# EXTENDED WARRANTIES, ADVERSE SELECTION, AND AFTERMARKETS.

ABSTRACT

This article investigates whether the contractual exclusion of third-party extended warranties should be legally permissible, using a model incorporating consumer heterogeneity. The welfare effects of competition in the market for extended warranties are shown to depend on the degree of competition in the product market. In contrast to the approach typically adopted by the courts, the article argues that manufacturers should not generally be permitted to practice requirements contracting in extended warranties, even when the product market is competitive.

INTRODUCTION

Competition authorities and economists have recently been wrestling with the question of whether manufacturers should have the right to monopolize aftermarkets for their products. An important class of such aftermarkets, characterized by adverse selection, is extended warranties and service contracts. This paper investigates the problem of whether the contractual exclusion of third-party extended warranties should be legally permissible. Consumer heterogeneity sometimes leads manufacturers to offer a menu of warranty contracts, typically implemented as a base warranty and an optional extended warranty. For most products on which a manufacturer offers an extended warranty, it faces competition from third-party service contracts. Some manufacturers, however, make it a condition of supplying a retailer that she carry only the manufacturer's extended warranty. A recent case is typical of the legal dilemma. Ford of Canada announced that as a condition for the use of its financing plan, dealers had to commit to exc lusive sales of Ford's extended warranty program. When the Federal Automobile Dealers Association threatened to take it before the Competition Bureau, Ford withdrew the condition, leaving dealers free to continue selling third-party extended service contracts. Should manufacturers generally have the right to impose such exclusivity conditions on retailers? This article argues that competition in extended warranties has an important effect on the base warranty, and therefore on welfare.

In both the United States and Canada, the legality of such requirements contracting is still uncertain, despite the magnitude of the extended warranty and service contract markets. Extended warranties are one of the most commonly purchased forms of insurance and marketed along with most large and many small appliances. Sears alone is reported to have sold over $1 billion worth of extended warranties in 1991 in the United States. Ford recorded profits in excess of $100 million from sales of extended warranties in 1988, despite fierce competition from independent insurers. [1] Around 40 percent of all automobiles and major appliances are sold with some form of extended warranty, typically with a large profit margin. [2] For retailers of products carrying extended warranties, the profits can be significant: Some analysts estimate that around half of operating profits for big consumer electronics chains come from sales of extended warranties. [3]

Although extended warranties are a form of insurance, they have quite unique features. Unlike most other forms of insurance, warranty contracts do not usually vary by amount of insurance but by duration instead. The reason for variation in the duration could be heterogeneity in consumer risk aversion or heterogeneity in the consumer's expected intensity of use. I model it as the latter in this article. This suggests that there will be opportunities for third degree price discrimination and the resulting welfare losses that come from distorting contracts for low intensity users to keep high intensity users from buying those contracts. The presence of third party extended warranties complicates the situation and may increase or decrease welfare. What is critical here is that because the warranty contract varies by duration and not by amount of coverage, the standard screening analysis employed in most insurance problems (beginning with Michael Rothschild and Joseph Stiglitz (1976) and Charles Wilson (1977)) doe s not apply directly to the warranty case.

In the next section, the relationship of this article to the previous literature on warranties and extended warranties is explained. The following section develops a model of warranties in which firms choose warranty duration and price. If firms cannot distinguish between heavy and light users, it is shown that they may offer separating or pooling warranties, where the separating warranty is simply characterized by the offer of an optional extended warranty. Sections on the monopolistic product market and the competitive product market examine the effects of competition in extended warranties. If the product market is monopolistic, then the presence of third-party insurers will destroy the price discrimination scheme of the manufacturer. This will be welfare enhancing only if "light" users continue to buy the product. Similarly, if the product market is characterized by competition, attempts to impose exclusivity restrictions on extended warranties may often be welfare-decreasing. This result contrasts with c urrent jurisprudence which is lenient to exclusivity restrictions if the product market is competitive. The article concludes with a brief discussion of policy implications, and with suggestions for further research.

RELATED LITERATURE

Economic analysis of warranties is divided by the principal function attributed to warranties: Some studies are premised on the notion that the function of warranties is to shift risk-bearing from risk-averse consumers to risk-neutral warrantors; while in others, warranties serve to provide a signal of product quality. [4] The seminal paper on warranties as a form of insurance is by Geoffrey Heal (1977), and it is this function of warranties which is of interest in the present article. Warranties typically seem to provide at best partial insurance, [5] which as will be seen may be the result of consumer heterogeneity, as observed by Winand Emons (1989) and V. Padmanabhan (1995), or of consumer moral hazard. Michael Spence (1977) was the first to show that warranties could be an informative signal for consumers if it is more costly for low quality manufacturers to provide long warranties than for manufacturers of high-quality products in a competitive market. Sandy Grossman (1981) extended the analysis to mono polistic markets and found that monopolists would always offer full warranties because not to do so would be a bad signal to consumers. Esther Gal-Or (1989), Linda Welling (1989), and Nancy Lutz (1989) derived conditions limiting the effectiveness of such signals in various environments. A series of related papers on two-sided moral hazard considered the problems of using warranties to minimize producer moral hazard in the presence of consumer moral hazard. [6] Steven Matthews and John Moore (1987) consider the problem of a monopolist who chooses both quality and warranty and explore the sensitivity of their results to different restrictions on the usual single crossing property which is employed in this article.

The focus of this article is the market for extended warranties, which, despite its importance, has attracted very little economic study. The two exceptions are work by Lutz and Padmanabhan. Their 1995 article considers a puzzle in the market for contact lenses, where, although warranties may be purchased as an option, the standard base warranty is zero. They attribute this to the presence of third-party insurers, whose extended warranties cause a negative externality on base warranties, since consumers with more coverage will be less careful in handling their lenses. They conclude that where consumer moral hazard is a problem and average product durability can be predicted with reasonable accuracy by an independent insurer, this externality will push base warranties to zero. This result applies to a particular type of product, since on the vast majority of products for which third-party extended warranties are available, base warranties are far from zero. Evidently some other paradigm is required to explain the variety of warranties and extended warranties available on different types of products. This article is based on the idea that warranties are heterogeneous because consumers are heterogeneous in their intensity of use and presents a model that helps to explain the variety of warranties which we observe. In contrast to Lutz and Padmanabhan's contact lens article, it relies on adverse selection rather than on moral hazard.

Lutz and Padmanabhan (1998), building on Matthews and Moore (1987), examine the effect of competition in extended warranty markets when the manufacturer is a monopolist in the product market and offers products of two different qualities to consumers with heterogeneous valuations. Warranties are modeled as a cash payment. They show that the presence of competitive extended warranties may cause manufacturers to change the quality of the product offered. This paper is closest in spirit to the model presented below, but differs in key results because warranties are modeled as a cash payment rather than as repair or replacement; and because product quality is determined endogenously.

The model used in this article differs from most other models in the literature because it focuses on the duration of the warranty. This follows from Philip Dybvig and Nancy Lutz (1993), who model warranties as a time-variable payment in case of breakdown. They show that the "block" form of warranties typically seen, with high coverage initially and zero coverage later, is optimal when there is double-sided moral hazard. The focus on duration is useful because it more closely resembles the choices that warrantors actually make, and because these choices are not isomorphic to models where warranties are a payment which varies by amount rather than duration of coverage.

THE MODEL

Consumers

The model specification is designed to allow consumer heterogeneity in the aspect which matters most for warranty length: the intensity of use, or the probability of breaking a product within a given period. In order to focus on this, consumers are assumed to be homogeneous in other ways, including their ex ante valuation of the product. In addition, consumers are risk averse, which is what makes the warranty valuable to them. They derive additive utility from money and the services of at most one unit of a durable product. By not buying the product, consumers obtain a reservation expected utility equal to their income, y.

Consumers of type i who buy the product obtain services of [k.sub.i] units per unit of time until the product breaks, so that if it breaks at time t, [k.sub.i]t is the cumulative amount of services obtained. The parameter k is introduced to reflect the fact that almost all products have some aspects in which intensity of use or roughness affects product durability. The good breaks with probability density f([k.sub.i]t) over the course of being used and all goods are broken by the time they have yielded K units of services,

[[[integral of].sup.1/[k.sub.i] K].sub.0] [k.sub.i]f([k.sub.i]t)dt = 1 for all I and K [epsilon] (0, [infinity]).

A product that breaks after yielding services of [k.sub.i]t units yields utility of u([k.sub.i]t) , where u is continuous and u,[greater than] 0 and u" [less than]0. Thus, expected product lifetime is inversely related to intensity of use [k.sub.t] but expected total product services is invariant to k. A broken product cannot be repaired and yields zero utility. Thus if a consumer of any type i were to buy the good (without a warranty) at price s, she would obtain von Neumann-Morgenstern expected utility independent of i

Y - s + [[[integral of].sup.1/[k.sub.i] K].sub.0] u([k.sub.i]t)kf([k.sub.i]t)dt.

Note that this expression integrates not over services through time but over cumulative services times the probability of breakdown at any time t. Buying the product without a warranty is essentially a lottery, with a range of possible outcomes, depending on t, the actual functioning life of the product. In order to keep the exposition simple, there is no discounting.

A warranty contract [[gamma].sub.j] consists of the pair {[T.sub.j], [p.sub.j]}. Purchasing a warranty at price p ensures that if the product breaks while the warranty is still in effect, that is, for t [less than or equal to] T, where T is the terminal date of the warranty, the product will be repaired or replaced. A repaired or replaced product is expected to provide x additional units of service. Ideally, a warranty should guarantee some amount of usage, but a limitation that firms face in setting warranties is that they generally cannot make them conditional on usage, which is unobservable; so they use simple time restrictions instead. [7] Thus, warranties typically expire after a set period of months or years. Following Dybvig and Lutz (1993), the warranty is restricted to a "block" form, so that there is for some period a constant level of benefits in case of breakdown, at the end of which period coverage drops discontinuously to zero. [8] A consumer of type i who buys the product with warranty thus obt ains expected utility

y - s - [p.sub.j] + [[[integral of].sup.[T.sub.j]].sub.0] u([k.sub.i]t + x)([k.sub.i]f([k.sub.i]t)dt + [[[integral of].sup.1/[k.sub.i]K].sub.[T.sub.j]] u([k.sub.i]t)[k.sub.i]f[k.sub.i]t)dt [9]

Consumers value warranty length because it increases the expected number of units obtained from the durable good, but their marginal valuation of the warranty is declining with T, since u" [less than] 0. The expected surplus a consumer of type i obtains from contract [[gamma].sub.j] is

[u.sub.i]([[gamma].sub.j]) = -[p.sub.j] + [[[integral of].sup.[T.sub.j]].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]f([k.sub.i]t)dt.

Firms

For the present the manufacturer is assumed to be a monopolist, although as shown in the last section of the article with suitable adjustments the model also applies to the case of competitive manufacturers. The manufacturer has marginal manufacturing cost c, and for each buyer of type i, warranty servicing cost w for breakdown in any state t [less than or equal to] T, and expected warranty servicing cost given [T.sub.i] of w [[[integral of].sup.[T.sub.i]].sub.0] [k.sub.i]f([k.sub.i]t)dt. [10] Its expected profits per unit sold to a buyer of type i is s + [p.sub.i] - c - w [[[integral of].sup.[T.sub.i]].sub.0] [k.sub.i]f ([k.sub.i]t)dt. Because consumers are assumed to have homogeneous valuations of the product itself, the manufacturer's maximization problem may be simplified by splitting it into two components. First the manufacturer sets s equal to the value to the consumer of the product without a warranty. Define [[gamma].sub.i] as the warranty contract which maximizes expected profits from the sale of the warranty contract only to buyers of type i, [[pi].sub.i]([[gamma].sub.i]) = [p.sub.i]- w [[[integral of].sup.[T.sub.i].sub.0] [k.sub.i]f([k.sub.i]t)dt. The monopolist's second problem, which will be focused on, is to choose [[gamma].sub.i].

