Hong Chen. and. Murray Frank 1. and. Hong Kong University of Science and Technology. March 30, Abstract

Transcription

1 Monopoly Priing When Customers Queue Hong Chen Faulty of Commere and Business Administration University of British Columbia, Vanouver, B.C. Canada and Murray Frank Faulty of Commere and Business Administration University of British Columbia, Vanouver, B.C. Canada and Shool of Business and Management Hong Kong University of Siene and Tehnology Clear Water Bay, Kowloon, Hong Kong Marh 3, 995 Abstrat It takes time to proess purhases and as a result a queue of ustomers may form. The priing and servie rate deisions of a monopolist who must take this into aount are haraterized. We nd that an inrease in the average number of ustomers arriving in the market either has no eet on the monopoly prie, or else auses the monopolist to redue the prie in the short run. In the long run the monopolist will inrease the servie rate and raise the prie. When ustomer preferenes are linear the equilibrium is soially eient. When preferenes are not linear equilibrium will not normally be soially eient. Journal of Eonomi Literature odes: L 2, L 5. Key Words: Monopoly, Queue, Customer Information, Servie Rate, Soial Welfare. For researh support, the authors would like to thank respetively: a Killam Faulty Researh Fellowship and a grant from the NSERC (Canada), and a grant from the SSHRC (Canada). We also thank Vojislav Maksimovi for very helpful omments. Please address orrespondene to M. Frank in Hong Kong; for mfrankusthk.ust.hk

2 Introdution The median delay in delivery of a purhase is more than a month in many industrial markets suh as airplane manufaturing, ship building, textile mill produts, steel, fabriated metals, noneletri mahinery, and eletri mahinery. There are at least two interesting features of market learing in suh industries. First, there is ommonly more variation in delivery lag than in posted prie. This suggests that delay and queueing phenomena play a ruial role in learing suh markets. Seond, the queue exists on the books of the rm and so is often not diretly observable by the potential ustomer who is onsidering plaing an order. Carlton and Perlo (994) provide a valuable review of the evidene on the importane of time and delay in market learing. In this paper we study a monopoly whih sets a prie, and in the long run also hooses a servie rate. The model diers from the standard theory of monopoly priing beause of the importane of delay. Asinmany of the industrial markets mentioned above, the queue of existing orders is not diretly observable by the ustomer. We derive the ustomer demand funtion, the optimal prie for the monopoly to set in the short run with a predetermined servie rate, as well as the optimal monopoly prie in the long run in whih the rm also piks the servie rate. The relationship between the market equilibrium and soial welfare maximization is analyzed. Fortunately we nd that many of the omparative stati eets derived in models without queues ontinue to hold. In some ases the magnitude of an expeted eet is altered. However there are also instanes in whih an eet derived from the standard timeless model an be drastially altered. This fundamental point does show up in some ways in our analysis. When there is an inrease in the number of ustomers oming to market, in the short run the monopolist will either leave the original prie unhanged, or else will ut the prie. The literature on monopoly priing with queues started with Naor (969). 2 He demonstrated that when the ustomers an observe the queue prior to joining, the monopolist will harge a higher prie than is soially eient. Edelson and Hildebrand (975) showed that when the ustomer preferenes are linear, and they make their purhase deisions without observing the urrent state of the queue, the monopoly equilibrium prie maximizes soial welfare. Hassin (986) showed that when the rm prefers to inform the ustomers of the queue length, then it is soially optimal to allow the rm to do so. But when the rm prefers not to inform the ustomers of the length of the queue, soial optimality mayormay not oinide with the rm's prot maximizing hoies. We allow for preferenes that are more general than the linear preferenes studied in these papers. This is not just a tehnial matter, it aets the eonomi interpretations of the results. Linear preferenes are not onsistent with ustomers who do disounting. We show that the equivalene of monopoly priing and soial welfare maximization disovered by Edelson 2 Cooper (99) reports that some years ago it was estimated that at that time more than 5, aademi artiles and books had been published relating to queues. Wol (989) provides a nie textbook treatment of the mathematis and operations researh literature on queues. For helpful reviews of related optimization-based approahes to queueing theory see Stidham (984) and Stidham and Webber (993).

