Transcription

1

2

3 1 Introducton Leasng contracts are extensvely used n durable goods markets. A thrd of the captal equpment n US corporatons s leased. In the automoble market the mportance of leases has been growng and has reached the pont where n 1996, one out of three new cars had been leased. If one consumer leases and another buys, should we expect them to behave derently n secondary markets or are the two consumers smply nancng a smlar utlzaton of a durable good n a derent way? Evdence n the car market, shows that buyers and lessees behave n a derent manner. Frst, the turnover of leased cars s hgher than that of sold cars. Second, o-lease used cars seem to be of better qualty than pre-owned cars of the same vntage. Evdence of the rst phenomenon s partcularly strong. Lessees buy ther cars at maturty only 25% of the tme. Snce most car leases have a two to three year duraton, ths means that a large fracton of leased cars are sold n the used market by the tme they are three years old. 1 In % of the so called premum used cars, whch nclude two to four year old cars, were o-lease cars. But back n 1993 only one out of four new cars was leased. For leased cars to account for 42% of trades n 1996, whle only beng 25% of the stock n 1993, they must have a propensty to be traded wthn the rst four years whch s 117% larger than that of sold cars. Further evdence s provded by Sattler (1995) who reports that 56% of 1989 car models n hs sample were held by ther rst owner ve years later. Moreover, the average length of perod that all new cars (leased and bought) are held before changng hands s almost 6 years. Ths means that new car buyers hold on to ther cars much longer than consumers who lease. Evdence that o-lease cars are better qualty s a lttle weaker but stll suggestve. In the automotve press t s a common ndng. For nstance, [T]he ndustry-wde assumpton [s] that the most desrable used vehcle s the consumer o-lease varety... 2 Moreover, advertsements of used cars for sale often specfy whether a car s o-lease. Ths would hardly be hghlghted f the percepton was that o-lease cars are no better than sold cars. Fnally, 1 Polk's analyss estmates 2.88 mllon o-lease vehcles returned to the used car market n 1997, 87 percent beng two- and three-year-old vehcles. In fact, nal 1997 numbers should show o-lease vehcles representng 65 percent of two-year-old vehcles on the market, and 57 percent of three-year-old vehcles. Source, Polk Corporaton: 2 Polk Corp, Press release

4 Desa and Puroht (1997), usng aucton data for a popular car model, nd that o-lease cars sell at a premum or, more precsely, that the prce declne s slower than for sold cars. How can we explan these phenomena? Can manufacturers benet from approprately desgnng leasng contracts? What are the consequences for socal welfare? We present a model to address these questons. A leasng contract, beyond specfyng the rental payments for the good, also speces the opton prce at whch the used good can be bought at maturty. Ths opton prce can be set ndependently of the prce n the used market and provdes an addtonal control varable for the manufacturer of the good. If the opton prce s set above the market clearng prce n the used market, lessees choose whch cars to keep on the bass of the opton prce. We show that n the absence of asymmetrc nformaton the addtonal control varable s useless; market allocatons and prots for manufacturers are the same for any menu of leasng contracts. We then ntroduce the possblty ofadverse selecton n the used market. We buld on the model of adverse selecton n durable goods markets presented n Hendel and Lzzer (1997b). The key ngredents of the model are the followng. Consumers are long lved and have heterogeneous valuaton for qualty. Cars are produced every perod and they deprecate. Thus, hgh valuaton consumers favor new cars and low valuaton consumers favor used cars. Ths generates the possblty of exchange n the second hand market. When there s adverse selecton, then leasng aects equlbrum allocatons n the market. We show that when leasng and sellng contracts are oered smultaneously on the market, then consumers who lease are hgher valuaton consumers. Thus, oerng both contracts serves to segment the market. Moreover, the percentage of o-lease cars that are returned at maturty s hgher than the percentage of pre-owned cars that are traded and the o-lease cars have hgher average qualty. These predctons of the model match the emprcally observed derences n behavor between lessees and buyers. We then nvestgate the welfare eects of leasng contracts. We show that a socal planner can use leasng contracts to amelorate the welfare dstorton caused by adverse selecton. 3 However, except for the case where there are only two types of consumers, no menu of leasng contracts can acheve even the second best allocaton. Leasng contracts are good tools to 3 Ths pont s related to the analyss n Guha and Waldman (1997) whch s dscussed n more detal later. 2

5 control the keepng behavor of new car consumers and therefore to adjust the volume of trade n the used car market. However, they cannot deal well wth the dstorton n the allocaton of used goods among used good buyers. Solvng ths dstorton requres that ner nformaton about the qualty of the used goods be obtaned from the rst users. A menu of leasng contracts s not capable of dong ths. We show that there exsts a mechansm that completely solves the adverse selecton problem; the rst best allocaton can be acheved by an ncentve compatble, ndvdually ratonal, and budget balanced mechansm. We then consder the ssue of the optmal choce of leasng contracts by a monopolst. We present an example that has the strkng feature that the manufacturer can rase the opton prce above the market clearng prce n the used market and ncrease ts prots wthout aectng the equlbrum allocaton n the market. The reason the monopolst prots from ncreasng the opton prce s the followng: Under sellng the opton of keepng s mplctly prced by the market clearng prce n the used market. Wth adverse selecton, ths prce reects the fact that the traded used good s of lower qualty than the good that s kept by the new car buyer. Thus, the new car buyer gets to keep a hgh qualty good that s prced as f t werealow qualty one. Leasng allows the manufacturer to rase the opton prce thereby reducng the compettve threat that the used good poses to the new good. 4 We go on to show that ths allows the manufacturer to protably expand output and that manufacturers wth unrelable cars benet more from leasng. Gven the ablty to control behavor n the used market, and gven the fact that adverse selecton s commonly perceved to have negatve eects on market allocatons, t s natural to ask why leasng contracts often nclude the opton of buyng the good at the end of the lease. If ths opton were very expensve, all used goods would be returned resultng n a pool of used cars that does not suer from adverse selecton. We show that allowng some lessees the opton of keepng the used good s the optmal polcy for the manufacturer despte the 4 Our analyss s thus n contrast wth McConnel and Schallhem (1983) who argue that In many cases leases grant the lessee an opton to purchase the leased asset at ts 'far market value' at the maturty date of the contract. However, the lessee can purchase the asset at ts market prce at maturty of the lease whether or not the contract contans such an opton. Thus, an opton to purchase the asset at ts far market prce s valueless and the equlbrum rental payments wll be the same whether or not the lease contract contans such an opton. The contrast s due to the fact that we study a world wth asymmetrc nformaton. 3

