12

Markdown Management

Nobody but a fungoid creature from another galaxy with no familiarity with earthly ways would ever pay list price for anything.

—Dave Barry, Dave Barry’s Money Secrets

“Waiting for the sale” is a time-honored strategy for savvy shoppers. The fashion-conscious may splurge on clothes shortly after the spring fashions arrive at Macy’s, Nordstrom, Bloomingdale’s, or Saks, but the budget-minded know they can save by waiting. And the longer they wait, the more they save; many items will be marked down by 70% or more by the end of the season. Waiting for the sale is not without risk. There is always the chance that the store will sell out of a desired item before it is marked down. Nonetheless, an increasing number of customers are willing to take the risk in order to realize the savings. Dave Barry may be overstating the case, but in many categories retail list price is becoming a ceiling, with most sales taking place at a discount. More and more customers refuse to buy at full price if they feel that a sale will be coming.

Retailers use a wide variety of mechanisms to discount their merchandise. Broadly speaking, these can be classified into promotions and markdowns. A promotion is a temporary reduction in price. Examples of promotions include a Memorial Day sale, Amazon’s Prime Day, a temporary two-for-one sale, an introductory low price, and a limited-time coupon for a 10% discount. The common link among these is that they are all temporary. In contrast, a markdown is a permanent reduction, usually to clear inventory before it becomes obsolete or needs to be removed to make way for new stock. When an item is marked down, both the seller and the buyer know that the price for that particular item will never go back up.

The difference between markdowns and sales promotions is illustrated in Figure 12.1, which shows the price over four months of a woman’s sweater (panel A) and a television set (panel B), both expressed as a percentage of list price. The television set was promoted three times, with its price returning to list following each promotion. On the other hand, once the price of the sweater was reduced, it never returned to earlier levels. After about three weeks at list price, it was marked down 20%, followed by three further markdowns. At the end of the four-month period, the sweater was selling for about 20% of its list price, while the television was priced at list.¹ Note that markdowns and promotions are not mutually exclusive; a temporary promotion can be applied to an item that is also marked down.

Figure 12.1 Example retail price tracks over four months for (A) a sweater and (B) a television set.

This chapter discusses the tactics behind markdown management. The point of markdown management is to determine the timing and magnitude of markdowns that move the inventory while maximizing revenue. For many years, retailers relied on their judgment or simple rules of thumb to determine when and how much to mark down their goods. However, over the past 20 years, an increasing number of retailers have begun to use more sophisticated analytic tools to help guide this process, and companies such as SAS and Oracle offer commercial markdown optimization software.²

We begin with at some background and how, for certain goods and services, markdowns segment the market and provide a simple method by which retailers can profit from intertemporal price differentiation. The markdown management process that retailers follow and some of the business issues that can constrain them are outlined. We formulate the markdown problem as a constrained optimization problem and consider the most common approaches to solving the problem. We examine the implications of strategic customers—those who anticipate future prices—on markdown policies. Finally, we look at the use of markdown management systems and the experiences of companies using markdown optimization.

12.1 BACKGROUND

The practice of marking down distressed inventory is probably as old as commerce itself. Nonetheless, in the United States, markdowns and promotions were relatively rare prior to 1950. Sales were associated with holidays such as Christmas and Thanksgiving, or they occurred as periodic price wars between rivals such as Macy’s and Gimbels in Manhattan. In fact, most retailers tried hard to avoid markdowns. In the 1950s, managers at the Emporium, a leading San Francisco department store, were told:

High markdowns benefit no one. Not the store. Not the manufacturer. Not the customer. The store loses in value of assets or inventory. The manufacturer loses in future sales and by loss of prestige of his product. The customer is getting merchandise that is not up to standard, at a low price it is true, but, remember—she would rather have fresh, new merchandise that she can be proud of and with real value at regular price than pay less for questionable merchandise. (Emporium 1952, 3)

Many retailers considered markdowns to be prima facie evidence of mistakes in purchasing, pricing, or marketing. The official line at the Emporium was “Like accidents, most markdowns are a direct result of carelessness” (Emporium 1952, 4). On the other hand, a few retailers have always recognized that markdowns are more than just a way to clean up after mistakes. As early as 1915, one of the leading textbooks on retailing noted:

This method of marking up goods and making reductions as necessary seems to be the most effective way to sell some goods, especially those that are subject to changes in style. Besides bringing handsome profits to the concern directly, the spectacular cuts in prices . . . furnish excitement for readers of the store’s advertising, and help to draw crowds who may buy other goods as well as those advertised. (Nystrom 1915, 242)

Since the 1960s, there has been a steady increase in both the size and the frequency of retail discounts. Figure 12.2 shows the growth in average discount from list price paid for items purchased at both department stores and specialty stores from 1967 through 1997. In 1967, the average price paid for all items purchased at department stores was at a 6% discount from list; by 1997, the average discount was 20%. The comparable figures for specialty stores were 10% and 28%. Both the fraction of goods sold at a discount and the average discount inexorably increased from 1967 through 1997. Sometime in the 1970s, purchasing goods at a discount went from being an exception to being the rule. Markdowns as a percentage of retail sales were 8% in 1971, rising to 35% in 1996. By 2006, 78% of all apparel sold by major chains in the United States was marked down (Levy et al. 2007).

Figure 12.2 Average discount from list price for goods sold at department and specialty stores from 1967 through 1997. Discounts include both markdowns and promotions. Data are based on stores with annual sales greater than $1 million to 1982 and greater than $5 million thereafter. Source: Data from National Retail Merchants Association 1968, 1977, 1987; National Retail Federation 1998.

The reasons for the steady rise in the average discount illustrated in Figure 12.2 have been much discussed. Here are some of the reasons that have been proposed.

• Increased customer mobility, leading to more intense competition among retailers.

• The rise of discount chains and outlets, many with “everyday low pricing” strategies.

• A “vicious,” or self-reinforcing, cycle under which customers expect discounts and will not purchase unless retailers provide them, thereby creating an expectation of future discounts.

• An increasing interest by consumers in variety rather than standard items. The market share of colored and patterned sheets relative to white sheets increased from 12% in 1970 to 60% in 1988. Sales of patterned men’s shirts also increased relative to standard white and blue (Pashigian 1988). Since nonstandard items have shorter shelf lives than standard items, merchants have to discount more often and more deeply to move old inventory and make room for new inventory.

• The increased use of markdown money, by which manufacturers reimburse retailers for some portion of the discounts provided to move slow-selling items. This has made it easier and more economical for retailers to resort to more aggressive discounting. Markdown money shifts some of the risk of unsold inventory from retailers to manufacturers.

It is worth noting that a few sectors have resisted the trend toward pervasive discounting. Discounting is still relatively rare in prescription drugs, movie tickets, and various staple food and home items. As a result, for these items, even relatively small promotions can often drive dramatic sales increases. Furthermore, over the last 25 years there have been a rising number of everyday-low-price retailers, such as Walmart and Dollar General, who discount rarely, if at all. Nonetheless, the consensus is that no matter the cause, the increase in retail discounting is likely to continue, and, for the majority of retailers, planning and executing discount programs will consume an increasing portion of their time and energy. Since retail is a notoriously thin-margin business, properly planned markdowns can play a make-or-break role in determining profitability.

12.1.1 Reasons for Markdowns

Let us return to the price histories for the sweater and the television set shown in Figure 12.1. Why was the sweater marked down but not the television set? If you asked the retailer, he would probably mention two reasons. First, the sweater is a fashion good—its perceived value decreases over time, and it needs to be marked down to reflect this declining value. On the other hand, the value of the television does not decline over time. The second reason is that the sweater needs to be sold to make room for next season’s line of clothes. If it is not sold by a certain date (called the out date), it will be removed from the store and sent to an outlet store or sold for 20% or less of its original price to a jobber—someone who will try to sell it for more. Markdowns are necessary to move the goods and clear the space for next season’s clothes. On the other hand, there is no pressing need to clear the televisions—that is, unless the current model is being discontinued or made obsolete by the imminent arrival of a new model.

As these examples show, from the seller’s point of view, markdown products typically share two characteristics:

1. Inventory (or capacity) is fixed.

2. The inventory must be sold by a certain out date or its value drops precipitously.

These two reasons provide the seller with motivation to mark down. However, there are additional reasons why the value of an item to buyers might decrease over time:

1. Time of use. An overcoat purchased in September can be worn during the upcoming winter months, unlike the same overcoat purchased in April. Many consumers assign value to wearing the coat during the upcoming winter, with that value declining as winter progresses.

