3

Models of Demand

This chapter introduces the concept of a price-response function. The price-response function specifies how the demand that a supplier will experience for a good will change as the seller changes his price for that good. I show how the price-response function can be derived from the distribution of willingness to pay among potential customers. We also look at the most common measures of price sensitivity and how they can be estimated from historic data, and at the most common forms used for price-response functions. The chapter also extends these measures to the case when a seller is offering multiple products that may compete with each other. This material provides a foundation for the analytic approaches to pricing that are described in the remainder of the book.

3.1 THE PRICE-RESPONSE FUNCTION

Neoclassical economic theory specifies that, under a set of (seemingly) reasonable assumptions, at any point in time, every consumer (individual or firm) has a maximum willingness to pay for every good on offer by every seller through every channel.¹ If the price of a good is greater than a buyer’s willingness to pay, she will not purchase. If the good is for sale at a price lower than her willingness to pay by more than one seller, she will purchase from the one such that the difference between her willingness to pay and the price—her so-called surplus—is maximized. She may not purchase from the seller with the lowest price because her willingness to pay may vary by seller. Thus, one customer might have a higher willingness to pay when purchasing from an online seller because of the convenience of delivery while another may have a higher willingness to pay to purchase in a store where she can examine the item in person before purchasing. A customer may prefer the service offered by a particular airline—or be a member of its loyalty program—so she is willing to pay a bit more to fly on that airline than for a flight at the same time offered by a competing airline. It is also the case that a customer may not know of all sellers who are offering a product or may check the prices of only a subset of those sellers.

TABLE 3.1

Willingness to pay and prices for Example 3.1

Example 3.1

Three customers are considering purchasing the same book from one of three potential sources: an online bookseller, a chain bookstore, and a local independent bookstore. Customer A strongly prefers the convenience of online shopping. Customer B is indifferent between channels and will purchase from the least expensive seller. Customer C likes to support her local independent bookstore. Their willingnesses to pay for purchasing the book from each seller and the price offered by each seller are shown in Table 3.1. In this case, Customer A will purchase online, Customer B will purchase from the chain, and Customer C will purchase from the independent bookseller.

Example 3.1 shows that, when there is a variation in willingness to pay among customers in a marketplace, different sellers can offer the same product at different prices and still all experience positive demand. The milk prices shown in Table 1.1 and the book prices shown in Table 1.2 are real-world examples of such price variation.

There are many different reasons for variation in willingness to pay. As in Example 3.1, different customers can have different preferences among different sellers or channels. One important driver of individual willingness to pay is propensity to shop—a customer who is willing to spend time comparing prices either online or offline will be more price sensitive than a customer who does not shop. Propensity to shop is related to the value that an individual places on personal time. Students and retirees are often found to be highly price sensitive—this can be explained in part by the relatively low opportunity cost of their time. Busy professionals tend to be less price sensitive at least in part because they assign a high opportunity cost to their time. Of course, there are people in all situations who enjoy bargain shopping for its own sake and get a thrill from finding a particularly low price, just as there are those who do not take the time to shop and are willing to accept a higher price in return.

A consequence of the distribution of willingness to pay across customers is that the demand experienced by a seller for a particular product will vary as a function of price; furthermore, the seller’s demand as a function of price can be derived from the underlying distribution of willingness to pay. Assume that the three customers of the independent bookseller in Example 3.1 represent his entire market and that his competitors do not respond to changes in his prices. In this case, if the independent bookseller raises his price from $18.50 to $19.01, his demand will drop from one to zero. On the other hand, if he lowers his price to $13.99, all three customers will purchase from him. We can represent the amount of demand a seller will realize at any price using a price-response function d = f(p) where p is price and d is the corresponding demand. The price-response function for the independent bookseller in Example 3.1 is given by

and is shown in Figure 3.1.²

The price-response function is a fundamental input to price and revenue optimization (PRO). There is a different price-response function associated with each element in the PRO cube—that is, there is a price-response function associated with each combination of product, market segment, and channel. Thus, a multichannel seller will generally have a different price-response function associated with online sales than with brick-and-mortar sales.

The price-response function is similar to the market demand function described in standard economic texts. However, there is a critical difference: The price-response function specifies demand for the product of a single seller as a function of the price offered by that seller. This contrasts with a demand curve, which specifies how an entire market will respond to changing prices. The distinction is critical because different firms competing in the same market face different price-response functions. In Example 3.1, the price-response function faced by the online seller differs from the function faced by the chain bookstore, which in turn differs from the price-response function faced by the independent bookstore. The differences in these price-response functions result from differences in willingness to pay across customers, which in turn stem from such factors as the varying effectiveness of seller marketing campaigns, perceived differences in quality or customer service, convenience, location, and loyalty, among other factors.

Figure 3.1 Price-response function for the bookseller in Example 3.1.

For our purposes, variation in willingness to pay among customers is the normal situation. However, most readers will be familiar with the concept of perfect competition introduced in every basic economics textbook. Perfect competition occurs when sellers offer identical products, all customers are indifferent among sellers, every seller is infinitesimally small, and customers and sellers have perfect information about all prices. In this case, customers will purchase only from sellers offering the lower price and the price will equilibrate to the long-run marginal cost. A commonly cited example is wheat. Each wheat farmer supplies an insignificant fraction of all of the wheat grown in the world. No matter how much an individual farmer produces—1,000 bushels or 10,000—it will be too little to have any effect on the global market price. If he tries to sell whatever he decides to produce above the market price, he will sell nothing, and if he prices below the market price, he will face demand far more than he can produce. In this situation, his motivation is to produce as much as he can at an incremental cost below the market price.