6 fact that lessees who have prvate nformaton about the qualty of the good wll only return the worse qualty cars. Ths s due to the followng phenomenon: If no keepng opton s allowed, all used car consumers purchase cars wth the same average qualty. But some of these consumers have valuatons for qualty that are almost as hgh as those of the lowest valuaton consumers who consume new goods, whereas other used car consumers have much lower valuatons. Thus a menu of leasng contracts, where one of the contracts ncludes an opton for keepng allows for a better segmentaton of the market. 2 Related Lterature Several roles for leasng have been suggested n the durable goods lterature. Bulow (1986) shows that leasng can be used by a monopolst to overcome the Coasan tme nconsstency problem. Waldman (1996) and Hendel and Lzzer (1997a) show that manufacturers may choose to lease because ths gves them addtonal market power n the used market. The monopolst exercses ths power by scrappng some of the used unts. These papers provde ratonales for leasng but do not contan analyses of adverse selecton and have no predctons on how unts that were sold and unts that were leased der n the secondary market. There s a large lterature n nance that focuses on the valuaton of leasng contracts. Most of ths lterature takes as gven the structure of the leasng contract and obtans the equlbrum lease value and the rental rate for a wde varety of leasng contracts. 5 There s also a lterature that addresses the tax ncentves for leasng. Tax ssues cannot fully explan the eects of leasng n the car market because, whle the tax advantages of leasng were lowered n the tax reform act of 1986, the mportance of leasng has ncreased. 6 Smth and Wakeham (1985) provde the most extensve analyss of the determnants of corporate leasng polcy. Ther analyss s nformal but very nsghtful. Some of the ssues that they consder are the followng: Repossessng an asset s easer for a lessor than for a secured debtholder n the event of bankruptcy. Thus, rms that have problems obtanng nancng mght be expected to lease. Sharpe and Nguyen (1995) present evdence that rms lkely to face hgh nancal contractng costs lease a hgher proporton of ther captal equpment. Ths 5 Examples are papers by Grenader (1995, 1996) and McConnel and Schallhem (1983). 6 See Sharpe and Nguyen (1995) for a dscusson of ths. 4

7 s an mplausble explanaton for the car market because t s unlkely that t can account for the observed hgher turnover of leased cars as opposed to sold cars. 7 Smth and Wakeham also argue that leasng mght befavored f the lessees plan to use the equpment for less than ts useful lfe and the lessor has a comparatve advantage n dsposng of the asset. However, they acknowledge that ths transacton cost explanaton for leasng s problematc snce t requres that the comparatve advantage of the manufacturer not be avalable under sellng. Many manufacturers (ncludng car manufacturers) however, allow dealers to accept trade-ns thereby makng avalable to buyers the same advantage that s avalable to lessees. Smth and Wakeham also dscuss the ratonales for a number of common provsons n leasng contracts. The most relevant to our analyss s ther dscusson of optons to purchase at the end of the lease. They argue that ths provson serves to gve an ncentve to the user to take care of the asset. We show that there s also a market segmentaton reason for the exstence of such provsons. Guha and Waldman (1996) s the only other paper that deals wth leasng and adverse selecton. The basc envronment they model s qute smlar to ours. The focus of ther paper s to show that leasng solves the lemons problem; leasng contracts oered by a compettve ndustry would lead to ecent allocatons. Most of ther analyss s concerned wth an envronment where there are two types (qualtes) of used cars and two types (valuatons for qualty) of consumers. Under addtonal assumptons, n ths world the socal optmum and the equlbrum outcome n a compettve ndustry are the same and they nvolve the absence of a keepng opton n the leasng contract; all users return the car at the end of the lease. They recognze that ths s a specal case and dscuss an example wth three types of cars where the socal optmum and the equlbrum outcome n a compettve ndustry stll concde but they both allow lessees to keep the hghest realzaton of qualty of the used good. As mentoned above, we show that leasng contracts can acheve a rst best allocaton only n the case where there s a sngle type of used car consumer. When there are many types of used car consumers there s a msallocaton of qualty n the secondary market 7 Because of the hgher turnover, consumers who lease spend more money on average than consumers who buy. Thus, t s unlkely that lessees are those who have hgher nancal contractng costs. 5

8 whch cannot be resolved through leasng contracts. We show that there exsts an ncentve compatble, ndvdually ratonal and balanced budget mechansm that mplements the rst best allocaton. Thus, n general, leasng contracts cannot even acheve the ncentve ecent allocaton. Other derences wth Guha and Waldman are the followng: Our analyss shows how rms wth market power can benet from the approprate desgn of leasng contracts. We show that sellng and leasng contracts can both be oered smultaneously n equlbrum, and how ths mxture generates outcomes that match observed patterns n the car market. In our paper the mx arses solely out of market segmentaton n the presence of adverse selecton. Guha and Waldman show that a mx of sellng and leasng can arse n a world where n addton to adverse selecton there s moral hazard on mantenance and some consumers have lower costs of mantenance. We wll hghlght the derences wth ther analyss at several ponts n the paper. Laont and Trole (1996) study the problem of nducng the rght amount of polluton and nvestment n polluton abatement. Despte the derence n the topc, ther model has some smlarty wth the one that we study. They look at a two perod problem where the second perod prvate nformaton s correlated wth the rst perod prvate nformaton and there s an nvestment n the rst perod. They show that the optmal mechansm nvolves an oer of two menus of opton contracts dependng on whether the agent decdes to nvest. In ther model the allocaton of polluton s not ex post ecent, whle n our model the allocaton n the optmal ncentve compatble mechansm s ex post ecent. The man derence between the models s the exstence of a second hand market, that s a crucal aspect of our model. Ths derence turns out to be mportant. 6