2. Fashionability. Some consumers place a high value on getting a fashionable item early in the season, while others may care much more about buying at the lowest price. This applies not only to clothing but also to short-life-cycle goods such as consumer electronics and video games, where some buyers value being among the first to own the latest gadget or game.

3. Deterioration. The quality of a good may deteriorate over time. Day-old bread does not taste as good as fresh bread. Clothes that have been on display for months can get a pawed-over look that makes them less appealing.

4. Obsolescence. Consumers may believe that a better-quality version of a product will be available at some future date. As time passes, it increasingly makes sense to wait for the new product rather than buy the current version. To induce customers to buy now rather than wait may require deeper and deeper discounts. This is an especially important effect in consumer electronic goods, computer equipment, and automobiles.

Note that the first three reasons involve customer segmentation and price discrimination. Time of use segments the market between those customers who need an overcoat now (and are willing to pay more for it) and those who can wait. Fashionability segments the market into fashionistas, who place a high value on being au courant, and value-conscious customers, who are willing to wait. And deterioration allows the seller to offer an inferior product at a lower price. As we have seen, customer segmentation and price differentiation are powerful ways for a seller to increase revenue, especially when his inventory is constrained. Thus, it should be no surprise that they play a major role in the use of markdowns as a selling strategy. It seems reasonable that prices would drop for items whose quality deteriorates over time—everyone’s willingness to pay for day-old bread is less than that for fresh bread.

Figure 12.3 Optimal prices, sales, and total revenue in (A) the single-period case and (B) the two-period case. In the two-period case, the optimal first-period price is $6.67, and the optimal second-period price is $3.33.

While deteriorating quality is often found in markdown products, we can use a simple model to show how markdowns can segment a market even when quality does not deteriorate.³ Consider a seller facing the linear price-response function d(p) = 1,000 – 100p, shown in Figure 12.3. Recall that this linear price-response function is equivalent to a population of 1,000 potential buyers whose willingness to pay is uniformly distributed between $0 and $10. Assume that the good being sold has a marginal cost of zero. (As we shall see, this is often a reasonable assumption in markdown optimization.) Section 5.3 shows that, given a single period, the price that maximizes the seller’s revenue is p* = $5, with corresponding sales of 500 units and total revenue of $2,500, which is equal to the shaded area in panel A of Figure 12.3. At this price, everyone in the population with a willingness to pay greater than or equal to $5 purchases the product, and those with a willingness to pay less than $5 either do not purchase or purchase from a competitor.

So far, so simple. But what happens if, after all the buyers with a willingness to pay higher than $5 had purchased the product, the seller had the opportunity to change the price? In this case, it should be apparent that the seller can gain additional revenue by charging a lower price in the second period. If the seller charges a second-period price higher than $5, he will not sell anything in the second period, since all customers with a willingness to pay higher than $5 have already purchased. But if he charges a lower price—say, $2.50—he can realize some additional revenue. In addition to the 500 units he sold at the first-period price of $5, he would sell an additional 250 units in the second period at a price of $2.50. His second-period buyers are those whose willingness to pay is between $2.50 and $5. By charging a lower price in the second period, the seller can increase his total revenue by 24%, from $2,500 to $3,125.

In fact, the seller can do even better. If he sets his first-period price at $6.67 and his second-period price at $3.33, he can achieve total revenue of $3,333.33. This solution is shown in panel B of Figure 12.3. He sells 333 units in the first period and 333 units in the second period, for a total contribution of 333 × $6.67 + 333 × $3.33 = $3,330—an increase of 33% over the single-price case! If he had more periods with more opportunities to change the price, he could make even more revenue by selling at progressively greater discounts in each period (see Exercise 1).

If the price-response function is linear, the optimal price in the second period of the two-period model will always be one-half of the optimal price in the first period. This result depends on a number of assumptions that are unlikely to be true in practice—particularly the global linearity of the price-response function and that customers do not anticipate the markdown. However, it is interesting that there are at least two cases of products that undergo a single markdown and the discount is 50% of the full price: baked goods and Broadway tickets. Day-old bread is often sold at half off the full price. And Broadway theaters have the option to sell tickets that otherwise would not be sold through a TKTS outlet in Times Square.⁴ Perhaps coincidentally, the price of a ticket purchased at TKTS is often one-half the full price.

How does a second-period markdown result in additional revenue? There was no change in market conditions between the two periods, and the pool of customers did not change. The additional revenue came entirely from customer segmentation. The seller has used time of purchase and decreasing prices as a way of segmenting customers between those with a high willingness to pay and those with a lower willingness to pay. In this case, the markdown is an example of time-based segmentation introduced in Chapter 6.

However, the simple model is based on several questionable assumptions. As previously noted, it assumes a single pool of customers who are all indifferent between buying in the first period and buying in the second period. In most retail situations, we would expect that at least some buyers would have a lower willingness to pay for the item in the second period, because they benefit from having the item sooner (as with a coat at the beginning of winter rather than at the end of winter), or because the item deteriorates (as with day-old bread), or because it becomes obsolete (as with a laptop computer). Of course, the presence of customers with decreasing willingness to pay will only increase the incentives for the seller to decrease price in the second period (see Exercise 2).

The second assumption underlying the two-period model is that customers do not anticipate the second-period markdown. If customers knew (or even suspected) that the second-period price would be lower, we would expect at least some of them to wait for the sale. This is cannibalization, and cannibalization reduces and can even eliminate the benefits of segmentation. One way to incorporate cannibalization in the two-period model is to specify a fraction of customers who will wait to buy in the second period even though their willingness to pay exceeds the first-period price. In other words, these customers would be willing to pay the first-period price, but they wait for the second period because they anticipate the markdown. A cannibalization fraction of 10% means that 10% of the customers who would be willing to pay at the first-period price wait to purchase in the second period at a lower price instead.

TABLE 12.1

Effect of cannibalization in the two-period model

Table 12.1 shows how optimal first- and second-period prices and total revenue change as the cannibalization fraction increases from 0 to 100%. As you might expect, increasing cannibalization results in decreased revenue. Furthermore, as cannibalization increases, both the first-period price and the second-period price increase. However, as long as the cannibalization fraction is less than 100%, the retailer still gains revenue relative to the single-period revenue of $2,500 by marking down in the second period. In this model, the retailer can improve revenue from following a markdown policy even if a majority of customers are willing to wait for the sale.

12.1.2 Markdown Management and Demand Uncertainty

The previous section shows how markdown management could improve profitability when demand is deterministic and the seller knows both the size of the total market and the distribution of customer willingness to pay. Of course, this assumption of perfect knowledge is highly unrealistic. It is easy to see that the seller’s uncertainty about demand provides an additional motivation for pursuing a markdown policy. Consider a merchant with an inventory of a particular style of sweater who is unsure if the sweater will be a top seller during the upcoming season. If the sweater is a top seller, he will be able to sell his entire inventory at $79; if it is not a top seller, then the market price will only be $59. If he has two periods in which to sell the sweaters and offers them at $59 during the first period, he will sell them all during the first period. But if he offers them at $79 during the first period, he may sell them all at the high price; if not, he can lower the price to $59 in the second period and sell the remaining stock. In this case, a markdown policy enables the seller to maximize revenue in the face of demand uncertainty.

The use of markdowns to resolve demand uncertainty is exemplified in dramatic fashion by the Dutch auction, a term that derives from the use of this mechanism in Dutch flower markets. The most famous of these markets is in the town of Aalsmeer, where more than 4 billion flowers are sold every year. In a Dutch auction, the seller sets a maximum and a minimum price for the product he is seeking to sell.⁵ During the period of the auction (5 minutes in the case of the Aalsmeer flower auction), the price drops steadily from the maximum price to the minimum price. Each potential buyer has a button that she can press at any time during the auction. When the first buyer pushes her button, she has bought the lot at the current price and the auction is over. If the price drops to the seller’s specified minimum without any takers, the lot goes unsold.

In some ways, the Dutch flower auction is like a fashion season in fast-forward. The two markets share important similarities: Both flowers and fashion goods are perishable. In both cases, the seller is often uncertain about the value that buyers will place on his good. For fashion goods, the uncertainty arises from the fickle tastes of customers and the inability of the seller to know whether any specific product will be considered hot during the current season. In the flower market, the uncertainty is due to the daily changing supply and demand for each type of flower. In both markets, the pattern of lowering prices over time can be seen as a way for sellers to extract the highest price for their constrained and perishable supply in the face of demand uncertainty.

