Saving and the Demand for Protection Against Risk
David Crainich1, Richard Peter2
Abstract: We study individual saving decisions in the presence of an endogenous future
consumption risk. The endogeneity arises because agents can buy insurance in our model.
We show that the slope of the reaction functions depends on the slope of absolute risk
aversion. If insurance is not Giffen, saving acts as a substitute for insurance in the Walrasian sense when absolute risk aversion is non-increasing. Conversely, if the substitution
effect of changes in the interest rate dominates the wealth effect, insurance acts as a substitute (complement) for saving in the Walrasian sense when absolute risk aversion is decreasing (increasing). Our findings complement earlier work where wealth effects were
absent and/or the temporal structure was different.
Keywords: risk; saving; insurance; substitute; complement
JEL-Classification: D11; D14; D81; D91; G22
Working Paper
Version: March 2016
1
CNRS (LEM, UMR 9221) and IÉSEG School of Management, Lille; e-mail: [email protected]
2
The University of Iowa, Iowa City; e-mail: [email protected]
1
1. Introduction
Risk management activities are usually analyzed in isolation in the literature. For
instance, the propensities to purchase insurance contracts, to make precautionary savings
or to perform prevention actions are in most cases dealt with separately. Ehrlich and
Becker’s (1972) classical contribution was the first analysis dedicated to the interaction
between various instruments - insurance and self-protection on the one hand, insurance
and self-insurance on the other hand - used to manage financial risks. This initiated a series of theoretical papers examining joint risk management decisions. Among these, Dionne and Eeckhoudt (1984) considered the relationship between saving and insurance
decisions and Menegatti and Rebessi (2011) examined simultaneous saving and selfprotection efforts. Note that the analysis of joint actions undertaken in order to protect
oneself against disease has been introduced the health economics literature (see for instance the analysis of self-protection activities and disease treatment in Henessy (2008) or
in Menegatti (2014)).
In this paper, we focus on the relationship between the demand for precautionary
savings and for insurance. Taken in isolation, the demand for precautionary savings has
been shown to depend – in the expected utility model - on the sign of the third derivative
of the utility function (see Leland (1968), Sandmo (1970) and Drèze and Modigliani
(1972)) while Kimball (1990) indicated that the intensity of this demand was measured by
the ratio of (minus) the third derivative to the second derivative of the utility function. In
these contributions, the future risk is exogenous. Surprisingly, the presence of endogenous
risks and their effect on saving has not been discussed much. This is despite the fact that
individuals have a variety of instruments to address future consumption risks (insurance,
self-protection, self-insurance etc.). We thus consider in this paper that individuals can
buy insurance contracts in the future (i.e. when they will be exposed to the risk) so that
the precautionary saving and the insurance decisions interact. Our contribution complements the one provided by Dionne and Eeckhoudt (1984). But instead of considering substitution or complementarity in the Hicksian sense (i.e. keeping the expected utility constant in order to cancel the wealth effect and isolating the price effect), we determine the
2
interaction of these two activities in the Walrasian sense. Besides, we also adopt a different temporal structure: instead of considering that saving and insurance decisions are contemporaneous, we assume that individuals make precautionary savings in the current period and purchase insurance contracts in the future.
The main result of our paper is that the nature of the interaction between insurance and precautionary savings depends on the way the Arrow-Pratt index of absolute
risk aversion changes with wealth. More precisely, if we assume that the demand for insurance falls when the price of insurance increases (i.e. insurance is not a Giffen good), we
show that an increase in the price of insurance leads to more precautionary savings (saving acts as a substitute for insurance in the Walrasian sense) if the utility function is DARA. Similarly, if higher interest rates increase savings, insurance is a substitute (complement) for saving in the Walrasian sense when absolute risk aversion is decreasing (increasing).
The paper is organized as follows. Section 2 describes the model and defines the
precautionary savings and the insurance purchase at the equilibrium. Sections 3 and 4
analyze the way these equilibrium values are modified by changes in the price of insurance and in the interest rate respectively. Section 5 concludes.