3 and Hildebrand (975) ontinues to hold when the ustomers have linear preferenes and the monopoly is able to hoose the servie rate as well as the prie. However we also show that one one moves beyond the linear preferene speiation that they analyzed, this welfare equivalene no longer holds in general. Unlike the usual over-harging by a monopoly, here the equilibrium may involve either over-harging or under-harging relative to soial welfare maximization. DeVany (976) is the only previous study of apaity, or servie rate hoie 3 in a monopoly queueing model. In his study the ustomers an observe the queue length. As we explain more fully in setion 2.3, in omparison to the rest of the literature, there are some quite dierent features in his formulation of the ustomer's problem. DeVany (976) makes some interesting observations about the monopoly equilibrium. First is his suggestion that the monopolist sets prie equal to marginal ost. Seond, while DeVany (976) does not expliitly solve a soial welfare maximization problem, he suggests that the monopolist hooses too little apaity for soial eieny. We will show that, at least when ustomers make the deision about joining the queue prior to observing its urrent length, the results are quite dierent. There have beenanumber of other monopoly priing and queueing models. Knudsen (972) allowed for more than a single queue at the rm. Donaldson and Eaton (98) showed that a monopolist may use a queue to separate out onsumers who have dierent valuations of time. Mendelson and Whang (99) analyzed the use of priority priing for dierent lasses of ustomers. 4 There have also been papers that analyze the formation of queues when the prie is exogenously onstrained to be below the market learing level. An interesting example is the study by Deaon and Sonstelie (985) of queues that arose when, by ourt order, a prie eiling was plaed temporarily on the Chevron gas stations in California. The rest of the paper is organized in the following manner. We start in setion 2 by presenting the analysis of the example in whih the ustomer preferenes are linear. We present this rst, both beause the derivations are simpler in this ase, and also beause this is the problem that has attrated most attention in the previous literature. In setion 3 we study the problem from the perspetive of the ustomers. The demand urve is derived from the ustomer's optimization problem. In setion 4 we analyze the short run situation in whih the rm has a predetermined servie faility, and the osts of operation are set to zero. Setion 5 extends the rm problem to the long run in whih the rm also has a servie rate hoie, and ostly prodution. Soial welfare properties of the monopoly equilibrium are presented in setion 6. Finally the onlusions are set out in 7. 3 What DeVany (976) terms \apaity" we refer to as \servie rate". We prefer the term \servie rate" beause it is more onsistent with the time averaging approah adopted both by DeVany (976) and by the urrent paper. 4 Queueing has also been onsidered in eonomi settings other than that of a monopoly. Luski (976), Levhari and Luski (978), Kalai, Kamien and Rubinovih (992), and Li and Lee (994) have begun to integrate the analysis of queues with oligopoly onsiderations. DeVany and Saving (983) and Davidson (988) introdued queueing into ompetitive models. Mendelson (985) studied queues that arise within the rm. Larson (987) provided an intriguing disussion of some of the psyhologial aspets of queueing. 2

4 2 The Example of Customers With Linear Preferenes The logial struture of the problem is quite simple. The rm is a monopoly. The rm move rst and selets the prie to harge. It is not allowed to harge dierent ustomers dierent pries. The monopoly selets and ommits to the prie in advane, knowing how that will aet the behavior of the ustomers. The ustomers see the prie posted by the rm, and know the rate at whih new potential ustomers arrive in the market, but annot observe the urrent state of the queue at the moment that they are onsidering ordering. Aordingly they an infer and respond to the system average, but not to the atual value. Without that knowledge it is as if the ustomers are all playing a simultaneous move game. We look for a Nash equilibrium in the ustomers strategies. In suh an equilibrium eah ustomer will be able to alulate the rate at whih ustomers join the queue at the rm. There is no disounting and we work with time averages. 5 All ustomers are idential apart from their moment of arrival. They eah will demand either nothing, or one unit of the good, from the rm. If the rm's oer is not suiently attrative then the ustomer will not plae an order. A ustomer who does not plae an order gets a return of v. Assume that the arrival times of potential ustomers are given by apoisson proess with rate. The rate at whih ustomers atually plae orders is, whih isto be determined. The information that is known to ustomer i is: R the reward from getting served by the rm, p the posted prie, v the value of their alternative opportunity, and the exponential rate of servie provided by the rm. Let i refer to a ustomer and let s i refer to his strategy hoie. Customer i will selet one of two feasible strategies: joining the queue, not joining the queue. We let s i = represent the deision not to join, and s i = represents the deision to join the queue. If ustomer i joins the queue the atual wait will be w i, whih is a random variable. The ost of waiting is denoted by C(w i ). The ustomer's utility funtion U() satises U > and U, and the ustomer's ost of delay funtion C() is nondereasing with C() =. Customer i piks either s i =,ors i =, in order to maximize the expeted utility, V i where, V i = ( U(v) if s i = U(R C(w i ) p) if s i =: In some papers analysis is arried out in terms of a \full prie" to the ustomer. The full prie onsists of the monetary prie, plus the ost of waiting. Using our notation P () = p+c(w i ()) is the full prie that ustomer i pays. In our model the rm selets a posted prie. The behavior of the ustomers together with the rate at whih orders are preessed, onverts the posted prie into a full prie. The posted prie is the rm's strategi variable. The full prie is an equilibrium outome. At what rate will potential ustomers be plaing orders? Obviously. To go further requires onsideration of the motivation for the ustomer's behavior. There are three 5 There is a tehnial aveat that should be added to all of our derivations. We are supposing that the system has existed long enough that we an work with the stationary distribution. 3

5 ases to be onsidered: all ustomers pik s i =, all ustomers pik s i =, some fration of the ustomers pik s i =. First, an it ever be an equilibrium for all of the ustomers to pik s i =? Clearly this is possible, but not espeially interesting. To rule this out we assume that at least when there are no other ustomers and the good is free, the ustomer will hoose s i =, i.e., U(R C(=)) >U(v), where = is the mean servie time. Seond, an it ever be an equilibrium for all of the ustomers to pik s i =? This will depend on the poliy of the rm. If the rm harges a low enough prie, servies the ustomers quikly enough, and the rm's produt is enough more valuable than the ustomer's alternative opportunity, then all the potential ustomers beome atual ustomers of the rm, and =. In this ase we know that the ustomer is making hoies suh that EU(R P ()) U(v), where as usual, E is the expetation operator. The expetation is taken with respet to the delay to be endured by the ustomer if making a purhase. If the ustomer buys from the rm, he aepts some risk sine he does not know how long a wait he faes. When solving the rm's problem, we will nd that the rm will never let the ustomer be stritly better o when buying from the rm. If that were the ase then the rm ould always raise its prots by raising its prie very slightly. Aordingly in equilibrium the inequality will atually be an equality. Third, suppose that if = then the ustomers would have a higher payo from taking the outside opportunity than they expet to get by plaing an order. Then at least some of the potential ustomers will not be joining the queue. The atual rate must be suh as to ause the onsumers to expet equal payos from plaing an order with the monopolist, or from taking the outside opportunity. In other words, the demand funtion is found by solving EU(R p C(w i ())) = U(v): () It is lear that () is a neessary ondition for an equilibrium when some but not all ustomers will be making purhases from the monopolist. In this ase we restrit attention to the symmetri equilibrium in whih all of the ustomers randomize. An important feature of the randomized solution is that it preserves the Poisson form for the atual arrival proess. We now suppose that both U and C are linear funtions. In this ase equation () takes the simpler form R p Ew i ()=v. The next task is to determine at what rate ustomers will plae orders with the rm. We are assuming that the arrival proess of potential ustomers is Poisson, that a proportion of them joins the queue, and that there is a single rm that has an exponential servie time with rate. Given these assumptions, solving for the expeted waiting time is straight forward. It is a standard result from the queueing literature, derived for instane in Wol (989) setion 5-5, that Ew i ()= for <, and Ew i () =for. If the arrival rate exeeds the servie rate, the expeted wait beomes innite beause more and more ustomers keep on getting added faster than they are being sent away with their 4 (2)