The Full-Information Benchmark

If the firm could identify the consumer type, [p.sub.i] would be set equal to the consumer's valuation of a warranty of duration [T.sub.i]:

[p.sub.i] = [[[integral of].sup.[T.sub.i]].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]f([k.sub.i]t)dt. (1)

The maximization problem for the firm is then simply

[[max.sub.[T.sub.i] [[[integral of].sup.[T.sup.i].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t) - w][k.sub.i]f([k.sub.i]t)dt (2)

for each type. The first order condition for this problem requires that

u([k.sub.i][T.sub.i] + x) - u([k.sub.i][T.sub.i]) = w [11] (3)

This equality yields an interior solution, called [[T.sup.*].sub.i], given two reasonable conditions: u(x) - u(0)[greater than] wand u(K + x) - u(K) [less than] w. Essentially these conditions require that the marginal utility of the warranty be greater than the cost of warranty servicing for low outcomes of t and less for high outcomes of t. The natural outcome of (3) is that the optimal full-information warranty length [[T.sup.*].sub.i] is inversely proportional to [k.sub.i], so that

[[T.sup.*].sub.i] = [k.sub.j]/[k.sub.i][[T.sub.j].sup.*].

A convenient graphical interpretation of this is given by Figure 1, which shows isoprofit and indifference curves for two types of consumers buying from the monopolist. Different levels of profits and consumer utility are given by vertical displacements of these curves. Utility is higher for lower indifference curves, while profit is higher for higher iso-profit curves. The slope of the indifference curve at any given t is just [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]t)][k.sub.i]f([k.sub.i]t), the marginal value to the consumer of extending the warranty; while the slope of the iso-profit curves is [wk.sub.i]f([k.sub.i]t), the marginal cost of extending the warranty. As a result of the linear additivity of utility of money in the model, the negative of consumer surplus from the warranty is given by the intercept of the consumer's indifference curve on the coordinate axis extending below the zero, while the firm's profit per consumer is just the intercept of the iso-profit curve above the zero. When the mon opolist has full information about consumer types, consumer surplus will be reduced to zero, as in Figure 1. Note that for both types the optimal price is the same, and only the duration varies, with the light user obtaining a longer warranty. By comparison, in the Stiglitz (1977) analysis of the insurance market, both types of user optimally obtain the same amount of insurance, but at different prices.

The intuition behind this difference is simple: For insurance, in a given period of time, different types of users present different levels of risk. In contrast, with warranties different types of users present the same level of risk over different periods, so that the optimal price will be the same for both types of consumers. This seemingly trivial transformation means that one cannot simply transfer the results of analysis of insurance to the warranty market. In the following section, the assumption is dropped that firms are able to identify consumers of different types ex ante.

ASYMMETRIC INFORMATION WITH A MONOPOLISTIC PRODUCT MARKET

When firms cannot identify user types ex ante, it becomes optimal in some circumstances to separate the types by offering a base warranty and an optional extended warranty. In this section, I compare the outcomes of permitting and excluding independent insurers from the extended warranty market when the product market is monopolistic. I assume that there are two consumer types, i = H,L, with 0 [less than] [k.sub.L] [less than] [k.sub.H] [less than or equal to] 1, and a unit mass of consumers with [lambda] representing the proportion of high-intensity users. The parameters [k.sub.H], [k.sub.L] and [lambda] are assumed to be common knowledge. I also assume the indifference curves of the two types of user exhibit a single crossing property:

[u([k.sub.H]t + x) - u([k.sub.H]t)][k.sub.H]f([k.sub.H]t) [greater than] [u([k.sub.L] + x) - u([k.sub.L]t)][k.sub.L] f([k.sub.L]t) (4)

for all t such that u([k.sub.L]t + x) - u([k.sub.L]t)[greater than] w . This inequality requires that the indifference curves for a heavy user are steeper than those for a lighter user, so that heavy users will value any warranty extension more than light users. [12] Warranties will never be offered for t such that u([k.sub.L]t + x) - u([k.sub.L]t) [less than] w because the marginal cost of the warranty is greater than its value to consumers.

First, the case in which the product manufacturer has exclusivity in its extended warranties is examined; then the case without exclusivity; and finally welfare outcomes in the two cases are compared. As before, [[T.sup.*].sub.i] denotes the optimal full-information warranty duration for consumers of type i.

Product Monopoly With Exclusivity in Extended Warranties

When the monopolist is unable to identify the consumer type, the problem of profit maximizing becomes:

[max[lambda][[pi].sub.H]([[gamma].sub.H])+(1 - [lambda])[[pi].sub.L]([[gamma].sub.L]) (5) [[gamma].sub.H],[[gamma].sub.L]

subject to four incentive compatibility and participation constraints:

PL: [U.sub.L]([gamma].sub.L]) [greater than or equal to] 0 (5) PH: [U.sub.H]([gamma].sub.H]) [greater than or equal to] 0 IL: [U.sub.L]([gamma].sub.L]) [greater than or equal to] [U.sub.L] [[gamma] .sub.H]) IH: [U.sub.H]([[gamma].sub.H] [greater than or equal to] [U.sub.H]([gamma].sub.L]

LEMMA 1: At the solution to this problem, the constraints PL and IH will bind; PH will bind only if light users are not served; and IL will only bind if there is pooling.

Proof: See Appendix.

This problem is similar to the familiar insurance problem with adverse selection described by Stiglitz (1977). However, it differs in that there is a range of separating solutions and a range of pooling solutions, as described in Propositions 1 and 2 below, depending on 1, the proportion of heavy users. I call the separating warranty contracts offered in response to this adverse selection problem [[T.sup.ME].sub.H], and [[T.sup.ME].sub.L], and the pooling warranty contracts [[T.sup.ME].sub.P]. The first superscript indicates a monopoly in the product market, while the second superscript indicates that the firm has exclusivity in the extended warranties for its own product.

For the following propositions, I define a critical value [[lambda].sub.1] [epsilon] [0,1] as the solution to:

[u([k.sub.L][T.sub.H] * +x) - u([k.sub.L][T.sub.H] *)][k.sub.L]([k.sub.L][T.sub.H] *) = [lambda]w[K.sub.H][T.sub.H] *) + (1 - [lambda])w[k.sub.L]f([k.sub.L][T.sub.H] *). (6)

This defines the [lambda] such that the tangency of the pooling iso-profit curve with the indifference curve of the light user is at t = [[T.sup.*].sub.H]. For an intuitive sense of the significance of this critical value, consider Figure 2. The indifference ("IC") and iso-profit ("IP") curves of each type of user are stacked vertically. The slope of the pooling iso-profit curve ("[IP.sub.p]") is given by the weighted average of the iso-profit curves for heavy and light users, and is therefore increasing with the proportion of heavy users. When the proportion of heavy users is high, the tangency between the pooling iso-profit curve and the light user's indifference curve will be to the left of [[T.sup.*].sub.H], and when [lambda] is lower, the tangency must be between [[T.sup.*].sub.H] and [[T.sup.*].sub.L].

PROPOSITION 1: For [[lambda].sub.1] [less than] [lambda] [less than or equal to] 1, the manufacturer's optimal warranty strategy will be a menu of separating contracts [[T.sup.ME].sub.H] and [T.sup.ME].sub.L] such that the heavy user obtains the full-information warranty duration [[T.sup.ME].sub.H] = [T.sup.*].sub.H] and the light user obtains a shorter warranty of duration

[[T.sup.ME].sub.L] [less than][[T.sup.*].sub.H][less than][[T.sup.*].sub.L].

Proof: See Appendix.

The separating warranty contracts are typically implemented as a base warranty bundled with the product and an optional extended warranty. These are depicted in Figure 2. The light user's warranty contract [T.sup.ME].sub.L] is just the base warranty, usually included automatically with the product; while [[T.sup.ME].sub.H] is the total warranty contract purchased by heavy users. The difference between [T.sup.ME].sub.H] and [T.sup.ME].sub.L] can thus be seen as an extended warranty. The first order condition of the monopolist's maximization, derived from (5), requires [[T.sup.ME].sub.L] satisfy

[u([k.sub.L]T + x) - u([k.sub.L]T)][k.sub.L]f([k.sub.L]T)

= [lambda][u([k.sub.H]T + x) - u([k.sub.H]T)][k.sub.H]f([k.sub.H]T + (1- [lambda])[wk.sub.L]f([k.sub.L]T). (7)

Thus the length of the base warranty is determined by the tangency of the light user's indifference curve with the weighted average of the indifference curve for the heavy user and the iso-profit curve for the light user. Note that while the heavy user's warranty duration with the separating contracts is equal to the optimal full information duration [[T.sup.*].sub.H], the light user's warranty is even shorter than that of the heavy user, the reverse of the full-information case.

PROPOSITION 2: For 0 [less than or equal to] [lambda] [less than or equal to] [[lambda].sub.1], the manufacturer's optimal warranty strategy will be a pooling contract [[[gamma].sup.ME].sub.P] with warranty duration such that [[T.sup.*].sub.H] [less than or equal to] [[T.sup.ME].sub.p] [less than or equal to] [[T.sup.*].sub.L].

Proof: See Appendix.

When [lambda] [less than or equal to] [[lambda].sub.1], the [[T.sup.ME].sub.L], given by (7) is greater than [[T.sup.*].sub.H], and the separating warranty characterized by Proposition 1 is no longer profit-maximizing, since it does not meet IL, the light user's incentive compatibility condition. Thus, for [lambda] [less than or equal to] [[lambda].sub.1], IL is also binding in equilibrium; but since both IL and IH are binding, the only possible solution is a single pooling warranty. This single warranty will have duration [[T.sup.ME].sub.p] defined by:

[u([k.sub.L]T + x) - u([K.sub.L]T)][k.sub.L]T)=[lambda]w[k.sub.H]f([k.sub.H]T) + (1 - [lambda])w[k.sub.L]f([k.sub.L]T).

This equation requires that the light user's indifference curve be tangent to the weighted average of the iso-profit curves for both types. To see that only pooling warranties are feasible in this range, note that for [lambda] [less than or equal to] [[lambda].sub.1], the iso-profit curve is steeper than the indifference curve for heavy users, rendering an extended warranty designed only for heavy users unprofitable. Similarly, an extension intended for light users only would be unprofitable since both types of users would purchase it.

There is thus a range of pooling warranties [[[gamma].sup.ME].sub.P] which are profit-maximizing for a sufficiently low proportion of heavy users. Stiglitz never obtains a pooling equilibrium in his well-known 1977 article on insurance, since the optimal amount of insurance is the same across types: the high-risk type is differentiated only by risk. With warranties, however, the "high risk" type actually has a shorter optimal full-information warranty, and it is this difference between warranties and insurance that leads to these different results. The possibility of pooling warranties obtained here can explain why extended warranties are not available on many products, despite user heterogeneity.

Product Monopoly Without Exclusivity in Extended Warranties

In this section, I consider a different regime in which independent insurers, having the same insurance costs w as the manufacturer, compete in the extended warranty market. The independent insurers can offer one or more contracts for each type i [[gamma].sub.1] = {[[tau].sub.i],[[pi].sub.i]} where [[tau].sub.i] is the period of extension of the warranty and [[pi].sub.i] is the price paid by the consumer for the extension. Expected surplus for a consumer buying the extended warranty would be:

[U.sub.i]([gamma],[gamma])=-p-[pi]+[[[integral of].sup.(T+[tau])].sub.0] u(kt+x)kf(kt)dt+[[[integral of].sup.1/k K].sub.(T+[tau])]u(kt)kf(kt)dt.

The independent contracts are thus supplementary to the contract offered by the manufacturer and increase the total period covered by a warranty. The industry may now be described as a sequential game: The monopolist offers one or more contracts; then independent insurers offer contracts of additional warranty protection; and finally consumers select contracts. The extended warranties must be continuous but not overlapping with a previous warranty, and a consumer might purchase a series of warranties. All firms are assumed to form rational expectations about the extended warranties that independent insurers will offer. The independent insurers are perfectly competitive, so that in equilibrium their prices will exactly reflect the costs of the contracts they offer. Following Rothschild and Stiglitz (1976), equilibrium in the competitive extended warranty market is defined as a set of extended warranty contracts such that, when consumers choose contracts to maximize expected utility, (i) no contract in the equi librium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit.

This study wishes to know how the manufacturer's warranty will differ in the presence of independent insurers. If the manufacturer has optimally offered a pooling warranty even without the threat of entry by independent insurers, then this warranty is unaffected by the possibility of entry, since any extension beyond the pooling warranty would be strictly unprofitable:

PROPOSITION 3: If 0 [less than or equal to][lambda] [less than or equal to] [[lambda].sub.1], then independent insurers cannot profitably enter.

Proof: Any extension designed only for heavy users will be unprofitable; any extension designed only for light users will attract both types; and any pooling extension will be unprofitable.