2. The Model
in
We consider two points in time,
with certainty, ∈ 1,2 . A consumption stream
+
̃ , where
of consumption, and
and
and . A decision-maker (DM) receives income
, ̃
is evaluated according to
is first-period utility of consumption,
is second-period utility
is the rate of pure preference for the present. We assume that
are increasing and concave which reflects risk aversion. Future income is subject to
a random loss of size
that occurs with probability . This risk at
the fact that the DM can purchase insurance to protect himself. Let
is endogenous due to
∈ 0,1 denote the
level of coverage and λ ≥ 0 be the loading factor such that 1 + λ is the per-unit price of
insurance. Consequently, the premium is given by 1 + λ
. Besides insurance, the DM
3
decides about his consumption allocation over time. Let
≥ 0. With these specifications, the DM’s objective function is given
random interest rate
by
denote savings with the non-
max %
−
#,$
+
'
+
+ 1 −
1+
− 1+λ
− 1−
+ 1+ − 1+λ
()
To compress notation, we use subscripts “L” and “N” to denote consumption in the loss
and the no-loss state, respectively. First-order conditions are given by
*
−
,
+
1− 1+λ
1+
,
+
,
+
− 1−
+ 1−
,
-
1+λ
,
-
= 0, 1
= 0. 2
The first equation describes that the optimal level of coverage is such that the marginal
rate of substitution between consumption in the loss state and in the no-loss state must be
equal to the slope of the line of insurance. The second equation implies that for optimal
saving marginal expected consumption utility must be equal across points in time. The
second-order conditions are satisfied, see the appendix.
3. Changes in the Price of Insurance
First, we investigate how a change in the price of insurance affects the demand for
insurance and the demand for saving. Note that we do not fix intertemporal expected utility as in Dionne and Eeckhoudt (1984). In this sense, we consider Walrasian demand rather than Hicksian demand to clarify whether and when there is gross substitution or
complementarity between insurance and saving. The cross-derivative of expected utility is
given by
0#$ =
1+
1− 1+λ
,,
+
− 1−
1+λ
With the help of the first-order condition (1), this can be rewritten as
where 1
=−
,,
0#$ =
⁄
,
1+
1−
1+λ
,
-
1- − 1+ ,
,,
-
.
denotes Arrow-Pratt risk aversion.3 From this we make
the following observation.
Recently, Bommier et al. (2012) address the conceptual difficulty of defining comparative risk aversion in
“certain x uncertain”-type choice situations. We admit that the measurement of risk aversion is much more
subtle in an intertemporal context. Still, we draw on standard terminology but try to point out caveats when
needed.
3
4
Remark 1: Reaction functions are decreasing (constant, increasing) if absolute risk aver-
sion is decreasing (constant, increasing).
The intuition is as follows: When savings are increased, this implies an increase in
second-period risk-free wealth. With decreasing absolute risk aversion, a given loss is less
painful at high wealth levels than at low ones. Consequently, insurance coverage should
be reduced to save on premium money. Conversely, when insurance coverage increases,
this implies a certain reduction in wealth due to the higher insurance premium but also a
contraction in future consumption risk. A reduction in second-period wealth stipulates
more saving according to consumption smoothing whereas a contraction in future consumption risk implies less saving because there is not so much need for precaution. DARA
is equivalent to the fact that the coefficient of absolute prudence exceeds the coefficient of
absolute risk aversion. As a consequence, the second effect dominates the first one and
optimal savings decrease upon an exogenous increase in insurance coverage. Overall, there
is a non-trivial interaction between saving and insurance as soon as risk aversion is not
constant across wealth. This indirect effect will have to be taken into account in the comparative statics analysis.
Let us begin with the direct effects, however. For insurance, we obtain that
0#λ =
−
1− 1+λ
,,
+
− 1−
1+λ
which we rewrite with the help of first-order condition (1):
0#λ =
− 1−
1+λ
1- − 1+ −
−
,,
-
,
+
,
+
+ 1−
+ 1−
,
-
.
