The Impact of Inventory Risk on Market Prices 1

Transcription

1 The Impact of Inventory Risk on Market Prices 1 Abstract This paper demonstrates and explains a surprising effect: that an increase in the number of competing firms may lead to an increase in the equilibrium market price in a situation when firms compete on both price and quantity (inventory/capacity). We attribute this effect to inventory risk: the potential mismatch between the firm s inventory and demand that occurs if the firm incorrectly judges how much additional demand it will generate if it lowers the price. As we show, such a risk increases in the number of competitors, making firms inventory management a lot more difficult and costly, hence triggering an increase in equilibrium prices. We use the concept of quantal response equilibrium to model competition between boundedly rational firms, illustrate the price increase, and then examine how it changes depending on the number of competitors, the degree of their rationality, and the composition of market demand and firms costs. 1 Corresponding author: Anton Ovchinnikov. anton.ovchinnikov@queensu.ca. Tel:

2 1 Introduction Standard economic theory suggests that an increase in the number of competitors in the market should lead to a decrease in market prices and an increase in order quantity (inventory/capacity 2 ). But the standard theory disregards two important operational elements of how firms compete. The first element is inventory: indeed, to be able to meet demand, firms often must procure inventory well in advance of knowing what the actual demand will be, and a mismatch between inventory and demand can be costly. The second element is rationality of competitors: the standard theory assumes that competing firms are fully rational, while in practice perfect rationality seldom holds (Chen et al. 2012). In this paper we show that, when accounting for these two operational elements, the logic of the standard theory reverts: that is, an increase in the number of competitors in the market may lead to an increase in market prices and a decrease in total quantity. This paper was motivated by a seemingly surprising result of the Supply Chain Game (Raz 2009) a classroom simulation in which multiple Beer-Game-like (Sterman 1989) supply chains compete in a market where retailers, in addition to quantity, also decide the retail price. We played this game multiple times with MBAs/Executives with three to eight competing supply chains and observed that the average price set by the retailers increased in the number of competitors. For example, combining the data for , the price with four competitors is 74% higher on average than that with three (p-value 0.035, unequal variances single-sided t-test). This result initially made no sense to us, as it directly contradicts the common logic and predictions of the standard theory, but its persistent recurrence with different students motivated us to ask whether such a price increase could indeed be an equilibrium behavior. Our model has two main components: the model of price and quantity competition, and the model of bounded rationality. For the model of competition we consider a market in which firms compete on price and quantity in the sense of the newsvendor model with pricing. The overall market demand is uncertain and consists of price-sensitive consumers who can be of two types: loyal consumers are spread equally across all firms and decide to purchase based on the price at their firm of choice, while bargain-hunting consumers visit the firm with the lowest price and decide to purchase based on that lowest price. Following the newsvendor logic, the firms determine their prices and quantities before knowing other firms decisions and demand realization, and doing so involves two types of costs: inventory cost and shortage cost. We capture this overall setup with a fairly stylized demand/profit model, but we experimented with several plausible generalizations. 2 In this paper we consider a single ordering decision for each firm, which makes the order quantity, capacity, and inventory identical. 1