6 good or servie ompleted. When the servie rate exeeds the arrival rate then equation (2) holds. The faster the servie, and the fewer the expeted number of ustomers, the shorter the expeted wait. In order for the problem to be of interest it must be the ase that at least some potential ustomers wish to plae orders. Aordingly we assume that R p v. If this were false then no ustomers would plae an order with the rm. Reall that = is the mean servie time, and so = is the ost of waiting if the arriving ustomer does not have towait for anyone else. When we turn to onsideration of the monopoly prie determination we will make the parallel assumption that it is feasible for the monopolist to set a prie that attrats at least one ustomer. In other words we will assume that R v >. Consider the situation in whih not all potential ustomers will atually wish to plae orders with the rm. Combining (2) with the linear version of (), we get 8 >< if p<r v = R p v >: if p R v. (3) This would be the demand urve if the potential arrival rate is suiently large. Sine the demand rate annot exeed that potential rate we have = minf; g: It is lear that if p R v =, then there is no inentive to plae an order. For p<r v =, the interpretation of (3) is quite attrative. The faster the servie oered by the rm, the more ustomers it will be able to sell to. The more impatient the ustomers, the fewer ustomers will buy from the rm. The more valuable the rm's produt, the more ustomers will buy from the rm. The higher the rm's posted prie, the fewer ustomers the rm will get. The higher the value of the alternative good to the ustomers, the fewer ustomers will buy from the rm. For p < R v =, substituting into (2) yields 8 R p v >< Ew i = >: if if < R p v (4) R p v. When there is a large pool of potential ustomers, given a prie, the expeted waiting time is independent of the servie rate of the rm. This observation atually holds for general servie time distributions. To see this fat note that when is large, R p Ew i ()=vfrom whih we an also obtain (4). 6 6 Another simple example is an exponential utility funtion and a linear ost funtion, U(x) = e x and C(w) = w; where >satisfying <. Under the same distributional assumptions, we haveeu(r P()) = e (R p) : Now substituting the above into the ustomer's deision rule, we get a demand funtion 5

7 Reall that the queue is not diretly observable by the ustomers. The monopolist ould hoose to inform the ustomers of their position in the queue. Would it be in the interests of the monopolist to do so? The rst issue is then whether the monopolist would wish to tell the ustomers the truth. If the monopolist is not onstrained to tell the ustomers the truth, he would be tempted to always tell the ustomers that they are very lose to the head of the queue. If the ustomers believed suh laims they would have a relatively low expeted ost of waiting. However, it is not apparent that suh laims should be believed. If laims about position in the queue annot be made redible then it is as if the monopolist is unable to make any laims at all. To go further we simply suppose that the monopolist has some mehanism to make the laims redible, suh as pledging his good name. 7 In that ase will he wish to inform the ustomers of their plae in the queue? It seems obvious that the answer is \no". If the ustomers know their plae in the queue, then only those who get positive surplus, or at least zero surplus will plae orders. If the ustomers do not know their plae in the queue, then plaing an order is like taking a risky gamble. As long as it is at least a fair bet, the ustomer will take it. After the fat, some of the ustomers realize positive surplus while others inur losses due to exessive waiting. Sine those who inurred losses would not have joined had they known the true situation, it seems that there will be a greater level of demand when the ustomers are not informed of their position in the queue. This intuition is only part of the story however. Proposition 2. Let ==. There exists a ritial point suh that if <, there will be at least as many purhase orders plaed when the ustomers annot observe their position in the queue, as when they an observe their position. But this statement is reversed, if >. This proposition is proved in the appendix. The proedure is to derive the demand under eah ategory of ustomer information, and then to ompare the number of ustomers in the two ases. Neither ase an be ruled out as being partiularly implausible. Both > and < are oneivable situations. What this proposition means is that, as long as the ow of potential ustomers is low relative to the speed of servie being oered by the rm, the intuition suggested above arries = minf; (R p g. The demand funtion has exatly the same eonomi interpretation as does v) e the demand funtion with a linear utility funtion, though the rates at whih hanges in response to the hange of R, p, v, and are dierent. The expeted waiting time is Ew i = minf ; ( e (R p v) )g: Again the eonomi interpretation is the same as in the linear ase. 7 The issues of redible versus inredible laims, and reputation building are very interesting. However to get into them here would take uswell away from the main fous of this paper. In a dynami setting we know from the supergame literature that one an onstrut equilibria in whih a reputation for honesty an be sustained provided the horizon is innite and the future is not disounted too muh, for instane see the disussion in Tirole (988). While we an readily onstrut suh examples, we do not think that there would be any further insight to be derived, and so we adopt the simpler approah of Hassin (986) and simply assume the existene of a ommitment tehnology. 6