12.1.3 Markdown Management Businesses

Three criteria are all required for a markdown opportunity to be present.

• The item for sale must be perishable.

• The supply must be limited.

• The desirability of the item must hold constant or decrease as it approaches its perishability date.

As we have seen, fashion goods, baked goods, Broadway theater tickets, and wholesale flower sales meet these criteria and commonly employ markdowns. Here are some other businesses in which markdowns are common.

• Holiday items. These include Halloween candy; Christmas trees, decorations, and cards; and fireworks for the Fourth of July. Such items usually begin to be marked down just before the holiday, with much deeper discounts once the holiday is over.

• Tours. A tour package typically includes both airline transportation and lodging. Although tours constitute less than 10% of leisure travel in North America, they represent a major portion of vacation travel in Europe and Asia. A typical tour package might be six days on the Costa del Sol (in Spain), in which the tour price covers both round-trip air transportation and hotel accommodations. Bus travel to and from the airport in Spain and meals might also be included. Tour operators start selling tours a year or so ahead. As time passes, the tour operator continually monitors the bookings for different tours. If a tour is performing worse than expected (i.e., bookings are low), the tour operator may lower the price. This is a markdown management problem because the price is always lowered (never raised) and the stock of inventory is perishable.

• Automobiles. Like fashion goods, most automobile manufacturers operate on an annual selling season. New models arrive every September and must be sold within a year to make room for next year’s model. Although some models may be considered hot and not require discounting, other models will not sell quickly enough at the list price to clear the lot. The manufacturer and the dealer will use a mixture of promotions and markdowns to move the inventory. In general, the average selling price of an automobile tends to decline fairly steadily over the season. This gives automobile pricing the flavor of a markdown management problem, although the markdowns are typically realized through various promotional vehicles rather than as a reduction in the list price.

• Clearances and discontinuations. Almost any seller can be in a situation where he has inventory he needs to clear by a certain date. When Maytag announces that it will be shipping a new and improved model of washing machine in three months, retailers have three months to sell their existing inventory before it becomes obsolete. Sequential markdowns are used to maximize revenue from this existing stock. The problem occurs at least occasionally for almost all retailers (and many wholesalers) but most frequently for those selling short-life-cycle goods such as computers and home electronics. In many cases, sellers of short-life-cycle goods mark down some portion of their total inventory every day.

Note that revenue management industries meet the first two criteria, but not the third. That is, while seats on a flight are limited and perishable, their desirability typically increases for most customers as departure approaches.

12.1.4 Constraints on Markdown Policies

While markdowns are frequently very profitable, business are often constrained in the markdowns that they can take, either by business rules or technical constraints. Some of the most common constraints are listed below.

• Price ladders. Discounted prices may need to be at discrete points (“rungs”) on a ladder either as discount percentages (for example, a five-rung discount ladder with 10%, 25%, 50%, 60%, and 80% as possible discounts from the list price) or as discrete price points (for example, a four-rung price ladder with potential prices of $29.99, $24.99, $19.99, and $10.99). In either case, markdowns need to be at one of the rungs on the ladder. Ladders keep pricing simple for both the retailer and the customer. In addition, they enable the retailer to take advantage of various penny-ending effects as discussed in Chapter 14.

• List price period. The list price may need to be in effect for some minimum period—typically several weeks—at the start of the season. The primary motivation for this rule is to maintain the value of the item in the minds of customers and to provide some integrity and rationale for the list price.

• Maximum and minimum price reductions. Since changing the price of an item in a brick-and-mortar store has some cost, minimum price reductions are used to avoid many small markdowns. Online retailers also use minimum price reductions to avoid incremental discounts of a few cents, which could confuse or annoy customers. On the other hand, most retailers want to avoid extremely large markdowns. An enormous initial markdown may suggest to customers that an item is inferior and may actually harm sales. Maximum reduction restrictions are often imposed on the first markdown taken.

• Collections price together. All items in a given line (e.g., Liz Claiborne sweaters) might need to have the same markdown cadence in a given store; that is, they must all be marked down the same percentage at the same time.

• Maximum number of markdowns. Often the total number of markdowns during a season is limited in a retail store to minimize the workload created by the need for re-tagging merchandise.

• Minimum final price. Typically, the final “fully marked down” price must exceed a certain percentage of the list price (say, 25%). This is to prevent extreme markdowns that devalue the merchandise. Instead of taking an extreme markdown, many retailers will send items to an outlet store or sell them to jobbers.

• Regional markdown coordination. In many cases, retail chain executives require that all stores in a specified region (e.g., Manhattan, London, Southern California) must have the same markdown cadence. The reason is to prevent customers from being motivated to shop at different locations in the same region to realize the best discount.

• Residual inventory. Sometimes, sufficient inventory must remain unsold to provide minimal stocking levels for outlet stores. This may seem a bit odd, since the rationale behind establishing outlet stores is to dispose of unsold merchandise from retail stores. But for some retailers, outlet stores have become significant profit centers in their own right, and these retailers want to ensure that there is sufficient unsold inventory from their retail stores to fully stock the outlet stores.

These restrictions can significantly reduce the benefits that a retailer might receive from markdowns on his inventory. Furthermore, they can complicate the computation of the optimal markdown policy by requiring the addition of constraints.

12.2 MARKDOWN OPTIMIZATION

We have seen why sellers might use a markdown policy. We now look at some approaches to determine how they should mark their inventory down over time. These models are based on the following assumptions.

• A seller has a fixed inventory (or capacity) without the opportunity to reorder.

• The inventory has a fixed expiration date (the out date), at which point unsold inventory either perishes or is sold at a small salvage value.

• Initially, the price of the good is set at list price. The seller can reduce the list price one or more times before the expiration date.

• Only price reductions are allowed. Once the price has been reduced, it cannot be increased again.

• The seller wants to maximize total revenue—including salvage value—from his fixed inventory.

Markdown optimization typically assumes that marginal costs are zero and that the goal of the seller is to maximize total revenue. This may seem surprising since the seller paid for the goods. However, at the point a seller is considering markdowns, his costs are sunk—his inventory has already been bought and paid for. Furthermore, since the seller cannot reorder, increased sales do not result in additional future orders. Therefore, the incremental cost of an additional sale is zero, and we can state the objective function in terms of maximizing revenue from a fixed inventory. This is an example of the principle articulated in Section 5.1.1: because the markdown prices do not change the costs paid by the seller, the costs should not influence the markdown prices.

Let p₁ denote the initial (list) price, and assume that we have T markdown opportunities before the out date. For example, a typical season for a fashion item might be 15 weeks. Assuming that the item might be marked down at most once per week, we would have p₁ as the price during the first week, p₂ as the price during the second week, and so on through the final price of p₁₅.⁶ We assume a salvage value of r that is received for all unsold items following the last markdown. This would be the price a retailer would receive from a jobber for unsold clothes or the price he would expect to sell them for at an outlet store. If inventory entirely perishes at the end of period T—as in the case of a tour operator or a Broadway show—then r = 0. It should be obvious that we would never set the price during any markdown week to be less than r.

Let x_i be the unsold inventory at the beginning of period t. Then x₁ is the initial inventory, which is given. We assume that demand in any period is a function of the price in that period, and we let d_t(p_t) denote the price-response function in period t. The price-response functions d_t(p_t) satisfy all the standard properties listed in Chapter 3—that is, they are downward sloping, continuous, and so on. Sales in period t are denoted by q_t. Since sales in any period cannot be more than inventory, we have q_t = min[d_t(p_t), x_t]. We initially consider deterministic models, in which x_i will uniquely determine d_t(p_t) and hence q_t. We then turn our attention to (more realistic) probabilistic models, in which both d_t(p_t) and q_t are random variables.

The starting inventory in each period is equal to the starting inventory from the prior period minus prior-period sales; that is, x_t+1 = x_t – q_t for t = 1, 2, . . . , T – 1. This process is illustrated in Figure 12.4. The retailer starts with an inventory x₁ and chooses a price p₁. He realizes sales of q₁ = min[d₁(p₁), x₁], and his inventory at the start of the second period is x₂ = x₁ – q₁. He chooses his second-period price p₂ and the process continues. At the end of the season, he has unsold inventory y = x_T – q_T, for which he receives a salvage value of r per unit. Thus, his total revenue from his fixed inventory is R = p₁q₁ + p₂q₂ + . . . + p_Tq_T + ry. In a deterministic model, the seller finds the set of prices that maximize R. In a probabilistic model, he finds the prices that maximize the expected value of R.

