Question

Patty delivers pizza using her own car, and she is paid according to the number of pizzas she delivers. The accompanying table shows Patty's total benefit and total cost when she works a specific number of hours. $$\begin{array}{ccc}\begin{array}{c}\text { Quantity of } \\\\\text { hours worked } \end{array} & \text { Total benefit } & \text { Total cost } \\\0 & \$ 0 & \$ 0 \\\1 & 30 & 10 \\\2 & 55 & 21 \\ 3 & 75 & 34 \\\4 & 90 & 50 \\\5 & 100 & 70\end{array}$$ a. Use marginal analysis to determine Patty's optimal number of hours worked. b. Calculate the total profit to Patty from working 0 hours, 1 hour, 2 hours, and so on. Now suppose Patty chooses to work for 1 hour. Compare her total profit from working for 1 hour with her total profit from working the optimal number of hours. How much would she lose by working for only 1 hour?

Short answer

Patty's optimal hours of work are where her marginal cost begins to exceed her marginal benefit. After calculating profit for all hours, compare the profit from working 1 hour to the optimal hours to find the loss from working only 1 hour.

Step-by-step solution

Determine Marginal Benefit

Marginal benefit (MB) is the additional benefit that Patty receives from working one more hour. To calculate this, subtract the total benefit of the previous hour from the total benefit of the current hour. Perform this for each transition between hours worked.

Determine Marginal Cost

Marginal cost (MC) is the extra cost incurred for working one more hour. Find this by subtracting the total cost of the previous hour from the total cost of the current hour, analogous to the marginal benefit calculations.

Identify Optimal Number of Hours

The optimal number of hours to work is the point where the marginal benefit equals or just exceeds the marginal cost (MB=MC or MB>MC and falls below MC after). Look at the MB and MC calculated from Steps 1 and 2 to make this determination.

Calculate Total Profit

Total profit is calculated by subtracting total cost from total benefit for each hour worked. Do this for each row in the table to understand Patty's profits at different hours worked.

Determine Profit from Working 1 Hour and Optimal Hours

Utilize the results from Step 4 to find the profit for working 1 hour and the profit for working the optimal number of hours determined in Step 3.

Calculate Profit Loss

Profit loss is the difference between the profit of working the optimal number of hours and the profit from working only 1 hour. Subtract Patty’s profit of working 1 hour from the profit of working the optimal number of hours to find this amount.

Key concepts

Marginal Benefit

The concept of marginal benefit (MB) is fundamental in microeconomics and pertains to the additional satisfaction or utility a consumer receives from consuming one more unit of a good or service. In the context of labor, it is the extra benefit an individual gets from working an additional hour. Calculating the MB in a given situation involves comparing the total benefits received between two consecutive levels of activity. In terms of our case with Patty, to find the MB of each hour worked, we subtract the total benefit of the previous hour from the total benefit of the current one, yielding the incremental advantage of that additional hour of effort.

• MB of 1st hour = Total Benefit at 1 hour - Total Benefit at 0 hours
• MB of 2nd hour = Total Benefit at 2 hours - Total Benefit at 1 hour
• ...and so on for each consecutive hour.

Understanding MB helps Patty decide up to which hour her work brings in more rewards than the preceding hour, indicating a profitable use of her time.

Marginal Cost

Marginal cost (MC) is a critical economic indicator that reflects the increase in total cost when the level of production or in this case, hours of work, are incremented by one unit. It effectively measures the cost of producing one more unit of a good or service or working for an additional hour. In Patty's scenario, to calculate the MC for each hour worked, one would deduce the total cost at one level of output from the total cost at the level just before it.

• MC of 1st hour = Total Cost at 1 hour - Total Cost at 0 hours
• MC of 2nd hour = Total Cost at 2 hours - Total Cost at 1 hour
• ...and similarly for subsequent hours.

Recognizing the MC is vital for Patty to know how much each extra working hour really costs her, which in turn influences how many hours she should ideally work to maximize her benefits.

Optimal Hours of Work

The optimal hours of work are determined by analyzing the intersecting points of MB and MC. For Patty, as a pizza delivery person, her optimal working hours are when the additional income (MB) from delivering one more pizza equals the extra costs (MC) incurred by that additional hour of work. If she works more hours beyond this optimal point, her MC will surpass the MB, leading to reduced profitability.

• If MB > MC, it is beneficial to work more hours.
• If MB = MC, Patty has reached her optimal work duration.
• If MB < MC after a certain point, she should consider working fewer hours.

Through marginal analysis, Patty can pinpoint precisely when she should stop working to ensure she doesn’t spend her hours in a way that would eventually cost her more than it benefits her.

Total Profit Calculation

Total profit calculation involves balancing total revenue against total costs. It is an essential step in understanding the financial viability of work or a business operation. For Patty, total profit for each hour worked is the difference between her total benefit (the income from delivering pizzas) and her total cost (including gas, vehicle wear and tear, and other expenses related to her delivery work).

• Total Profit = Total Benefit - Total Cost
• Profit at 0 hours = \(0 - \)0
• Profit at 1 hour = \(30 - \)10
• Profit at 2 hours = \(55 - \)21
• ...and this pattern continues for each subsequent hour worked.

Calculating profit for each hour helps Patty assess which hours add to her earnings and which don’t. Making smart decisions based on these calculations can significantly affect her total earnings versus the time and resources spent.