INTERNATIONAL ECONOMIC REVIEW
Vol. 51, No. 1, February 2010
DOUBLE-SIDED ADVERSE SELECTION IN THE PRODUCT MARKET
AND THE ROLE OF THE INSURANCE MARKET∗
BY S. HUN SEOG1
Seoul National University
I investigate the interrelation between a product market and an insurance market when adverse-selection problems
exist both in consumers and in firms. Firms offer warranties for product failures. Consumers may further purchase first-
party insurance for the residual risks of product failures. Given that the insurance market exists, two types of equilibria
are possible: (a) Different firm types offer different pooling warranties attracting both good and bad consumer types
or (b) good firms attract only bad consumers and bad firms attract both types of consumers. I discuss the existence and
the efficiency implication of the insurance market.
Observing that warranty can replace insurance, Grossman (1981) raises an interesting ques-
market? For example, people purchase health insurance and burglar/theft insurance instead of
purchasing warranties from doctors and burglar alarm manufacturers, respectively. Although
both warranty and insurance provide the same risk-sharing mechanism for consumers, the in-
teraction between the two has not been fully investigated.
This article is concerned with the interaction between the product market and the insurance
market where consumers experience product failures. Firms in the product market offer war-
purchase first-party insurance for the residual risks of product failure, if any, in the insurance
market. I suppose that the types of firms (or products) and consumers affect the risk of product
failure. A double-sided adverse-selection problem exists because the types of consumers and
firms are private information. I investigate the properties of the warranty and the first-party
insurance in an equilibrium. I also discuss the endogenous existence of an insurance market, the
efficiency implication of the introduction of an insurance market, and others issues regarding
warranty and insurance.
This article is related to two strands of literature. First, this article is related to the warranty
literature. The roles of warranty have been investigated since Heal (1977) and Spence (1977).
Among others, adverse selection is one of the main considerations for warranty.2Firm-sided
adverse-selection consideration can be found in Spence (1977) and Grossman (1981). Spence
(1977) argues that warranty can signal product quality when the signaling costs are higher for
low-quality firms than for high-quality firms. Grossman (1981) shows that low-quality firms also
have incentives to offer full warranties, both in competitive and in monopolistic circumstances,
∗Manuscript received January 2007; revised January 2008.
1The author thanks the participants in the American Risk and Insurance Association meeting in 2001 and Asia-
Pacific Risk and Insurance Association meeting in 2002 for their comments. The author also thanks Thi Nha Chau
for her support. Please address correspondence to: S. Hun Seog, KAIST Business School, Korea Advanced Institute
of Science and Technology (KAIST), 207-43 Cheongryangri-Dong Dongdaemun-Gu, Seoul, 130-012, Korea. Phone:
+82-2-958-3527. Fax: 782-2-958-3160. E-mail: firstname.lastname@example.org.
2Risk aversion, moral hazard, and price discrimination are other important contexts in which warranty is discussed.
See, for example, Salop (1977), Cooper and Ross (1985), and Dybvig and Lutz (1993).
C ?(2010) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social
and Economic Research Association
because firms can maximize consumer’s welfare (in the competitive market) or economic rents
(in the monopolistic market).
He shows that separating, pooling, or no equilibrium is possible, depending on the competition
level, the proportion of high risks, and the existence of the insurance market.3
(1976) initiated the research on adverse selection in the competitive insurance market. They
develop a model in which insurers offer self-selecting contracts to consumers with different risk
types. In their model, firms conjecture that other firms will not change their contracts after the
introduction of new contracts (called RS conjecture throughout this article). A Rothschild and
Stiglitz equilibrium (an RS equilibrium) is a set of contracts (called RS contracts) such that (i)
expected utility maximizing consumers select contracts; (ii) each contract in the equilibrium set
makes a nonnegative expected profit; and (iii) under RS conjecture, no new contract, if offered,
will make a positive profit. Note that RS equilibrium may not exist if the proportion of low-risk
consumers is high.
