5

Optimization

Chapter 4 introduces the concept of a price-response function, which specifies the demand that a seller will experience for a product at any positive price. This chapter shows how to use a price-response function to generate the optimal price, where optimal can mean either profit-maximizing or revenue-maximizing. We begin by considering the elements of profit, including the appropriate use of costs and ancillary revenues in maximizing profit. In Section 5.2, we derive conditions that must be satisfied for a price to be optimal (again, we consider both revenue-maximizing and profit-maximizing prices) and that can be used to calculate optimal prices. Sections 5.3 and 5.4 present conditions that guarantee that a unique optimal price exists and extend the optimization approach to the case when a seller is setting prices for multiple products simultaneously. Section 5.5 describes a data-driven approach to optimizing a single price—that is, an approach in which we do not assume any particular form for price response. Section 5.6 discusses the issue of competitive response, and finally, Section 5.7 shows how an efficient frontier can help a seller who is interested in two different objective functions—say, maximizing profit and maximizing revenue—visualize the trade-off between them.

5.1 ELEMENTS OF CONTRIBUTION

The goal of pricing optimization is typically to maximize total contribution, defined as incremental contribution times sales. Unit contribution is the expected incremental revenue from each transaction minus the expected incremental cost associated with each transaction. In the case where we are selling a single unit at a known price and we know the incremental cost associated with the unit, contribution per unit can be written as p – c, where p is price and c is incremental cost. Throughout the rest of the book, we typically write unit contribution in this form. However, in many cases, calculating the revenue and costs associated with a customer commitment may not be so straightforward. For example, in many freight industries, the customer commitment is a contract to carry all of the freight tendered for the next year at an agreed-on set of rates. At the time the commitment is made, how much freight will actually be tendered and on which lanes is unknown to both the shipper and the carrier, and so both the revenue and the cost of service associated with the contract are uncertain. In this case, expected revenue and expected cost of the contract need to be modeled explicitly.

Most industries are not as extreme as freight; however, in many cases, the calculation of the revenue and associated cost may not be entirely straightforward. This section addresses two particular issues in estimating contribution—calculating incremental costs and incorporating ancillary revenue.

5.1.1 Incremental Costs

Cost seems like it should be the simplest component of pricing and revenue optimization. After all, a company may not know the price-response functions it faces, but surely it should know its costs. Unfortunately, things are not quite so simple. With any exposure to the field of accounting, one quickly realizes that the simple question “How much did it cost us to provide this product to this customer through this channel on Thursday?” does not have a unique well-defined answer. Rather, the cost of providing even a simple product depends on what the answer will be used for. As a consequence, accountants have derived a host of product-costing methodologies, including fully allocated costs, partially allocated costs, marginal costs, avoided costs, and financial costs, to name just a few.

While each costing methodology has its place, pricing and revenue optimization decisions are based on the incremental cost of a customer commitment, where incremental cost is defined as follows.

The incremental cost of a customer commitment is the difference between the total costs a company would experience if it makes the commitment and the total cost it would experience if it does not.

Calculating incremental cost for a customer commitment depends critically on the context and nature of the commitment. Here are some examples.

• A drugstore sells bottles of shampoo and replenishes its stock weekly. Because selling a unit will result in an additional unit order, the incremental cost for selling a bottle of shampoo is the wholesale unit cost.

• An airline is considering whether to sell a single seat to a passenger on a future flight. The incremental cost is the additional meal and fuel cost that would be incurred from flying an additional passenger plus any commissions or fees the airline would pay for the booking.

• A retailer buys a stock of fashion goods at the beginning of the season. Once he has bought the goods, he cannot return them and cannot reorder. During the season he wants to set and update the prices that will maximize the total revenue he will receive from the fashion goods. Since the cost of the goods is sunk and selling a unit will not drive any additional future sales, his incremental cost per sale is zero.¹

• A distributor is bidding for a year-long contract with a hospital to be the preferred provider for disposable medical supplies. From previous experience, the distributor knows that this hospital is an expensive customer requiring high levels of customer support and having wide variances in orders and high rates of product returns. The expected incremental cost of the contract includes not only the expected cost of the items the hospital will purchase but also the expected cost of customer service, operating costs and holding costs driven by the wide variance in orders, and the expected costs of returns. In this case, these costs need to be associated with the contract, but some of them, such as the cost of customer service, may not be specifically associated with any particular product.

These examples illustrate some of the key characteristics of incremental cost. These characteristics can be summarized as follows.

• Incremental cost is forward looking. It is based on the effect a customer commitment will have on future costs. Costs that have already been taken or that are driven by decisions already made prior to the customer commitment are sunk and do not enter into the calculation of incremental cost.

• Incremental cost is marginal. It is the expected cost of making this customer commitment. The incremental cost of making this commitment may not be the same as the average cost of similar commitments made in the past.

• Incremental cost is not fully allocated cost. Only costs that change as the result of a customer commitment are part of the incremental cost. Overhead, or fixed costs of staying in business, generally should not be allocated to any specific commitment. As a result, the incremental cost of a customer commitment is usually less than the fully allocated cost. Further, the sum of the incremental costs of all commitments will be less than the total operating cost of the company because an unallocated residual fixed cost will remain after all the incremental costs have been totaled.

• The elements of incremental cost can depend on the type, size, and duration of the commitment. Specifically, if the customer commitment is a multiunit order, then order-based or setup costs need to be allocated across all the units in the order.

• The incremental cost of a commitment can be uncertain. Trucking companies such as Roadway Express and Yellow Freight sell contracts to shippers. Each contract covers the next year and commits the trucking company to carry all the freight tendered by the customer at an agreed-on tariff. The incremental cost associated with one of these contracts is likely to be highly uncertain at the time the commitment is made. First, the amount, timing, and origin and destination of the freight the customer will tender over the next year is uncertain. In addition, the cost of moving the customer’s freight will depend on the freight the company will move for other customers, which is also uncertain. This cost will depend in part on what portion of the customer’s freight is backhaul business, which can utilize excess capacity on existing truck trips, and what portion will be headhaul freight, which will require adding new truck trips.

In each case, the calculation of incremental cost requires understanding the nature of the customer commitment and then estimating the additional costs that would be generated by making the commitment—or, equivalently, the costs that would be avoided by not making the commitment. A simple rule of thumb is this: If changing a price would not change a particular cost, then that cost should not influence the price.