9 3 The Model We assume that there s an nnte horzon. Tme s dscrete, and there s a unt mass of consumers who lve forever (no new consumers are born). Preferences resemble those n Mussa and Rosen (1978). In any gven perod each consumer demands at most one unt of the good, a consumer of type consumng a sequence of unts wth qualtes (q 0 ;q 1 ;:::) at prces (p 0 ;p 1 ;:::) obtans utlty u(; q; p) = 1 t=0 t (q t, p t ) t where t s ether zero or one dependng on whether he consumes n perod t or not. We assume that consumers are heterogeneous and we represent the dstrbuton of consumer tastes by the cumulatve dstrbuton functon F :[; ]! [0; 1]. Wth the excepton of the example n secton 7 and some of the dscusson n secton 6, we assume that F () s strctly ncreasng and contnuous. Consumer heterogenety s essental for the used markets to have an allocatve role (no trade would be possble f consumers were homogeneous). The value of s assumed to be unobservable so that tradng s anonymous. Goods last two perods. We denote by v the qualty of a new good and by w the qualty of a used one. Both v and w are assumed to be random varables. However, snce the realzaton of v wll be rrelevant, we shall smply denote by v the expectaton of the qualty of the new product. Let G() be the dstrbuton of w wth support [w l ;w h ]. We assume that G s strctly ncreasng and w h v (the good deprecates). Denton 1 We denote by z the type such that 1, F ( z ) z. Thus, z s the type such that there s a mass z of consumers wth hgher valuaton; for example, 1=2 s the medan type. Goods can be sold or leased. Only new goods are leased and the length of a lease s one perod. There are potentally many leasng contracts. Each contract s characterzed by the rental prce P L and the prce of keepng the good at the end of the lease (or opton prce) P k. The prce at whch a new good s sold s denoted by P n. A specal case of a leasng contract s n fact the sellng contract. If the prce of the opton of keepng s set at zero (P k =0), the rental prce P L s n fact a sellng prce. All cars that are returned at the end of the lease are then sold on the used market. 8 Because of the presence of asymmetrc nformaton, the 8 Waldman (1997) and Hendel and Lzzer (1997a) show that a monopoly manufacturer may have an ncen- 7

10 used cars that are returned by consumers who chose derent contracts are derent goods. The prce of a used good that was leased under contract s denoted by P u. We assume that the type of contract s observable by used good buyers,.e. the latter know whether they are buyng a good that was leased under contract or contract j. A customer who consumes a new good at date t knows the realzaton of w at date t +1 for that unt of the good. A buyer who decdes to buy a used good does not know the value of w for that good. We assume that there s an nnte number of consumers of each type so the realzed dstrbuton of qualtes w s G(). 9 We shall denote by y the output of new goods. We shall only analyze steady state equlbra. The mass of consumers who get to consume ether a new or a used product at any date s at most 2y. We wll rst x the menu of leasng contracts and/or y to determne equlbrum behavor n the market. We then dscuss the ncentves for a manufacturer and a socal planner to optmally choose output and the structure of contracts. Remarks We analyze the case of many possble leasng contracts for two reasons. Frst, we want to capture the derences observed n the car market between the behavor of lessees and the behavor of buyers. In the buyng/leasng case we beleve t s realstc to assume that used car consumers can dstngush between the pre-owned and o-lease unts. As dscussed n the ntroducton the automotve press does hghlght the derences between o-lease cars and pre-owned cars. Moreover, used car buyers can easly dstngush between the two types of used cars by checkng the ttle of the car. The second reason to be nterested n the case of many contracts s to explore the welfare eects of adverse selecton and the role of leasng contracts n ameloratng the dstortons. We want to nd out what allocatons can be mplemented through leasng contracts thereby dscoverng ther lmtatons. For much of the analyss the reader can, wth no loss, thnk of the case where there are only two contracts: leasng and sellng. The analyss of the examples only focuses on these contracts. To focus on the eects of asymmetrc nformaton we gnore the tme consstency problem faced by manufacturers of durable goods. Ths would ntroduce an addtonal dmenson tve to scrap some of the used unts. We gnore ths ncentve to focus on the ssues at hand. 9 Ths s lke a jont dstrbuton of and another varable dstrbuted ndependently of ; that plays no role n the model, t just captures that there s a densty of each specc type : 8