Since Rothschild and Stiglitz (1976), diverse approaches to adverse selection have been stud-
ied. Among others, alternative self-selecting equilibrium models are developed under different
assumptions of insurance firm’s conjecture and behavior.4Wilson (1977), Miyazaki (1977), and
greater sophistication and have more flexible budget constraints. A Wilson–Miyazaki–Spence
equilibrium (WMS equilibrium) is a set of contracts (called WMS contracts) such that (i) ex-
pected utility maximizing consumers select contracts; (ii) contracts in the equilibrium set, in
sum, make nonnegative expected profits; and (iii) no new contract, if offered, will make a non-
negative profit even when all contracts that lose money as a result of the new contract offer are
withdrawn. A WMS equilibrium is known to be Pareto superior to an RS equilibrium, if they
This article integrates the warranty model and the insurance model. I extend the existing
models in two aspects. First, I consider the case in which both product qualities and consumer
types may affect the risk of product failure. I investigate the case of double-sided adverse
selection as an extension to the one-sided adverse-selection models. Second, I consider both
the product market and the insurance market, whereas most existing papers consider only one
market.5Considering only one market ignores the fact that warranty and first-party insurance
may compete with each other.6I set up a two-stage model. In the first stage, firms offer self-
selecting warranty contracts. In the second stage, observing the results of the first stage, insurers
offer self-selecting insurance contracts for the residual risks. In this setting, I seek a modified
RS equilibrium to investigate how first-party insurance and warranties affect each other.
I find that two types of equilibria are possible, given that the insurance market exists. In
one type of equilibrium, different firm types offer different pooling warranties attracting both
types of consumers. Consumers subsequently purchase insurance in the insurance market. In
the other type of equilibrium, good firms attract only bad consumers and bad firms attract both
types of consumers. Consumers of the bad firms subsequently purchase insurance. Consumers
of the good firms, however, do not purchase insurance because they are offered full warranties.
In both equilibria, the combined contracts of warranty and insurance for the consumers of the
bad firms make the WMS contracts, even if the RS conjecture is applied.
of the warranty, not the coverage of the warranty. However, it turns out that the results are analogous to those of the
usual self-selection approach.
(Dionne, 1983; Dionne and Lasserre, 1985; Cooper and Hayes, 1987; Hosios and Peters, 1989; D’Arcy and Doherty,
1990; Dionne and Doherty, 1994).
5An exception is Hollis (1999), in which the insurance market is also considered.
6In my treatment, the insurance market includes the secondary warranty market.
DOUBLE-SIDED ADVERSE SELECTION
Based on the analysis, I discuss the endogenous existence of an insurance market and the
efficiency implication of the introduction of an insurance market. An insurance market exists as
long as the consumer-sided adverse-selection problem exists. In addition, the insurance market
are few warranties, the prospects of the insurance market, and the warranties in the automobile
The remainder of the article is composed as follows. I describe the model in Section 2. Section
3 studies benchmark cases in which only one-sided adverse selection exists. In Section 4, I
investigate the double-sided adverse-selection problems, given that the insurance market exists.
Section 5 provides discussions. Section 6 concludes.
Consumers purchase products in the product market in the first stage and then purchase
insurance in the insurance market in the second stage. I assume that each consumer purchases
one product, if he does. Consumers face the risk of product failures after they make their
purchases. In the first stage, firms offer self-selecting warranty contracts to cover the loss due
to the product failure. In the second stage, observing the results of the first stage, insurers offer
self-selecting insurance contracts for the residual risks.7,8
There are two types of firms in the product market. Good firms produce good products
with a low probability of product failure, whereas bad firms produce bad products with a high
probability of product failure. The occurrence of product failure also depends on the types of
consumers. There are two types of consumers: good consumers, who use the products carefully,
and bad consumers, who use the products less carefully. The types of firms and consumers are
private information. I denote good (bad) firms as g-firms (b-firms) and good (bad) products as
g-products (b-products). I also denote good (bad) consumers as G-consumers (B-consumers).
