On Automobile Repairs and the Provision of Loaner Vehicles
by Dealers1

byAmitrajeet A. Batabyal2

1

Batabyal thanks Yoshiro Higano and two anonymous referees for their helpful comments on a previous version of this paper. Inaddition, he acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.
2

Department of Economics, Rochester Institute of Technology, 92 Lomb Memorial Drive, Rochester, NY 14623-5604, USA. Internet
aabgsh@rit.edu

1

On Automobile Repairs and the Provision of Loaner Vehicles
by Dealers
Abstract
In this note, we analyze two hitherto unstudied questions from the perspective of anautomobile dealer that uses a particular temporal decision rule to provide loaner vehicles tocustomers who come to its facility with automobile repair requests. First, on the assumption that thebasic criterion for the above mentioned decision rule is satisfied, we determine the expected repair
times of automobiles for which our dealer provides customers with loaner vehicles. Second, weascertain the average number of loaner vehicles that are out with customers.

Keywords: Automobile, Dealer, Loaner Vehicle, Repair, Uncertainty
JEL Codes: L62, C44, D81

2

1. Introduction
Readers who either own or lease an automobile will recognize that like any other machine,
an automobile will break down from time to time. When this happens, the owner and/or the lessor
will need to bring his or her automobile to a repair shop which is, very frequently, the repair facility
associated with the business operations of an automobile dealer.
When a customer brings his or her malfunctioning automobile to a dealer’s repair facility,
this dealer will, under certain circumstances, provide the pertinent customer with a loaner vehicle.
Now, the circumstances under which a dealer will provide repair seeking customers with loaner
vehicles vary, but, generally speaking, the two key factors that are germane here are the cost to thedealer and the time it will take to repair the automobiles in question. For leased automobiles or forautomobiles with unexpired warranties, the underlying automobile manufacturer and the dealer
typically have existing cost sharing agreements in place and hence in these instances, from a dealer’sperspective, this cost issue is largely a deterministic matter.
In contrast, for automobiles that are no longer on a warranty of some sort, the dealer has
considerable discretion in determining whether to provide customers with loaner vehicles at its, i.e.,the dealer’s, expense. Here, the key variable is time. Put differently, customers with lengthy repair
jobs tend to get loaner vehicles and customers with relatively short repair jobs do not get loaner
vehicles. Instead, this latter group of customers is generally asked either to wait in the dealer’s
premises or to arrange for its own transport to and from this dealer’s repair facility.
Readers with automobiles will agree that the above description of events is a faircharacterization of automobile repairs and the provision of loaner vehicles by dealers in mostsettings in the United States of America. Despite the great practical relevance of the twin tasks of3

repairing malfunctioning automobiles and providing loaner vehicles to customers, to the best of ourknowledge, questions associated with these twin tasks have not been studied previously in either theeconomics or the regional science literatures. What has been studied thus far, but in isolation, arethe two topics of automobile repairs and the behavior of automobile dealers in particular
circumstances. Therefore, we now briefly survey this literature in the next two paragraphs and thenwe proceed to the two specific questions of this note.
Focusing first on automobile repairs, the impact that the regulation and the provision ofconsumer information have on automobile repairs has been analyzed by Webbink (1978).
Kolodinsky (1995) argues that when it comes to automobile repairs, the traits of individuals aresalient in explaining what she calls the “complaint behavior”of consumers. In their empiricalanalysis of warranty data, Majeske et al. (1997) seek to correct the biases that arise when a single
individual contributes many automobile repairs to the repair population. Finally, in the context ofautomobile recalls, Rupp and Taylor (2002) have shown that the largest owner repair responses areassociated with newsworthy hazardous defects in new domestic vehicles in their initial model year.
Focusing next on automobile dealers, Eckard (1985) has studied the effects that state level
regulations on the entry of new automobile dealers have on the prices of new automobiles.
Mathewson and Winter (1989) have analyzed the efficiency effects of laws that curb the ability of
automobile manufacturers to either add new dealers in specific market areas or to terminate existing
dealers. Between customers and dealers, who benefits most from automobile manufacturer incentive
programs such as customer rebates and dealer discount options? Busse et al. (2006) show that
customers benefit more from rebates but that dealers profit more from discount options. Finally,
Arnold and Penard (2007) have examined the effects that dealer search costs and online

