Optimal prevention and background risk in a
two-period model
- Christophe COURBAGE (The Geneva Association, Geneva, Switzerland)
- Béatrice REY (Université Lyon 1, Laboratoire SAF)
2011.25
Laboratoire SAF – 50 Avenue Tony Garnier - 69366 Lyon cedex 07 http://www.isfa.fr/la_recherche
Optimal prevention and background risk in a
two-period model
Christophe Courbage ∗ and Béatrice Rey +
∗
The Geneva Association, Switzerland, and
+
ISFA, Université de Lyon, Université Lyon 1, France
Abstract
In this paper, we look at the impact of background risk on optimal prevention with
respect to another risk. We carry out our analysis in a two-period framework and use
various configurations of background risks defined either in the first or second period, as
state-independent or state-dependent. We show that results differ depending on both the
nature of the background risk and the properties of the utility function.
JEL: D81
Keywords: Prevention, Background risk, Stochastic dominance
1. Introduction
Prevention is an essential tool to protect against risks. It is an activity that reduces
the probability of a loss should it occur1 . The literature on prevention dates back from
the earlier work of Ehlrich and Becker (1972) who showed that contrary to intuition,
prevention and insurance could be complements. Their pioneer work has led to extensive
literature on the role of individual preferences in explaining optimal prevention. In particular, a series of papers investigated the role of risk aversion on the demand for prevention
(see e.g. Dionne and Eeckhoudt, 1985; Briys and Schlesinger, 1990; Julien et al.,1999).
More recently, other works highlighted the concept of prudence as a driver of optimal prevention (see Chiu, 2005; Eeckhoudt and Gollier, 2005; Menegatti, 2009). Most of these
works have been carried out in the presence of only one source of risk. However, economic
decision makers often confront several sources of risk. They usually make decisions about
one risk whilst simultaneously facing other risks. A large amount of literature has shown
1
Following Ehrlich and Becker’s (1972) terminology, this activity refers also to self-protection.
1
that the introduction of a background risk affects many economic decisions and individual
behaviours with respect to other risks (see e.g. Doherty and Schlesinger, 1990; Gollier
and Pratt, 1996; Eeckhoudt et al., 1996; Hau, 1999; Rey, 2003). Surprisingly, no work,
to the best of our knowledge, has addressed the issue of prevention in the presence of
background risk. This paper tries to fill this gap.
To carry out our analysis, we rely on a recent work of Menegatti (2009) who modelises
prevention activities in a two-period framework. This stems from the fact that often the
decision to engage in prevention activity precedes its effect on the probability, contrary to
the common analysis of prevention in a one-period framework which implicitly assumes
that the decision to engage in prevention activity and its effect on the probability are
simultaneous. Not only is the use of a two-period framework justified on realistic grounds,
but also it makes it possible to obtain tractable first-order conditions and to find simple
conditions on individual preferences to sign the effect of a background risk on prevention.
Another advantage of a two-period framework is that it makes it possible to consider
various typologies of background risk. The background risk can occur either in the first
period when the individual decides his level of prevention, or in the second period when
the first risk the individual wants to protect against occurs. A third possibility is to have
a state-dependent background risk as proposed by Fei and Schlesinger (2008), i.e. that
the background risk can occur in the second period either only in the loss state, making
the loss itself risky, or in the no-loss state.
In this paper, we first look at how the introduction of a background risk modifies the
optimal level of prevention considering these three forms of background risk. We show
that results differ depending on both the configuration of the background risk and prudence and/or risk aversion. These background risks have a different impact on prevention
as they differently affect either the marginal cost of prevention or its marginal benefit.
For example, we show that a prudent individual does not necessarily exert more effort in
the face of a background risk. Secondly, we generalise our results and investigate how an
increase in the background risk impacts the optimal level of prevention. So as to define
increase in background risk, we use the framework of nth-order stochastic dominance.
Stochastic dominance encompasses very general form of risk increase including the wellknown concept of mean-preserving increase in risk introduced by Rothschild and Stiglitz
(1970) as well as increase in downside risk as defined by Menezes et al. (1980). We show
that results are driven by the signs of the successive derivatives of the utility function
to any order n. Depending on the type of increase in risk considered, more restrictive
conditions than prudence are required, such as temperance or edginess. As an illustration, in the case of an increase in downside second-period background risk, only prudent
individuals will increase prevention.