The proportion of b-firms is r, and the proportion of B-consumers is t. Once a failure occurs, it
causes a fixed loss of L to consumers. The probability of a product failure is denoted by pijfor
ij-consumer, i.e., i-consumer who purchases j-product. I assume that pGg≤ pGb, pBg≤ pBb.
In the product market, I assume that bad firms are competitive and that the supply from good
firms is short of the demand from either type of consumers.9In order to reflect the shortage of
supply, I assume that g-firms have limited capacities. The capacity, however, is not necessarily
small, so that lowering margin per warranty can be a viable competitive strategy for g-firms.10
Production cost is normalized to be zero. I extend Grossman’s model in two ways. First, I con-
sider heterogeneous consumers in addition to heterogeneous firms. Second, I also consider the
insurance market. In the insurance market, I follow Rothschild and Stiglitz (1976). The insur-
ance market is competitive. Insurers are risk neutral, and the insurance premium is actuarially
warranty for zero price. I treat We= (W, W−L) as the endowment of the model.11Let us denote
for the indirect expected utility of i-consumer who purchases warranty C in the product market.
As will be clear, Vi(C) is uniquely determined once C is given, because consumers are assumed
7If there is no insurance market, the second stage is ignored.
8 The assumption that insurers can observe the results of the first stage is critical to my results (especially for the
firm-separating equilibrium). This assumption is in line with the conventional assumption in the self-selection literature
that insurers can observe the consumer’s purchase of insurance from competitors. There has been a long-time criticism
that this assumption is too strong and unrealistic. This article is not free of this criticism, as pointed out by a referee.
9The second assumption is slightly different from Grossman (1981) in that Grossman assumes the supply from good
firms is short of total demand. I change the assumption because there are two types of consumers in my model.
10The limited capacity can also be applied to b-firms. It is, however, irrelevant because the b-firms’ margins are zero
in an equilibrium, so that lowering margins will simply decrease their profits.
11Weis the wealth position that consumers can obtain without considering warranty. This treatment assumes that
consumers can always purchase products with zero warranty. This assumption allows us to focus on the trade of risk.
A description for πg(K?) and πg(Cs
g) in (ND1)?and (ND2)?:
(ND1)?states that g-firms should prefer offering W?∗
b-firms). Suppose the pooling warranty K?(actually) offered by b-firms is described by (qk, sk),
where qkis the price for warranty and skis the warranty level. Because b-firms make a zero
profit, qk= p?bsk. On the other hand, a B-consumer will have the intermediate wealth pair (W −
qk, W − L − qk+ sk) right after purchasing warranty from a b-firm. Now the B-consumer will
IBb. Because the B-consumer will be fully protected and the insurance market is competitive,
pBbL)/(pBb-p?b). If the g-firm mimics b-firms and offers K?, the expected profit per warranty
becomes πg(K?) = qk− p?gsk= (p?b− p?g)(W?∗
(ND2)?states that g-firms should prefer offering W?∗
expected profit when the g-firm offers firm-separating warranties attracting both types of con-
optimal (separating) warranty attracting both types of consumers, given C?∗
Bb(=K?+ A?B) to offering K?(mimicking
Bb1− W + pBbL)/(pBb− p?b).
Bbto offering other firm-separating war-
b. That is, Cs
maxπg(Cg) = (1 − t?)(XGg− pGgYGg) + t?(XBg− pBgYBg)
= (1 − t?)[W − W?
+t?[W − W?
VB(Cg) ≥ VB(C?
∗), attracting B-consumers
VG(Cg) ≥ VG(C?
∗), attractingG-consumers(infact, binding)
Bg1) = (1 − pBg)u(W?
Now, I can solve the problem as follows:
Gg1) + pBgu(W?
L = (1 − t?)[W − W?
+δgG[(1 − pGg)u(W?
Bg1) − u(W?∗
Gg1+ L)] + t?[W − W?