8 through. However the proposition also tells us that the intuition is inomplete. 8 When the ow of ustomers is large relative to the servie speed of the rm, the answer is reversed. Why is that? Suppose that the ow of ustomers is large relative to the servie speed oered by the rm. In this ase if the ustomer is not being told his position in the queue, the proportion of potential ustomers who atually plae orders is fairly low. Suppose instead that the rm is orretly informing the ustomers as to their plae in the queue. Now if the queue is urrently long and the ustomer is told this, the ustomer walks away. But they were probably going to walk away anyhow. Suppose the queue is urrently quite short. Now the ustomers beomes very likely to plae orders that they would not otherwise have plaed. Bad information (from the ustomer's perspetive) auses little loss of orders, good information auses gains. The timing of these gains is preisely when they are most valuable, when the rm is faing the possibility of under utilizing its faility. If the ow of new ustomers into the market is high enough relative to the servie speed of the rm, then this eet dominates. The next question is to ask how hanges when the basi onditions hange. The answer has a fairly simple intuition. Anything that makes it more likely that the ustomer will plae an order redues the need for the monopolist to say anything. This basi intuition is reeted in the following proposition that is proved in the appendix. Proposition 2.2 The ritial point has the following omparative statis. inreases as R and inrease, and as, p, and v derease. Before losing this setion we wish to emphasize that these propositions are propositions about the eet of information revelation on demand, for a given prie. They are not propositions about protability of information revelation. From Hassin (986) we already know that a simple extension to a statement about protability is not true. The reason for the diulty in extending the result is that, is a funtion of p. Following Hassin (986) we know that for the ase of predetermined servie rates and ustomers with linear preferenes, if (R v)= 2, then the rm will always nd it more protable to reveal the queue length. If that ondition is not satised then there will be a, suh that if < then the rm will nd it more protable not to reveal the queue length, while if > then it will be more protable for the rm to reveal the queue length. 2. Short Run Monopoly Priing In this setion we show how the rm faed with suh ustomers will selet a prie to harge in the short run. This situation is short run in the sense that the servie rate is predetermined and annot be altered by the monopolist. The rm's problem is now max p<r v = = p minf; g: (5) R p v 8 A further qualiation onerns ustomer risk aversion. Telling the ustomers their position in the queue redues one soure of unertainty that they fae. We leave suh further ompliations for future study. 7

9 When is great enough, the objetive funtion is onave in(;r v =), and so from the standard rst order ondition, we nd the optimal priing for the rm p m = R v ((R v)=) =2 : (6) For this solution to make sense we need p m and this happens when (R v). We assume that this always holds. If this did not hold, then the ustomer would be unable to justify spending the neessary expeted time in the queue. When <, the demand funtion may take the form = or the form given by equation (3) depending on the prie. The maximum prie onsistent with having all potential ustomers atually plae an order ( =)is p =R v [ ] + ; (7) where x + = maxfx; g. The notation [ ] + is used here so that the formula an be expressed without expliitly needing to restate the assumption that <. The prie p is the lowest prie that it ould ever make sense for the rm to harge. If the rm hooses to set a higher prie than that given by (7), then it will give up on serving the entire market. If the rm hooses to only serve some of the potential ustomers, the maximization is over (p ;R v =). We will refer to p m as \rst order ondition priing", and p as \market apture priing". The optimal prie is determined by a simple omparison between (p m ) and (p ). If the arrival rate of potential ustomers is low enough, the rm would like to attrat even more ustomers than are oming to market. But that is not possible. In that ase it might aswell harge as high a prie as possible to those ustomers who do ome to market. The highest prie that is \possible" is the prie that leaves eah of the ustomers just indierent between plaing an order and not doing so. This will be the lowest prie that it ever makes sense for the monopolist to harge. 9 Aordingly the optimal prie is given by p = maxfp m ;p g: Substituting the optimal prie into the demand funtion, we an obtain the atual demand in response to the optimal prie: = minf; R v =2 g: (8) The next question is to ask how the prie hanges as,, R,,andvhange. Diret alulation shows Proposition 2.3 When rst order ondition priing is optimal for the monopolist, hanges to the basi onditions in the market produe the following responses. p = R v =2 < ; 2 p v = + =2 <; 2 (R v) p R = =2 >; 2 (R v) p = (R v) =2 p > ; 2 =: 9 Atually one ould say that it is that prie minus a vanishingly small amount so that all ustomers stritly prefer to plae an order. 8