3 In particular, we analyzed a logit demand model which relaxes the assumption that all bargainhunters visit the lowest price firm. We also analyzed two alternative profit models for the situation when demand exceeds supply: one when unsatisfied demand is lost, and another when unsatisfied demand is backlogged and must be satisfied later at a higher cost (expedited shipment, overtime) or at a discount. All these model variants lead to qualitatively similar results. For the model of bounded rationality we utilize the concept of quantal response equilibrium (QRE) as introduced by McKelvey and Palfrey (1995). The fundamental principle underlying QRE is that a player may select any admissible strategy, but the strategies that lead to higher utility (profits in our case) are selected with higher probability. Specifically, we use the logit QRE, which is a parsimonious way of capturing the degree of players rationality with a single rationality parameter: when this parameter is small, the distribution of players strategies is more tightly centered around the strategy with the highest profit (implying a near-perfect rationality), but when the parameter is large, the distribution is spread out (implying significant irrationality). This has two implications: players are not always able to maximize their own profits, but also they are not sure what their opponents will do; both are clearly resemblant of how people/managers actually behave in general, and in inventory competition situations in particular (see Ovchinnikov et al. 2014). Using this combined model of price and quantity competition with boundedly rational firms, we show that an increase in the number of competitors may lead to an increase in market prices. We attribute this effect to inventory risk: the potential mismatch between the firm s inventory and demand that occurs if the firm incorrectly judges how much additional demand it will generate if it lowers the price (in our setup this implies judging whether the firm s price will be lowest or not). Intuitively, each firm needs to choose between two kinds of strategies. One is to charge a high price, hoping that it will not be lowest. In this case the firm s demand will be small, as it will consist only of its own loyal customers, which implies that the firm needs to procure little inventory. An alternative is to charge a low price, hoping that it will be lowest. In that case the firm s demand will be large, as it will consist of not only its own loyal customers but also of all the bargain-hunters, which, in turn, implies that the firm should procure a lot of inventory. The inventory risk comes into play when the firm is misjudging whether it will indeed be the lowest price firm. With the former strategy, if the firm s high price will in fact turn out to be lowest (which can happen because of the firm s own error in optimization, or bounded rationality of other players), then the firm will incur a significant shortage cost. With the latter strategy, if the firm s low price will be undercut by some other firm (which, again, can happen because of its own bounded rationality or that of 2

4 another firm), then the firm will liquidate the additional inventory it procured in anticipation of the demand from bargain-hunters, which is also costly. The magnitude of such a mismatch between the firm s inventory and demand, and hence the magnitude of the associated shortage/ordering costs increases in the number of competitors, increasing inventory risk. To compensate for this risk, the firm increases its price, and in equilibrium so do all other firms as well. The connection between the bargain-hunting behavior of consumers, firm s pricing and the resultant inventory risk exists in many practical situations. Consider Groupons: online discount coupons which firms distribute via the website of the same name, Hall (2011) describes a baker, Mrs. Brown, who vastly under-estimated the popularity of the deal and was besieged by 8,500 people who signed up for the bargain. To address this abnormal surge in demand Brown incurred additional costs as she had to bring in agency staff on top of [her] usual staff, noting also that [m]y poor staff were having to slog away at all hours - one of them even came in at 3am. This example illustrates the connection between the firm s pricing and inventory risk. Further, the larger is the number of competitors (and hence the smaller is the firm s loyal demand in relation to the bargain-hunting demand), the bigger is the risk, which, in equilibrium (depending on various model parameters) could lead to higher prices. Performing sensitivity analysis of the price-increase effect to the model parameters, we find that it is strongest when the degree of rationality is medium, the fraction of bargain-hunters is high, the shortage cost is high, and the difference in price sensitivity between the loyal and bargain-hunting consumers is low. In the reverse situation, an increase in the number of competitors may have no effect on market prices. Interestingly, we find that market prices are rather insensitive to the uncertainty in the total market demand. Managerially, our results have important implications for both competitive strategy and regulation. The regulators must be careful when promoting measures to increase competition in the industries with low consumer loyalty and low supply chain responsiveness: such measures may have negative welfare implications as they could hurt both firms (resulting in lower, possibly negative, profits) and consumers (resulting in higher prices and lower quantities). The firms must be careful in thinking of which markets to enter: being a low-cost firm and being able to charge a low price is not enough; to succeed in markets with price and inventory competition firms must also have agile, scaleable supply chains in order to be able to effectively deal with the inventory risk. 3

