Why variable inventory costs intensify retail competition, and a case for manufacturer returns

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1 Why variable inventory costs intensify retail competition, and a case for manufacturer returns Abstract We study how the extent to which the inventory cost is sunk or remains variable a ects retail competition when retailers do not observe each others stocks before choosing their prices. We nd intense price competition when inventory costs remain variable, and that markups increase as the sunk fraction increases. We then study whether a manufacturer should endogenously reduce the extent to which retailers inventory cost is sunk by o ering the possibility of returning unsold stocks for a fraction of the wholesale price. The answer solves a trade-o : returns intensify retail competition by giving retailers the incentive to order excess stocks, but such ine cient over-production must be paid up-front by the manufacturer. Therefore the optimal return decreases as the manufacturing marginal cost increases. We nally study how the inability to observe retail stocks a ects manufacturer pro tability relative to a situation where stocks are observed (where returns do not change pro ts). 1 Introduction More often than not retailers must order stocks in advance to make them readily available to consumers on their shelves. Yet the extent to which retailers inventory cost is sunk or remains variable varies signi cantly from one product to another as unsold inventory may some scrap value. Variations in the value to retailers of unsold stocks can be either exogenous or arise endogenously. As an example of the latter, manufacturers often let retailers return unsold goods for a fraction of the wholesale price a practice that makes inventory cost less sunk and that has been widely documented in a variety of contexts and for a multitude of products. Several other tools can have a similar e ect, when they require the retailer to pay a price per unit sold and a lower price per unit that remains unsold such as a wholesale price charged per unit delivered and an additional royalty paid per unit actually sold (the analysis of this paper naturally extends to all such tools). 1

2 This paper carries two main messages. First, that the extent to which inventory costs are sunk or remain variable has a signi cant a ect on retail market outcomes if realistically retailers do not observe rivals inventory before choosing prices (while, as supported by the currently held view discussed below, it has no e ect if instead retailers observe inventory choices). Second, we show how a return policy can be pro tably used by an upstream manufacturer (and by extension other policies, such as the above mentioned per unit royalties). A current view is that the extent to which inventory costs are sunk or remain variable has no e ect on retail market outcomes if retailers inventory choices precede price competition. As Kreps and Scheinkman (1983) found in their study of what has became a workhorse model, if retailers choose prices after having observed the stock choices of their rivals even situations that look very Bertrand-like, i.e. where most of the cost remains variable at the pricing stage, will still give the Cournot outcome (Kreps and Scheinkman, 1983). When it was introduced, their model intended to describe industries where rms make long-term capacity choices, such as the size of a plant, that are either observed or can be accurately inferred by competitors over time such as cement or chemicals. The assumption that stocks are observable seems however hard to justify in many retail settings, which provides the motivation for the present analysis. In practice a retailer is unlikely to know the stocks other retailers have of any particular good when choosing his price. Unlike the size of a plant, retail stocks are not only hard to observe but also, given their transient nature, historical data cannot be used to provide a reliable estimate. This view seems to be shared by the operations research literature. For example, standard newsvendor settings assume that retailers choose prices and stocks without knowing the choices of rival retailers. Our rst result challenges the above mentioned irrelevance result by showing that if retailers do not observe each other s stocks before choosing prices then the extent to which inventory cost is sunk a ects substantially the level of stocks ordered and retail markups. When the whole inventory cost remains variable both retailers charge a price equal to the wholesale price, as in Bertrand competition, yet each retailer also orders enough stocks to serve on its own the whole market. This is intuitive since if each retailer believes the other retailer is following such a strategy then he has no incentive to deviate: he cannot expect to make a larger pro t by setting a higher price (as all consumers would purchase from the other retailer) but also gains or loses nothing if he orders less than the prescribed equilibrium inventory. When part of the cost is sunk each retailer selects instead a price that is above the wholesale price and uses that margin to cover the cost of any stock that remains unsold. While each 2

3 retailer still orders enough inventory to serve the market at his chosen price, his exact price choice cannot be predictable (i.e., a pure-strategy equilibrium does not exist) since otherwise he would be undercut by his rival and sell nothing (while still incurring the sunk part of the cost and thus making a loss). Therefore when part of the cost is sunk, retailers must use a mixed-strategy, sometimes charging a high price (and accordingly choosing a low stock) and other times charging a low price as if holding a sale (and accordingly ordering a large stock). Fortunately in the absence of retailer di erentiation the unique regular equilibrium is elegantly characterized. In equilibrium the expected price and retail margin of each retailer increases as the fraction of the wholesale price that is sunk increases, as this increases the downside of being left with unsold stock which leads retailers to order smaller stocks more often and results in less intense price competition. This naturally creates retail price dispersion and, like in Varian s (1980) theory of sales, even if there is no uncertainty about aggregate demand the retailers face a residual demand that is endogenously uncertain (a consequence of the mixed-strategy of his rival). Padmanabhan and Png (1997) (PP hereafter) have found empirically that retail margins were signi cantly reduced when an upstream manufacturer o ered retailers the possibility to return unsold stocks. PP used the following argument to explain their ndings: If retailers buy from the manufacturer at a linear wholesale price and observe the stocks of each other before choosing their prices, without returns, retail competition is of the Cournot style following Kreps and Scheinkman (1983). They then conjectured that o ering retailers the possibility to return unsold stock at the wholesale price, i.e. full returns, would create intense Bertrand price competition among retailers since it eliminates retailers cost of holding excessive stocks. However this insight cannot be reached if the (PP s) model is solved correctly, wrote Wang (2004), while showing that when retailers observe each others stocks then in equilibrium they do not order excess stocks but instead use their stock choice as a precommitment to soften price competition and the Cournot outcome is still obtained therefore returns do not change the equilibrium strategies or pro ts of that game. 1 Our second result shows that PP s original idea can be successfully formalized if instead, but realistically, we drop that rather standard assumption that retailers observe each others stocks before choosing prices. Naturally, if stocks are not observed then they cannot provide a precommitment to soften price competition. In that case if a producer makes the retailers 1 Reintroducing demand uncertainty in their model, Padmanabhan and Png (2004) concluded that returns could help to "manage price competition between retailers but that this e ect holds only in the presence of demand uncertainty" under particular parameter conditions. The magnitude of that e ect remains however small. 3