In contrast, if the manufacturer's optimal pre-entry contract separates the two types by the offer of two incentive-compatible contracts, the optimal strategy will be changed by the possibility of entry. The manufacturer cannot make any economic profits in the market for extended warranties because of competition. The monopolist will therefore choose between not serving the light users at all and offering the most profitable single base warranty, where the second superscript indicates that the manufacturer does not have exclusivity in the extended warranty market. This warranty, depicted in Figure 3, will have duration defined by the tangency of the indifference curve of light users and the iso-profit curve of pooled users. Thus, the condition determining the duration of this base warranty will be just the same as (8). This leads to:

PROPOSITION 4: If [[lambda].sub.1] [less than] [lambda] [less than or equal to] 1, competition from independent insurers will lead to an increase in the base warranty duration (a) from [[T.sup.ME].sub.L] to [[T.sup.MN].sub.P] if both types continue to be served or (b) from [[T.sup.ME].sub.L] to [[T.sup.*].sub.H] if the manufacturer does not serve the light users.

Proof: See Appendix.

In addition to the pooling warranty offered by the manufacturer, for any [[T.sup.MN].sub.P] [less than] [[T.sup.*].sub.H], the competitive insurers (and perhaps the manufacturer) will offer an extended warranty to provide optimal coverage for heavy users, so that ex post there is separation of user types. The heavy user's contract will extend the total period of coverage to [[T.sup.*].sub.H] at the fair-odds price w [[[integral of].sup.[[T.sub.H].sup.*]].sub.[[T.sup.MN].sub.P]] [K.sub.H]f([k.sub.H]t]dt. The extended warranty purchased by the heavy user can be seen in Figure 3 as the difference between [psi] and [[[gamma].sup.MN].sub.P].

Welfare Effects of Exclusivity in Extended Warranties

Provided that both types continue to be served, permitting competition in the extended warranty market has a non-negative welfare effect when the product market is monopolistic. Transfers of money in this model have no welfare effect; the only variable of interest is the length of warranties. The closer they are to the full information lengths [[T.sup.*].sub.H] and [[T.sup.*].sub.L], the larger is welfare. Thus, because competition in extended warranties lengthens the warranty obtained by the light user (while that of the heavy user remains the same), welfare increases.

If, however, light users form a sufficiently small part of the market, then the light user may not be served at all. In this case, the base warranty is equal to the full-information warranty for the heavy user, and the light user will not buy the product. The welfare result here is the familiar one of price discrimination. If the profits from light users are not large enough, then without the possibility of price discrimination, they may not be served and welfare decreases. But if light users continue to be served, the elimination of price discrimination by independent insurers increases welfare.

ASYMMETRIC INFORMATION WITH A COMPETITIVE PRODUCT MARKET

In this section, I undertake the same exercise as above, but with the assumption that the product manufacturers are in a perfectly competitive industry In this case, the consumer gains a surplus while firms make zero profits. With full information, the consumer surplus in the competitive case is exactly equal to the firm's profit in the monopoly case. As before, consumers are assumed to be of two a priori unidentifiable types.

Exclusivity in Extended Warranties

Each firm is initially assumed to have exclusivity in the sale of extended warranties for its own product. This analysis is similar to that of Rothschild and Stiglitz (1976). As in their models, if the proportion of heavy users is sufficiently high, then a separating equilibrium will be obtained, and if the proportion of heavy users is below a critical value, then no equilibrium may exist at all. Equilibrium in this competitive base warranty market is defined as a set of contracts such that, when consumers choose contracts to maximize expected utility, (i) no contract in the equilibrium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit. Each contract consists of the sale of the product plus a warranty for the price p + s. The base warranty is restricted to start at the time the consumer purchases the product; and the profitability of any base warranty will also depend on the set of extended warranties that are offered subsequently.

With heterogeneous consumers, there are two possible types of solutions again depending on [lambda]. To write Proposition 5, I define, in the appendix, a new critical value [[lambda].sub.2] [epsilon][0,1]. Here [[lambda].sub.2] is the critical value of [lambda] at which light users are indifferent between pooling and separating contracts.

PROPOSITION 5: (a) For [lambda] [greater than] [[lambda].sub.2], separating warranties will be offered in equilibrium such that the heavy user obtains a warranty of duration [[T.sup.*].sub.H], while the light user obtains a shorter warranty of duration [[T.sup.CE].sub.L], and firms make zero profits on each type of warranty; and (b)for [lambda][less than or equal to] [[lambda].sub.2], no equilibrium will exist.

Proof: The proof of this proposition is identical to that provided by Rothschild and Stiglitz (1976) and Wilson (1977).

The separating warranty contracts will be such that the heavy user's duration is again the full-information one, while the light user obtains a short warranty, constrained by the heavy user's incentive compatibility. Both warranties will be priced at cost. This equilibrium is pictured in Figure 4, in which as will be seen, [[[gamma].sup.CE].sub.L] is constrained to be priced at fair cost and to be the best possible contract for the light user subject to the incentive compatibility constraint of the heavy user, who obtains the warranty contract [[[gamma].sup.CE].sub.H].

If [lambda] [less than or equal to] [[lambda].sub.2], the well-known arguments of Rothschild and Stiglitz will show that there is no equilibrium, since both pooling and separating candidate equilibria can be broken profitably. Their argument is restated briefly, with reference to Figure 5. Both heavy and light users will prefer the zero-profit pooling warranty [[[gamma].sup.CN].sub.P] to the separating warranties [[[gamma].sup.CE].sub.H] and [[[gamma].sup.CE].sub.L]. However, this pooling warranty is vulnerable to the defection of one firm to the contract [xi], which will attract only light users. But then only heavy users will buy [[[gamma].sup.CN].sub.P], which will be withdrawn. Thus, neither a pooling nor a separating equilibrium is possible.

The Effect of Independent insurers on a Competitive Product Market

This section examines how competition in the extended warranty market will affect manufacturer warranties. The result here does not depend on the proportion of heavy and light users:

PROPOSITION 6: In the presence of competitive independent insurers, competitive manufacturers will offer a single base warranty [[[gamma].sup.CN].sub.P].

Proof: See text.

First, consider the effect of non-exclusivity in the range of [lambda] [greater than] [[lambda].sub.2]. It is clear that the presence of competitive extended warranties renders the manufacturers' separating warranties loss-making, since heavy users will purchase the warranties designed for light users plus an extension from the independent insurers. [13] (In Figure 4, all users would buy the base warranty [[[gamma].sup.CE].sub.L], and heavy users would also buy the extension [Psi].) Therefore, the manufacturers can only offer pooling warranties. The only feasible competitive pooling warranties fall on the zero profit pooling curve, designated [IP.sub.p] in Figure 4. Independent insurers can offer supplementary extended warranties only to heavy users, since any extended warranty designed for light users will only attract all users.

We now examine the effect of independent insurers when l[less than or equal to][l.sub.2] Consider the base warranty [[[gamma].sup.CN].sub.P] shown in Figure 5. With exclusivity in extended warranties, this base warranty can be undermined by the offer of a contract such as [xi] which will attract only light users; thus only the heavy users buy [[[gamma].sup.CN].sub.P], leading to equilibrium breakdown. However, the presence of independent insurers has the effect of generating a stable equilibrium. Given the base warranty [[[gamma].sup.CN].sub.P], independent insurers can offer the extended warranty [Y.sub.H] = [psi] - [[[gamma].sup.CN].sub.P]. Contract [xi], were it offered, would allow independent insurers to offer the two contracts [Y.sub.H] = [psi] - [xi] and [Y.sub.L] = [phi] - [xi]. This would be possible since these two contracts together would make zero profits. (Note that [Y.sub.L] is the best extended warranty contract that can be offered to light users since anything below the pooling line would attr act other companies to offer yet further extended warranties.) Since light users would prefer the total contracts [phi] to [[[gamma].sup.CN].sub.P], and heavy users would prefer [psi] to [psi], all consumers would buy the contract [xi] plus the extensions. But then [xi] would become a loss-making pooling contract. Thus, no base warranties could be offered which would break the pooling equilibrium. The pooling warranty [[[gamma].sup.CN].sub.P] also cannot be broken by any extended warranty since it makes zero profits regardless of the extended warranties.

Welfare Effects of Exclusivity in Extended Warranties

If [lambda] [greater than] [[lambda].sub.2], that separating warranties are offered when manufacturers have exclusivity in their warranties, the loss of exclusivity may have positive or negative welfare effects. If the proportion of heavy users is very high, then the new base pooling warranty [[[gamma].sup.CN].sub.P] may be even shorter than the separating base warranty [[[gamma].sup.CN].sub.L]. In this case, welfare will fall with the introduction of competition in extended warranties. However, if the proportion of heavy users is not so high, then the new base pooling warranty will be longer than the base warranty with exclusivity, in which case welfare would increase. Thus even in the perfectly competitive market, there may be a welfare gain from forbidding exclusivity conditions on extended warranties. If [lambda] [greater than] [[lambda].sub.2], then it is only the presence of independent insurers that permits equilibrium in the market, and it is thus difficult to make any determination of how welfare would be changed.

CONCLUSIONS

This article suggests some direction for the regulation of the extended warranty market. Manufacturers may very well wish to make supplies to a dealer or retailer conditional on that dealer's carrying only the manufacturer's extended warranty. Should this be permitted? There are two common approaches to this problem. First, the "Chicago" approach which would permit tying regardless of market structure. A second approach is the one generally adopted by the U.S. courts: If the manufacturer has market power in the "tying" (product) market, then it should not be permitted to practice tying (or as in this case requirements contracting) into the "tied" (extended warranty) market; but if the manufacturer has no market power in the first market, then tying should be permitted. Neither of these approaches is optimal. If the product market is characterized by monopoly or oligopoly, then competition in extended warranties is beneficial, provided that light users are not excluded from the market, and the monopolist shoul d then be prevented from excluding the independent insurers from the extended warranty market (i.e., tying). On the other hand, even if the product market is itself characterized by vigorous competition, then tying in the secondary market for extended warranties may lead to a decrease in welfare! Thus, courts should be sympathetic to retailers who object to supplier requirements that exclude third-party warranties. The conclusion that emerges is inline with other economics analysis of aftermarkets, which suggests that case by case judgements are required rather than any per se rule. [14] Table 1 presents the results of the article in a summary format.

The practice of requiring dealers to carry only the manufacturer's replacement parts may be interpreted analogously to a restriction on extended warranties, since the manufacturer is thus restricting competition in service after the base warranty. This type of requirements contracting has been a contentious issue in the courts at least since 1936, when the U.S. Supreme Court upheld a lower court decision approving General Motors' requirement that GM dealers install only GM replacement parts. [15] There have been a variety of cases on similar grounds since then, most of which have revolved around two issues: (1) whether third-party replacement parts might be inferior and result in damage to the manufacturer's reputation and (2) whether the requirement would harm competition. [16] The analysis in this article helps to clarify another potentially important issue: Third-party replacement parts may affect welfare by preventing schemes of price discrimination by manufacturers. It was exactly this reasoning that the successful plaintiffs used in the 1992 U.S. Supreme Court case on Kodak. [17]

There are some opportunities for extending this research. In particular, it would be interesting to try to test empirically whether the legal status of manufacturer restrictions on independent extended service contracts in different jurisdictions varied sufficiently to cause different outcomes in manufacturer base warranties. While this article discusses extended warranties, its analysis applies rather more widely. For example, a simple variation on the model presented here would have manufacturers choosing different qualities, instead of different warranties, in order to separate heavy and light users. In general, the model may be applied to any case in which ancillary services provided by a third-party can effectively bridge the gap between contracts that were intended to separate types of users who impose different costs on the firm, and suggests that it may often be welfare-reducing to allow the first firm to exclude the third party. What is most surprising is that this result will sometimes hold even whe n the first firm is in a competitive industry.

APPENDIX

LEMMA 1: Step 1. Consider any pair of contracts {[[gamma].sub.H], [[gamma].sub.L]}, satisfying the four incentive compatibility and participation constraints, and choosing [[gamma].sub.L] to be such that [U.sub.L] ([[gamma].sub.L]) = 0. By (4), this requires [T.sub.H] [greater than or equal to] [T.sub.L] and [P.sub.H] [greater than or equal to] [p.sub.L]. Now consider a second pair of contracts {[[gamma].sub.H], [[gamma].sub.L]}, where [T.sub.L] = [T.sub.L] and [P.sub.L] [less than] [p.sub.L] [less than] [p.sub.L], such that {[[gamma].sub.H], [[gamma].sub.L] also satisfy the four incentive compatibility and participation constraints. Compared to [[gamma].sub.L], profits may always be increased by choosing [[gamma].sub.L]. Therefore, PL is always binding.

