The Profitability of Purchase Limits During Shortages

Unexpected events can create shortages. Pandemics can close factories or disrupt distribution (Morris 2020). In addition to pandemics, natural disasters (e.g., hurricanes, tornados, droughts, fires), factory disasters, strikes, civil unrest, wars, trade disputes, pirates at sea, and other unexpected events can abruptly disrupt supply, rapidly escalate demand, or do both. Most products can suddenly become scarce, including water, gasoline, toilet paper, meat, toys, chlorine, computer chips, and so on. For example, pandemics can disrupt supply chains and unexpectedly reduce the inventory of cleaning and disinfectant products or food products (e.g., cream cheese, avocados, soybeans, milk, pasta) (Sager 2022; Wong 2022). Factory strikes can suddenly cause stockouts of some packaged goods, and sudden droughts can reduce coffee bean supplies.

Shortages differ from ordinary stockouts. Shortages involve the entire supply chain and are a result of an unexpected event (called a shock) that causes the quantity demanded to temporarily exceed the quantity supplied at posted market prices (Goldfarb 2013; Hobbs 2021). By contrast, ordinary stockouts result from a single store's strategic inventory policy to minimize inventory-carrying costs given stochastic demand (Arrow, Harris, and Marschak 1951; Jing and Lewis 2011; Silver, Pyke, and Peterson 1998; Van Delft and Vial 1996).

Managing Shortages

We study shortage management from a profit-seeking perspective. Simply increasing inventory is not a solution for an unexpected shortage because carrying excessive inventory is prohibitively costly and increases the risk of negative demand shock resulting from unexpected closures, curfews, harsh weather, regulation, new superior products, and so on. For example, lawn product demand plunges during hurricanes. Demand at malls, as well as suitcase and dress clothing demand, plunges during pandemics like COVID-19. Excessive inventory can be as problematic as insufficient inventory.

The extant literature examines various tools for managing shortages, including (1) raising prices to decrease demand (Shleifer and Vishny 1991), (2) selling products using first-come-first-served queues (Polterovich 1993), (3) demarketing scarce products (demarketing in shortages may lack traditional benefits; see Gerstner, Hess, and Chu 1993; Miklós-Thal and Zhang 2013), (4) limiting sales to priority groups (e.g., loyal customers, first responders, backlogged orders) (Cattani and Souza 2002), and (5) allocating supply randomly using lottery-like mechanisms (Kerr 1995). Finally, centralized governments or agencies with coercive enforcement capabilities can employ political solutions (Shleifer and Vishny 1991). However, government objectives might differ from private firm objectives (Flowers and Stroup 1979; Tobin 1952).

Purchase limits are emerging as a popular tool for managing shortages (Robinson 2022), but this tool receives less attention in the literature than others. Purchase limits set a maximum number of units a consumer can buy in a single transaction (or period). During the COVID-19 pandemic, many major retailing chains—including Walmart, Target, CVS, Walgreens, Costco, Aldi, ShopRite, Hy-Vee, Ride Aid, Kroger, Wegmans, and H-E-B—implemented purchase limits on scarce products such as meat cuts, paper towels, hand sanitizers, face masks, disinfectant wipes, liquid bleach, disposable gloves, bathroom tissues, and rubbing alcohol (Browne 2020; Noel, Porter, and Buckley 2020; Radke 2020; Redman 2020; Snouwaert 2020; Webster 2020). Gas stations limited fuel purchases in May 2021 when a cyberattack on Colonial Pipeline, a major U.S. fuel provider, caused a gasoline shortage. Purchase limits can apply to almost all products (e.g., batteries, generators, water, electricity) (Eliasson 2015).

Despite their popularity, the goal of purchase limits is not obvious for profit-maximizing stores because it can be as profitable to sell one unit to two consumers as it is to sell two units to one customer. Public sector services (e.g., water, electrical utilities) claim egalitarian goals such as the reduction of hoarding, waste, panic buying, arbitrage, and unfair distribution. Stores seeking consumers’ goodwill could also have equity-related goals. However, purchase limits can also create ill will from first arrivals with greater needs who have expended greater efforts to find available supplies, and ill will may offset goodwill (Kang and Bunge 2020). It is also unclear whether goodwill creates greater future revenue because consumers must adopt nonoptimal future buying behavior after the shortage to reward some stores and punish others. Besides, given empty shelves, current consumers may be unaware of whether past sales were equitable. Finally, limits entail nontrivial administrative, enforcement, and opportunity costs. For example, in the days before a hurricane, gas stations without limits can quickly sell available gasoline inventory and close early in the week, allowing the station and employees to prepare for the hurricane.

Thus, the advantages of purchase limits may go beyond altruistic motives. We focus on the profitability of purchase limits during shortages for profit-maximizing multiproduct sellers. We examine which sellers should adopt limits and when. Our recommendations depend on observable factors such as store size (i.e., average basket size), shortage severity, competition, and seasonality.

Endogenous Scarcity

Although our focus is on exogenous shortage shocks, product scarcity can be endogenous. Prior research has studied how firms can exploit endogenous scarcity. First, firms can use scarcity as a signaling tool. Stock and Balachander (2005) examine how product scarcity can signal high quality in different conditions on marginal costs and reservation prices, and Balachander, Liu, and Stock (2009) provide empirical support that the scarcity of a new car at introduction acts as a signal and results in higher consumer preference. Accordingly, Cachon, Gallino, and Olivares (2019) find that more inventory (less scarcity) of a vehicle submodel signals low popularity and can decrease sales. Queue lengths can also reveal scarcity that signals higher service quality and causes herding behavior (Veeraraghavan and Debo 2009).

Second, scarcity can help pricing. Restricting capacity can create spot-market scarcity and enable higher advance prices (Shugan and Xie 2000; Xie and Shugan 2001). Amaldoss and Jain (2005b) show that scarcity contributing to product exclusivity can cause demand to increase with prices when some consumers desire uniqueness, and Amaldoss and Jain (2005a) also show that highly profitable snobs, who like scarce conspicuous products, can have upward-sloping demand curves but only when snobs have followers. Further, Tereyağoğlu and Veeraraghavan (2012) find that scarcity can increase profits by exploiting snobs when scarcity is credible. However, the addition of limited edition products increases price competition, so only high-quality brands benefit from scarcity (Balachander and Stock 2009). In nonprofit art markets, Tereyağoğlu, Fader, and Veeraraghavan (2017) find that nondecreasing price policies can increase revenues given future seat scarcity.

Third, scarcity may influence firm strategies in other contexts. For example, creating scarcity with limited editions incentivizes leaders to initially purchase products and causes subsequent buying frenzies to increase, rather than decrease, total sales (Amaldoss and Jain 2008). Debo and Van Ryzin (2009) show that asymmetric inventory allocations to ex ante identical retailers can increase the expected satisfied demand compared with symmetric inventory allocations because scarcity at one retailer can trigger herding behavior at the other retailer. In addition, Zhu and Ratner (2015) find that overall scarcity in a certain set of product options can induce more consumers to choose preferred options because scarcity can polarize consumers’ product evaluations for those options. Momot, Belavina, and Girotra (2020) study how social network information helps select a desired customer base for exclusive, scarce products.

Research Objectives

We analyze how profit-maximizing stores can manage exogenous shortage shocks using prices and purchase limits. Our model considers key exogenous observable factors that influence the price-and-limit strategy, including store size, shortage severity, competition, and seasonality. Moreover, we provide public policy implications by examining the condition when imposing purchase limits can improve consumer surplus.

Our analysis addresses the following research questions:

When should stores impose purchase limits rather than increase price? When should stores not impose purchase limits (i.e., simply adopt a first-come-first-served rule)?

When should stores impose purchase limits with price increases?

When and how do the exogenous factors (i.e., the store size, shortage severity, competition, and seasonality) influence stores’ price-and-limit strategies?

Does competition still matter given insufficient supplies to poach competitors’ consumers?

How do purchase limits affect consumer surplus? Can purchase limits improve both profits and consumer surplus? If so, when and why?