5 2 Literature Review Newsvendor quantity competition is a well-researched area. Starting with the fundamental works of Parlar (1988) and Lippman and McCardle (1997), the operations management literature contains primarily theoretical work, e.g., Netessine and Rudi (2003). Behavioral works are scarce and recent: Becker-Peth et al. (2013) considered contract design in the leader-follower framework, Ovchinnikov et al. (2014) considered the standard quantity competition. Newsvendor models with pricing is also a well-researched area, starting with the fundamental work of Petruzzi and Dada (1999). But the combination of newsvendor model with pricing and competition has been proven to be a notoriously difficult problem. To our knowledge Zhao and Atkins (2008) is the only paper that directly addresses this problem; more so, we are not aware of any behavioral work that considers price-setting newsvendor competition. These papers provide useful frameworks for building our model. Lippman and McCardle (1997) emphasized the importance of demand allocation rules, while Zhao and Atkins (2008) emphasized consumer search behavior. Our model incorporates both these elements: loyal customers are split equally across firms and do not search (as their name suggests, they are loyal to their firm of choice), while bargain-hunters (as, again, their name suggests) all search for the firm with the lowest price. More so, we consider both the case with lost sales (as in Zhao and Atkins 2008) and the case with backlog, when demand must be satisfied at a higher cost (as in Mrs Brown s Groupon misadventure, when she had to pay for overtime and bring temps). Behavioral papers (Bolton and Katok 2008, Becker-Peth et al. 2013, Bolton et al. 2012, Lau et al. 2012, and Ovchinnikov et al. 2014) point out to the predictably irrational individual order variability, which in a competitive setting implies that a player is able to predict only the distribution of the opponents actions. This observation is a critical motivator for our model with boundedly rational newsvendor firms. Models of bounded rationality have been used by a number of authors for the newsvendor decisions (Su 2008, Wu and Chen 2014), as well as for other operational decisions (Chen et al. 2012); our treatment of bounded rationality is quite similar to theirs. In terms of our results vis-à-vis those of other authors, we are not aware of any work that studied how the equilibrium outcome of price and quantity competition changes in the number of competitors, hence none of these papers are directly comparable to ours. Zhao and Atkins (2008) provide some insight into a possibility of seeing a result similar to ours, noting that competing vendors may set higher prices when inventory competition dominates (Figure 1 in their paper), and arguing that this result is quite robust. It is critical to note, though, that the baseline in their paper is a single isolated newsvendor (a monopolist). Another important distinction is that they 4

6 found that the equilibrium never has lower total quantity and higher prices (a situation that hurts both firms and consumers and has overall negative welfare implications). In contrast, we show that the total quantity may decrease when the number of competitors increases, which, together with a price increase, implies that consumers welfare declines. More so, we also show that the firm s profit may become negative, which suggests that the overall welfare declines as well; equivalently, this suggests that some firms might leave the market, restoring a positive-profits equilibrium, lowering prices and increasing quantities, but with fewer competitors. Our price-increase result compliments similar results in Economics and Industrial Organization. Anderson et al. (1995) and Chen and Riordan (2008) showed that a price-increase can be driven by product differentiation. Rosenthal (1980) considered a setting in which (as in our paper) there is a mixture of loyal and bargain-hunting consumers and showed that a price-increase can be driven by the inability of sellers to impose different prices to these two classes of consumers. A fundamental difference between our paper and his is that the total size of the market in his paper increases with the number of competitors, while we assume, in line with other newsvendor competition papers, that the overall market demand is constant and is (re)allocated between the competitors. In this sense the insights of our papers are also technically similar, but conceptually very different: e.g., in our context the price-increase result implies that prices may go up if the number of retailers in a city increases, while his price-increase result implies that prices in a large city (with more demand and more retailers) will be higher than in a small city. Another important difference is that neither of these papers consider inventories; thus our work offers an additional explanation for a possible price-increase, one that is particularly salient in operational settings: inventory risk. 3 Model As mentioned in the introduction, our model has two components: the profit model that describes the payoff (profit) of each competing firm given its own strategy and those of other firms, and the bounded rationality model that describes how the firms choose their strategies in equilibrium. 3.1 Profit Model Consider n firms that compete on price and quantity in the sense of a newsvendor model with pricing; that is, they commit to the selling price and the order quantity (inventory/capacity) before they learn about the uncertain demand and actions of other firms. We assume that market demand consists of a mixture of two types of consumers: loyal consumers (fraction λ) and bargain hunters (fraction 1 λ). Loyal consumers are spread equally across all 5