Under perfect competition, the seller has no pricing decision—his price is set by the operation of the larger market. The seller can only improve profit by finding ways to decrease his costs or increase the amount he can produce at an incremental cost below the market price. At any price below the market price, the demand seen by a seller would be equal to the entire demand in the market—the amount D in Figure 3.2. At any price above the market price, he sells nothing.

A seller in a perfectly competitive market has no need for pricing and revenue optimization—indeed, he has no need of any pricing capability whatsoever, because the price is set by the marketplace. This is true of pure commodity markets such as zinc oxide and Brent crude oil and stocks sold on the New York Stock Exchange—in these cases, no one decides on a selling price, because the price is determined by the interplay of supply and demand. Sellers decide only how much to produce and sell. In this case, the price-response function is as shown in Figure 3.2.

Figure 3.2 Price-response function in a perfectly competitive market.

The vast majority of marketplaces, however (and all of the markets that we consider), are not perfectly competitive. Rather, as a supplier moves his price up or down, he will see his demand change—as in the price-response function shown in Figure 3.1 for the independent bookseller. The primary drivers of the relationship between price and demand are variation in preferences, information, propensity to shop, and differential access to alternatives across customers.

3.1.1 The Price-Response Function and Willingness to Pay

The demand for a product from a seller is the result of thousands, perhaps millions, of individual buying decisions on the part of potential customers. Each potential customer observes the seller’s price and decides whether to purchase the product. Those who do not purchase may purchase the same or a similar product from the competition, or they may simply decide to do without. The price-response function specifies the number of customers in a segment who decide to purchase a seller’s product through a channel at every price. We usually cannot directly track the thousands or even millions of individual decisions that ultimately manifest themselves in demand for our product—although online sellers can often track most or all of the customers who viewed the prices of their products. Section 4.2.2 shows that information on the number of customers who viewed the price of a product can be an important input to understanding price sensitivity. In any case, the price-response function results from the distribution of willingness to pay across the consumer population.

We assume that, at any time, every customer in the population has a willingness to pay for every product being sold through every channel. Each customer compares her willingness to pay to the prices being offered by one or more competing sellers. If more than one alternative is priced below her willingness to pay, she will purchase the option with the highest surplus, defined as the difference between her willingness to pay and the price. As in Example 3.1, not every customer will choose the cheapest alternative.

Assume that we have a population of customers of size D. The number of customers who will purchase at a price p is equal to the number of customers whose willingness to pay for the product is greater than or equal to p. (For simplicity, we assume that if a customer’s willingness to pay is exactly equal to the price, she will purchase.) Define the function f(x) as the density function for willingness to pay across the population, and define D as the potential demand—the number of customers who are potentially interested in purchasing the product and are aware of its price. Then, for any price p, is the fraction of the population with willingness to pay ≽ p, and the demand for the product at price p, which we denote d(p), can be written as

where F̄(p) represents the complementary cumulative distribution function of the density function f(p). F̄(p) is defined by

F̄(p) is the probability that a random draw from the distribution with density function f(x) will have a value greater than or equal to p. In our case, it is the probability that a randomly chosen customer will have a willingness to pay greater than or equal to p.

The implication of Equation 3.1 is that a price-response function can be generated from any willingness-to-pay distribution and that every price-response function corresponds to a willingness-to-pay distribution. Conversely, we can derive the density function for the willingness-to-pay distribution from the price-response function according to

which is nonnegative because the price-response function is downward sloping.

Example 3.2

The total potential market for a spiral-bound notebook is D = 20,000, and willingness to pay is distributed uniformly between $0 and $10.00 as shown in Figure 3.3. This means that

We can apply Equation 3.1 to derive the corresponding price-response function:

The price-response function d(p) = (20,000 – 2,000p)⁺ is a straight line with d(0) = 20,000 and demand equal to zero for prices greater than or equal to $10.00, as shown in Figure 3.3.

One of the advantages of Equation 3.1 is that it partitions the price-response function into a potential-demand component D and a willingness-to-pay component f(x). This is often a convenient way to model a market. For example, we might anticipate that total demand varies seasonally for some product while the willingness-to-pay distribution remains constant over time. Then, given a forecast of total demand D_t over season t, we can calculate expected demand over the season as

D_t in Equation 3.3 is a measure of potential demand while F̄(p) is the fraction of potential demand that converts to actual demand at price p. This approach allows us to decompose the problem of forecasting total potential demand from the problem of estimating price response. It also allows us to model influences on willingness to pay and total demand independently and then to combine them. For example, we might anticipate that a targeted advertising campaign will not increase the total population of potential customers D_t but that it will shift the willingness-to-pay distribution. On the other hand, if we open a new retail outlet, we might anticipate that the total demand potential for the new outlet will be determined by the size of the population served, while the willingness to pay will have the same distribution as existing stores serving populations with similar demographics. We also find this decomposition useful in the approaches to customized pricing described in Chapter 13.

Figure 3.3 Uniform willingness-to-pay distribution.

