Question

Jameel runs a driver education school. The more driving instructors he hires, the more driving lessons he can sell. But because he owns a limited number of training automobiles, each additional driving instructor adds less to Jameel's output of driving lessons. The accompanying table shows Jameel's production function per day. Each driving lesson can be sold at \(\$ 35\) per hour. Determine Jameel's labor demand schedule (his demand schedule for driving instructors) for each of the following daily wage rates for driving instructors: \(\$ 160, \$ 180,\) \(\$ 200, \$ 220, \$ 240,\) and \(\$ 260\)

Short answer

Answer: To answer this question, refer back to the labor demand schedule obtained in step 4 of the solution. Find the number of driving instructors that maximize profit at a wage rate of 180.

Step-by-step solution

Identify variables in the problem

First, let's identify two important aspects of this exercise: the production function, which shows how many driving lessons Jameel can sell for different numbers of driving instructors, and the price per driving lesson (found to be \(35\) per hour). We also need to consider the different daily wage rates for the driving instructors as given in the exercise.

Define profit function

We need to calculate the profit for each level of driving instructors. To do that, we need to define a profit function as follows: Profit = (Number of driving lessons sold * Price per driving lesson) - (Number of driving instructors * Wage rate per driving instructor) Let's apply this formula to each level of driving instructors and calculate the profit for each wage rate given.

Calculate profit for each level of driving instructors and wage rates

Let's create a table with the number of driving instructors (from \(0\) to maximum given in the production function) and calculate the profit for each wage rate. Here, for each number of driving instructors, calculate Profit using the formula: Profit = (Number of driving lessons sold * 35) - (Number of driving instructors * Wage rate per driving instructor)

Determine labor demand schedule

After calculating the profit for each level of driving instructors and wage rates, we need to determine the labor demand schedule. This means finding the number of driving instructors that Jameel should hire for each given wage rate. For each wage rate, we should select the number of driving instructors that yield the highest profit. If multiple levels of driving instructors result in the same profit, Jameel can choose any of them (or choose the range between them). For each given wage rate (160, 180, 200, 220, 240, 260), find the number of driving instructors that maximize the profit based on the calculated profit table in step 3. This will give us the labor demand schedule for Jameel's driving education school.

Key concepts

Production Function

At the heart of Jameel's driver education school lies his production function, which is a vital economic concept. It illustrates the relationship between inputs, in this case, the number of driving instructors, and the resulting output, which is the number of driving lessons sold daily. In this scenario, as Jameel hires more driving instructors, he can provide more driving lessons. However, the production function often faces diminishing returns. This means each additional driving instructor hired adds fewer driving lessons than the last one. This is due to the limited number of training automobiles available, which restricts the efficiency of additional instructors. Understanding the production function helps businesses anticipate how their outputs change with varying inputs, paving the way toward strategic decisions in resource allocation.

Profit Maximization

Profit maximization is a key objective for Jameel as he operates his driver education school. It involves finding the optimal number of driving instructors that will maximize profits. The profit function is defined as total revenue minus total cost. Here, total revenue is computed by multiplying the number of driving lessons sold by the price per lesson, which is set at $35.
Total cost depends on the wage rate and the number of driving instructors employed. By adjusting these variables and examining the resulting profits, Jameel seeks the number of instructors that yield the highest profit at given wage rates. This process involves comparing potential profits across different scenarios to choose the most financially rewarding setup.

Wage Rate

The wage rate is the daily amount Jameel pays to his driving instructors. It significantly affects his hiring decisions and overall profitability. In this exercise, Jameel's labor demand schedule is influenced by various wage rates: $160, $180, $200, $220, $240, and $260. Higher wage rates increase total costs, thus potentially decreasing profit if not balanced by sufficient revenue.

• A lower wage rate allows Jameel to hire more instructors, increasing the number of driving lessons without drastically escalating costs.
• Conversely, a higher wage rate could reduce the number of instructors Jameel can afford, impacting lesson quantity unless counteracted by a surge in demand or pricing.

Analyzing how these rates interact with his production allows Jameel to make informed staffing decisions.

Driving Instructors

Driving instructors are essential human resources in Jameel's driver education school. They are the individuals responsible for delivering lessons and ensuring students learn essential driving skills. The number of instructors hired influences the aggregate number of lessons provided. Increasing the number of instructors can boost productivity and sales, but only up to the point where the number of instructors does not exceed the capacity of available training automobiles. Beyond that, additional instructors yield diminishing returns. Jameel must balance hiring enough instructors to meet demand without overreaching capacity limits to maintain efficiency and profitability.

Driver Education

Driver education is the main service offered by Jameel's school. It's the process through which students learn to drive, gaining skills and knowledge under the guidance of experienced driving instructors. A critical part of this service involves a quality learning environment facilitated by a well-structured instructor-to-learner ratio. Jameel's focus is not just on maximizing profits but also providing valuable, effective education to his students. This entails having enough instructors to cater to student needs without overcrowding, which could compromise the quality of instruction.
Balancing economic efficiency with high-standard education ensures satisfaction and safety for all learners involved.