Large Losses and Equilibrium in Insurance Markets
Lisa L. Poseya
Paul D. Thistleb
ABSTRACT
We show that, if losses are larger than wealth, individuals will not insure if the loss probability is
above a threshold. In a two risk type model of an insurance market with adverse selection, if the
high risks’ loss probability is above the threshold, then no trade occurs at the Rothschild-Stiglitz
equilibrium.
Keywords: adverse selection, contracts, no trade
JEL Classification: D82, D86, G22
a
Department of Risk Management, Pennsylvania State University, 369 Business Building, University Park, PA
16802, 814-865-0615, [email protected]
Corresponding author: Department of Finance, University of Nevada, Las Vegas, 4505 Maryland Parkway, Box
456008, Las Vegas, Nevada, 89154-6008, 702-895-3856, [email protected]
b
Large Losses and Equilibrium in Insurance Markets
1. Introduction.
Individuals can face large potential losses that exceed their wealth. With a median net worth for
US households of $68,828 in 2011 (U.S. Census Bureau, 2014), it is easy to see that a large
medical expense or liability judgment could easily wipe out a household’ s net worth.1 We show
that, when losses exceed wealth, risk averse individuals will not insure if the probability of loss
exceeds a threshold. We examine the implications of this for insurance markets with adverse
selection. We show that if high risks do not buy insurance then there is no trade in the insurance
market. The Rothschild-Stiglitz equilibrium contracts for both high and low risks are the null
contracts. This is the reverse of the “ adverse selection death spiral” since it is high risks rather
than low risks that drop out of the market.
Shavell (1986) and Sinn (1982) consider the case where losses exceed wealth. Shavell
shows that, for a fixed loss probability, there is a wealth threshold where individuals do not
insure if their wealth is below the threshold and fully insure if it is above the threshold. Sinn
notes that whether the premium exceeds the maximum willingness to pay depends on the
probability of loss but does not consider threshold values. Neither author considers the case of
asymmetric information.
Hendren (2013, 2014) gives conditions under which the unique equilibrium contract in an
insurance market with adverse selection is the null contract. The “ no trade condition” is that
lower risks are never willing to pay to be in a pool with higher risks. This leads to an adverse
selection death spiral. If the no trade condition does not hold, then either there is an equilibrium
1
In 2009, the average cost of an auto crash fatality was $6,000,000 and the average cost of an injury was
$126,000 (AAA, 2011).
2
where insurance is bought or there is no equilibrium. As Hendren (2014) points out, the “
equilibrium of market unraveling” (no trade) and the “ unraveling of market equilibrium” (no
equilibrium) are mutually exclusive.
We analyze the standard Rothschild-Stigliz (1976) model of adverse selection. The no
trade condition does not hold in this model, so we would expect active trade in equilibrium. We
relax Hendren’ s implicit assumption that losses do not exceed wealth and show that if the high
risks’ loss probability exceeds the threshold, the unique Rothschild-Stiglitz equilibrium contracts
are the null contracts – there is no trade – or else there is no equilibrium. The no trade
equilibrium is the result of high risks dropping out of the market. High risks do not buy
insurance at an actuarially fair price because it covers losses they would not be able to pay if
uninsured and is therefore too expensive. Low risks would be willing to buy insurance at their
actuarially fair price if a minimum level of coverage were provided. The need to screen high
risks limits the coverage for low risks to a level that is below this minimum and the low risks
choose not to insure.
A key factor in our analysis is the assumption that final wealth cannot be negative so a
portion of any loss greater than wealth will be externalized. For example, if an individual has a
liability loss greater than their wealth, the injured third party may not receive full compensation.
Similarly, the medical expenses of uninsured or underinsured patients with insufficient wealth
may ultimately be borne by medical providers. Our results may provide an explanation for why
health insurance markets are not well-functioning for low income populations.2
2
We thank Nathan Hendren for this insight.
3
2. The Decision to Buy Insurance
Assume individuals have utility function u, with
, and
and
finite, have
initial wealth w, and face a potential loss l with probability p. The loss is larger than wealth,
. If an individual incurs a loss, they are left with final wealth of zero due to bankruptcy
protection. Insurance is actuarially fairly priced so if it is purchased, full insurance is optimal and
yields expected utility
. If insurance is purchased, expected utility is
. If
, the individual cannot pay for full coverage. If
, they
will not buy insurance since doing so gives final wealth of zero with certainty, while not doing so
gives final wealth of zero with less than certainty. This implies there is a critical value
such that the individual will not buy insurance if
< w/l
.
The net benefit of buying insurance when p ≤ w/l is
(1)
Observe that
,
,
,
and
. If
. If
(2)
, there is never a benefit to buying insurance and
then
, at least in some neighborhood of zero, and there is a net
benefit to buying insurance if the probability of loss is low enough. Then there is a unique
such that
.
This proves the following result:
Proposition 1: Assume
. Then there is a , where
individuals buy full insurance if
and do not buy insurance if
The threshold probability
is increasing in
and decreasing in .