Step 2. Consider two contract combinations {[[gamma].sub.H],[[gamma].sub.L]} and {[[gamma].sub.H],[[gamma].sub.L]}, where [T.sub.H] = [T.sub.H] and [p.sub.H] [less than] [p.sub.H], and with both contract combinations again satisfying the four incentive compatibility and participation constraints. By (4), this requires [T.sub.H] [greater than or equal to] [T.sub.L] and [p.sub.H] [greater than or equal to] [p.sub.L] and [T.sub.H] [greater than or equal to] [T.sub.L] and [p.sub.H] [greater than or equal to] [p.sub.L]. For any [[gamma].sub.H] such that [U.sub.H]([[gamma].sub.H])[greater than] [U.sub.H]([[gamma].sub.L]), profits may be increased by increasing [p.sub.H] until IH is binding.

Step 3. The implication of PL and IH binding is that PH cannot bind if both types are served. Since by (4) the indifference curves for the high type are steeper than those for low types, IH binding requires PH not binding unless [[gamma].sub.L] = {0,0}.

Step 4. If only one contract is offered, by definition IL must be binding. However, if separating contracts are offered, IL cannot be binding. This is evident given (4) and IH.

PROPOSITIONS 1 AND 2: Proof: First to show that [[lambda].sub.1], is well-defined, it is required that 0 [less than or equal to] [[lambda].sub.1] [less than or equal to] 1. It is required that

w[k.sub.1]f([k.sub.L][T.sub.H]*)[less than or equal to][u[k.sub.L][T.sub.H]*+x)-u([k.sub.L][T.sub.H]*)][k.sub.1]f([k.sub. L][T.sub.H]*)[less than or equal to]w[k.sub.H]f([k.sub.H][T.sub.H]*).

The first inequality is obviously true. By (5), the second inequality is equivalent to

[u([k.sub.L][T.sub.H]*+x)-u([k.sub.L][T.sub.H]*)][k.sub.1]f([k.sub.L] [T.sub.H]*)[less than or equal to][u([k.sub.H][T.sub.H]*+x)-u[k.sub.H][T.sub.H]*)[k.sub.H]f([k.sub.H ][T.sub.H]*).

This is true by (4). Thus 0 [less than or equal to] [[lambda].sub.1] [less than or equal to]1.

For [lambda] [greater than] [[lambda].sub.1], the problem for the monopolist is to maximize

[max.sub.[T.sub.H],[p.sub.H],[T.sub.L],[p.sub.L]] [lambda][[p.sub.H] - w[[[integral of].sup.[T.sub.H]].sub.0] [k.sub.H]f([k.sub.H]t)] + (1 - [lambda])[[p.sub.l] - w[[[integral of].sup.[T.sub.L]].sub.0] [k.sub.L]f([k.sub.L]t)]

subject to the two binding constraints

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The six first order conditions for the six unknowns are

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [delta][pi]/[delta][T.sub.H] = -[lambda]w[k.sub.H]f([k.sub.H][T.sub.H]) + [[beta].sub.1][u([k.sub.H][T.sub.H] + x) - u([k.sub.H][T.sub.H])][k.sub.H]f([k.sub.H][T.sub.H]) = 0

(c) [delta][pi]/[delta][p.sub.L] = 1 - [lambda] + [[beta].sub.1] - [[beta].sub.2] = 0

(d) [delta][pi]/[delta][p.sub.H] = [lambda] - [[beta].sub.1] = 0

plus the two binding constraints. By (c) and (d), [[beta].sub.1] = [lambda] and [[beta].sub.2] = 1 are obtained. It is obvious from (b) that [[T.sup.ME].sub.H] = [[T.sub.H].sup.*]. The application of IH and PL binding then immediately identifies [[T.sub.L].sup.ME].

For [lambda] [less than] [[lambda].sub.1], the constraints change: in particular, the incentive compatibility constraints for the light user becomes binding, since the separating solution has [[T.sub.L].sup.ME] [greater than] [[T.sub.H].sup.ME], so that IL is no longer met. Thus IL becomes binding for such values of [lambda], so that

[[[integral of].sup.[T.sub.L]].sub.0]] [u([k.sub.L]t + x]) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt - [p.sub.L] = [[[integral of].sup.[T.sub.H]].sub.0]] [u([k.sub.L]t + x) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt - [p.sub.H].

Now the problem can be simplified to the pooling problem, since the two binding incentive compatibility constraints require that [[gamma].sub.L] = [[gamma].sub.H]:

[max.sub.[T.sub.p],[p.sub.p] [p.sub.p]] - w[[[integral of].sup.[T.sub.p]].sub.0] [[lambda][k.sub.H]f([k.sub.H]t) + (1 - [lambda])[k.sub.1]f([k.sub.L]t)]dt (9) subject to the binding constraint [p.sub.p] = [[[integral of].sup.[T.sub.p]].sub.0] [u([k.sub.L]t + x) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt. Substituting this constraint into the objective function and taking the FOC yields the result [lambda]w[k.sub.H]f([k.sub.H]T) + (1 - [lambda])w[k.sub.L]f([k.sub.L]T) = {u([k.sub.L]T + x) - u([k.sub.L]T)}[k.sub.L]f([k.sub.L]T), which defines the optimal [[T.sup.ME].sub.P]. PL defines the optimal [[p.sup.ME].sub.p], thus identifying the optimal pooling solution. Here [[lambda].sub.1] is the largest [lambda] for which this optimal [[T.sup.ME].sub.P] [greater than or equal to] [T.sub.H] *.

For completeness, I show that this pooling warranty is more profitable than any separating warranty if [lambda] [less than] [[lambda].sub.1]. A separating warranty would require an optional extended warranty to be offered to either the heavy or the light user. No profitable extended warranty can be sold to the heavy user, since the slope of the iso-profit curve is steeper for the heavy user than the indifference curve for t [greater than or equal to] [T.sub.H] *. No profitable extended warranty can be offered to the light user only, since by (4), the slope of the indifference curve for the heavy user is steeper than the slope of the light user's indifference curve and any extended warranty beyond [T.sub.p] satisfying PL will be purchased by both types, making it a pooling warranty. But we have already found the optimal pooling warranty.

PROPOSITION 4: Proof: Following the entry of any independent insurers, the manufacturer makes zero profits on any optional extended warranty. The manufacturer will therefore choose between the profit-maximizing pooling contract, in which case both types buy the product, or the profit-maximizing contract for heavy users only.

(a) The case where both types are served is considered first. The objective function is the same as (9). The first order condition is given by (8) which solves for the solution [[T.sup.MN].sub.p]. Compared with the first order condition for the separating solution (7), only the first term on the RHS is different. Since

[u([k.sub.H]T + x) - u([k.sub.H]T)][k.sub.H]f([k.sub.H]T) [greater than] w[k.sub.H]f([k.sub.H]T) T [less than] [[T.sub.H].sup.*], the RHS of (8) is less than the RHS of (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] t [less t [[T.sub.H].sup.*].

But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore [[T.sup.MN].sub.p] [greater than or equal to] [[T.sup.ME].sub.L]

(b) The case in which only the heavy user is served is now considered. In this case, the monopolist sets the full-information warranty contract [[gamma].sub.H] * and makes profits of [lambda][[pi].sub.H] ([[gamma].sub.H] *) but loses all profits on the light user. Thus, the monopolist will choose to serve only the heavy user if

[lambda]{[[pi].sub.H]([[gamma].sub.H] *) - [[pi].sub.H]([[[gamma].sup.MN].sub.P])} [greater than] (1 - [lambda]){[[pi].sub.L]([[[gamma].sup.MN].sub.P]) + (s - c)}.

Definition: [[lambda].sub.2] is defined as the [lambda] which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [[T.sup.CE].sub.L] is defined by

W[[[integral of].sup.[[T.sub.H].sup.*]].sub.0][k.sub.H]f([k.sub.H]t)dt-[[[integral of].sup.[[T.sup.*].sub.H]].sub.[[T.sup.CE].sub.L]][u([k.sub.L]t+x)-u( [k.sub.L]t)][k.sub.1]f([k.sub.L]t)dt = w[[[integral of].sup.[[T.sup.CE].sub.L]].sub.0][k.sub.1]f([k.sub.L]t)dt.

Aidan Hollis is Assistant Professor, Department of Economics at the University of Calgary. The author is indebted to two anonymous referees, Nancy Gallini, Frank Mathewson, Mike Peters, Ralph Winter, and participants at seminars at the University of Toronto, the University of Western Ontario, l'Universite de Montreal, and the Institut d'Analisi Economica in Barcelona for helpful comments.

(1.) Cited in Lutz and Padmanabhan (1995).

(2.) Plotkin (1985) and Therrien (1991).

(3.) Therrien (1991).

(4.) Two other explanations of the function of warranties are provided by Salop (1977), who observed that with imperfect consumer information on product qua1ity, the warranty may be used as a tool of price discrimination; and by Braverman, Guasch, and Salop (1983) who noted that a warranty may be used as part of a two-part tariff to extract consumer surplus.

(5.) See, e.g., Bryant and Gerner (1982), Gerner and Bryant (1981), and Priest (1981).

(6.) Cooper and Ross (1985), Emons (1988), Dybvig and Lutz (1993), Soberman (1995), and Lutz and Padmanabhan (1998).

(7.) Automobiles are a partial exception to this rule, since it is possible to limit warranties by the distance driven. Even in this case, however, the odometer does not specify the type of terrain or the style of driving, which influences the expected rate of car breakdown. So in the case of automobiles, the analysis should not refer to time of driving but to intensity of use.

(8.) Note that this analysis ignores the possibility of any kind of deductible. Many types of warranties do not have explicit deductibles, although some of them do: for example, car warranties often have a deductible, and some warranties cover only labor and not parts. Including deductibles in the analysis would not change its flavor or conclusions but would make it considerably more complicated.

(9.) The model of the warranty employed here does not perfectly match the observed form of repair or replacement warranties, since making x a constant allows for the warranty to be exercised only once. Repair or replacement warranties actually have the form x(T - t) where x(z) [greater than] 0 and x' [greater than] 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] z [grea to] 0 and x(z) = 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] z [less than] 0. In this form to the consumer of the warranty is declining with t because after repair or replacement, the product is returned with only the balance of the warranty. As it turns out, only the marginal condition of warranty value at t = T is of interest, so that the model may be simplified by using a constant without any change in the results.

(10.) More generally, the warranty servicing cost could be written as w(kt), with the restriction that it be continuous and non-decreasing.

(11.) The second order condition is

[d.sup.2][pi]/[dT.sup.2] = [k.sup.2]{[u([kT + x) - u(kT0 - w]f'(kT) + [u'(kT = x) - u'(kT)]f(kT)}.

The second term is negative, and the first term is equal to zero when the first order condition is satisfied so that this is a local maximum.

(12.) This assumption will not apply globally, since the probability density of heavy users breaking their product will eventually become lower than that of light users when most heavy users have already broken their product. However, the assumption is not unreasonable: we do not observe warranties which are so long that the probability of light users breaking the good is higher than that of heavy users, perhaps since the cost of those warranties is higher than the utility gains to consumers.

(13.) The potential free-riding by independent insurers, which is what drives Proposition 6, is analogous to the "externality" which Lutz and Padmanabhan (1995) find with consumer moral hazard. They find that competitive provision of extended warranties leads to zero base warranties because consumers will take less care of the product than is consistent with a competitively priced warranty. Analogously, in the adverse selection story here, it is not possible for the manufacturer to offer separating warranties. However, the outcome with adverse selection is that the manufacturer will offer a single base warranty which in general will be of strictly positive length.

(14.) See, e.g., Borenstein, MacKie-Mason, and Netz, 1996.

(15.) Pick Manufacturing Co. v. General Motors Corp. et al., 80 F. 2d 641 (1935), affirmed per curiam, 299 U.S. 3 (1936).

(16.) In re General Motors Corp. and General Motors Sales Corp., 34 FTC 58, 86 (1941); Dicto Graph Products, Inc., v. Federal Trade Commission, 217 F. 2d 821, 828 (1954): Englander Motors, Inc. v. Ford Motor Co., 267 F. 2d 11(1959); Alles Corp. v. Senco Products, Inc., 329 F. 2d. 567 (1964).

(17.) "Eastman Kodak v. Image Technical Services, Inc. 504 U.S. 451(1992).