11 n the derences between sellng and leasng whch would obscure the force that we want to hghlght. Thus, when we dscuss manufacturers' ncentves we shall assume that they have the ablty to commt. When goods deprecate, the commtment outcome can be an equlbrum of the game where a monopolst cannot commt. 10 We neglect the ssue of moral hazard on the part of consumers. Ths s a potentally mportant phenomenon snce the level of mantenance that s chosen by new car consumers may depend on the terms of the leasng contract. A lessee who s facng a hgh opton prce and therefore antcpates not exercsng the opton to buy at the end of the lease mght take less good care of the car. We justfy ths neglect on three grounds. Frst, most cars are stll under warranty by the tme the lease expres. Ths reduces the dsncentve to mantan. Second, as mentoned n the ntroducton, the common percepton s that cars comng o leases are better qualty cars. Ths s consstent wth our analyss and suggests that, f there s a moral hazard problem on the part of consumers, ths problem s of secondary mportance relatve to the forces that we dscuss. 11 The thrd reason we neglect moral hazard s that nterestng phenomena arse even n a world wth pure adverse selecton; ncludng mantenance decsons by the consumer would add unnecessary complexty to our analyss. 12 Results n sectons 4 and 5 descrbe equlbrum allocatons n the market for used goods gven some prces n the new good market. The characterzaton does not rely on any assumpton on the market structure of the producers of the good. We do not deal wth optmal behavor of the manufacturer untl secton 7 where we present a smple monopolst example and secton 8 where we descrbe some features of the optmal menu of contracts under both monopoly and competton. 10 See Bond and Samuelson (1987). 11 Some readers may have an objecton n the opposte drecton; leasng could be a way toovercome the moral hazard problem. Manufacturers are n a better poston to verfy the qualty of the car at the end of a lease. However, f ths were true, t s not clear why manufacturer could not do ths for sold cars that are traded n at dealers. In fact, some manufacturers do certfy some pre-owned cars. 12 Guha and Waldman (1997) contans a dscusson of the ssue of moral hazard under leasng. 9

12 4 An Irrelevance Result Before dscussng adverse selecton, we consder the case where there s no asymmetrc nformaton. Ths benchmark wll allow us to better evaluate the mpact of adverse selecton on the optmal structure of leasng contracts. We show that n the absence of asymmetrc nformaton leasng contracts have no eects, the ablty to manpulate the keepng prce has nether an eect on prots nor on the equlbrum allocaton. Snce the qualty of the used cars s observable an allocaton n the used market must specfy whchtype consumes whch qualty. Wemust therefore dene a functon # :[w l ;w h ]! [; ] that speces that a consumer of type #(w) consumes a used good of qualty w. Denote prces n the used market by a functon P u :[w l ;w h ]!< + that descrbes the prce of every qualty of used good. Used markets must clear for every qualty w. Let us brey descrbe the equlbrum under a sellng contract. 13 In equlbrum, types n [; 2y ] do not buy cars at all, types n [ 2y ; y ] buy used cars, wth lower types consumng lower qualtes, and types n [ y ; ] buy new cars at every date. Thus there s a hundred percent volume of trade n the used market (all used unts change hands); a new good buyer at date t wll be a new good buyer at all dates and therefore never holds on to a used good. In order to see why ths s the case, consder a consumer who bought a new car at tme t and s an owner of a used car when used markets open at date t +1. Keepng the used car at date t +1 (thus postponng the purchase of a new car) s the same as buyng a used car at tme t (keepng the used car has the same opportunty cost as buyng one). Snce the consumer preferred buyng a new car at date t, he must prefer to sell the used car at date t +1 n order to buy a new car agan. To obtan market clearng n the used market for every qualty gven that the supply of used cars s y and the dstrbuton of used car qualtes s G, the functon #() must be ncreasng (hgher types consume hgher qualty) and must satsfy F (#(w)), F ( 2y )=yg(w) 8w 2 [w l, w h ] (market clearng for every qualty w). Equlbrum prces under sellng can be shown to be the followng: P u (w) = Z w w l #(s)ds + 2y w l (1) 13 For detals see Hendel and Lzzer (1997b) Secton

13 Z wh P n = y (v, w h )+P u (w h )+ P u (w)dg(w) (2) w l The expresson for P u (w) s obtaned through ncentve condtons for used car consumers. The expresson for P n can be nterpreted by observng that P n, P u (w h )= y (v, w h )+E(P u (w)) Thus, the extra cost of buyng a new car relatve to the best avalable substtute (the hghest qualty used car) s equal to the extra qualty of the new car (v, w h ), evaluated by the margnal consumer of the new car y, plus the expected dscounted resale value of the used car E(P u (w)). Denote now by ^P L the mpled rental prce of a new car n a sellng-only regme. By equaton 2 ^P L must satsfy: ^P L = y (v, w h )+P u (w h ) (3) Under a leasng contract the manufacturer speces a rental prce P L, and a functon P k :[w l ;w h ]!< + whch determnes the future prce of keepng a car of qualty w. Notce rst that f the manufacturer sets P k (w) =P u (w) then the equlbrum s the same as under sellng and the equlbrum P L s ^P L. The queston we want to ask now s the followng: can the manufacturer prot from manpulatng P k (w)? The answer s gven n the followng proposton. Not only does the manufacturer not gan, but nothng s aected by manpulatng P k (), except that any gaps between P k (w) and P u (w) get captalzed nto P L : Proposton 1 () Leasng has no eect on prots and on the equlbrum allocaton. () The equlbrum P u () satses equaton 1 regardless of P k (). P L satses the followng equaton. Z P L = ^P wh L + maxfp u (w), P k (w); 0gdG(w) (4) w l Where ^P L s gven n equaton 3 and P u (w) s gven n equaton 1 Proof: We shall show that for any P k () the equlbrum allocaton under sellng s stll an equlbrum under leasng, that P L satses 4 and that prots to a manufacturer are the same. We wll omt the proof that the equlbrum s unque. 11