TABLE 5.1

Incremental cost elements for Acme Hotel

The methodology for calculating incremental costs is closely related to activity-based costing (ABC). Activity-based costing is a management accounting approach to allocating costs to their underlying causes in order to give a clearer view of the real drivers of cost within an organization. In our case, the underlying cost-drivers are various aspects of the customer commitment. As an example, consider the hypothetical incremental cost elements associated with the Acme Hotel shown in Table 5.1. Three categories of cost are represented: a transaction cost that is generated by every booking, a daily cost that depends on the number of days in the rental, and a commission. The commission is 15% for bookings through online booking platforms such as Expedia and Orbitz and 0 for bookings through Acme Hotel’s own website.

Example 5.1

Both Mr. Smith and Ms. Jones want to stay for two nights in the Acme Hotel with costs shown in Table 5.1. The nightly rate available to both Smith and Jones is $150.00/night. Mr. Smith is booking through an online booking platform, while Ms. Jones is booking through the Acme Hotel website. The incremental cost for Mr. Smith’s two-night stay is $31.26 + 2(.15 × $150 + $12.42) = $101.10, while the expected unit cost for the same rental to Ms. Jones is $31.26 + 2 × $12.42 = $56.10.

Note that, in calculating the incremental costs, only cost elements that vary with aspects of the customer stay are included. Costs such as front-desk salary and corporate overhead are not included because they will not change, depending on whether Mr. Smith or Ms. Jones makes a booking.

5.1.2 Ancillary Products and Services

In many industries, particularly service industries, ancillary products and services can be an important source of revenue. An airline passenger may also purchase a beer, a meal, or duty-free goods aboard her flight. Sporting events and concerts sell food, beverages, and “merch” (T-shirts and other souvenirs). Insurance is a highly profitable sideline for automobile sellers, rental car companies, online travel booking companies, and mortgage lenders. Paid warranties are very profitable add-ons to sales of automobiles, appliances, and consumer electronics. These additional sources of contribution from a customer above and beyond the price she pays are called ancillary products or services. An ancillary product is one that would not be sold without the sale of the main product—thus, no one without a ticket to the soccer game can buy a beer in the stadium. This differentiates ancillary products from product complements that may be sold together but can also be sold individually, such as hot dogs and hot-dog buns.

Ancillary contribution is the net contribution from sales of ancillary products or services. To maximize expected contribution, a revenue management company needs to include the estimated ancillary contribution for each booking request. This can be challenging because the amount of ancillary contribution that will be received from a prospective booking is typically unknown at the time a pricing or availability decision needs to be made.

To incorporate ancillary contribution into revenue management, we typically use the expected ancillary contribution per customer, which we will label π^C. If each attendee at a soccer match purchases, on average, 1.4 beers and the incremental contribution (not price) from each beer is $1.00, then π^C = $1.40. The expected contribution per customer is p + π^C – c. In general, higher values of π^C will tend to lead to lower optimal prices—thus, an event that expects to make high profits from ancillary revenue should set its ticket prices lower: it can make up for the lost revenue per ticket by the ancillary contribution from the additional customers attracted by the lower price.

Expected ancillary contribution may vary among elements of the pricing and revenue optimization (PRO) cube, and this can provide a rationale for varying price by product, customer segment, and/or channel. For example, many borrowers who take out a consumer loan in the UK also take out insurance against defaulting on the loan because of job loss or injury. Insurance is highly profitable for most lenders, which means that it is a sensible tactic for lenders to offer lower prices to customer segments that the lenders have determined are more likely to take out insurance. As a more extreme example, hotels associated with casinos commonly estimate the gambling revenue that they can expect different customer segments to generate. They then charge lower prices (even 0) or provide expanded availability to customer groups that they believe will generate high levels of gambling revenue.

Example 5.2

Both Mr. Smith and Ms. Jones have previously stayed in hotels in the Acme chain many times. The customer relationship management (CRM) system for Acme shows that Mr. Smith has never ordered room service or eaten in the hotel restaurant in any of his previous stays. Ms. Jones, however, has always ordered dinner from room service, from which the hotel makes an average contribution of $15 per meal. On this basis, Acme forecasts that Mr. Smith will take no meals at the hotel and that Ms. Jones will order two room-service dinners. In this case, referring back to Example 5.1, the total expected contribution from each of them would be as follows:

In Example 5.2, incorporating expected ancillary contribution and incremental cost into the calculation reveals that Ms. Jones is expected to be about 38% more profitable than Mr. Smith, even though they are paying the same room rate. The hotel operator could make use of this information by providing a coupon for Ms. Jones for a lower rate. Alternatively, if there is only one remaining room available at the property, a profit-maximizing hotelier would prefer to provide availability to Ms. Jones rather than Mr. Smith. Thus, revenue management decisions such as those discussed in Chapter 8 should be based on maximizing expected contribution.

While ancillary revenue is important in some applications, it is not explicitly included in most of the calculations that follow. However, in cases where expected ancillary contribution can be significant, the contribution term p – c should be understood as including the additional term π^c, representing expected ancillary contribution.

5.2 THE BASIC PRICE OPTIMIZATION PROBLEM

The difference between the price at which a product is sold and its incremental cost is called its unit margin. The sum of the margins of all products sold during a time period is called the total contribution. We denote the total contribution achieved at a price p by Π(p). In most cases, the seller’s goal is to maximize total contribution. The unit margin from a product priced at p is m(p) = p – c, where m(p) is unit margin and c is incremental cost. The basic single-price optimization problem is

The fundamental trade-off faced by the seller seeking to maximize total contribution is illustrated in Figure 5.1. If the seller’s price is equal to incremental cost, his total contribution will be 0. As the seller increases the price from c, unit margin increases faster than demand decreases so that total contribution increases. At some price, the rate at which demand decreases balances the rate at which unit margin increases—this is the price that maximizes expected total contribution. In most cases, the total-contribution function Π(p) is hill shaped, with a single peak, as shown in Figure 5.1. The top of this peak is the maximum total contribution the supplier can realize in the current time period, and p* is the price that maximizes the total contribution. Note the fundamental lack of symmetry in the total-contribution curve: The supplier can lose money by pricing too low (below incremental cost), but he cannot lose money by pricing too high—the worst that can happen is demand gets driven to zero. This means that a risk-averse seller would tend to overprice.

Figure 5.1 Total contribution as a function of price.