4 inventory cost variable by o ering them a full return policy then retailers set prices at the wholesale price level, like in Bertrand, since as PP conjectured the retailers choose not be constrained by their stocks. This is an important nding since, judging by the number of citations, the original idea proposed by PP remains compelling and has had a profound impact on how academics think about return policies. Yet, and unlike what PP had conjectured, full returns may still be unpro table (relative to a situation with no returns). This is also intuitive, because returns can only result in intense price competition when they induce retailers to order excess stocks, but the manufacturer must then pay up-front for the production cost of stocks that are eventually returned. Our third result is to characterize the optimal distribution policy, i.e. not only the wholesale price but also the fraction of the wholesale price a manufacturer should o er as a return. This requires solving the manufacturer s trade-o between the bene t of more intense retail competition, which leaves him with more room to increase the wholesale price, and the higher cost of more excess stocks. We nd that the optimal wholesale price increases with the manufacturing marginal cost, while the optimal fraction of the wholesale price that is returned decreases with that cost. For example, consistent with this result, generous return policies are common in the distribution of books, CDs and software, all products that have a low cost relative to the price paid by consumers. Finally we ask if a manufacturer bene ts from unobservable stocks. Our fourth and nal result is that a manufacturer bene ts when the manufacturing marginal cost is low, but unobservable stocks hurt him when that cost is high a result which also holds in the presence of downstream di erentiation. This follows from directly comparing our results with those of Wang (2004), for which we summarize the main di erences and ndings in picture 1. Wang (2004) showed that when retailers stock levels are observed before prices are chosen then the retail outcome is always Cournot (independently of the return policy). In that case in equilibrium all stocks that are ordered are also eventually sold to consumers but the manufacturer cannot squeeze the retail margins since retailers use their stock choices as a quantity precommitment to soften price competition and protect those margins. Here we have found that when stocks levels are unobservable then the manufacturer may use a return policy to intensify retail competition but at the cost of excess stocks. An observable stocks regime should eventually become more pro table to a manufacturer when the number of retailers is large. In that case Cournot retail competition with many retailers already drives retail markups close to zero without the need of producing excessive stocks. 4

5 2 Literature review In addition to the work described above, there is a large literature in economics and marketing that explains the use of returns by an upstream monopolist. Yet all those theories crucially rely on demand uncertainty. We brie y review this large literature below. With a single retailer, if demand is uncertain, it has been shown that returns can for example transfer risk from a retailer to the manufacturer (e.g. Kandel, 1996, Marvel and Peck, 1995). In that case a return policy and a linear wholesale price can also coordinate a supply chain or improve pro ts over outright sales (e.g. Pasternack, 1985, Emmons and Gilbert, 1998 and Marvel and Peck, 1995). If demand is realized after production takes place but before the wholesale price is determined, then returns allow a manufacturer to achieve the same pro t whether she remains the owner of the stock until demand is realized or transfers that ownership to the retailer which would otherwise decrease the manufacturer s pro t, despite increasing the industry total surplus (Biyalogorsky and Koenigsberg, 2010). If a retailer needs to invest on a marketing-mix activity that a ects consumer demand but the terms of trade may be renegotiated once demand is realized, then transferring the ownership of the stock to the retailer and allowing for returns can reduce double marginalization by converting what would be an ex-post negotiation over the wholesale price into a negotiation over the return price (Iyer and Villas-Boas, 2003). When retailing is perfectly competitive, but demand is still uncertain, buybacks can also lead to optimal levels of inventory and price dispersion (Marvel and Wang, 2007). They may also be pro tably used as a substitute for a price oor. By inhibiting the price cutting that would otherwise occur when demand is low, returns protect the retailers margins in those states. This increases the retailers incentives to hold larger inventories in equilibrium, even if retail margins are lower when demand is high (Deneckere, Marvel and Peck, 1996). Such logic, that returns policies attenuate retail price competition when demand is low but intensify competition when demand is high, also extends to a retail oligopoly (Butz, 1997, Padmanabhan and Png, 2004). In a retail oligopoly, contracts with a linear wholesale and return prices can coordinate the supply chain only if the manufacturer retains controls over the retail price (Pasternack, 1985). Yet supply chain coordination can still be achieved if the wholesale and return price depend on the prices chosen by retailers (Bernstein and Federgruen, 2005), or if in addition a xed fee is charged and demand uncertainty or retailer di erentiation are large (Narayanan, Raman and Singh, 2005 and Krishnan and Winter, 2007). 2 2 In Narayanan et al. (2005) the uncertainty is over the level of demand, i.e. any level of demand between zero 5