4intermediaries have on the price setting process in the automobile market.
As noted above, given the absence of prior research on the related tasks of repairing
malfunctioning automobiles and providing loaner vehicles to customers, in this note, we analyze twohitherto unstudied questions from the viewpoint of an automobile dealer that provides loaner
vehicles using a particular temporal decision rule to customers who come to its facility withautomobile repair requests. First, on the assumption that the primary criterion for the above decision
rule is satisfied, we determine the expected repair times of automobiles for which our dealer
provides loaner vehicles. Second, we ascertain the average number of loaner vehicles that are out
with these customers. Before proceeding further, we would like to reiterate that this contribution ofours is a note and hence it is not our objective here to carry out an exhaustive analysis of the many
facets of automobile repairs and the provision of loaner vehicles by dealers. This notwithstanding,in the concluding section 3, we do provide a reasonably detailed discussion of two ways in which
the analysis contained in this note might be extended.
The rest of this note is organized as follows. First, in section 2.1, we describe a simple modelof the interaction between customers who arrive at an automobile dealer’s facility with repair
requests and, say, the dealer’s service manager who must decide which customers to give loaner
vehicles to. Second, in section 2.2, we compute the mean repair time of automobiles for which theservice manager provides loaner vehicles. Third, in section 2.3, we calculate the average number ofloaner vehicles that are out with customers. Finally, section 3 concludes and then discusses twoways in which the research delineated in this note might be extended.
2. The Theoretical Framework
2.1. Preliminaries

5Consider customers with malfunctioning automobiles who arrive at an automobile dealer’srepair facility in accordance with a stationary Poisson process with time independent rate

3Wesuppose that the repair times of the various malfunctioning automobiles are uniformly distributedon the interval

The service manager begins the process of repairing malfunctioning

automobiles immediately upon their arrival in the dealer’s repair facility.4 The exact time it will taketo complete a particular repair job can be determined only upon the arrival of the relevant
automobile.
Consistent with the discussion in section 1, we suppose that the service manager uses thefollowing decision rule to allocate loaner vehicles to individual customers. If the repair time for aparticular automobile takes longer thantime units wherethen the customer in question

is provided with a loaner vehicle until his or her automobile repair job is completed. Given thisdescription of our basic setup, our next task is to determine the expected repair time of automobiles
for which our service manager will have to provide a loaner vehicle, given that the above temporalcondition is satisfied.
2.2. The expected repair time
Recall that the arrival process of customers with malfunctioning automobiles is Poisson withrate

Therefore, we deduce that the arrival process of customers who will require loaner

vehicles is also Poisson but now with rate

whereNow, under the condition that3

See Taylor and Karlin (1998, pp. 267-332) or Tijms (2003, pp. 1-32) for textbook treatments of the Poisson process.
4We are implicitly assuming that there are ample repair facilities and that this is why the service manager is able to begin the processof repairing malfunctioning automobiles immediately upon their arrival in the dealer’s premises. If this were not the case then it iscertainly likely that at least some arriving customers with malfunctioning vehicles would have to wait to receive repair service. It
is possible to account for this last contingency by analyzing a “finite repair capacity” model. However, this would complicate oursubsequent mathematical analysis without providing any fundamental new insight. Therefore, in the remainder of this paper, we
suppose that the above stated assumption is valid.

6the automobile of a customer requiring a loaner vehicle has a repair timelarger thaninsymbols, we can write the conditional expectation we seek as
Our first task is to obtain a closed-form expression for the probability
anyforNow, from elementary properties of conditional probabilities and the fact that we areworking with the uniform distribution, it follows that for any

we can write(1)

Using equation (1), we can simplify the conditional expectationthat we seek. This

simplification gives us(2)

The reader should note that instead of proceeding as we have above, we could follow aslightly different procedure to obtain the equation (2) expression for the conditional expectation
In this slightly different procedure, we would first note that

(3)

Then, using equation (3),we would infer that(4)7It is clear that either of the two procedures delineated above gives us the same closed-form
that we seek.5

expression for the conditional expectation

The right-hand-side (RHS) of either equation (2) or (4) tells us that the expected repair times
of automobiles for which our dealer’s service manager will have to provide a loaner vehicle, giventhat this manager’s decision rule is satisfied, is a simple additive function of the cutoff time
the right endpoint

andof the interval over which the uniformly distributed repair times are defined.