Key Findings

We model a property of the scarce product not considered by previous research: the store traffic associated with having inventory of the scarce product. The vast promotion literature shows that a promoted product or, in our case, a scarce product, increases store traffic, which, in turn, increases the sales of other products at multiproduct stores (Walters and Mackenzie 1988). These sales occur for many reasons, including unplanned purchases, impulse buying, and opportunistic purchases. Profits from other items can compensate for lost margins on the promoted product even when there are cherry pickers (Fox and Hoch 2005). Analogously, profit-maximizing stores can strategically adopt purchase limits to increase the scarce product availability and store traffic during shortages. This simple idea leads to some counterintuitive findings.

Briefly, a price-and-limit strategy (deciding price levels and whether to impose purchase limits) involves a trade-off between increasing the scarce product's profit margin and increasing store traffic (profits from other items). This trade-off plays out in surprising ways as the store size, shortage severity, competitive intensity, and seasonality vary. Our key findings include the following:

During shortages, stores can use purchase limits strategically to increase profits.

For moderate shortages, large (multiproduct) stores with large average shopping baskets use low prices to increase current traffic and impose limits to increase future traffic.

For moderate shortages, small stores with low traffic profits set high prices without limits because purchase limits are unnecessary to save inventory and only decrease sales at high prices.

For severe shortages, large stores use low prices to leverage limited supplies of the scarce product to maximize current traffic because traffic profits offset lower margins. Stockouts are inevitable at low prices, so purchase limits are unnecessary administrative costs.

For severe shortages, small stores adopt high prices to enjoy high margins while imposing limits to increase future traffic (note that high prices alone are insufficient to save inventory).

In moderate shortages, limits can make low prices more profitable for large stores, but in severe shortages, limits can make high prices more profitable for small stores.

Competition can decrease prices, which can cause stores to use limit strategies in moderate shortages to save inventory but abandon limit strategies in severe shortages because stockouts are inevitable.

Purchase limits can increase both store profits and consumer surplus. In moderate shortages, without limits, stores might raise prices to save inventory while enjoying high margins. In severe shortages occurring during off-peak demand periods, current purchase limits enable stores to save inventory and later decrease prices to increase traffic in a future peak period. Note that we consider shortages occurring during both peak and off-peak periods.

Basic Model

“The art of model building consists of stripping a situation of all nonessential complexity in order to analyze the key driving components in the most transparent manner possible” (Wernerfelt 1991, p. 232). To retain parsimony and transparency, our model focuses on the essential components that drive marketing strategies involving purchase limits for retail stores facing shortages.

Stores

We consider two profit-maximizing stores, s 1 and s 2 . When a shortage occurs, the stores make two decisions for two periods ( t = 1 , 2 ) : their product price (p) and whether to impose purchase limits (i.e., limit the purchase quantity to one). The stores adopt purchase limits only when limits strictly improve profits because of the administrative costs associated with implementing purchase limits. The two periods capture how the stores react to a shortage in period 1 and how the market evolves in period 2. The static strategy avoids unnecessary complexity (Cui, Raju, and Zhang 2008) and enhances interpretability (Farrell and Shapiro 1988; Fudenberg and Tirole 2000). Later, our model extension enables the stores to adopt dynamic strategies.

The profit-maximization assumption is key because it eliminates other explanations based on egalitarian goals (e.g., reducing hoarding, waste, arbitrage, unfair distribution). For example, purchase limits might prevent arbitragers from buying at legal prices and price gouging on black markets, but arbitragers do not necessarily influence store profits. Note that purely altruistic stores may not even consider purchase limits, and they may merely allocate supply according to consumer need or coordinate with other stores for welfare-maximizing allocations. We focus on using purchase limits for profits in a noncooperative game, which may provide a benchmark for research on altruistic goals.

Although our model enables the stores to adjust prices in a shortage, it does not consider any price gouging tools. For example, the stores do not sell scarce products at exorbitant prices via e-commerce platforms like eBay and Amazon. Most states have price gouging laws that allow only 10%–15% price increases. These state laws apply to different product categories during government-declared emergencies. For example, Illinois laws focus on petroleum-related products, and Maine laws focus on necessities (Zwolinski 2008). Consistent with these laws, the stores do not price gouge (i.e., prices do not exceed the highest preshortage price) in our model.

To focus on the price-and-limit strategy during a shortage, we implicitly assume that the stores optimally manage other variables. For example, the stores optimally engage in marketing activities to generate store traffic (e.g., advertising, store promotions, channel coordinating activities, online marketing) using diverse marketing tools (e.g., in-store displays, coupons, loss leaders, creative merchandising, experiential marketing, social media involvement). In our model, purchase limits are not used to compensate for bad decisions and the resulting suboptimal traffic.

Finally, we also implicitly assume that the stores engage in an optimal inventory policy prior to a shortage. Given inventory-carrying costs, the stores consider the likelihoods of a shortage and demand loss because excessive inventory can cause substantial losses when demand decreases. For example, a blackout during a hurricane can cause substantial spoilage in freezers. According to Ike Boruchow, a Wells Fargo retail analyst, 5%–10% of pre-COVID-19 demand may be decimated permanently (Thomas 2020). Thus, the optimal inventory policy perfectly manages neither shortages nor low demand. In our model, the preshortage inventory level is exogenously determined (lower preshortage inventory implies a more severe shortage).

Consumers

Store j serves two segments, L j and H j , in two periods ( t = 1 , 2 ) where L j and H j ’s willingness to pay (WTP) for the product is u L and u H > u L , respectively, for j = s 1 , s 2 . The high-end segment H j includes loyal consumers (loyals), and the low-end segment L j includes switchers (Hansen, Singh, and Chintagunta 2006; Narasimhan 1988; Simester 1997). Loyalty is defined as a behavioral construct reflecting the proclivity to shop at the same store (Deka 2016) for a variety of reasons, including high transportation costs, high switching costs, habit, fondness for shopping experiences at one store, and store attributes (e.g., locality, layout, parking, nearby restaurants).

Both segments need at most two units per period. A unit of the product can represent a literal unit or be interpreted more generally (i.e., a unit could be two packages or whatever quantity the stores set as a limit so that purchase limits are binding). We are agnostic about how many quantities should be included in a unit because our results only depend on limits reserving sufficient supply for future periods. Moreover, including heavy users who need more than two units per period also has little impact on our underlying mechanism. With purchase limits, they would behave like other consumers because all consumers are limited to buying one unit in a period. Without purchase limits, heavy users would appear as multiple consumers, merely expanding the size of the market.

In each period, consumers decide which store they visit on the basis of product availability and price. Loyals visit the focal store (j) when the product is in stock and has a price no greater than their WTP (u_H − p_j ≥ 0); otherwise, loyals exit the market. Loyals do not consider visiting store ∼j (i.e., the competitor). However, switchers consider visiting store ∼j if the product is unavailable at the focal store, or its price is higher than their WTP (u_L − p_j <0); otherwise, switchers visit the focal store (Armstrong, Vickers, and Zhou 2009). The shopper behavior may merely reflect prominence, including proximity to the shopper's home or business, convenient location on frequently traveled routes, or available nonscarce products desired by the shopper. For example, consumers may visit a prominent gas station on a frequently traveled route with their own routine (e.g., earning free car wash credits and buying coffee, sandwiches, lottery tickets, or cigarettes). Switchers visit store ∼j only if the product is in stock, the price is no greater than their WTP (u_L − p_∼j ≥ 0), and its offer dominates store j's offer. In each visit, consumers buy the product (maximum two units), if available, complying with the limit policy while they spend α on other items.

Consumers know a store's product availability and price when they consider visiting that store (Su and Zhang 2009), consistent with the consumer rationality premise in the game theory literature (i.e., consumers infer from market conditions what stores will do and act accordingly). Beyond rationality, mobile apps provide product availability and price. For example, during gasoline shortages, consumers can use mobile apps such as GasBuddy (Lewis and Marvel 2011). Alternatively, drivers can observe yellow out-of-gas bags on pumps before visiting gas stations. OurStreets and similar apps can help locate grocery products and find prices during shortages, and other mobile apps report the availability and prices of pharmaceuticals, transportation, and other products in short supply.

