Purchase insurance for future risk:
a two-period model for insurance
and saving/borrowing
Harris Schlesinger
Boyi Zhuang
October 2014
Abstract
The purpose of this article is to examine the demand for insurance in a two-period model
in which the individual purchases insurance for the future potential risk. Then we add the
endogenous saving/borrowing and other factors into the model and discuss the impacts. We
also show the comparative statics results for both the insurance-only model and the full
model with other factors in it.
0
Introduction
Standard insurance literatures usually use a one-period model in which people make
insurance decisions at the same time when the potential losses occur. Under this framework,
Mossin made a famous statement in his 1968 paper which states that “a risk-averse
individual will choose to fully insure at an actuarially fair premium.” The intuition behind
this theorem is straightforward: a risk-averse person will always prefer a path with certain
outcome than the one with same expected outcome but a positive variance (risk). This
theorem has been investigated for many different types of insurance (e.g. Schlesinger (1981)
for deductible insurance and Schlesinger (2006) for upper-limit insurance).
However, the real world insurances normally do not work that way. Almost all types
of insurance (i.e. auto insurance, real estate insurance, health insurance, etc.) require people
to purchase the insurance in advance of the time when the potential loss can occur (or the
time when the insurance company will reimburse the individual if a loss indeed occurs).
Moreover, even though the amount of risk behind some types of insurance do not vary too
much (i.e. auto liability insurance, health insurance, etc.), the insurance companies tend to
offer several different levels of insurance coverage. And individuals choose among the
different coverage, a phenomenon that sometimes is difficult to be explained by the
standard one-period insurance model.
Meanwhile, people also have other choices beside insurance to smooth their
consumption and protect themselves from uncertainties in the future. Saving and borrowing
are commonly used in daily life. Surprisingly, very few has been done to study the demand
for insurance with endogenous saving and borrowing. Hubbard, Skinner, and Zeldes (1995)
performed empirical test to show that social insurance will decrease people’s precautionary
1
saving. Kantor, S. E., & Fishback, P. V. (1996) used workers’ compensation as social
insurance to show that it reduces both private insurance purchases and precautionary saving.
The empirical results in Engen and Gruber (2001) suggested that there is a “crowed out”
effect of unemployment insurance on household saving. All the three papers above used
empirical data and did not allow individuals to make insurance and saving decisions freely.
Dionne and Eeckhoudt (1984) presented a two-period model which analyzes the optimal
choice of insurance and saving. However, they only described saving implicitly and did not
allow borrowing, and some of their results are ambiguous.
In this paper we extend the standard single-period model to a two-period one and
examine the demand for insurance under different circumstances. We also investigate the
impacts of endogenous saving/borrowing and extend the result of Mossin’s Theorem into
this setting. We then add positive interest rate to form a model that feels more like what
happens in the real world.
Our paper is organized as follows. In the first section we present the benchmark model
with only insurance available for the individual. We begin with fair insurance price and
same initial wealth levels in both period followed by different initial wealth levels across
time. Then we extend the model with endogenous saving/borrowing choices and a positive
loading for the insurance price, along with a positive interest rate. Section 2 provides the
comparative statics for both the benchmark model and the full model. Section 3 concludes
the paper.
1. The models
Two-period model with same initial wealth
2
The individual is endowed with certain incomes of same amount, 𝑊, in both current and
future periods. He also faces a potential loss in the second period. We assume only two
states are possible, so either a loss of size 𝐿 occurs with probability 𝜋, or no loss occurs
with probability 1 − 𝜋.
The individual chooses the amount of insurance, 𝛼, he wants to purchase at time 1,
which will reimburse him at time 2 if a loss state occurs. The price of insurance is denoted
by 𝑃. So his current consumption is,
𝐶1 = 𝑊 − 𝛼𝑃
where
𝑃 = (1 + 𝜆)𝜋𝐿
and his future consumptions with or without a total loss are, respectively,
𝐶2 = 𝑊 − 𝐿 + 𝛼𝐿
𝐶2 = 𝑊
The individual’s objective is then to maximize the expect utility of the two period,
𝑉 ≡ 𝜋[𝑈(𝑊 − 𝛼𝑃) + 𝑈(𝑊 − 𝐿 + 𝛼𝐿)] + (1 − 𝜋)[𝑈(𝑊 − 𝛼𝑃) + 𝑈(𝑊)]
with respect to 𝛼.