Inspecting equation (2) or (4), it is straightforward to verify that, consistent with our intuition, wheneither the cutoff timeor the right endpoint

increases, the expected repair time under study alsoincreases. We now proceed to the second and final task of this note and that is to ascertain the mean
number of loaner vehicles that are out with customers.
2.3. Mean number of loaner vehicles
To obtain a closed-form expression for this average number, we shall apply the prominentmodel from queuing theory6 to our problem. This application tells us that the average number ofloaner vehicles that are out with customers is given by the product of the rate of the arrival processof customers who will require loaner vehicles

and the expected repair time of automobiles forwhich our service manager will have to provide a loaner vehicle given in the RHS of either equation(2) or (4). In symbols, we get(5)5

We thank an anonymous referee for pointing out this slightly different procedure for deriving the closed-form expression for theconditional expectation
6See Taylor and Karlin (1998, pp. 564-565) or Tijms (2003, pp. 9-15) for textbook accounts of the8model in queuing theory.

Alternately, using equation (4), we get(6)Inspecting the RHS of either equation (5) or (6), it is straightforward to verify twocomparative statics results that conform well with our intuition. First, as the rateof the Poisson

arrival process of all customers goes up, the average number of loaner vehicles that are out withcustomers also goes up. Second and in contrast, as the cutoff time

for providing customers withloaner vehicles goes up, the mean number of loaner vehicles that are out with customers goes down.
The answers provided to the two questions of this note in the equation pairs (2) and (4) and (5) and(6) have a significant bearing on the costs incurred by the automobile dealer under study. Hence, it
is reasonable to suppose that an automobile dealer of the sort studied in this note will want to use
these answers not only for planning purposes but also to aid the goal of conducting business
operations so as to minimize the cost of such operations.
3. Conclusions
In this note, we analyzed two previously unstudied questions from the perspective of anautomobile dealer that uses a particular temporal decision rule to provide loaner vehicles tocustomers who come to its facility with automobile repair requests. First, on the assumption that thebasic criterion for the above decision rule is satisfied, we determined the expected repair time ofautomobiles for which our dealer would have to provide loaner vehicles. Second, we ascertained theaverage number of loaner vehicles that are out with customers.
The analysis in this note can be extended in a number of different directions. Here are two9

suggestions for extending the research delineated here. First, it would be instructive to use the results
obtained in this note to set up an optimization problem in which the cutoff timeis determined

endogenously. In this regard, one way to proceed is as follows. Following the suggestions made byan anonymous referee, letdenote the time independent marginal (unit) unit cost accruing to ourdealer’s service manager from providing a loaner vehicle. Similarly, letdenote the time

independent marginal (unit) cost accruing to an arriving customer if (s)he is unable to obtain aloaner vehicle. Then, from either equation (5) or (6) it follows that the expected cost to our dealer’s
service manager from the provision of loaner vehicles isSome thought

and the analysis leading up to equations (2) and (4) tell us that the expected cost to an arriving
customer who is unable to obtain a loaner vehicle isOne—but clearly not theonly—way to determine the cutoff time endogenously would be to choose

to minimize the(possibly weighted) sum of the two expected cost expressions that we have just identified.
Second, to model the fact that the automobile repair requests of arriving customers are more
likely to occur in some time periods than in others, it would be useful to examine questions of thesort studied in this note with the proviso that the customers arrive at an automobile dealer’s repair
facility in accordance with a non-stationary Poisson process. Studies of automobile repairs and theprovision of loaner vehicles by dealers that incorporate these features of the problem into theanalysis will provide additional insights into a set of issues that are both practically relevant andtheoretically interesting.

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