We focus on consumers who need the scarce product because those who only buy other items are irrelevant to our analysis of the scarce (focal) product. For example, consumers who buy coffee, sandwiches, or lottery tickets rather than scarce fuel at gas stations are not our focus. This approach is consistent with the promotion literature that focuses on consumers who seek a promoted product and possibly make purchases of other items on the same shopping trip. In this vein, our model abstracts out all other nonfocal issues (e.g., channel members’ reactions to changes in the store strategy and promotion strategies for other items); note that there may be other interesting variables that are marginally endogenous.

Store Traffic

We define the store traffic, ζ t j , as the number of shoppers who visit store j in period t for j = s₁, s₂ and t = 1, 2. Having inventory of the scarce product can increase store traffic beyond normal traffic. Lack of inventory, by contrast, decreases store traffic. Previous empirical findings suggest that when marketing tools increase store traffic (e.g., selling loss leaders, offering quantity discounts, bundling at low prices, offering attractive seasonal items, frequent shopper programs), sales of nonfocal products increase at multiproduct stores (Walters and Mackenzie 1988). We capture this relationship, called a lift, by consumers spending α per visit on other items.

The lift occurs for many reasons: (1) unplanned purchases (Bell, Corsten, and Knox 2011; Bucklin and Lattin 1991; Stilley, Inman, and Wakefield 2010), (2) impulse buying (Beatty and Ferrell 1998; Kollat and Willett 1967; Muruganantham and Bhakat 2013; Rook and Hoch 1985), (3) one-stop shopping (Kahn and Schmittlein 1992; Rhodes 2015), and (4) opportunistic purchases (Zhang et al. 2018). These purchases are major sources of store revenue (Agee and Martin 2001; Clover 1950; Harmancioglu, Finney, and Joseph 2009; Iyer 1989; Johnson 2017; Stern 1962). Although there is a distinction among these different types of purchases, we use the terms interchangeably to reflect the relationship between store traffic and sales of nonfocal products at multiproduct stores.

Four other store management research streams provide additional empirical and theoretical support for the lift: (1) loss leader research in which stores increase traffic with loss leaders (Chen and Rey 2012; DeGraba 2006; Ellison 2005; Hess and Gerstner 1987; In and Wright 2014; Lal and Matutes 1994; Li, Gu, and Liu 2013; Simester 1997; Walters and Mackenzie 1988), (2) promotion research in which stores run promotions to increase traffic (Ailawadi et al. 2006), (3) advertising research in which stores advertise one product to attract consumers who may buy other items (Lal and Matutes 1994), and (4) inventory research in which stockouts decrease sales of other items (Anderson, Fitzsimons, and Simester 2006).

A primary driver of our findings is purchase limits’ traffic enhancing function. This function has gone largely unnoticed by the popular press, which focuses instead on altruistic motives, so it makes our findings entirely new. Although promotions are also traffic enhancing, the mechanisms differ. Promotions sacrifice profit margins to increase current traffic, which can cause stockouts during shortages, but limits can save inventory for future sales without reducing margins. Furthermore, limits can be more beneficial for small stores, which benefit less from store traffic, than large stores.

Shortage

Our model considers two shortage levels to capture varying shortage severity: moderate and severe. With no shortage, the stores can cover the total demand in each period. However, during a shortage, the stores cannot obtain any scarce product from suppliers in period 2. The inventory level in period 1 is different in moderate and severe shortages. In the moderate shortage, the stores have four units (i.e., they can only meet period 1 demand), whereas the stores have two units in the severe shortage (i.e., they cannot even meet period 1 demand). The two levels sufficiently illustrate how the role of purchase limits changes with the shortage severity.

Store Size

The stores provide one common product and other (different or identical) products. We define store size as the number of products a store carries whose sales are sensitive to store traffic. It is well known that unplanned buying (or average shopping basket size) increases with store size (Bell, Corsten, and Knox 2011). Thus, α (i.e., consumers’ aggregate spending on nonfocal items in each store visit) represents the exogenous store size.

By the definition of store size, consumers buy more products in larger stores that have larger α, which is consistent with the retailing literature that suggests α reflects store-specific factors such as the number of product categories, the breadth of the assortment within a category, and the types of categories. For example, large retailers like Walmart, Costco, and gas stations that sell many items (e.g., food, lottery tickets, drinks) may have higher α than small specialty retailers, point-of-sale kiosks, and gas stations that have small or no convenience stores. Usually, brick-and-mortar stores have larger α than online stores for structural reasons. For example, the layout and environment of brick-and-mortar stores discourage single-product purchases (Iyer 1989; Mohan, Sivakumaran, and Sharma 2013). However, online stores are trying to increase their α.

Competition

We consider two competing stores, s₁ and s₂. Without loss of generality, s₂'s size is greater than s₁'s size, α s 2 > α s 1 > 0 , where α_j is store j's α for j = s₁, s₂ (note that α s 1 and α s 2 can be arbitrarily close; e.g., Walmart competes with stores of similar size such as Target). Therefore, s₂ has greater consumer spending on nonfocal items than s₁. For example, a single-product store might compete with a multiproduct store (Rhodes and Zhou 2019). During shortages, when the scarce product is out of stock at the focal store, consumers may consider the competitor of different size. However, we focus on symmetric stores’ competition while relaxing key assumptions in the “Model Extension” section of this article.

Analysis

No Shortage

This subsection develops a no-shortage benchmark in which the stores have ample inventory. First, we derive store j's traffic (i.e., the number of consumers who visit store j), sales volume for the product (i.e., units sold), and profits in period t given the stores’ prices (p_j, p_∼j), denoted ζ t j , D t j , and π t j , respectively; let π T j = π 1 j + π 2 j (i.e., store j's total profits) for j = s₁, s₂ and t = 1, 2. Subsequently, we derive an equilibrium. We normalize the marginal cost of the product to 0.

When store j sets a high price (u_L < p_j ≤ u_H), only loyals (H_j) visit store j. However, if store j sets a low price (p_j ≤ u_L), then its loyals and switchers (H_j and L_j) visit store j. Moreover, its competitor's switchers (L_∼j) visit store j if and only if p_∼j > u_L and p_j < p_∼j for j = s₁, s₂. Thus, given the stores’ prices, store j's traffic is

ζ t j ( p j ; p ∼ j ) = { 0 if p j > u H , ψ H j if u L < p j ≤ u H , ψ j + ψ L ∼ j I ( ⋅ | γ 0 j ) otherwise ,

(1)

for j = s₁, s₂ and t = 1, 2, where ψ i j is the size of store j's segment i for i = L, H; ψ j = ∑ i ψ i j ; γ 0 j = { ( p j , p ∼ j ) | p ∼ j > u L , p j < p ∼ j } ; I ( ⋅ | γ 0 j ) = 1 if ( p j , p ∼ j ) ∈ γ 0 j , and 0 otherwise. For simplicity, we normalize the size of each segment to 1.

With no shortage, consumers who visit store j buy two units of the product and spend α j on other items. Thus, given ζ t j in Equation 1, store j's sales volume and profits are

D t j = 2 ζ t j , π t j = p j D t j + α j ζ t j ,

(2)

for j = s₁, s₂ and t = 1, 2. From the profit function in Equation 2, we derive the equilibrium and obtain the optimal prices for the monopolistic stores. All proofs and the condition for a unique pure strategy equilibrium are in the Web Appendix.

Figure 1 shows the equilibrium in the right panel and the optimal prices for the monopolistic stores in the left panel (for the exact conditions, see the Web Appendix). The equilibrium prices depend on the store sizes, which reflect the stores’ average shopping basket sizes. When both stores are small, they set high prices to target loyals and avoid competition (UHE₀). When both stores are large, they set low prices to increase store traffic and profits from unplanned purchases (ULE₀). When the stores sufficiently differ in size, they differentiate. The larger store sets a low price, whereas the smaller store sets a high price (UAE₀). When both stores are midsize, regardless of their relative size, one sets a low price, and the other sets a high price (ME₀). Comparing competitive and monopolistic stores, we find that, in a unique equilibrium, competition can cause only the larger store to reduce its price because it has a greater incentive to increase traffic than the smaller store.