Assume the utility is concave, then the first-order condition is
𝑑𝑉
= −(1 + 𝜆)𝜋𝐿𝑈 ′ [𝑊 − 𝛼(1 + 𝜆)𝜋𝐿] + 𝜋𝐿𝑈′(𝑊 − 𝐿 + 𝛼𝐿) = 0
𝑑𝛼
Notice that if the premium is actually fair (i.e. 𝜆 = 0 ), the first-order condition
becomes
𝑈′[𝑊 − 𝛼𝜋𝐿] = 𝑈′(𝑊 − 𝐿 + 𝛼𝐿)
3
which yields the optimal amount of insurance under fair premium: 𝛼 =
1
1+𝜋
. Notice that
we do not require any particular utility form as long as the utility function is concave.
It is interesting to observe that the individual actually purchases less than full
insurance when the insurance is sold at fair price, which is different from previous
insurance literatures. The outcome of the insurance only occurred in the loss state in period
2. Without further adjustment from saving, the wealth level for the other state in period 2
is always 𝑊, which will not be affected by any choices the individual makes during period
1. While purchasing insurance provides protection for the potential loss, it also brings new
risk to the individual because the insurance could be totally worthless in the second period.
So the best the individual can achieve is a type of second best where the marginal loss of
utility for purchasing insurance is equal to the expected marginal gain of utility which the
insurance can provide.
And the probability of loss, 𝜋, affects people’s decision even when the premium is
fair. The individual buys less insurance as the probability of loss increases. At first glance,
the finding appears unrealistic. However, as 𝜋 goes up, the price of insurance also goes up,
the agent needs to sacrifice more consumption at time 1 for the same amount of insurance
purchased. Because the insurance price is fair, the amount of expected consumption gain
in period 2 will increase by the same amount. Due to the concavity of the utility, the
marginal cost of purchasing insurance will be higher than the marginal benefit. As a result,
people will decrease their insurance purchases.
Two-period model with different initial wealth
4
Now we allow different endowments for the two time periods in the model. Holding other
factors the same, his current consumption is then,
𝐶1 = 𝑊1 − 𝛼𝑃
and his future consumptions with or without a total loss are, respectively,
𝐶2 = 𝑊2 − 𝐿 + 𝛼𝐿
𝐶2 = 𝑊2
The individual’s expect utility of the two period then become
𝑉 ≡ 𝜋[𝑈(𝑊1 − 𝛼𝑃) + 𝑈(𝑊2 − 𝐿 + 𝛼𝐿)] + (1 − 𝜋)[𝑈(𝑊1 − 𝛼𝑃) + 𝑈(𝑊2 )]
and the first-order condition is
𝑑𝑉
= −(1 + 𝜆)𝜋𝐿𝑈 ′ [𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿] + 𝜋𝐿𝑈′(𝑊2 − 𝐿 + 𝛼𝐿) = 0
𝑑𝛼
1
We can solve the optimal amount of insurance under fair premium: 𝛼 = 1+𝜋 (1 +
𝑊1 −𝑊2
𝐿
) . Again the individual does not necessary purchase full insurance when the
insurance is sold at fair price. Still, the outcome in one of the two states in second period
is not affected by individual’s choices, so we will again have a type of second best.
However, more factors have influence on people’s decision for insurance when the
premium is fair. The difference between the two endowments and the potential loss size
enter the equation for the optimal insurance. Compare to the optimal insurance level in the
first model, we have a multiplier of 1 +
𝑊1 −𝑊2
𝐿
. So the individual will buy more/less
insurance when the initial wealth is higher/lower in period 1, because the marginal cost of
purchasing insurance becomes lower/higher than the expected marginal benefit. And the
more initial wealth the individual has in period 1 compare to period 2, the more insurance
he will purchase, because he is willing to sacrifice more of his wealth in period 1 for the
5
potential gain in wealth in period 2. On the other hand, the higher endowment in period 2
enables the individual to bear more potential loss, so the individual will purchase less
insurance.