6 There is also work in the context of asymmetric information on an uncertain demand. For example, returns may be used by a manufacturer to signal to retailers private information on demand (Kandel, 1996), or elicit from retailers their private information on demand or provide them with incentives to acquire such information (Arya and Mittendorf, 2004 and Taylor and Xiao, 2009). The role of returns in learning demand of a new product has also been studied, assuming that the retail price is xed and demand is random but identical across periods (Sarvary and Padmanabhan, 2001). Finally, as mentioned in the opening paragraph, our results also apply for example to per unit royalties. While we refrain from providing here a review of the large literature on royalties, we note that most of the work in that literature with a single manufacturer has focused on either agency problems with a single buyer or on situations where there are multiple buyers assuming that the outcome of downstream competition is either Bertrand or Cournot. In the former case the focus is on how royalties can help control moral hazard, adverse selection and risk-sharing (see e.g. Bousquet et al., 1998 and Cachon and Lariviere, 2005). In the later the focus has been in understanding the choice of di erent licensing mechanisms in the presence of competition for example when the seller s innovation reduces the costs of downstream buyers (Kamien and Tauman, 1986, Wang, 1998 and San Martin and Saracho, 2010). This paper suggests that Bertrand and Cournot forms of competition may not provide good benchmarks if realistically we assume that retailers order stocks before choosing their prices and cannot observe each others stocks. Our work would therefore also provide new tools to advance our analysis and understanding of royalties. However we leave further extensions to future research. 3 A retail model with unobservable stocks There are two risk neutral retailers that purchase stocks at a constant per unit wholesale price w, a part r of which is recoverable if a unit is not sold. We consider a two stage setting similar to Kreps and Scheinkman (1983) (they are equivalent when r = 0): Stage one: Each Retailer i, knowing w and r, simultaneously chooses his stock s i. The only cost of retailing is the payment of the wholesale price w, so i pays ws i. Stage two: Each retailer i chooses a price p i. Consumers observe both prices and purchase or and some higher level is equally likely. In Krishnan and Winter (2007) there is uncertainty over both the level of demand and consumers preferences over retailers while using a rst-order approach which means the results do not apply to cases without su cient retail di erentiation. 6

7 not. The value of sold and unsold stock is realized at a value of p i and r respectively per unit sold and not sold. One important element of the information structure still needs to be speci ed: do retailers observe or not the stocks ordered by the other retailer before choosing prices? Kreps and Scheinkman (1983) assumed that retailers observe the stocks of each other before stage two. Instead we assume here that each retailer does not observe the stock of his rival before choosing his price. When retailers do not observe stocks before choosing prices, stage one and stage two of the game become a simultaneous move game where retailers choose a pair (s i ; p i ). As we argued in the introduction, in many retail settings both the observability and variability of cost assumptions may be more realistic than the alternatives. This approach builds on Levitan and Shubik (1978) and Gertner (1985), but in addition we allow for a part of retailers costs to be variable: for each unit a part of the cost w r is sunk and a part r remains variable. Demand assumptions are also similar to those in Kreps and Scheinkman (1983). The consumer demand D(p), with D 0 < 0, has a choke price c < p < 1 where p = inf(p jd(p) = 0). Market revenue pd(p) is strictly concave on [0; p], so rst-order conditions on the market pro t (p determine the market monopoly price p m (c). c)d(p) Retailers are undi erentiated and therefore all consumers would like to purchase from the retailer with the lowest price and do so if the stock of that retailer is su ciently large. When both charge the same price, they share the market equally. If prices are di erent then, for any given pairs (s 1 ; p 1 ) and (s 2 ; p 2 ), if p i < p j retailer i faces the total consumer demand, while if p i > p j then i faces a residual demand from those consumers who were unable to buy from retailer j because of a stockout, i.e. since D(p j ) > s j. Formally the sales of retailer i are given by 3 8 min fs >< i ; D(p i )g if p i < p j q i = min fs i ; max f0; D(p i ) s j gg if p i > p j n n oo >: min s i ; D(p i ) min s j ; D(p i) 2 if p i = p j A pure-strategy for retailer i is a pair (s i ; p i ), where p i 0 and s i 0. To focus on the payo relevant strategies, and make it possible to de ne uniquely the equilibrium, we make the innocuous assumption that each retailer i can choose a price for which there is positive demand 3 Here we have e cient rationing of demand but this is not essential as our results hold under a large family of demand rationing rules introduced by Davidson and Deneckere (1986) that range from e cient to proportional rationing. E cient rationing can be interpreted as a situation where if there is price dispersion then those consumers with the highest valuation enter the market rst and purchase from the lowest priced retailer. 7