7 firms and decide to purchase based only on the price at their firm of choice regardless of the prices set by other firms. In contrast, bargain-hunters search for the firm with the lowest price, all 3 visit such a firm, and decide to purchase based on that firm s price. Bargain-hunters, as their name suggests, are more price-sensitive than loyals. We assume that behaviors of both consumer types are governed by linear demand functions with multiplicative 4 uncertainty. That is, if firm i charges price p i then it generates demand λ n (a p i) + ϵ from its loyal customers, where a is a nominal market size, ϵ = Uniform[1 B, 1 + B] is a random demand parameter, and ( ) + max[, 0]. Additionally, if p i happens to be the lowest price, then firm i also generates demand (1 λ)(a δp i ) + ϵ from bargain-hunters, where δ 1 captures how much more price-sensitive bargain-hunters are compared to loyals. Note that each firm faces two sources of demand uncertainty: market uncertainty (How big is the total market demand?) and competition-induced uncertainty (Will it get the bargain-hunting demand?) Thus, the total demand of firm i, given its own price p i and other firms price vector p i equals: λ n (a p i) + ϵ, if p i > min[p i ]; D i (p i, p i, ϵ) = λ n (a p i) + ϵ + (1 λ) (a δp i ) + ϵ, if p i < min[p i ]; λ n (a p i) + ϵ + 1 λ K (a δp i) + ϵ, if K s.t. i, j K p i = p j < min o K [p o ]. (1) where the latter term reflects a possible tie: in this case the bargain-hunting demand is split equally among the K firms with the identical lowest price. Remark 1 This demand function is a special case of a more general logit demand model in which a bargain-hunter s utility from purchasing at price p i is u i = a δp i + η, and the bargain-hunting demand of firm i is (1 λ)(a δp i ) + ϵ exp{u i /α} j {1,...,n} exp{u j/α}, where α is a parameter that measures the variance of the random noise η. When α 0 such a logit model converges to the demand model in (1), and when α it also converges to the model in (1) but with λ = 1. All our qualitative results hold for such a more general logit demand model, but the intuition behind the inventory risk phenomenon is easier to illustrate on the model in (1), hence we use it throughout. Each firm incurs an identical purchasing cost c per unit of inventory and shortage cost s per unit of unsatisfied demand. Regarding the latter, as mentioned in the introduction, the operations management literature distinguishes between two fundamentally different scenarios: 3 The assumption that all bargain hunters visit the same firm is not critical for our results, see Remark 1 4 Our results are insensitive to this specific assumption; they hold for additive demand uncertainty model, and, in fact, even when demand from each customer type is deterministic, i.e., ϵ 1. 6

8 The lost sales scenario, in which the firm simply incurs a penalty s for each unit of unsatisfied demand, resulting in the profit of: Π i (p i, q i, p i, q i ) = E ϵ { pi min[d i (p i, p i, ϵ), q i ] c q i s [D i (p i, p i, ϵ) q i ] +} (2) where q i the order quantity of firm i and q i is the vector of other firms quantities. Note that this formulation implicitly assumes that consumers who experienced a stockpot do not search for the inventory at other firms, an extreme case of Zhao and Atkins (2008) model. But given the market structure with two types we believe that such an assumption is reasonable: loyal customers do not search because, as their name suggests, they consider purchasing from only their firm of choice, and bargain-hunters do not search because other firm s prices are not a bargain. The backlog scenario, in which the firm does not lose the sale, but in order to satisfy the demand that exceeds its order quantity q i it incurs additional cost s (expedited shipment, overtime workers, etc.), resulting in the profit of: { Π i (p i, q i, p i, q i ) = E ϵ pi min[d i (p i, p i, ϵ), q i ] c q i + (p c s) [D i (p i, p i, ϵ) q i ] +} (3) Alternatively, the firm may offer a discount s% to compensate for consumers waiting, in which case the last term in the profit equation should be changed to (p(1 s) c) [D i (p i, p i, ϵ) q i ] +. Interestingly, the qualitative results of this paper are not impacted by these alternative formulations: prices increase and quantities decrease in the number of competitors. The backlog model is somewhat less intuitive to interpret (because q i is only the initial quantity, which the firm may increase at an additional cost); hence for the remainder of the paper we use the lost sales model. 3.2 Model of Bounded Rationality: QRE Our model of bounded rationality relies on the observation that even when making standard newsvendor ordering decisions (in isolation or in competition) repetitively in an unchanged circumstances, human decision makers routinely changed their decisions, while the rational action was to order the same (optimal) quantity every time; e.g., see Bolton and Katok (2008), Bolton et al. (2012) for the isolated newsvendors and Ovchinnikov et al. (2014) for the competitive case. That is, decision makers are boundedly rational, and our model captures that through the concept of quantal response equilibrium (QRE) introduced by McKelvey and Palfrey (1995). In particular, we use the logit QRE model, in which the distribution A i according to which player i chooses action 7