For the potential loss size L, there are two cases. The first one is when 𝑊1 > 𝑊2. The
higher the loss size is, the more expensive the insurance will be. To match the marginal
cost and the expected marginal benefit, the extra amount of insurance (compare to model
1 where the endowments are the same) the individual will purchase become smaller. The
second case is when 𝑊1 < 𝑊2 . When the potential loss size is small, the individual will
consider it as bearable, due to the higher initial wealth in period 2, so he will purchase even
less insurance. When the potential loss size becomes larger, the individual will be willing
to substitute some of his consumption in period 1 to insure part of his potential loss in
period 2, so he will purchase more insurance, but still less than what he would if the initial
endowments were the same.
Two-period model with saving/borrowing
In order to smooth out the consumptions for both periods, we now introduce an endogenous
saving or borrowing into the model. In addition to consumption and insurance purchases,
the individual also decides how much he wants to save or borrow at time period 1, and will
withdraw or pay back that amount at time period 2. Assume the endowments are the same
across time, his current consumption is,
𝐶1 = 𝑊 − 𝛼𝑃 − 𝑆
where 𝑆 stands for saving if 𝑆 > 0, and borrowing if 𝑆 < 0. His future consumptions with
or without a total loss are, respectively,
6
𝐶2 = 𝑊 + 𝑆 − 𝐿 + 𝛼𝐿
𝐶2 = 𝑊 + 𝑆
The individual’s expect utility of the two period then become
𝑉 ≡ 𝜋[𝑈(𝑊 − 𝛼𝑃 − 𝑆) + 𝑈(𝑊 + 𝑆 − 𝐿 + 𝛼𝐿)] + (1 − 𝜋)[𝑈(𝑊 − 𝛼𝑃 − 𝑆) + 𝑈(𝑊
+ 𝑆)]
and the first-order conditions with respected to 𝛼 and 𝑆 are, respectively,
𝑑𝑉
= −(1 + 𝜆)𝜋𝐿𝑈′[𝑊 − 𝛼 (1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝐿𝑈′(𝑊 + 𝑆 − 𝐿 + 𝛼𝐿) = 0
𝑑𝛼
𝑑𝑉
= −𝑈′[𝑊 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝑈′(𝑊 + 𝑆 − 𝐿 + 𝛼𝐿) + (1 − 𝜋)𝑈′(𝑊 + 𝑆) = 0
𝑑𝑆
It is easy to see that second order conditions hold so we have a set of maximum
solution for optimal insurance and saving/borrowing. We solve the first-order conditions
1
for fair premium and get 𝛼 = 1 and 𝑆 = − 2 𝜋𝐿. This time, just as Mossin (1968) states,
individual purchases full insurance when the premium is actually fair.
When the initial endowments are different across periods, i.e. 𝑊1 ≠ 𝑊2 , it is
straightforward to show that 𝛼 = 1 and 𝑆 =
𝑊1 −𝑊2 −𝜋𝐿
2
solve both first order conditions.
And the individual is able to equate his final wealth in all three states to
𝑊1 +𝑊2 −𝜋𝐿
2
.
By introducing the endogenous saving/borrowing term into the model, the individual
now has more control over his wealth levels and consumptions across time and states, and
the outcomes in every state are affected by the individual’s decisions. So now we have a
first best solution for the optimal insurance and the saving/borrowing.
Purchasing more insurance will further increase the marginal cost and further decrease
the expected marginal benefit. With the saving/borrowing term, the individual can adjust
his wealth levels in both period 1 and period 2. So he will be able to smooth the
7
consumptions and equate the wealth levels in all three states. By adjusting his
saving/borrowing according to the insurance premium and the difference between the
initial wealth across time, the individual can equate his consumption in first period to the
consumption in the second period no matter whether the loss occur or not, so he will
purchase more insurance than if there were no saving/borrowing term.
