1 LEASING, LEMONS, AND MORAL HAZARD by Justin P. Johnson Johnson Graduate School of Manageent Cornell University Sage Hall Ithaca, NY and Michael Waldan Johnson Graduate School of Manageent Cornell University Sage Hall Ithaca, NY Deceber 2007 *We would like to thank Ari Gerstle and Kaeshwari Shankar for research assistance.
2 ABSTRACT A nuber of recent papers have analyzed leasing in the new-car arket as a response to the adverse-selection proble in the used-car arket originally explored in Akerlof (1970). In this paper we consider a odel characterized by both adverse selection as in these earlier papers and oral hazard concerning the aintenance choices of new-car drivers. We show that this approach provides explanations for a nuber of epirical findings concerning real-world new- and used-car arkets including that leasing has becoe ore popular over tie, that very high-incoe new-car drivers lease ore, and that used cars leased when new sell for ore than used cars purchased when new. We also copare and contrast our approach to new-car leasing with alternative approaches.
3 I. INTRODUCTION In the last twenty-five years there has been a draatic change in the way new cars are arketed. In the early 1980s consuer leasing of new cars was alost unheard of, while by the id-2000s roughly one-fourth of new cars arketed directly to consuers are leased. The obvious questions are, what is the role of leasing in the new-car arket and why has new-car leasing grown so draatically? Recently, both Hendel and Lizzeri (2002) and Johnson and Waldan (2003) argued that leasing in the new-car arket is a response to the adverse-selection or leons proble in the used-car arket originally explored in Akerlof (1970). This paper analyzes a odel characterized by both adverse selection as in these earlier papers and oral hazard concerning aintenance, where the resulting analysis provides explanations for a nuber of epirical regularities concerning real-world new- and used-car arkets. In trying to understand leasing in the new-car arket, Hendel and Lizzeri approach the proble ainly fro the perspective of a onopolist while our own earlier paper focuses on copetition. However, the basic logic of the analyses is siilar. Adverse selection in the used-car arket arises fro the private inforation of used-car sellers, but this eans leasing should avoid the proble because with leasing used cars are returned to dealers and dealers have no private inforation. In the onopoly case, this results in leasing because avoiding used-car-arket inefficiency increases consuers willingness to pay and thus also increases profits. In the copetitive case, this results in leasing because avoiding inefficiency iproves consuer welfare so consuers prefer it (reeber, in the copetitive case consuers receive any increase in social welfare because firs earn zero profits in equilibriu). Both of the articles discussed above focus ostly on benefits associated with lease contracts. But as discussed in earlier papers such as Sith and Wakean (1985) and Mann (1992), there is an iportant cost associated with leasing which is that leasing is likely to be associated with oral hazard concerning consuer aintenance. In particular, these authors argue that a leased car will be inadequately aintained because with high probability the consuer will return the car to the anufacturer at the end of the lease contract. As we will show, given asyetric inforation, oral hazard also applies to cars that are purchased when new because aintenance expenditures are not reflected in the price that used cars sell for on the secondhand arket. The other iportant point to note is that real-world lease contracts typically contain a variety of provisions, such as those concerning andatory aintenance, that are included in the contract to aeliorate the oral-hazard proble.
4 2 In this paper we consider a odel characterized by both adverse selection and oral hazard concerning aintenance. There is adverse selection because used-car quality is privately observed by the individual who drove the car when it was new, while leasing reduces the proble because it liits the ability of new-car drivers to act on that private inforation. There is oral hazard because in choosing aintenance new-car drivers ignore how their choices affect the welfare of used-car purchasers. Finally, in the leasing case this oral-hazard proble is partially avoided by having the lease contract specify aintenance levels for new-car probles identified during routine aintenance activities. Analysis of this odel yields that the introduction of oral hazard leaves unchanged ost of the ajor results of our earlier paper, while at the sae tie a nuber of interesting new findings eerge. There are three new findings. First, the odel provides a possible explanation for the draatic growth in new-car leasing over tie. Specifically, we find that leasing can increase when there is a significant increase in the degree to which new cars are reliable or trouble free, which can explain the increase in leasing since new-car reliability has substantially increased over tie. Second, consistent with epirical findings in Aizcorbe and Starr-McCluer (1997), we find that under plausible conditions high-incoe new-car drivers have a higher frequency of leasing. Third, although not a direct iplication of the odel because in our odel new cars do not vary in ters of their reliability, our general approach predicts that the frequency of leasing should be higher for car odels that are ore reliable or trouble free. This result is consistent with evidence reported in Desai and Purohit (1999) and Mannering, Whinston, and Starkey (2002) which both show in cross-sectional analyses that leasing is positively related to new-car reliability. As indicated, we also find results consistent with earlier analyses of this topic. For exaple, as found in our own earlier analysis and Hendel and Lizzeri s, used cars leased when new sell for ore on the secondhand arket than siilar cars purchased when new. Desai and Purohit (1998) present evidence along these lines. Another exaple is that, consistent with our own earlier analysis, leasing should be ore coon for consuer groups for who the cost of abiding by the standard restrictions found in lease contracts is sall. 1 For exaple, holding all else equal, because of axiu ileage restrictions in lease contracts, consuers who anticipate driving a sall nuber of iles should be ore likely to lease their vehicles than purchase the. This, in turn, is in fact exactly what Mannering, Winston, and Starkey (2002) find. 1 We failed to point out this result in our earlier analysis, but it is a clear iplication of that analysis.