5.2.1 Optimality Conditions

The problem in Equation 5.1 is an unconstrained optimization problem, and standard optimization theory tells us that the maximum will occur at the price for which the derivative of Π(p) is equal to 0. The derivative of Π(p) with respect to price is

To find the price that maximizes total contribution, we set Π'(p) = 0, or

Thus, to maximize total contribution, p* must satisfy

We can also establish conditions for a price to maximize total revenue, defined by R(p) = pd(p) Let p̂ denote the price that maximizes expected revenue; then the equivalent condition to Equation 5.3 for the revenue-maximizing price is

Equation 5.4 implies that the price that maximizes revenue is the same as the price that maximizes total contribution if costs were set to 0. This means maximizing revenue is equivalent to maximizing profit in industries in which incremental costs are either zero or very close to zero. This includes some service industries such as movie theaters or sporting events, as well as television and online advertising. Incremental costs are zero (or at least very small) in many cases in which a company has already purchased a fixed amount of nonreplenishable inventory, such as in the case of fashion-goods retailers, who purchase inventory for an entire season at one time. Once the inventory has been purchased, the incremental cost of a sale is zero—and the seller should set a price that maximizes revenue from his fixed stock of inventory.

Example 5.3

A widget maker is looking to set the price of widgets for the current month. His unit cost is $5.00 per widget and his demand for the current month is governed by the linear price-response function d(p) = (10,000 – 800p)⁺. (Recall that the notation (x)⁺ indicates the maximum of 0 and x.) In this case, d'(p) = – 800. Substituting into the condition 5.3, we obtain the condition on the optimal price that (10,000 – 800p*)⁺ = 800(p* – $5.00), which means that p* = $8.75. At the optimal price, total widget sales will be 3,000 and total contribution will be $11,250. If the widget maker were looking to maximize revenue, he would choose the price p̂ such that (10,000 – 800p̂)⁺ = 800p̂ or p̂ = $6.25 with corresponding sales of 5,000 and revenue of $31,250.

Conditions 5.3 and 5.4 can be used to define three conditions that can be used to find profit-maximizing and revenue-maximizing prices.

Marginal revenue equals marginal cost. We can rewrite Equation 5.3 as

The term on the left-hand side of Equation 5.5 is marginal revenue—the derivative of total revenue with respect to price. This is the rate at which revenue changes as price increases. Typically, marginal revenue is greater than zero at low prices but less than zero at higher prices. The term on the right-hand side of Equation 5.5 is marginal cost: the rate at which cost is changing with price. Note that marginal cost is always less than or equal to zero—an increase in price results in lower demand, which in turn leads to lower total costs. Equation 5.5 is a mathematical expression of the well-known principle that total contribution is maximized when marginal revenue equals marginal cost.

The equivalent condition for revenue maximization is total revenue is maximized when marginal revenue equals zero. This makes sense—as we increase price from zero, total revenue initially increases with price, implying that marginal revenue is positive. However, at some point, demand begins to decrease at a rate such that total revenue will decrease as price is increased. The point at which revenue stops increasing and begins to decrease is the point of maximum revenue and the point at which marginal revenue equals 0.

Figure 5.2 shows the marginal revenue and marginal cost curves for the widget maker in Example 5.3. Marginal revenue decreases linearly. Marginal cost is constant at − $4,000. The contribution-maximizing price lies where these two curves cross—in this case at p* = $8.75. The revenue-maximizing price occurs where the marginal revenue line crosses $0, in this case at p̂ = $6.25.

Figure 5.2 Marginal revenue and marginal cost.

Equation 5.5 provides further useful guidance on price changes that could improve total contribution. If marginal revenue is greater than marginal cost, then the seller can increase his contribution by increasing price. If, on the other hand, marginal revenue is lower than marginal cost, he should decrease his price to increase contribution.

Hazard rate equals inverse margin. Equation 5.5 can be rearranged as –d'(p*)/d(p*) = 1/(p* – c). Recalling that –d'(p*)/d(p*) = h(p), where h(p) is the hazard rate associated with the price-response function, we find that the contribution-maximizing price satisfies

Stating that in words, at the contribution-maximizing price, the hazard rate is equal to the inverse margin. The equivalent condition for revenue maximization is

which means that at the revenue-maximizing price, the hazard rate is equal to the inverse price.

Elasticity equals inverse unit margin. Recall from Section 3.2.3 that elasticity is equal to price times hazard rate; that is, ε(p)=h(p)p. We can derive an elasticity-based condition for contribution maximization by multiplying both sides of Equation 5.6 by p*, which gives us

In words, at the contribution-maximizing price, the elasticity is equal to the inverse unit margin rate, where unit margin rate is defined as the unit margin as a fraction of total price.² Equation 5.7 can be rearranged to derive another formula for the optimal price:

Equation 5.8 can be used to calculate the optimal price when both the elasticity and the incremental cost are known.

Example 5.4

A seller faces a constant-elasticity price-response function with an elasticity of 5 and his cost is $3.00. This means that his profit-maximizing price is given by

The equivalent condition for revenue maximization is

which means that, at the revenue-maximizing price, the elasticity is equal to 1.

5.2.2 Price Elasticity and Profitability

We pause to consider the implications of the relationship between price elasticity and unit margin specified in Equation 5.7. Equation 5.7 can be used to estimate the implicit elasticity—that is, the belief that a seller must have about the elasticity of his price response if he believes that his current price is maximizing expected profit.

Example 5.5

A seller believes he is pricing optimally, and his unit margin rate is 20%. This can only be true if the elasticity at his current price is equal to 5.

Imputed price elasticity can serve as a reality check on the credibility of a company’s current pricing.

An extremely important implication of Equation 5.7 is that higher unit margin rates are associated with products that have lower price elasticity.

Example 5.6

Norton’s Salt is an extremely common brand of table salt sold in bulk in grocery stores. The unit cost for Norton’s Salt is $1.50 per one-pound container and it has a price elasticity of 6. La Cloque Rouge Fleur du Sel is a specialty salt harvested by hand from certain salt ponds in the south of France and sold in fancy packaging for use in gourmet cooking. The unit cost for La Cloque Rouge is $4.20 per one-pound package and it has a price elasticity of 1.5. The contribution-maximizing price for Norton’s is × $1.50 = $1.80, and the contribution-maximizing price for La Cloque Rouge is × $4.20 = $12.60. Thus, the unit margin is $0.30 per pound for Norton’s and $8.40 per pound for La Cloque Rouge.

Example 5.6 illustrates the importance of price elasticity in determining unit margin. In general, high elasticity is associated with highly commoditized products, which have easy substitution and lots of competition. These products will have low unit margin rates, which means that they typically need to be sold in large volumes to cover fixed production costs and cost of capital. Low elasticity is associated with highly specialized or unique products and services, luxury products, and less competitive markets. These products will typically demonstrate higher unit margin rates, which means that they can profitably be produced in lower volumes. The primary job of marketing and branding can be described as seeking to lower price elasticity by creating product differentiation. In the example, La Cloque Rouge may be chemically indistinguishable from Norton’s (or not), but its branding, market positioning, and packaging have distinguished it to the extent that it enjoys a much lower elasticity and, hence, can command a higher unit margin.