Two-period model for precautionary saving or borrowing only
What if there is not any insurance offered in the economy, but people can still use saving,
or borrowing, if necessary, to smooth their consumption across periods and protect
themselves from future risk? Let the individual face the same potential risk in second
period, and decide how much he would like to save or borrow in the first period. Again we
begin with same endowments across periods. The expected utility for two periods is as
follow:
𝑉 ≡ 𝜋[𝑈(𝑊 − 𝑆) + 𝑈(𝑊 + 𝑆 − 𝐿)] + (1 − 𝜋)[𝑈(𝑊 − 𝑆) + 𝑈(𝑊 + 𝑆)]
It is easy to show that if the probability of loss is 0 and the potential loss will never
occur, the optimal saving for the individual will be 0. On the other hand, if the probability
of loss is 1 and the loss state will always appear, the individual will save half of the loss
size. And the optimal amount of saving will increase as the probability of loss increases,
1
so we have 0 < 𝑆 < 2 𝐿 when 0 < 𝜋 < 1.
For the case when endowments are different across periods, we have
𝑊1 −𝑊2 +𝐿
2
𝑊1 −𝑊2
2
<𝑆<
when 0 < 𝜋 < 1. In this case, the individual might borrow instead of save even
when he faces a potential loss in the future, given that his endowment in first period is
higher than his endowment in second period.
8
Compare the results above with the ones we get from the model for insurance and
endogenous saving/borrowing, we can see that the optimal amount of saving or borrowing
1
is − 2 𝜋𝐿 for same endowments across periods and
𝑊1 −𝑊2 −𝜋𝐿
2
for different endowments, so
with insurance as an additional choice, the individual will decrease his saving, or increase
his borrowing in both cases. The combination of insurance and saving/borrowing offers the
individuals the ability to further smooth the consumption across periods and better protect
themselves from the potential shocks in the future.
Saving/borrowing with non-zero interest rate
How will things change when saving yields a positive return and borrowing has a cost?
Will people still purchase full insurance under this circumstance? First, assume the initial
endowments are the same across two periods, then the individual’s consumption levels for
first period and for two possible outcomes of second period are:
𝐶1 = 𝑊 − 𝛼𝑃 − 𝑆
𝐶2 = 𝑊 + 𝑅𝑆 − 𝐿 + 𝛼𝐿
𝐶2 = 𝑊 + 𝑅𝑆
where 𝑅 > 1 is the interest rate. Here we assume the interest rate for saving and borrowing
is the same.
The individual’s expect utility of the two period then becomes
𝑉 ≡ 𝜋[𝑈(𝑊 − 𝛼𝑃 − 𝑆) + 𝑈(𝑊 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)] + (1 − 𝜋)[𝑈(𝑊 − 𝛼𝑃 − 𝑆) + 𝑈(𝑊
+ 𝑅𝑆)]
and the first-order conditions with respected to 𝛼 and 𝑆 are, respectively,
𝑑𝑉
= −(1 + 𝜆)𝜋𝐿𝑈′[𝑊 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝐿𝑈′(𝑊 + 𝑅𝑆 − 𝐿 + 𝛼𝐿) = 0
𝑑𝛼
9
𝑑𝑉
= −𝑈′[𝑊 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝑅𝑈′(𝑊 + 𝑅𝑆 − 𝐿 + 𝛼𝐿) + (1 − 𝜋)𝑅𝑈′(𝑊 + 𝑅𝑆)
𝑑𝑆
=0
Now assume the individual will purchase full insurance (𝛼 = 1) with positive interest
rate. then we have
(1 + 𝜆)𝑈′[𝑊 − (1 + 𝜆)𝜋𝐿 − 𝑆] = 𝑈′(𝑊 + 𝑅𝑆)
1
Solve for 𝑆 we have 𝑆 = − 1+𝑅 𝜋𝐿, plug this result into the first order derivative with
respect to 𝑆,
𝑑𝑉
𝑅
𝑅
= −𝑈′ (𝑊 −
𝜋𝐿) + 𝑅𝑈′ (𝑊 −
𝜋𝐿) > 0
𝑑𝑆
1+𝑅
1+𝑅
because 𝑅 > 1. So purchasing full insurance is not an optimal solution with a positive
interest rate. The individual will purchase less than full insurance.