OPEN IN VIEWER

Figure 1.

Price strategy for no shortage in the basic model.

Moderate Shortage

This subsection examines how the stores react to a moderate shortage in which the stores have four units in period 1 and do not obtain additional supply in period 2.

For a moderate shortage, the stores determine their prices (p_j) and whether they impose a limit (limit the purchase quantity to one). Let l_j denote store j's purchase limit strategy, where l j = 1 if store j imposes a limit, and 0 otherwise, for j = s₁, s₂. The set of store j's strategy can be reduced to R M j = { ( p j , l j ) | ( p j > u L , l j = 0 ) ∨ ( p j ≤ u L , l j = 1 ) } for j = s 1 , s 2 . Given (p_j, l_j) ∈ R M j and ( p ∼ j , l ∼ j ) ∈ R M ∼ j , store j's traffic is

ζ t j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 1 if u L < p j ≤ u H , l j = 0 , 2 + I ( ⋅ | γ M j ) otherwise ,

(3)

for j = s₁, s₂ and t = 1, 2, where γ M j = { ( p j , l j , p ∼ j , l ∼ j ) | p ∼ j > u L , p j < p ∼ j , ( p j , l j ) ∈ R M j , ( p ∼ j , l ∼ j ) ∈ R M ∼ j } ; I ( ⋅ | γ M j ) = 1 if ( p j , l j , p ∼ j , l ∼ j ) ∈ γ M j , and 0 otherwise. Store j's sales volume is

D 1 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 2 if u L < p j ≤ u H , l j = 0 , 2 + I ( ⋅ | γ M j ) otherwise , D 2 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 4 - D 1 j ( ⋅ ) otherwise ,

(4)

for j = s₁, s₂. From Equations 3 and 4, store j's profits are π t j = p j D t j + α j ζ t j for j = s₁, s₂ and t = 1, 2. We derive the equilibrium from the profit function while obtaining the optimal price-and-limit decisions for the monopolistic stores.

Figure 2 shows the equilibrium in the right panel and the optimal decisions for the monopolistic stores in the left panel. Comparing the moderate shortage model results with the benchmark, we find that when a moderate shortage occurs, the stores may increase prices without purchase limits to exploit the high-end segment (i.e., loyals). If the stores adopt purchase limits, their profits decrease because high prices induce only the loyals to visit the focal store, and purchase limits reduce sales (note that the focal store's supply covers the loyals’ demand in a moderate shortage).

OPEN IN VIEWER

Figure 2.

Price-and-limit strategy for moderate shortage in the basic model.

P₁ compares the stores’ price-and-limit strategies in monopolistic and competitive markets when a moderate shortage occurs.

P₁: Suppose that a moderate shortage occurs. In a unique equilibrium, competition can cause only the larger store ( s 2 ) to lower its price and impose purchase limits.

P₁ has the surprising finding that, for a moderate shortage, competition can cause the larger store to impose purchase limits (it would not without competition). Although we might expect that limiting purchases would make a store less competitive, limits enable the larger store to decrease the price and increase both current and future store traffic, making the larger store more competitive (note that this differs from loss leader competition in which loss leaders do not necessarily increase future traffic from the same consumers but often reduce it). However, only the larger store can adopt this strategy because it benefits from unplanned purchases in both periods that justify profits lost from the low price. The smaller store enjoys high margins without purchase limits.

Corollary 1:

For moderate shortages, purchase limits increase the profits of both stores and consumer surplus simultaneously when 2 Δ u < α j < 4 Δ u for j = s 1 , s 2 , where Δ u = u H – u L .

Stores can improve profits using purchase limits as an additional marketing tool. Interestingly, purchase limits can also increase consumer surplus simultaneously (Corollary 1). For a moderate shortage, purchase limits can increase profits at low prices by preventing stockouts and increasing future store traffic. However, at high prices, limits only reduce sales with no benefit because high prices alone can prevent stockouts. Thus, purchase limits can cause the stores to only lower prices, increasing consumer surplus; note that switchers have no benefits, but loyals can save u H − u L per transaction or ( u H − u L ) / α × 100 % on their shopping basket with nonscarce items. Corollary 1 indicates that purchase limits may act as substitutes for price increases during a shortage.

Severe Shortage

This subsection studies how the stores react to a severe shortage in which the stores have two units in period 1 and do not obtain additional units in period 2. Moreover, we examine how the store size, shortage severity, and competition influence the stores’ strategies.

For a severe shortage, the stores determine their prices ( p j ) and whether they impose a limit ( l j ) . Given that the stores impose a limit only when it improves profits, the set of store j's strategy can be reduced to R S j = { ( p j , l j ) | ( p j > u L , l j = 1 ) ∨ ( p j ≤ u L , l j = 0 ) } for j = s₁, s₂. Thus, given ( p j , l j ) ∈ R S j and ( p ∼ j , l ∼ j ) ∈ R S ∼ j , store j's traffic is

ζ 1 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 1 if u L < p j ≤ u H , l j = 1 , 2 + I ( ⋅ | γ S j ) otherwise , ζ 2 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 1 if u L < p j ≤ u H , l j = 1 , 0 otherwise ,

(5)

for j = s₁, s₂, where γ S j = { ( p j , l j , p ∼ j , l ∼ j ) | p ∼ j > u L , p j < p ∼ j , ( p j , l j ) ∈ R S j , ( p ∼ j , l ∼ j ) ∈ R S ∼ j } ; I ( ⋅ | γ S j ) = 1 if ( p j , l j , p ∼ j , l ∼ j ) ∈ γ S j , and 0 otherwise. Store j's sales volume is

D 1 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 1 if u L < p j ≤ u H , l j = 1 , 2 otherwise , D 2 j ( p j , l j ; p ∼ j , l ∼ j ) = { 0 if p j > u H , 2 - D 1 j ( ⋅ ) otherwise ,

(6)

for j = s₁, s₂. From Equations 5 and 6, store j's profits are π t j = p j D t j + α j ζ t j for j = s₁, s₂ and t = 1, 2. We derive the equilibrium from the profit function while obtaining the optimal price-and-limit decisions for the monopolistic stores. Figure 3 shows the equilibrium in the right panel and the monopolistic stores’ optimal decisions in the left panel.

OPEN IN VIEWER

Figure 3.

Price-and-limit strategy for severe shortage in the basic model.

P₂ compares the limit strategies for severe shortages with those for moderate shortages.

P₂: Suppose that the smaller store (s₁) and larger store (s₂) are sufficiently small and large, respectively, α s 1 < 2 Δ u < α s 2 . The smaller store imposes purchase limits in severe shortages but not in moderate shortages. By contrast, the larger store imposes purchase limits in moderate shortages but not in severe shortages.

P₂ shows that the smaller and larger stores’ limit policies can switch as the shortage severity increases. For a moderate shortage, the smaller store sets a high price without purchase limits because there is sufficient supply at a high price to serve loyals in both periods. By contrast, for a severe shortage, the smaller store sets a high price with limits to prevent stockouts in period 1 that would reduce traffic and profits in period 2 (i.e., high prices and limits are complements).

Interestingly, the opposite occurs for the larger store. For a moderate shortage, the larger store sets a low price to attract its competitor's consumers. At a low price, limits prevent stockouts and save inventory to increase traffic and profits in period 2. By contrast, for a severe shortage, the larger store still sets a low price for store traffic but does not set limits because limits fail to prevent stockouts of the scarce product in period 1 (i.e., producing no benefits in period 2).

The intuition is that, in less severe shortages, the smaller store finds that high prices sufficiently save inventory to increase future store traffic. In severe shortages, the smaller store also needs purchase limits to save inventory. By contrast, the larger store finds low prices better because traffic is more valuable. Then, in less severe shortages, the larger store needs purchase limits to save inventory for future traffic. However, purchase limits are insufficient to save inventory in severe shortages (i.e., ineffective). Therefore, given any administrative costs, limits are suboptimal. Note that the larger store benefits more from capturing some of the smaller store's price-sensitive consumers at low prices than from selling a few scarce products at high prices.