Customer willingness to pay is not constant but shifts with changes in taste and changes in circumstance. A customer’s willingness to pay for a cold soft drink increases as the weather gets warmer—a fact that the Coca-Cola company considered exploiting with vending machines that changed prices with temperature (see Chapter 14 for a discussion of the temperature-sensitive vending machine idea). Willingness to pay to see a movie is higher for most people on Friday night than on Tuesday afternoon. A sudden windfall or a big raise may increase an individual’s maximum willingness to pay for a new car. To the extent that such changes are random and uncorrelated among customers, they will not change the overall willingness-to-pay distribution, since increasing willingness to pay on one person’s part will tend to be balanced by another’s decreasing willingness to pay. On the other hand, systematic changes across a population of customers will change the overall distribution and cause the price-response function to shift. Such systematic changes may be due to seasonal effects, changing fashion or fads, or an overall rise in purchasing power for a segment of the population. These systematic changes need to be analyzed and incorporated into estimating price response.

A disadvantage of the willingness-to-pay formulation is its assumption that customers are considering purchasing only a single unit. This is a reasonable assumption for relatively expensive and durable items. However, for many inexpensive or nondurable items, a reduction in price might cause some customers to buy multiple units. A significant price reduction on a washing machine will induce additional customers to buy a new washing machine, but it is unlikely to induce many customers to purchase two. However, a deep discount on socks may well induce customers to buy several pairs. This additional induced demand is not easily incorporated in a willingness-to-pay framework—willingness-to-pay models are most applicable to big-ticket consumer items and industrial goods.

3.1.2 Properties of Price-Response Functions

The price-response functions used in PRO analysis have a time dimension associated with them. This is in keeping with the dynamic nature of PRO decisions—we are not fixing a single price that will last in perpetuity but setting a price that will stay in place for some finite period of time. The period might be minutes or hours (as in the case of a fast-moving e-commerce market), days or weeks (as in retail markets), or longer (as in long-term contract pricing). At the end of the period we have the opportunity to change prices. The demand we expect to see at a given price will depend on the length of time the price will be in place. Thus, we can speak of the price-response function for a copy machine model over a week or over a month, but without an associated time interval there is no single price-response function. Given this, every price-response function we consider has the following three properties (unless noted otherwise):

1. It will be nonnegative. That is, demand will be greater than or equal to zero at every price.³

2. It will be downward sloping. That is, demand decreases (or at least does not increase) as price increases.

3. It will be continuous. That is, it will have no gaps or jumps. The price-response function for the independent bookseller in Figure 3.1 has a small number of discrete steps because there are only three potential customers. As the number of customers increases, the price-response function will approach a more continuous function. While price-response functions are not truly continuous because of the discrete nature of customer demand and because prices can only occur at discrete intervals such as $.01, we often treat them as continuous for mathematical convenience.⁴

Under these three conditions, a price-response function can be written in the form shown in Equation 3.3. Furthermore, every price-response function obeying the three conditions above corresponds to a probability density function on willingness to pay and vice versa. The correspondence between price-response functions and probability density functions is a useful property to which we return when discussing specific price-response functions in Section 3.3.

The statement that price-response functions are downward sloping is a form of the so-called law of demand—demand decreases when prices are increased and vice versa. As we have seen, the law of demand is a natural and inevitable consequence of the fact that the price-response function is based on a distribution of customer willingness to pay. However, it should be noted that downward-sloping price-response functions do not mean that high prices will always be associated with low demand. A hotel revenue management system will experience higher average rates when occupancy is high and lower average rates when occupancy is low. What the downward-sloping property does indicate is that, in any time period, demand would have been lower if prices had been higher, and vice versa. This is consistent with both economic theory (in which consumers maximize their utilities subject to a budget constraint) and to real-life experience.

3.2 MEASURES OF PRICE SENSITIVITY

It is often useful to have a measure of price sensitivity—the rate at which demand will change as the price changes. The three most common measures for price sensitivity are slope, hazard rate, and elasticity. We define each of these measures in terms of the price-response function and show how they can be estimated from data. In order to estimate these properties from data, we assume that we have observations of demand at two different prices d̂(p₁) and d̂(p₂). Here, the caret indicates that the demand is actually being observed from data rather than derived from a model. For the measures described in this section to be accurate, p₁ and p₂ need to be relatively close to each other—say within 5% or so, and a sufficient number of observations are needed to achieve statistical significance.

3.2.1 Slope

The slope of the price-response function at a given price is measured by its derivative d'(p). Unless the price-response function is linear, the slope of the price-response function will change as the price changes. The empirical estimate of the slope of the price-response function between two prices is given by the difference in demand divided by the difference in

prices:

Because the price-response function is downward sloping, d'(p) will always be less than or equal to zero, which means that a proper estimator dˆ(p₁, p₂) of the slope should also be less than or equal to zero.

The slope can be used as a local estimator of the change in demand that would result from a small change in price. Assume that we have observed the demand at prices p₁ and p₂. Then, we can use Equation 3.4 to estimate (p₁, p₂) and, by the definition of the slope, for some price p close to p₁, we can write

That is, we can use the estimate of the slope to estimate the demand at a nearby price.

Example 3.3

A semiconductor manufacturer has estimated that the slope of its price-response function at the current price of $0.13 per chip is –1,000 chips/week. From Equation 3.5, he would estimate that a 2-cent increase in price to $0.15 per chip would result in a reduction in demand of about 2,000 chips per week and a 3-cent decrease in price would result in approximately 3,000 chips/week in additional demand.