5 3 One focus of this paper is investigating the extent to which a odel of the new- and used-car arkets based on adverse selection and oral hazard can explain evidence fro the autoobile arket. Adverse selection and oral hazard are potentially iportant in other arkets as well, but we have not yet done any systeatic study of the extent to which this paper s results are consistent with evidence fro other arkets. For exaple, Gilligan (2004) finds evidence in favor of adverse selection in the arket for used business aircraft, and it would be interesting to know whether evidence concerning leasing in that industry is consistent with our theoretical results. 2 II. MODEL AND PRELIMINARY ANALYSIS This section constructs a odel of the new- and used-car arkets characterized by adverse selection. We then analyze the odel and show the role of leasing in reducing the proble. In the next section we investigate what happens when costs of leasing, including a oral-hazard cost, are incorporated into the odel. A) The Model There are two ain differences between this section s odel and the odel analyzed in Propositions 1 to 3 in our earlier paper. The first is that here we allow ore than two consuer groups. The ain reason for this is that when we extend the odel in Section III this assuption allows us to develop results concerning how leasing should vary with consuer incoe. The second is that new cars now coe in varying quality levels. Consider an infinite-period odel in which a perfectly copetitive industry produces cars. Cars last two periods, where a car zero periods old is referred to as new while a one-period-old car is referred to as used. There are M 1 types of cars, where a car of type, = 1,..., M 1, is of quality when new and costs c to anufacture. It is further assued N N k N q q > q and c > ck if < k. Due to stochastic depreciation there is uncertainty concerning used-car quality. In particular, the quality of a used car of type is a rando draw fro a distribution described by a probability density function 2 There are a nuber of epirical studies that look for evidence of adverse selection in durable-goods arkets. Bond (1982,1984) investigates the arket for used pickup trucks and finds evidence of adverse selection in the arket for older trucks only, while Genesove (1993) finds soe evidence for adverse selection in dealer-auction arkets for used cars. More recently, in addition to Gilligan s paper, Porter and Sattler (1999) and Schneider (2005) find evidence inconsistent with adverse selection in studies of the autoobile arket while Eons and Sheldon (2005) and Pierce (2006) find evidence in favor of adverse selection in studies of the autoobile arket.
6 4 h ( q U ) and cuulative distribution function ( U U H q ), where h ( q ) > 0 for all U h ( q ) = 0 for all used car of type. U U q outside of this interval. Let E( q ) U q L q < N < q and be the unconditional expected quality of a There are M groups of infinitely-lived consuers, where consuer group consists of a continuu of nonatoic consuers of ass x, = 1,..., M. In each period an individual in group drives either zero or one car, where v is the valuation per unit of quality that a group individual places on driving a car in any period, v 1 > v2 >... > vm. It is also assued that firs and all consuers are risk neutral and have a discount factor δ, 0 < δ < 1. New cars can be purchased or leased for a single period, where a lease contract contains both lease and buyback prices. The buyback price is the price at which a consuer who leases a new car can buy the car at the end of the lease period. In particular, if a consuer leases a new car in period t, then at the beginning of period t + 1 the consuer either returns the car or purchases the car at the buyback price. If the car is returned, then the fir sells the car on the secondhand arket in period t + 1. The inforation assuptions follow. First, the quality of any specific used car is known only by the individual who drove the car when it was new. 3 Second, as in Hendel and Lizzeri (2002) and Johnson and Waldan (2003), buyers on the secondhand arket can observe whether a used car was purchased or leased when it was new, and also secondhand-arket buyers know the car s type. The tiing of oves in each period follows. First, each individual who drove a new car in the previous period learns the quality of his or her used car. Second, each consuer who leased a new car in the previous period decides whether to return the car to the anufacturer or buy it. Third, each fir chooses which type of car to produce and announces a purchase price for a new car. Further, when leasing is an option, the fir also announces a lease price and a buyback price. Fourth, consuers ake their purchasing and leasing decisions. Fifth, a secondhand arket opens up in which prices equate supply and deand. 4 3 However, we assue that the individual who drove the car when it was new cannot purchase the car on the secondhand arket (the consuer does have the option of purchasing a different used car on the secondhand arket). Without this assuption leasing would not reduce adverse selection because it would not suppress the private inforation of new-car drivers. 4 In odeling the secondhand arket we assue trades are ade anonyously. That is, a used-car buyer does not know the identity of the individual who drove the car in the previous period. As a result, given asyetric inforation, a consuer who sells used cars cannot establish a reputation concerning his or her behavior in the secondhand arket. We ipose this assuption because this type of reputation foration does not see to be iportant in real-world used-car arkets.
