Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Sally Leonard, a real estate broker. is relocating in a large metropolitan area where she has received job offers from realty company A and realty company B. The number of houses she expects to sell in a year at each firm and the associated probabilities are shown in the following tables. The average price of a house in the locale of company \(\mathrm{A}\) is $$\$ 308,000$$, whereas the average price of a house in the locale of company \(\mathrm{B}\) is $$\$ 474,000$$. If Sally will receive a \(3 \%\) commission on sales at both companies, which job offer should she accept to maximize her expected yearly commission?

Step-by-Step Solution

Answer

Sally should accept the job offer from the company with the highest expected commission. To calculate this, multiply the expected number of houses sold (found by summing the product of the number of houses and their corresponding probabilities for each company) by the average house price and the 3% commission rate for each company. Based on these calculations, compare the expected commission for both companies and decide the best job offer for Sally to maximize her yearly commission.

Step 1: Calculate the expected number of houses sold at each company

The first step is to determine the expected number of houses Sally will sell in a year for both companies. To calculate the expected value (E), we need to multiply the number of houses sold (x) by their corresponding probabilities (P(x)) and sum the results for each company. For company A, let's denote the number of houses sold as \(x_A\) and the corresponding probabilities as \(P_A(x_A)\). For company B, let the number of houses sold be \(x_B\) and the corresponding probabilities as \(P_B(x_B)\). Next, we'll find the expected value for the number of houses sold in each company: \(E_A(x_A) = \sum_{i=1}^n x_A * P_A(x_A) \) \(E_B(x_B) = \sum_{i=1}^n x_B * P_B(x_B) \) Using the data given in the exercise, we can now calculate these expected values.

Step 2: Calculate the expected commission for each job offer

Once we have the expected number of houses sold at each company, we move on to calculating the expected commission for each job offer. To do this, we need to multiply the expected number of houses sold by the average house price, then multiply that by the 3% commission rate. For company A, we denote the expected commission as \(E_C^{A}\), the average house price as \(P_A\) , and the commission rate as \(c\). Then, the formula for the expected commission for company A would be: \(E_C^{A} = E_A(x_A) * P_A * c \) Similarly, for company B, we denote the expected commission as \(E_C^{B}\) and the average house price as \(P_B\). Then, the formula for the expected commission for company B would be: \(E_C^{B} = E_B(x_B) * P_B * c \) Now, we can plug in the values from the exercise and calculate the expected commission for each job offer.

Step 3: Compare the expected commission and choose the best job offer

With the expected commission calculated for each job offer, now we must compare them to determine which job offer Sally should accept to maximize her expected yearly commission. If the expected commission for job offer A is greater than job offer B, then Sally should accept job offer A. Otherwise, Sally should accept job offer B.

Key Concepts

Probability

Probability is a fundamental concept in understanding how likely an event is to occur. It ranges from 0 to 1, where 0 means the event will not happen, and 1 means it will happen for sure. In Sally's case, probability helps us estimate how many houses she might sell in a year at each realty company. When Sally considers job offers, each company provides different scenarios with probabilities associated with the number of houses sold. These probabilities allow us to calculate an expected value, which gives Sally a clearer picture of average outcomes over time. By multiplying each number of houses by its respective probability and summing these products, Sally can find the expected number of houses she might sell. For example, if there is a 30% chance of selling 10 houses and a 70% chance of selling 15, the expected number sold would be:

• 0.3 (probability) × 10 (houses) = 3
• 0.7 (probability) × 15 (houses) = 10.5

Summed together, the expected value is 13.5 houses. This calculation helps Sally weigh her options and anticipate what the future might hold at each company.

Expected Commission

The expected commission represents the average amount of money Sally could earn from selling houses with a given commission rate over time. Understanding expected commission helps Sally make informed decisions about which job offer will maximize her income. After determining the expected number of houses she could sell at each firm, Sally calculates her expected commission. This involves multiplying this expected value by the average house price and then by the commission rate (3% in this case). Let's say Sally expects to sell 20 houses at company A with an average price of $308,000. The formula for calculating her expected commission ( \(E_C^{A}\)) would be:

• Expected Houses Sold × Average House Price × Commission Rate

\[E_C^{A} = 20 \text{ houses} \times 308,000 \text{ dollars} \times 0.03 = 184,800 \text{ dollars}\]Performing this calculation for both companies enables Sally to compare the expected earnings. By doing so, she can figure out which offer is more lucrative based on the potential sales performance and associated earnings at each company.

Decision Making in Economics

Decision making in economics involves choosing the best option based on anticipated outcomes and benefits. For Sally, this decision comes down to comparing the expected commissions from each job offer to determine which is more financially beneficial. Sally's decision-making process uses expected value to evaluate the potential financial upside of each opportunity. She calculates expected commissions, allowing her to objectively weigh the pros and cons of each job offer. Critical factors that influence her decision include:

• Differences in average house prices at each company.
• Variations in sales probability and expected houses sold per year.
• The uniform commission rate applied in both scenarios.

By breaking down these elements with the help of probability and expected commission calculations, Sally can make an informed decision. The goal is to select the job offer that aligns with her financial goals and provides the best possible outcome. This is a primary example of economic decision-making, where an individual uses numerical analysis and forecasts to make the most beneficial choice.