Figure 12.4 Dynamics of markdown optimization.

12.2.1 A Deterministic Model

Since we know that sales in each period need to be less than starting inventory in that period, one might think that we need a set of T constraints specifying that q_i ≤ x_i for i = 1, 2, . . . , T. However, with a little reflection, it should be clear that we require only one constraint—namely, that the sum of sales for all periods needs to be less than or equal to the starting inventory. With this in mind, we can write the deterministic markdown optimization problem as

subject to

We can simplify the formulation even further by substituting for y in the objective function, which gives a new objective function:

Since rx₁ is simply a constant, we can eliminate it from the objective function and write the markdown management problem (MDOWN) as

subject to

The interpretation of the markdown management problem is straightforward. The objective function 12.1 states that the goal of the retailer is to maximize total revenue. This is expressed as maximizing the total increment over the salvage value received for each unit sold. Constraint 12.2 states that the retailer cannot sell more than his first-period inventory. The constraints in 12.3 guarantee that the markdown prices are decreasing over time, and constraint 12.4 guarantees that the final price is greater than the salvage value.⁷

Example 12.1

A retailer has a stock of 160 men’s sweaters, and he has four months to sell the sweaters before he needs to clear the shelf space. The merchant wishes to establish a list price at the beginning of the first month and then mark the sweaters down at the beginning of each of the next three months. Sweaters unsold at the end of the fourth month will be sold to an outlet store for $5 apiece. Furthermore, by dint of his long experience, the seller knows that demand in each of the four months can be represented by independent linear price-response functions:

d₁(p₁) = (120 – 1.5p₁)⁺,

d₂(p₂) = (90 – 1.5p₂)⁺,

d₃(p₃) = (80 – 1.5p₃)⁺,

d₄(p₄) = (50 – 2p₄)⁺,

where p_i is the price in month i for i = 1, 2, 3, 4. Formulating this problem as in MDOWN (Equations 12.1 through 12.4) gives the results shown in Table 12.2. If the retailer had to maintain a single price during the entire four months, his price-response function would be the sum of the four individual price-response functions—that is, d(p) = d₁(p) + d₂(p) + d₃(p) + d₄(p)—and his price optimization problem would be to find the price p that maximizes pd(p) + $5[160 – d(p)]⁺. The optimal solution to this problem is p = $34.72 with d(p) = 133, so the total contribution is 133 × $34.72 + 27 × $5 = $4,752.76. The optimal markdown policy resulted in a revenue increase of about 7% over the use of the optimal single price.

If a seller really knew the price-response functions for each week in the selling season, he could plug them into MDOWN and solve for the optimal markdown schedule at the beginning of the season, and his work would be done. Since the functions are deterministic, sales will be exactly as predicted, and there would be no need to deviate from his initial planned markdown schedule. In reality, of course, things are not so simple. Without access to the ever-elusive crystal ball, sales are likely to deviate from expectations. The most effective approach is to incorporate this uncertainty explicitly in calculating prices—an approach we take in the next section. However, the deterministic problem formulation can be used as the basis of a simple dynamic markdown algorithm.

TABLE 12.2

Optimal solution for example deterministic markdown problem

Deterministic Markdown Pricing Algorithm

1. Solve MDOWN to determine the initial price of p₁.

2. Observe sales q₁ during the first period. Starting inventory for the second period is x₂ = x₁ – q₁.

3. Solve MDOWN for p₂, p₃, . . . , p_T using a starting inventory of x₂. Put p₂ in place as the price for the second period.

4. At the beginning of each subsequent period, the current inventory is the previous period’s starting inventory minus last-period sales—that is, x_t = x_t–1 – q_t. Solve MDOWN for p_t, p _{t +1}, . . . p_T, using a starting inventory of x_t. Use the value of p_t as the price during the upcoming period t.

Example 12.2

Consider the previous example and assume that during the first month, the retailer sells 70 sweaters instead of the 56 he anticipated. This means he has a starting inventory of only 90 sweaters at the beginning of month 2. Using x₂ = 90, he can use the price-response functions d₂(p₂), d₃(p₃), and d₄(p₄) to recalculate prices for months 2, 3, and 4. The new optimal prices are p₂ = $34.00, p₃ = $30.67, and p₄ = $16.50. He therefore sets his price for month 2 at $34.00. At the end of month 2, he can observe remaining inventory and solve once more to determine the price he will charge in month 3.

This algorithm does not explicitly incorporate future uncertainty into its calculation of the current price. However, it is adaptive in the sense that it recalculates the price in each period, recognizing that sales to date are likely to be different from those originally anticipated. While this is less than ideal, this general approach can lead to improvements over human judgment. One study computed the results of a deterministic algorithm similar to MDOWN applied to 60 fashion goods at a women’s specialty apparel retailer in the United States and compared the predicted results to what actually occurred when the store used its usual markdown policy. To conform with the business practices of the retailer, the algorithm only changed the discount level if the new discounted price was at least 20% lower than the previous price. Furthermore, each style had to be offered at list price for at least four weeks prior to the initial markdown. The deterministic algorithm increased total revenue by about 4.8% relative to standard store practice. Much of this increase came from taking smaller markdowns sooner than usual store policy (Heching, Gallego, and van Ryzin 2002).

12.2.2 A Simple Approach to Including Uncertainty

A seller is facing his last markdown opportunity. He has a fixed inventory x and must set the price p for which it will be sold during the upcoming period. At the end of that period, any remaining unsold inventory will be sold for the salvage price r (which may be 0). If he knows the price-response function for this period, his problem is a simple single-period special case of MDOWN. But now we add uncertainty to the mix. Specifically, we assume that demand given price, D(p), is a random variable whose distribution depends on p. What price should the seller set to maximize expected revenue? The problem faced by the seller is

where the first term on the right-hand side of Equation 12.5 is expected revenue from first-period sales and the second term is salvage revenue.

Equation 12.5 is not too difficult to solve if we can model how the expectation E[min{D(p), x}] depends on p. One simple possibility is to assume that D(p) follows a uniform distribution between 0 and a – bp for some a and b. Then, if a = 200 and b = 10, demand would be uniformly distributed between 0 and 180 when p = $2.00 and uniformly distributed between 0 and 150 for p = $5.00. Expected first-period revenue, salvage revenue, and total revenue for different prices are illustrated in Figure 12.5 for the case of a = 200, b = 10, a salvage value r = $5.00, and an inventory of x = 60. In this case, the optimal price is $13.29, with corresponding expected sales revenue of $440.89, expected salvage revenue of $134.23, and expected total revenue of $575.02.

We can extend this formulation of the two-period markdown problem to create a simple algorithm that incorporates uncertainty. The idea is similar to the deterministic algorithm in the previous section, in which we solved the single-period deterministic problem to establish a markdown price for the next period, observed sales during that period, and then solved the same problem again at the beginning of the next period, using the new inventory value to get a new price. This time we follow a similar philosophy using the probabilistic formulation in Equation 12.5, but with a twist. At the beginning of each period, we estimate a probability distribution on future demands as if we are going to hold the price constant for all remaining periods. That is, for any price in period t, we define the random variable

Figure 12.5 Expected revenue as a function of price in the single-period markdown model with salvage.

Example 12.3

A seller faces two pricing periods, with the out date occurring at the end of period 2. D₁($5.00), the demand in period 1 given a price of $5.00, follows a normal distribution with mean of 50 and variance of 25, and D₂($5.00) follows a normal distribution with mean of 30 and variance of 10. Then D̂($5.00) = D₁($5.00) + D₂($5.00) follows a normal distribution with mean of 80 and variance of 35.

In the simple probabilistic algorithm, we calculate the price for the upcoming period as if it were the final markdown period. Then we find the current price p_t by solving Equation 12.5 using D̂(p_t) as our probabilistic forecast of demand. This is equivalent to assuming that the price we choose for the current period will be held constant for the remainder of the selling season. This leads to the simple probabilistic algorithm.

Simple Probabilistic Markdown Pricing Algorithm

1. Set an initial price of p₁.

2. Observe sales x₁ during the first period. Starting inventory for the second period is x₂ = x₁ – q₁.

3. At the start of each period t, calculate the distribution of D̂_t(p_t) = D_t(p_t) + D_{t + 1} (p_t) + . . . + D_T (p_T).

