لخّصلي

EQUILIBRIUM IN COMPETITIVE INSURANCE
MARKETS: AN ESSAY ON THE ECONOMICS OF
IMPERFECT INFORMATION*
MICHAEL ROTHSCHILD AND JOSEPH STIGLITZ

INTRODUCTION
Economic theorists traditionally banish discussions of infor-
mation to footnotes. Serious consideration of costs of communication,
imperfect knowledge, and the like would, it is believed, complicate
without informing. This paper, which analyzes competitive markets
in which the characteristics of the commodities exchanged are not
fully known to at least one of the parties to the transaction, suggests
that this comforting myth is false. Some of the most important con-
clusions of economic theory are not robust to considerations of im-
perfect information.
We are able to show that not only may a competitive equilibrium
not exist, but when equilibria do exist, they may have strange prop-
erties. In the insurance market, upon which we focus much of our
discussion, sales offers, at least those that survive the competitive
process, do not specify a price at which customers can buy all the in-
surance. they want, but instead consist of both a price and a quan-
tity-a particular amount of insurance that the individual can buy
at that price. Furthermore, if individuals were willing or able to reveal
their information, everybody could be made better off. By their very
being, high-risk individuals cause an externality: the low-risk indi-
viduals are worse off than they would be in the absence of the high-risk
individuals. However, the high-risk individuals are no better off than
they would be in the absence of the low-risk individuals.
These points are made in the next section by analysis of a simple
model of a competitive insurance market. We believe that the lessons
gleaned from our highly stylized model are of general interest, and
attempt to establish this by showing in Section II that our model is
robust and by hinting (space constraints prevent more) in the con-
clusion that our analysis applies to many other situations.

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I. THE BASIC MODEL
Most of our argument can be made by analysis of a very simple
example. Consider an individual who will have an income of size W
if he is lucky enough to avoid accident. In the event an accident occurs,
his income will be only W - d. The individual can insure himself
against this accident by paying to an insurance company a premium
ai, in return for which he will be paid '2 if an accident occurs. Without
insurance his income in the two states, "accident," "no accident," was
(W W - d); with insurance it is now (W - a1, W - d + a2), where a2
= a2 - al. The vector a = (a,, a02) completely describes the insurance
contract.'
I.1 Demand for Insurance Contracts
On an insurance market, insurance contracts (the a's) are traded.
To describe how the market works, it is necessary to describe the
supply and demand functions of the participants in the market. There
are only two kinds of participants, individuals who buy insurance and
companies that sell it. Determining individual demand for insurance
contracts is straightforward. An individual purchases an insurance
contract so as to alter his pattern of income across states of nature.
Let W1 denote his income if there is no accident and W2 his income
if an accident occurs; the expected utility theorem states that under
relatively mild assumptions his preferences for income in these two
states of nature are described by a function of the form,
(1)
fV(p, W1, W2) = (1 - p)U(WI)

• pU(W2),
  where U( ) represents the utility of money income2 and p the
  probability of an accident. Individual demands may be derived from
  (1). A contract a is worth V(p, a) = V(p, W - aI, W - d + a2). From

THE ECONOMICS OF IMPERFECT INFORMATION

all the contracts the individual is offered, he chooses the one that
maximizes V(p, a). Since he always has the option of buying no in-
surance, an individual will purchase a contract a only if V(p, a) ?
V(p, 0) = V(p, W, W - d). We assume that persons are identical in
all respects save their probability of having an accident and that they
are risk-averse (U" < 0); thus V(p, a) is quasi-concave.
I.2 Supply of Insurance Contracts
It is less straightforward to describe how insurance companies
decide which contracts they should offer for sale and to which people.
The return from an insurance contract is a random variable. We as-
sume that companies are risk-neutral, that they are concerned only
with expected profits, so that contract a when sold to an individual
who has a probability of incurring an accident of p, is worth
(2)
7r(p, a) = (1 -p)ai
-pa2
= al -p(ai

• a2)-
  Even if firms are not expected profit maximizers, on a well-organized
  competitive market they are likely to behave as if they maximized
  (2).3
  Insurance companies have financial resources such that they/are
  willing and able to sell any number of contracts that they think will
  make an expected profit.4 The market is competitive in that there is
  free entry. Together these assumptions guarantee that any contract
  that is demanded and that is expected to be profitable will be sup-
  plied.