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This article investigates whether the contractual exclusion of third-party extended warranties should be legally permissible, using a model incorporating consumer heterogeneity. The welfare effects of competition in the market for extended warranties are shown to depend on the degree of competition in the product market. In contrast to the approach typically adopted by the courts, the article argues that manufacturers should not generally be permitted to practice requirements contracting in extended warranties, even when the product market is competitive.

INTRODUCTION

Competition authorities and economists have recently been wrestling with the question of whether manufacturers should have the right to monopolize aftermarkets for their products. An important class of such aftermarkets, characterized by adverse selection, is extended warranties and service contracts. This paper investigates the problem of whether the contractual exclusion of third-party extended warranties should be legally permissible. Consumer heterogeneity sometimes leads manufacturers to offer a menu of warranty contracts, typically implemented as a base warranty and an optional extended warranty. For most products on which a manufacturer offers an extended warranty, it faces competition from third-party service contracts. Some manufacturers, however, make it a condition of supplying a retailer that she carry only the manufacturer's extended warranty. A recent case is typical of the legal dilemma. Ford of Canada announced that as a condition for the use of its financing plan, dealers had to commit to exc lusive sales of Ford's extended warranty program. When the Federal Automobile Dealers Association threatened to take it before the Competition Bureau, Ford withdrew the condition, leaving dealers free to continue selling third-party extended service contracts. Should manufacturers generally have the right to impose such exclusivity conditions on retailers? This article argues that competition in extended warranties has an important effect on the base warranty, and therefore on welfare.

In both the United States and Canada, the legality of such requirements contracting is still uncertain, despite the magnitude of the extended warranty and service contract markets. Extended warranties are one of the most commonly purchased forms of insurance and marketed along with most large and many small appliances. Sears alone is reported to have sold over $1 billion worth of extended warranties in 1991 in the United States. Ford recorded profits in excess of $100 million from sales of extended warranties in 1988, despite fierce competition from independent insurers. [1] Around 40 percent of all automobiles and major appliances are sold with some form of extended warranty, typically with a large profit margin. [2] For retailers of products carrying extended warranties, the profits can be significant: Some analysts estimate that around half of operating profits for big consumer electronics chains come from sales of extended warranties. [3]

Although extended warranties are a form of insurance, they have quite unique features. Unlike most other forms of insurance, warranty contracts do not usually vary by amount of insurance but by duration instead. The reason for variation in the duration could be heterogeneity in consumer risk aversion or heterogeneity in the consumer's expected intensity of use. I model it as the latter in this article. This suggests that there will be opportunities for third degree price discrimination and the resulting welfare losses that come from distorting contracts for low intensity users to keep high intensity users from buying those contracts. The presence of third party extended warranties complicates the situation and may increase or decrease welfare. What is critical here is that because the warranty contract varies by duration and not by amount of coverage, the standard screening analysis employed in most insurance problems (beginning with Michael Rothschild and Joseph Stiglitz (1976) and Charles Wilson (1977)) doe s not apply directly to the warranty case.

In the next section, the relationship of this article to the previous literature on warranties and extended warranties is explained. The following section develops a model of warranties in which firms choose warranty duration and price. If firms cannot distinguish between heavy and light users, it is shown that they may offer separating or pooling warranties, where the separating warranty is simply characterized by the offer of an optional extended warranty. Sections on the monopolistic product market and the competitive product market examine the effects of competition in extended warranties. If the product market is monopolistic, then the presence of third-party insurers will destroy the price discrimination scheme of the manufacturer. This will be welfare enhancing only if "light" users continue to buy the product. Similarly, if the product market is characterized by competition, attempts to impose exclusivity restrictions on extended warranties may often be welfare-decreasing. This result contrasts with c urrent jurisprudence which is lenient to exclusivity restrictions if the product market is competitive. The article concludes with a brief discussion of policy implications, and with suggestions for further research.

RELATED LITERATURE

Economic analysis of warranties is divided by the principal function attributed to warranties: Some studies are premised on the notion that the function of warranties is to shift risk-bearing from risk-averse consumers to risk-neutral warrantors; while in others, warranties serve to provide a signal of product quality. [4] The seminal paper on warranties as a form of insurance is by Geoffrey Heal (1977), and it is this function of warranties which is of interest in the present article. Warranties typically seem to provide at best partial insurance, [5] which as will be seen may be the result of consumer heterogeneity, as observed by Winand Emons (1989) and V. Padmanabhan (1995), or of consumer moral hazard. Michael Spence (1977) was the first to show that warranties could be an informative signal for consumers if it is more costly for low quality manufacturers to provide long warranties than for manufacturers of high-quality products in a competitive market. Sandy Grossman (1981) extended the analysis to mono polistic markets and found that monopolists would always offer full warranties because not to do so would be a bad signal to consumers. Esther Gal-Or (1989), Linda Welling (1989), and Nancy Lutz (1989) derived conditions limiting the effectiveness of such signals in various environments. A series of related papers on two-sided moral hazard considered the problems of using warranties to minimize producer moral hazard in the presence of consumer moral hazard. [6] Steven Matthews and John Moore (1987) consider the problem of a monopolist who chooses both quality and warranty and explore the sensitivity of their results to different restrictions on the usual single crossing property which is employed in this article.

The focus of this article is the market for extended warranties, which, despite its importance, has attracted very little economic study. The two exceptions are work by Lutz and Padmanabhan. Their 1995 article considers a puzzle in the market for contact lenses, where, although warranties may be purchased as an option, the standard base warranty is zero. They attribute this to the presence of third-party insurers, whose extended warranties cause a negative externality on base warranties, since consumers with more coverage will be less careful in handling their lenses. They conclude that where consumer moral hazard is a problem and average product durability can be predicted with reasonable accuracy by an independent insurer, this externality will push base warranties to zero. This result applies to a particular type of product, since on the vast majority of products for which third-party extended warranties are available, base warranties are far from zero. Evidently some other paradigm is required to explain the variety of warranties and extended warranties available on different types of products. This article is based on the idea that warranties are heterogeneous because consumers are heterogeneous in their intensity of use and presents a model that helps to explain the variety of warranties which we observe. In contrast to Lutz and Padmanabhan's contact lens article, it relies on adverse selection rather than on moral hazard.

Lutz and Padmanabhan (1998), building on Matthews and Moore (1987), examine the effect of competition in extended warranty markets when the manufacturer is a monopolist in the product market and offers products of two different qualities to consumers with heterogeneous valuations. Warranties are modeled as a cash payment. They show that the presence of competitive extended warranties may cause manufacturers to change the quality of the product offered. This paper is closest in spirit to the model presented below, but differs in key results because warranties are modeled as a cash payment rather than as repair or replacement; and because product quality is determined endogenously.

The model used in this article differs from most other models in the literature because it focuses on the duration of the warranty. This follows from Philip Dybvig and Nancy Lutz (1993), who model warranties as a time-variable payment in case of breakdown. They show that the "block" form of warranties typically seen, with high coverage initially and zero coverage later, is optimal when there is double-sided moral hazard. The focus on duration is useful because it more closely resembles the choices that warrantors actually make, and because these choices are not isomorphic to models where warranties are a payment which varies by amount rather than duration of coverage.

THE MODEL

Consumers

The model specification is designed to allow consumer heterogeneity in the aspect which matters most for warranty length: the intensity of use, or the probability of breaking a product within a given period. In order to focus on this, consumers are assumed to be homogeneous in other ways, including their ex ante valuation of the product. In addition, consumers are risk averse, which is what makes the warranty valuable to them. They derive additive utility from money and the services of at most one unit of a durable product. By not buying the product, consumers obtain a reservation expected utility equal to their income, y.

Consumers of type i who buy the product obtain services of [k.sub.i] units per unit of time until the product breaks, so that if it breaks at time t, [k.sub.i]t is the cumulative amount of services obtained. The parameter k is introduced to reflect the fact that almost all products have some aspects in which intensity of use or roughness affects product durability. The good breaks with probability density f([k.sub.i]t) over the course of being used and all goods are broken by the time they have yielded K units of services,

[[[integral of].sup.1/[k.sub.i] K].sub.0] [k.sub.i]f([k.sub.i]t)dt = 1 for all I and K [epsilon] (0, [infinity]).

A product that breaks after yielding services of [k.sub.i]t units yields utility of u([k.sub.i]t) , where u is continuous and u,[greater than] 0 and u" [less than]0. Thus, expected product lifetime is inversely related to intensity of use [k.sub.t] but expected total product services is invariant to k. A broken product cannot be repaired and yields zero utility. Thus if a consumer of any type i were to buy the good (without a warranty) at price s, she would obtain von Neumann-Morgenstern expected utility independent of i

Y - s + [[[integral of].sup.1/[k.sub.i] K].sub.0] u([k.sub.i]t)kf([k.sub.i]t)dt.

Note that this expression integrates not over services through time but over cumulative services times the probability of breakdown at any time t. Buying the product without a warranty is essentially a lottery, with a range of possible outcomes, depending on t, the actual functioning life of the product. In order to keep the exposition simple, there is no discounting.

A warranty contract [[gamma].sub.j] consists of the pair {[T.sub.j], [p.sub.j]}. Purchasing a warranty at price p ensures that if the product breaks while the warranty is still in effect, that is, for t [less than or equal to] T, where T is the terminal date of the warranty, the product will be repaired or replaced. A repaired or replaced product is expected to provide x additional units of service. Ideally, a warranty should guarantee some amount of usage, but a limitation that firms face in setting warranties is that they generally cannot make them conditional on usage, which is unobservable; so they use simple time restrictions instead. [7] Thus, warranties typically expire after a set period of months or years. Following Dybvig and Lutz (1993), the warranty is restricted to a "block" form, so that there is for some period a constant level of benefits in case of breakdown, at the end of which period coverage drops discontinuously to zero. [8] A consumer of type i who buys the product with warranty thus obt ains expected utility

y - s - [p.sub.j] + [[[integral of].sup.[T.sub.j]].sub.0] u([k.sub.i]t + x)([k.sub.i]f([k.sub.i]t)dt + [[[integral of].sup.1/[k.sub.i]K].sub.[T.sub.j]] u([k.sub.i]t)[k.sub.i]f[k.sub.i]t)dt [9]

Consumers value warranty length because it increases the expected number of units obtained from the durable good, but their marginal valuation of the warranty is declining with T, since u" [less than] 0. The expected surplus a consumer of type i obtains from contract [[gamma].sub.j] is

[u.sub.i]([[gamma].sub.j]) = -[p.sub.j] + [[[integral of].sup.[T.sub.j]].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]f([k.sub.i]t)dt.

Firms

For the present the manufacturer is assumed to be a monopolist, although as shown in the last section of the article with suitable adjustments the model also applies to the case of competitive manufacturers. The manufacturer has marginal manufacturing cost c, and for each buyer of type i, warranty servicing cost w for breakdown in any state t [less than or equal to] T, and expected warranty servicing cost given [T.sub.i] of w [[[integral of].sup.[T.sub.i]].sub.0] [k.sub.i]f([k.sub.i]t)dt. [10] Its expected profits per unit sold to a buyer of type i is s + [p.sub.i] - c - w [[[integral of].sup.[T.sub.i]].sub.0] [k.sub.i]f ([k.sub.i]t)dt. Because consumers are assumed to have homogeneous valuations of the product itself, the manufacturer's maximization problem may be simplified by splitting it into two components. First the manufacturer sets s equal to the value to the consumer of the product without a warranty. Define [[gamma].sub.i] as the warranty contract which maximizes expected profits from the sale of the warranty contract only to buyers of type i, [[pi].sub.i]([[gamma].sub.i]) = [p.sub.i]- w [[[integral of].sup.[T.sub.i].sub.0] [k.sub.i]f([k.sub.i]t)dt. The monopolist's second problem, which will be focused on, is to choose [[gamma].sub.i].

The Full-Information Benchmark

If the firm could identify the consumer type, [p.sub.i] would be set equal to the consumer's valuation of a warranty of duration [T.sub.i]:

[p.sub.i] = [[[integral of].sup.[T.sub.i]].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]f([k.sub.i]t)dt. (1)

The maximization problem for the firm is then simply

[[max.sub.[T.sub.i] [[[integral of].sup.[T.sup.i].sub.0] [u([k.sub.i]t + x) - u([k.sub.i]t) - w][k.sub.i]f([k.sub.i]t)dt (2)

for each type. The first order condition for this problem requires that

u([k.sub.i][T.sub.i] + x) - u([k.sub.i][T.sub.i]) = w [11] (3)

This equality yields an interior solution, called [[T.sup.*].sub.i], given two reasonable conditions: u(x) - u(0)[greater than] wand u(K + x) - u(K) [less than] w. Essentially these conditions require that the marginal utility of the warranty be greater than the cost of warranty servicing for low outcomes of t and less for high outcomes of t. The natural outcome of (3) is that the optimal full-information warranty length [[T.sup.*].sub.i] is inversely proportional to [k.sub.i], so that

[[T.sup.*].sub.i] = [k.sub.j]/[k.sub.i][[T.sub.j].sup.*].