8 only if he has some stock to sell, i.e. retailer i can choose a pair (s i ; p i ) for which D(p i ) > 0 only if s i > 0. The pro t of retailer i is then i = (p i r)q i (w r)s i. (1) This game has no pure-strategy equilibria (unless r = w), since if retailer i were to choose an exact price and stock then either his rival j would nd it pro table to just undercut i and sell to the whole market (in which case i would still incur the sunk part of the cost and thus make a loss), or if it did not then it would be i nding it pro table to expand its capacity (in which case it would instead be j that makes a loss). The absence of a pure-strategy equilibrium means that retailers are unable to predict rival s choices, even if there is complete information about aggregate demand and all costs involved. In a mixed strategy equilibrium each retailer i must be indi erent between ordering a low quantity and choosing a high price (in which case i is likely to be undercut by his rival j and not sell his stock, but he will make a high margin on this small quantity if he sells), and ordering a large quantity and hold a sales promotion (in which case i is likely to undercut j and sell all his stock, but then make a low margin on this large quantity). Retailer i predicts the frequency and intensity of j s promotions, but he cannot predict if retailer j will hold a promotion or set a high price. This retail game has a unique regular Nash equilibrium, i.e. where strategies have no gaps or mass points inside the support of prices (although they could have mass points at the boundaries). 4 To characterize the equilibrium, we rst show that retail competition always drives retail pro ts to zero: even if each retailer charges a positive markup over the wholesale price, the amount he makes when he is the lowest priced retailer exactly o sets the amount needed to cover the sunk part of his inventory cost when he is the higher priced retailer. Second, in a regular equilibrium each retailer must order enough to serve the market at the price he has chosen and therefore the lowest priced retailer covers the whole market, i.e. s i (p i ) = D(p i ). These two properties identify a single distribution of prices: when retailer i sets a price above w there is a probability F (p i ) that the other retailer j chooses a lower price and leaves retailer i with unsold stock. From the 4 While it is possible that other equilibria could exist, we have found none and this equilibrium is the unique symmetric equilibrium, and also the unique equilibrium if demand is linear. 8

9 zero pro t condition and since s i (p i ) = D(p i ) we have that 5 (1 F (p i ))(p i w)d(p i ) = F (p i )(w r)d(p i ), F (p i ) = p i w p i r. We have the following formal result: Proposition 1. For a given w and r, the unique regular equilibrium outcome of retail competition has each retailer i choosing his price using a cumulative distribution function (with a pdf f) F (p i ) = p i w p i r for all p i 2 [w; p] and F (p i ) = 1 for p i > p, and for each chosen price p i he orders the stock s i (p i ) = D(p i ). An increase in r intensi es retail competition since, by shifting F leftward, it leads retailers to order larger stocks and charge (on average) lower margins. As r! w the price of both retailers converges to the wholesale price w and therefore retailers mark ups converge to zero. Proof. See all proofs in the Appendix. When most of the cost is sunk the retailers buy on average small stocks and charge larger margins to make up for the loss they make when they are unable to sell their stocks. As the fraction of the cost that is sunk decreases, the downside of being left with unsold stock also decreases. This leads retailers to order (on average) larger stocks and charge lower margins which is formally captured by a shift in F leftward. 4 Endogenous use of returns by a manufacturer Given the previous result, we would expect manufacturers to use such marketing tools that change retail competition to their advantage. One tool that achieves this are returns, but as we also mentioned in the introduction the insights apply to other tools with a similar structure (such as per unit royalties). This idea was rst suggested by Padmananabhan and Png (1997). We study a model similar to theirs but with a di erent information structure on stocks. 6 Like Kreps and Scheinkman (1983), but unlike PP, we assume no retail di erentiation (see however section 5 on how the ndings extend to di erentiation). PP assumed that retailers observe the 5 Note that retailer i orders s i = 0 with probability w r, understood as the probability that a particular retailer p r delists the good. Here this is represented by a mass point at p. 6 Also, like Kreps and Scheinkman (1983) we assume no retail di erentiation, which PP had in their model. See section 5 on how the ndings extend to di erentiation. 9

10 stocks of each other before choosing prices, following Kreps and Scheinkman (1983). Wang (2004) has however shown formally that in such setting returns have no e ect on strategies and pro ts (that result holds with and without di erentiation). Here we will instead assume, like we did above, that stocks are unobserved. There are now two retailers and a manufacturer with a constant marginal cost c. Like PP we consider a game with three stages, where the manufacturer s choice of a public retail policy, w and r, precedes stages one and two of the game described in section 3. 7 Stage zero: The manufacturer sets a distribution policy with a wholesale price w and return price r at which retailers can buy and return any quantity. Stage one: Each retailer i simultaneously decides how much to order and pays ws i. Stage two: Each retailer i chooses a price p i and consumers, after observing both prices, decide to purchase or not. Unsold stocks are returned to the manufacturer and refunded at the return price r per unit. A pure-strategy for the manufacturer is a pair (w; r), with w 0 and 0 r < w. A purestrategy for retailer i is a pair (s i ; p i ), where p i 0 and s i 0. is = P [(w c)s i r(s i q i )] i The manufacturer s pro t and the retailers pro t remain given by (1). 4.1 Full returns versus no returns With full returns, as r! w, the cost of holding an excessive stock is zero and therefore each retailer i orders s i = D(w), enough stock to serve the whole market at the wholesale price, and as F (w)! 1 both prices converge to the wholesale price. It follows that with full returns (F R) the manufacturer s pro t becomes F R (w; c) = (w 2c)D(w) = m (2c) Note the equivalence to the pro t of a monopolist that sells directly to consumers at a price of w but has a marginal cost 2c instead of c. Thus the optimal wholesale price, wf R (c) = pm (2c), exceeds the monopoly price. The bene t from intense retail competition created by full returns 7 Discriminatory distributions policies can, for example, attract antitrust scrutiny. Also, in the ten product categories empirically studied by PP, no manufacturer used non-linear pricing. One explanation is that arbitrage by retailers across multiple markets served by the same manufacturer can restrict the use of non-linear policies. 10