It is important to recognize that the accuracy of the approximation in Equation 3.5 degrades for larger changes in prices and that the slope cannot be used as an accurate predictor of demand at prices far from the current price. It is also important to realize that the slope of the price-response function depends on the units of measurement being used for both price and demand.

3.2.2 Hazard Rate

The hazard rate of a price-response function at a particular price is equal to minus one times the slope of the price-response function at that price divided by the demand. That is, h(p) = –d'(p)/d(p). Because the slope of demand is negative, the hazard rate is positive. Larger hazard rates mean higher price sensitivity. When the price-response function can be represented in the form d(p) = DF̄(p), the hazard rate can be written h(p) = d(p)/F̄(p); that is, it is equal to the density function of the willingness-to-pay distribution at the price divided by the complementary cumulative distribution function. Given observations of demand at two prices, the corresponding hazard rate estimator is

3.2.3 Price Elasticity

Perhaps the most commonly used measure of price sensitivity is price elasticity, defined as the ratio of the percentage change in demand to the percentage change in price; that is,

where ε(p₁, p₂) is the elasticity of a price change from p₁ to p₂. The numerator in Equation 3.6 is the percentage change in demand, and the denominator is the percentage change in price. Reducing terms gives

The downward-sloping property guarantees that demand always changes in the opposite direction from price. Thus, the minus sign on the right-hand side of Equation 3.7 guarantees that ε(p₁,p₂) ≥ 0. An elasticity of 1.2 means that a 10% increase in price would result in a 12% decrease in demand, and an elasticity of 0.8 means that a 10% decrease in price would result in an 8% increase in demand.

ε(p₁,p₂), as defined by Equation 3.7, is called the arc elasticity between the two prices p₁ and p₂. It requires both prices to be calculated and, unless the price-response function displays constant elasticity, it will be different for any choice of p₂, which means that it is not uniquely defined for a choice of p₁. We can derive a point elasticity at p₁ by taking the limit of Equation 3.7 as p₂ approaches p₁:

In words, the point elasticity of demand at a particular price is equal to –1 times the slope times the price, divided by demand. It is also equal to the hazard rate times the price. ε(p) is also called the own-price elasticity of demand to distinguish it from cross-price elasticities, which we discuss below. Since d'(p) ≤ 0, point elasticity will be greater than or equal to zero.⁵ The point elasticity is a useful estimate of the change in demand resulting from a small change in price.

Example 3.4

A semiconductor manufacturer is selling 10,000 chips per month at $0.13 per chip. He believes that the price elasticity for his chips is 1.5. Thus, a 15% increase in price from $0.13 to $0.15 per chip would lead to a decrease in demand of about 1.5 × 15% = 22.5%, or from 10,000 to about 7,750 chips per month.

One of the appealing properties of elasticity is that, unlike slope, its value is independent of the units being used. Thus, the elasticity of electricity is the same whether the quantity of electricity is measured in kilowatts or megawatts and whether the price units are dollars or euros.

Like slope, point elasticity is a local property of the price-response function. However, the term price elasticity is often used more broadly and somewhat loosely. Thus, statements such as “gasoline has a price elasticity of 1.22” are imprecise unless they specify both the time period and the reference price. In practice, the term price elasticity is often used simply as a synonym for price sensitivity. Items with “high price elasticity” have demand that is very sensitive to price, while “low price elasticity” items have much lower sensitivity. Often, a good with a price elasticity greater than 1 is described as elastic, while one with an elasticity less than 1 is described as inelastic.

Elasticity depends on the time period under consideration, and, as with other aspects of price response, we must specify the time frame we are talking about. For most products, short-run elasticity is lower than long-run elasticity because buyers have more flexibility to adjust their buying behavior to higher prices in the long run. For example, the short-run elasticity for gasoline has been estimated to be 0.2, while the long-run elasticity has been estimated to be 0.7. In the short run, the only options consumers have in response to high gas prices are to take fewer trips and to use public transportation. But if gasoline prices stay high, consumers will start buying higher-miles-per-gallon—or even electric—cars, depressing overall demand for gasoline even further. A retailer raising the price of milk by 20 cents may not see much change in milk sales for the first week or so and conclude that the price elasticity of milk is low. But he will likely see a much greater deterioration in demand over time. The reason is that customers who come to shop for milk after the price rise will still buy milk, since it is too much trouble to go to another store. But some customers will note the higher price and switch stores the next time they shop.

TABLE 3.2

Estimated price elasticities for various goods and services

On the other hand, the long-run price elasticity of many durable goods—such as automobiles and washing machines—is lower than the short-run elasticity. The reason is that customers initially respond to a price rise by postponing the purchase of a new item. However, they will still purchase at some time in the future, so the long-run effect of the price change is less than the short-run effect.

It is important to specify the level at which we are calculating elasticity. Market elasticity measures total market response if all suppliers of a product increase their prices—perhaps in response to a common cost change. Market elasticity is generally much lower than the price-response elasticity faced by an individual supplier within the market. The reason is simple: If all suppliers raise their prices, the only alternatives customers have is to purchase a substitute product or to go without. On the other hand, if a single supplier raises its price, its customers have the option of purchasing from the competition.