When the initial endowments are different across periods, i.e. 𝑊1 ≠ 𝑊2 , it is
straightforward to show that the individual will also purchase less than full insurance.
Because there is a positive return for saving and a cost for borrowing, the individual
will adjust his saving or borrowing levels. As a result, the overall wealth in period 1 and
the overall wealth in period 2 will change accordingly. So the marginal cost for purchasing
insurance will not be equal to the expected marginal benefit any more under full insurance.
Hence the individual will purchase less insurance than what he would when the interest
rate is zero, and buy less than full insurance.
Insurance sold with a positive loading
Until this point, we only considered insurance with fair price. So what happens if the price
is unfair?
10
First, let us consider insurance with same initial wealth in both periods. We know that 𝛼 =
1
1+𝜋
solves the first order condition when 𝜆 = 0, so when 𝜆 > 0,
|𝛼=
1 (1
1+𝜋
+ 𝜆)𝜋𝐿𝑈 ′ [𝑊 − 𝛼(1 + 𝜆)𝜋𝐿] > 𝜋𝐿𝑈′(𝑊 − 𝐿 + 𝛼𝐿)
because 𝑈′[𝑊 − 𝛼(1 + 𝜆)𝜋𝐿] also increases when 𝜆 increases. In order to make the RHS
equal to LHS, we need 𝑈′[𝑊 − 𝛼(1 + 𝜆)𝜋𝐿] < 𝑈′(𝑊 − 𝐿 + 𝛼𝐿), which can be achieved
by reducing the amount of insurance purchased. So we know that the optimal insurance
1
when the price is unfair will be some value of 𝛼 < 1+𝜋.
It is straightforward to see that if the insurance is sold with a positive loading, the
optimal insurance will also decrease in the case when the initial endowments are different
across the two periods.
Even though unlike the Mossin’s theorem, the individual does not purchase full
insurance under fair price, he still decreases his insurance purchases when the price is unfair.
And this is easy to understand as when price increases, marginal cost increases and
expected marginal benefit decreases. So the individuals will buy less than what he would
buy if the insurance were sold at fair price
Now let us consider the case when there is endogenous saving/borrowing in the model.
First, assume the individual still purchases full insurance under unfair price. So he will
only adjust his optimal saving/borrowing. then we have
(1 + 𝜆)𝜋𝐿𝑈′[𝑊 − (1 + 𝜆)𝜋𝐿 − 𝑆] = 𝜋𝐿𝑈′(𝑊 + 𝑆)
plug this result into the first order derivative with respect to 𝑆,
𝑑𝑉
1
= (1 −
)𝑈′(𝑊 + 𝑆)
(1 + 𝜆)
𝑑𝑆
Because 𝜆 > 0 and the first order derivative of the utility is positive,
11
(1 −
1
)𝑈′(𝑊 + 𝑆) > 0
(1 + 𝜆)
So the two first order conditions cannot both be satisfied by only adjusting the level of
saving/borrowing, which leads to the conclusion that purchasing full insurance is never
optimal even with endogenous saving/borrowing.
The proof for different endowments across time is similar. In conclusion, the Mossin’s
Theorem can be proved for the two-period model with endogenous saving/borrowing.
Individual purchases full insurance if and only if the insurance is sold at fair price.
2. Comparative statics analysis
Changing wealth and its effects on optimal insurance for the benchmark model
What will happen to the optimal insurance level if the level of wealth changes? Some
previous literatures state insurance as inferior good (e.g. Mossin (1968), etc.). However, in
real world wealthy people sometimes purchase more insurance than poor people.