11 is countered by the production cost of excess stocks, so only when c = 0 does the manufacturer extract the vertically integrated monopoly pro t. In the case of no returns (NR) we have r = 0 and the manufacturer s (expected) pro t is given by the wholesale margin multiplied by the expected stocks to be sold to both retailers since all sales are nal, i.e., NR (w; c) = 2(w c) w f(p)d(p)dp, (2) In this case retail competition is less intense (as the prices exceed the wholesale price) but there is no cost associated with excess stocks. We nd that: Proposition 2. In the unique regular equilibrium the manufacturer pro t is higher with full returns than with no returns if the manufacturing marginal cost c is lower than some threshold value c > 0 and the opposite holds if c > c c. Moreover the optimal wholesale price with no returns is lower than with full returns. A prediction of this model is that if a manufacturer switches from a policy without returns to a policy with full returns then the wholesale price should increase. Moreover a manufacturer should only do this once c becomes su ciently low. So this predicts a positive relationship between such transitions with simultaneous reductions in the manufacturing cost and increases in the wholesale price. To illustrate these results, suppose that demand is described by a general linear form D(p) = a bp (in this case the regular equilibrium is also the unique equilibrium). Then a is the measure of market size, p = a=b is the choke price, and the monopoly price lies at half the distance between c and p, i.e. p m (c) = (p + c)=2. In the case of no returns the pro t in (2) simpli es to NR (w; c) = 2 (w c) p w(1 (ln wp ), and wnr (c) p + (1 )c with = 0:4=1:4. In the case of full returns w F R = pm (2c) is p 2 + c if c < p 2 and p if c p 2. Therefore w F R > w NR since the former is slightly below a third of the distance between the marginal cost and the choke price. The comparison between the equilibrium pro ts when demand is linear is illustrated in g. 2 below, where the horizontal axis represents c as a share of the choke price, i.e., k = c=p, and the vertical axis represents the percentage of the monopoly pro t captured by the manufacturer. 11

12 Profit k Fig 2. F R (grey) and NR (black) as a percentage of m. Note that a change in policy from full to no-returns (and vice-versa) can have a signi cant e ect on pro t when k is either high or low. Also, reductions in the marginal cost c and increases in the consumers willingness to pay the latter captured by an increase in p that reduces k are likely to make full returns more pro table than no returns. The dashed line in gure 2 represents the manufacturer s expected pro t when she uses the optimal wholesale and return prices. Naturally this pro t is larger than both the pro t with full and no returns. We derive it in the next subsection. 4.2 The optimal wholesale and return prices Finding the optimal return level requires solving a trade-o between the bene t of more intense retail competition and the production cost of excess stocks associated to it. Given the retailers optimal responses, each retailer orders enough to serve the market at the price he chooses, and the equilibrium price paid by consumers is therefore the lowest of the two retail prices, denoted by p l = min fp i ; p j g. The manufacturer makes (w r) on each unit that is purchased by consumers and gets (w c r) on each unit that is produced but ends being returned. So for any p l the quantity expected to be returned is f(p 0 ) p l 1 F (p l ) D(p0 )dp 0. The probability the lowest of the two prices is p l is the rst (or smallest) order statistic of a sample of size 2 from the continuos probability distribution F, which is exactly 2f(p l )(1 F (p l )). Therefore the manufacturer s (expected) pro t with unobservable stocks (UN) for a 12

13 general distribution policy (w; r) is f(p 0 ) UN = 2f(p l )(1 F (p l )) (w c)d(p l ) + (w c r) w p l 1 F (p l ) D(p0 )dp 0 dp l, For tractability we consider the situation where demand is linear. Proposition 3. If demand is linear, the optimal w and r with unobservable stocks are w (c) = p + c 2 = p m (c) and r (c) = p 2 = pm (0). From this result it follows that the returned fraction of the wholesale price is 8 r (c) w (c) = p p + c, and since the optimal wholesale price is the integrated monopoly price we have (w c) = (p c)=2 and (w c r ) = c=2 and therefore the manufacturer s (expected) pro t simpli es to UN = p c h a 2 2 (1 k)2i {z } Expect. quantity sold c h a )i 2 2 (1 + 2k ln k k2 where k = c {z } p. (3) Expect. quantity returned When the manufacturer s marginal cost is low the optimal distribution policy involves large returns and therefore a large fraction of the total production is also returned. As the marginal cost increases, the manufacturer nds it optimal to reduce the percentage of the wholesale price that is returned to reduce the unsold fraction of the total quantity produced note from (3) that the ratio of the quantity returned to the quantity sold decreases as k increases, converging to zero as c converges to the choke price, i.e. as k! 1. 5 Observable versus unobservable stocks Comparing the situation where stocks are observed before price competition to the situation where they are not will help us understand the incentives of a manufacturer to promote or hinder the exchange of this information. Wang (2004) showed that if retailers observe each others stocks before choosing prices then the market outcome is independent of returns and coincides with the Cournot outcome with 8 The ratio decreases with c, from 100% to 50%. As discussed in the conclusion, in practice these percentage should be lower due to additional retailing costs. 13