,
-
,
As is well known, insurance can be a Giffen good such that an increase in price does not
necessarily reduce the optimal level of coverage. The explanation is that a price increase
impoverishes the consumer; more specifically, she has less wealth in the bad state of the
world where higher insurance coverage increases her consumption. This is the origin of
the positive effect undermining the common intuition that an increase in price should
reduce the level of coverage. This was first discussed by Hoy and Robson (1981); later,
Briys, Dionne and Eeckhoudt (1989) and Hau (2008) state the necessary and sufficient
conditions for insurance to be non-Giffen. For our purposes it is sufficient to note from
above that 0#λ is negative under CARA and IARA, whereas under DARA the sign is am-
5
biguous. We will distinguish between the cases where insurance is Giffen and where it is
not throughout the analysis.
For saving we derive
0$λ = −
1+
,,
+
+ 1−
,,
-
>0
due to the fact that marginal utility is diminishing. Here the reason is that a higher perunit price of insurance reduces the consumer’s wealth in the second period so that the
marginal benefit of saving is increased. As the insurance premium is paid in the second
period, the marginal cost of saving is unaffected and consequently the net effect on saving
is positive.4
Let us now apply the implicit function rule to determine how direct and indirect
effects play out together. We obtain that
8
1
7
=
− 0$$ 0#λ + 0#$ 0$λ ,
5 8λ 9
68
5 = 1 − 0## 0:λ + 0#$ 0;λ ,
48λ 9
where 9 is the determinant of the Hessian matrix. It is positive in any case as we know
from the second-order conditions (see the appendix). The first line states the net effect of
changes in the price of insurance on the demand for insurance whereas the second line
gives the net effect of changes in the price of insurance on the demand for saving, i.e., the
cross-price effect. From the discussion above we get the following Proposition.
Proposition 1: If insurance is not Giffen, the optimal level of insurance decreases and the
demand for saving increases when the price of insurance increases under non-increasing
absolute risk aversion. The cases where absolute risk aversion is increasing or where insurance is Giffen are indeterminate.
The intuition is quite simple. If insurance is not a Giffen good, the direct effect of a
price increase on the demand for insurance is negative. The direct effect on the demand
for saving is positive in any case. Now, a decrease in insurance exerts a non-negative substitution effect on saving when risk aversion is non-increasing (Remark 1). Likewise, an
When the premium is paid in the first period, an increase in the per-unit price of insurance reduces wealth
in the first period so that the marginal cost of saving is larger, whereas the marginal benefit remains unchanged. Consequently, the direct effect on saving would be negative in this case. This underlines that our
approach is quite different from the treatment in Dionne and Eeckhoudt (1984).
4
6
increase in saving exerts a non-positive substitution effect on the demand for insurance
when risk aversion is non-increasing. As a result the net effect is negative for the demand
for insurance and positive for the demand for saving. This extends the finding by Dionne
and Eeckhoudt (1984) to the case of Walrasian demand.
When preferences exhibit increasing absolute risk aversion, reaction functions are
upward sloping (Remark 1). Consequently, the indirect effect from saving on insurance is
positive, countervailing the negative direct effect, and the indirect effect from insurance
on saving is negative, countervailing the positive direct effect. In this case, the net effects
depend on the relative strength of both effects and are indeterminate. Similarly, when
insurance is a Giffen good, preferences must necessarily exhibit DARA implying downward sloping reaction functions. Then, the negative substitution effect from saving on
insurance countervails the positive direct effect of a price increase in the Giffen case, and
the negative substitution effect from insurance on saving countervails the positive direct
effect. Again, net effects are ambiguous.
We can also analyze this graphically. In Figure 1, we can see the initial reaction
functions
and
, which are downward sloping. This implies that we are in the
case of DARA. An increase in the price of insurance λ ↑ is associated with a decrease in
the level of coverage for a given level of saving. Therefore, insurance is not a Giffen good.
Finally, the direct effect on saving implies an upward shift of the reaction function. As we
can see, the new optimum is such that saving increases whereas the level of insurance
coverage decreases. The reason is that the direct effect and the indirect effect for each decision variable move in the same direction here. The other cases can be analyzed with a
similar graphical treatment.
7
4. Changes in the Interest Rate
The indirect effects exposed in Remark 1 still hold. Let us now examine the direct
effects of changes in the interest rate on the demand for: 1) savings and; 2) insurance coverage.