P₃ shows the effects of competition on the stores’ price-and-limit strategies.

P₃: Suppose that a severe shortage occurs. In a unique equilibrium, competition can cause only the larger store ( s 2 ) to lower its price without purchase limits.

P₁ and P₃ show that competition has different effects on the stores’ limit strategies depending on the shortage severity. Competition can cause only the larger store to lower its price for traffic. Given the low price, the larger store imposes purchase limits for moderate shortages to save inventory. However, purchase limits do not prevent stockouts in severe shortages. Thus, the larger store does not benefit from purchase limits. From P₁, P₂, and P₃, we see that purchase limits and higher prices are strategic substitutes in moderate shortages but synergistic complements in severe shortages.

Corollary 2:

For severe shortages, purchase limits can increase both stores’ profits but do not increase consumer surplus.

Corollaries 1 and 2 show how the shortage severity moderates the effects of purchase limits on store profits and consumer surplus. For moderate shortages, purchase limits can cause the stores to decrease prices to maximize store traffic and increase store profits, and the price decrease improves consumer surplus (Corollary 1). For severe shortages, at a low price, purchase limits fail to prevent stockouts in period 1, and store traffic suffers in period 2. Therefore, limits are ineffective. However, at a high price, the stores serve only their high-end segment or loyals (i.e., the demand decreases to two units per period), so purchase limits (i.e., limiting the purchase quantity to one) save inventory for future sales to increase future traffic and total profits. The profits from the scarce product do not change, but the profits from other items increase. Thus, given purchase limits, the stores do not decrease prices (i.e., limits do not increase consumer surplus). This implies that limits and high prices are not perfect substitutes. Purchase limits spread demand better than high prices.

Model Extension

In this section, we extend the basic model and show that main results are robust to the inclusion of other factors. We also provide new results to improve our understanding of shortage management.

To analyze how stores of similar size develop their shortage strategies under additional realistic conditions, we extend the basic model with symmetric stores in three ways. First, the basic model focuses on the immediate reactions to a shortage. We relax the stores’ commitment to the initial strategy and consider dynamic strategies. In the dynamic game, the forward-looking stores adjust their price-and-limit strategy in each period on the basis of changing inventory and market conditions.

Second, we allow consumers’ spending on nonfocal items per visit (α) to vary across periods. Let α_t denote α in period t for t = 1, 2, where α₁ can be greater or less than α₂. We refer to that intertemporal variation as seasonality because it depends on whether the shortage begins in a peak or off-peak period. There are many other reasons for that variation: (1) The purchase cycle for the scarce product may not match the average shopping trip cycle. (2) Consumers may fear future shortages of other items. (3) The economy may become stronger or weaker, so consumers may increase or decrease their budget for other items. (4) Store merchandising efforts may shift from the scarce product to others. (5) Transaction costs (e.g., long queues, congestion, store hours) may change. (6) Cherry pickers can enter or leave the market. Note that cherry pickers do not necessarily have lower α. Fox and Hoch (2005) find that cherry pickers systematically buy 67% larger market baskets than shoppers visiting only a single store, consistent with economic incentives associated with transaction costs.

Third, the basic model assumes that consumers know the stores’ product availability and prices. Our extension considers nontrivial search costs to obtain the competitor's information consumers are not familiar with. Given search costs, the stores can no longer attract the competitor's switchers. When switchers decide not to visit the focal store, they will not incur search costs to obtain the competitor's information because they expect that the symmetric competitor cannot serve them as well. Thus, search costs demarcate competitive and noncompetitive markets.

No Shortage

This subsection develops a no-shortage benchmark in which the stores have sufficient units to meet demand. For the benchmark, we obtain the demand and store traffic functions from D t j and ζ t j in the basic benchmark model by replacing p j with p jt , where p jt is store j's price in period t for j = s 1 , s 2 and t = 1 , 2 . Then, we obtain store j's profits in period t, π t j = p jt D t j + α t ζ t j . Note that with no shortage, period 2 repeats period 1 with α 2 instead of α 1 . From the profit function, we derive the subgame perfect Nash equilibrium for the competitive stores and the optimal prices for the noncompetitive stores (i.e., markets with nontrivial search costs).

Figure 4 shows the equilibrium and the optimal prices for the competitive and noncompetitive stores in the right and left panel, respectively (for the exact conditions, see the Web Appendix). The equilibrium prices depend on consumer spending on nonfocal items in periods 1 and 2 (i.e., α₁ and α₂). When the stores are small (for small α₁ and α₂), they charge high prices to target their loyals and avoid competition (UHSE₀). Suppose that the stores are large. When consumers buy many other items only in period 2 (for small α₁ and large α₂), similarly, the stores set high prices in period 1, but they set low prices in period 2 to increase store visits and profits from unplanned purchases (USE1₀). Conversely, when consumers buy many other items only in period 1 (for large α₁ and small α₂), the stores set low prices in period 1 but high prices in period 2 (USE2₀). Last, when consumer spending on other items is large in both periods, the stores set low prices in both periods (ULSE₀).

OPEN IN VIEWER

Figure 4.

Price strategy for no shortage in the extended model.

Comparing the competitive stores with the noncompetitive stores, we find that competition causes the stores to decrease prices in either period 1 or period 2 depending on α₁ and α₂, unless α_t is too small or large for t = 1, 2. For example, competition can cause a large store with large α₁ to lower its price in period 2 because, without competition, the store already sets a low price in period 1, given large α₁, so the store decreases its price in period 2 to attract the competitor's consumers when facing competition (see Figure 4).

Moderate Shortage

This subsection studies how the stores react to a moderate shortage in the extended model in which the stores have four units in period 1 and do not obtain additional supply in period 2.

For a moderate shortage, store j determines its price ( p jt ) and whether it imposes a limit in period t for j = s 1 , s 2 and t = 1 , 2 . Let l jt denote store j's purchase limit strategy in period t, where l jt = 1 if store j imposes a purchase limit in period t, and 0 otherwise. Similarly, we can obtain the profit function as shown in the basic model. Then, we derive the subgame perfect Nash equilibrium for the competitive stores and the optimal price-and-limit decisions for the noncompetitive stores (i.e., markets with nontrivial search costs).

Figure 5 shows the equilibrium and optimal decisions in the right and left panel, respectively.

OPEN IN VIEWER

Figure 5.

Price-and-limit strategy for moderate shortage in the extended model.

We find that in moderate shortages, small stores set high prices maximizing their profit margin of the scarce product and do not impose purchase limits in both periods (note that in moderate shortages, purchase limits decrease profits at a high price because only loyals visit the focal store, and limits reduce sales). Suppose that the stores are large. When consumer spending on nonscarce items is small in period 1 but large in period 2, similarly, the stores set high prices without limits in period 1 to obtain the maximum profits from the scarce product. However, they set low prices without limits in period 2 to increase traffic and sales from nonscarce items. By contrast, when consumer spending on nonscarce items is large in period 1 but small in period 2, the stores set low prices with limits in period 1 to increase traffic and save inventory for future sales. In period 2, given low profits from nonscarce items, the stores set high prices without limits to clear inventory at high profit margins. Last, the large stores with large α 1 and α 2 set low prices with limits in period 1 but without limits in period 2 to increase traffic and profits from nonscarce items throughout the shortage. In period 2, the stores sell out the scarce product with or without purchase limits (i.e., purchase limits provide no benefits to the stores).

P₄ compares the price-and-limit policy in competitive and noncompetitive markets.

P₄: Suppose that a moderate shortage occurs. Competition can cause a store to:

Reduce its price only in period 1 with purchase limits when consumer spending is large in period 2 and not too small in period 1, Δ u < α 1 < 2 Δ u < α 2 .

Reduce its price only in period 2 without purchase limits when consumer spending is large in period 1 and not too small in period 2, Δ u < α 2 < 2 Δ u < α 1 .

