Your company, Premier Car Rental, has a reputation for renting very nice, well-maintained cars. You have been given the task of analyzing the strength of recent demand at a group of local rental offices and recommending possible changes in the rate structure.
Your market can be divided into two kinds of customers, salespeople and tourists. The salespeople are interested in using cars to visit and socialize with clients around the city. The tourists are interested in some spots around the city, but want to go to outlying areas as well. They drive more. Based on the company's experience, business falls off pretty sharply if you charge more than $1 per mile driven, and is estimated to fall to zero (for both salespeople and tourists) if the rate became $1.50. For a typical salesperson, the number of miles driven during a one-week period increases by 100 for each $0.10 reduction in the charge per mile. A typical tourist increases his miles driven (during a one-week period) by 200 for each $0.10 reduction in the charge per mile.
During a typical week the group of offices that you are analyzing has a total of 300 customers (drivers): 200 salespeople and 100 tourists.
a) Draw a demand curve to scale for a typical salesperson, with the mileage charge (in dollars) on the vertical axis and the number of miles driven per week on the horizontal axis. Write an equation for the demand curve in slope-intercept form.
b) Then do the same for a typical tourist. Draw a demand curve for a typical tourist, and write an equation in slope-intercept form.
c) Now consider the total demand for 300 customers, 200 salespeople and 100 tourists. For possible mileage charges of $0, $0.10, $0.20, $0.30, etc., figure out the total number of miles that would be driven by 200 salespeople during a week. Add in the total number of miles that would be driven by 100 tourists. Draw a total demand curve for the 300 people, and write an equation in slope-intercept form. Then write an equation for marginal revenue in slope-intercept form. (Assume no price discrimination.)
d) The rental cars depreciate at a rate of $0.30 per mile driven. For the company, this is the marginal cost of miles driven.
e) Write a simple equation for marginal cost for the company. (Customers pay for all gasoline consumed.) Draw marginal revenue and marginal cost curves on the same diagram that you have for the total demand curve. Using the equations for marginal revenue and marginal cost, determine the total number of miles that would be best (from the company's point of view) for people to drive each week.
f) If there were no weekly charge, what would be the best mileage charge for Quality Car Rental (QCR) from QCR's point of view? Explain your answer by using the equations, and also show the answer on your diagram for total demand, marginal revenue, and marginal cost. Using the chapter 11 formula that relates price, marginal cost, and elasticity, find the elasticity of demand for the total number of miles being driven.
g) If the mileage charge were kept as is, and a weekly rate were charged on top of it, how high could the weekly rate be without a typical salesperson leaving the market? If the same mileage charge and weekly rate apply to all customers, what would weekly profits be?
h) If the mileage charge were reduced by $0.20 per mile, how high could the weekly rate then be without a typical salesperson leaving the market? What would then happen to weekly profits? Explain your answers.
i) Determine the best mileage charge and weekly rate by using calculus. Write an equation for total profits, where total revenue is obtained from both the mileage charge (multiplied by the total number of miles driven) and the weekly rate (multiplied by the number of customers). The weekly rate, in turn, equals the area of a triangle of consumer surplus that would otherwise accrue to a typical salesperson. Profits equal total revenue minus total cost.
j) From a math standpoint, the tricky part is to take the above profit equation and, in every case where a quantity appears in the equation (either a total quantity of miles, or a quantity of miles for one customer), to substitute in an expression involving P, where P refers to the mileage charge. (You will have to make use of the equations for demand curves.) Then take the derivative of profits with respect to P and set it equal to zero. Solve for P. You can then figure out the weekly rate to charge.
k) Some customers may have Costco membership cards. How do you think their elasticity of demand for renting cars (for a week) would compare to that of non-Costco members? Depending on your response, would it then be worthwhile to offer discounts on the weekly rate for such people, in order to attract more customers? Explain your answers.