10 When the market apture priing is optimal for the monopolist, hanges to the basi market onditions produe the following responses. p v = < ; p = p = < ; p R =>; ( ) > ; p 2 = ( ) 2 < : Most of the interpretations seem very natural. Inreased ost of waiting auses the rm to ut the posted prie. Inreased value of the rm's good auses the rm to raise its prie. Inreased value of the outside opportunity for the ustomer auses the rm to ut its prie. Inreased speed of servie by the rm allows it to raise the prie that it harges. Finally, as long as there are enough ustomers arriving, the prie that the rm harges is independent of the arrival rate. What limits the rm's ability to raise prie is the rate at whih itan proess the ustomers, and the fat that exessive delay will indue potential ustomers to avoid beoming atual ustomers. Perhaps the most urious of these results is given by p <. It has following interpretation. Suppose that it is stritly optimal for a rm to sell to all of the potential ustomers, so that it is still optimal for the rm to sell to all potential ustomers, if inreases by a small amount. The rm would be better o if there were more ustomers. If the number of potential ustomers does inrease, then the rm will ut the posted prie in order to ontinue to apture all available ustomers. To ontinue to get them all the rm must oset the inreased ost of waiting that the extra ustomers impose on eah other. It is natural to ask how sensitive is the result to the details of our model. If the onsumers have a quantity hoie, as well as a deision of whether or not to join the queue, then the analysis is more omplex. There are osetting eets, and in general the sign of p beomes ambiguous. There is one eet that reets the attempt to apture more ustomers, and another eet that reets the attempt to get more money from eah ustomer. It is then an empirial question whether the inreased demand translates into a prie fall, no eet on prie, or a prie rise. Sometimes Keynesian analysis is haraterized as the analysis of situations in whih prie alone does not lear the market and there may be demand or supply shortages. It is often asserted that at less than \full apaity" any inrease in demand translates into inrease in prodution, while at or above full apaity any inrease in demand would be translated into prie inreases. Here we found somewhat dierent eets. As we have just pointed out it is possible for inreased demand to lead to no hange in prie, or even to indue a prie ut. It should be emphasized that this analysis was all arried out for xed servie rate. In the long run we would expet adjustments to the servie apabilities to ome into play, as they do in setion 5. The seond aveat is that our analysis is all partial equilibrium. To get into the various impliations of general equilibrium analysis would go well beyond the sope of this paper. 9

11 2.2 Choosing a Servie Rate In this subsetion we turn to the long run problem faing the monopolist. The example is extended to allow the rm to hoose the rate of servie to oer. We let q> be the marginal ost of inreasing the speed of servie, and r is the marginal ost of the atual prodution. For simpliity we do not have any xed osts. The optimization problem takes the following form: max p; s.t. (p; )=(p r) q = minf; R p v g; r<p<r v =; : The objetive funtion (9) is not onave. If the rm is to be viable it is lear that it must over the physial osts of prodution and so p>r. If the rm is to have any ustomers, it must be the ase that p<r v =. Aordingly if the problem is to be of interest it must be the ase that r<r v =. The objetive funtion is bounded by (9) (p; ) (p r q) (p r) R p v : () Suppose that R v = r q. The seond term on the right-hand-side of () is nonnegative. Realling that p<r v =, and using () we see that (p; ) < ([R v =] r q): If the oeient onis less than or equal to, it is optimal to set =. Operation would yield negative prot. For the rest of this subsetion, we will onsider more interesting ase, R v = r>q. Whatever the ustomer arrival rate, the monopolist will selet a proessing rate to aommodate all of the potential ustomers. We an distinguish innite and nite potential ustomer arrival rates. Suppose that =. In this ase, if we take p = R v =, then for > small enough, p r q>. It is therefore lear that the optimal servie rate is =. Next suppose that <. If the rm hooses to operate and so >, we laim that == must hold. Why is that? Suppose that this were not true, so that R p v = =(R p v) <. The objetive funtion is exatly the same as the right-hand-side of (). The optimal p must be larger than r + q as otherwise the objetive funtion beomes negative. Hene it is desirable for to be as large as possible. Therefore, the optimal must be =(R p v) = as laimed. With onstant returns to sale in the hoie of servie rate, the marginal ost of selling to a ustomer is onstant. The marginal revenue is also onstant up to the point when all the ustomers are being served, due to the idential unit demand assumption. If the marginal ost exeeds the marginal revenue, the rm is not viable. If the marginal revenue exeeds the

12 marginal ost, the monopolist would like to sell an innite amount. If there are an innite number of ustomers arriving at eah moment, he does so. If there is only a nite arrival rate, then that determines the rate at whih the monopolist will hoose to sell. This allows us to simplify the optimization problem to max r<p<r v = (p r) q( + R p v ): The objetive funtion of this problem is onave, so using the rst order ondition, we nd the optimal solution. The solution to the monopolist's problem has p = R v (q=) =2 ; and =+(=q) =2 : The rm will hoose to operate if and only if (R v q r) 2(q) =2 > : () This an be veried by substituting the values for (p ; ) bak into the objetive funtion and heking for nonnegative prots. While the interpretations of p and seem quite sensible, two elements are worth omment. First, the physial osts of prodution do not aet the prie that the monopolist harges in the long run. Neither does it aet the optimal speed of operation. It does aet whether the rm should hoose to operate or not. This feature of the solution is really driven by the assumption that the ustomers demand either zero or one unit from the monopolist. The same sort of thing happens in monopoly priing without queues, but with unit demand ustomers (see exerise.2 in Tirole (988)). The seond aspet that is worth omment onerns the reation when there is an inrease in the number of potential ustomers oming to market. It auses the rm to add apaity and to raise the prie harged. Further, >. That is to say the optimal speed to proess the ustomers is greater than the rate at whih potential ustomers are arriving in the market. The reason for this is that if they were equal and all ustomers plaed orders, the expeted wait would grow to innity. So not all ustomers would be willing to plae order, and so some potential sales would be being lost. By onsidering the partial derivatives of the left-hand-side of (), we know that if a rm hooses to operate, it will still hoose to operate, when R and inrease, and when v, and q derease. When a rm hooses to operate, the optimal servie rate inreases as and inrease and as q dereases, and the optimal prie inreases as R and inrease and as v, and q derease. 2.3 Soial Welfare When Customers have Linear Preferenes Soial welfare is dened to be the sum of produer and onsumer surplus. Given the presene of market power, one might be inlined to expet the equilibrium to be soially ineient. However Edelson and Hildebrand (975) derived the surprising result that the monopoly prie