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I.3 Information about Accident Probabilities
We have not so far discussed how customers and companies come
to know or estimate the parameter p, which plays such a crucial role
in the valuation formulae (1) and (2). We make the bald assumption
that individuals know their accident probabilities, while companies
do not. Since insurance purchasers are identical in all respects save
their propensity to have accidents, the force of this assumption is that
companies cannot discriminate among their potential customers on
the basis of their characteristics. This assumption is defended and
modified in subsection 11.1.
A firm may use its customers' market behavior to make infer-
ences about their accident probabilities. Other things equal, those with
high accident probabilities will demand more insurance than those
who are less accident-prone. Although possibly accurate, this is not
a profitable way of finding out about customer characteristics. In-
surance companies want to know their customers' characteristics in
order to decide on what terms they should offer to let them buy in-
surance. Information that accrues after purchase may be used only
to lock the barn after the horse has been stolen.
It is often possible to force customers-to make market choices in
such a way that they both reveal their characteristics and make the
choices the firm would have wanted them to make had their charac-
teristics been publicly known. In their contribution to this symposium,
Salop and Salop call a market device with these characteristics a
self-selection mechanism. Analysis of the functioning of self-selection
mechanisms on competitive markets is a major focus of this paper.
I.4 Definition of Equilibrium
We assume that customers can buy only one insurance contract.
This is an objectionable assumption. It implies, in effect, that the seller
of insurance specifies both the prices and quantities of insurance
purchased. In most competitive markets, sellers determine only price
and have no control over the amount their customers buy. Nonethe-
less, we believe that what we call price and quantity competition is
more appropriate for our model of the insurance market than tradi-

.

THE ECONOMICS OF IMPERFECT INFORMATION
tional price competition. We defend this proposition at length in
subsection 11.2 below.
Equilibrium in a competitive insurance market is a set of con-
tracts such that, when customers choose contracts to maximize ex-
pected 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 nonnegative profit. This notion of
equilibrium is of the Cournot-Nash type; each firm assumes that the
contracts its competitors offer are independent of its own actions.
I.5 Equilibrium with Identical Customers
Only when customers have different accident probabilities, will
insurance companies have imperfect information. We examine this
case below. To illustrate our, mainly graphical, procedure, we first
analyze the equilibrium of a competitive insurance market with
identical customers.5

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the states: no accident, accident, respectively. The point E with
coordinates (W1, VV2) the typical customer's uninsured state. In-is
difference curves are level sets of the function of equation (1). Pur-
chasing the insurance policy a = (al, a2) moves the individual from
E to the point (V1 - al, W2+ a2).
Free entry and perfect competition will ensure that policies
bought in competitive equilibrium make zero expected profits, so that
if a is purchased,
(3)
al(l-p)-a2p
= 0.
The set of all policies that break even is given analytically by (3) and
diagrammatically by the line EF in Figure I, which is sometimes re-
ferred to as the fair-odds line. The equilibrium policy a maximizes
the individual's (expected) utility and just breaks even. Purchasing
a locates the customer at the tangency of the indifference curve with
the fair-odds line. a* satisfies the two conditions of equilibrium: (i)
it breaks even; (ii) selling any contract preferred to it will bring in-
surance companies expected losses.
Since customers are risk-averse, the point a* is located at the
intersection of the 450-line (representing equal income in both states
of nature) and the fair-odds line. In equilibrium each customer buys
complete insurance at actuarial odds. To see this, observe that the
slope of the fair-odds line is equal to the ratio of the probability of not
having an accident to the probability of having an accident ((1 -
p)/p), while the slope of the indifference curve (the marginal rate of
substitution between income in the state no accident to income in the
state accident) is [U'(W1) (1 - p)]/[U'(W2)p], which, when income
in the two states is equal, is (1 - p)/p, independent of U.
I.6 Imperfect Information: Equilibrium with Two Classes of
Customers
Suppose that the market consists of two kinds of customers:
low-risk individuals with accident probability pL, and high-risk in-
dividuals with accident probability pH > p L. The fraction of high-ris'k
customers is X, so the average accident probability is p = XpH + (1
X)pL. This market can have only two kinds of equilibria: pooling
equilibria in which both groups buy the same contract, and separating
equilibria in which different types purchase different contracts.
A simple argument establishes that there cannot be a pooling
equilibrium. The point E in Figure II is again the initial endowment
of all customers. Suppose that a is a pooling equilibrium and consider
7r(p, a). If ir(p, a) < 0, then firms offering a lose money, contradicting