7 5 Finally, to siplify the analysis we focus on equilibria in which contracting and trading options are stationary. When only buying and selling of new and used cars is allowed, we look for equilibria in which new- and used-car prices are tie invariant. When firs are also able to offer lease contracts, we focus on equilibria in which each fir offers the sae contract in each period and in which arket prices are invariant to tie. Furtherore, to siplify the exposition, we do not consider equilibria in which firs offer ters that are not accepted and equilibria characterized by trades aong identical consuers. B) Analysis To siplify the analysis, we ipose the following restrictions on the paraeters. First, we assue equation (1) is satisfied for all (, k) pairs such that < M and k < M. N U N U (1) v q + δ v E( q ) < c < v ( q E( q )) M M This assuption ensures that group M individuals never purchase or lease new cars of any type while individuals in groups 1 through M 1 never purchase used cars. At the end of the section we discuss how the analysis changes when consuers in ore than one group are potential buyers on the M 1 secondhand arket. Second, we assue x > x. This assuption ensures secondhand-arket M = 1 prices are positive. Third, to ensure that individuals in each group, = 1,..., M 1, purchase or lease type- new cars rather than type- k new cars, k, we assue equation (2) is satisfied for all N q* [ q L, q k ] and qˆ such that H ( qˆ) H ( q = k *). We ipose this restriction for tractability reasons. N N N q N q k qˆ k q* k (2) v q c + δ v qdh ( q)] [ v q c + δ v qdh ( q)] [ δc Fourth, our earlier assuption that for each car type used-car quality is bounded between q L and N q ensures that there is necessarily an adverse-selection proble when there is asyetric inforation and firs sell new cars. See Appendix A. We begin with a second-best analysis concerning what happens when firs only sell new cars. Suppose there is a social-welfare-axiizing social planner who can only control the frequency with which a new-car driver sells the car on the secondhand arket in the following period (as is standard we define social surplus as the suation of econoic profits and consuer surplus). The equilibriu in this case is as follows. First, for every < M, a group driver purchases a new car of type in any period he or she does not own a used car at the beginning of the period. Second, for every < M there exists a value q ~ such that consuer i of type who owns a used car at the beginning of period t k k k
8 6 keeps the car if its quality exceeds M consuer if the used car s quality is less than q ~. Third, in the sae situation the consuer sells the car to a group cars but in each period soe purchase used cars on the secondhand arket. q ~. Fourth, group M consuers never purchase new In other words, new cars are driven by higher valuation consuers and, aong these consuers, there is a positive correlation between the new car an individual buys and the valuation the individual places on quality. Further, owners of used cars choose to keep higher quality used cars and sell the lower quality used cars on the secondhand arket. The next step is to consider what happens when firs only sell new cars and there is no social planner deterining the frequency with which used cars are traded on the secondhand arket. In ost respects this case does not look very different than the social-planner case just discussed. That is, consuers in groups 1 through M 1 buy new cars in any period in which they do not own used cars at the beginning of the period. Further, when such a consuer does own a used car at the beginning of a period, he or she keeps the car when it is high quality and sells the car when its quality is low. But there is an iportant difference. In the social planner case the frequency with which used cars are traded on the secondhand arket is socially efficient. That is no longer the case here. As first explored by Akerlof (1970), given asyetric inforation concerning used-car quality, a car s price on the secondhand arket reflects the average quality of siilar used cars offered for sale rather than the car s actual quality. The result is that fewer used cars are traded on the secondhand arket than is socially efficient. Proposition 1 foralizes what happens in this case. Below U q it is the realization for quality in period t of the used car that individual i purchased and drove in period t 1 when it was a new car, PU q E ( ) denotes the expected quality of secondhand-arket used cars of type, and PU p is the secondhand-arket price for those cars. All proofs are in the Appendix. Note, to siplify the stateents of the propositions, we assue that a consuer who is indifferent between selling his or her used car and driving the car will sell it (siilarly, a consuer who is indifferent between returning a leased car when it becoes used and driving the car will return it). Proposition 1: Suppose firs sell new cars. Then there are one or ore equilibria each of which is characterized by a function q *, q * < q~ for = 1,..., M 1, such that i) through iii) describe the equilibriu.