14 wholesale price w. When demand is linear the optimal wholesale price is p m (c), and thus the optimal wholesale price is the same whether stocks are observed or not. Moreover, when stocks are observable and demand is linear, each retailer purchases one third of the monopoly quantity. With public information on stock there will be no excess stocking but the manufacturer cannot squeeze the retail margins since retailers use stocks as a quantity precommitment to soften price competition and protect their margins. The manufacturer s equilibrium pro t with observable stocks (OB) becomes OB = 2 p 3 c 2 a 3 (1 k) = 2 3 m (c). (4) As we have seen, unobservable stocks allow the manufacturer to squeeze retail margins at the expense of overstocking. If the marginal cost is low the manufacturer uses a generous return policy to intensify retail competition without su ering much from the cost of producing excess stocks. Naturally the net e ect of stock information on the manufacturer s pro t depends on the cost of producing the excess stocks. Comparing (4) with (3)( g. 3 illustrates this e ect): Proposition 4. When demand is linear and the manufacturer uses the optimal wholesale and return prices, there is a critical value of the normalized cost e k such that for k < e k the manufacturer s pro t when stocks are not observable is larger than when they are observable, and the opposite holds for k > e k (where k = c=p). a k Fig 3. UN (black line) and OB (grey line) as a fraction of a. Suppose now that the manufacturer must commit to make stocks observable or not before learning his marginal cost, as for example costs change from period to period. Can we still tell 14

15 which information regime is preferred? Again we can answer this question in the speci c case where demand is linear. Suppose c is a random variable drawn from a continuous probability distribution with density g and full support in [0; bc]. It can be shown that if g is uniformly distributed and bc < p the manufacturer is better o if stocks are not observable, and is indi erent if bc p (this result if of course dependent on a linear demand). More generally, it can be further shown that if g has full support in [0; p] then the manufacturer is better o with observable stocks if g is monotone increasing, she is better o with unobservable stocks if g is monotone decreasing (and thus indi erent between the two regimes if c is uniformly distributed, i.e. g is constant). This last observation reinforces Proposition 4, showing that it extends to situations with uncertain marginal costs: if lower marginal costs are relatively more likely than higher marginal costs then the manufacturer is better o if stocks are unobservable and the opposite result holds when high manufacturing marginal costs are relatively more likely. 5.1 An extension to retail di erentiation The model we have studied so far has the advantage of being directly comparable with Kreps and Scheinkman s (1983), an important workhorse model in marketing, economics and operations research. Yet Both PP (1997) and Wang (2004) considered a model where retailers are di erentiated. While we have been unable to provide a closed form solution to the mixed strategy equilibria in their speci c model with di erentiated retailers, we were able to con rm that several main ndings extend to that particular setting: when stocks are unobservable then i) with full returns there exist equilibria with Bertrand prices but necessarily excess stocks, ii) retail competition is changed by the introduction of a full return policy, and iii) a manufacturer is better o than when stocks are observable if the manufacturing cost is su ciently low. Consider a model which is similar in all aspects to the one studied above with the exception that the demand directed to each retailer i is like in PP (1997) and Wang (2004) (in that case their and our models are similar, except for the information on stocks). They considered the demand system, derived in Dixit (1979) from a representative consumer, which gives rise to inverse demands given by p i = a q i dq 2, with d 2 (0; 1) and substitutability increasing in d. This leads to the direct demands q i = p i + p 2, with = a(1 d)=(1 d 2 ), = 1=(1 d 2 ) and = d=(1 d 2 ). To ensure that strategies spaces are compact, we have that retailers cannot order stocks larger than some value S and prices larger than some value P (with both chosen to be su ciently large). We nd: 15

16 Proposition 5. For any r < w the retail stage game with product di erentiation only has equilibria in (non-degenerate) mixed strategies. For r = w there exist pure-strategy equilibria with prices equal to, and stocks necessarily larger than, in the retail Bertrand outcome with a wholesale price w. From this result we conclude that when stocks are not observed before retailers choose their prices the irrelevance result in Wang (2004) no longer holds: First, the retail outcome is never Cournot. Second, o ering full-returns can change the equilibrium outcome. Unfortunately the method used in section 3 to characterize the mixed strategies equilibrium does not apply in this model with di erentiation. There we used the fact that we could collapse the joint randomization of prices and quantities to a randomization over a single variable (interchangeably price or quantity). Here this cannot be done. To characterize the equilibria we would need to nd not only a distribution of prices but in addition a distribution of stocks for each price in the support of that distribution which we have not been able to do. Yet, by revealed preferences and continuity, we obtain the analogous to Proposition 4 in the context of retail di erentiation. Corollary 1. In the game with product di erentiation, if with unobservable stocks when retailers are o ered full returns they play an equilibrium with Bertrand prices and su ciently large stocks (which are invariant in w), then for c su ciently low the manufacturer s pro t when stocks are not observable must be larger than when they are observable. 6 Conclusion In practice retailers are unlikely to perfectly observe the stocks of their rivals before choosing their prices. The two main messages of this paper are that in this situation i) retail competition will be a ected by the extent to which inventory costs remain variable and ii) a manufacturer can pro tably use policies that change the fraction of the retailers cost that is variable to induce retailers to compete more aggressively in prices, thus squeezing the retail margins. In our model some level of returns always increases manufacturer pro tability if retailers do not observe the stocks of their rivals, while this would have no e ect if instead stocks were observable. We have assumed that there were no retailing costs in addition to the wholesale price. We could introduce two types of retail costs: a cost of selling the product (e.g. sales and post-sales service) and a cost of stocking the product. With the former type of cost the analysis and intuition 16