A convenient graphical interpretation of this is given by Figure 1, which shows isoprofit and indifference curves for two types of consumers buying from the monopolist. Different levels of profits and consumer utility are given by vertical displacements of these curves. Utility is higher for lower indifference curves, while profit is higher for higher iso-profit curves. The slope of the indifference curve at any given t is just [u([k.sub.i]t + x) - u([k.sub.i]t)][k.sub.i]t)][k.sub.i]f([k.sub.i]t), the marginal value to the consumer of extending the warranty; while the slope of the iso-profit curves is [wk.sub.i]f([k.sub.i]t), the marginal cost of extending the warranty. As a result of the linear additivity of utility of money in the model, the negative of consumer surplus from the warranty is given by the intercept of the consumer's indifference curve on the coordinate axis extending below the zero, while the firm's profit per consumer is just the intercept of the iso-profit curve above the zero. When the mon opolist has full information about consumer types, consumer surplus will be reduced to zero, as in Figure 1. Note that for both types the optimal price is the same, and only the duration varies, with the light user obtaining a longer warranty. By comparison, in the Stiglitz (1977) analysis of the insurance market, both types of user optimally obtain the same amount of insurance, but at different prices.

The intuition behind this difference is simple: For insurance, in a given period of time, different types of users present different levels of risk. In contrast, with warranties different types of users present the same level of risk over different periods, so that the optimal price will be the same for both types of consumers. This seemingly trivial transformation means that one cannot simply transfer the results of analysis of insurance to the warranty market. In the following section, the assumption is dropped that firms are able to identify consumers of different types ex ante.

ASYMMETRIC INFORMATION WITH A MONOPOLISTIC PRODUCT MARKET

When firms cannot identify user types ex ante, it becomes optimal in some circumstances to separate the types by offering a base warranty and an optional extended warranty. In this section, I compare the outcomes of permitting and excluding independent insurers from the extended warranty market when the product market is monopolistic. I assume that there are two consumer types, i = H,L, with 0 [less than] [k.sub.L] [less than] [k.sub.H] [less than or equal to] 1, and a unit mass of consumers with [lambda] representing the proportion of high-intensity users. The parameters [k.sub.H], [k.sub.L] and [lambda] are assumed to be common knowledge. I also assume the indifference curves of the two types of user exhibit a single crossing property:

[u([k.sub.H]t + x) - u([k.sub.H]t)][k.sub.H]f([k.sub.H]t) [greater than] [u([k.sub.L] + x) - u([k.sub.L]t)][k.sub.L] f([k.sub.L]t) (4)

for all t such that u([k.sub.L]t + x) - u([k.sub.L]t)[greater than] w . This inequality requires that the indifference curves for a heavy user are steeper than those for a lighter user, so that heavy users will value any warranty extension more than light users. [12] Warranties will never be offered for t such that u([k.sub.L]t + x) - u([k.sub.L]t) [less than] w because the marginal cost of the warranty is greater than its value to consumers.

First, the case in which the product manufacturer has exclusivity in its extended warranties is examined; then the case without exclusivity; and finally welfare outcomes in the two cases are compared. As before, [[T.sup.*].sub.i] denotes the optimal full-information warranty duration for consumers of type i.

Product Monopoly With Exclusivity in Extended Warranties

When the monopolist is unable to identify the consumer type, the problem of profit maximizing becomes:

[max[lambda][[pi].sub.H]([[gamma].sub.H])+(1 - [lambda])[[pi].sub.L]([[gamma].sub.L]) (5) [[gamma].sub.H],[[gamma].sub.L]

subject to four incentive compatibility and participation constraints:

PL: [U.sub.L]([gamma].sub.L]) [greater than or equal to] 0 (5) PH: [U.sub.H]([gamma].sub.H]) [greater than or equal to] 0 IL: [U.sub.L]([gamma].sub.L]) [greater than or equal to] [U.sub.L] [[gamma] .sub.H]) IH: [U.sub.H]([[gamma].sub.H] [greater than or equal to] [U.sub.H]([gamma].sub.L]

LEMMA 1: At the solution to this problem, the constraints PL and IH will bind; PH will bind only if light users are not served; and IL will only bind if there is pooling.

Proof: See Appendix.

This problem is similar to the familiar insurance problem with adverse selection described by Stiglitz (1977). However, it differs in that there is a range of separating solutions and a range of pooling solutions, as described in Propositions 1 and 2 below, depending on 1, the proportion of heavy users. I call the separating warranty contracts offered in response to this adverse selection problem [[T.sup.ME].sub.H], and [[T.sup.ME].sub.L], and the pooling warranty contracts [[T.sup.ME].sub.P]. The first superscript indicates a monopoly in the product market, while the second superscript indicates that the firm has exclusivity in the extended warranties for its own product.

For the following propositions, I define a critical value [[lambda].sub.1] [epsilon] [0,1] as the solution to:

[u([k.sub.L][T.sub.H] * +x) - u([k.sub.L][T.sub.H] *)][k.sub.L]([k.sub.L][T.sub.H] *) = [lambda]w[K.sub.H][T.sub.H] *) + (1 - [lambda])w[k.sub.L]f([k.sub.L][T.sub.H] *). (6)

This defines the [lambda] such that the tangency of the pooling iso-profit curve with the indifference curve of the light user is at t = [[T.sup.*].sub.H]. For an intuitive sense of the significance of this critical value, consider Figure 2. The indifference ("IC") and iso-profit ("IP") curves of each type of user are stacked vertically. The slope of the pooling iso-profit curve ("[IP.sub.p]") is given by the weighted average of the iso-profit curves for heavy and light users, and is therefore increasing with the proportion of heavy users. When the proportion of heavy users is high, the tangency between the pooling iso-profit curve and the light user's indifference curve will be to the left of [[T.sup.*].sub.H], and when [lambda] is lower, the tangency must be between [[T.sup.*].sub.H] and [[T.sup.*].sub.L].

PROPOSITION 1: For [[lambda].sub.1] [less than] [lambda] [less than or equal to] 1, the manufacturer's optimal warranty strategy will be a menu of separating contracts [[T.sup.ME].sub.H] and [T.sup.ME].sub.L] such that the heavy user obtains the full-information warranty duration [[T.sup.ME].sub.H] = [T.sup.*].sub.H] and the light user obtains a shorter warranty of duration

[[T.sup.ME].sub.L] [less than][[T.sup.*].sub.H][less than][[T.sup.*].sub.L].

Proof: See Appendix.

The separating warranty contracts are typically implemented as a base warranty bundled with the product and an optional extended warranty. These are depicted in Figure 2. The light user's warranty contract [T.sup.ME].sub.L] is just the base warranty, usually included automatically with the product; while [[T.sup.ME].sub.H] is the total warranty contract purchased by heavy users. The difference between [T.sup.ME].sub.H] and [T.sup.ME].sub.L] can thus be seen as an extended warranty. The first order condition of the monopolist's maximization, derived from (5), requires [[T.sup.ME].sub.L] satisfy

[u([k.sub.L]T + x) - u([k.sub.L]T)][k.sub.L]f([k.sub.L]T)

= [lambda][u([k.sub.H]T + x) - u([k.sub.H]T)][k.sub.H]f([k.sub.H]T + (1- [lambda])[wk.sub.L]f([k.sub.L]T). (7)

Thus the length of the base warranty is determined by the tangency of the light user's indifference curve with the weighted average of the indifference curve for the heavy user and the iso-profit curve for the light user. Note that while the heavy user's warranty duration with the separating contracts is equal to the optimal full information duration [[T.sup.*].sub.H], the light user's warranty is even shorter than that of the heavy user, the reverse of the full-information case.

PROPOSITION 2: For 0 [less than or equal to] [lambda] [less than or equal to] [[lambda].sub.1], the manufacturer's optimal warranty strategy will be a pooling contract [[[gamma].sup.ME].sub.P] with warranty duration such that [[T.sup.*].sub.H] [less than or equal to] [[T.sup.ME].sub.p] [less than or equal to] [[T.sup.*].sub.L].

Proof: See Appendix.

When [lambda] [less than or equal to] [[lambda].sub.1], the [[T.sup.ME].sub.L], given by (7) is greater than [[T.sup.*].sub.H], and the separating warranty characterized by Proposition 1 is no longer profit-maximizing, since it does not meet IL, the light user's incentive compatibility condition. Thus, for [lambda] [less than or equal to] [[lambda].sub.1], IL is also binding in equilibrium; but since both IL and IH are binding, the only possible solution is a single pooling warranty. This single warranty will have duration [[T.sup.ME].sub.p] defined by:

[u([k.sub.L]T + x) - u([K.sub.L]T)][k.sub.L]T)=[lambda]w[k.sub.H]f([k.sub.H]T) + (1 - [lambda])w[k.sub.L]f([k.sub.L]T).

This equation requires that the light user's indifference curve be tangent to the weighted average of the iso-profit curves for both types. To see that only pooling warranties are feasible in this range, note that for [lambda] [less than or equal to] [[lambda].sub.1], the iso-profit curve is steeper than the indifference curve for heavy users, rendering an extended warranty designed only for heavy users unprofitable. Similarly, an extension intended for light users only would be unprofitable since both types of users would purchase it.

There is thus a range of pooling warranties [[[gamma].sup.ME].sub.P] which are profit-maximizing for a sufficiently low proportion of heavy users. Stiglitz never obtains a pooling equilibrium in his well-known 1977 article on insurance, since the optimal amount of insurance is the same across types: the high-risk type is differentiated only by risk. With warranties, however, the "high risk" type actually has a shorter optimal full-information warranty, and it is this difference between warranties and insurance that leads to these different results. The possibility of pooling warranties obtained here can explain why extended warranties are not available on many products, despite user heterogeneity.

Product Monopoly Without Exclusivity in Extended Warranties

In this section, I consider a different regime in which independent insurers, having the same insurance costs w as the manufacturer, compete in the extended warranty market. The independent insurers can offer one or more contracts for each type i [[gamma].sub.1] = {[[tau].sub.i],[[pi].sub.i]} where [[tau].sub.i] is the period of extension of the warranty and [[pi].sub.i] is the price paid by the consumer for the extension. Expected surplus for a consumer buying the extended warranty would be:

[U.sub.i]([gamma],[gamma])=-p-[pi]+[[[integral of].sup.(T+[tau])].sub.0] u(kt+x)kf(kt)dt+[[[integral of].sup.1/k K].sub.(T+[tau])]u(kt)kf(kt)dt.

The independent contracts are thus supplementary to the contract offered by the manufacturer and increase the total period covered by a warranty. The industry may now be described as a sequential game: The monopolist offers one or more contracts; then independent insurers offer contracts of additional warranty protection; and finally consumers select contracts. The extended warranties must be continuous but not overlapping with a previous warranty, and a consumer might purchase a series of warranties. All firms are assumed to form rational expectations about the extended warranties that independent insurers will offer. The independent insurers are perfectly competitive, so that in equilibrium their prices will exactly reflect the costs of the contracts they offer. Following Rothschild and Stiglitz (1976), equilibrium in the competitive extended warranty market is defined as a set of extended warranty contracts such that, when consumers choose contracts to maximize expected utility, (i) no contract in the equi librium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit.

This study wishes to know how the manufacturer's warranty will differ in the presence of independent insurers. If the manufacturer has optimally offered a pooling warranty even without the threat of entry by independent insurers, then this warranty is unaffected by the possibility of entry, since any extension beyond the pooling warranty would be strictly unprofitable:

PROPOSITION 3: If 0 [less than or equal to][lambda] [less than or equal to] [[lambda].sub.1], then independent insurers cannot profitably enter.

Proof: Any extension designed only for heavy users will be unprofitable; any extension designed only for light users will attract both types; and any pooling extension will be unprofitable.