9 7 i) In every period t, all group consuers, = 1,..., M 1, who do not own a used car at the beginning of the period purchase a new car of type at price c. ii) In every period t, each consuer i in group, = 1,..., M 1, who owns a used car at the U beginning of the period characterized by q > ) q * keeps the used car and drives it (purchases a new car and sells the used car). it ( iii) In every period t, soe group M consuers purchase a used car on the secondhand arket. The ain result in Proposition 1 is that, as discussed above, due to adverse selection there is less trade on the secondhand arket than in the social planner s case. Another interesting aspect of the proposition is that there are potentially ultiple equilibria. The logic is that the return to selling a used car on the secondhand arket is positively related to the secondhand-arket price, which is itself positively related in equilibriu to the frequency with which a consuer who owns a used car at the beginning of a period decides to sell the car. So, for exaple, for a fixed paraeterization there can be both an equilibriu with low secondhand-arket prices and low values for q *, and an equilibriu with high secondhand-arket prices and high values for q *. See Wilson (1980) and Ki (1985) for earlier analyses of secondhand-arket odels with this type of ultiplicity. We now consider what happens when firs can sell or lease new cars. Note, below L p is the B price for a new car leased to a group consuer, p is the buyback price in the lease contract signed LU by group consuers, E ( ) is the expected quality of type- used cars leased when new and q then sold on the secondhand arket, LU p is the price these used cars sell for, π + denotes social surplus in this case, and π denotes the highest equilibriu value for social surplus when firs only sell new cars. Proposition 2: Suppose firs can both sell and lease new cars. Then there exists a unique equilibriu characterized by all new-car drivers leasing as opposed to purchasing new cars, where i) through iv) + describe the equilibriu. Also, π > π '. i) In every period t, all group consuers, = 1,..., M 1, who did not lease a new car in the previous period, lease a new car of type at a price L p and buyback price B p. ii) In every period t, each consuer i in group who leased a new car in the previous period U characterized by q > ( ) q~ purchases the car at the buyback price and drives it (returns the used car and leases a a new car). it
10 8 iii) In every period t, soe group M consuers purchase a used car on the secondhand arket. iv) B LU p > p for all, = 1,..., M 1. Proposition 2 has a nuber of interesting results. First, when a car is leased there is no adverseselection proble in the following period when the car becoes used. That is, the used car is sold on the secondhand arket given exactly the sae set of realizations for used-car quality as in the earlier socialplanner analysis. The logic is that leasing and optially setting the buyback price suppresses the private inforation of new-car drivers with the result that adverse selection is avoided. Second, all new cars are leased rather than sold. This result is a direct raification of the first result. Since leasing avoids adverse selection, new-car drivers prefer to lease rather than purchase because it iproves their welfare. A third interesting result concerns buyback prices. As we found in our earlier paper, a lease B contract s buyback price is higher than the car s secondhand-arket price if returned, i.e., p > p for all, = 1,..., M 1. The logic is that it is only efficient for consuers in any group to keep N used cars with quality close to q and in order to achieve this result buyback prices are set higher than secondhand-arket prices. As we discussed in detail in our earlier paper, this finding atches well with evidence we collected fro advertiseents that appeared in the Sunday New York Ties during 1998 (see LU also the recent epirical analysis in Pierce (2006)). As a final point, without the paraeter restriction N U k v ( q E( q )) > c for all (, k) pairs such that < M and k < M, there would be the possibility of ultiple consuer groups purchasing used cars on the secondhand arket. We have analyzed this case and, given an equilibriu exists, results are quite siilar. 5 The only significant difference is that the atching of used cars to used-car buyers is potentially less efficient than in the analysis found above. In Propositions 1 and 2, since only one group purchases used cars in equilibriu, the atching of used cars sold and the purchasers of these cars is (trivially) efficient. In contrast, when ultiple consuer groups purchase used cars, there is potentially inefficient atching of used cars sold and the purchasers of these cars on the secondhand arket. III. ADVERSE SELECTION AND MORAL HAZARD In the previous section we showed that, given asyetric inforation, new-car leasing iproves social welfare because it avoids adverse selection. In that analysis, however, when firs had the option 5 In our analysis of this case we were unable to show that an equilibriu necessarily exists. This is because of possible discontinuities in the secondhand-arket prices of used cars that arise in this case.
11 9 of leasing the result was that all new cars were leased, but the real-world autoobile arket clearly exhibits a ix of selling and leasing. In this section we do two things. First, we derive an equilibriu that exhibits a ix of selling and leasing. Second, we discuss how this odel provides explanations for various epirical regularities concerning real-world new- and used-car arkets. A) The Model In order to generate an equilibriu that exhibits a ix of selling and leasing, we introduce costs associated with leasing. The first cost we introduce is, as in our earlier paper, a cost to consuers of abiding by the standard restrictions found in lease contracts such as those concerning axiu ileage. The second is a cost due to oral hazard concerning consuer aintenance. 6 The idea is that if a consuer leases a new car, then the consuer underinvests in aintenance because if the consuer returns the car he or she does not bear the consequences of inadequate aintenance. We also incorporate the idea that in real-world lease contracts soe aspects of this proble are avoided through provisions in the contract that stipulate andatory aintenance activities. To be specific, we introduce the following changes. First, z it denotes the cost consuer i incurs in period t if he or she drives a leased car in period t. In particular, each z it is a rando draw fro the probability density function f (z) and cuulative distribution function F (z), where f ( z) > 0 if 0 < z < Z and f ( z) = 0 for all z outside of this interval. We also assue each z it is privately observed by individual i at the end of period t 1. Note, for tractability reasons we assue z it is a new draw fro the probability density function f (z) each period rather than being constant fro period to period. Second, consider consuer i who purchases a new car in period t. During period t the consuer privately observes the realization of a rando variable λ it and also the consuer s realization of z it+ 1. λ it captures both the inherent deterioration of the car as it ages and the car s return to aintenance, where λ it is a rando draw fro a distribution described by the probability density function g (λ) and cuulative distribution function G (λ), g ( λ) > 0 for all 0 < λ < 1 and g ( λ) = 0 for all λ outside of this interval. Based on these realizations, the consuer then chooses a aintenance 6 See Mann (1992) for an earlier foral analysis of leasing and oral hazard concerning consuer aintenance. Other analyses that consider durable goods and consuer aintenance, but not in ters of the leasing decision, include Schalensee (1974), Ki (1985), and Shapiro (1995).