9 pair (p i, q i ) from the set of all admissible actions S i given the opponents distributions A i is given by the following probability function: where β is the rationality parameter. exp{e A i Π i (p i, q i, p i, q i )/β} P i (p i, q i A i, A i ) = ˆp i,ˆq i A i exp{e A i Π i (ˆp i, ˆq i, p i, q i )/β} Our formulation is consistent with the idea of attraction models, e.g., Camerer and Ho (1999), in the sense that the actions that lead to higher payoffs will be chosen with larger probabilities. It is easy to see that when β is large, the distribution will be more flat : in the limit when β, A i will converge to a uniform distribution over all possible actions/strategies/pairs (p i, q i ) that is, players exhibit little rationality making random decisions regardless of the resultant profits. But when β is small, in the limit when β 0, the QRE distribution will converge to the profitmaximizing choice made with probability 1 a case with full rationality. See Chen et al. (2012) for detailed discussion of the logic behind QRE and its use in the OM models of bounded rationality, as well as Wu and Chen (2014) for an application to newsvendor ordering decisions. (4) 4 Model Analysis In this section we use the model presented above to first illustrate our main result, and then study its sensitivity to some of the key model parameters. We perform a series of numerical simulations in which we solve for the QRE varying the number of competitors, n = 2, 3, 4. For the base case we assume the following parameter values: rationality parameter, β = 100, market size, a = 100, demand uncertainty parameter B = 0.05, fraction of loyal customers, λ = 0.5, bargain-hunters price sensitivity, δ = 1.5, purchase cost, c = 20, shortage cost, s = 10. To implement QRE we discretized the firm s prices and quantities s.t., p i, q i {0, 10,..., 90, 100} and as as in many QRE implementations (Chen et al. (2012), Wu and Chen (2014)) we look only for symmetric equilibria. We performed equilibrium calculations in Mathematica; the source code is available upon request. We note that when prices are continuous it is easy to show that a symmetric pure strategy equilibrium does not exist 5. But with discrete prices this argument fails: when a minimum price change is limited by the discritization step (10 in our case, and likewise in practice there often 5 In any (symmetric) candidate equilibrium with positive profit by slightly decreasing its price a firm looses only a small profit from its loyal consumers but captures all bargain-hunting demand, and with a zero profit, increasing its price a firm will lose all bargain-hunting demand but will make a positive profit from its loyal consumers; see Proposition 1 in Rosenthal (1980) 8

10 (a) (b) Figure 1: In (a): average equilibrium price and quantity, in (b): expected total equilibrium quantity (left axis) and profit (right axis) over all firms, for n = 2, 3, 4. exists a small set of reasonable prices) such a deviation can be not profitable and a pure strategy equilibrium may well exist. That being said, in a noticeable number of cases when the rationality parameter was very small (i.e., when QER equilibrium converges to a pure strategy equilibrium when one exists), our QRE algorithm did not converge, suggesting that a pure strategy equilibrium did not exist. Thus we report only cases with β 1, for which the computation always converged to an equilibrium. To check if the obtained equilibrium is unique, we implemented each computation twice: first with a starting vector of uniformly distributed prices and quantities, and second with a random vector. In all cases the computation converged to the same distribution of strategies, which, although strictly speaking, does not guarantee that the equilibrium is unique, strongly suggests that it might. With this approach we proceed to our main result. 4.1 Main result: price increases as the number of competitors grows Proposition 1 An increase in the number of competitors can lead to an increase the average equilibrium price and a decrease in the average equilibrium quantity. Figure 1 (a) presents the firm s average equilibrium price and quantity. As the Figure shows, the price increases and the quantity decreases as the number of competitors grows from 2 to 4. Figure 1 (b) illustrates two additional points: not just the average quantity for each firm decreases as it naturally might since the same market demand is allocated across more firms but the total quantity over all firms decreases as well. The average profit (per firm) and the total profit (over all firms) are also decreasing. This result is different from that of Zhao and Atkins (2008) who observed that in their model while price increase was possible (vis-a-vis a monopoly; they 9