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I

the definition of equilibrium. If i-(p, a) > 0, then there is a contract
that offers slightly more consumption in each state of nature, which
still will make a profit when all individuals buy it. All will prefer this
contract to a, so a cannot be an equilibrium. Thus, i-(p, a) = 0, and
a lies on the market odds line EF (with slope (1 - p)/p).
It follows from (1) that at a the slope of the high-risk indifference
curve through a, U[H, is (pL/l - pL) (1 - pH/pH) times the slope of
UL, the low-risk indifference curve through a. In this figure UH is a
broken line, and UL a solid line. The curves intersect at a; thus there
is a contract, A in Figure II, near a, which low-risk types prefer to a.
The high risk prefer a to A. Since A is near a, it makes a profit when
the less risky buy it, (r(pL, f)(pL, a) > 7r(p, a) = 0). The exis-
tence of A contradicts the second part of the definition of equilibrium;
a cannot be an equilibrium.
If there is an equilibrium, each type must purchase a separate
contract. Arguments, which are, we hope, by now familiar, demon-
strate that each contract in the equilibrium set makes zero profits.
In Figure III the low-risk contract lies on line EL (with slope (1 -
and the high-risk contract on line EH (with slope (1 -pL)/pL),
pH)/pH).As was shown in the previous subsection, the contract on
EH most preferred by high-risk customers gives complete insurance.

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This is a'H in Figure III; it must be part of any equilibrium. Low-risk
customers would, of all contracts on EL, most prefer contract d which,
like aH, provides complete insurance. However, i3 offers more con-
sumption in each state than aH, and high-risk types will prefer it to
aH. If f and aH are marketed, both high- and low-risk types will
purchase A.The nature of imperfect information in this model is that
insurance companies are unable to distinguish among their customers.
All who demand ,3must be sold A. Profits will be negative; (aHfi) is
not an equilibrium set of contracts.
An equilibrium contract for low-risk types must not be more
attractive to high-risk types than aH; it must lie on the southeast side
of UH, the high-risk indifference curve through aH. We leave it to the
reader to demonstrate that of all such contracts, the one that low-risk
types most prefer is aoL,the contract at the intersection of EL and UH
in Figure III. This establishes that the set (aHH,aL) is the only possible
equilibrium for a market with low- and high-risk customers. 6 How-
ever, (aH, aL) may not be an equilibrium. Consider the contract y in
Figure III. It lies above UL, the low-risk indifference curve through
a L and also above UH. If y is offered, both low- and high-risk types

THE ECONOMICS OF IMPERFECT INFORMATION

will purchase it in preference to either aH or aL. If it makes a profit
when both groups buy it, y will upset the potential equilibrium of (a H,
aL). y's profitability depends on the composition of the market. If
there are sufficiently many high-risk people that EF represents
market odds, then y will lose money. If market odds are given by EF'
(as they will be if there are relatively few high-risk insurance cus-
tomers), then y will make a profit. Since (aH, aL) is the only possible
equilibrium, in this case the competitive insurance market will have
no equilibrium.
This establishes that a competitive insurance market may have
no equilibrium.
We have not found a simple intuitive explanation for this non-
existence; but the following observations, prompted by Frank Hahn's
note (1974), may be suggestive. The information that is revealed by
an individual's choice of an insurance contract depends on all the
other insurance policies offered; there is thus a fundamental infor-
mational externality that each company, when deciding on which
contract it will offer, fails to take into account. Given any set of con-
tracts that breaks even, a firm may enter the market using the infor-
mational structure implicit in the availability of that set of contracts
to make a profit; at the same time it forces the original contracts to
make a loss. But as in any Nash equilibrium, the firm fails to take
account of the consequences of its actions, and in particular, the fact
that when those policies are no longer offered, the informational
structure will have changed and it can no longer make a profit.
We can characterize the conditions under which an equilibrium
does not exist. An equilibrium will not exist if the costs to the low-risk
individual of pooling are low (because there are relatively few of the
high-risk individuals who have to be subsidized, or because the
subsidy per individual is low, i.e., when the probabilities of the two
groups are not too different), or if their costs of separating are high.
The costs of separating arise from the individual's inability to obtain
complete insurance. Thus, the costs of separating are related to the
individuals' attitudes toward risk. Certain polar cases make these
propositions clear. If pL = 0, it never pays the low-risk individuals
to pool, and by continuity, for sufficiently small pL it does not pay
to pool. Similarly, if individuals are risk-neutral, it never pays to pool;
if they are infinitely risk averse with utility functions