5.3 EXISTENCE AND UNIQUENESS OF OPTIMAL PRICES

We have shown that a profit-maximizing price can be determined by solving any one of the optimality conditions in Section 5.2.1 for the value of p*. However, we have not shown that such a price actually exists. We have also not shown that, if such a price does exist, it is unique—if hundreds of different prices satisfy one of the optimality conditions, then that condition is not very useful. In addition, as those who remember their calculus will realize, a price that satisfies the first-order conditions is not necessarily a maximizer; it could be a point that minimizes profit (or more rarely an inflection point in which profit is neither maximized nor minimized). This section presents conditions under which a price that meets the optimality conditions listed in Section 5.2.1 will be the price that uniquely maximizes profit.

For what follows, we assume three additional properties of price-response functions in addition to the three specified in Section 3.1.2.

1. Differentiable. The price-response function has a unique derivative at each point.

2. Increasing elasticity. Elasticity increases (or is constant) with increasing price.

3. Elasticity greater than 1. Elasticity exceeds 1 for some finite price.

Of these conditions, number 2 may seem the least obvious; however, it is a property of all commonly used price-response functions as well as a real-world phenomenon. You can confirm that all of the price-response functions in Table 3.3 have elasticities that increase with price—except for the constant-elasticity function, which has the same elasticity at every price. The intuitive reason that elasticity should increase with price is that demand decreases as price gets higher so that a small change in price has a much larger percentage impact at a high price than at a low price.³

Finally, we require that the elasticity is greater than 1 for some price. This eliminates the case in which elasticity increases with price but never exceeds 1. In this case, as discussed above, the seller could always increase both revenue and profit by increasing his price. Since this is unrealistic, it must be the case that elasticity exceeds 1 for some price and thus, since elasticity is an increasing function of price, it will exceed 1 for all prices greater than that.

We can now make some statements about existence and uniqueness of optimal prices. If properties 1–3 above hold for a price-response function that also obeys the four conditions listed in Section 3.1.2 and unit cost is greater than or equal to zero, then

1. There exist unique profit-maximizing and revenue-maximizing prices p* and p̂ with p* ≥ p̂.

2. The profit-maximizing price p* satisfies p* ≥ c so that the seller does not realize negative contribution at the profit-maximizing price.

These statements can be proved mathematically, but they can also be seen clearly in Figure 5.3. Elasticity is a continuous, increasing function of price denoted by ε(p). The revenue-maximizing price p̂ occurs where the elasticity is equal to 1. In the figure, this occurs where the elasticity curve intersects 1. The intersection of the elasticity curve with (p – c)/p occurs when ε(p) = (p – c)/p, which, according to condition 5.7, occurs at the profit-maximizing price p*. It is clear that, for any nonincreasing elasticity curve, there must be one and only one intersection between the elasticity curve and each of the two downward sloping curves, which demonstrates the existence and uniqueness of the two prices p̂ and p*.

Figure 5.3 Existence and uniqueness of the revenue-maximizing price p̂ and contribution-maximizing price p*.

It is also obvious from Figure 5.3 that the intersection of the elasticity curve with (p – c)/p must occur at a price p*>c. This means that the unit margin at the profit-maximizing price is greater than or equal to 0. Furthermore, (p – c)/p ≥ 1, which means that the revenue-maximizing price will be less than or equal to the profit-maximizing price, with equality occurring only in the case that c = 0. Note that the revenue-maximizing price may be less than the unit cost (as it is in Figure 5.3), in which case a revenue-maximizing seller would lose money on each sale.

One way to check the optimality of p* is to use the marginality test, which states that a particular price is maximizing contribution only if raising the price by a penny or lowering the price by a penny results in reduced contribution. The principle of marginality should be fairly intuitive—if we could increase contribution by changing the price, then the current price would not be optimal—but it is a useful check nonetheless. Table 5.2 shows the results of varying the price in Example 5.3 from $8.74 to $8.76. As expected, the price of $8.75 results in the highest margin of the three alternatives. However, the pattern of change among the prices is instructive: At $8.74 we are selling eight more units per month, but this is not enough to make up for the lost margin per unit. At $8.76 we are making a penny more per unit sold, but this is offset by the loss of sales of eight units. The optimal price, $8.75, exactly balances the gain in units sold from the potential loss of margin from raising prices further.

TABLE 5.2

Impact of price on marginal contribution near the optimal price

5.4 OPTIMIZATION WITH MULTIPLE PRICES

The optimality conditions presented in Section 5.2.1 pertain to a seller selling a single product. Of course, in many cases, a seller will be setting prices for many products, representing multiple cells in the PRO cube. If demand for products is independent, then the seller can set the price for each product independently using any one of the approaches described so far. However, in many cases, product demands will not be independent—some products may be substitutes for each other, and some may be complements. In this case, the optimal price for a particular product will depend on the prices of other products, and optimal prices for different products cannot be calculated independently.

Assume that a seller is setting n prices given by the vector p = (p₁, p₂, . . . , p_n) and demand mappings such that d_i(p) is the demand for product i as a function of the entire price vector p. Then the profit optimization problem is to find the vector of prices that solves the optimization problem

subject to p_i ≥ l_i, where l_i is a lower bound to be imposed on the price of product i and c_i is the incremental cost of product i. Lower bounds are required because an unconstrained optimum might result in some prices that are substantially lower than cost, or even less than 0. Note that the demand for each product depends not only on its own price but also on the price of all other products.

The first-order necessary conditions for the optimal prices in the multiprice case can be written as

j

The condition for prices to be optimal can be written as

where V_i(p*) is defined by

With a little algebra, we can derive the optimality condition that must hold for each product:

If the price of product i does not influence the demand for any product other than itself, then V_i(p*) = 0 and Equation 5.10 reduces to Equation 5.3. In this case, the optimal price of product i can be found independently of the prices of any other products using methods described above. If V_i(p*) > 0, it means that, as a whole, product i has more complements than substitutes and its price should be lower than its independent optimal price because its sales stimulate other sales. If V_i(p*) < 0, then substitution dominates complementarity and the optimal price of product i is higher than its individual optimal price. The idea is that, as we raise the price, some of the loss from lost sales is recaptured through sales of substitutes.