12 10 level, e it, e it [ 0, e], for which he or she bears the cost k ( e it ), where k (e) is a strictly increasing and convex function with k ( 0) = k'(0) = 0. The quality of the used car owned by the consuer at the beginning of period t + 1 is then given U by the continuously differentiable function q λ, E ), where is the car s type, E = e y, and ( it it it it + y [ 0, Y ]. We assue for all, = 1,..., M 1, ( λ, E) is strictly increasing in λ for any given E, strictly increasing in E for any given λ, and strictly concave in E. We also assue for all, U L U N = 1,..., M 1, q (λ,0) = q and q (λ, E) q for all ( λ, E) pairs. In words, independent of the realization of λ used-car quality equals the iniu value if both y and the aintenance expenditure equal zero, and also used-car quality can never exceed new-car quality. Note, in this specification y captures the potential iportance of oral hazard. That is, the saller is y the larger is the oral-hazard proble in the sense that not aintaining a used car translates into worse outcoes for used-car quality. Third, consider consuer i who leases a new car in equilibriu. With probability q U 1 α the consuer is free to choose any aintenance level as above. However, with probability α a iniu C aintenance level, e (λ), is specified in the lease contract, where refers to the car s type. In words, this corresponds to a setting in which with probability α the proble with the car is identified during routine aintenance and the lease contract specifies a iniu aintenance level, while with probability 1 α the consuer privately observes the proble so the lease contract cannot specify a iniu aintenance level. One interesting aspect of this oral-hazard proble is that it applies to both newly leased and newly purchased cars. The logic is that, because asyetric inforation eans the secondhand-arket price an individual receives for a used car is not directly a function of its quality, an individual who purchases a new car and anticipates selling it in the subsequent period underinvests in aintenance. A related point is that the oral-hazard proble will soeties be ore severe for new cars that are purchased than those leased. The reason is that, as discussed above, when a new car is leased part of the oral-hazard proble is avoided by requiring iniu aintenance levels for probles identified during routine aintenance, but this option is not available when a new car is purchased. The tiing of oves in each period is now as follows. First, each consuer i who drove a new car in the previous period finds out the realization of that car s current quality. Second, each consuer who leased a new car in the previous period decides whether to return the car or purchase it at the
13 11 buyback price. Third, each fir chooses which type of car to produce and announces a new-car purchase price. Further, when leasing is an option, each fir announces a lease price, a buyback price, and a iniu aintenance level for each λ for new-car probles identified during routine aintenance. Fourth, consuers ake their purchasing and leasing decisions. Fifth, a secondhand arket opens up in which prices equate supply and deand. Sixth, each consuer i who purchased or leased a new car observes λ it and z it+ 1, and then subject to contractual provisions chooses a new-car aintenance level. B) Analysis As in the previous section, we ipose a nuber of paraeter restrictions. First, as in the previous analysis, we restrict the analysis to paraeterizations such that group M individuals never purchase or lease new cars while individuals in groups 1 through M 1 never purchase used cars. To ensure this we assue equation (1) is still satisfied for all (, k) pairs such that < M and k < M, U where now E ( ) refers to the expected quality of a type- used car given a aintenance expenditure q U q k equal to e and E ( ) is defined siilarly. Second, we again ipose a restriction that ensures that individuals in each group, = 1,..., M 1, purchase or lease type- new cars rather than type- k new cars, k. Specifically, we assue that, for all decreasing functions λ *( z), equation (3) is satisfied for all (, k) pairs such that < M and k < M. N (3) v q c + ax [ k( e) + [ v q N k 0 Z 1 λ*( z) c k + Z 0 e 1 λ*( z) e δ v q ax [ k( e) + δ v ( λ, e)] dg( λ) df( z) q ( λ, e)] dg( λ) df( z)] δc Third, we restrict the analysis to paraeterizations such that for each of groups 1 through M 1 soe individuals purchase new cars in equilibriu which is ensured by assuing Z is sufficiently large (an earlier assuption ensures there is soe leasing in equilibriu). Fourth, we continue to assue M 1 x > x which again ensures that secondhand-arket prices are positive. See Appendix A. M = 1 Siilar to the analysis of the previous odel, we begin with a brief discussion of what happens when firs only sell new cars and there is a social planner who only controls the frequency with which a new-car driver sells the car on the secondhand arket in the following period. In particular, as in the previous odel, the social planner chooses a frequency for each new-car driver to sell the car on the secondhand arket in the following period and then each driver keeps used cars of high quality and sells low-quality used cars. In this odel this idea translates into a function λ * such that an individual in k k
14 12 group, = 1,..., M 1, who owns a used car at the beginning of period t + 1 keeps the used car and drives it (purchases a new car and sells the used car to a group M individual) if λ > ) λ *. it ( We now consider what happens without a social planner when firs only sell new cars. Given our paraeter restrictions, in this case individuals in groups 1 through M 1 do not purchase used cars while individuals in group M do not purchase new cars. There are four additional conditions that characterize what happens in this case. First, every individual in group, = 1,..., M 1, purchases a new car of type in every period he or she does not own a used car at the beginning of the period. Second, there is a function λ, = 1,..., M 1, such that an individual in group who owns a used K car at the beginning of period t + 1 keeps the used car and drives it (purchases a new car and sells the K K used car to a group M individual) if λ > ( ), where λ < λ * for all = 1,..., M 1. Third, it λ in every period in which an individual in group, = 1,..., M 1, drives a new car and anticipates K selling the car in the following period, i.e., λit