiently and helps to avoid unnecessary labor costs which do not increase profits.

Production Function Analysis

Production function analysis is a tool used to understand how different inputs affect the total output of a company. In Marty’s Frozen Yogurt scenario, the inputs include workers and the output is measured in cups of frozen yogurt produced daily.

By analyzing the production function, Marty can identify how changes in the number of workers impact production levels. This can help in determining the most efficient scale of operations. For instance, if doubling the number of workers only increases production by 50%, it might indicate diminishing returns.

This analysis helps in identifying the point of diminishing returns, which is the stage where each additional input contributes less to the production compared to the previous input.

Comprehensive production function analysis assists companies in optimizing resource allocation, planning strategic expansions, or adjustments to ensure that they are maximizing output without unnecessary costs. It is a foundational principle for making informed economic decisions.