The problem of setting optimal prices in the face of substitutes and complements requires finding the values of that simultaneously satisfy Equation 5.10 for all i. This can be a complex undertaking, but I propose an iterative algorithm that can be used if we know all elasticities and cross-derivatives.

Optimal Multipricing Algorithm

0. Set all prices for i = 1, 2, . . . ,n. Choose a convergence criterion Δ > 0 and set the iteration counter k = 1.

1. Loop through all products 1, 2, . . . ,n.

a. For product i, calculate that satisfies

b. Calculate the change in price between iterations:

3. If max _i δ_i < Δ, stop and set p* = p^k. Otherwise set k ← k + 1 and go to step 1.

If complementary and substitution effects are not too strong (as is usually the case), the multipricing algorithm will converge to the optimal prices. However, the algorithm has a number of obvious implementation challenges, notably its assumption that we know not only own-price elasticity but the cross-price derivatives for all combinations of products.

5.5 A DATA-DRIVEN APPROACH TO PRICE OPTIMIZATION

Applying the optimality conditions described in Section 5.2.1 to find the profit-maximizing or contribution-maximizing prices requires that we know the price-response function for a product. This is a strong requirement. Assume that we know nothing about the price-response function for a product, except that it satisfies the conditions listed in Section 5.2.1. Furthermore, we assume that the following two additional conditions hold:

1. We are able to test the demand response at different prices.

2. The product sells sufficiently rapidly that we can quickly observe demand responding to different prices.

Figure 5.4 Regions for raising, lowering, and holding price in the learning-and-earning algorithm.

In this case, we can use a multi-armed bandit approach to determine the optimal price that maximizes a given objective function. This is a test-and-learn (or learning-and-earning) approach in which we observe the payoff from different decisions—in this case, different prices—and gradually favor the prices that generate the highest total contribution. (More details on multi-armed bandit approaches are given in Appendix B.)

To see how this approach can be used to update prices for a seller who seeks to maximize profit, consider Figure 5.4, which graphs elasticity ε(p) and the inverse unit margin curve p/(p – c) for some product for every p ≥ c. As in Figure 5.3, the unique profit-maximizing price is where the two curves cross. Note that, if p/(p – c) > ε(p), then p < p*, and we could move closer to the optimal price (and increase profitability) by increasing the price. If p/(p – c) < ε(p), then the current price is too high, and we could move closer to the optimal price by lowering the price.

The seller does not know the price-response function and thus does not know the elasticity at any price. However, he has the capability to set different prices and observe demand at those prices. This suggests the following approach to finding the optimal price.

Learning-and-Earning Algorithm for Price Optimization

0. Choose initial parameters δ > 0, Δ > 0, and ? > 0.

1. Set k = 0, and start with an initial test price p₀ > 0.

2. For some small δ, test demand at prices p_k and p_k + δ, which we denote by d(p_k) and d(p_k + δ), respectively. Estimate local elasticity by

3. Compare (p_k) to (p_k – c) / p_k. Update depending on the relationship between the two:

a. If (p_k) is close to (p_k – c) / p_k, say |(p_k – c) / p_k – (p_k)| < Δ, then p_k is close to p*. Stop the algorithm and set the price to p_k.

b. If (p_k – c) / p_k – (p_k) > Δ, the current price is too low. Set p_k+1 ← p_k + ? and k ← k + 1 and go to step 2.

c. If (p_k) – (p_k – c) / p_k > Δ, the current price is too high. Set p_k+1 ← p_k – ? and k ← k + 1 and go to step 2.

The learning-and-earning algorithm described above has the great advantage that it makes no assumption about the functional form of the price-response function. Rather, it estimates the elasticity at the current price and determines whether to raise the price, lower the price, or hold the price the same. While it does not require a form for the price-response function, it does require that the following values be specified:

• δ is the distance between prices that are being tested to estimate elasticity.

• Δ is the stopping condition—the difference between the estimated elasticity and the inverse unit margin.

• ? is the amount by which the price is adjusted if the elasticity is not sufficiently close to the inverse unit margin.

These values are called hyperparameters and they determine how quickly (and if at all) the approach will converge to the optimal price. For example, if ? is set too small, the price will approach the optimal price very slowly—which means that a lot of time will be spent learning rather than earning. However, if ? is set too large, the price may not converge at all: it may bounce back and forth between a price that is below the optimum and one that is above the optimum. For this reason, it is usually ideal to reduce ? from iteration to iteration so that the steps get smaller over time. One common approach is to set ? as the initial step size and then set the step size on iteration k according to ?_k = ?/k. (More details can be found in Sutton and Barto 2018, sec. 2.5.) In general, setting and updating hyperparameters involves quite a bit of tuning—starting with plausible values and adjusting with experience to improve performance.

In theory, the data-driven algorithm would be run once until it converged to an optimal price, which would then remain in place forever. A finite period of learning would be followed by an infinite future of earning. However, things are rarely that simple. In reality, it is likely that the price-response function (and hence the local elasticity) will change over time. Reasons for change include competitive actions, seasonality, life-cycle changes, introduction of substitutes, and changes in consumer tastes, among other factors. With such events, the current price is likely to be no longer optimal. The most common cure for this issue is to initiate testing for a price that is believed to be optimal with some small probability in each period. If elasticities have changed, then initiate the learning-and-earning algorithm for price optimization again to determine the optimal price under the new conditions.

5.6 COMPETITIVE RESPONSE AND OPTIMIZATION

Chapter 4 shows that if we have access to competitive prices, we can incorporate them in price-response estimation. However, this would seem to take us only partway to nirvana. After all, if we are taking competitive prices into account in setting our prices, we should anticipate that our competitors will take our price into account when they set their prices. If we drop a price, we should anticipate the possibility that competitors will match it, possibly erasing much of the additional demand we might otherwise expect. The results of changing a price would certainly be different, depending on whether our competitors decided to match it.

Attempts to predict competitive response and incorporate it into current pricing decisions generally fall within the realm of game theory, and there is a vast literature on the application of game theory to pricing. While game theory is an important tool in strategic pricing, it is less relevant to the tactical decisions of pricing and revenue optimization. Most pricing and revenue optimization systems do not seek to forecast competitive response and incorporate it explicitly into their recommendations. There are many reasons for this, but I believe that one is particularly important: pricing and revenue optimization is based on playing the best response (i.e., finding the expected contribution-maximizing price) to whatever the competitors are currently doing. Theoretical examinations of markets find that this is almost always a good thing to do and often the best possible action. For example, if price response in a market is well modeled by a multinomial logit function and sellers all set their prices to maximize profit based on the prices set by others, the market will converge to a unique set of equilibrium prices for all sellers. Game theory is not required in this case to determine the profit-maximizing price for any seller (Gallego et al. 2006).