Table 3.2 shows elasticities that have been estimated for various goods and services. Note that a staple such as salt is very inelastic—customers do not change the amount of salt they purchase very much in response to market price changes. On the other hand, the price elasticity of the market-response function faced by any individual seller of salt is quite large, since salt is a fungible commodity in a highly competitive market. This effect can be seen in Table 3.2 in the difference between the short-run elasticity for automobile purchases (1.2) and the much larger elasticity (4.0) faced by Chevrolet models. The table also illustrates that long-run elasticity is greater than short-run elasticity for airline travel (where customers respond to a price rise by changing travel plans in the future and traveling less by plane), but the reverse is true for automobiles (where consumers respond to price rises by postponing purchases).

3.2.4 Cross-Product Measures of Sensitivity

The measures we have considered so far have been own-product measures of price sensitivity—that is, they measure the sensitivity of demand for a product to its own price. When a seller is offering more than a single product, he may also be interested in how changing the price of one product will affect the demand for other products. If lowering the price of one product increases the sales of another, then the two products are said to be complements—an example is hot dogs and hot-dog buns. If lowering the price of one product decreases the demand of another, then the two products are said to be substitutes—think Coke and Pepsi. If changing the price of one product has no effect on the demand for another, then the two products are said to be independent. Cross-product measures quantify both the direction and the magnitude of the effects.

Assume that a seller is offering n products with prices p = (p₁,p₂, . . . , p_n) that have corresponding demands d(p) = (d₁(p), d₂(p), . . . , d_n(p)).⁶ Here, the notation d_i(p) is used to make clear that the demand for product i can depend not only on its own price but on the prices being offered for all the other products. In this case, we let ∂d_i(p)/∂p_j denote the derivative of the demand for product i with respect to price j. The sign of the derivative depends on the relationship between the two products. If ∂d_i(p)/∂p_j > 0, then products i and j are substitutes. If ∂d_i(p)/∂p_j < 0, then the two products are complements. If ∂d_i(p)/∂p_j = 0, they are independent.

We define the cross-price elasticity between product i and product j as

The cross-price elasticity between product i and product j is the percentage change in demand for product i resulting from a 1% change in the price of product j. To estimate ε_ij(p), we could run a price test in which we hold all of the prices constant, except for the price of product j. Let p₁ be the vector of original prices and let p₂ be the vector in which all prices are the same, except that price p_j has been replaced by a perturbed price p_j + δ. In this case, we can calculate an estimator of the cross-price elasticity by

Some typical values that have been estimated for cross-price elasticities are .66 for butter and margarine and .28 for beef and pork (Frank 2015). This would suggest that a 1% increase in the price of butter would lead to a 0.66% increase in demand for margarine.

Note that estimating cross-price elasticity is substantially more difficult than estimating own-price elasticity for two reasons. First, there are a lot more cross-elasticities than own-price elasticities—a seller offering n products has n own-price elasticities and n² – n cross-price elasticities. With 50 products, this would mean 50 own-price elasticities and 2,450 cross-price elasticities.⁷ The amount of price-testing required to estimate all 2,450 cross-price elasticities would be well beyond the capability of most sellers. Second, the vast majority of cross-price elasticities are quite small, almost always less than 1 and thus significantly closer to zero than own-price elasticities, which means that more samples are required to generate statistically reliable estimates.

For a seller who is offering a large number of products, calculating cross-price elasticities between all pairs of products is usually impossible. This means that the seller needs to select clusters of products among which he believes that cross-price elasticity may be significant and estimate cross-price elasticities among products in the cluster under the assumption that cross-price elasticity between a product in the cluster and one outside the cluster is negligibly small. For substitutes, the clusters consist of similar products, often defined in the same category or product family (e.g., big-screen televisions, men’s jeans). For complements, the appropriate clusters consist of items that are typically purchased together (e.g., hot dogs and hot-dog buns). Clusters of substitutes are usually derived from a seller’s preexisting product hierarchy, while clusters of complements can be generated by examination of sales records to determine what products are typically purchased together.

A measure related to cross-price elasticity that is sometimes useful is the diversion ratio, which measures what fraction of the change in demand for product i that occurs with a change in product i’s price is borne by product j. That is, if we raise the price for product i, what fraction of the reduced demand for product i will translate into increased demand for product j? If i and j are perfect substitutes—and the only two substitutes for each other—we would expect the number to be close to 1. If i and j are independent, then the diversion ratio between them should be close to 0. The formula for the diversion ratio between i and j is

where ε_ii (p) is the own-price elasticity of product i. The diversion ratio can be used to determine which product is the buyer’s second choice for a particular product. It is also used in evaluating merger proposals with the thought that a high diversion ratio for the products of two companies means that they are viewed as close substitutes by the market.

3.3 COMMON PRICE-RESPONSE FUNCTIONS

In this section we consider five common price-response functions. For each one I provide an overview of the function, explain how to calculate the different measures of price response for that function, and discuss its appropriate application. (More details and additional price-response functions can be found in van Ryzin 2012.) Information about the functions is summarized in Table 3.3.