Schlesinger (2000) pointed out with greater wealth often comes greater risk, so people with
more wealth no necessary purchase less insurance. Dionne and Eeckhoudt (1984) show
that the derivative of optimal insurance with respect to wealth can be either positive, null
or negative, and they claim that this finding reduces the importance of interiority property
observed in the standard insurance literature. However, the sign is uncertain and there is
no clear condition for when the sign will change. So we want to see if our model can
provide additional information about this issue.
Other things remain the same, the individual’s objective is to maximize the expect
utility of the two period,
12
𝑉 ≡ 𝜋[𝑈(𝑊 − 𝛼𝑃) + 𝑈(𝑊 − 𝐿 + 𝛼𝐿)] + (1 − 𝜋)[𝑈(𝑊 − 𝛼𝑃) + 𝑈(𝑊)]
with respect to 𝛼. As long as the utility function is concave in consumption, the second
order condition with respect to 𝛼 is always negative, any 𝛼 satisfies first order condition
will be global optimal. The first order condition is then
−𝑃𝑈 ′ (𝑊 − 𝛼𝑃) + 𝜋𝐿𝑈′(𝑊 − 𝐿 + 𝛼𝐿) = 0
Differentiate the first order condition with respect to 𝑊, we have:
𝑑𝛼
𝑃𝑈 ′′ (𝑊 − 𝛼𝑃) − 𝜋𝐿𝑈′′(𝑊 − 𝐿 + 𝛼𝐿)
=
𝑑𝑊 𝑃2 𝑈 ′′ (𝑊 − 𝛼𝑃) + 𝜋𝐿2 𝑈′′(𝑊 − 𝐿 + 𝛼𝐿)
Because the second order derivative of utility is always negative, the denominator is always
negative, and the sign of this equation depends only on the numerator. Because we know
𝑃𝑈 ′ (𝑊 − 𝛼𝑃) = 𝜋𝐿𝑈′(𝑊 − 𝐿 + 𝛼𝐿) from first order condition, and 𝑃𝑈 ′ (𝑊 − 𝛼𝑃) and
𝜋𝐿𝑈′(𝑊 − 𝐿 + 𝛼𝐿) will always be positive, so the sign of the derivative is the same as the
sign of:
−[
𝑃𝑈 ′′ (𝑊 − 𝛼𝑃) 𝜋𝐿𝑈 ′′ (𝑊 − 𝐿 + 𝛼𝐿)
−
]
𝑃𝑈 ′ (𝑊 − 𝛼𝑃) 𝜋𝐿𝑈 ′ (𝑊 − 𝐿 + 𝛼𝐿)
And this equation is simply 𝐴𝑅𝐴(𝑊 − 𝛼𝑃) − 𝐴𝑅𝐴(𝑊 − 𝐿 + 𝛼𝐿)
If the insurance is sold at fair price, in which case the consumption level in first period
is equal to the consumption level in second period, given a loss occurs. Then the above
derivative will always be zero, so changing initial wealth will have no effects on optimal
insurance.
If the insurance is sold with a positive loading, people will decrease their insurance
purchases. As a result, the consumption level in first period will be higher than the
consumption level in second period, given a loss occurs. So the signs of the derivative will
depend on the types of absolute risk aversion the individual exhibits.
13
If the individual has a CARA utility, this equation will always be zero. The marginal
benefit and marginal cost always change at the same rate, and the individual will not
become more or less risk averse as his wealth level changes, so the amount of insurance he
purchases will remain the same. Under CRRA utility, changing the wealth level would have
no effects on the level of optimal insurance.
If the individual has an IARA utility, as the wealth level increases, the marginal cost
increases at a higher rate than the marginal benefit does, and the individual becomes more
risk averse. So optimal insurance increases as wealth level increases.
If the individual has a DARA utility, as the wealth level increases, the marginal cost
increases at a lower rate than the marginal benefit does, and the individual becomes less
risk averse. So optimal insurance decreases as wealth level increases. When the wealth
level increases, people know they would have more to consume in both periods and become
less risk averse, so they will decrease the amount of insurance purchased.