4. Find p_t that maximizes (p_t – r)E[min{D̂_t(p_t), x_t}] + rx_t. Set this value as the price for the current period.

This approach was tested for six items in eight stores of the Chilean fashion retail chain Falabella and compared against actual sales for the 1995 autumn-winter season. The recommendations from the probabilistic algorithm generated revenue 12% higher than that obtained by the product managers (Bitran, Caldentey, and Mondschein 1998). It is tempting to compare the 12% improvement in this study with the 4.8% improvement reported from the deterministic approach described in Section 12.2.1 and attribute the improvement to the incorporation of uncertainty in the algorithm. However, this would ignore the fact that the improvements came from applications in two very different settings and that the improvements were reported based on differences from previous store practices, which may have been very different in the two situations.

12.2.3 Exhaustive Search

The previous two sections described approaches to setting markdown prices using explicit optimization. In both cases, the optimization problems were specified without incorporating any of the business considerations described in Section 12.1.4. One way to incorporate such business considerations into the markdown problem would be by adding constraints to the optimization problems in 12.1 through 12.4 and 12.5. As long as the number of the constraints is relatively small and remains stable over time, this is not much of a problem. However, if the number of constraints is large or is changing over time, then it can become a problem. Specifically, explicit optimization approaches tend to work well when the number of constraints is small and the constraints have a simple structure. As the number and complexity of constraints increases, then explicit optimization approaches tend to take more time—often much more time—to find a solution. In the case where business considerations require many constraints to be imposed and these considerations change over time, exhaustive search may be a superior alternative to explicit optimization.

To understand how exhaustive search works, assume that our current price is p₁, our current inventory is I, and we have three periods left and four possible prices at which we can sell in each period with p₁ > p₂ > p₃ > p₄. We also assume that there is a salvage price r < p₄ at which all unsold inventory can be sold, with the possibility that r = 0. We want to determine what price to charge in the next period (t = 2), knowing that we will be able to change the price at most twice more. Possible markdown price paths are shown in Figure 12.6. Each arrow represents a potential price transition. To execute exhaustive search, we calculate the demand at each node in the graph in Figure 12.6 at the specified price so that the demand at price-level i in week t is given by d_t (p_i). These demands can be estimated using a demand model such as the one described in Section 12.3.

Once these demands have been calculated, we move forward in time and estimate the expected revenue for each possible path. For each pathway, we start with the initial inventory. When we arrive at a node, we calculate sales as the minimum of the demand at that node and the remaining inventory. Revenue at a node is the product of the sales and the price associated with that node. Each path terminates at the salvage node and any remaining inventory is sold at the salvage price—which will be zero if the inventory is totally perishable. Note that, for exhaustive search as for explicit optimization, we calculate a markdown price for all future periods, but we only need to commit to the price for the upcoming period. If sales are higher or lower than expected, we can recalculate the future markdown path based on updated inventory.

Figure 12.6 Possible markdown paths with three periods and four prices on a price ladder.

Example 12.4

Consider a retailer with 95 units left in stock of a television that is being replaced by a newer model in three weeks and needs to be marked down. The television set currently sells at $500, and the seller has a pricing ladder with four prices: $500, $450, $400, and $350. Unsold televisions will be sent to an outlet where any remaining inventory will be sold for $300—the salvage price. The demand for the television set is independent of week with d($500) = 10, d($450) = 20, d($400) = 40, and d($350) = 60, where each number is demand per week that will be realized at that price. We can calculate his optimal policy using enumeration. Starting in the first period, if he sets the initial price at $500, he will sell 10 televisions for first-period revenue of $5,000 and have a remaining inventory of 85; if he sets the initial price at $450, he will sell 20 televisions for revenue of $9,000 and have a remaining inventory of 75, and so on. When we have completed this process, we can add up the revenue per period and the salvage revenue for each policy to calculate the total revenue per policy as shown in Table 12.3. As can be seen from the table, the optimal policy is to drop the price immediately to $450 and then drop it again to $400 for the second and third weeks. This policy sells all of the television sets and realizes expected revenue of $39,000. Note that the only immediate decision facing the seller is the next price to set. In this case, he would set the price to $450, observe demand during the week, and perform another optimization at the end of the week to determine the next price to charge.

The complexity of the exhaustive search method comes from the dimensionality of the problem. The markdown schedule shown in Figure 12.6 has 20 possible markdown policies; a full 13-week schedule with 10 rungs each week would have 497,420 possible markdown schedules. While this does not necessarily present a particular computational challenge in itself, it can begin to present an obstacle when markdowns need to be computed for millions of items across hundreds of stores. To deal with this challenge, systems based on exhaustive search use various methods to prune the tree—that is, eliminate branches before evaluating them further. For example, if one combination of prices for the first two weeks results in less revenue and less remaining inventory than another combination, then the first combination can be eliminated from further consideration.⁸

The strength of the exhaustive search method arises from the fact that additional constraints tend to make exhaustive search easier because they eliminate the need to evaluate pathways that do not meet those constraints. For example, assume that in Example 12.4, the seller wanted to impose a constraint that there should be at most one markdown before the end of the season. By eliminating markdown plans with multiple markdowns, this would reduce the number of alternatives that need to be evaluated from 20 to 16. In general, adding constraints makes explicit optimization more difficult but exhaustive search easier.

TABLE 12.3

Revenues by week, salvage revenue, and total revenue for markdown policies applied to Example 12.4

NOTE: Optimal values are shown in italics.

12.3 ESTIMATING MARKDOWN SENSITIVITY

Key inputs to all of the optimization approaches described in Section 12.2 are estimates of demand (or expected demand) as a function of the price in each period—the functions denoted d_i (p_i) and E[d(p)] in Section 12.2. Any of the techniques for measuring price sensitivity described in Chapter 4 can be used to estimate the parameters of these functions. However, markdown management systems often use forecasting approaches that focus on the change in demand from an expected path that was induced by a markdown.

Sales and prices from an actual markdown schedule are illustrated in Figure 12.7. This figure shows price (as a percentage of list) and weekly sales for a woman’s blouse across a 30-week selling season that started in week 23 and lasted through week 52. In this case, there were three markdowns: a 30% markdown in week 32, a further 20% taken off in week 40, and another 20% reduction in week 48. The final price was 30% of the list price. Sales of the blouse grew for the first few weeks and then began to decline until the first markdown was taken, at which point sales surged. Each of the three markdowns was accompanied by a jump in sales followed by a decline. This pattern is typical—markdowns can induce sudden and substantial sales gains. Markdown optimization systems typically seek to represent this phenomenon by separately modeling the natural time track of demand—the product life cycle—and the incremental effect of markdowns and then combining the two.

Figure 12.7 Example of weekly sales and price under a markdown policy. Source: Adapted from Levy and Woo 1999.

Figure 12.8 Example of weekly sales and baseline product life cycle forecast corresponding to Figure 12.7.

The first step in estimating markdown sensitivity is to calculate a baseline product life cycle forecast. This is a forecast of how sales would have proceeded through the season without any markdowns—that is, how the item would have sold if the price remained constant at list price. The baseline product life cycle can be estimated by aggregating sales of fashion goods for periods in which no markdowns have been taken. Figure 12.8 shows the baseline product life cycle along with actual sales for the markdown schedule in Figure 12.7. One way to estimate price sensitivity is to attribute all sales in excess of the baseline product life cycle to the effect of markdowns and to fit an appropriate price-response function. For the example in Figure 12.8, the sales in excess of the baseline for weeks 32 through 39 would be attributed to the 30% markdown taken in week 32, the sales in excess of the baseline for weeks 40 through 47 would be attributed to the 50% total discount in place during that period, and the sales in excess of the baseline for the remainder of the season would be attributed to the 70% total discount initiated in week 48.

TABLE 12.4

Baseline forecast and actual sales for weeks 32 through 39 in Example 12.5

Example 12.5

For the example in Figure 12.8, the baseline product life cycle forecast of sales and the actual sales for weeks 32 through 39 are as shown in Table 12.4. Forecast sales of 1,219 at the list price and actual sales of 4,253 at the 30% discount provide two points that can be used to establish a price-response function.

Note from Table 12.4 that the seller experienced a (4,253 – 1,219)/1,219 = 249% increase in sales from a 30% decrease in price, corresponding to a markdown elasticity of 249/30 = 8.3. Such unusually high levels of elasticity are often found in markdown and promotions situations. They do not imply that the elasticity of sales with respect to list price is anywhere near as high under usual circumstances. Typically, markdown elasticities are higher than list price elasticities both because some customers wait to purchase in anticipation of the markdown and because advertised markdowns increase excitement and sales.