Reduce its price with purchase limits in period 1 but without purchase limits in period 2 when consumer spending is midrange for both periods 1 and 2, Δ u < α t < 2 Δ u for t = 1 , 2.

P₄ shows that when competition decreases prices, the price reduction timing and purchase limit strategy depend on consumer spending on nonscarce items in periods 1 and 2. For example, suppose that the shortage occurs in an off-peak period (i.e., consumer spending on other items is smaller in period 1 than in period 2). Without competition, the stores set high prices without limits in off-peak period 1 to obtain large profits from the scarce product, but they set low prices in peak period 2 to increase traffic and profits from nonscarce items. When the stores compete, a store decreases its price to attract the competitor's consumers in period 1 and increases traffic while imposing limits to save some units for future sales in peak period 2. The price reduction does not occur when spending is very small in off-peak period 1 because the profit gain from other items does not exceed the profit loss from the decrease in the scarce product’s profit margin.

Conversely, suppose that the shortage occurs in a peak period (i.e., consumer spending on other items is larger in period 1 than in period 2). Without competition, the stores charge low prices with limits in peak period 1 to increase store traffic and profits from other items, whereas they set high prices without limits in off-peak period 2 to maximize the profits from the scarce product. When the stores compete, they adopt the same low-price-limit strategy in period 1, but a store decreases its price without limits to increase traffic by attracting the competitor's consumers in period 2. The price reduction in off-peak period 2 requires that consumer spending is not too small in that period because otherwise the benefits from increased traffic are not sufficient to compensate for the decreased margin of the scarce product.

Finally, suppose that consumer spending on other items is midrange without seasonality such that, without competition, the stores set high prices without limits in both periods to maximize the profits from the scarce product. When the stores compete, a store decreases its price in both periods, given the competitor's high price, to attract the competitor's switchers and maximize traffic profits. With the low-price policy, the store adopts different limit strategies in each period. In period 1, the store imposes limits to keep some units for future sales. However, in period 2, the store has no incentive to impose limits because, given limited supply in period 2, the store clears inventory regardless of whether it imposes limits.

Severe Shortage

This subsection examines how the stores react to severe shortages in which the stores have only two units in period 1 and do not obtain additional units in period 2. We also study how key observable factors (the store size, shortage severity, competitive intensity, and seasonality) affect the stores’ dynamic price-and-limit policies.

Given a severe shortage, store j determines its price-and-limit strategy ( p jt , l jt ) in period t for j = s 1 , s 2 and t = 1 , 2 . Similarly, we can obtain the profit function as shown in the basic model. Then, we derive the subgame perfect Nash equilibrium for the competitive stores and the optimal price-and-limit decisions for the noncompetitive stores (i.e., markets with nontrivial search costs).

Figure 6 shows the equilibrium and optimal decisions in the right and left panel, respectively. We show that in severe shortages, small stores set high prices in both periods to maximize the profit margin of the scarce product. In period 1, small stores impose limits to save inventory, increase future traffic, and obtain future sales from other items. However, small stores do not use limits in period 2 because, given limited supply, limits produce no benefits. Suppose that the stores are large. When consumers buy many items in peak period 2 but not in off-peak period 1, similarly, the stores set high prices with limits in period 1 to achieve the high margin while saving inventory for future traffic. In period 2, however, the stores set low prices without limits to increase traffic and exploit consumers’ large spending on other items. Finally, when consumers buy many items in peak period 1 but not in off-peak period 2, the stores set low prices in period 1 to increase store traffic and profits from nonscarce items. In severe shortages, limits cannot save inventory at low prices, so the stores do not impose limits in period 1. The stores have no scarce products to sell (i.e., require no strategy) in period 2.

P₅: Small stores impose purchase limits in severe shortages but not in moderate shortages. By contrast, large stores impose purchase limits in moderate shortages but not in severe shortages when consumer spending is large only in period 1. When consumer spending is also large in period 2, however, large stores impose purchase limits regardless of the shortage severity.

OPEN IN VIEWER

Figure 6.

Price-and-limit strategy for severe shortage in the extended model.

P₅ is consistent with P₂. Small stores focus on profit margins rather than store traffic. Thus, small stores set high prices regardless of the shortage severity. However, small stores’ limit strategy varies with the shortage severity. In moderate shortages, small stores impose no purchase limits to increase the sales of the scarce product at the high price, whereas in severe shortages, small stores impose purchase limits to save inventory for future sales and obtain some additional sales from other items in period 2.

By contrast, when the shortage occurs in a peak period, large stores set low prices in peak period 1, regardless of the shortage severity, to increase traffic and sales from other items. However, like small stores, large stores have different purchase limit strategies depending on the shortage severity, but large stores’ purchase limit strategies are the opposite of small stores’ purchase limit strategies. In moderate shortages, large stores impose limits to save inventory for future traffic, whereas in severe shortages, large stores do not impose limits because limits cannot save inventory at low prices. Interestingly, when consumer spending is also large in period 2 (i.e., with a long season), for severe shortages, large stores charge high prices with limits in period 1 because it is more profitable to save inventory and increase future traffic than sell out the scarce product in period 1. For moderate shortages, large stores charge low prices with limits in period 1 to maximize store traffic in both periods (note that in moderate shortages, purchase limits can save inventory without price increases). In period 2, regardless of the shortage severity, large stores focusing on store traffic set low prices without purchase limits.

P₆: Suppose that a severe shortage occurs. Competition does not affect small stores’ price-and-limit strategies. However, for large stores, competition can cause a store to decrease its price without purchase limits in period 1 when consumer spending is small in period 2 and not very large in period 1. When consumer spending is large in period 2, even with competition, both large stores set high prices with limits in period 1 and lower prices without limits in period 2.

P₄ and P₆ show that competition can cause different limit strategies, depending on the shortage severity. Given sufficient demand in a severe shortage, small stores charge high prices with limits in period 1 and no limits in period 2, regardless of competition. Large stores without competition adopt the same strategy when consumer spending is small in period 2 and not very large in period 1. However, with competition, a store decreases its price in period 1 because it can attract the competitor's consumers, so increasing store traffic becomes more profitable than selling a few scarce products at a high price. The low price prevents purchase limits from saving inventory in period 1, so the store does not impose limits, consistent with P₃. Interestingly, when consumer spending is large in period 2, large, even competitive, stores set high prices and impose limits in period 1 to prevent stockouts. In period 2, the large stores decrease prices without limits to further increase store traffic with inventory saved from period 1.

Corollary 3:

Regardless of the shortage severity, purchase limits can increase the symmetric stores’ profits and consumer surplus simultaneously.

Corollary 3 is consistent with Corollary 1 for moderate shortages, and moreover, it shows that purchase limits can simultaneously increase store profits and consumer surplus, even in severe shortages. Suppose that consumer spending is small in off-peak period 1 and large in peak period 2 during a severe shortage. If purchase limits are prohibited, the stores cannot prevent stockouts in period 1. Then, given low consumer spending in period 1, the stores set high prices to maximize the profits from the scarce product. However, purchase limits allow the stores to save inventory at high prices in off-peak period 1, and the stores decrease prices in peak period 2 to further increase store traffic, thereby improving both store profits and consumer surplus.

Example: Food Shortages After COVID-19 Omicron Outbreak

To illustrate our analytical analysis, we provide a numerical example considering the shortages created by the highly contagious Omicron variant of COVID-19, which disrupted global supply chains in 2022. After the Omicron outbreak, retailers faced shipping delays as a result of sick truck, railway, and dock workers. On average, the U.S. out-of-stock levels on general grocery products, such as foods, beverages, household cleaning, and personal hygiene products, increased to 12% (15% on foods only), compared with the usual out-of-stock levels of 7%–10% (Cavale and Walljasper 2022). Some major supermarket chains (e.g., Publix U.S., Coles Australia, ParknShop Hong Kong) imposed purchase limits on their scarce products, including pet foods, meats, and staples (Crea 2022; Hong Kong Free Press 2022; Xiao 2022).