11 (n=2) (n=4) Figure 2: Illustration for the inventory risk: the degree of disparity between the firm s demand if it is not lowest price (lower error bars) and the firm s demand if its price is lowest (upper error bars), as a function of p i. did not consider an increase in the number of competitors) the total quantity increased as well. Additionally, notice that with n = 4 the expected profit over all firms (and hence the expected profit for each firm as well) is negative. Remark 2 The negative average profit result implies that the equilibrium with four competitors is not sustainable. In a practical situation this means that one of the firms might exit the market, restoring a positive-profit equilibrium with three firms. So what causes such a counterintuitive equilibrium? We attribute it to the inventory risk. Consider Figure 2, which for the case with n = 2 (left) and n = 4 (right) presents four quantities as a function of the firm s price, p i : the expected equilibrium demand (solid line), the firm s demand if its price is not lowest (firm s loyal demand, lower error bars), the firm s total demand if its price is lowest (firm s own loyals plus all bargain-hunters, upper error bars), and the probability of the firm s price being lowest (dotted line, right axis). Two observations are evident. First, the degree of disparity between the firm s demand if it happens to be the lowest price firm and the firm s own demand from loyal customers is very large. For example, for n = 2 the lowest-price demand is up to three times larger than the loyal demand, and for n = 4 it is up to five times larger 6. That is, an increase in the number of 6 At p i = 0, with n = 2 the firm s expected loyal demand is 25 (1/n = 50% of overall demand, of which 50% are loyal). But if p i is lowest then the firm s total demand is 75 = 25 (loyal demand) + 50 (all bargain hunting customers, (1 λ)a), which is 3 loyal demand. Similarly for n = 4 lowest-price demand at p i = 0 is 62.5 and the firm loyal demand is 12.5, i.e., a 5 difference. 10

12 (n = 2) (n = 3) (n = 4) Figure 3: QRE distributions for n = 2, 3, 4; in each 3D plot the left axis is for q i and the right is for p i. competitors increases (in relative terms) the additional demand that the firm captures if it is the lowest-price firm. Second, the probability of getting this additional demand decreases in the number of competitors compare the dashed lines in Figure 2. The combination of these two factors, an increase in the relative additional demand with a simultaneous decrease in the probability of seeing that additional demand, makes the firm s inventory management a lot more difficult. For example, consider a low price strategy for the firm, e.g., charging p i = 30 (note that with c = 20, this is the lowest reasonable price). With n = 2 competitors, the firm s expected demand is close to maximal and the probability of seeing that demand is large, hence the firm chooses to order a lot of inventory; the logic is reversed in the high-price strategy. The medium price strategy is hardly profitable because regardless of how much inventory the firm will order it will incur a massive cost of either liquidation or shortage: there is no inventory policy that will easily satisfy demand with so much variability. But when n = 4 the low price strategy becomes unprofitable: from Figure 2, the price needs to be as low as p i = 10 for the expected demand to be close to maximal, but the firm will then lose money since c = 20. At p i = 30 the situation is similar to the medium price with n = 2. Hence the firms find the high price strategy more appealing, which reduces the inventory risks, but leads to an increase in prices and a decrease in quantities the effect we describe. Observe how these low and high price strategies reveal themselves in the 3D plots in Figure 3: for n = 2 the peak is at the low prices, while for n = 4 the peak is at the high prices. 4.2 Sensitivity Analysis Next we discuss the sensitivity of the price-increase effect to various model parameters: rationality of the players, β, shortage cost, s, fraction of loyal customers, λ, demand uncertainty, B, and bargain-hunters price-sensitivity δ. In each case we varied the respective parameter while keeping other parameters as their baseline values as per Section 4.1. Figure 4 presents the results, which 11