This paper is organised as follows. In the next section, we introduce the benchmark
model when there is no background risk. Section 3 deals with the effect of the background
risk on prevention, considering three types of background risk. Section 4 generalises results
to the case of an increase in background risk in terms of nth-order stochastic dominance.
Finally, a short conclusion is provided in the last section.
2. The benchmark model
As specified in the introduction, we follow Menegatti (2009) by modelising prevention
in a two-period setting, i.e. we assume that the effort is undertaken in the first period
while its effect on the probability occurs in the second period.
2
The agent chooses the effort level e in order to maximise his total intertemporal utility,
V (e). His total utility is given by
V (e) = u(w1 − e) + p(e)v(w2 − l) + (1 − p(e))v(w2 )
(1)
where u and v are the utility function in the first and second period respectively. We
assume that utility is increasing in wealth in each period (u0 (x) > 0 ∀x and v 0 (x) > 0
∀x) and the individual is risk averse in both periods ( u00 (x) < 0 ∀x and v 00 (x) < 0 ∀x).
p(e) is the probability that the event generating the loss l (l > 0) occurs, and w1 and w2
are the safe wealth in period 1 and period 2 respectively.
As usual2 , we assume that the probability p(e) is such that p0 (e) < 0 and p00 (e) > 0
for all levels of e, i.e. that an increase in effort reduces the probability that the loss will
occur at a decreasing rate.
The first-oder condition (FOC) for a maximum writes as V 0 (e) = 0, which is equivalent
to:
u0 (w1 − e) = p0 (e)(v(w2 − l) − v(w2 ))
(2)
Note that the second order condition (V 00 (e) < 0 ∀e) is always satisfied with the
assumptions made on u, v and p(e) so that the optimal level of prevention e∗ is fully
determined by the FOC (V 0 (e∗ ) = 0).
The left-hand term of eq. (2), u0 (w1 −e), represents the marginal cost of prevention, i.e.
the loss of first-period utility due to exerting effort, while the right-hand term, p0 (e)(v(w2 −
l) − v(w2 )), represents the marginal benefit of prevention, i.e. the expected gain of second
period utility due to a reduction in the probability of loss.
In the next section, we ask how the introduction of background risk on wealth
impacts the level of effort. The introduction of a background risk can take various
forms. The background risk can be in the first period, when the individual decides on
his level of prevention. It can also be in the second period, when the first risk occurs.
A third possibility is to have a state-dependent background risk, i.e. assuming that the
loss in the second period is itself risky or that the background only occurs in the no-loss
state. Another possibility occurs when a risk is introduced in the two periods.
In each of these cases, the effect of a background risk on prevention activities is different
depending on individual preferences.
3. No background risk versus background risk
In presence of other(s) risk(s) the agent expected utility has the following general
expression:
V (e) = E[u(w1 − e + ˜1 )] + p(e)E[v(w2 − l + ˜2b )] + (1 − p(e))E[v(w2 + ˜2g )].
(3)
where E denotes the expectation operator over the random variables ˜k (k = 1, 2b, 2g)
assumed actuarially neutral (E(˜k ) = 0 ∀k). Next sections studies different cases presented
above.
3.1 State-independent background risk
2
See Dionne and Eeckhoudt (1985), or Julien et al. (1999) for example.
3
We first analyze the effect the introduction of one risk in the first period:
˜2b = ˜2g = 0. For instance, let us assume that the individual is confronted with another
source of risk when deciding on his level of prevention, making his current environment
more uncertain. In this case, the problem becomes
V1 (e) = E[u(w1 − e + ˜1 )] + p(e)v(w2 − l) + (1 − p(e))v(w2 ).
(4)
The optimal level of prevention e∗1 is given by
V10 (e∗1 ) = −E[u0 (w1 − e∗1 + ˜1 )] + p0 (e∗1 )(v(w2 − l) − v(w2 )) = 0.
Comparing the two optimal values,
V10 (e∗ )
e∗1
0
(5)
∗
and e , we have
= −E[u (w1 − e∗ + ˜1 )] + u0 (w1 − e∗ ).