As for the case when the initial wealth levels are different across time, the results are
more straightforward. Increasing the initial wealth in period 1 will increase individual’s
insurance purchases while increasing the initial wealth in period 2 will decrease his
insurance purchases, regardless of the types of risk aversion the individual exhibits. The
intuition behind is simple as increasing the wealth level in period 1 decrease the marginal
cost while increasing the wealth level in period 2 decrease the expected marginal benefit,
so the individual will change his decision for insurance purchases accordingly.
Comparative statics for the full model
14
Let us consider the model with endogenous saving/borrowing, different initial wealth
levels in the two periods, positive loading, and positive interest rate. The objective function
is
𝑉 ≡ 𝜋[𝑈(𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆) + 𝑈(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)]
+ (1 − 𝜋)[𝑈(𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆) + 𝑈(𝑊2 + 𝑅𝑆)]
and the first order conditions for optimal insurance purchases and saving/borrowing are,
respectively,
𝑑𝑉
= −(1 + 𝜆)𝜋𝐿𝑈′[𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝐿𝑈′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿) = 0
𝑑𝛼
𝑑𝑉
= −𝑈′[𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] + 𝜋𝑅𝑈′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿) + (1
𝑑𝑆
− 𝜋)𝑅𝑈′(𝑊2 + 𝑅𝑆)
Denote the first order conditions as 𝑓(𝛼, 𝑆, 𝑊1 , 𝑊2 , 𝑅, 𝐿, 𝜋) and 𝑔(𝛼, 𝑆, 𝑊1 , 𝑊2 , 𝑅, 𝐿, 𝜋) ,
respectively. In order to simplify notation, we also denote
𝐷 = 𝑓𝛼 𝑔𝑆 − 𝑓𝑆 𝑔𝛼
which is always positive by the second order condition.
Then we can obtain the effects of changes in the initial wealth levels on the
equilibrium value of optimal insurance. Combine the partial second order derivatives with
first order conditions (more details about this calculations are provided in the Appendix),
we get
𝑑𝛼
= 𝐴1 [𝐴𝑅𝐴(𝑊2 + 𝑅𝑆) − 𝐴𝑅𝐴(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)]}
𝑑𝑊1
𝑑𝛼
= 𝐴2 [𝐴𝑅𝐴(𝑊2 + 𝑅𝑆) − 𝐴𝑅𝐴(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)]}
𝑑𝑊2
where 𝐴1 and 𝐴2 are positive polynomials.
15
Because we have proved that people purchase less than full insurance with a positive
loading, so (𝑊2 + 𝑅𝑆) > (𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿), and the signs of the derivatives depend on
the types of absolute risk aversion the agent has. As long as the agent’s preference exhibits
DARA/CARA/IARA, increasing wealth levels in either period, or both period will
decrease/no change/increase the amount of optimal insurance. This finding also proves that
when endogenous saving/borrowing presences, DARA is both the necessary and the
sufficient condition for insurance to be an inferior good.
With similar manipulations, we can show that under CARA or DARA, the optimal
insurance decreases when interest rate increases, and increases when the potential loss size
increases. As for the change of probability of loss, things are a bit different. Under DARA,
the effects on optimal insurance are ambiguous. But we can prove that the optimal
insurance decreases as the probability of loss increases when the individual exhibits CARA
or IARA.
3. Conclusion
The two-period model where people purchase the insurance in advance of the time when
the potential loss might occur better describes the time structure of the real world insurance.
In addition, the endogenous saving/borrowing enriches people’s options and enables them
to smooth their consumption across time and to achieve a better total outcome.
There are several interesting findings generated from this structure. First of all, we
show that people purchase less than full insurance even under fair price. And the amount
of insurance they purchased decreases as the probability of loss increases. Even though the
16
Mossin’s Theorem does not completely hold under this circumstance, people still decrease
their insurance purchases when the price is unfair.
Once we introduce the endogenous saving/borrowing into the model, people have the
ability to smooth their consumption across time and states, and are able to achieve the firstbest outcome. Mossin’s Theorem is proved for this setting. However, if there is a positive
interest rate for saving/borrowing, people will purchase less than full insurance even under
fair price.