3.3.1 Linear Price-Response Function

We have seen that a uniform distribution of willingness to pay generates a linear price-response function. The general formula for the linear price-response function is

where D > 0 and b > 0 and the notation (x)⁺ indicates the maximum of 0 and x. The linear price-response function is shown in Figure 3.4. The satiating price—that is, the price at which demand drops to zero—is P = 1/b. The slope of the linear price-response function is D'(p) = Db for 0 < p < P and D'(p) = 0 for p ≥ P. The hazard rate is h(p) = b/(1 – bp) for 0 < p < P and undefined for p ≥ P. The elasticity of the linear price-response function is ε(p) = bp/(1 − bp), which ranges from 0 at p = 0 and approaches infinity as p approaches P, dropping again to 0 for p > P. For the linear price-response function, the parameter D represents demand when p = 0 and (1 – bp)⁺ is the complementary cumulative distribution function (Equation 3.2) of a uniform distribution with f(p) = 1/b for 0 ≤ p ≤ P.

TABLE 3.3

Price-response functional forms and properties

* The slope, hazard rate, and elasticity for the linear price-response function are valid only for p < D/b. At prices above D/b, demand is equal to 0.

** denotes the density function, and the cumulative normal distribution function at the value p, with mean μ and standard deviation δ.

Figure 3.4 Linear price-response function.

We use the linear price-response function in many examples because it is a convenient and easily tractable model of market response. However, it is not a realistic global representation of price response. The linear price-response function assumes that the rate of change in demand from a 10-cent increase in price will be the same for all prices. This is unrealistic, especially when a competitor may be offering a close substitute. In this case, we would usually expect the effect of a price change to be greatest when the base price is close to the competitor’s price.

3.3.2 Exponential Price-Response Function

The exponential price-response function has the form d(p) = De^–bp with D > 0 and b > 0. The slope of the exponential price-response function is D'(p) = –bAe^–bp, which can also be written D'(p) = –bD(p). The hazard rate is constant with h(p) = b, and the elasticity is linear in price: ε(p) = bp. The exponential price-response function has the property that the logarithm of demand is linear in price—that is, ln[d(p)] = ln[D] – bp—thus the parameters of the exponential price-response function can be estimated by applying linear regression using the logarithm of demand as the dependent variable and price as the independent variable.

The exponential price-response function is finite but not satiating—there is no price at which demand goes to 0. The exponential price-response function corresponds to the case in which willingness to pay follows an exponential distribution.

3.3.3 Constant-Elasticity Price-Response Function

As the name implies, the constant-elasticity price-response function has a point elasticity that is the same at all prices. That is,

for all p > 0 and some constant elasticity ε > 0. The constant-elasticity price-response function is

where A > 0 is a parameter chosen such that d(1) = A. For example, if we are measuring price in dollars and d($1.00) = 10,000, then A = 10,000. The slope of the constant-elasticity price-response function is D'(p)=εAp^–(ε+1). The hazard rate associated with the constant-elasticity price-response function is h(p) = εp, and the elasticity is, of course, constant at ε.

An important property of the constant-elasticity price-response function is that the logarithm of demand is linear in the logarithm of price; that is, ln[d(p)] = ln[A] – εln[p]. Thus, given historic observations of price and demand, linear regression can be applied to the logarithms of the demands and the logarithm of prices to estimate the parameters of the constant-elasticity price-response function.

Examples of a constant-elasticity price-response function for different values of the elasticity parameter are shown in Figure 3.5. Note that constant-elasticity price-response functions are neither finite nor satiating. Demand does not drop to zero at any price, no matter how high, and demand continues to approach infinity as price approaches zero, which would imply that the potential demand is infinite. Because of this property, the constant-elasticity price-response function does not correspond to any well-defined distribution of willingness to pay.

It is interesting to analyze the impact of price on revenue for a seller facing constant-elasticity price response. Let R(p) denote total revenue at price p. Then R(p) = p × d(p), and

Figure 3.5 Constant-elasticity price-response functions.

Taking the slope of this function gives

Since d(p) > 0, the derivative of revenue with respect to price for a constant-elasticity price-response function will be always greater than 0 for ε < 1(inelastic demand) and will always be less than 0 for ε > 1 (elastic demand). This means that the seller facing constant inelastic price response can always increase revenue by increasing price. In fact, profit will also increase, so a seller facing constant inelastic demand should raise his price as high as he can—to infinity, if possible.

On the other hand, if ε < 1, a seller can maximize revenue by setting the price as close to zero—without actually becoming zero—as possible. If ε = 1, then total revenue will not change as the seller changes his price. Because these properties are not found in real-world markets, the constant-elasticity function is problematic as a global price-response function. For this reason, constant elasticity is better suited to represent local price response than to use as a global price-response function. In real-world markets, price elasticity typically increases with increasing price for all goods and services.

3.3.4 Logit Price-Response Functions

Both the linear and the constant-elasticity price-response functions are useful local models of price response. However, they both have rather severe limitations as global models of customer behavior. This can be seen by considering their corresponding willingness-to-pay distributions. The linear price-response function assumes that willingness to pay is uniformly distributed between 0 and some maximum value. The constant-elasticity price-response function assumes that the distribution of willingness to pay drops steadily as price increases, approaching, but never reaching, zero. Neither of these functions seem to represent customer behavior realistically.