13 are summarized in the set of observations below. Observation 1 The price-increase effect is most pronounced when firms are neither too rational, nor too irrational. As Figure 4 suggests, when firms are fully rational, or very irrational, the price-increase effect disappears. This happens because of the following logic. Nearly fully rational firms choose symmetric equilibrium, which means that there is no competition-induced inventory risk the bargain hunting demand is split equally between the firms. Firms that are very irrational place equal probability on all prices, which leads to the average price of 50 in the limit, and that is what Figure 4 illustrates. Observation 2 The price-increase effect becomes less pronounced as the shortage cost decreases, but it still exists even at s = 0. The magnitude of price increase is naturally larger when shortage cost is large: the mismatch between inventory and demand is then more costly. But even when s = 0, the effect is not eliminated completely because the firm still carries the risk of overstocking (leftovers). Observation 3 Consumer behavior parameters, fraction of loyals, λ, and price-sensitivity of bargain hunters, δ, have a similar effect on the increase in prices: price increase is large when λ and δ are small, but vanishes when these parameters are large. This observation is again related to the inventory risk: when most consumers in the market are loyal (high λ) or bargain-hunters are very price sensitive (high δ) the additional demand the firm generates from bargain-hunters as compared to loyals is small. E.g., when λ 1 all consumers are loyal and there is no inventory risk at all. Similarly, when δ c/a even at the lowest reasonable price p i = c the firm does not generate any bargain-hunting demand, which is identical to having only loyals. When faced with only loyals, the firm s optimal price is (a c)/2 = 60 and is independent of the competitors actions, which is what Figure 4 shows. [The price is not exactly equal to 60 because of demand uncertainty.] When λ and δ are low, then bargain-hunters present most opportunity but also most risk, because the size of loyal demand is small (e.g., with λ = 0 there are no loyal customers at all). An increase in price compensates for this risk in two ways: it increases the margin on every unit sold, and decreases the order quantity and thus the associated losses from leftover inventory when the firm does not capture bargain-hunting demand. Observation 4 The average equilibrium price is rather insensitive to the magnitude of uncertainty in the total market demand, B. 12

14 As Figure 4 suggests, equilibrium prices tend to increase somewhat as the total market demand becomes more uncertain, but the increase is very small. This happens because the magnitude of competition-induced uncertainty (and hence inventory risk) is much higher (3-5 as per the earlier discussion) than the magnitude in the market uncertainty. More so, as the number of competitors increases, each firm s market uncertainty decreases while the competition-induced uncertainty increases. 5 Discussion and Conclusions The goal of this short paper is to demonstrate and explain a surprising effect: that an increase in the number of competing firms may lead to an increase in the equilibrium market price in a situation when firms compete on both price and quantity/inventory/capacity. While a similar effect was highlighted previously by several authors in the economics literature in stylized settings, we offer a new, richer understanding of this effect in operational settings and link it to inventory risk: the potential mismatch between the firm s inventory and demand that occurs if the firm incorrectly judges how much additional demand it will generate if it lowers the price (in our set up this implies judging whether the firm s price will be lowest or not). As we show, such a risk increases in the number of competitors, making firms inventory management a lot more difficult and costly, hence triggering an increase in the equilibrium prices. We use the concept of quantal response equilibrium (QRE) to model competition between boundedly rational firms, illustrate the price increase and then examine how it changes depending on the number of competitors, the degree of their rationality, and the composition of market demand and firms costs. While for the most part of the paper we analyzed a specific and rather simplistic demand and profit model, we experimented with many plausible generalizations of our model and in all cases observed qualitatively similar results. Our results have important implications for both regulation and competitive strategy. For government officials our results suggest that they must be careful when promoting measures to increase competition in industries with low consumer loyalty and low supply chain responsiveness. These industries could, for example, include functional products (commodity foods, basic apparel) manufacturers and retailers which according to Fisher (1997) should have efficient rather than responsive supply chains. As we demonstrate such measures may have an adverse effect, hurting both firms and consumers. Our results suggest that the regulators should examine the structure of demand and supply when deciding on promoting competition. For competitive strategy, our results suggest that firms must be careful in thinking of which 13