The comparative statics show how optimal insurance changes as the environment
factors (initial wealth in either or both periods, interest rate, loss size and probability of
loss) change. Some new and intuitive results are found.
This paper only consider the type of coinsurance, and assume it will either be a no
loss state or a total loss state. Further research can be done for other types of insurance and
more complicate loss setting.
17
References:
Dionne, G., & Eeckhoudt, L. (1984). Insurance and saving: some further results. Insurance:
Mathematics and Economics, 3(2), 101-110.
Engen, E. M., & Gruber, J. (2001). Unemployment insurance and precautionary
saving. Journal of monetary Economics, 47(3), 545-579.
Hubbard, R. G., Skinner, J., & Zeldes, S. P. (1995). Precautionary saving and social
insurance (No. w4884). National Bureau of Economic Research.
Kantor, S. E., & Fishback, P. V. (1996). Precautionary saving, insurance, and the origins of
workers' compensation. Journal of Political Economy, 419-442.
Mossin, J. (1968). Aspects of rational insurance purchasing. The Journal of Political
Economy, 76(4), 553-568.
Schlesinger, H. (1981). The optimal level of deductibility in insurance contracts. Journal
of risk and insurance, 465-481.
Schlesinger, H. (2000). The theory of insurance demand. In Handbook of insurance (pp.
131-151). Springer Netherlands.
Schlesinger, H. (2006). Mossin's Theorem for Upper‐Limit Insurance Policies.Journal of
Risk and Insurance, 73(2), 297-301.
18
APPENDIX
A. Changing initial wealth levels and its effects on optimal insurance
First we have the derivatives of optimal insurance over initial wealth levels in the two
periods, respectively:
𝑓𝑆 𝑔𝑊1 − 𝑓𝑊1 𝑔𝑆
𝑑𝛼
=
𝑑𝑊1
𝐷
𝑓𝑆 𝑔𝑊2 − 𝑓𝑊2 𝑔𝑆
𝑑𝛼
=
𝑑𝑊2
𝐷
Substitute in the second order derivatives and simplify, we get:
𝑑𝛼
1
= {𝜋𝑅𝐿𝑈 ′′ [𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆][((1 + 𝜆)𝜋𝑅 − 1)𝑈′′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)
𝑑𝑊1 𝐷
+ ((1 + 𝜆)𝑅 − (1 + 𝜆)𝜋𝑅)𝑈′′(𝑊2 + 𝑅𝑆)]}
𝑑𝛼
1
= {𝜋𝐿𝑈 ′′ [𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆][((1 + 𝜆)𝜋𝑅 − 1)𝑈′′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)
𝑑𝑊2 𝐷
+ ((1 + 𝜆)𝑅 − (1 + 𝜆)𝜋𝑅)𝑈′′(𝑊2 + 𝑅𝑆)]}
From the first order condition with respect to insurance we have:
sf
(1 + 𝜆)𝜋𝐿𝑈 ′ [𝑊1 − 𝛼(1 + 𝜆)𝜋𝐿 − 𝑆] = 𝜋𝐿𝑈′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)
substitute this equation into the first order condition with respect to saving/borrowing, we
have:
[1 − (1 + 𝜆)𝜋𝑅]𝑈′(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿) = [(1 + 𝜆)𝑅 − (1 + 𝜆)𝜋𝑅]𝑈′(𝑊2 + 𝑅𝑆)
So we can divide the derivatives with either side and then time it back. Doing these
yields:
19
𝑑𝛼
= 𝐴1 [𝐴𝑅𝐴(𝑊2 + 𝑅𝑆) − 𝐴𝑅𝐴(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)]}
𝑑𝑊1
𝑑𝛼
= 𝐴2 [𝐴𝑅𝐴(𝑊2 + 𝑅𝑆) − 𝐴𝑅𝐴(𝑊2 + 𝑅𝑆 − 𝐿 + 𝛼𝐿)]}
𝑑𝑊2
where 𝐴1 and 𝐴2 are positive polynomials.
20