8
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I. 7 Welfare Economics of Equilibrium
One of the interesting properties of the equilibrium is that the
presence of the high-risk individuals exerts a negative externality on
the low-risk individuals. The externality is completely dissipative;
there are losses to the low-risk individuals, but the high-risk indi-
viduals are no better off than they would be in isolation.
If only the high-risk individuals would admit to their having high
accident probabilities, all individuals would be made better off
without anyone being worse off.
The separating equilibrium we have described may not be Pareto
optimal even relative to the information that is available. As we show
in subsection 11.3below, there may exist a pair of policies that break
even together and that make both groups better off.
II. ROBUSTNESS
The analysis of Section I had three principal conclusions: First,
competition on markets with imperfect information is more complex
than in standard models. Perfect competitors may limit the quantities
their customers can buy, not from any desire to exploit monopoly
power, but simply in order to improve their information. Second,
equilibrium may not exist. Finally, competitive equilibria are not
Pareto optimal. It is natural to ask whether these conclusions (par-
ticularly the first, which was an assumption rather than a result of the
analysis) can be laid to the special and possibly strained assumptions
of our model. We think not. Our conclusions (or ones very like) must
follow from a serious attempt to comprehend the workings of com-
petition with imperfect and asymmetric information. We have ana-
lyzed the effect of changing our model in many ways. The results were
always essentially the same.
Our attempts to establish robustness took two tacks. First, we
showed that our results did not depend on the simple technical
specifications of the model. This was tedious, and we have excised
most of the details from the present version. The reader interested
in analysis of the effects (distinctly minor) of changing our assump-
tions that individuals are alike in all respects save their accident
probabilities, that there are only two kinds of customers, and that the
insurance market lasts but a single period, is referred to earlier ver-
sions of this paper.7 An assessment of the importance of the as-

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sumption that individuals know their accident probabilities, while
insurance companies do not (which raises more interesting issues),
is given in subsection 11.1 below.
Another approach to the question of robustness is the subject
of the next three subsections. In them we question the behavioral
assumptions and the equilibrium concepts used in Section I.
11.1 Information Assumptions
Suppose that there are two groups of customers and that not all
individuals within each group have the same accident probability. The
average accident probability of one group is greater than that of the
other; individuals within each group know the mean accident prob-
ability for members of their group, but do not know their own accident
probabilities. As before, the insurance company cannot tell directly
the accident probability of any particular individual, or even the group
to which he belongs. For example, suppose that some persons occa-
sionally drink too much, while the others almost never drink. Insur-
ance firms cannot discover who drinks and who does not. Individuals
know that drinking affects accident probabilities, but it affects dif-
ferent people differently. Each individual does not know how it will
affect him.
In such a situation the expected utility theorem states that in-
dividuals make (and behave according to) estimates of their accident
probabilities; if these estimates are unbiased in the sense that the
average accident probability of those who estimate their accident
probability to be p actually is p, then the analysis goes through as
before.
Unbiasedness seems a reasonable assumption (what is a more
attractive alternative?). However, not even this low level of correctness
of beliefs is required for our conclusions. Suppose, for example, that
individuals differ both with respect to their accident probabilities and
to their risk aversion, but they all assume that their own accident
probabilities are p. If low-risk individuals are less risk-averse on av-
erage, then there will not exist a pooling equilibrium; there may exist
no equilibrium at all; and if there does exist an equilibrium, it will
entail partial insurance for both groups. Figure IV shows that there

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will not exist a pooling equilibrium.If there were a pooling equilib-
rium,it wouldclearlybe with completeinsuranceat the marketodds,
since both groups'indifferencecurves have the slope of the market
odds line there. If the low-riskindividuals are less risk-averse,then
the two indifference curves are tangent at F, but elsewhere the
high-riskindividuals'indifference curve lies above the low-risk in-
dividuals' indifference curve. Thus, any policy in the shaded area
betweenthe two curveswill be purchasedby the low-riskindividuals
in preferenceto the pooling contract at F.
Othersuch cases can be analyzed,but we trust that the general
principle is clear. Our pathologicalconclusions do not require that
people have particularly good information about their accident
probabilities.They will occurunder a wide variety of circumstances,
includingthe appealingcase of unbiasedness.Neither insurancefirms
nor their customershave to be perfectly informed about the differ-
ences in riskpropertiesthat exist amongindividuals:What is required
is that individualswith different risk propertiesdiffer in some char-
acteristicthat can be linked with the purchaseof insuranceand that,
somehow,insurancefirms discoverthis link.
11.2 Price Competition Versus Quantity Competition
One can imagine our model of the insurancemarket operating
in two distinct modes. The first, price competition, is familiarto all