λ, the aintenance expenditure equals zero. Fourth, for each group, = 1,... M 1, there is a single price that applies to all type- used cars sold on the secondhand arket, where this price equals v ultiplied by the average quality of the type- K secondhand-arket used cars (which itself depends on λ ). The intuition behind these results is as follows. Because of asyetric inforation, all used cars sold by a particular consuer group sell for the sae price and this price reflects the average quality of all the used cars this consuer group offers on the secondhand arket. In turn, due to adverse selection, just as in Akerlof s analysis and the analysis of the previous section, fewer cars are traded on the secondhand arket than axiizes social welfare. As discussed earlier, there is also oral hazard in this odel even in the absence of leasing. As just discussed, with asyetric inforation the secondhand-arket price for a used car sold by a particular consuer in soe group reflects the average quality of type- secondhand-arket used cars. Hence, oral hazard arises because when a consuer sells a used car the price the consuer receives is not a function of the previous aintenance expenditure. As a result, because new-car drivers correctly anticipate when they will sell their used cars in the following period, the previous period s aintenance expenditures for cars offered on the secondhand arket equal zero. PU LU PU LU + We now allow leasing. Below E ( q ), E ( q ), p, p, π, and π ' have the sae P definitions as previously. Also, e λ, z ) is the aintenance level chosen by new-car driver i in ( it it+ 1 group who purchased a new car in period t, while e λ, z ) is the aintenance level chosen by L ( it it+ 1
15 13 new-car driver i in group who leased a new car in period t and the new-car proble was not C ( it discovered during routine aintenance. Finally, e λ ) is the contractually specified iniu aintenance level for a type- new car leased when new given the new-car proble is identified during routine aintenance. Proposition 3: Suppose firs can both sell and lease new cars. Then every equilibriu is characterized K L B by functions z, λ (z), λ *( z), p, p, e P ( λ, z), e L C ( λ, z), and e (λ), such that i) + through vi) describe the equilibriu. i) In every period t, each consuer i in group, = 1,..., M 1, who drove a used car in the previous period purchases a new car of type at price c and chooses the aintenance level P ( it it+ 1 e λ, z ) if + z z. it ii) In every period t, each consuer i in group, = 1,..., M 1, who drove a used car in the L + previous period leases a new car of type at price p if z it < z, where the contract is C characterized by the buyback price p and aintenance function e (λ). For new-car B probles (not) identified during routine aintenance, the aintenance level chosen is at least C L (equal to) e λ ) e ( λ, z )). ( it ( it it+ 1 iii) In every period t, each consuer i in group, = 1,..., M 1, who purchased a new car in K the previous period keeps the car if λ > λ ( z ), takes the actions described in i) if K + it 1 it λit 1 λ ( zit ) and z it z, and the actions described in ii) if λit 1 λ ( zit ) and + PU PU z z. Any used car sold is sold to a group M consuer for p = v E( q ). it < iv) Suppose the new-car proble in period t 1 was not identified during routine aintenance. In every period t, each consuer i in group, = 1,..., M 1, who leased a new car in the previous period buys back the car if λ > λ *( z ), returns the car and takes the actions it 1 it described in i) if λit 1 λ *( zit ) and z it z, and returns the car and takes the actions described in ii) if λit 1 λ *( zit ) and z it < z *. Any returned used car is sold to a group M LU LU consuer for p = v E( q ). M v) Suppose the new-car proble in period t 1 was identified during routine aintenance. In every period t, each consuer i in group, = 1,..., M 1, who leased a new car in the previous period either buys back the car, returns the car and takes the actions described in i) (this requires + K M
16 14 z it z + ), or returns the car and takes the actions decribed in ii) (this requires LU LU returned used car is sold to a group M consuer for p = v E( q ). M z + it < z vi) For all, = 1,..., M 1, ( λ, z) = ( > ) 0 if λ ( > ) λ ( z), ( λ, z) = ( > ) 0 if e P λ ( > ) *( z), and e ( λ) > 0 if λ is sufficiently sall. λ C K e L ). Any Proposition 3 is characterized by a ix of selling and leasing. For each group that drives new cars in equilibriu those with low leasing costs lease new cars and avoid adverse selection, while those with high leasing costs purchase new cars and avoid the leasing cost. The equilibriu is also characterized by oral hazard and specifically the idea that leasing allows a consuer to avoid oral hazard when the new-car proble is identified during routine aintenance. That is, vi) tells us that when a consuer purchases a new car and anticipates selling the used car in the following period, the consuer spends nothing on aintenance. Siilarly, if a consuer leases a new car, the new-car proble is not identified during routine aintenance, and the consuer anticipates returning it to the anufacturer in the following period, then again nothing is spent on aintenance. However, if a consuer leases, the newcar proble is identified during routine aintenance, and λ is sall so there is a large return to aintenance, then a positive aount is spent on aintenance whether or not the consuer anticipates returning the car. Finally, there can be ultiple equilibria here for the sae reason ultiple equilibria were possible in Proposition 1, where each equilibriu has the properties described in Proposition 3. That is, there can be an equilibriu in which there are low secondhand-arket prices for used cars purchased when new and a sall proportion of these cars are traded on the secondhand arket, and an equilibriu in which these prices are high and a high proportion of these cars are traded. Further, now there is an additional reason why ultiple equilibria can exist. That is, for each group, = 1,..., M 1, there is a possibility of ultiple lease contracts in equilibriu (each of which is consistent with i) through vi)), where each group consuer is indifferent across all these contracts. C) Further Results In this subsection we discuss five further results that follow fro our approach to odeling the new- and used-car arkets that serve as explanations for various real-world epirical regularities.