In contrast, if the manufacturer's optimal pre-entry contract separates the two types by the offer of two incentive-compatible contracts, the optimal strategy will be changed by the possibility of entry. The manufacturer cannot make any economic profits in the market for extended warranties because of competition. The monopolist will therefore choose between not serving the light users at all and offering the most profitable single base warranty, where the second superscript indicates that the manufacturer does not have exclusivity in the extended warranty market. This warranty, depicted in Figure 3, will have duration defined by the tangency of the indifference curve of light users and the iso-profit curve of pooled users. Thus, the condition determining the duration of this base warranty will be just the same as (8). This leads to:

PROPOSITION 4: If [[lambda].sub.1] [less than] [lambda] [less than or equal to] 1, competition from independent insurers will lead to an increase in the base warranty duration (a) from [[T.sup.ME].sub.L] to [[T.sup.MN].sub.P] if both types continue to be served or (b) from [[T.sup.ME].sub.L] to [[T.sup.*].sub.H] if the manufacturer does not serve the light users.

Proof: See Appendix.

In addition to the pooling warranty offered by the manufacturer, for any [[T.sup.MN].sub.P] [less than] [[T.sup.*].sub.H], the competitive insurers (and perhaps the manufacturer) will offer an extended warranty to provide optimal coverage for heavy users, so that ex post there is separation of user types. The heavy user's contract will extend the total period of coverage to [[T.sup.*].sub.H] at the fair-odds price w [[[integral of].sup.[[T.sub.H].sup.*]].sub.[[T.sup.MN].sub.P]] [K.sub.H]f([k.sub.H]t]dt. The extended warranty purchased by the heavy user can be seen in Figure 3 as the difference between [psi] and [[[gamma].sup.MN].sub.P].

Welfare Effects of Exclusivity in Extended Warranties

Provided that both types continue to be served, permitting competition in the extended warranty market has a non-negative welfare effect when the product market is monopolistic. Transfers of money in this model have no welfare effect; the only variable of interest is the length of warranties. The closer they are to the full information lengths [[T.sup.*].sub.H] and [[T.sup.*].sub.L], the larger is welfare. Thus, because competition in extended warranties lengthens the warranty obtained by the light user (while that of the heavy user remains the same), welfare increases.

If, however, light users form a sufficiently small part of the market, then the light user may not be served at all. In this case, the base warranty is equal to the full-information warranty for the heavy user, and the light user will not buy the product. The welfare result here is the familiar one of price discrimination. If the profits from light users are not large enough, then without the possibility of price discrimination, they may not be served and welfare decreases. But if light users continue to be served, the elimination of price discrimination by independent insurers increases welfare.

ASYMMETRIC INFORMATION WITH A COMPETITIVE PRODUCT MARKET

In this section, I undertake the same exercise as above, but with the assumption that the product manufacturers are in a perfectly competitive industry In this case, the consumer gains a surplus while firms make zero profits. With full information, the consumer surplus in the competitive case is exactly equal to the firm's profit in the monopoly case. As before, consumers are assumed to be of two a priori unidentifiable types.

Exclusivity in Extended Warranties

Each firm is initially assumed to have exclusivity in the sale of extended warranties for its own product. This analysis is similar to that of Rothschild and Stiglitz (1976). As in their models, if the proportion of heavy users is sufficiently high, then a separating equilibrium will be obtained, and if the proportion of heavy users is below a critical value, then no equilibrium may exist at all. Equilibrium in this competitive base warranty market is defined as a set of contracts such that, when consumers choose contracts to maximize expected utility, (i) no contract in the equilibrium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit. Each contract consists of the sale of the product plus a warranty for the price p + s. The base warranty is restricted to start at the time the consumer purchases the product; and the profitability of any base warranty will also depend on the set of extended warranties that are offered subsequently.

With heterogeneous consumers, there are two possible types of solutions again depending on [lambda]. To write Proposition 5, I define, in the appendix, a new critical value [[lambda].sub.2] [epsilon][0,1]. Here [[lambda].sub.2] is the critical value of [lambda] at which light users are indifferent between pooling and separating contracts.

PROPOSITION 5: (a) For [lambda] [greater than] [[lambda].sub.2], separating warranties will be offered in equilibrium such that the heavy user obtains a warranty of duration [[T.sup.*].sub.H], while the light user obtains a shorter warranty of duration [[T.sup.CE].sub.L], and firms make zero profits on each type of warranty; and (b)for [lambda][less than or equal to] [[lambda].sub.2], no equilibrium will exist.

Proof: The proof of this proposition is identical to that provided by Rothschild and Stiglitz (1976) and Wilson (1977).

The separating warranty contracts will be such that the heavy user's duration is again the full-information one, while the light user obtains a short warranty, constrained by the heavy user's incentive compatibility. Both warranties will be priced at cost. This equilibrium is pictured in Figure 4, in which as will be seen, [[[gamma].sup.CE].sub.L] is constrained to be priced at fair cost and to be the best possible contract for the light user subject to the incentive compatibility constraint of the heavy user, who obtains the warranty contract [[[gamma].sup.CE].sub.H].

If [lambda] [less than or equal to] [[lambda].sub.2], the well-known arguments of Rothschild and Stiglitz will show that there is no equilibrium, since both pooling and separating candidate equilibria can be broken profitably. Their argument is restated briefly, with reference to Figure 5. Both heavy and light users will prefer the zero-profit pooling warranty [[[gamma].sup.CN].sub.P] to the separating warranties [[[gamma].sup.CE].sub.H] and [[[gamma].sup.CE].sub.L]. However, this pooling warranty is vulnerable to the defection of one firm to the contract [xi], which will attract only light users. But then only heavy users will buy [[[gamma].sup.CN].sub.P], which will be withdrawn. Thus, neither a pooling nor a separating equilibrium is possible.

The Effect of Independent insurers on a Competitive Product Market

This section examines how competition in the extended warranty market will affect manufacturer warranties. The result here does not depend on the proportion of heavy and light users:

PROPOSITION 6: In the presence of competitive independent insurers, competitive manufacturers will offer a single base warranty [[[gamma].sup.CN].sub.P].

Proof: See text.

First, consider the effect of non-exclusivity in the range of [lambda] [greater than] [[lambda].sub.2]. It is clear that the presence of competitive extended warranties renders the manufacturers' separating warranties loss-making, since heavy users will purchase the warranties designed for light users plus an extension from the independent insurers. [13] (In Figure 4, all users would buy the base warranty [[[gamma].sup.CE].sub.L], and heavy users would also buy the extension [Psi].) Therefore, the manufacturers can only offer pooling warranties. The only feasible competitive pooling warranties fall on the zero profit pooling curve, designated [IP.sub.p] in Figure 4. Independent insurers can offer supplementary extended warranties only to heavy users, since any extended warranty designed for light users will only attract all users.

We now examine the effect of independent insurers when l[less than or equal to][l.sub.2] Consider the base warranty [[[gamma].sup.CN].sub.P] shown in Figure 5. With exclusivity in extended warranties, this base warranty can be undermined by the offer of a contract such as [xi] which will attract only light users; thus only the heavy users buy [[[gamma].sup.CN].sub.P], leading to equilibrium breakdown. However, the presence of independent insurers has the effect of generating a stable equilibrium. Given the base warranty [[[gamma].sup.CN].sub.P], independent insurers can offer the extended warranty [Y.sub.H] = [psi] - [[[gamma].sup.CN].sub.P]. Contract [xi], were it offered, would allow independent insurers to offer the two contracts [Y.sub.H] = [psi] - [xi] and [Y.sub.L] = [phi] - [xi]. This would be possible since these two contracts together would make zero profits. (Note that [Y.sub.L] is the best extended warranty contract that can be offered to light users since anything below the pooling line would attr act other companies to offer yet further extended warranties.) Since light users would prefer the total contracts [phi] to [[[gamma].sup.CN].sub.P], and heavy users would prefer [psi] to [psi], all consumers would buy the contract [xi] plus the extensions. But then [xi] would become a loss-making pooling contract. Thus, no base warranties could be offered which would break the pooling equilibrium. The pooling warranty [[[gamma].sup.CN].sub.P] also cannot be broken by any extended warranty since it makes zero profits regardless of the extended warranties.

Welfare Effects of Exclusivity in Extended Warranties

If [lambda] [greater than] [[lambda].sub.2], that separating warranties are offered when manufacturers have exclusivity in their warranties, the loss of exclusivity may have positive or negative welfare effects. If the proportion of heavy users is very high, then the new base pooling warranty [[[gamma].sup.CN].sub.P] may be even shorter than the separating base warranty [[[gamma].sup.CN].sub.L]. In this case, welfare will fall with the introduction of competition in extended warranties. However, if the proportion of heavy users is not so high, then the new base pooling warranty will be longer than the base warranty with exclusivity, in which case welfare would increase. Thus even in the perfectly competitive market, there may be a welfare gain from forbidding exclusivity conditions on extended warranties. If [lambda] [greater than] [[lambda].sub.2], then it is only the presence of independent insurers that permits equilibrium in the market, and it is thus difficult to make any determination of how welfare would be changed.

CONCLUSIONS

This article suggests some direction for the regulation of the extended warranty market. Manufacturers may very well wish to make supplies to a dealer or retailer conditional on that dealer's carrying only the manufacturer's extended warranty. Should this be permitted? There are two common approaches to this problem. First, the "Chicago" approach which would permit tying regardless of market structure. A second approach is the one generally adopted by the U.S. courts: If the manufacturer has market power in the "tying" (product) market, then it should not be permitted to practice tying (or as in this case requirements contracting) into the "tied" (extended warranty) market; but if the manufacturer has no market power in the first market, then tying should be permitted. Neither of these approaches is optimal. If the product market is characterized by monopoly or oligopoly, then competition in extended warranties is beneficial, provided that light users are not excluded from the market, and the monopolist shoul d then be prevented from excluding the independent insurers from the extended warranty market (i.e., tying). On the other hand, even if the product market is itself characterized by vigorous competition, then tying in the secondary market for extended warranties may lead to a decrease in welfare! Thus, courts should be sympathetic to retailers who object to supplier requirements that exclude third-party warranties. The conclusion that emerges is inline with other economics analysis of aftermarkets, which suggests that case by case judgements are required rather than any per se rule. [14] Table 1 presents the results of the article in a summary format.

The practice of requiring dealers to carry only the manufacturer's replacement parts may be interpreted analogously to a restriction on extended warranties, since the manufacturer is thus restricting competition in service after the base warranty. This type of requirements contracting has been a contentious issue in the courts at least since 1936, when the U.S. Supreme Court upheld a lower court decision approving General Motors' requirement that GM dealers install only GM replacement parts. [15] There have been a variety of cases on similar grounds since then, most of which have revolved around two issues: (1) whether third-party replacement parts might be inferior and result in damage to the manufacturer's reputation and (2) whether the requirement would harm competition. [16] The analysis in this article helps to clarify another potentially important issue: Third-party replacement parts may affect welfare by preventing schemes of price discrimination by manufacturers. It was exactly this reasoning that the successful plaintiffs used in the 1992 U.S. Supreme Court case on Kodak. [17]

There are some opportunities for extending this research. In particular, it would be interesting to try to test empirically whether the legal status of manufacturer restrictions on independent extended service contracts in different jurisdictions varied sufficiently to cause different outcomes in manufacturer base warranties. While this article discusses extended warranties, its analysis applies rather more widely. For example, a simple variation on the model presented here would have manufacturers choosing different qualities, instead of different warranties, in order to separate heavy and light users. In general, the model may be applied to any case in which ancillary services provided by a third-party can effectively bridge the gap between contracts that were intended to separate types of users who impose different costs on the firm, and suggests that it may often be welfare-reducing to allow the first firm to exclude the third party. What is most surprising is that this result will sometimes hold even whe n the first firm is in a competitive industry.

APPENDIX

LEMMA 1: Step 1. Consider any pair of contracts {[[gamma].sub.H], [[gamma].sub.L]}, satisfying the four incentive compatibility and participation constraints, and choosing [[gamma].sub.L] to be such that [U.sub.L] ([[gamma].sub.L]) = 0. By (4), this requires [T.sub.H] [greater than or equal to] [T.sub.L] and [P.sub.H] [greater than or equal to] [p.sub.L]. Now consider a second pair of contracts {[[gamma].sub.H], [[gamma].sub.L]}, where [T.sub.L] = [T.sub.L] and [P.sub.L] [less than] [p.sub.L] [less than] [p.sub.L], such that {[[gamma].sub.H], [[gamma].sub.L] also satisfy the four incentive compatibility and participation constraints. Compared to [[gamma].sub.L], profits may always be increased by choosing [[gamma].sub.L]. Therefore, PL is always binding.