Gg1) + pGgu(W?
Gg2) − (1 − pGb)u(W?∗
Gg1) − pBgu(W?
Gb1) − pGgu(W?∗
Bg1) − (1 − pBg)u(W?
where δgB, δgG, and µgBare Lagrange multipliers.
Gg1= −(1 − t?)(1 − pGg) + δgG(1 − pGg)u?(W?
Gg1) − µgB(1 − pBg)u?(W?
Gg1) = 0(A.7)
Gg2= −(1 − t?)pGg+ δgGpGgu?(W?
Gg2) − µgBpBgu?(W?
Gg2) = 0(A.8)
Bg1= −t + δgBu?(W?
Bg1) + µgBu?(W?
Bg1) = 0(A.9)
Bg1) − u(W?∗
Bb1) = 0,δgB> 0
> 0,δgB= 0.
DOUBLE-SIDED ADVERSE SELECTION
W (C )
• πj= X: iso-profit line for j-firm with profit X, for j = g, b.
• πj(C): expected profit per warranty of j-firm offering warranty C.
• φ(i; C): fair odd line for i-consumer purchasing C, for i = G, B.
• Wi(C): final wealth position of i-consumer purchasing C.
• We: endowment.
• W1, W2: wealth in the no loss state and wealth in the loss state, respectively.
NONEXISTENCE OF POOLING EQUILIBRIUM
Arranging the equations, I have:
(1 − pGb)u?(W?
t?(1 − pBg)u?(W?
Gg1) + (1 − t?)(1 − pGg)u?(W?
Bg1) − δgB(1 − pBg)u?(W?
Bg1) − δgBpBgu?(W?
Gg2) + (1 − t?)pGgu?(W?
W?Gg1, W?Gg2, and W?Bg1satisfying (A.11).
g) = (1 − t?)[W − W?
Gg1+ L)] + t?[W − W?
Bg1− pBgL], with
PROOF OF PROPOSITION 5. Suppose, on the contrary, that a PE exists. Because consumers do
not know the firm types, i-consumers will think of their risks as pi. Note also that the insurance
premium should be based on p’is. Now, I know that the result should be a WMS outcome with
the probability of loss is pBand pG. First, suppose that the WMS equilibrium coincides with the
RS equilibrium. That is, firms offer the pooling warranty of We(no warranty). Note that both
types of firms make zero profit with this warranty. Let us call the final wealth pairs as We
make a positive profit if it offers We
B-consumers only. When pBg= pBand pGg< pG, it is easy to see that a g-firm can increase
profit by offering a warranty on the zero-profit line of b-firms that is slightly different from We,
attracting both types of consumers. These results show that offering Wecannot make a PE.
Second, suppose that the WMS equilibrium is different from the RS equilibrium, so that the
pooling warranty is one, say Cp, on the zero-profit line of b-firms that is different from Weas
depicted in Figure A1. The resulting final wealth pairs are WB(Cp) and WG(Cp), respectively,
where WB(Cp) is the fully protected position. Because the iso-profit line of g-firms passing
through Cpis steeper than the zero-profit line of b-firms, g-firms can separate themselves from
b-firms by offering another warranty that is below the iso-profit line but above the zero-profit
G, respectively, where We
Bis the fully protected position. Now, when pBg< pB, a g-firm can
Binstead of offering We. Note that only B-consumers will
B. Because pBg< pB, the g-firm will make a positive profit by offering We
line of b-firms. One such warranty is denoted by D??in Figure A1. The warranty will attract
both types of consumers. Thus, the g-firm can increase the profit margin per warranty as well as
increase the number of sales to the maximum. One tricky case can be found when pgis close to
pb. In this case, it is not obvious if G-consumers prefer the warranty like D??from g-firms to Cp.
In order for the warranty to be a separating one that attracts G-consumers, g-firms may have to
lower their margins. The margin change, however, is small because pgis close to pb. Therefore,
g-firms can increase their profits by making the maximum number of sales.
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