(6)
This equation is negative if and only if u000 (x) ≥ 0 ∀x. Therefore, the introduction of a
background risk in the first period reduces the level of prevention (e∗1 ≤ e∗ ) if and only if
the individual is prudent in the first period. If this result is rather surprising at first sight,
it can be easily explained. Indeed, the introduction of the background risk in the first
period increases the marginal cost of prevention of a prudent individual in comparison to
the situation without background risk since E[u0 (w1 − e∗ + ˜1 )] ≥ u0 (w1 − e∗ ) if and only if
u000 ≥ 0, while it leaves unchanged the marginal benefit of prevention. The prudent agent
(in the first-period) thus decreases his optimal effort as it increases the marginal cost of
prevention without modifying its marginal benefit.
This results can be also explained in terms of harms disaggregation as introduced
by Eeckhoudt and Schlesinger (2005) who showed that a prudent agent prefers larger
wealth in the period where he bears the risk. Hence, by reducing prevention in the first
period, the agent increases his wealth in the period he faces the background risk, which
is appreciated by a prudent individual. ajouter commentaire 2 du referee B. ??.
Yet, when the additional risk is introduced in the second period, this is no longer the
case. This case corresponds to ˜1 = 0 and ˜2b = ˜2g , denoted ˜2 . The optimal level of
prevention e∗2 is given by
V20 (e∗2 ) = −u0 (w1 − e∗2 ) + p0 (e∗2 )(E[v(w2 − l + ˜2 )] − E[v(w2 + ˜2 )]) = 0.
(7)
∗
Evaluating this condition at e , we obtain
V20 (e∗ ) = −p0 (e∗ )(v(w2 − l) − v(w2 ) − E[v(w2 − l + ˜2 )] + E[v(w2 + ˜)]).
(8)
It is easy to verify that this equation is positive if and only if v 000 (x) ≥ 0 ∀x. Indeed,
let’s define the function g such that g(x) = v(x) − v(x − l) for all x. Saying that the
previous equation is positive is equivalent to say that the expression g(w2 ) − E[g(w2 + ˜2 )]
is negative which is equivalent to g 00 (x) ≥ 0 i.e. v 000 (x) ≥ 0 ∀x. Thus, for an individual
prudent in the second period, the introduction of a background risk increases the level of
prevention (e∗2 ≥ e∗ ). Contrary to the preceding case, the introduction of a background
risk in the second period affects the marginal benefit of prevention, while it does not
affect the marginal cost. Indeed, for an individual prudent in the second period, the
introduction of the background risk in the second period increases the marginal benefit of
prevention since p0 (e∗ )(E[v(w2 − l + ˜2 )] − E[v(w2 + ˜2 )]) ≥ p0 (e∗ )(v(w2 − l) − v(w2 )) if and
only if v 000 ≥ 0, without impacting the marginal cost. The prudent agent (in the second
period) thus increases his optimal effort as it increases the marginal benefit of prevention
without modifying its marginal cost. ajouter commentaire 3 rapporteur B ?? .
Results can be summarised summarized ?? as follow:
4
Proposition 1
The introduction of a background risk in the first period reduces the level of prevention
∗
( e1 ≤ e∗ ) if and only if the individual is prudent in the first period.
The introduction of a background in the second period increases the level of prevention
∗
( e2 ≥ e∗ ) if and only if the individual is prudent in the second period.
Therefore, as stressed earlier, the impact of a background risk on optimal prevention
differs depending on the period when the background risk is introduced. This is explained
by the fact that when the background risk is introduced in the first period, it impacts the
marginal cost of prevention, while when the background risk is introduced in the second
period, it affects the marginal benefit of prevention.
While we have considered that the additional risk can be introduced either in the
first period or in the second period, a third possibility is that the risk occurs only in
certain state of nature, either in the state of loss making the loss more uncertain, or in
the state of no-loss. State dependent risk has been introduced by Fei and Schlesinger
(2008) to address the effect of a background risk on the demand for insurance.
3.2. State-dependent risk
First, we assume that the background risk appears in the state of nature where there
is the loss (“bad” state of nature): ˜1 = ˜2g = 0.
The problem becomes:
V2b (e) = u(w1 − e) + p(e)E[v(w2 − l + ˜2b )] + (1 − p(e))v(w2 ).