The philosophy of pricing and revenue optimization is to make money by many small adjustments, searching for and exploiting many small and transient opportunities of profit as they appear in the marketplace. It is about making a little more money from each transaction. It is not about finding the great pricing move that will stagger the competition. Many of the price adjustments called for by pricing and revenue optimization are likely to fall below the radar screen of the competition and may not trigger any explicit response whatsoever. Furthermore, if competitors respond to our prices in the future as they have in the past, the price-response functions estimated according to the methods described in Chapter 4 will already incorporate the response of competitors.

This does not mean that competitive response need never be considered in pricing. The potential competitive responses to major bet-the-company bids or substantial changes in pricing strategy need to be carefully considered. For example, in the early 1990s, Hertz Rent-a-Car initiated the use of a sophisticated pricing optimization system. Previously Hertz, like other national car rental companies, had changed prices only rarely. Whenever one of the companies dropped a price, the others were likely to follow immediately—often with even larger drops—precipitating an industry-wide fare war. Hertz took great pains to communicate to the industry that its new system would be changing prices much more commonly than before—some prices would go up, some would go down, but all would change much more frequently. The communication was successful: Hertz was able to initiate its revenue management program without inciting retaliatory price wars. Once Hertz initiated revenue management, it was able to generate additional revenue through thousands of small adjustments to prices that its competitors were unable to match.

Similarly, online pricing enables much more rapid access to competitive prices via scraping competitor websites. For a third-party seller on Amazon, there is great value to winning the buy box, or becoming a featured offer. However, in most cases a seller can only win the buy box if he offers the lowest price (or at least matches the lowest price) for an item. Because winning the buy box is effective in driving sales, sellers are inspired to match (or undercut) other sellers, which can leading to pricing dynamics that drive the price to a floor. Once the price has reached the floor, it is costly for a seller to raise his price and risk losing the buy box. This situation is similar to a classical Bertrand game, in which two sellers competing to sell an identical item in a marketplace are both forced to lower their price to just above marginal cost.⁴

5.7 OPTIMIZATION WITH MULTIPLE OBJECTIVE FUNCTIONS

In many cases, sellers are not interested in only a single objective. Many times a manager with profit and loss responsibility has been enjoined to maximize both profitability and market share. However, the optimization approaches that we have considered so far in this section can only deal with a single objective function: either maximize profit or maximize revenue. As we have seen, in most cases doing both simultaneously is not possible—the price that maximizes profit is not the one that will maximize revenue and vice versa, except in the special case in which incremental cost is zero. In this section, we discuss how a seller can understand the trade-offs between different objective functions and incorporate them into decision making. We use the example of maximizing revenue versus maximizing profit, but the same concepts can be used for any two competing objectives such as profit and market share.

Example 5.7

The CEO of the widget-making company in Example 5.3 decides that the firm’s goal for the next month will be to maximize revenue from widget sales as part of the long-term strategy to increase market share. The revenue-maximizing price can be found by solving Equation 5.4. The resulting revenue-maximizing price is equal to $6.25, with corresponding sales of 5,000 units and revenue of $31,250. The corresponding per-unit margin is $1.25, and the total-contribution margin is 5,000 × $1.25 = $6,250. We can compare this to the maximum margin contribution of $11,250 and conclude that to maximize total revenue, the company needs to give up $11,250 – $6,250 = $5,000 of contribution margin to gain $5,000 in additional revenue.

The decision that management needs to make in Example 5.7 is whether it is worth giving up a total contribution of $5,000 per month to buy an additional 2,000 units of demand. While this trade-off is clear, it may be that management is interested in some goal other than pure profit maximization or pure revenue maximization. Figure 5.5 shows how profit and revenue change as a function of price for an example seller. In this figure, revenue is maximized at the price p̂ = $4.00 and profit is maximized at the price p* = $6.00, with p̂ ≤ p*. (Recall that the two prices will be equal only if the incremental cost is zero.) Now, assume that the seller is interested in both revenue and profit. Then, as evident from Figure 5.5, he would only consider prices in the interval between p̂ and p*. For prices below p̂, he could increase both revenue and profit by increasing price. For prices above p*, he could increase both revenue and profit by decreasing price. Thus, if profit and revenue are the only two goals he is interested in, he would only consider prices greater than p̂ and less than p*.

Figure 5.5 Profit and revenue as a function of price.

This leaves open the question of which price in the range [p̂, p*] he should choose. If the seller knows his trade-off between profit and revenue, then it is straightforward to accommodate this trade-off. Let us say, for example, that he is willing to give up $1.00 in profit in order to drive an additional $5.00 in sales. This trade-off could reflect management’s judgment about how the stock market values growth relative to profit in determining his share price. In this case, he would seek to maximize R(p) + 5Π(p), where R(p) and Π(p) are revenue and profit, respectively, at price p. More generally, let us assume that the seller’s trade-off is $1 of profit in order to gain dollars of revenue. We call α the seller’s revenue-profit trade-off. The seller is seeking to maximize the objective function pd(p) + αd(p)(p – c), which is equivalent to maximizing d(p)[(1 + α)p – αc]. This means that we can write his objective function as

This means that the seller can find the price that satisfies his revenue-profit trade-off by finding the profit-maximizing price with the modified unit cost If α = 0, the seller cares only about revenue, and he acts as if his unit cost is zero. As α → ∞, the seller cares only about maximizing expected profit.

Example 5.8

The widget maker in Example 5.3, with a unit cost of $5.00 and facing the price-response function of d(p) = (10,000 – 800p)⁺, decides that he has a revenue-profit trade-off of 10; that is, he would be willing to forgo $1 of profit to gain $10 in revenue. In this case, he can determine the corresponding optimal price by setting which gives p = $8.52. The corresponding revenue is $27,127.68 and the total contribution is $11,207.68.

If a seller knows his revenue-profit trade-off, then he can use any of the optimal price criteria listed in Section 5.2.1, substituting the modified cost for the actual cost. For example, the optimal price with a revenue-profit trade-off of α must satisfy the condition

Note that when α = 0, Equation 5.11 becomes the revenue-maximizing condition in Equation 5.9, and as α approaches infinity, it approaches the profit-maximizing condition in Equation 5.7. Furthermore, both the direct optimization approach and the data-driven approach described in Section 5.5 can easily be adapted to work with the modified cost in order to find the price that achieves the desired trade-off.