THE ECONOMICS OF IMPERFECT INFORMATION
students of competitive markets. Associated with any insurance
contract axis a number q (a) = a/a2, which, since it is the cost per unit
coverage, is called the price of insurance. Under price competition,
insurance firms establish a price of insurance and allow their cus-
tomers to buy as much or as little insurance as they want at that price.
Thus, if contract a is available from a company, so are the contracts
2a and (%/)a; former pays twice as much benefits (and costs twicethe
as much in premiums) as a; the latter is half as expensive and provides
half as much coverage.
Opposed to price competition is what we call price and quantity
competition. In this regime companies may offer a number of different
contracts, say al, a2, ... , an. Individuals may buy at most one con-
tract. They are not allowed to buy arbitrary multiples of contracts
offered, but must instead settle for one of the contracts explicitly put
up for sale. A particular contract specifies both a price and a quantity
of insurance. Under price and quantity competition it is conceivable
that insurance contracts with different prices of insurance will exist
in equilibrium; people who want more insurance may be willing to pay
a higher price for it (accept less favorable odds) than those who make
do with shallower coverage. Under price competition customers will
buy insurance only at the lowest price quoted in the market.
The argument of Section I depends heavily on our assumption
that price and quantity competition, and not simply price competi-
tion, characterizes the competitive insurance market. This assumption
is defended here. The argument is basically quite simple. Price
competition is a special case of price and quantity competition.
Nothing in the definition of price and quantity competition prevents
firms from offering for sale a set of contracts with the same price of
insurance. Since the argument above characterized all equilibria under
price and quantity competition, it also characterized all equilibria
when some firms set prices and others set prices and quantities. Thus,
it must be that price competition cannot compete with price and
quantity competition.8
This argument hinges on one crucial assumption: regardless of
the form of competition, customers purchase but a single insurance
contract or equivalently that the total amount of insurance purchased

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by any one customer is known to all companies that sell to him. We
think that this is an accurate description of procedures on at least
some insurance markets. Many insurance policies specify either that
they are not in force if there is another policy or that they insure
against only the first, say, $1,000 of losses suffered. That is, instead
of being a simple bet for or against the occurrence of a particular event,
an insurance policy is a commitment on the part of the company to
restore at least partially the losses brought about by the occurrence
of that event. The person who buys two $1,000 accident insurance
policies does not have $2,000 worth of protection. If an accident occurs,
all he gets from his second policy is the privilege of watching his
companies squabble over the division of the $1,000 payment. There
is no point in buying more than one policy.
Why should insurance markets operate in this way? One simple
and obvious explanation is moral hazard. Because the insured can
often bring about, or at least make more likely, the event being insured
against, insurance companies want to limit the amount of insurance
their customers buy. Companies want to see that their customers do
not purchase so much insurance that they have an interest in an ac-
cident occurring. Thus, companies will want to monitor the purchases
of their customers. Issuing contracts of the sort described above is the
obvious way to do so.
A subtler explanation for this practice is provided by our argu-
ment that price and quantity competition can dominate price com-
petition. If the market is in equilibrium under price competition, a
firm can offer a contract, specifying price and quantity, that will at-
tract the low-risk customers away from the companies offering con-
tracts specifying price alone. Left with only high-risk customers, these
firms will lose money. This competitive gambit will successfully upset
the price competition equilibria if the entering firm can be assured
that those who buy its contracts hold no other insurance. Offering
insurance that pays off only for losses not otherwise insured is a way
to guarantee this.
It is sometimes suggested that the term "competitive" can be
applied only to markets where there is a single price of a commodity
and each firm is a price taker. This seems an unnecessarily restrictive
use of the term competitive. The-basic idea underlying competitive
markets involves free entry and noncollusive behavior among the
participants in the market. In some economic environments price
taking without quantity restrictions is a natural result of such mar-
kets. In the situations described in this paper, this is not so.