This simple shortage example reveals how stores can use purchase limits to increase profits and how purchase limits influence consumer surplus. Consider two monopolistic stores that sell a meat product and other items for two periods. The stores differ in shopping basket size. The larger store (e.g., a supermarket) provides many other items whose sales are sensitive to store traffic, but the smaller store (e.g., a butcher) only carries a few products. Each store serves low- and high-end consumer segments whose WTP is $8 and $12 per unit, respectively. A consumer needs at most two units in each period. Besides the focal meat product, consumers spend $40 on other items per visit at the larger store but only $2 at the smaller store. For simplicity, we assume that the marginal costs and segment sizes are 0 and 1, respectively.

Our discrete demand model has four key features: (1) Two segments allow heterogeneity in the simplest, most transparent way to examine price changes. (2) Having consumers use two units in each period provides sufficient complexity to explore how one-unit limits could change consumer behavior. (3) Fixed WTP prevents price gouging, consistent with legal limits on price increases during emergencies in many states. However, some stores might price gouge illegally if some consumers were willing to pay more during shortages. (4) In our competitive model, price sensitive low-end consumers, who may have lower transportation costs or smaller opportunity costs for time, switch stores only if the focal store cannot serve them because of stockouts or high prices.

Without shortages, an $8 price serves both segments making focal product profits of $8 × 4 = $32 per period, whereas a $12 price only serves the high-end segment making focal product profits of $12 × 2 = $24 per period. Moreover, the $8 price generates greater store traffic and sales from other products. Thus, both stores adopt the $8 price. At this price, consumer surplus is $12 − $8 = $4 for every high-end consumer purchase, amounting to $4 × 2 × 2 × 2 = $32 in total because two high-end consumers (i.e., one from the larger store and one from the smaller store) buy two units over two periods. Note that without shortages, neither store gains from purchase limits that simply decrease sales without corresponding benefit.

Suppose that a moderate shortage occurs in which each store only has four units (period 1 demand) for two periods. At $12, the high-end consumer buys two units in each period. At $8, the low- and high-end consumers buy two units each in period 1, which leaves no inventory for period 2. Scarce product profits are $12 × 4 = $48 at $12 and $8 × 4 = $32 at $8. The profit difference is $48 − $32 = $16. The store traffic across periods is the same with either price. Thus, without limits, both stores price at $12.

With a one-unit-limit strategy, at the $12 price, the stores sell fewer units with no benefit; thus, the limit strategy is dominated. With a one-unit-limit strategy, at the $8 price, the stores can sell one unit to both low- and high-end consumers in each period. Therefore, purchase limits, like high prices, save inventory for period 2 and increase period 2 store traffic by two consumers. The resulting benefit is $40 × 2 = $80 for the larger store and $2 × 2 = $4 for the smaller store. The $80 gain exceeds the $16 loss from lower margins but the $4 does not. Thus, only the larger store uses a low-price-limit strategy. Purchase limits lower the opportunity cost of selling at a lower price ($8). Moreover, consumer surplus increases with limits by ($12 − $8) × 2 = $8 because the high-end consumer pays only $8 each period (i.e., saves 10% of the shopping basket for other items) at the larger store. The consumer surplus is lost in the counterfactual scenario in which purchase limits are prohibited.

For moderate shortages, we see that limits can increase the larger store's profits because limits enable the larger store to maximize traffic with a low price ($8). The smaller store, which gains less from additional store traffic, prefers a high price ($12) at which purchase limits only decrease sales and profits. These results indicate that purchase limits can act as substitutes for price increases and mitigate the negative effects of shortages on consumer surplus given a high price alternative.

In competitive markets, store strategies do not change. If the smaller store lowers the price to $8, it can increase traffic by two low-end consumers in each period, at best, and obtain an additional $2 × 2 × 2 = $8 from other items. However, the gain does not offset the loss of ($12 − $8) × 4 = $16 from the lower margin. Thus, the smaller store sets the same high price ($12) without limits. Given the smaller store's high price, the larger store's low-price-limit strategy attracts additional store traffic from the smaller store's low-end consumer, so the larger store has no incentive to raise prices. In the moderate shortage, the increased traffic would not entirely deplete the larger store's inventory, given purchase limits, which allows the larger store to enjoy additional period 2 traffic. Interestingly, competition decreases consumer surplus because in period 2, the larger store's high-end consumer has a smaller probability (i.e., 1/3) of obtaining the scarce product given limited supply and high traffic.

Suppose that a severe shortage occurs in which each store has two units (less than period 1 demand) for two periods. Thus, scarce product profits are $12 × 2 = $24 at the $12 price and $8 × 2 = $16 at the $8 price with a difference of $24 − $16 = $8. At $12, purchase limits move sales to period 2 because the high-end consumer buys one unit in each period instead of both units in period 1. The increase in traffic makes limits profitable. At $8, purchase limits produce no benefit because no inventory is left for period 2, even with limits (note that at $8, all units are sold, so purchase limits are unnecessary and avoidable administrative costs), and the total traffic remains at two consumers. Thus, both stores price at $12 and impose limits (i.e., purchase limits are complements to price increases that add profits from increased store traffic). In the counterfactual scenario in which purchase limits are prohibited, the larger store decreases the price to $8 because it cannot obtain the additional period 2 traffic profits if the price is $12. However, the smaller store has small traffic profits and sacrifices traffic to enjoy the high margin at $12. Therefore, purchase limits increase both stores’ profits because limits increase period 2 traffic at the high price, whereas purchase limits reduce consumer surplus as a result of the larger store's price increase.

With competition, the smaller store's strategy remains unaffected. Although lowering the price to $8 can entice the larger store's low-end consumer to switch to the smaller store, it only increases traffic profits by $2 at most, which cannot cover the $8 loss from the lower scarce product margin. However, given the smaller store's $12 price, the larger store decreases the price to $8 because the lower price increases traffic profits by $40, which more than offset the $8 loss. Facing certain stockouts in period 2, the $8 price leverages the minimal inventory of the larger store to attract as many consumers as possible in period 1. Thus, competition can increase the larger store's profits and consumer surplus.

It is interesting to note that availability increases store traffic, but availability is not guaranteed. Consumers may randomly find the scarce product sold out. For example, if two consumers visit a store that has only one unit, the probability of buying a unit is 1/2. The risk of buying none might reduce traffic and cause the store to increase the price, which would, in turn, decrease consumer surplus.

Discussion and Conclusions

Shortages result from unexpected events or shocks—pandemics, piracy, civil unrest, hurricanes, droughts, fires, and strikes—that suddenly disrupt supply or spike demand. As supply chains grow more complex, and many producers adopt just-in-time inventory policies, shortages become more common. One possible response is the adoption of a purchase limit strategy that restricts per transaction quantities. Despite their popularity, there is little extant research on purchase limits. We develop a shortage model that considers purchase limits in scenarios with varying observable factors: store size, shortage severity, competitive intensity, and seasonality. We show that stores can use purchase limits strategically to increase their profits during shortages.

When shortages occur, stores decide whether to impose purchase limits. Store traffic can decrease if a store does not have inventory of a scarce product because some consumers shop elsewhere. Having inventory of a scarce product can increase traffic and revenue from selling other items. Purchase limits can bring back store traffic that was temporarily lost from shortages, particularly for stores facing competitors. This function of purchase limits has gone completely unnoticed in the popular press, which only considers altruistic motives (e.g., reducing hoarding, waste, panic buying, arbitrage, unfair distribution), making our findings entirely new.

Although purchase limits are somewhat similar to traffic enhancing store promotions, limits have different mechanisms: (1) Limits can occur with both low and high prices (i.e., increase traffic without sacrificing margins). (2) Limits can be profitable even when they do not increase current store traffic because limits can shift demand to future periods. (3) In severe shortages, limits can be more beneficial for small stores that benefit less from traffic than large stores. (4) Store traffic does not necessarily come from competitors’ sales, but it can come from getting the same consumer to revisit the store.

Stores can also adjust their prices during shortages. Higher prices increase profit margins and can save inventory for future sales but decrease current store traffic and sales of nonscarce items. By contrast, lower prices increase current store traffic, but decrease profit margins, and can create stockouts of scarce products.