17 remain unchanged, since it can be thought as a inward shift of the demand curve changing only the critical values. On the other hand a cost of holding stock would make the manufacturer choose to o er less generous return policies since using returns to intensify retail competition becomes then a less e ective tool. 9 For the manufacturer the returned stock may also be a source of additional revenue (e.g. manufacturer scrap value or recycling) or cost (e.g. transportation and disposal). We have omitted these from the present version, but it is simple to introduce those elements. If the scrap value of excess stock is a fraction of the manufacturing cost then the manufacturer will nd it pro table to o er a more generous return policy to intensify retail competition. The opposite holds in the case where returned stocks create additional costs. The latter can motivate a manufacturer to develop secondary uses for her product, such as for example export returned apparel to developing countries. Even if those products were sold below their original production cost, and therefore those alternative markets are in isolation unpro table, the additional revenue they generate makes a return policy less onerous. This allows the manufacturer to o er more generous return policies in the main markets to intensify retail competition, which in turn increases her overall pro tability. 9 Note that the wholesale price is then only a fraction of a retailer s cost, which means that even a full return policy cannot completely eliminate the retailer s cost of holding excessive stocks. 17

18 References Arya, A. and Mittendorf, B. (2004) "Using Return Policies to Elicit Retailer Information", RAND Journal of Economics, 35: Berck, P., Brown, J., Perlo, J. and Villas-Boas, S. (2008) "Sales: Tests of theories on causality and timing", International Journal of Industrial Organization 26: Bernstein, F. and Federgruen, A. (2005) "Decentralized Supply Chains with Competing Retailers Under Demand Uncertainty", Management Science, 51: Biyalogorsky, E and Koenigsberg, O. (2010) "Ownership Coordination in a Channel: Incentives, Returns and Negotiations", Quantitative Marketing and Economics, 8: Bousquet, A., Cremer, H., Ivaldi, M., Wolkovicz, M. (1998) "Risk sharing in licensing", International Journal of Industrial Organization, 16: Butz, D. (1997), "Vertical Price Controls with Uncertain Demand", Journal of Law and Economics, 40: Cachon, G. and Lariviere, M. (2005), "Supply chain coordination with revenue-sharing contracts: Strengths and limitations", Management Science, 51: Davidson, C. and Deneckere, R. (1986), "Long-run competition in capacity, short-run competition in price, and the Cournot model", RAND Journal of Economics, 17: Deneckere, R., Marvel, H. and Peck, J. (1996), "Demand Uncertainty, Inventories, and Resale Price Maintenance", Quarterly Journal of Economics, 111: Gertner, R. (1985), "Simultaneous move price-quantity games and non-market clearing equilibrium" Massachusetts Institute of Technology. Glicksberg, I. (1952), "A further generalization of the Kakutani xed theorem, with application to Nash equilibrium points." Proceedings of the American Mathematical Society, 3: Iyer, G. and Villas-Boas, M. (2003), "A Bargaining Theory of Distribution Channels", Journal of Marketing Research, 40: Kandel, E. (1986), "The Right to Return", Journal of Law and Economics, 39: Kamien, M.I., Tauman, Y., (1986) "Fees versus royalties and the private value of a patent", Quarterly Journal of Economics, 101: Kreps, D. and Scheinkman, J. (1983), "Precommitment and Bertrand competition yield Cournot outcomes", Bell Journal of Economics, 14: Krishnan, H. and Winter, R. (2007), "Vertical Control of Price and Inventory", American Economic Review, 97:

19 Levitan, R. and Shubik, M. (1978), "Price Duopoly and Capacity Constraints", International Economic Review, 13: Narayanan, V.G., Raman, A. and Singh, J. (2005), "Agency Costs in a Supply Chain with Demand Uncertainty and Price Competition", Management Science, 51: Marvel, H., and Wang, H. (2007), "Inventories, Manufacturer Returns Policies, and Equilibrium Price Dispersion under Demand Uncertainty", Journal of Economics & Management Strategy, 16: Marvel, H., and Peck, J. (1995), "Demand Uncertainty and Returns Policy", International Economic Review, 36: Narayanan, V., Raman, A. and Singh, J. (2005) "Agency Costs in a Supply Chain with Demand Uncertainty and Price Competition", Management Science, 51: 120. Padmanabhan, V. and Png, I. (1997), "Manufacturer s Returns Policies and Retail Competition", Marketing Science, 16: Padmanabhan, V. and Png, I. (2004), "Reply to: Do Return Policies Intensify Retail Competition", Marketing Science, 23: Pasternack, B. (1985), "Optimal Pricing and Return Policies for Perishable Commodities", Marketing Science, 4: San Martín, M. and Saracho, A. (2010) "Royalty licensing", Economics Letters, 107: Taylor, T. A., and Xiao, W. (2009) "Incentives for retailer forecasting: rebates vs. returns", Management Science, 55: Wang, H. (2004), "Do Return Policies Intensify Retail Competition" Marketing Science, 23: Varian, H. (1980), "A Model of Sales", American Economic Review, 70: Proofs Proof of Proposition 1. In step 1 we show that there cannot exist a pure-strategy equilibrium. In step 2 we show that there cannot either exist a mixed strategy equilibrium where the expected pro t of both retailers is strictly positive. In step 3 we show that in any equilibrium the expected pro t of both retailers must be zero. In step 4 we show that the proposed strategies form the unique regular equilibrium. We perform the comparative statics in step 5. For a given distribution policy (w; r), the Nash equilibrium of the retail subgame is a pair (s 1 ; p 1 ) and (s 2 ; p 2 ) such that for each retailer i we have that (s i ; p i ) maximizes i given (s j ; p j ) (for simplicity we have dropped the dependence of (s i ; p i ) on w and r). A mixed strategy of retailer i is a c.d.f. F i of prices and a c.d.f. G p i of stocks when i s price is p. Given the strategy of retailer j, and since each retailer never orders more than D(p) when setting p, the expected 19