How would we expect customers’ willingness to pay to be distributed? Consider the case where we are selling a compact car. Assume that competing models are generally similar to ours and are selling at a market price of about $22,000. We would expect customers’ willingness to pay to be concentrated near the market price. Many customers might be indifferent between our car and competing models. Some might be willing to pay a small premium for our model, and others might be willing to pay a small premium for the competing model. When the market price is $22,000, not many customers will be willing to pay $32,000 (or more) for our car. On the other hand, as we drop our price below the general market price, we will initially attract a lot of customers who will be happy to purchase our car at a discount from the market price. There would be very few customers for whom we would need to offer a $12,000 discount before they would even consider purchasing our model instead of an alternative.

This suggests that we would generally expect customer willingness to pay to follow a bell-shaped distribution with a peak around the general market price. This is, in fact, a characteristic of many price-response functions used in practice. A bell-shaped distribution of willingness to pay generates a price-response function of the general reverse S-shaped (or reverse sigmoid) form shown in Figure 3.6. When a seller prices very low, he sees lots of demand, but demand changes slowly as he increases prices because, even if his price increases by a small amount, his offering is still a great bargain. In the region around market price, demand is very sensitive to his price—small changes in price can lead to substantial changes in demand. At high prices, demand is low and changes slowly as he raises price further. Empirical research has shown that this general form of price-response function fits observed price response in a wide range of markets.

Figure 3.6 Reverse S-shaped, or reverse sigmoid, price-response function.

The most popular reverse S-shaped price-response function is the logit:

Here, a, b, and A are parameters with A > 0 and b > 0. Parameter a can be either greater than or less than 0, but in most cases we will have a > 0. Note that the potential market can be written as d(0) = Ae^a/(1 + e^a). Thus, A scales the total size of the market, while b relates to prices sensitivity, with larger values of b corresponding to greater price sensitivity. The price-response function is steepest at the point p̆ = a/b. In Figure 3.6, a/b = $1.00. This point can be considered to be approximately the market price.

The logit price-response function is shown for different values of b in Figure 3.7. Here we have fixed p̆ = $13,000, so a = $13,000b for each function. Demand is very sensitive to price when price is close to p̆. Higher values of b represent more price-sensitive markets. As b grows larger, the market approaches perfect competition. In other words, the price-response function increasingly approaches the perfectly competitive price-response function in Figure 3.2.

Logit willingness to pay follows a bell-shaped curve known as the logistic distribution. The density function for the logistic distribution is similar to the normal distribution, except it has somewhat fatter tails—that is, it approaches zero more slowly at very high and very low values. An example of the logistic willingness-to-pay distribution is shown in Figure 3.8. The highest point (mode) of the logistic willingness-to-pay distribution occurs at p̆ = a/b, which is also the point at which the slope of the price-response function is steepest. This is a far more realistic willingness-to-pay distribution than the distributions associated with either the constant-elasticity or linear price-response functions. For that reason, the logit is usually preferred to linear or constant-elasticity price-response functions when the effects of large price changes are being considered.

Figure 3.7 Logit price-response functions with A = 20,000 and p̆ = $13,000.

Figure 3.8 Willingness-to-pay distribution corresponding to the logit price-response function in Figure 3.7 with b = 0.0005. The y-axis has been scaled by a factor of 20,000.

3.3.5 The Probit Price-Response Function

Given that the most intuitively realistic density function for willingness to pay is bell-shaped, it is reasonable to ask, What about the most common bell-shaped curve of them all—the normal distribution? In fact, normally distributed willingness to pay corresponds to the so-called probit price-response function. The probit price-response function can be written as d(p) = A[1 – Φ (p; μ, σ)], where A > 0 is a measure of the overall market size and Φ (p; μ, σ) is the cumulative normal distribution (which has no closed-form representation) with parameters μ and σ. We must have σ > 0, but μ can be greater or less than (or equal to) zero.

Those with some familiarity with probability theory will recognize that μ is the mean and σ is the standard deviation of the willingness-to-pay distribution. The slope of the probit is greatest when p = μ, which is intuitive because the mean of the willingness-to-pay distribution is an estimate of the market price, and we would anticipate a high level of price sensitivity at that price. Note that, when price is equal to the mean of the probit, demand will be equal to d(μ) = A/2.

The slope of the probit is given by

d'(p) = –Aϕ(p; μ, σ).

The hazard rate for the probit is given by h(p) = ϕ(p; μ, σ) /[1 – ϕ(p; μ, σ)], and the elasticity is ε(p) = pϕ(p; μ, σ) /[1 – ϕ(p; μ, σ)]. The demand at p = 0 is given by d(0) = Aϕ(0; μ, σ).

While the probit function has some theoretical appeal because it corresponds to a normal distribution of willingness to pay, the fact that it has no closed-form representation means that the logit function is far more commonly used in practice. Research has shown that the probit and logit functions give very similar results in most cases. However, the probit function is difficult to work with because there is no closed-form version. Also, the two distributions (and their corresponding price-response functions) behave in a very similar fashion in the region around the market price. In fact, “By judicious adjustment of the coefficients, logit and probit models can be made to virtually coincide over a fairly wide range . . . and it is practically impossible to choose between them on empirical grounds” (Cramer 2003, 26). Since this is so, and the logit is much easier to work with, the logit price-response function is much more widely used than the probit.

3.4 SUMMARY

• A key input into any PRO problem is a price-response function that specifies how demand for a product changes as a function of price. The price-response function is typically nonnegative, continuous, and downward sloping.