13 is also soially eient when the ustomers annot observe the length of the queue before plaing their orders. They all this ase the \no balking" ase. When the ustomers an observe the queue length (\balking" permitted) soial eieny is not obtained by monopoly priing. We illustrate the welfare equivalene result for our model, and then extend it to show equivalene in the hoie of servie rate. The onsumer surplus per unit of time for a given arrival rate and prie p is CS = [EU(R p C(w())) U(v)].The produer's surplus in this ase is the same as the rm's revenue and so it is given by PS = p. The soial welfare is SW = CS + PS, or equivalently, SW =[EU(R p C(w())) U(v)+p] (2) The soial planner's problem is to maximize SW over p and subjet to the onstraints that p and. To ahieve this maximum, it is lear that <must hold beause C() =. With linear preferenes, it is lear from (2) that SW =[R Ew() v]=[r v ]; for <and otherwise, SW =. As one might expet, in this ase soial welfare does not depend on the prie, it only depends on the alloation. The prie is a pure transfer. It an be diretly veried that SW is a onave funtion of. Hene the soial optimal demand rate, is given by the rst-order ondition = minf R v =2 ; g: The reason for the \min" in this expression is to reet the presene of the onstraint. The following result was derived by Edelson and Hildebrand (975) in following up an earlier equivalene result of Edelson (97). Proposition 2.4 (Edelson and Hildebrand (975) Equivalene) The soial welfare maximizing solution, is the same as the monopoly solution for in (8). DeVany (976) suggests that the monopolist supplies too little output for soial eieny. This seems to be driven by his assumptions onerning the ustomer's problem. There are some unusual features of the ustomer's deision making problem in DeVany (976). The ustomer arrival rate depends on the posted prie for reasons that are never explained. It is apparently not due to the ustomers' taste for the good, nor is it due to the ustomer's opportunity osts, sine these features are expliitly aounted for elsewhere in the analysis. This extra role of the posted prie appears to be double ounting, and it does aet the onlusion to be reahed about welfare. Perhaps more minor diulties are:. If the arrival rate is to depend on a prie, why the posted prie rather than the expeted full prie? 2. When the ustomer nds an unusually long line at the monopolist, he is supposed to go to another rm with a stohasti queue. The length of the seond queue is assumed to be unorrelated with the rst queue despite the fat that the postulated behavior of the ustomers reates just suh a orrelation. 2

14 Next onsider the long run in whih the rm and the soial planner are both permitted to hoose the servie rate as well as the prie. When the servie rate is not predetermined, the rm's prot is PS =(p r) q, where r is the marginal ost of the atual prodution and q> is the marginal ost of inreasing the speed of servie. The soial welfare in this ase is SW =(R v r =( )) q; (3) whih again does not depend on the prie. The soial planner's problem is to hoose and subjet to and <, suh that SW is maximized. We note that SW is onave in >although it is not onave jointly in and. Letting SW= = yields Substituting (4) into (2) yields = +(=q) =2 : (4) SW =(R v r q) 2(q) =2 : (5) The soial welfare funtion SW given by (5) is a onvex funtion of and so maximization of (5) subjet to must be ahieved at either =or=. By omparing the values of SW at = and =,we nd that the optimal = if (R v r q) > 2(q) =2, and = otherwise. Substituting =into (4) would yield =, and substituting = would yield =+(=q) =2. Proposition 2.5 The soial welfare maximizing solution is to set = =if (R v r q) 2(q) =2, and to set =and =+(=q) =2 if R v r q>2(q) =2. This is the same as the monopoly solution in subsetion 2.2. This result ontrasts sharply with DeVany (976), the only previous results on monopoly servie rate hoie that we know of in the literature. In DeVany (976) the monopoly hooses too little of, what he terms apaity, and what we term servie rate. This result shows that the Edelson and Hildebrand (975) welfare equivalene does diretly extend to the long run situation in whih the rm has a hoie of servie rate. 3 The More General Customer Problem Having worked out the example of linear ustomer preferenes, it is natural to ask, to what extent do these results extend to more general forms of ustomer preferenes? The rest of the paper is direted at answering this question. We will show that the omparative statis are mostly unaeted, but the soial welfare results are sensitive to the linearity assumption. Now onsider the more general ase of ustomer demand. We no longer restrit U(x) =x. We still assume that U >, and, U. It is lear that if R p C(=) v, or equivalently, p R C(=) v, there is no inentive for a ustomer to plae an order and 3

15 hene the demand rate is zero. To avoid this trivial ase, through this setion we will only onsider the ase p<r C(=) v. Reall that the servie time is exponentially distributed with a mean servie rate. When the servie rate is stritly greater than the arrival rate, the stationary waiting time is exponentially distributed with rate ; otherwise, the expeted waiting time is innite. Sine no one would like to wait for an innite amount of time, it is lear that <must prevail for the ustomer's problem. When >,wehave Z ueu(r p C(w i ())) = ( ) = U(R p) Z U(R p C(x))e ( )x dx (6) U (R p C(x))C (x)e ( )x dx; (7) where prime is used to denote a rst derivative. Equation (7) is obtained through integration by parts. In the above, we assumed the expetation exists. In partiular, we suppose lim x! U(R p C(x))e ( )x =: The demand funtion is given by = if and only if Z ( ) U(R p C(x))e ( )x dx U(v): (8) Inequality (8) implies that >. When, not all potential ustomers will be served in this market. The general form of the demand funtion is summarized as follows. Proposition 3. Let = (R; p; v; ) be the unique solution from equation () with EU(R p C(w i ())) given by (6) and (7). Then the demand funtion is = minf; (R; p; v; )g: It should be pointed out that if we take C(w) =w, then is also a funtion of. Proposition 3.2 Case. queue. Then Suppose that not all the ustomers will be hoosing to join the R = R > ; p = p < ; v = v < ; = =: Case 2. Suppose that all the ustomers will be hoosing to join the queue and that (8) holds with equality. Then inreasing or R, and dereasing, p, orvwork as above. Case 3. Suppose that all the ustomers will be hoosing to join the queue and that equation (8) is a strit inequality. Then =; R = p = v = =: In a tehnial sense these are ases in whih the left derivatives are not equal the right derivatives. 4