20 pro t of retailer i if he chooses a pair (s i ; p i ) is e i (s i ; p i ) = Z D(p) 0 0 i (p i ; s i ; p j ; s j )dg p j (s jj p j )df j (p j ). If i uses a mixed strategy, then his expected pro t must be equal for all pairs (s i ; p i ) that are in the support of his strategy. Step 1. By contradiction there is no pure-strategy equilibrium. (The formal proof is not essential for the remainder of the argument, therefore we here only provided the intuition, but everything can be stated formally.) If both retailers charge the same price and make positive pro ts, then one retailer would nd it pro table to deviate by slightly undercutting his rival and order a stock equal to the market demand at that price. Also, there cannot be an equilibrium where both retailers earn zero pro ts and charge the same price because, with the assumption that s i > 0 if D(p i ) > 0, one retailer could deviate to a slightly higher price and make a positive pro t (since in any equilibrium each retailer must order less than D(p i )). It can be shown in a similar way that pure-strategy equilibria with di erent prices also cannot exist because a similar pro table deviation always exists for one retailer. Step 2. We now show that there cannot exist a mixed strategy equilibrium where the expected pro t of both retailers is strictly positive, i.e. where both e i > 0 and e j > 0. De ne p as the o i lowest price that retailer i uses with a positive probability, and p = min np i ; p j. If both pro ts were positive then p > w. We rst must show that p = p i = p j and then that both retailers choose to serve the whole market at the bottom of their price distribution, i.e. at p both order D(p). and (p Both retailers cannot have a mass point at p. Suppose they do. If both pro ts are positive, w) > 0, both retailers should order at least D( p 2 ) when charging p (since the pro t is strictly increasing in the stock at least up to D( p 2 ), as they can be sure to sell any lower amount for p independently of the stock choice of the other retailer). But then j could increase his pro t by deviating from p to p " (with " arbitrarily small) and serve the whole demand at that price for sure (instead of sharing it with i with a strictly positive probability), which means retailer j would have no mass at p. We also have that p m (w) > p. Suppose that p m (w) < p. In equilibrium the sum of the pro t of both retailers cannot exceed the monopoly pro t, so one retailer could pro tably deviate to p m (w) and D(p m (w)) to make monopoly pro t for sure. Moreover if p m (w) = p and j does not have a mass point at p (since both cannot have one), then the probability i ties with j at p is zero. But then it would be optimal for retailer i to have F i (p) = 1 with s i (p) = D(p m (w)) and obtain 20

21 the monopoly pro t. However this leaves retailer j with no pro ts (and therefore a contradiction). It follows then that p i = p j. Suppose not, that p i < p j while p m (w) > p. Then i increases n o his pro t by raising its price from p i to any price below min p j ; p m (w) and sell to the whole market at that price. However we have seen that p i = p < p m (w), thus it must be that p i = p j. Neither retailer has a mass point at p. If there is then the one with a mass point (say i, as both cannot have a mass point) would face a zero probability of a tie at p while p w > 0 and thus at that price his pro t increases up to D(p). But if i chooses D(p) at p, then j can make a higher pro t if instead of charging p he charges p " and orders D(p ") and avoids the strictly positive probability of a tie with i (as in case of a tie he would not sell more than D(p)=2, while it sells D(p) for sure for any price p < p). Since neither retailer has a mass point at p (and therefore the probability of a tie at p is zero), the pro t of a retailer i at p is (p since p w > 0. Thus at p both retailers order D(p). w)s i (p), which is strictly increasing in the stock up to D(p) Further, suppose i is the retailer that charges the highest price with a positive probability, and therefore the highest price at which any of the two retailers orders a strictly positive stock and still makes a strictly positive pro t (if it is the same for both then i can be either). Call that highest price p i. By the de nition of p i we have p j p i < p. Thus the probability i s demand at p i is D(p i ) must be zero if j makes a strictly positive pro t, and therefore chooses a strictly positive quantity for every price in his strategy). It follows that the stock chosen by i if he charges p i must be less than D(p i ). Given that at p both retailers order D(p) and at p i we have that s i (p i ) < D(p i ), there must exist some p < p 0 p i which is the minimum price such that at least one of the two retailers orders with some positive probability an amount 0 < s < D(p 0 ). Both retailers cannot have a mass point at this price p 0. If they did then both must be ordering at least D(p 0 )=2 at this price. The reason is that a retailer only makes a sale when he chooses p 0 if he is the lowest priced retailer (as both order D(p) for lower prices), and therefore if the expected pro t at p 0 is positive then it must be strictly increasing up to D(p 0 )=2. Then one retailer, say i, would nd some price p 0 " at which j does not have a mass (since the number of points of positive mass in any probability distribution must be countable) and at which he is able to sell D(p 0 ") > D(p 0 ) with a probability that is strictly higher by at least the value of the mass point of j at p 0 but for a price that is only marginally lower than p 0. This provides a pro table deviation for i from p 0 and we obtained a contradiction. For a similar reason we cannot have one retailer, say player i, having a mass point at p 0 while j does not and yet f j (p 0 ) > 0 as j would have a similar pro table deviation. 21