14 Equaton 4 says that the rental prce of the new good has to be adjusted upward from ^P L to captalze all the gans that lessees can make n the used market by keepng the car and mmedately sellng n the used market when P k <P u. Thus, keepng decsons are unaected by changng P k (): when P k (w) <P u (w) the true cost of keepng s P u (w) and the eectve rental prce of consumng a used cars s unchanged; when P k (w) > P u (w) consumers who dd not plan to keep at P u also do not keep at P k, hence nobody pays a P k (w) >P u (w). It s also clear that, gven that the supply of used cars s unchanged, the equlbrum P u () s unchanged and stll satses equaton 1. Thus, the equlbrum allocaton s unchanged. To obtan prots, from the prevous analyss we get that n the used market the lessor sells cars of qualty w at P u (w) when P u (w) <P k (w) and at P k (w) when P u (w) >P k (w). We thus obtan that prots for a lessor who produces y unts and leases these unts are: = y 1, (P L + ( Z P u (w)dg(w) + fw:p u (w)p k (w)g Z P k (w)dg(w))) fw:p u (w)>p k (w)g Substtutng nto ths expresson P L from equaton 4 we obtan that prots are not aected by the structure of the leasng contract. The proposton shows that the ablty of the manufacturer to manpulate the opton prce has no role n a world wth symmetrc nformaton. The ntuton for ths s that under observable qualty consumers wll nd a way to explot all the gans from trade and they wll undo through tradng any change n P k. An mportant feature of the equlbrum allocaton n the case of no asymmetrc nformaton s that t s rst best ecent, all types of consumers are matched wth the rght qualty of the good. Moreover, t s easy to see that a compettve ndustry that produces the good at a margnal cost of c would produce the optmal quantty of the good. Ths s mportant n reference to our welfare results n secton 6. We could allow for a whole menu of functons P k (). However, snce qualty s observable, we can focus wthout loss of generalty on a sngle functon. Ths s because, n a world wth observable qualty, all consumers of new cars wll consume new every perod. Snce they all consume the same bundle they wll all choose the cheapest way to consume t. 12

15 5 Adverse Selecton 5.1 Consumer Behavor We now turn to an analyss of the eects of leasng under adverse selecton. Let w u denote consumers' belefs about the average qualty of used cars that were leased under contract and were returned at the end of the lease. Clearly, ratonal consumers wll always beleve that w u E(w),.e. they wll never beleve that lessees keep the bad qualty cars and return the good qualty cars; at best they return all cars regardless of qualty, n whch case, w u wll have that w u = E(w). Because of adverse selecton, unless P k s very hgh, typcally we < E(w). The average suppled qualty, whch we shall denote by w A, s determned by the keepng behavor of the lessees who chose contract. Of course, equlbrum requres w u = w A 8,.e. n equlbrum expectatons are correct. The appendx dscusses the determnaton of w A. Consumers take as gven the followng varables: fw u ;PL ;Pk ;Pu g 2I. Dene V u () to be the dscounted utlty of a consumer of type who buys a used car from leasng contract every perod. 14 Then, () = wu, P u 1, V u (5) The determnaton of the utlty of consumers who contemplate leasng s more complex snce such consumers must decde what to do when the lease perod s over. They can exercse the opton of keepng the used car by payng P k or return the car at no cost and lease another new car. These decsons clearly depend on the prvate nformaton of the consumer, namely the qualty ofthe car w and the ntensty ofvaluaton for qualty. It s easy to show that optmal behavor s characterzed by a cut-o rule; f the qualty of the car s lower than some level x (), aconsumer of type wll return the car, f the car s of hgher qualty the consumer wll exercse the opton. Assume a consumer of type chooses contract and behaves optmally condtonal on ths choce. We denote by V L () the dscounted utlty for a consumer who has no car, chooses to 14 In what follows we shall suppress P L ;P u ;P k and w u as arguments of the value functons V () and cut-o functons x(). 13

16 lease a new car, decdes for contract and then behaves optmally condtonal on ths choce. V L () =v, P L + [G(x ())V L () +(1, G(x ()))(E(wjw x ()), P k Ths expresson has a smple nterpretaton. The rst term (v, P L + V L ())] ) s the ow of utlty enjoyed durng the lease perod. In the next perod there are two possble events: ether (1) the qualty ofthe car turns out to be below the cut-o (wth probablty G(x ())) and the consumer starts a new lease, n whch event he gets utlty V L (), or (2) the qualty s hgher than the cut-o (wth probablty 1, G(x ())), n whch event he enjoys a ow of utlty of E(wjw x ()), P k, and n the next perod he starts a new lease whch wll then yeld a utlty ofv L (). To determne the value of x (), observe that the consumer must be nderent between (1) keepng the used good of qualty x () at a cost of P k, and (2) ntatng a new lease rght away. Opton (1) yelds a utlty ofx (), P k V L (). Therefore we have: x (), P k + V L (). Opton (2) yelds a utlty of =(1, )V L () (6) Proposton 2 Gven the choce of contract, optmal behavor for consumers who lease s characterzed byacontnuous ncreasng functon x :[ l ; h ]! [w l ;w h ] such that a consumer of type keeps all used cars of qualty above x () and returns all those wth qualty below x (), where x () s gven by equaton 6. Moreover, V L () = v + (1, G(x ()))(E(wjw x ()), P k ), P L 1, G(x ()), 2 (1, G(x ())) We omt the proof of ths proposton snce t can be obtaned by a smple modcaton of the arguments n Hendel and Lzzer (1997b), Proposton 3. Because the cut-o functon s ncreasng hgher types are less lkely to exercse the opton of keepng. If cars are sold nstead of leased, equaton 7 must be moded to take nto account the fact that when the consumer buys the car, he does not pay anythng f he decdes to keep and he gets P u f he decdes to sell. Ths yelds: V n () = [(v + (1, G(x()))E(wjw x())] + G(x())P u, P n 1, G(x()), 2 (1, G(x())) (7) (8) 14