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1.3 Restrictions on Firm Behavior and Optimal Subsidies
An important simplification of the analysis of Section I was the
assumption that each insurance company issued but a single contract.
We once thought this constraint would not affect the nature of equi-
librium. We argued that in equilibrium firms must make nonnegative
profits. Suppose that a firm offers two contracts, one of which makes
an expected profit of say, $S, per contract sold, the other an expected
loss of $L per contract. The firm can make nonnegative expected
profits if the ratio of the profitable to the-unprofitable contracts sold
is at least A,where A = LIS. However, the firm can clearly make more
profits if it sells only the contracts on which it makes a profit. It and
its competitors have no reason to offer the losing contracts, and in
competitive equilibrium, they will not be offered. Since only contracts
that make nonnegative profits will be offered, it does not matter, given
our assumptions about entry, that firms are assumed to issue only a
single contract. If there is a contract that could make a profit, a firm
will offer it.
This argument is not correct. The possibility of offering more
than one contract is important to firms, and to the nature and exis-
tence of equilibrium. Firms that offer several contracts are not de-
pendent on the policies offered by other firms for the information
generated by the choices of individuals. By offering a menu of policies,
insurance firms may be able to obtain information about the accident
probabilities of particular individuals. Furthermore, although there
may not be an equilibrium in which the profits from one contract
subsidize the losses of another contract, it does not follow that such
a pair of contracts cannot break what would otherwise be an equi-
librium.
Such a case is illustrated in Figure V. EF is again the market odds
line. A separating equilibrium exists (&H, -L). Suppose that a firm
offered the two contracts, aHf and aLf; aH' makes a loss, aLf makes
a profit. High-risk types prefer aH' to &H, and low-risk types prefer
aL' to L. These two contracts, if offered by a single firm together,
do not make losses. The profits from aLt subsidize the losses of aHf.
Thus, (KHf, aLI) upsets the equilibrium (&H, -L).
This example points up another possible inefficiency of sepa-
rating equilibria. Consider the problem of choosing two contracts (aH,
aL) such that aL maximizes the utility of the low-risk individual
subject to the constraints that (a) the high-risk individual prefers aH
to aL and (b) the pair of contracts aH and aL break even when bought
by high- and low-risk types, respectively, in the ratio Xto (1 - X). This

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is a kind of optimal subsidy problem. If the separating equilibrium,
when it exists, does not solve this problem, it is inefficient. Figure V
shows that the separating equilibrium can be inefficient in this sense.
We now show that if there are enough high-risk people, then the
separating equilibrium can be efficient.
The optimal subsidy problem always has a solution (a H*, aL*).
The optimal high-risk contract aH* will always entail complete in-
surance so that V(pH, aH*) = U(W - pHd + a), where a is the per
capita subsidy of the high risk by the low risk. This subsidy decreases
=income for each low-risk person by zya (where My X/(1 - X)) in each
state. Net of this charge aL* breaks even when low-risk individuals
buy it. Thus, aL* = (a, + ya, a2 - ya), where a1 = a2pL/(l - pL).
To find the optimal contract, one solves the following problem:
Choose a and a2 to maximize
U(X)
(1 - pL) + U(Z)pL,
subject to
U(Y)

U(X)
(1 - pH) + U(Z)pH
a >O
where

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X = Wo - ya -
a2pL/(l

pL),
Y= Wo-pHd
and

• a,
  Z = Wo - d - ya +
  a2.
  The solution to this problem can be analyzed by standard Kuhn-
  Tucker techniques. If the constraint a > 0 is binding at the optimum,
  then the solution involves no subsidy to the high-risk persons; (aH*,
  aL*) is the separating equilibrium. It is straightforward but tedious
  to show that a sufficient condition for this is that
  (pH
  pL(1

pL)
-p)L)

U'(Y)[U'(Z)

• U'(X)]
  U (X)U'(Z)
  a2*.
  where X, Y, and Z are determined by the optimal a*,
  right-hand side of (4) is always less than
  U'(Wo-d)[U(Wo
  -d)