In addition to product life cycle and markdown sensitivity, demand forecasts for markdown management typically include seasonality. Seasonality represents the natural variation in sales for different product categories over the course of year. This can be driven by the seasonality of the items themselves—bathing suits sell more in the summer than in the winter—as well as events such as holidays and Black Friday. Seasonality can be modeled using a factor that depends on the week of the year.

We can put these elements together to derive a simple demand model. Let t = 1, 2, . . . , 52 represent the week of the year and τ represent the number of weeks since a product was introduced.

In this equation,

d(t) = Demand for an item (what we are trying to forecast) in week t

LC(τ) = Natural life-cycle demand for an item as a function of weeks since it was introduced

SEAS(t) = Seasonality as a function of week of the year

PD(t) = Price discount effect in week t

The price discount effect represents the increase in demand that will be realized from a markdown in a period. It can be modeled in several ways, but one approach is to set

PD(t) = e^{β[(p–pt)/p]},

where p is the full price, p_t is the actual (marked-down) price, and β > 0 is the markdown elasticity. Note that p_t = p corresponds to no markdown, in which case PD(t) = 1, which means that demand is entirely determined by the product life cycle and seasonality. As the markdown discount increases, p_t decreases and PD(t) accordingly increases, reflecting the ability of the markdown to drive additional demand.

Equation 12.6 incorporates the three most important elements that determine demand in a markdown situation. However, other factors may need to be incorporated to make the model fully realistic. For one thing, additional promotions—such as taking an additional 10% off of everything—may occur during the markdown period. Such promotions may require an additional factor to be added. Also, when inventory of an item gets low, sales often drop as a result of the broken assortment effect: When inventory of an item is low in a physical store, the remaining items may become harder to find. Furthermore, some items—like a sweater—may be priced at the item level, but, as inventory becomes low, certain sizes will sell out before the inventory is totally depleted. Finally, some consumers may want to purchase a certain minimum quantity of certain items but are unable to do so if inventory is too low. The result is that sales will drop independently of price if inventory drops below a certain level. Both promotions and broken assortment effects can be modeled by adding factors to Equation 12.6.⁹

12.4 STRATEGIC CUSTOMERS AND MARKDOWN MANAGEMENT

In his 1885 novel, Au Bonheur des Dames (The Ladies’ Paradise), Emile Zola chronicled the changes that a new form of commerce was bringing to Paris. The action in the novel revolves around the new department store Au Bonheur des Dames. The fictional store is based closely on Le Bon Marché, a pioneering store that brought mass merchandising and promotions to Paris for the first time. In the novel, two friends, Madame Marty and Madame Bourdelais, are shopping together when Madame Marty, on a whim, buys a red parasol for 14 francs, 50 centimes. Madame Bourdelais chides her, “You shouldn’t be in such a hurry, in a month’s time you could have got it for ten francs. They won’t catch me like that!” (Zola [1885] 1995, 245).

Madame Bourdelais is the first recorded example of a strategic customer—a customer who anticipates future pricing action on the part of a seller and acts accordingly. Specifically, modern customers have learned from past experience that most fashion goods and other markdown items will be marked down. This creates a motivation to wait for the sale—a phenomenon that has become so pervasive that, as noted above, only a small fraction of fashion goods are still sold at list price. Waiting for the sale is not just a phenomenon for brick-and-mortar stores; it is also commonplace in e-commerce where retailers not only apply markdowns but also employ regularly scheduled promotions such as Cyber Monday and Amazon’s Prime Day. One survey showed that 77% of customers in India who purchase online “sometimes” or “always” wait for a sale (Gautam, Shubham, and Mishra 2018). Customers have learned that goods being sold on the internet are just as likely to be marked down as they are in physical stores.

The fact that customers can anticipate a markdown and that some of them are willing to wait for it complicates markdown pricing. So far, the discussion has assumed that customers do not anticipate a markdown—they purchase when the price first goes below their willingness to pay. If customers truly behaved this way, the logic behind a markdown strategy would be unassailable—the model of sequential downward pricing introduced in Section 12.1 would maximize revenue. But a customer who believes that she could purchase the item in a future period at a lower price and is willing to wait might do so even if the current price is less than her willingness to pay. Countering her tendency to wait is the possibility that the item may not be available in the next period if it sells out in the current period.

We can formulate the decision faced by a customer in the following way. Assume that the current price of the item is p_H and that the customer believes, based on the past actions of the sellers, that, in the next period, the item will be marked down to some lower price p_L < p_H. Her willingness to pay for the item is w and, for simplicity, we assume that she is indifferent between purchasing the item today or purchasing in the next period. We further assume that w > p_H so that she would be willing to purchase in the first period if she knew that the item would not be available in the second period. The final element in her decision is whether the item is likely to be available for purchase if she waits. Denote her judgment of the probability that the item will be available in the second period as ρ. Then, a risk-neutral customer will purchase in the first period if and only if her surplus from a first-period purchase is greater than her expected surplus from waiting—that is, if w – p_H ≥ ρ(w – p_L). Rearranging terms, we see that she will purchase in the first period if ρ ≤ (w – p_H)/(w – p_L).

Example 12.6

Sharon is considering whether to buy a ticket for a Broadway show for full price at $400 or go to the TKTS outlet on the day of the show in hope that a ticket will be available at half price for $200. Her willingness to pay for a ticket is $500. She will purchase now if her probability that the ticket will be available at TKTS is less than or equal to ($500 – $400)/($500 – $200) = 1/3. If Sharon believes that the probability that a ticket for the show will be available at TKTS is greater than 1/3, then she will take the chance and seek to buy at half price.

Example 12.6 illustrates the important role that consumer expectations play in determining the profitability of a markdown strategy. Specifically, the higher the likelihood that a good will be sold at a markdown in the future, the more customers will be willing to wait. In Example 12.6, the ticket seller wants customers to believe that tickets will be scarce in the future so they will buy now. However, to segment the market and sell at differential prices, the seller would like to sell second-period seats at a discount. If the seller sells too many seats at a discount, it encourages high-willingness-to-pay customers to wait rather than purchase at full price. How do we determine the right policy in this case?

We can explore this issue with a simple game-theoretic model. Assume that a seller sells to two classes of customers: those with a high willingness to pay (w_H) for his product and those with a low willingness to pay (w_L), with w_H > w_L > 0. Assume that the number of high-willingness-to-pay customers is D_H and the number of low-willingness-to-pay customers is D_L. The seller can sell in either period at one of two prices: a high price p_H or a low price p_L, with p_H > p_L > 0. For the moment, we assume that p_H and p_L are given—that is, they are not determined by the seller. Finally, we assume that w_H > p_H > w_L > p_L, which means that the high-willingness-to-pay customers would be willing to purchase at either price, while the low-willingness-to-pay customers would only be willing to pay at the lower price. The seller has two decisions: how much of the good to purchase (x) and the price to charge in each of the two periods. We assume that customers are aware of the seller’s decisions and act to maximize their expected utilities.

Assume that the seller has x units on hand. What prices should he charge in each period? There are three possibilities:

1. Charge the high price in both periods—denote this policy by (p_H, p_H).

2. Charge the low price in both periods: (p_L, p_L).

3. Adopt a markdown policy: (p_H, p_L).

We do not need to consider the policy of raising the price (p_L, p_H) since everyone would purchase in the first period and it achieves the same revenue as (p_L, p_L).

Under the first policy, only the high-willingness-to-pay customers will purchase and the seller will realize revenue of R(p_H, p_H) = p_H min(x, D_H). Under the second policy, both high- and low-willingness-to-pay customers will purchase at the lower price and the corresponding revenue will be R(p_L, p_L) = p_L min(x, D_H + D_L). Let us assume that x ≤ D_H + D_L so that supply is less than or equal to total demand. In this case, R(p_H, p_H) ≥ R(p_L, p_L) if x ≤ (p_H/p_L)D_H. In other words, if supply is sufficiently scarce, it is better to price high than price low. If supply is sufficiently abundant, then it is better to price low than price high.