17 15 Corollary 1 to Proposition 3: Holding all other paraeters fixed, if y is sufficiently sall, then there exist values α L and α H, 0 < α L < α H < 1, such that an increase fro a value α < α L to a value α > α H increases the percentage of new cars leased. The corollary states that, if y is sufficiently sall, i.e., the function that translates new-car aintenance into used-car quality is such that the oral-hazard proble is potentially large, then increasing the proportion of new-car probles identified during routine aintenance fro a sufficiently low value to a sufficiently high value increases leasing. The logic is as follows. Since when a new-car proble is identified during routine aintenance a iniu aintenance level is specified in the lease contract, as the proportion of probles identified during routine aintenance rises the advantage of using leasing to address these probles also rises. The end result is that increasing this proportion fro a sufficiently low value to a sufficiently high value causes leasing to increase. 7 This first result provides an explanation for the growth in leasing during the last twenty-five years. The typical proble not identified during routine aintenance occurs when a part in the car becoes defective or worn out unpredictably. Hence, the above result suggests that, if new cars becoe sufficiently ore trouble free or reliable, then the proportion of probles not caught by routine aintenance will fall which in turn should increase leasing. This is an explanation for the growth in leasing during the last twenty-five years because new cars have, on average, becoe substantially ore trouble free or reliable during this tie period. For exaple, Consuer Reports based on survey evidence reports a significant increase in new-car reliability over this tie period. 8 Note that an alternative approach for odeling increased reliability is to assue that increased reliability increases built-in durability and thus a car s quality when used relies less on the aintenance 7 Given the logic is straightforward, one ight ask why the result only holds given y sufficiently sall and a large enough oveent in α. Basically, the odel is such that the return to purchasing a new car can increase (but does not have to) as α increases and this can stop the result fro holding generally. Assuing y sufficiently sall liits how uch the return to purchasing can increase as α increases, while the large change in α places a lower bound on how uch the return to leasing increases. Hence, these two conditions together are sufficient to ensure that the proportion of cars leased rises. 8 See, for exaple, the discussion in Consuer Reports (2003). Also, very recently there has been a decline in new-car leasing fro roughly thirty percent of total new-car transactions to roughly twenty-five percent. One possible reason for this recent decline is that lease ters in the late 1990s and early 2000s were overly generous because anufacturers systeatically overestiated residual values, and that lease ters have becoe less generous recently as anufacturers realized their istake and started to ore accurately estiate residual values This arguent is consistent with a discussion in Global Vehicle Rearketing 2003: North Aerican Markets (2003).
18 16 decisions of new-car drivers, i.e., increased reliability decreases the iportance of the oral-hazard proble. Although we do not show it forally, this alternative approach for odeling increased reliability provides a copleentary avenue using our fraework by which increased reliability can increase the proportion of new cars that are leased. To see this, suppose that as in Corollary 1 the starting value for α is low. With a sufficiently sall α the costs associated with oral hazard will be larger for cars that are leased rather than purchased when new. This is because with sall α leasing s advantage in addressing oral hazard through andatory aintenance will be saller than its disadvantage due to used cars being sold with higher probability on the secondhand arket when they were previously leased rather than purchased. This eans that ignoring any increase in α due to increased reliability but assuing the initial value for α is low, leasing will be associated with higher oral-hazard costs but an increase in reliability that lowers the oral-hazard proble should reduce the agnitude of these extra costs. In turn, since leasing rather than purchasing a new car reduces the adverse-selection proble, an increase in reliability that reduces the extra oral-hazard costs associated with leasing should increase the proportion of new cars that are leased. A second result is that in a cross-sectional analysis ore reliable new cars should have a higher frequency of leasing. This result is not a direct iplication of our analysis since in our analysis all new cars have the sae α. However, it clearly follows fro our basic approach in that ore reliable new cars should have fewer probles not identified during routine aintenance, which in turn eans a lower oral-hazard proble if they are leased and thus ore leasing. This result is consistent with evidence reported in Desai and Purohit (1999) and Mannering, Winston, and Starkey (2002). Desai and Purohit conduct an epirical analysis of how leasing varies by car odel. They develop a easure of reliability using inforation reported in Consuer Reports and show that ore reliable car odels are leased ore. Mannering, Winston, and Starkey report siilar results in an epirical study of the probability a household leases rather than purchases a new car as a function of various consuer attributes. The third result captured in Proposition 3 is that individuals who have higher costs of abiding by the standard restrictions found in lease contracts should purchase new cars, while those with lower costs should lease. The only study that we know of that addresses this issue is the Mannering, Winston, and Starkey study just entioned. One of their results is that the probability of leasing is lower if the household has attributes that indicate that the household expects to drive a high nuber of iles. In other