As indicated in the expression below, the effect of a change in the interest rate on
the demand for savings is ambiguous:
0$= =
,
+
+ 1−
This expression corresponds to:
,
-
+
1+
,,
+
+ 1−
,,
-
0$= = β [ Eu '( w% + (1 + r ) s ) + (1 + r ) sEu ''( w% + (1 + r ) s )]
where w% = w2 + (1 + λ )α pL + L% with L% =d ( p(1 − α ) L,1 − p ; −(1 − p)(1 − α ) L, p) .
Using the technique exposed by Gollier (2001)5, we obtain:
% ( w% + (1 + r ) s )]]
EU sr = β [ Eu '( w% + (1 + r ) s )[1 − R( w% + (1 + r ) s ) + wA
5
See section 16.2 page 239.
8
where R ( w ) denotes the coefficients of relative risk aversion evaluated in w . It is
then sufficient that the coefficient of relative risk aversion is lower than unity for savings
to be increasing with the interest rate.6 The ambiguous relationship between savings and
the interest rate can be explained as follows. The interest rate is the opportunity cost of
consumption (as opposed to saving). As a consequence, raising the interest rate creates
incentives to reduce consumption and, therefore, to increase savings (substitution effect).
But an increase in the interest rate also makes individuals wealthier (as long as their initial
saving is higher than zero) in period 2. Since the marginal utility of consumption is decreasing, this reduces the incentives to save at period 1 in order to consume at period 2
(wealth effect).
In what follows, we will consider both cases (i.e. that savings are increasing or decreasing with the interest rate). We now evaluate the effect of a change in the interest
rate on the demand for insurance. This effect is given by the following expression:
EUα r = β Ls( p(1 − (1 + λ ) p)vL'' − (1 − p)(1 + λ ) pvL'' )
Using the first-order condition (1), we obtain:
EUα r = β Ls(1 − p) p(1 + λ )vN' [ AN − AL ]
From this, we conclude that the demand for insurance falls when the interest rate
rises when the Arrow-Pratt coefficient of absolute risk aversion is decreasing. This can be
explained as follows: a higher interest rate makes individuals wealthier in period 2. As a
result, their propensity to purchase insurance coverage falls (resp. rises) if their absolute
risk aversion falls (resp. rises) with wealth.
As in section 3, we now use the implicit function theorem to determine the overall
effect of an increase in the interest rate on savings and on the demand for insurance. We
have:
8
1
7 =
− 0## 0$> + 0#$ 0#= ,
9
58
68
1
5
4 8 = 9 − 0$$ 0#> + 0$$ 0;= .
Descriptively, relative risk aversion is often assumed to exceed unity. An alternative sufficient condition is
that partial risk aversion is below one (Chiu et al., 2012), which finds much more support empirically
(Binswanger, 1981; Bar-Shira et al., 1997).
6
9
From these expressions and the discussion above, we get the following Proposition.
Proposition 2: If the demand for savings is such that the substitution effect dominates the
wealth effect (so that the direct effect of the interest rate on savings is positive), the optimal level of saving increases when the interest rate increases. The optimal insurance coverage decreases (resp. increases; resp. is constant) when the interest rate increases under
decreasing (resp. increasing; resp. constant) absolute risk aversion. The cases where the
direct effect of the interest rate on savings is negative are indeterminate.
The case of DARA utility functions is depicted in Figure 2.
Figure 2
Since the direct effect on saving of an increase in the interest rate is supposed to be
positive, an upward shift of s (α ) results from an increase in r . This shift leads to a reduction in the insurance coverage (indirect effect) since savings and insurance move in opposite directions when the utility function is DARA (see remark 1). In addition, an increase
in the interest rate reduces the demand for insurance (direct effect) when the utility function is DARA. Again, this raises savings as an indirect effect (see remark 1). In this situa-
10
tion, the direct and indirect effects have the same impacts on the variables (increasing
savings and reducing insurance coverage.
Consider now the case of IARA utility functions depicted in Figure 3.
s
indirect effect
direct
effect
direct
effect
indirect
effect
Figure 3
We suppose again that the direct effect on saving of an increase in the interest rate
is positive, so that an upward shift of s (α ) results from an increase in r . Since insurance
and savings now move in the same direction because the utility function is IARA (see remark 1), this shift raises the insurance coverage (indirect effect). The IARA assumption is
also such that an increase in the interest rate raises the insurance coverage (direct effect).