But is a markdown policy ever optimal? Whether a markdown is optimal depends on the expectations of the high-willingness-to-pay customers. Assume that the seller adopts the markdown policy (p_H, p_L). Then, as noted above, high-willingness-to-pay customers will purchase in the first period if they feel that the probability that the item will be available at the lower price in the second period is lower than (w_H – p_H)/ (w_H – p_L). If all of the high-willingness-to-pay customers wait until the second period to purchase, they will be competing with the low-willingness-to-pay customers for the scarce supply. Assuming that every customer has an equal probability of purchasing the scarce supply, the probability that any customer will be able to purchase is equal to the supply divided by the total demand, or x/(D_L + D_H). This means that a markdown policy will only be effective if

If condition 12.7 holds, under a markdown policy (p_H, p_L), the high-willingness-to-pay customers will all purchase in the first period at the high price, while the low-willingness-to-pay customers will wait to purchase in the second period at the lower price. If condition 12.7 does not hold, all of the high-willingness-to-pay customers will wait and seek to purchase at the lower price. If x ≤ D_H, there is no motivation for the seller to adopt a markdown policy since he can sell all of his inventory in the first period at the high price. Define Z as the quantity on the right side of the inequality (Equation 12.7)—that is, Z = (D_L + D_H) (w_H – p_H)/(w_H – p_L). Now assume that D_H < x < Z. In this case, by adopting a markdown policy, the seller can achieve revenue of p_H D_H + p_L (x – D_H). This is clearly higher than the revenue of p_H D_H that he would achieve from the high-price policy (p_H p_H). It is also higher than the revenue of p_Lx that he would achieve from the low-price policy (p_L p_L). So, in this case, a markdown policy is optimal. Finally, when x ≥ Z, the revenue from a high-price policy is p_H D_H and from a low-price policy is p_Lx; which of these two policies generates higher revenue will depend on the specific values of p_H p_L, x, and D_H. The conditions under which these different policies are optimal are summarized in Table 12.5.

Example 12.7

A theater owner has 100 unsold seats left and has established the price of tickets for his show at $100 per seat. He has the option to sell unsold seats for $50 on the day of the show. Alternatively, he could drop the price to $50 immediately. He knows that there are two types of customers for the show: theater lovers who are willing to pay up to $150 for a seat and bargain hunters who are willing to pay $75 for a seat. He believes there are 50 theater lovers and 200 bargain hunters in the market. This means that Since his remaining inventory of 100 seats is greater than the theater-lover demand of 50 but less than Z, his optimal policy is a markdown policy. Under this policy, he will achieve revenue of 50 × $100 + 50 × $50 = $7,500. Note that, if he had 130 seats left, his optimal policy would be to discount immediately, in which case his revenue would be 130 × $50 = $6,500.

In Example 12.7, increasing the available seating capacity from 100 to 130 actually reduces the revenue that the theater owner can generate. This occurs because, with the greater capacity, high-willingness-to-pay customers realize that the seller will ultimately reduce the price and that their chance of scoring a ticket at a discount is sufficiently high to make it worthwhile for them to wait. This illustrates that it is in the seller’s interest to communicate when supply is constrained—“Only two seats left at this price” or “Available only while supply lasts”—to encourage buyers to purchase now under the belief that the future price will not be discounted.

TABLE 12.5

Optimal policies and associated revenue with strategic customers as a function of inventory

NOTE: Z = (D_L + D_H) (w^H - p^H)/ (w_H - p_L).

This model of strategic customers is quite simple and is based on possibly unrealistic assumptions—namely, that both the seller and all customers have full information about willingness to pay, supply, and demand. However, despite these simplifications, the model provides a number of key insights. First of all, the optimality of a markdown policy depends on the relationship of supply (or capacity) to high-willingness-to-pay demand. If high-willingness-to-pay demand is less than (or not much greater than) supply, it may make sense to keep the price high—as with a hot toy at Christmas. If supply is much greater than high-willingness-to-pay demand—and customers know it—it may be better simply to lower price now rather than delay sales. It is the intermediate case in which it makes sense for the supplier to adopt a markdown policy.

The simple strategic customer model assumes that all customers know what policy the seller will follow—in particular whether he is going to mark down the price in the second period. This assumption is not as unrealistic as it seems—for seasonal fashion goods, sellers often have rather predictable markdown cadences and customers can use the seller’s past markdown behavior to determine the likelihood that the seller will mark down current inventory. This suggests that it is in the interest of the seller to create doubt in the customer’s mind over whether he will be marking down current inventory—specifically to reduce the probability in the customer’s mind that inventory will be put on sale. Research has shown that there can be cases in which a seller should pursue a mixed strategy—that is, sometimes not mark down inventory even when it would maximize short-run revenue for him to do so, or even withhold inventory from a markdown to reduce the probability that a customer who waits for a sale will be able to purchase at a discount. The point of both of these strategies is to artificially reduce the probability that customers who wait will be able to find inventory on sale, therefore encouraging them to purchase now at full price.

12.5 MARKDOWN MANAGEMENT IN ACTION

Automated markdown management and optimization systems were first developed and deployed at scale in the 1990s. One motivator was a dramatic expansion in retail capacity in the 1990s, which was followed by the need to focus on improving operations. Another reason was the arrival of the 800-pound gorilla of brick-and-mortar retailing: Walmart. With its wildly successful everyday-low-pricing policy, Walmart forced its competitors to improve every facet of their operations, from purchasing and logistics through pricing and promotions. While retailing has always been a thin-margin business, the need to wring every penny of profit from the market has never been greater.

The first widely publicized success of markdown optimization was at ShopKo. In 2000, ShopKo, which operated 141 discount stores under the ShopKo name and another 229 smaller, rural discount stores under the Pamida brand, installed a markdown optimization system developed by Spotlight Solutions. A pilot program showed a 14% increase in revenue from using the markdown optimization software. The success of the ShopKo pilot was widely reported, including in a Wall Street Journal article, helping to spur interest in the potential for such solutions. In addition to ShopKo, retailers that have announced the adoption of markdown optimization systems include the Gap, JCPenney, Home Depot, Bloomingdale’s, Sears, and Circuit City.

Markdown managements systems were initially developed for traditional retailers, who typically order their inventory well in advance—often six months or so—and then have it delivered by ocean shipping. For these retailers, there is usually no opportunity to replenish stock during the season (at least at a regional level), which means that ordering inventory for the season is a one-shot deal. As we have seen, this creates a strong incentive to order substantial amounts of inventory at the beginning of a season and then rely on a markdown strategy to maximize revenue from the inventory. One of the most successful innovations in retailing in the past 30 years is fast fashion, pioneered by the Spanish retailer Zara. In contrast to traditional retailers, Zara and other fast-fashion retailers, such as H&M and Forever 21, use a much more rapid supply chain to allow multiple replenishments during a season. This allows them to purchase much less initial inventory, thereby reducing the need for markdowns—the fraction of items that go on markdown at Zara is less than 20%, compared to 40% at traditional retailers. Furthermore, Zara realized 85% of list price on marked-down inventory compared to 25% on the part of traditional retailers. Nonetheless, Zara sponsored the development of a markdown optimization system tailored to its specific needs that is credited for increasing revenue from marked-down inventory by 6% (Caro and Gallien 2012).

It should be noted that markdown optimization is equally applicable to online and offline fashion retailers. Online retailers have the same need to clear excess inventory as their offline counterparts, and many of them use markdown optimization systems to maximize the return from their inventory.

In theory, markdown optimization increases revenue by optimizing the timing and depth of each markdown decision. In practice, a substantial portion of the benefits from markdown optimization comes from two sources. First, human decision makers typically wait too long to begin marking down inventory and then take very large markdowns. The reason seems to be that the people making markdown decisions tend to be the same people who bought the merchandise in the first place and are reluctant to appear to admit a mistake by marking down. As the pioneering American retailer E. A. Filene put it:

One of the few certainties in retailing is that some of the merchandise bought enthusiastically in the wholesale market, where it looks eminently saleable, will, when it reaches the store, prove stubbornly unsaleable. But despite this inevitability, buyers, like the rest of humanity, are reluctant to admit and address their errors. They find endless excuses for the slow sale of merchandise: the weather’s still too hot; the weather’s still too cold. Easter’s late this year; Easter was early this year. It hasn’t been advertised; the ad was lousy; the ad ran on the wrong day; the ad ran in the wrong newspaper.

The danger in these rationalizations is that they usually increase markdowns. A coat that does not sell early in the fall may still tempt customers if it is marked down early by as little as 25 percent. But as the season ends, it must be marked down far more drastically to tempt a customer who, having done without a new coat for so long, may otherwise reasonably decide to wait until the next season. (Quoted in Harris 1982, 132)

Markdown optimization systems have generally resulted in retailers’ taking smaller discounts earlier than they used to do, usually substantially increasing overall revenue (Namin, Ratchford, and Soysal 2017); E. A. Filene’s observation from 1910 has been vindicated, over 100 years later.