The
U'(Wo)]
U'(Wo)2
so that there exist values of y (and thus of X) large enough to satisfy
(4).
11.4 Alternative Equilibrium Concepts
There are a number of other concepts of equilibrium that we
might have employed. These concepts differ with respect to as-
sumptions concerning the behavior of the firms in the market. In our
model the firm assumes that its actions do not affect the market-the
set of policies offered by other firms was independent of its own of-
fering.
In this subsection we consider several other equilibrium concepts,
implying either less or more rationality in the market. We could, for
instance, call any set of policies that just break even given the set of
individuals
who purchase them an informationally
consistent
equi-
librium. This assumes that the forces for the creation of new contracts
are relatively weak (in the absence of profits). Thus, in Figure III, aH
and any contract along the line EL below a L is a set of informationally
consistent separating equilibrium contracts; any single contract along
the line EF is an informationally consistent pooling equilibrium
contract. This is the notion of equilibrium that Spence (1973) has
employed in most of his work. The longer the lags in the system, the
greater the difficulty of competing by offering different contracts, the
more stable is an informationally consistent equilibrium. Thus, while

QUARTERLYJOURNAL OF ECONOMICS
this seems to us a reasonable equilibrium concept for the models of
educational signaling on which Spence focused, it is less compelling
when applied to insurance or credit markets (see Jaffee and Russell's
contribution to this symposium).
A local equilibrium is a set of contracts such that there do not
exist any contracts in the vicinity of the equilibrium contracts that
will be chosen and make a positive profit. If we rule out the subsidies
of the last subsection, then the set of separating contracts, which
maximizes the welfare of low-risk individuals, is a local equilibri-
um.
The notion that firms experiment with contracts similar to those
already on the market motivates the idea of a local equilibrium. Even
if firms have little knowledge about the shape of utility functions, and
about the proportions of population in different accident probabilities,
one would expect that competition would lead to small perturbations
around the equilibrium. A stable equilibrium requires that such
perturbations not lead to firms making large profits, as would be the
case with some perturbations around a pooling point.
These two concepts of equilibrium imply that firms act less ra-
tionally than we assumed they did in Section I. It is possible that firms
exhibit a greater degree of rationality; that is, firms ought not to take
the set of contracts offered by other firms as given, but ought to as-
sume that other firms will act as they do, or at least will respond in
some way to the new contract offered by the firm. Hence, in those
cases where in our definition there was no equilibrium, because for
any set of contracts there is a contract that will break even and be
chosen by a subset of the population, given that the contracts offered
by the other firms remain unchanged, those contracts that break the
equilibrium may not break even if the other firms also change their
contracts. The peculiar provision of many insurance contracts, that
the effective premium is not determined until the end of the period
(when the individual obtains what is called a dividend), is perhaps
a reflection of the uncertainty associated with who will purchase the
policy, which in turn is associated with the uncertainty about what
contracts other insurance firms will offer.
Wilson (1976) introduced and analyzed one such nonmyopic
equilibrium concept. A Wilson equilibrium is a set of contracts such
that, when customers choose among them so as to maximize profits,
(a) all contracts make nonnegative profits and (b) there does not exist
a new contract (or set of contracts), which, if offered, makes positive
profits even when all contracts that lose money as a result of this entry
are withdrawn. In the simple model of Section I, such equilibria always

\THE ECONOMICS OF IMPERFECT INFORMATION

exist. Comparing this definition with the one of subsection 1.4 above
makes it clear that, when it exists, our separating equilibrium is also
a Wilson equilibrium. When this does not exist, the Wilson equilib-
rium is the pooling contract that maximizes the utility of the low-risk
customers. This is A in Figure VI. 3 dominates the separating pair (aL,
aH). Consider a contract like oy,which the low risk prefer to d. Under
our definition of equilibrium it upsets F. Under Wilson's it does not.
When the low risk desert f for -y, it loses money and is withdrawn.
Then the high risk also buy oy.When both groups buy y, it loses money.
Thus, Mydoes not successfully compete against F.
Although this equilibrium concept is appealing, it is not without
its difficulties. It seems a peculiar halfway house; firms respond to
competitive entry by dropping policies, but not by adding new policies.
Furthermore, although counterexamples are very complicated to
construct, it appears that a Wilson equilibrium may not exist if groups
differ in their attitudes towards risk. Finally, in the absence of col-
lusion or regulation, in a competitive insurance market, it is hard to
see how or why any single firm should take into account the conse-
quences of its offering a new policy. On balance, it seems to us that
nonmyopic equilibrium concepts are more appropriate for models of
monopoly (or oligopoly) than for models of competition.