The shortage severity affects the optimal price level for imposing purchase limits. In moderate shortages, stores have enough supply to cover demand at high prices, so limits only decrease sales. At low prices, stockouts can occur without limits. Purchase limits can save inventory for future sales. Thus, large multiproduct stores, where the average shopping basket size is large, should maintain low prices and impose limits to maximize traffic. However, small stores should increase prices and not impose limits because, given fewer sales from other items, additional store traffic fails to offset lost margins on the scarce product.

In severe shortages, stockouts can occur at any price levels. Purchase limits can prevent stockouts but only at high prices. Thus, small stores should increase prices and impose limits to increase future store traffic. Large stores should adopt the same strategy if there is no competition. However, in competitive markets, large stores reduce prices to poach small stores’ consumers and obtain larger profits from the additional traffic. Low prices exhaust all available inventory with or without limits. Thus, purchase limits are only unnecessary administrative costs that provide no benefit. Briefly, large stores may cleverly leverage their available limited inventory to increase store traffic before facing stockouts. Note that consumers may have a small chance of obtaining the scarce product, which may be more than sufficient to cover travel costs.

In general, several key exogenous observable factors affect store strategies. First, sufficiently different store sizes can cause different pricing strategies. Large multiproduct stores may decrease prices because they value store traffic that brings profits from other items. However, small stores may increase prices because they carry few products and value high profit margins. Second, the shortage severity has different effects on the price-and-limit strategy depending on the store size. Without competition, for moderate shortages, large stores set low prices and impose limits, whereas small stores charge high prices but do not impose limits. By contrast, large and small stores charge high prices and impose limits in severe shortages. Third, competition can lower prices, which ironically cause different limit strategies in moderate and severe shortages. Low prices cause stores to impose limits in moderate shortages but not in severe shortages. Limits are not beneficial when supply is so low that inventory cannot be saved for future sales. Finally, purchase limits can increase both profits and consumer surplus for moderate shortages because purchase limits can make only low prices more profitable. For severe shortages, purchase limits can also increase consumer surplus when the shortages occur in off-peak period 1 followed by peak period 2 because purchase limits can save inventory for peak period 2, which causes stores to lower prices in that period to increase traffic.

Implications

Our findings provide both managerial and public policy implications. First, during a shortage, purchase limits can be more profitable than higher prices or first-come-first-served rationing, even though the latter has no enforcement costs, requires minimal administrative costs as sales occur via existing procedures, and reduces inventory-carrying costs by accelerating sales. Limits increase future traffic and sales from other items, which can more than compensate for these costs. Without a shortage, the profit advantage of purchase limits can disappear for normal store-level supply and demand fluctuations because limits only drive consumers to competitors that have ample supplies.

Second, purchase limits can improve consumer welfare. Although some may believe that the improvement occurs through equitable allocations, the allocations based on purchase limits do not necessarily improve consumer surplus because those limits might take supply from those who have the greatest valuation for the scarce product. Moreover, most consumers may not even know how scarce products are allocated at the store level because they only observe stockouts. We find, instead, that purchase limits can improve consumer surplus because purchase limits act as substitutes for price increases and cause stores to lower prices. Recall that both purchase limits and higher prices can save inventory to enhance future store traffic. This implies that purchase limits can cause profit-seeking stores to unwittingly achieve egalitarian goals and improve equity, inadvertently achieving societal benefits during shortages. Therefore, during shortages, governments can possibly consider encouraging purchase limits (e.g., providing information on availability) because purchase limits provide incentives for lower prices even without price gouging laws.

Third, stores may benefit from increased transaction costs in a shortage, although conventional wisdom is that stores should reduce transaction costs. Shortages can increase transaction costs for many reasons, such as congestion and long lines (note that single-product shoppers may only shop at high priced online gray markets or small convenience stores during shortages). The increased transaction costs may lead consumers to purchase multiple products rather than a single product, which increases consumer spending on nonscarce items, especially in large stores. Thus, the increased transaction costs make purchase limits more important as a marketing tool to maintain store traffic throughout shortage periods.

Fourth, competition still matters during shortages. Given limited supply of the scarce product, poaching competitors’ consumers to sell the scarce product may not be profitable. However, stores can increase traffic profits from nonscarce items by poaching competitors’ consumers. Stores can leverage limited supplies by providing only the possibility of obtaining the scarce product. Moreover, we find that competitive limit strategies depend on shortage severity. During moderate shortages, purchase limits can allow low prices (poaching) by preventing stockouts, whereas purchase limits are unnecessary administrative costs at low prices during severe shortages because limits cannot prevent stockouts.

Fifth, store size influences shortage strategies. For moderate shortages, large stores set low prices to attract competitors’ consumers while imposing purchase limits to save inventory to increase future traffic and profits. For severe shortages, large stores do not impose limits because limits do not prevent stockouts at low prices. Small stores do the opposite (Conerly 2022). They set high prices without limits for moderate shortages and high prices with limits for severe shortages.

Finally, our competitive shortage model with discrete demand (loyals and switchers) produces pure strategy equilibria that provide clear, simple implications on stores’ price-and-limit policies to manage exogenous shortage shocks. In price promotion research, zero-sum competition often causes a price war that produces mixed-strategy equilibria with random price promotions. However, in our shortage model, a price war does not occur given increased monopoly power (note that the monopoly power varies with the shortage severity). Such competition with limited inventory is a unique feature of shortages.

Important Features of Shortages

Shortages involve many important features including the following:

Shortages can decrease competition given less inventory to compete with.

Shortages affect the entire supply chain and all stores, unlike other ordinary stockouts.

Shortages can disrupt habitual shopping behavior.

Stores deviate from the usual equilibrium and develop temporary shortage strategies.

Shortage strategies resulting from unexpected events differ from planned scarcity strategies.

Having inventory of scarce products can attract new consumers.

Selling one unit of a scarce product to many consumers makes the same revenue as selling many units to one arbitrager, although the latter requires lower transaction costs.

Both supply- and demand-side factors can influence shortage strategies.

Legal constraints may restrict storage strategies (e.g., price gouging).

Observing empty shelves is insufficient to infer whether allocation is equitable.

Future Research

Future research could explore more complex limiting mechanisms, such as need-based limits and limits linked to available inventory. Properly done, purchase limit strategies that emphasize need may increase customer satisfaction and future purchases (e.g., Seiders et al. 2005). Perhaps, limits may also influence scarce product usage (e.g., Nevskaya and Albuquerque 2019).

It is also worth exploring whether institutional arrangements can more effectively plan for shortages than single sellers (Carson et al. 1999). Supply chain managers may foresee possible shortages and develop appropriate institutional ties to manage them, like just-in-time inventory policies that shift inventory to the lowest cost channel member (Rubel, Naik, and Srinivasan 2011).

There are other factors worthy of future research. First, we focus on economic issues and leave analyses of equality, goodwill, and fairness (Guo 2015) to future research. Second, future research could consider different types of scarce products (e.g., perishable goods that are difficult to inventory). Third, purchase limit strategies with price discrimination based on customer behavior before a shortage could be investigated (Shaffer and Zhang 2000). Fourth, future research could allow loyals and switchers to have different preferences for large and small stores such that those stores have different segment sizes. Fifth, we analyze situations when stockouts diminish store traffic, but stockouts may increase consumer needs for substitutes of the scarce product and reduce cannibalization (Moorthy 1988), which may help management of substitute products. Sixth, future research could investigate strategic bundling (Stremersch and Tellis 2002) of scarce products with abundant products. Seventh, beyond scarce products, the effect of shortages on store strategies for other items could be considered. Stores may be able to raise the prices of nonscarce products given increased transaction costs. Finally, we only use search costs to demarcate competitive and noncompetitive markets, but future research could develop more complex search models for shortages.

Supplemental Material

sj-pdf-1-mrj-10.1177_00222437221101808 - Supplemental material for The Profitability of Purchase Limits During Shortages

Supplemental material, sj-pdf-1-mrj-10.1177_00222437221101808 for The Profitability of Purchase Limits During Shortages by Jihwan Moon and Steven M. Shugan in Journal of Marketing Research