(9)
The optimal level of prevention e∗2b is given by
V2b0 (e∗2b ) = −u0 (w1 − e∗2b ) + p0 (e∗2b )(E[v(w2 − l + ˜2b )] − v(w2 )) = 0.
(10)
Comparing this optimal effort level to the optimal value without background risk, we have
V2b0 (e∗ ) = −p0 (e∗ )(v(w2 − l) − E[v(w2 − l + ˜2b )]).
(11)
This equation is positive for all utility function v such that v 00 (x) ≤ 0 ∀x. Hence, the
introduction of a background risk in the loss state of nature increases the optimal effort
level of prevention (e∗2b ≥ e∗ ).
In the case where the risk appears in the good state of nature, ˜1 = ˜2b = 0, we obtain
the opposite result. Indeed, in this case, the problem writes as follows
V2g (e) = u(w1 − e) + p(e)v(w2 − l) + (1 − p(e))E[v(w2 + ˜2g )].
(12)
Denoting e∗2g the optimal effort level of prevention, it is easy to show that e∗2g ≤ e∗ if and
only if v 00 (x) ≤ 0 ∀x. We then have the following proposition.
Proposition 2
For all risk averse individuals in the second period, the introduction of a background
risk in the loss state of nature increases the optimal effort level of prevention ( e∗2b ≥ e∗ ),
while the introduction of a background risk in the no-loss state of nature decreases the
optimal effort level of prevention ( e∗2g ≤ e∗ ).
5
We can explain these results in a similar way as before. The introduction of a statedependent background risk only impacts the marginal benefit of prevention and leaves
unchanged its marginal cost. More precisely, the introduction of the background risk in
the bad state of nature increases the marginal benefit of prevention for all risk averse
individuals in comparison to the case without background risk since (p0 (e∗ )(E[v(w2 − l +
˜)] − v(w2 )) ≥ p0 (e∗ )(v(w2 − l) − v(w2 ))) if and only if v 00 (x) ≤ 0. On the contrary, in
the case where the background risk appears in the good state of nature, it decreases the
marginal benefit of prevention for all risk averse individuals in comparison to the case
without background risk since (p0 (e∗ )(v(w2 − l) − Ev(w2 + ˜)) ≤ p0 (e∗ )(v(w2 − l) − v(w2 )))
if and only if v 00 (x) ≤ 0.
3.3. Global risk
In this section, we examine the case of a global risk that is the case where a risk occurs
in period one and in period 2: ˜1 = ˜2b = ˜2g , where risk in the two period is independent.
We denote it ˜. The agent expected utility writes as:
V̂ (e) = E[u(w1 − e + ˜)] + p(e)E[v(w2 − l + ˜)] + (1 − p(e))E[v(w2 + ˜)].
(13)
Let’s denote by ê the optimal level of prevention, it verifies V̂ 0 (ê) = 0. Evaluating the
first order condition at e∗ , we have:
V̂ 0 (e∗ ) = −E[u0 (w1 − e∗ + ˜) + p0 (e∗ )(E[v(w2 − l + ˜)] − v(w2 + ˜)).
(14)
Replacing the term p0 (e∗ ) by its expression given in section 2, we obtain:
V̂ 0 (e∗ ) = −E[u0 (w1 − e∗ + ˜)] + u0 (w1 − e∗ )
(E[v(w2 − l + ˜)] − E[v(w2 + ˜)])
.
(v(w2 − l) − v(w2 ))
(15)
The effect of the global risk is ambiguous. Indeed, the sign of V̂ 0 (e∗ ) is not always
positive or negative if the agent is prudent in all periods as it is commonly assumed.
Nevertheless, we can stress sufficient conditions to obtain some interesting results. In
that follows, in order to simplify mathematical expressions, we denote by N
the expression
D
(E[v(w2 − l + ˜)] − E[v(w2 + ˜)])
.
(v(w2 − l) − v(w2 ))
V̂ 0 (e∗ ) >, =, < 0 is then equivalent to
N
u0 (w1 − e∗ )
>, =, < 1.
E[u0 (w1 − e∗ + ˜)] D
(16)
If the agent is prudent in period one (u000 > 0) and is prudent in period two (v 000 > 0) then
u0 (w1 − e∗ )
N
< 1 and
> 1 so that the effect of the global risk is ambiguous.