17 We now want to determne how optmal behavor parttons the set of consumers as ether (1) non-buyers, (2) buyers of used cars of type, or (3) lessees of contract. The next proposton descrbes ths classcaton. All of the statements n the proposton are consequences of the followng standard self selecton result. Lemma 3 Suppose that consumers ; 0 choose qualtes q and q 0 wth q>q 0. Then > 0. Proof: Denote the prce of consumng qualty q by P and the prce of consumng qualty q 0 by P 0. Then, P >P 0 snce otherwse both types would choose qualty q. Because type chooses qualty q, q, P q 0, P 0. Analogously, 0 q 0, P 0 0 q, P. Combnng the two nequaltes we obtan 0 (q, q 0 ) P, P 0 (q, q 0 ). Thus, snce P, P 0 > 0, > 0. >From now on we shall restrct attenton to the case where P k loss of generalty snce, whenever P u P u. Ths s wthout >P k, all lessees wll exercse the opton regardless of the realzaton of qualty because there s an arbtrage opportunty. Ther keepng behavor x () would thus be determned by P u. Denote by u the set of types who consume used cars of type, n the set of types who choose to lease a new car accordng to contract. Proposton 4 () Suppose that w u >w u j. Then, f 2 u and 0 2 u j, >0. () Suppose that P k >P k j. Then, f 2 n and 0 2 n j, >0. () If 2 n and 0 2 u j, >0. Proof: See Appendx. Remarks: Proposton 4 says that the set of types s parttoned nto two classes of non overlappng ntervals. In the rst class are consumers wth relatvely low valuatons. These consumers buy used cars, wth the lowest types buyng used cars of low expected qualty, and hgher types buyng used cars of hgher expected qualty. In the second class are consumers wth relatvely hgh valuaton. These consumers are lessees, those wth hgher valuatons choose contracts wth hgher opton prces. Because the envronment facng a consumer s statonary, f a partcular type of behavor s optmal for a consumer at date t t wll be optmal at all future dates where the consumer faces the same choce. Thus, f contract s best for a consumer of type at date t t wll be optmal at all future dates. 15

18 5.2 Equlbrum Equlbrum requres that supply equals demand n the used market and n the new market, and that expected qualty w u equals the average traded qualty w A for each type of car. As we explaned n the ntroducton we stll take as gven the supply n the new market wthout specfyng the market structure, we deal wth optmal suppler's behavor later n the paper. The appendx states precsely how aggregate demand, supply and average traded qualty are determned from ndvdual behavor. 15 and Lzzer (1997b) to show that an equlbrum exsts. It s easy to adapt the proof n Theorem 7 of Hendel For the case n whch all unts are sold, Hendel and Lzzer (1997b, Secton 4.3) showed that the volume of trade n the used market s always strctly between zero and one hundred percent. When P k > P u, keepng a used car becomes less appealng, mplyng that the volume of trade wll always be postve n ths case as well. It s clear though that P k can be set so hgh as to dscourage keepng altogether. Thus, under leasng we may have that the volume of trade s 100%. We wll show later that ths s not optmal for a manufacturer. We now dscuss some other features of equlbrum. Proposton 5 Suppose that P k >P k j. Then, () If any consumer chooses contract, P L <P L j. () w A > wj A, and G(x ()) > G(x j ( 0 )) > n ;0 2 n j ; average qualty and volume of trade are hgher for the contract wth the hgher opton prce. () Contract generates a hgher present value of revenue per unt. Proof: Part () s mmedate; f P L j P L no consumer would choose contract. Part () s a consequence of Proposton 2 and Proposton 4. We want to show that, f buys contract and 0 buys contract j, x () >x j ( 0 ). By Proposton 4, > 0. Because chose contract, V L () Vj L (). Thus, from equaton 6, x () >x j (). By proposton 2, x j () >x j ( 0 ). Thus, x () >x j ( 0 ). 15 The determnaton of the equlbrum quanttes s complcated by the need to account for the relatve frequency wth whch derent types of lessees keep ther cars: as derent types have derent x ()s. For example, f a consumer of type always kept the car he would lease (and consume new) only half as often as a consumer of type 0 who always returned hs used car. 16

19 To prove that G(x j ( 0 )) > n j, observe that the lowest x j( 0 ) can be s n the case where P k j =0. But then contract j s a sellng contract and ths s an mmedate consequence of the postve volume of trade result n Hendel and Lzzer (1997). Ths concludes the proof of (). To prove part (), observe that the average dscounted qualty consumed by a consumer who chooses contract s strctly hgher than the average qualty consumed by a consumer who chooses contract j. If the present value of the expendture by the consumer n contract was not also hgher, nobody would choose contract j. Moreover, P u >P u j snce the qualty of used goods n contract s hgher by part (). Thus, the present value of revenue per unt from contract s hgher Equlbrum Implcatons A specal case of Proposton 5 s one where the only two optons are a sngle sellng contract and a sngle leasng contract. Thus, the results of Proposton 5 are consstent wth the observed phenomena n the used car market dscussed earler. As we dscussed n the ntroducton, there s a common percepton n the automotve press and among consumers that the average qualty of o-lease used cars s hgher than the average qualty of used cars that are sold by consumers who had bought ther car. Moreover the average age of o-lease used cars s much smaller than the average age of non leased used cars that are sold n the used market for the rst tme. Thus, turnover s hgher for leased cars. These phenomena are consstent wth equlbrum behavor n our model by proposton 5 part (), for P k > P u ; the latter s the cost of keepng under a sellng contract, whch s perceved as lower than P k. 16 In secton 8.2 we solve for the optmal contract oered by a monopolst; we wll see t ndeed nvolves a much hgher P k than P u. Observe that the menu ofcontracts serves to segment the market snce derent contracts appeal to derent types. For the smple case where there are only two contracts, a leasng and a sellng contract, part () says that the rental rate P L n the leasng contract s lower than the mplct rental rate n a sellng contract. It may be puzzlng that anybody would choose the sellng contract f the rental rate s lower n the leasng contract. The reason some 16 Guha an Waldman (1996) report that P k s on average 18% hgher than P u. 17