19 17 words, leasing is less popular aong households for who on average the cost of abiding by axiu ileage restrictions is likely to be higher. Our fourth result concerns how leasing varies with consuer valuation on quality or siilarly L consuer incoe. Below r denotes the proportion of group individuals who lease. Corollary 2 to Proposition 3: Holding all other paraeters fixed, if y is sufficiently sall and α is sufficiently large, then L L r 1 > r, = 2,..., M 1, given v 1 sufficiently large. Corollary 2 to Proposition 3 states that, if y is sufficiently sall and α is sufficiently large, then the highest valuation group leases ore than lower valuation groups if the highest group s valuation is sufficiently high. In other words, if we interepret higher valuations for quality as higher incoes, then very high-incoe new-car drivers lease ore than lower incoe groups. The logic behind this result is as follows. y sufficiently sall and α sufficiently large eans that the potential oral-hazard proble is large and leasing avoids a significant proportion of the oralhazard proble through andatory aintenance. In turn, when the potential oral-hazard proble is large and leasing avoids uch of it, the frequency of leasing is priarily deterined by oral-hazard considerations. Given this, consider very high-incoe new-car drivers whose valuation for quality is so high that, independent of whether they lease or purchase, they drive new cars alost every period. For these consuers the oral-hazard proble is severe because they alost never keep their used cars so their unconstrained incentive is to alost always invest zero in aintenance. The result is that, because leasing eliinates oral hazard for probles identified during routine aintenance, leasing is coon aong these consuers. The above corollary is consistent with epirical findings in Aizcorbe and Starr-McCluer (1997). Aizcorbe and Starr-McCluer analyze data fro the Federal Reserve Board s Survey of Consuer Finances and the Bureau of Labor Statistics Consuer Expenditure Survey and find a positive relationship between leasing and consuer incoe. 9 For exaple, using the Consuer Expenditure Survey, they find that in % of households with incoes below $10,000 leased a car, 1.0% of households with incoes between $10,000 and $25,000 leased, 1.6% of households with incoes 9 Mannering, Winston, and Starkey (2002) show that leasing is positively related to consuer incoe, but they do not provide detailed evidence concerning how leasing varies with incoe class as Aizcorbe and Starr-McCluer do.
20 18 between $25,000 and $50,000 leased, 4.1% of households with incoes between $50,000 and $100,000 leased, and 11.4% of households with incoes above $100,000 leased. One interesting aspect of their findings is that the proportion of new cars leased is not everywhere increasing with incoe (this follows given that for lower incoe groups the percentage of households acquiring new cars rose with incoe). Rather, siilar to our theoretical analysis, leasing is positively related to incoe in the sense that very high-incoe households lease ore than lower incoe groups. 10 Another interesting aspect of our fourth result is that the standard arguent concerning oral hazard and leasing found, for exaple, in Sith and Wakean (1985), akes exactly the opposite prediction. That is, the standard arguent suggests oral hazard is a larger proble for high-incoe individuals because they drive new cars basically every period, but since in that arguent oral hazard only applies to leased cars the prediction is that leasing should be lower not higher with high-incoe individuals. In other words, the standard arguent concerning oral hazard and leasing does not explain the finding in the data that high-incoe consuers lease ore not less. Our fifth result follows fro the following corollary. Corollary 3 to Proposition 3: LU PU p > p for all, = 1,..., M 1. The fifth result is that for used cars of type, = 1,..., M 1, the secondhand-arket price is higher for cars leased rather than purchased when new. This result follows fro the idea that leasing reduces adverse selection. In other words, because leasing reduces adverse selection, type- used cars offered on the secondhand arket leased when new are of higher average quality and sell for ore than type- used cars offered on the secondhand arket purchased when new. One ight argue that in cases in which 10 Another possibility for why high-incoe households lease ore often is that high-incoe households are in possession of ore new cars at any point in tie (note: this would yield a positive relationship between leasing and incoe because in 1992 used cars were alost never leased). However, other evidence reported by Aizcorbe and Starr-McCluer suggests that although this is a contributing factor to their finding that high-incoe households lease ore, it is not the sole factor and thus high-incoe households ust also have a higher propensity to lease when they acquire new cars. For exaple, using the Survey of Consuer Finances, Aizcorbe and Starr-McCluer find that the top two incoe groups are in possession of the sae average nuber of vehicles in 1992, while the percentage of these vehicles acquired as new was 52.9% for the second-highest incoe group and 66.0% for the highestincoe group. This factor by itself can explain why the highest-incoe group has a higher percentage of households in possession of a leased vehicle, but it cannot fully explain why, as reported above, the highest-incoe group s percentage is ore than double the second-highest incoe group s percentage. Clearly, the highest-incoe group ust also have a higher propensity to lease when acquiring new vehicles.