16 For the derivation of this proposition see the appendix. It an also be shown that, is inreasing and onave in R, and is dereasing and onave in p. Unlike the linear problem we do not have an expliit dependene on beause we do not have an expliit funtional form to work with. For many reasonable waiting ost funtions, suh as C(x) = x, it an be veried in a manner paralleling the above analysis that the higher the waiting ost (the larger the value of ), the lower the demand rate. The interpretation of this proposition is quite natural. In ase not all ustomers will be joining the queue. We nd that the higher the value of servie, the higher the demand rate. The higher the prie harged for servie, the lower the demand rate. The higher the value of the outside opportunity, the lower the demand rate. The higher the rate at whih the rm an proess orders, the higher the rate at whih ustomers will plae orders. Case 2 is just the boundary. Whih way the results go depends on whih of the other two ases the system moves to. Case 3 is a very favorable market for onsumers. The rm ould learly improve prots by raising the prie, and so we would not expet suh a situation to arise. 4 Privately Optimal Priing by the Monopolist Having haraterized the demand behavior of the ustomers, we are now in a position to analyze the deisions of the rm that would like to make a prot by selling to these ustomers. In this part of the paper we suppose that the rm has been exogenously endowed with a proessing rate at no ost, and that there are onstant osts of prodution whih for simpliity are set equal to zero. This ase an be interpreted as short run analysis in whih the proessing rate is predetermined. In setion 5 we analyze the long run in whih the monopolist hooses. The rm's problem is max (p) =p (9) p s.t. p<r v C(=); p>: Depending on whether inequality (8) holds, either =, or else is determined by equation (). One again we will have the distintion between rst order ondition priing and market apture priing. First onsider the ase when inequality (8) does not hold so that is given by (). The rst order ondition for optimality is The seond order suient ondition is +p =: (2) p 2 p + p2 : (2) p2 In the appendix we show that the seond order ondition is satised. 5

17 Next onsider the ase when inequality (8) does hold for p = R v C(=). In this ase, the rm has the hoie of selling to all potential ustomers or of harging a high enough prie that only some fration of the ustomers plae purhase orders. If the rm hooses to take the whole market ( =)we again all this market apture priing. In this ase it is lear that the optimal priing p is uniquely determined by hoosing that value of p that sets ondition (8) to be an equality. If the rm deides not to sell to all potential ustomers, then is given by (). As a result the optimal prie an be determined as in the rst ase, by the rst order ondition (2). Let p m denote this optimal prie. Overall, whether the rm should hoose to take the whole market depends on whether (p ) (p m ). Given that is from () the objetive funtion is onave. Aordingly it inreases when p p m and dereases when p p m. On the other hand, it is lear that the optimal prie must be no lower than p. Reall that at p, the rm an take the whole market. Therefore, if p m p, then p m is the overall optimal prie, otherwise, p is optimal sine (p) dereases for p p.to summarize, the optimal prie is p = maxfp m ;p g: Having haraterized the monopoly prie, we now turn to onsider how the monopoly prie p varies with all other parameters. The omparative statis are broken into two ases depending on whether (8) holds or not. Proposition 4. When rst order ondition priing is optimal for the monopolist, hanges to the basi onditions in the market produe the following responses. p R > ; p v < ; p > ; p =: When market apture priing is optimal for the monopolist, hanges to the basi onditions in the market produe the following responses. p R > ; p v < ; p > ; p < : Overall we see that the same basi eets arise here as were found in the linear ase. When the waiting ost funtion is C(x) = x, it an be veried similarly that the higher the waiting ost, the lower the prie that the rm harges. 2 5 The Firm's Choie of Servie Rate Having studied the situation in whih the servie rate is predetermined, we now turn to a onsideration of the hoie of servie rate. There are three types of ost faing the rm. There is a onstant marginal ost of speeding up the servie rate, q. There is a xed ost F. 2 Wehave not presented the ase when p m = p. In this ase, the left partial derivatives do not agree with the right partial derivatives; some are the same as the rst ase, while others are the same as the seond. As a result it is similar to the disussion in setion 3 when (8) holds with an equality. 6

18 The ost of prodution is a onstant marginal ost r. Some disussion of nonlinear ost of speeding up servie is inluded at the end of this setion. Now the rm's problem is max p; (p; )=(p r) q F (22) s.t. r<p<r v C(=); ; where = minf; g is given by Proposition 3.. First, note that (p; ) (p; ) := (p r) q F: Using Proposition 3.2 yields = p r q: (23) Reall the onstraint p < R v C(=) in (22). If R v C(=) r q, the partial derivative (23) is less than or equal to zero and so it is optimal to hoose =. Therefore, we will assume that R v C(=) r>q. As in the linear ase, it is easy to see that the optimal servie rate is =, if=. Proposition 5. Suppose that <. If the optimal >, then ==. In other words, if the rm hooses to operate, then it is optimal for the rm to serve the whole market. Suppose to the ontrary, the optimization problem (22) is equivalent to the one with an objetive funtion dened above. At the optimum we must have p r q > ; otherwise < sine <always holds. It follows from (23) that = >. In other words it is desirable to make as large as possible. While inreasing also inreases, the optimal must make =. As a result of this proposition, the optimization problem simplies to max r<p<r v C(=) (p r) q F; where = (R; p; v; ) is a funtion of p; and it is found by solving (R; p; v; ) =. It is shown in the appendix that 2 p = 2 : (24) 2 p2 As a result we know that the objetive funtion is onave, and so the rst order ondition gives the optimal prie. Proposition 5.2 () If the rm hooses to operate, the optimal prie p is the unique solution to q p =; (25) 7