20 types choose to buy s that n the sellng contract the prce of the opton of keepng s lower. In a world wth adverse selecton, consumers value ths opton because they plan to keep the hgh realzatons of qualty of the used car. One can then ask the opposte queston: f the opton s so valuable, why does anybody choose the leasng contract wth the hgh opton prce? The two questons together can be rephrased to ask why nether contract domnates the other from the pont of vew of consumers. The reason s that derenttypes of consumers value the opton derently. Hgh valuaton consumers do not value the opton very much snce they are not lkely to keep the car, and therefore choose the leasng contract. Low valuaton consumers, on the other hand, value the opton more because they are more lkely to keep the car. We shall come back n sectons 7 and 8.2 to the ssue of how the prcng of the keepng opton nuences equlbrum behavor and the prots of a manufacturer. 6 Welfare We shall now dscuss the welfare consequences of ntroducng leasng contracts. In ths secton we contnue to gnore manufacturers' ncentves to study what allocatons can be acheved through leasng contracts and the lmts of these contracts. It turns out that leasng contracts are powerful nstruments when only two types of consumers are present. In the more general case, however, they do not acheve ecency as shown by the followng proposton. Proposton 6 () If there are two types of consumers then the ex post ecent allocaton can be obtaned through a sngle leasng contract. () If there s more than one type ofconsumer buyng used cars then the ex-post ecent allocaton cannot be acheved through leasng contracts. Proof: Part () s due to the fact that when there are only two types of consumers ex post ecency s solely an ssue of volume of trade. Let L and H be the two types, wth L < H. Gven y, the ex post ecent allocaton requres that the hgh types consume the hghest possble qualty. Let be the measure of the set of hgh types. If y, the ex post ecent allocaton nvolves hgh types never keepng. Ths can be acheved by settng a very hgh P k. If >y, let x be the cut-o rule such that, f t s adopted by all hgh types, all 18

21 hgh types consume new and no low type consumes new. 17 It s easy to see that t s possble to nd a P k that generates such keepng behavor by the hgh types as an optmal response. Thus, for any y t s possble to construct contracts that generate ex post ecent allocatons. It s then also possble to choose the ecent y. To prove part (), denote by 1 and 2 two types of consumers such that 1 < 2 and both are used car consumers. Let w 1 ;w 2 denote the qualtes consumed by the two types. If these qualtes are the result of equlbrum wth leasng contracts, we have that w 1 = E(wjcontract 1), w 2 = E(wjcontract 2) for some contracts 1,2 (possbly dentcal). The form these expected qualtes take s E(wjcontract ) = E(wjw x ) for some x. Thus, there s an overlap n the support of the qualtes consumed by the derent types of consumers. Ths means that there s always a postve probablty that low types of consumers end up consumng realzatons of qualty that are hgher than the realzatons of qualty consumed by a consumer of type 2. Ths s a volaton of ex post ecency. Guha and Waldman (1997) suggest that leasng can lead to rst best allocatons. Throughout ther analyss of the pure adverse selecton case they assume that there are only two types of consumers. 18 Part () of proposton 6 restates ther result and generalzes t to any dstrbuton of qualty of used cars. 19 Guha and Waldman also show that a compettve ndustry would provde the optmal leasng contract. Part () of Proposton 6 shows that the assumpton of two types s crucal n ther analyss; f there s more than one type consumer who buys used cars t s mpossble to acheve ex post ecent allocatons solely through leasng contracts. Leasng contracts make t possble to manpulate the volume of trade but t s not possble to solve the allocatve dstorton n the used market. Proposton 6 says that leasng contracts cannot acheve the ex post ecent allocaton except n a very specal case. Ths s because the leasng contracts cannot completely resolve the msmatchng of qualtes and types of consumers n the used market. 17 See secton 7 for an example that shows how ths works. 18 In ther dscusson of moral hazard they also dscuss the case of an addtonal dmenson of heterogenety. Consumers can also der by how costly t s for them to mantan the car. 19 They assume that there are at most three types of used cars. 19

22 Another mportant queston s whether leasng contracts can acheve ecency subject to ncentve constrants. The followng result shows that the answer to ths queston s negatve. We show that t s possble to acheve the rst best allocaton even wth an ncentve compatble, ndvdually ratonal and balanced budget mechansm. Ths mples that leasng contracts are ncomplete tools. Proposton 7 There exsts an ncentve compatble budget balanced mechansm that mplements the rst best allocaton. Proof: The rst best allocaton s descrbed n proposton 1. The socal planner can mplement ths through the followng mechansm. Use a leasng contract wth a very hgh P k so that nobody keeps and P L = ^P L, from equaton 3. When a consumer returns the car he must report the qualty of the car. Snce hs utlty does not depend on hs report t s (weakly) ncentve compatble for hm to tell the truth. The allocaton n the used market can be decentralzed wth the functon P u (w) descrbed n equaton 1. Ths replcates the rst best allocaton. Budget balance can easly be shown to hold for the optmal choce of output gven a margnal cost c. Ths s because the revenues that are receved by the socal planner n the proposed mechansm are the same as the revenues n the world wth observable qualty. Remarks: Frst note that ths result s n stark contrast wth what happens n statc adverse selecton models. The contrast s due to the fact that n our model, at the tme the new car consumer enters nto the contract, he does not yet know the qualty of the car. Thus, hs payment does not depend on ths nformaton. Snce the ex post ecent allocaton requres hm to consume new goods every perod, he should never keep the used good. Thus, extractng nformaton about qualty s not costly for the socal planner. Second, observe that there s a major derence between a leasng contract and the mechansm descrbed n proposton 7. In a leasng contract there s no nformaton communcated from the lessee to the used car consumers beyond that whch s nferred from the equlbrum behavor. By proposton 6 the nformaton transmtted va leasng contracts s much too coarse to lead to ecent allocatons. 20