The methods above can be used by a seller to determine the price that meets his revenue-profit trade-off assuming that he can quantify that trade-off. In many cases, a seller cannot provide a value for his revenue-profit trade-off. In this case, an efficient frontier can help the seller visualize the potential trade-offs between revenue and profit. An efficient frontier, such as the one shown in Figure 5.6, shows all of the possible combinations of revenue and profit (or any two other objectives) that a seller could achieve by varying the price of an item. Let R(p) and Π(p) denote the revenue and profit, respectively, realized at price p. The dotted curve in Figure 5.6 shows all of the profit and revenue combinations that the seller could realize at different prices. However, the solid part of the curve is the only section of interest. The rightmost point on the solid part of the curve represents the maximum revenue that can be achieved and the corresponding profit: it is the point (R(p̂), Π(p̂)). The leftmost point on the solid part of the curve represents the maximum contribution that can be achieved and the corresponding revenue: it is the point (R(p*), Π(p*)). The solid curve connecting these two points is the efficient frontier, which consists of the only achievable combinations of profit and revenue that the firm would consider. The dashed points represent combinations of profit and revenue that the seller could achieve at some price, but in every case there is a price on the efficient frontier that generates both more profit and more revenue.

Figure 5.6 A single-product efficient frontier.

The efficient frontier is a convenient and frequently used approach to help visualize the trade-offs involved in pricing. In the case of a single price, the efficient frontier can be generated by starting at the revenue-maximizing price and calculating both profit and revenue for prices at regular intervals up to the profit-maximizing price. These can then be graphed to generate an efficient frontier.

Example 5.9

A seller has an incremental cost of $4.00 and faces a linear price-response function given by His revenue-maximizing price is $6.00, and his profit-maximizing price is $8.00. Revenue and profit for each price at $0.10 intervals between $6.00 and $8.00 are shown in Table 5.3. Plotting these out gives the efficient frontier shown in Figure 5.6.

The situation is more interesting when we consider two or more products. In the case of multiple products, the feasible set of revenue and profit combinations is a region rather than a curve as in the example of a single price. An example is shown in Figure 5.7. In this figure, the shaded area is the set of feasible revenue and profit combinations—that is, the combinations of revenue and profit that could be achieved by setting different prices for two different products. In fact, it is only points on the boundary of the feasible region that are of interest to a seller who cares only about revenue and profit. To see this, consider a seller whose current set of prices results in the revenue and profit combination represented by point A in Figure 5.7. This seller could realize the same revenue but higher profit by setting the prices that result in the revenue and profit combination at point B. Alternatively, the seller could realize the same profit but more revenue by pricing at point C. Finally, the seller could realize more revenue and more profit by pricing at one of the points on the boundary of the frontier between B and C. In any case, a seller who cared only about revenue and profit would not (knowingly) set prices that resulted in the revenue and profit combination at point A.

TABLE 5.3

Efficient frontier for Example 5.9

Figure 5.7 An efficient frontier with multiple products. Source: Adapted from Phillips 2018.

In the case of multiple products, the efficient frontier is the set of undominated revenue and pricing pairs, which means that for a price on the efficient frontier, additional profit can be achieved only by reducing revenue and vice versa. Points not on the efficient frontier are dominated, which means that either one or both of profit and revenue can be increased without decreasing the other. Let p = (p₁, p₂, . . . , p_n) denote the vector of prices offered by the seller. Then, in Figure 5.7, point D corresponds to the revenue and profit achieved at the profit-maximizing set of prices—that is, (R(p*), Π(p*)). Point E is the revenue and profit combination achieved at the revenue-maximizing price, (R(p̂), Π(p̂)). The boundary of the feasible region between these points is the efficient frontier.

As in the case of a single price, the multiprice efficient frontier can help a seller visualize the trade-off between revenue and profit that he faces and can help him choose accordingly. In fact, the slope of the tangent line to the efficient frontier at a point is the ratio between incremental revenue and incremental profit at that point. Thus, in Figure 5.8, line A has a slope of about 0.33. This means that a seller choosing point D values an additional dollar of profit about 3 times as much as an additional dollar of revenue. The slope of line B is about 2.5, which means that a seller choosing point E values each additional dollar of profit only about 1/2.5 or 40% of the value of an additional dollar of revenue.

Figure 5.8 Points on the efficient frontier that might be chosen by two different sellers.

Source: Adapted from Phillips 2018.

There are two ways in which the efficient frontier can be calculated with multiple prices. The first is to solve the optimization problem with an objective function of maximizing αR(p) + (1 – α)Π(p), where 0 ≤ α ≤ 1 is a parameter that determines the trade-off between incremental profit and incremental revenue. To trace out the efficient frontier, the seller could solve the optimization problem αR(p) + (1 – α)Π(p) for different values of α—say, α = (0,.1,.2, . . . , .9, 1.0)—and then plot the results. The points plotted will be points on the efficient frontier such as the one in Figure 5.8.

An alternative is to maximize profit with different revenue constraints. The idea would be to first maximize revenue and profit to determine both the profit- and revenue-maximizing vectors of prices (p* and p̂, respectively) and the profit and revenue corresponding to each vector of prices. Then the maximum revenue that the seller can achieve is R̂ = R(p̂), and the revenue he would achieve at the profit-maximizing price is R* = R(p*) < R̂. The seller would then choose n equally spaced values of revenue between R* and R̂, where the ith value of revenue is given For each value of i = (1, 2, . . . , n –1), the supplier would solve the optimization problem

Maximize Π(p),

subject to R(p) ≤ Rⁱ.

The resulting values of Π(p) and R(p) are points on the efficient frontier.

5.8 SUMMARY

• The cost used in pricing and revenue optimization is the incremental cost of a customer commitment. It is the difference between the total costs a company would incur from satisfying the commitment versus not making it. The incremental cost will vary with the duration and size of the commitment and is not a fully allocated cost. A rule of thumb is that if a change in demand resulting from a change in price would not change the cost, then the cost should not be included in determining the price.

• For a single product, the following are equivalent conditions for a price to be a profit maximizer:

• Marginal revenue equals marginal cost.

• The derivative of total contribution with respect to price is zero.

• The hazard rate equals the inverse unit margin.

• Elasticity equals the inverse unit margin.

• For a single product, the revenue-maximizing price is always less than or equal to the profit-maximizing price. They are only equal if the unit cost is equal to zero. The following are equivalent conditions for a price to be revenue maximizing:

• Marginal revenue equals zero.