A second important benefit from markdown management systems has been their ability to tailor markdown schedules to the particular characteristics of individual regions or stores. In many cases, retailers had simply used the same markdown cadence for all stores in a chain: a particular style of sweater would be marked down by the same amount at the same time in Boston as in Los Angeles. Markdown management systems have been an important catalyst for enabling different markdown schedules for different stores. Canadian apparel retailer Northern Group Retail Ltd. implemented a markdown management solution from ProfitLogic (now part of Oracle) that it credited for helping move the company away from chainwide discounting to an approach “more attuned to regional needs, weather patterns, and other trends” (McPartlin 2004, 50). In another example, a men’s retail clothing chain traditionally put swimsuits on sale in September in all its stores nationwide. This made sense in northern locations such as Boston, Chicago, and New York, where the summer season was ending, but it made little sense in Orlando and Miami, where the tourist season was just beginning. The adoption of a markdown management system was the catalyst that enabled the chain to establish different markdown schedules in New York and Miami.

12.6 SUMMARY

• Under a markdown policy, the price of an item is sequentially decreased until either it sells or a selling period expires. Markdowns are widely used in a variety of industries—notably fashion goods and short-life-cycle products. The use of markdowns has been growing steadily in the United States, at least since World War II.

• A markdown policy is effective when the item for sale is perishable, supply is limited, and the desirability of the item decreases as it approaches its expiration date.

• A markdown policy is also a way for a seller to maximize expected revenue when supply is constrained and he is uncertain about the distribution of customer willingness to pay.

• Markdowns enable a seller to segment his customers between those who are willing to pay more to buy early and those who are willing to wait in order to save money.

• When price-response functions are known, the markdown optimization problem can be formulated and solved as a mathematical program.

• When price-response functions are uncertain, the markdown optimization problem can be formulated and solved as a dynamic program. This requires starting in the last period and working backward in time to determine the optimal policy for the present.

• Business constraints such as a limit on the number of markdowns that can be taken or minimum time intervals between sequential markdowns can significantly add to the complexity of the markdown problem. When the markdown problem is highly constrained, it can be more efficient to solve it using exhaustive search than to apply explicit optimization.

• Another consideration in markdown management is the presence of strategic consumers who anticipate the possibility of a markdown and may wait to take advantage of a lower price. When strategic consumers are present, it may be profitable for a seller to destroy unsold inventory rather than mark it down or to adopt a mixed strategy and randomize the decision to mark down.

• Many of the benefits of real-world markdown management implementations have come from taking smaller discounts earlier than customary practice and from the ability to derive markdown cadences tailored to the specific situations of individual stores within a chain.

12.7 FURTHER READING

More discussion of markdown optimization can be found in Smith 2009, which describes results at some retailers. Valkov 2006 describes other successful implementations. Namin, Ratchford, and Soysal 2017 compares the results of different markdown approaches to human-managed markdowns. Ramakrishnan 2012 has an extensive discussion of practical considerations in modeling markdowns, including the incorporation of business constraints and the use of exhaustive search. Hamermesh, Roberts, and Pirmohamed 2002 presents a case study of the challenges faced by ProfitLogic, one of the earliest commercial providers of markdown optimization software.

Markdown pricing in the presence of strategic customers has been a topic of considerable research over the last decade or so. For a survey, see Gönsch et al. 2013. On using a mixed strategy, see Gallego, Phillips, and Sahin 2008. On withholding inventory, see Liu and van Ryzin 2008. Özer and Zheng 2016 considers the effect of potential customer regret on the decision of whether to mark down inventory.

12.8 EXERCISES

1. Extend the two-period markdown model in Section 12.1.1 to three periods. That is, assume that the price-response function is d(p) = 1,000 – 100p, marginal cost is 0, and customers purchase as soon as price falls below their willingness to pay. What three prices, p₁, p₂, and p₃, will maximize total revenue? What if there are four periods and four prices? What is the general formula for n prices?

2. Now extend the two-period markdown model in Section 12.1.1 to the case where customers have a lower willingness to pay for the good in the second period. Specifically, assume that each customer’s willingness to pay in the second period is 75% of her willingness to pay in the first period. All other assumptions remain the same.

a. What is the optimal price and corresponding total revenue for the seller, assuming he can charge only a single price in both periods?

b. What are the optimal prices and corresponding total revenue, assuming he can charge different prices in the two periods?

3. A department store has 700 pairs of purple capri stretch pants that it must sell in the next four weeks. The store manager knows that demand by week for the next four weeks will be linear, with the following price-response functions:

Week 1: d₁(p₁) = 1,000 – 100p₁

Week 2: d₂(p₂) = 800 – 100p₂

Week 3: d₃(p₃) = 700 – 100p₃

Week 4: d₄(p₄) = 600 – 100p₄

Assume that the demands in the different weeks are independent—that is, that customers who do not buy in a given week do not come back in subsequent weeks.

a. What is the optimum price the retailer should charge per pair if he can only set one price for all four weeks? What is the corresponding revenue?

b. Assume he can charge a different price each week. What are the optimum prices by week he should charge? What is the corresponding revenue?

4. Consider a seller who wants to sell a product in two periods when all customers are strategic: they all fully anticipate the actions of the seller and choose to purchase in the first period, wait to purchase in the second period, or do not purchase at all. There are two classes of customers: low-willingness-to-pay customers have a maximum willingness to pay of $4, while high-willingness-to-pay customers have a maximum willingness to pay of $10. All customers maximize expected value and, if demand exceeds supply in either period, the limited supply will be allocated randomly to all customers without regard to their willingness to pay. In either period, the seller can set a price of either $2 or $7. The seller must choose what price to charge in each period as well as the total amount of the product to purchase. The unit cost of the product is $1 and the salvage value is $0.

a. Assume that there are 10,000 low-willingness-to-pay customers and 5,000 high-willingness-to-pay customers. How much should the seller order and put on sale to maximize profitability? What price should he charge in period 1 and in period 2? What is the resulting revenue?

b. Now, assume that the seller can charge any price he wants in both periods—that is, he is not constrained to $2 or $7. What price would he charge and how much would he order?

5. The questions in this problem are variations on Example 12.4. The best way to answer them is to build a spreadsheet model that calculates the revenue for the different policies under different parameter values. Each subproblem starts with the values as in the example—that is, the subproblems are independent, not cumulative.

a. Consider the markdown problem in Example 12.4. What is the optimal policy if everything else is the same as in the example, but the salvage value is $0 per television?

b. What is the optimal policy if the starting inventory is 120 instead of 95?

c. What is the optimal policy if the seller wants to have at least 10 television sets remaining at the end of the third week to provide sufficient inventory for his outlet store?

d. What is the optimal policy and corresponding revenue if each markdown generates additional demand of 5 television sets on top of the standard demand? Assume that the additional demand occurs once, in the period when the markdown occurs. That is, if the seller adopts the policy ($500, $450, $450), he will generate an additional 5 units of demand in week 2; if he adopts the policy ($500, $450, $350), he will generate an additional 5 units of demand in week 2 and in week 3.

NOTES

1. These examples and others can be found in Warner and Barski 1995.

2. In this section, I use the terms markdown management and markdown optimization interchangeably.

3. This model was first proposed by in Lazear 1986.

4. TKTS is a nonprofit consortium of theaters operating a discount-ticket booth. Obtaining tickets at the discount prices requires waiting in line, and there is no guarantee that the show you want will be available when you get to the counter. Thus, TKTS creates an inferior product that can be sold at a lower price with minimal cannibalization of full-price tickets.

5. The term Dutch auction has also been used for a somewhat different method for selling multiple products in financial markets and on eBay, leading to some confusion.

6. There is no requirement that the periods need to be of equal length. If, for example, the retailer had a policy that all items had to be sold at list price for at least three weeks before being marked down for the first time, the first period would be three weeks long, and there would be only 13 total pricing periods for the season.

7. Strictly speaking, constraint 12.4 is not required, since the optimal solution to MDOWN will never result in a markdown price that is less than the salvage value.

8. For a more in-depth description of exhaustive search applied to markdown management, including additional examples of path reduction rules, see Ramakrishnan 2012.

9. The broken assortment effect was first described by in Smith and Achabal 1998. For details on how additional effects such as promotions and the assortment effect can be incorporated in modeling markdown demand, see Ramakrishnan 2012.