Step 2. Consider two contract combinations {[[gamma].sub.H],[[gamma].sub.L]} and {[[gamma].sub.H],[[gamma].sub.L]}, where [T.sub.H] = [T.sub.H] and [p.sub.H] [less than] [p.sub.H], and with both contract combinations again satisfying the four incentive compatibility and participation constraints. By (4), this requires [T.sub.H] [greater than or equal to] [T.sub.L] and [p.sub.H] [greater than or equal to] [p.sub.L] and [T.sub.H] [greater than or equal to] [T.sub.L] and [p.sub.H] [greater than or equal to] [p.sub.L]. For any [[gamma].sub.H] such that [U.sub.H]([[gamma].sub.H])[greater than] [U.sub.H]([[gamma].sub.L]), profits may be increased by increasing [p.sub.H] until IH is binding.

Step 3. The implication of PL and IH binding is that PH cannot bind if both types are served. Since by (4) the indifference curves for the high type are steeper than those for low types, IH binding requires PH not binding unless [[gamma].sub.L] = {0,0}.

Step 4. If only one contract is offered, by definition IL must be binding. However, if separating contracts are offered, IL cannot be binding. This is evident given (4) and IH.

PROPOSITIONS 1 AND 2: Proof: First to show that [[lambda].sub.1], is well-defined, it is required that 0 [less than or equal to] [[lambda].sub.1] [less than or equal to] 1. It is required that

w[k.sub.1]f([k.sub.L][T.sub.H]*)[less than or equal to][u[k.sub.L][T.sub.H]*+x)-u([k.sub.L][T.sub.H]*)][k.sub.1]f([k.sub. L][T.sub.H]*)[less than or equal to]w[k.sub.H]f([k.sub.H][T.sub.H]*).

The first inequality is obviously true. By (5), the second inequality is equivalent to

[u([k.sub.L][T.sub.H]*+x)-u([k.sub.L][T.sub.H]*)][k.sub.1]f([k.sub.L] [T.sub.H]*)[less than or equal to][u([k.sub.H][T.sub.H]*+x)-u[k.sub.H][T.sub.H]*)[k.sub.H]f([k.sub.H ][T.sub.H]*).

This is true by (4). Thus 0 [less than or equal to] [[lambda].sub.1] [less than or equal to]1.

For [lambda] [greater than] [[lambda].sub.1], the problem for the monopolist is to maximize

[max.sub.[T.sub.H],[p.sub.H],[T.sub.L],[p.sub.L]] [lambda][[p.sub.H] - w[[[integral of].sup.[T.sub.H]].sub.0] [k.sub.H]f([k.sub.H]t)] + (1 - [lambda])[[p.sub.l] - w[[[integral of].sup.[T.sub.L]].sub.0] [k.sub.L]f([k.sub.L]t)]

subject to the two binding constraints

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The six first order conditions for the six unknowns are

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [delta][pi]/[delta][T.sub.H] = -[lambda]w[k.sub.H]f([k.sub.H][T.sub.H]) + [[beta].sub.1][u([k.sub.H][T.sub.H] + x) - u([k.sub.H][T.sub.H])][k.sub.H]f([k.sub.H][T.sub.H]) = 0

(c) [delta][pi]/[delta][p.sub.L] = 1 - [lambda] + [[beta].sub.1] - [[beta].sub.2] = 0

(d) [delta][pi]/[delta][p.sub.H] = [lambda] - [[beta].sub.1] = 0

plus the two binding constraints. By (c) and (d), [[beta].sub.1] = [lambda] and [[beta].sub.2] = 1 are obtained. It is obvious from (b) that [[T.sup.ME].sub.H] = [[T.sub.H].sup.*]. The application of IH and PL binding then immediately identifies [[T.sub.L].sup.ME].

For [lambda] [less than] [[lambda].sub.1], the constraints change: in particular, the incentive compatibility constraints for the light user becomes binding, since the separating solution has [[T.sub.L].sup.ME] [greater than] [[T.sub.H].sup.ME], so that IL is no longer met. Thus IL becomes binding for such values of [lambda], so that

[[[integral of].sup.[T.sub.L]].sub.0]] [u([k.sub.L]t + x]) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt - [p.sub.L] = [[[integral of].sup.[T.sub.H]].sub.0]] [u([k.sub.L]t + x) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt - [p.sub.H].

Now the problem can be simplified to the pooling problem, since the two binding incentive compatibility constraints require that [[gamma].sub.L] = [[gamma].sub.H]:

[max.sub.[T.sub.p],[p.sub.p] [p.sub.p]] - w[[[integral of].sup.[T.sub.p]].sub.0] [[lambda][k.sub.H]f([k.sub.H]t) + (1 - [lambda])[k.sub.1]f([k.sub.L]t)]dt (9) subject to the binding constraint [p.sub.p] = [[[integral of].sup.[T.sub.p]].sub.0] [u([k.sub.L]t + x) - u([k.sub.L]t)][k.sub.L]f([k.sub.L]t)dt. Substituting this constraint into the objective function and taking the FOC yields the result [lambda]w[k.sub.H]f([k.sub.H]T) + (1 - [lambda])w[k.sub.L]f([k.sub.L]T) = {u([k.sub.L]T + x) - u([k.sub.L]T)}[k.sub.L]f([k.sub.L]T), which defines the optimal [[T.sup.ME].sub.P]. PL defines the optimal [[p.sup.ME].sub.p], thus identifying the optimal pooling solution. Here [[lambda].sub.1] is the largest [lambda] for which this optimal [[T.sup.ME].sub.P] [greater than or equal to] [T.sub.H] *.

For completeness, I show that this pooling warranty is more profitable than any separating warranty if [lambda] [less than] [[lambda].sub.1]. A separating warranty would require an optional extended warranty to be offered to either the heavy or the light user. No profitable extended warranty can be sold to the heavy user, since the slope of the iso-profit curve is steeper for the heavy user than the indifference curve for t [greater than or equal to] [T.sub.H] *. No profitable extended warranty can be offered to the light user only, since by (4), the slope of the indifference curve for the heavy user is steeper than the slope of the light user's indifference curve and any extended warranty beyond [T.sub.p] satisfying PL will be purchased by both types, making it a pooling warranty. But we have already found the optimal pooling warranty.

PROPOSITION 4: Proof: Following the entry of any independent insurers, the manufacturer makes zero profits on any optional extended warranty. The manufacturer will therefore choose between the profit-maximizing pooling contract, in which case both types buy the product, or the profit-maximizing contract for heavy users only.

(a) The case where both types are served is considered first. The objective function is the same as (9). The first order condition is given by (8) which solves for the solution [[T.sup.MN].sub.p]. Compared with the first order condition for the separating solution (7), only the first term on the RHS is different. Since

[u([k.sub.H]T + x) - u([k.sub.H]T)][k.sub.H]f([k.sub.H]T) [greater than] w[k.sub.H]f([k.sub.H]T) T [less than] [[T.sub.H].sup.*], the RHS of (8) is less than the RHS of (7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] t [less t [[T.sub.H].sup.*].

But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore [[T.sup.MN].sub.p] [greater than or equal to] [[T.sup.ME].sub.L]

(b) The case in which only the heavy user is served is now considered. In this case, the monopolist sets the full-information warranty contract [[gamma].sub.H] * and makes profits of [lambda][[pi].sub.H] ([[gamma].sub.H] *) but loses all profits on the light user. Thus, the monopolist will choose to serve only the heavy user if

[lambda]{[[pi].sub.H]([[gamma].sub.H] *) - [[pi].sub.H]([[[gamma].sup.MN].sub.P])} [greater than] (1 - [lambda]){[[pi].sub.L]([[[gamma].sup.MN].sub.P]) + (s - c)}.

Definition: [[lambda].sub.2] is defined as the [lambda] which satisfies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where [[T.sup.CE].sub.L] is defined by

W[[[integral of].sup.[[T.sub.H].sup.*]].sub.0][k.sub.H]f([k.sub.H]t)dt-[[[integral of].sup.[[T.sup.*].sub.H]].sub.[[T.sup.CE].sub.L]][u([k.sub.L]t+x)-u( [k.sub.L]t)][k.sub.1]f([k.sub.L]t)dt = w[[[integral of].sup.[[T.sup.CE].sub.L]].sub.0][k.sub.1]f([k.sub.L]t)dt.

Aidan Hollis is Assistant Professor, Department of Economics at the University of Calgary. The author is indebted to two anonymous referees, Nancy Gallini, Frank Mathewson, Mike Peters, Ralph Winter, and participants at seminars at the University of Toronto, the University of Western Ontario, l'Universite de Montreal, and the Institut d'Analisi Economica in Barcelona for helpful comments.

(1.) Cited in Lutz and Padmanabhan (1995).

(2.) Plotkin (1985) and Therrien (1991).

(3.) Therrien (1991).

(4.) Two other explanations of the function of warranties are provided by Salop (1977), who observed that with imperfect consumer information on product qua1ity, the warranty may be used as a tool of price discrimination; and by Braverman, Guasch, and Salop (1983) who noted that a warranty may be used as part of a two-part tariff to extract consumer surplus.

(5.) See, e.g., Bryant and Gerner (1982), Gerner and Bryant (1981), and Priest (1981).

(6.) Cooper and Ross (1985), Emons (1988), Dybvig and Lutz (1993), Soberman (1995), and Lutz and Padmanabhan (1998).

(7.) Automobiles are a partial exception to this rule, since it is possible to limit warranties by the distance driven. Even in this case, however, the odometer does not specify the type of terrain or the style of driving, which influences the expected rate of car breakdown. So in the case of automobiles, the analysis should not refer to time of driving but to intensity of use.

(8.) Note that this analysis ignores the possibility of any kind of deductible. Many types of warranties do not have explicit deductibles, although some of them do: for example, car warranties often have a deductible, and some warranties cover only labor and not parts. Including deductibles in the analysis would not change its flavor or conclusions but would make it considerably more complicated.

(9.) The model of the warranty employed here does not perfectly match the observed form of repair or replacement warranties, since making x a constant allows for the warranty to be exercised only once. Repair or replacement warranties actually have the form x(T - t) where x(z) [greater than] 0 and x' [greater than] 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] z [grea to] 0 and x(z) = 0 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] z [less than] 0. In this form to the consumer of the warranty is declining with t because after repair or replacement, the product is returned with only the balance of the warranty. As it turns out, only the marginal condition of warranty value at t = T is of interest, so that the model may be simplified by using a constant without any change in the results.

(10.) More generally, the warranty servicing cost could be written as w(kt), with the restriction that it be continuous and non-decreasing.

(11.) The second order condition is

[d.sup.2][pi]/[dT.sup.2] = [k.sup.2]{[u([kT + x) - u(kT0 - w]f'(kT) + [u'(kT = x) - u'(kT)]f(kT)}.

The second term is negative, and the first term is equal to zero when the first order condition is satisfied so that this is a local maximum.

(12.) This assumption will not apply globally, since the probability density of heavy users breaking their product will eventually become lower than that of light users when most heavy users have already broken their product. However, the assumption is not unreasonable: we do not observe warranties which are so long that the probability of light users breaking the good is higher than that of heavy users, perhaps since the cost of those warranties is higher than the utility gains to consumers.

(13.) The potential free-riding by independent insurers, which is what drives Proposition 6, is analogous to the "externality" which Lutz and Padmanabhan (1995) find with consumer moral hazard. They find that competitive provision of extended warranties leads to zero base warranties because consumers will take less care of the product than is consistent with a competitively priced warranty. Analogously, in the adverse selection story here, it is not possible for the manufacturer to offer separating warranties. However, the outcome with adverse selection is that the manufacturer will offer a single base warranty which in general will be of strictly positive length.

(14.) See, e.g., Borenstein, MacKie-Mason, and Netz, 1996.

(15.) Pick Manufacturing Co. v. General Motors Corp. et al., 80 F. 2d 641 (1935), affirmed per curiam, 299 U.S. 3 (1936).

(16.) In re General Motors Corp. and General Motors Sales Corp., 34 FTC 58, 86 (1941); Dicto Graph Products, Inc., v. Federal Trade Commission, 217 F. 2d 821, 828 (1954): Englander Motors, Inc. v. Ford Motor Co., 267 F. 2d 11(1959); Alles Corp. v. Senco Products, Inc., 329 F. 2d. 567 (1964).

(17.) "Eastman Kodak v. Image Technical Services, Inc. 504 U.S. 451(1992).

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Author: | Hollis, Aidan |
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Publication: | Journal of Risk and Insurance |

Geographic Code: | 1USA |

Date: | Sep 1, 1999 |

Words: | 10218 |

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