The Demand Function

scores of fares, ranging from first-class roundtrip tickets at $2,400 and greater to discount tickets below $250. On average, half the tickets sold for fares below $400, some 20 percent of tickets were priced above $800, with the remainder priced in between. Some travelers cashed in frequent flier miles. Some purchased at discounts from third-party providers; others received lower fares for restricted tickets requiring Saturday stayovers. In general, early buyers paid less, but fares fluctuated day-to-day depending on demand.

The question here is: How can demand analysis help the airlines win the game of yield management?

In Chapter 2, we presented a simple model of profit maximization. There the manager began with demand and cost functions and used them to determine the profit-maximizing price and output level for a given product or service. In this chapter, we will take a closer look at demand and the role it plays in managerial decision making.

The notion of demand is much richer than the simple formulation given in Chapter 2. For instance, up until now we have studied the dependence of demand on a single factor: price. We begin this chapter by considering the multiple determinants of demand. Next, we look more closely at the responsiveness of demand to these factors, a concept captured in the basic definition of elasticity. In the remaining sections, we present a richer formulation of demand and show how it can be used to guide managers in their goal of maximizing profits. Toward this end, we will refine our optimization techniques to account for more complicated demand conditions—those that include the possibilities of market segmentation and price discrimination.

DETERMINANTS OF DEMAND

The Demand Function

To illustrate the basic quantitative aspects of demand, let’s start with a concrete example: the demand for air travel.2 Put yourself in the position of a manager for a leading regional airline. One of your specific responsibilities is to analyze the state of travel demand for a nonstop route between Houston, Texas, and a rapidly growing city in Florida. Your airline flies one daily departure from each city to the other (two flights in all) and faces a single competitor that offers two daily flights from each city. Your task is complicated by the fact that the number of travelers on your airline (and therefore the revenue your company earns) has fluctuated considerably in the past three years. Reviewing this past experience, you realize the main determinants of your airline’s traffic are your own price and the price of your competitor. In addition, traffic between the two

2We are not ready yet to analyze the complicated problem of setting multiple fares described in the opening of this chapter. That must wait until the concluding section.

cities was brisk during years in which the Texas and Florida economies enjoyed rapid expansion. But, during the slowdown of 2008, air travel fell between the two cities.

Your immediate goal is to analyze demand for coach-class travel between the cities. (The small aircraft used on this route does not accommodate firstclass seating.) You begin by writing down the following demand function:

Q f1P, P°, Y2. [3.1]

This expression reads, “The number of your airline’s coach seats sold per flight (Q) depends on (is a function of) your airline’s coach fare (P), your competitor’s fare (P ), and income in the region (Y).” In short, the demand function shows, in equation form, the relationship between the quantity sold of a good or service and one or more variables.

The demand function is useful shorthand, but does not indicate the exact quantitative relationship between Q and P, P , and Y. For this we need to write the demand function in a particular form. Suppose the economic forecasting unit of your airline has supplied you with the following equation, which best describes demand:

Q 25 3Y P° 2P. [3.2]

Like the demand equations in Chapter 2, Equation 3.2 predicts sales quantity once one has specified values of the explanatory variables appearing on the right-hand side.3 What does the equation say about the present state of demand? Currently your airline and your competitor are charging the same one-way fare, $240. The current level of income in the region is 105.4 Putting these values into Equation 3.2, we find that

Q 25 3(105) 1(240) 2(240) 100 seats.

A comparison of this prediction with your airline’s recent experience shows this equation to be quite accurate. In the past three months, the average number of coach seats sold per flight (week by week) consistently fell in the 90- to 105-seat range. Since 180 coach seats are available on the flight, the airline’s load factor is 100/180 55.5 percent.

3Methods of estimating and forecasting demand are presented in Chapter 4. 4This value is an index of aggregate income—business profits and personal income—in Texas and Florida. The index is set such that real income (i.e., after accounting for inflation) in 2005 (the socalled base year) equals 100. Thus, a current value of 105 means that regional income has increased 5 percent in real terms since then. In the depth of the Texas recession, the index stood at 87, a 13 percent reduction in real income relative to the base year.