• The derivative of revenue with respect to price is zero.

• The hazard rate equals the inverse of price.

• Elasticity equals one.

• Assume that the price-response function for a single product satisfies the following conditions:

• It is decreasing in price.

• It is continuous.

• It is differentiable.

• It has elasticity that increases with price.

• It has elasticity greater than 1 for some finite price.

Then there exists a unique profit-maximizing price that is greater than the unit cost. There is also a unique revenue-maximizing price that may or may not be greater than the unit cost.

• The relationship between elasticity and unit margin implies that products facing lower elasticity will have higher unit margin rates than those facing higher elasticity. This means that—all else being equal—sellers should deploy their marketing, advertising, and product improvement budgets in a way that is targeted to reducing price elasticity for their products.

• A data-driven approach can be used to set prices that approach the profit-maximizing price p* by using the following facts: at prices higher than p*, elasticity is greater than the inverse unit margin, and at prices lower than p*, elasticity is lower than the inverse unit margin. The data-driven approach tests demand at different prices, estimates the elasticity, and then uses the relationship between elasticity and inverse unit margin to calculate a new price.

• Often, sellers are interested in both profit and revenue. If a seller can quantify his trade-off between profit and revenue, he can determine the price that best meets the trade-off by solving an optimization problem with an objective function that appropriately weights profit and revenue. If the seller cannot quantify his trade-off, an efficient frontier can help him visualize the possible combinations of revenue and profit that he can achieve.

5.9 FURTHER READING

Activity-based costing (ABC) and the related discipline of activity-based management (ABM) arose in the mid-1980s from the realization that typical cost accounting systems often resulted in a distorted view of the value of different activities primarily because of improper allocation of fixed costs. ABC is a management accounting approach designed to help management understand the incremental cost implications of doing more or less of certain activities (e.g., carrying fewer or more passengers), as opposed to financial account systems where all costs are allocated according to various regulations (e.g., SEC or FASBI) to provide standardized and consistent reports to regulators and shareholders. The idea was popularized in Kaplan and Cooper 1997. For an updated view of ABC, see Kaplan and Anderson 2007. While some organizations (although by no means all) have adopted ABM, the significance for pricing optimization and revenue management is understanding that the costs that are relevant to pricing decisions are only those that would change given the change in demand resulting from the pricing decision.

The Bertrand model was probably the earliest example of game-theoretic reasoning in economics: it was articulated by the French mathematician Joseph Louis François Bertrand in 1883 and developed into a mathematical model by Francis Edgeworth in 1925. Since then, there has been an extensive economics literature on the application of game theory to pricing. For thorough treatments of game theory with a focus on economics, standard refences are Fudenberg and Tirole 1991, Myerson 1991, and Gibbons 1992. For a survey of game-theory models in pricing, see Kopalle and Shumsky 2012.

The efficient frontier was introduced by Harry Markowitz (1959) to illustrate the trade-off between risk and return in an investment frontier. For an explanation of how it can be derived and used in this context, see Luenberger 1998, chap. 6. The efficient frontier has been widely used in many situations in which there is a trade-off between two competing alternatives. For more detail on its use in pricing and revenue management, see Phillips 2012b, and for the use of efficient frontiers in consumer credit pricing, see Phillips 2018, chap. 7.

5.10 EXERCISES

1. Consider a seller seeking to maximize contribution. Under what relative values of his current price p, his cost c, and his point elasticity ε(p) should the seller raise his price to increase contribution? Under what conditions should he lower his price? Keep his price the same?

2. A retailer is currently charging a price of $147.52 for a Hewlett-Packard OfficeJet printer that costs him $112.00 per unit. He determines that the point price elasticity of this model of printer is 5.1 at its current price.

a. If he wants to maximize net contribution, is he better off raising his price, lowering his price, or keeping it the same?

b. If the elasticity of 5.1 is valid over a range of at least $20.00 on either side of his current price, what is his optimal (contribution-maximizing) price?

3. An auto manufacturer can manufacture compact cars for an incremental cost of $5,000 apiece. She faces a logit price-response function for sales in the next month, with parameters C = 40,000, b = 0.0005, and p̂ = $12,000. What price will maximize the total contribution? How many cars will she sell during the month?

4. Frank owns a hot-dog stand. It costs him $1 to make each hot dog and he faces a linear price-response function d(p) = (100 – 8p)⁺ for each day.

a. Find Frank’s contribution-maximizing price and the corresponding contribution. (For this and following questions, allow non-integer units of sales for simplicity.)

b. Suppose a big hot-dog franchise offers him a contract. According to the contract, the franchise will provide an unlimited number of prepared hot dogs per day for a fixed cost of K dollars per day. If Frank accepts this contract, what is his optimal price?

c. For what values of K should Frank accept the contract if he is maximizing profit?

d. Now the franchise offers Frank a new contract that charges a daily fixed cost of $25. However, they will provide at most 40 hot dogs to Frank every day. Should Frank accept this offer? Explain your answer.

NOTES

1. The problem of finding optimal prices for a good with limited stock whose value is decreasing over time is a markdown pricing problem discussed in detail in Chapter 12.

2. The unit margin rate L(p) = (p – c)/p is also known as the Lerner index, which was named after the economist Abba Lerner, who proposed that it be used as a measure of market power. If the price being charged by a firm is denoted by p, then L(p) = 0 would indicate that a firm has no market power (price equals incremental cost) and L(p) = 1 would indicate very high market power. Note that the condition for a profit-maximizing price can be written as ε(p*) = 1/L(p*).

3. Elasticity increasing with price is equivalent to a property of the underlying willingness-to-pay distribution called increasing general failure rate (IGFR). Any price-response function corresponding to a willingness-to-pay distribution with an increasing general failure rate will have unique profit- and revenue-maximizing prices. Such distributions include the normal, the uniform, the exponential, and the logistic. For more details, including a list of distributions that do and do not satisfy the IGFR property, see Lariviere 2006.

4. The Bertrand model of competition is based on two sellers selling an identical product. Both sellers have the same incremental cost, c. Customers observe the prices of both sellers and are indifferent among sellers: if one seller is cheaper, they will all purchase from that seller, and if the sellers offer the same price, demand will split equally between the sellers. Bertrand argued that, if either seller charged a price p > c, the other seller’s best response would be to charge some price p', with p > p' > c. Since this holds true for both sellers, their prices would inevitably converge toward cost. In fact, prices would equilibrate at c + δ, where δ is the smallest increment of price allowed (e.g., $0.01).