This shifts α ( s ) curve to right and again, since saving and insurance go in the same direction, this increases the demand for saving (indirect effect). The case depicted represents
once again a situation where the direct and indirect effects have the same positive impacts
on saving and insurance coverage.
When the direct effect of the interest rate on savings is negative, the direct and indirect effects move in opposite direction, so that the impact of an increase in the interest
rate on both savings and the demand for insurance are indeterminate.
11
6. Discussion and Conclusion
Risk management tools are usually analyzed in isolation. This is the case of precautionary
savings which is defined as the extra saving due to risky future income. In the literature, the
risk in question is usually considered as background since it is assumed that it cannot be modified through prevention actions, diversified or insured against. These paper thus deal with the
way the introduction of an exogenous risk affects savings. Less attention has been dedicated
to the interaction between savings and other economics decisions that could be used to deal
with a future risk (that would be then endogenous). This is a question we address in this paper
since we analyze the relationship between the demand for precautionary savings and insurance.
Our contribution complements Dionne and Eeckhoudt (1984) who examined a similar question. However, we consider substitution or complementarity in the Walrasian sense and not in
the Hicksian sense (as Dionne and Eeckhoudt did). Besides, we also assume – as in it ususlly
the case in the literature in risk theory - that individuals make precautionary savings in the
current period and purchase insurance contracts in the future (while saving and insurance decisions are simultaneous in Dionne and Eeckhoudt (1984)).
To determine the interaction between savings and insurance, we examine the effect of an increase in the price of insurance on the propensity to save and the effect of an increase in the
interest rate on the demand for insurance. Assuming that saving and insurance are both normal goods, we show that these two risk management tools are substitutes if individuals’ utility
functions are DARA (Decreasing Absolute Risk Aversion).
12
13
References
Bar-Shira, Z., R. Just, and D. Zilberman, 1997, Estimation of Farmers’ Risk Attitude: An
Econometric Approach, Agricultural Economics, 17(2-3): 211-222.
Binswanger, H., 1981, Attitudes Towards Risk: Theoretical Implications of an Experiment
in Rural India, The Economic Journal, 91(364): 867-890.
Bommier, A., A. Chassagnon, and F. Le Grand, 2012, Comparative Risk Aversion: A Formal Approach with Applications to Saving Behavior, Journal of Economic Theory,
147(4): 1614-1641.
Briys, R., G. Dionne, and L. Eeckhoudt, 1989, More on Insurance as a Giffen Good, Jour-
nal of Risk and Uncertainty, 2(4): 415-420.
Chiu, H., L. Eeckhoudt, and B. Rey, 2012, On Relative and Partial Risk Attitudes: Theory
and Implications, Economic Theory, 50(1): 151-167.
Dionne, G. and L. Eeckhoudt, 1984, Insurance and Saving: Some Further Results, Insur-
ance: Mathematics and Economics, 3(2): 101-110.
Gollier, C., 2001, The Economics of Risk and Time, Cambridge MA: MIT Press.
Hau, A., 2008, When is a Coinsurance-Type Insurance Policy Inferior or even Giffen?
Journal of Risk and Insurance, 75(2): 343-364.
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14
Appendix
We denote the objective function by 0 for expected intertemporal consumption utility.
For the second-order conditions, note that
0## =
? 1− 1+λ
,,
+
+ 1−
@ 1+λ A
due to risk aversion of second-period utility. Furthermore, also
0$$ =
,,
+
1+
,,
+
+ 1+
,,
-
,,
-B
C0
C0
because of risk aversion of first- and second-period utility. The cross-derivative of expected utility is given by
0#$ =
1+
1− 1+λ
,,
+
− 1−
1+λ
,,
-
and is ambiguous. To calculate the determinant of the Hessian matrix, we also need the
square of the cross-derivative, which is
0#$ =
1+
+ 1−
1− 1+λ
1+λ
,,
-
,,
+
.
−2 1−
1− 1+λ
1+λ
From there it is easy to see that the determinant of the Hessian is obtained as
0## 0$$ − 0#$ =
,,
0## +
1+
so that the objective function is globally concave in
1−
,, ,,
+ -
and .
1−2 1+λ
,, ,,
+ -
> 0,