15 markets to enter and which strategies to use. Our results imply that being a low-cost firm and being able to charge a low price is not enough: to succeed in markets with price and quantity competition firms must also have agile, scaleable supply chains in order to be able to efficiently deal with the inventory risk. For example, operating an online operation in addition to brick-andmortar stores could decrease the shortage cost (a firm can send items directly to customers in the event of a stock-out), being able to transfer inventory from one market to another, being able to scale production fast and at a low cost (recall Mrs Brown s bakery example) would all be effective strategies for dealing with inventory risk. Finally, our results also suggest that firms should pay careful attention to the degree of rationality of their competitors for it directly contributes to the inventory risk. In particular, firms may be better off by exiting some markets if there are too many irrational competitors; interestingly, the resultant equilibrium with fewer competitors may be better for consumers. In terms of future work, our paper opens up a number of interesting opportunities. One set of such opportunities deals with behavioral extensions. To the best of our knowledge even the first-degree phenomenon of a price-increase is yet to be established in a behavioral experiment; in fact Dufwenberg and Gneezy (2000) showed that the average price decreased when the number of competitors increased from two to four in a traditional winner-takes-all pricing game (an extreme case of our model with λ, c, s = 0). Additional behavioral extensions could include measuring the degree of rationality of human decision makers in price and inventory competition problems, exploring heterogeneity in rationality and best response to irrationality in the sense of Becker-Peth et al. (2013) and Ovchinnikov et al. (2014). Another set of possible extensions is to directly consider the firm s operational strategies mentioned above and how the equilibrium will change if the firms are asymmetric in their costs, customer loyalty, rationality, or may have different signals about other players operational characteristics. We deliberately kept this paper short and simple, but all these extensions are of interest for future research. References [1] Anderson S. P, de Palma, A. and Y. Nesterov Oligopolistic Competitoin and the Optimal Provision of Products. Econometrica 63(6) [2] Becker-Peth M., Katok E. and U. W. Thonemann Designing Buyback Contracts for Irrational But Predictable Newsvendors. MS 59(8)

16 [3] Bolton G. E., Ockenfels A. and U. W. Thonemann Managers and Students as Newsvendors. MS, forthcoming. [4] Bolton G. E. and E. Katok Learning by Doing in the Newsvendor Problem: A Laboratory Investigation of the Role of Experience and Feedback. MSOM. 10(3) [5] Camerer C. and T. Ho Experience-weighted Attraction Learning in Normal Form Games. Econometrica. 67(4) [6] Chen Y. and M. H. Riordan Price-increasing Competition. RAND J of Econ 39(4) [7] Chen Y., Su, X. and X. Zhao Modeling Bounded Rationality in Capacity Allocation Games with the Quantal Response Equilibrium. MS, 58(10) [8] Dufwenberg, M. and U. Gneezy Price competition and market concentration: an experimental study. Int J of Ind Org. 18 () [9] Hall J Groupon demand almost finishes cupcake-maker. The Telegraph. November 22, 2011 [10] Fisher, M. L What Is the Right Supply Chain for Your Product? HBR Mar-Apr [11] Ovchinnikov A., Moritz B. and B. Quigora How to Compete Against a Behavioral Newsvendor. Working paper, Queen s University. [12] Lau N., Bearden J. N. and S. Hasija Distributional Features of Newsvendor Behavior. Working paper, INSEAD. [13] Lippman S. A. and K. F. McCardle The Competitive Newsboy. OR. 45(1) [14] McKelvey R. D. and T. R. Palfrey Quantal Response Equilibria for Normal Form Games. Games and Economic Behavior, 10, p-38. [15] Netessine S. and N. Rudi Centralized and Competitive Inventory Models with Demand Substitution. OR. 51(2), [16] Parlar M Game theoretic analysis of the substitutable product inventory problem with random demands. Naval Research Logistics. 35(3), [17] Petruzzi N.C. and M. Dada Pricing and the Newsvendor Problem: A Review with Extensions. OR 47(2)

17 [18] Raz G Supply Chain Game (Simulation). Darden Business Publishing. OM-1386R. [19] Rosenthal R. W A Model in which an Increase in the Number of Sellers Leads to a Higher Price. Econometrica 48(6) [20] Sterman, J. D Modeling Managerial Behavior: Misperceptions of Feedback in a Dynamic Decision Making Experiment. MS [21] Su X Bounded Rationality in Newsvendor Models. MSOM 10(4) [22] Wu D. Y. and K-Y. Chen Supply Chain Contract Design: Impact of Bounded Rationality and Individual Heterogeneity. POM 23(2) [23] Zhao X. and D. R. Atkins Newsvendors Under Simultaneous Price and Inventory Competition. MSOM 10(3)

18 (Sensitivity to rationality, β) (Sensitivity to shortage cost, s) (Sensitivity to fraction of loyals, λ) (Sensitivity to bargain-hunters price-sensitivity, δ) (Sensitivity to demand uncertainty, B) Figure 4: Sensitivity of the average equilibrium price (vertical axis) to various model parameters. 17