19 and the optimal servie rate is given by = (R; p ;v;), where p = p (R; v; ;q) and = (R; v; ;q):=(r; p (R; v; ;q);v;). The optimal prie p inreases as R and inrease and as v and q derease. The optimal servie rate inreases as and v inrease and as q dereases, and remains unhanged as R varies. (2) The rm will hoose to operate if and only if (p (R; v; ) r) q (R; v; ) F>: Inreasing R and, and dereasing r, v, q and F all make it more likely that the rm will hoose to operate. The proof of the proposition in the appendix. This proposition shows that the major results found in the linear ase are not exeptional. The interpretations are natural. In the long run, inreases in the value of the good, and in the number of potential ustomers oming to market both trigger inreases in the monopoly prie. The more valuable the outside opportunity of the ustomers, the lower the monopoly prie. The more expensive it is to speed up servie, the lower the posted prie. The more expensive it is to speed up servie the slower the servie will be. An inrease in the rate of potential ustomers oming to market indues a more than proportionate inrease in the servie rate the monopolist will hoose. This last observation, as with the others are diret generalizations of what was found for the linear ustomer ase. As in the ase of ustomers with linear preferene, the optimal level of the prie and the servie rate, do not depend on the prodution ost r. Only the deision whether or not to operate depends on r. We again aution that, this independene is driven by the unit demand ustomer assumption. In this setion of the paper we have inluded various osts of operating and have allowed for some generalizations on the demand side of the model. However, there are still further generalizations that are possible but not analyzed here. Three of these in partiular an be mentioned: the ost of inreasing the speed of proessing orders ould be nonlinear, the servie rate distribution might not be exponential, ustomers might not have idential unit demands. The ost of inreasing the servie rate ould be nonlinear. Let q() be the servie rate ost funtion. If for all >, R v = q (), then it is optimal not to operate. If R v = > q () for all >, then the same result as the above holds. A suient ondition for having this ase is that q is onave and q () <R v =. The ase when R v = > q () holds only for some > is more ompliated. The next issue onerns the distributional assumption for the servie rate. In our analysis we have assumed that the servie rate is exponential. This is a ommon assumption to make, but it is not the only possibility. Some of the results in this subsetion ould be sensitive to this distributional assumption. The reason for this is that in general, we do not always have = =. We do not pursue this tehnial issue any further here. The nal issue is the maintained hypothesis of idential unit demand ustomers. While this type of ustomer demand is widely used, as we have already pointed out, it is a somewhat speial ase. It has several advantages for our purposes. It makes the struture of the problem partiularly similar to the standard queueing framework. It also allows diret omparison 8

20 with Naor (969) and with Edelson and Hildebrand (975). In our analysis unit demand ustomers are onvenient, sine they permit us to use the number of ustomers arriving in the market as a diret measure of the potential sales for the monopolist. Furthermore the unit demand assumption is quite a reasonable approximation for some ases, possible examples inlude purhases of airplanes, ars, pianos or other high value disrete, durable goods. More generally the number of units demanded will depend on the posted prie, and perhaps even on the time spent in the queue in some ases. An important related issue onerns ustomer heterogeneity. Inmuh of the analysis, the results will arry over if we replae the idential ustomer, with the mean of an appropriate distribution. However, not all results will neessarily generalize so diretly. We leave all these ompliations for future study. 6 Soial Welfare When analyzing soial welfare in the ase of ustomers with linear preferenes, we found that monopoly priing maximized soial welfare. To gobeyond the ase of ustomers with linear preferenes beomes analytially messy. The entral question that needs to be answered by suh an extension is, whether Edelson and Hildebrand (975) equivalene ontinues to hold one we movebeyond linear speiations? The answer to this question is, \no". To see this we diretly illustrate the point using pieewise linear preferene funtions. Some expressions for more general preferenes are set out in the appendix. They are messy and do not seem to oer further insight. The entral idea needed to understand the welfare properties of monopoly priing in our ontext, onerns the presene or absene of onsumer surplus in an equilibrium. In any equilibrium for the market the onsumer's surplus is zero. Why will an equilibrium have no onsumer surplus? An equilibrium onsists of a posted prie by the monopoly, and a purhase-no purhase deision by eah potential ustomer. Suppose that there was positive onsumer surplus at a andidate equilibrium. If there were more potential ustomers, then at least one of them (or more rigorously a small positive proportion of them) would hoose to join the queue expeting a positive surplus. If there were no more potential ustomers, then the monopolist ould stritly inrease his expeted prots by raising the posted prie by at least some small >. Doing so will ost him no lost sales, and will give him extra revenue on eah sale. So for a andidate solution to be an equilibrium there must be no onsumer surplus. While there will be zero onsumer surplus in the market equilibrium under our assumptions, the same will not normally be true of the soially optimal solution. Hene in general there is no equivalene between soial optimality, and monopoly priing with ustomers who queue. We use the tratable ase of pieewise linear preferenes to make this point. We now suppose that the ustomer's utility funtion takes the form U(x) = ( x if x A A + a(x A) if x A, 9 (26)