, such that
4
3. Rothschild-Stiglitz Equilibrium
Now consider the implication of the decision not to buy insurance for equilibrium in the
Rothschild-Stiglitz (1976) framework. Loss probabilities,
private information, where
risk and
premium and
.
and
for high and low risks are
is the proportion of the population that is high
the proportion that is low risk. Contracts are denoted
where
is the
is the indemnity net of premium. Firms simultaneously offer contracts, and then
individuals decide which, if any, contract they will buy. In equilibrium contracts must break
even, satisfy the self-selection constraint
which, if offered, would earn positive profits.
, and there must be no other contract
Let
denote the Rothschild-Stiglitz
equilibrium contracts. The Rothschild-Stiglitz equilibrium exists if the proportion of high risks is
large enough,
.
Now consider the no trade condition.
probability. If
Let
be the pooled loss
, as in Hendren’ s case, the no trade condition is:
(3)
and
.
(4)
Condition (3) says that low risks are unwilling to pay the actuarially fair price of pooled
coverage. As noted in Hendren (2014), in the canonical two type case with
can only hold for risk averse individuals if
. Hendren’ s analysis for
either there is an equilibrium where non-null contracts are traded (if
not exist (if
implies that
) or equilibrium does
).3 We show that if losses are larger than wealth, Hendren’ s no trade condition
does not apply.
3
, condition (4)
See Hendren’ s (2013) Theorem 1 and subsequent discussion.
5
The Rothschild-Stiglitz equilibrium when
are uninsured, they are at the point
is illustrated in Figure 1. If individuals
. The traditional fair odds lines for high and low
risks, P H and P L as well as the pooled fair odds line P P, depicted as emanating from
, are relevant only above the horizontal axis. The wealth combinations actually faced
by policyholders level off at the horizontal axis, at
, for all
FIGURE 1
Equilibrium with Large Losses
PL
.
6
Let
be the value of
at which the fair pooled price line is just tangent to the low risk
indifference curve through E0.
Proposition 2: Assume
,
equilibrium contracts are the null contracts,
. Then the Rothschild-Stiglitz
.
Proof: The curves UH0 and UL0, the high and low risk indifference curves through E0, are the
individual rationality constraints. Low risks prefer any policy along P L above point B, and prefer
no insurance to any policy along P L below B. Since
, the high risk fair odds line lies
completely below the indifference curve UH0, so high risks do not insure. Since the self-selection
constraint UH0 is flatter than UL0, it must intersect the fair odds line P L below B, for example, at
point A. Low risks prefer to remain uninsured. If the pooled fair odds line, labeled P P in Figure 1,
lies below the indifference curve UL0, i.e., if
, then the equilibrium is
. ||
Since both high and low risks obtain the null contract, this is a pooling equilibrium.
However, the more important characteristic of the equilibrium is that no trade occurs.
The existence of equilibrium depends on the location of the pooled fair odds line. In
Figure 1, P P lies below the low risk indifference curve UL0. Since the proportion of high risks is
large enough, the equilibrium exists. If the proportion of high risks were too small, the pooled
fair odds line would intersect the low risk indifference curve and the Rothschild-Stiglitz
equilibrium would fail to exist. Thus, there is either an equilibrium with no trade or there is no
equilibrium.
It is interesting to consider how the equilibrium changes as pH changes (we assume
equilibrium exists).4 Suppose that
. Then the high risk fair odds line is tangent to the
indifference curve UH0 at full insurance and high risks are fully insured. The self-selection
constraint UH0 still intersects the low risk fair odds line at point A so the equilibrium contracts
4
The critical value
, increases as pH approaches pL.
7
are the full insurance contract for high risks and the null contract for low risks. For
high risk contract is actuarially fair full insurance. As
the
falls the high risk contract moves up the
45% full insurance line. The intersection between the high risk indifference curve that is the
self-selection constraint and the low risk fair odds line moves up the low risk fair odds line. As
long as the intersection is below B low risks will not insure. There is a probability,
which the intersection is at B.
Then for
, at
, we have the usual Rothschild-Stiglitz
equilibrium where high risks fully insure and low risks receive partial coverage.
4. Conclusion
We show that, if losses are larger than wealth, there is a threshold where individuals will not
insure if the loss probability is above the threshold. In a market with adverse selection and
, if the high risks’ loss probability is above the threshold, then no trade occurs at the
Rothschild-Stiglitz equilibrium. Hendren’ s no trade condition does not apply when losses are
larger than wealth. However, Hendren’ s broader point, that the unraveling of insurance markets
involves either no equilibrium or no trade and that these possibilities are mutually exclusive,
remains valid.
8
Refer ences
American Automobile Association, 2011, Crashes vs congestion: What is the cost to society?
https://newsroom.aaa.com/wp-content/uploads/2011/11/2011_AAA_CrashvCongUpd.pdf
Hendren, N. 2013, Private information and insurer rejections, Econometrica, 81(5):1713-1762.
Hendren, N. 2014, Unravelling vs unravelling: A memo on competitive equilibriums and trade in
insurance markets, Geneva Risk and Insurance Review, 39:176-183.
Rothschild, M. and J. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay on
the economics of imperfect information, Quarterly Journal of Economics, 90(4):629-649.
Shavell, S., 1986, The judgement proof problem, International Review of Law and Economics,
6:45-58.
Sinn, H-W., 1982, Kinked utility and the demand for human wealth and liability insurance,
17:149-162.
U.S. Census Bureau, 2014, Household Wealth in the U.S.:2000-2011.