23 The nal result on welfare hghlghts the fact that 100% trade n the used market (.e., f no consumer keeps used cars) s only optmal f qualty can be allocated n the rght way n the used market. Proposton 8 If the socal planner s constraned to usng menus of leasng contracts, the allocaton that maxmzes consumer surplus does not nvolve full trade n the second hand market. Proof: Suppose that the we start from an allocaton wth 100% trade. Ths mples that all types n [ 2y ; y ] consume used goods every date and all types n [ y ; h ] consume new goods every date. Take the set of consumers wth types between y and y +. Instead of forcng them to trade, allow them to keep the hgh realzatons of qualty of the used (ths can be mplemented wth a leasng contract wth a slghtly lower opton prce than the one that generates no keepng). The resultng equlbrum allocaton wll nvolve some consumers wth slghtly lower valuaton consumng new cars (types n [ y, ; y )). We thus have a transfer of qualty from types n [ y ; y + ) to types n [ y, ; y ). Ths s bad from the welfare perspectve. However, the qualty of the used good falls because types n [ y, ; y + ) keep some of the realzatons of qualty of the used good. Thus, there s also a transfer n qualty from types n [ 2y ; y, ) to types n [ y, ; y ). Ths transfer s good for welfare. In other words, types n [ y, ; y ) gan at the expense of types n [ y ; y + ) and [ 2y ; y, ). The welfare loss s neglgble snce types n [ y, ; y ) and types n [ y ; y + ) are very close. The welfare gan however s substantal because types n [ y, ; y ) and those n [ 2y ; y, ) are not close on average. Thus ths type of devaton from full trade s benecal. The logc of ths result s the followng. As we saw n proposton 6, t s not possble to ecently match the rght used cars wth the rght types of consumers by only usng leasng contracts. In ths world, allowng some lessees to keep the hgh draws of the used good has good welfare consequences because t permts better matchng of qualty totypes: t reduces the qualty consumed by the lowest valuaton consumers ncreasng the qualty consumed by consumers wth ntermedate valuaton. 21

24 7 Leasng and Market Power: Opton Prcng and Optmal Output. We shall now dscuss n detal a smple example where the optmal soluton for a monopoly manufacturer s partcularly easy to characterze. We dscuss results and ntuton n ths secton, whle dervatons are relegated to the Appendx. The pont of the example s to show that the opton of keepng the used good s msprced n a sellng contract. We explan the source of ths msprcng and show how the optmal prcng of ths opton n a leasng contract allows the manufacturer to protably expand output. Consder two types of consumers, low valuaton wth L =1=3 and hgh valuaton wth H =1. There s a mass of 1/2 of each type of consumer. We also assume that = :8, v =2, and that G(), the dstrbuton of qualty of used goods, s unform on [1, s; 1+s] wth s 2 [0; 1]. The parameter s represents the extent of asymmetrc nformaton; a low (hgh) s means that there s lttle (a lot of) uncertanty about the qualty ofaused car. The role of the assumpton s 1 s to guarantee that w h v. 7.1 Sellng vs. Leasng We shall rst x output at y =1=3, solve for the equlbrum under sellng, and then solve for the optmal leasng contract. Fxng a partcular y s a good startng pont because the features of the soluton wll be very smlar for all choces of y but the dscusson should be easer to follow for a partcular numercal value of y. We wll later obtan optmal output under both sellng and leasng for the case of zero margnal cost. 20 Sellng: Snce y = 1=3, there are not enough new cars for the hgh types to consume new cars every perod. Thus, n equlbrum some hgh types must consume some used cars. The equlbrum has the followng features: (1) Hgh types buy new cars and none of the low types buy new cars. (2) The keepng behavor of hgh types s x( H )=1,.e. the hgh types keep the upper half of the realzatons of qualty of the used cars. The way we nd the equlbrum (for detals see Appendx B) s to solve the P u that clears the used market at 20 An output of 1/3 would be optmal for some weakly convex cost functon. However, output under leasng would be derent than under sellng. 22

25 the w A mpled by x( H )=1; then solve the P n that makes x( H )=1optmal, and nally show that the new market clears as well (snce x( H )=1; hgh types keep half of ther cars, hence the new market clears, as = 1, the latter s the mass of hgh types): Prots for the manufacturer n ths case are: S = P n y = :53333, s=30. Leasng: In Appendx B we show that the manufacturer benets from settng P k >P u despte the fact that under the optmal leasng contract the equlbrum allocaton s the same as under sellng. The optmal contract here nvolves leasng only; because there are only two types, more complcated menus serve no purpose. We saw that under sellng market clearng requres that x( H )=1, mplyng that P n must be such that ths keepng behavor s optmal for the hgh types. The mportant consequence of ths s that under sellng, V n ( H ) >V u ( H ) (see V n and V u n Appendx B). Ths s n stark contrast wth what happens when there s symmetrc nformaton n the used market. In the case of symmetrc nformaton the new car buyers must be nderent between the new good and the best alternatve, the hghest qualty used good. Under adverse selecton, the new car buyers must be nderent between buyng a new car and keepng a used car of qualty x( H ). Ths nderence s determned by the cost of keepng a used car of qualty x( H ). Under sellng, ths cost s P u (the amount that would be receved n the used market by sellng the used car), but P u must equal L w A to clear the used market. Ths prce s too low because t evaluates a lower qualty good (w A <x( H ))byalower valuaton consumer ( L < H ). Ths means that the opton of keepng the used car s not fully prced for the hgh types. Thus, n a world wth adverse selecton, the competton for the new good s a used good that s prced too low. In contrast, n a world wthout nformaton asymmetres the qualty of the used good s observed n the market and therefore adequately prced. By rasng the opton prce above the market clearng prce n the used market (P k > P u ) the manufacturer can better prce the keepng opton and extract more rents from the hgh valuaton consumers. The equlbrum s solved n a smlar way to the one descrbed for sellng, the only derence s that the manufacturer has two nstruments to make x( H )=1; they are P k and P L : Actually, there are many pars of (P L ;P k ) that generate x( H )=1as optmal keepng behavor. The followng expresson descrbes the market clearng P L that corresponds to any 23