• Three common key measures of price sensitivity are the slope, hazard rate, and elasticity of the price-response function, where the slope is defined as the derivative of the price-response function, the hazard rate is the negative of the slope divided by demand, and elasticity is the percentage change in demand that would result from a 1% change in price. All three measures are local properties of the price-response function, in that they can be used to estimate the effects of small changes in price but not large changes.

• In most cases, price-response functions can be considered as the measure of the number of people whose maximum willingness to pay (or reservation price) is greater than a certain price. In this case, a price-response function corresponds to a particular distribution of willingness to pay across a population. For example, a linear price-response function corresponds to a uniform distribution on willingness to pay.

• Linear and constant-elasticity price-response functions are both commonly used in analysis. However, both tend to be unrealistic when applied to large changes in price. In such cases, a reverse S-shaped model, such as the logit or the probit, is likely to be more appropriate.

3.5 FURTHER READING

Consumer theory specifies a set of axioms that govern consumer preferences. Assuming that these axioms hold, each consumer will have a utility function that specifies her relative utility among different potential bundles of goods. Given this utility function, the prices of all goods, and her budget, we can derive her willingness to pay for any specific good. Details of the axioms and the derivation of willingness to pay can be found in standard microeconomic textbooks such as Nicholson 2002 and Frank 2015.

A succinct mathematical treatment of different price-response functions and their properties can be found in van Ryzin 2012.

3.6 EXERCISE

The average ticket prices for concerts held by six different artists in 2018 were as follows:⁸

Given that the incremental costs of mounting a show are roughly the same for all performers, what do you think accounts for the wide variation in the average ticket price commanded by the artists? Do you believe that all of these artists are seeking to maximize expected revenue from their concerts? If not, why not, and what might they be trying to do? During this same period, the average ticket price to see Springsteen on Broadway, in which Bruce Springsteen performed alone in a Broadway theater, was $508.66. Why do you think this price is so much higher than the other ticket prices listed?

NOTES

1. The existence of a willingness to pay (which may be less than zero) for every good in the market at every point in time can be derived from a set of simple and apparently intuitive axioms on customer preference. An example is the axiom of completeness, which states that, given any two products A and B, a consumer either prefers A to B or prefers B to A or is indifferent between them. Another example is the axiom of transitivity, which states that if a consumer prefers A to B and prefers B to C, she must also prefer A to C. With the addition of a few other axioms, we can derive the result that, for any product at any time, a consumer has a maximum willingness to pay for the product. A discussion of the axioms and a description of how they lead to a maximum willingness to pay for every product in the market can be found in any standard microeconomics textbook discussion of consumer theory. Similarly, the existence of a business’s maximum willingness to pay is a consequence of the standard theory of the firm in which firms act to maximize profit. Introductions to consumer theory and the theory of the firm can be found in any standard microeconomics textbook such as Frank 2015 or Mankiw 2017. Deviations from these assumptions are considered irrational and their implications are discussed in Chapter 14.

2. Here we have arbitrarily assumed that, if the surplus for two channels is tied, a customer will purchase from the leftmost in the table of the two channels with the same surplus.

3. This means that what we are selling is a good—something people are willing to buy—rather than an illth—something people are willing to pay to get rid of. This is not a restrictive assumption—we can convert an illth with negative price to a good with positive price by exchanging the buyer and the seller. Thus, instead of assuming that people sell trash (an illth) at a negative price, it is more natural to assume that people buy trash removal (a good) at a positive price.

4. In reality, strict continuity of price-response functions often does not hold. In particular, prices for most items sold in the United States do not vary by less than 1 cent and most items are sold in discrete units. We could easily have the situation where d($5.11) = 5,000 and d($5.10) = 5,010. In this case, price response is not technically continuous or invertible—there is no price such that d(p) = 5,005. However, as long as these jumps are small relative to overall demand, we can act as if the price-response function is continuous between consecutive prices and round to the nearest penny once we have solved for the right price, knowing that our final price will be off by at most 1 cent.

5. Here we use the convention that own-price elasticities should be positive, hence the minus sign in Equations 3.6, 3.7, and 3.8. This means that the products obeying the law of demand will have positive own-price elasticities. This enables us to associate a term like “more elastic” with higher values of the elasticity, which seems intuitive. However, note that, in the literature, elasticity is sometimes calculated without the minus sign, which means it will be reported as negative.

6. Throughout the book, I use boldface variables to denote a vector. Thus, in the text, p = (p₁, p₂, . . . , p_n) is a vector with n component prices, and p_i is the ith component price of the vector.

7. The number of potential cross-price elasticities with n products is the number of combinations of different products, or n² – n. For products whose price is small relative to a consumer’s total income, the cross-price elasticity of product i with respect to product j will be the same as the cross-price elasticity of product j with respect to product i. This means that, for example, the percentage change in the demand for margarine resulting from a 1% change in the cost of butter will be the same as the percentage change in the demand for butter resulting from a 1% change in the cost of margarine. In this case, the number of cross-price elasticities that need to be computed is equal to n(n – 1)/2. On the other hand, if the price of one or both of the products is large relative to consumer income, then the cross-price elasticities are likely to be quite different. As an example, a 10% drop in housing prices is likely to stimulate far more home purchases with accompanying homeowner’s insurance than a 10% drop in homeowner’s insurance rates is likely to stimulate new home purchases.

8. Concert ticket prices are as posted by Pollstar (https://www.pollstar.com) in August 2018.