0
∗
E[u (w1 − e + ˜)]
D
As we have understood in the previous sections, for a prudent individual the
6
effect of the additional risks plays in a opposite erection so that the total effect
is undetermined without restrictive condition on the amplitude of bla bla bla
.
We can derive the following results:
if v 000 = 0 (utility function corresponding to the quadratic utility) and if u000 > 0 then
ê < e∗ ,
if u000 = 0 and if v 000 > 0 then ê > e∗ .
Note that these results are in accordance to the ones obtained in the previous sections.
Indeed, considering the quadratic utility function (v 000 = 0) permits to neutralize the effect
of the risk in the second period. By analogy, considering (u000 = 0) permits to neutralize
the effect of the risk in the first period.
dire quelques mots sure l’analogie avec le cas claque cf commentaries 6 du
rapporteur B.
4. Increase in the background risk
expliquer pourquoi on ne traite pas le cas de la section 3.3 .
In the previous section, we looked at the impact of introducing a background risk on
prevention. Yet, an interesting and connected question is to wonder how an increase in
background risk modifies prevention decision.
To carry out our analysis we use the concept of nth-order stochastic dominance to define change in background risk. Let F and G denote two cumulative distribution functions
of wealth, defined over a probability support contained
within the open interval
Rz
R z ]a, b[. Define F1 = F and G1 = G. Now define Fk+1 (z) = a Fk (t)dt and Gk+1 (z) = a Gk (t)dt for
k ≥ 1. The distribution F dominates the distribution G via nth-order stochastic dominance (denoted nSD) if Fn (z) ≤ Gn (z) for all z, and if Fk (b) ≤ Gk (b) for k = 1, 2, .., n − 1.
If the random wealth variables ˜ and β̃ have distributions F and G respectively, β̃ is said
to be riskier than ˜ in terms of nth-order stochastic dominance, or equivalently that ˜
nSD β̃ .
From Ingersoll (1987), we know that if the random variable β̃ is riskier than ˜ in
terms of nth-order stochastic dominance, then E[f (β̃)] ≤ E[f (˜)] for all functions f
with derivatives3 f 0 , f 00 , f 00 , .., f (n) such that (−1)k+1 f (k) ≥ 0 for k = 1, 2, .., n. Note
that preferences over nth-order stochastic dominance represent the common preferences
of all decision-makers whose preferences satisfy risk apportionment of degrees 1 to n in
the terminology of Eeckhoudt and Schlesinger (2006). These decision-makers prefer to
disaggregate risk across equiprobable states of nature.
When the first n−1 moments of ˜ and β̃ are equals, nth-order stochastic dominance coincides with the Ekern’s (1980) concept of increase in nth-degree risk. As an example, β̃ is
an increase in second-degree risk over ˜ if ˜ dominates β̃ via second-order stochastic dominance and both random variables have equal mean. This is what Rothshild and Stiglitz
(1970) define as a “mean-preserving increase in risk”. Similarly, Menezes et al. (1980)
describe an increase in third-degree risk, which is also called an “increase in downside
risk”.
3
We denote by f (k) the k th derivative of f .
7
As before we consider three scenarios: an increase in the background risk in the first
period, an increase in the background risk in the second period and an increase in the
state-dependent background risk.
8
4.1 State-independant background risk increase
Let us start with the case where the background risk is introduced in the first period,
i.e. in the case where the individual is confronted with another source of risk when
deciding on his level of prevention making his current environment uncertain. The agent
maximisation problem is then defined by eq. (??). Let’s now introduce another background risk βe that is riskier than ˜ in terms of nth-order stochastic dominance, then the
agent problem is the following:
e + p(e)v(w2 − l) + (1 − p(e))v(w2 )
V1β (e) = E[u(w1 − e + β)]
(17)
The optimal level of prevention e∗1β is given by
0
e + p0 (e∗ )(v(w2 − l) − v(w2 )) = 0.
V1β
(e∗1β ) = −E[u0 (w1 − e∗1β + β)]
1β
(18)
Comparing the two optimal values, e∗1β and e∗1 , we have
0
e + E[u0 (w1 − e∗ + ˜)].
V1β
(e∗1 ) = −E[u0 (w1 − e∗1 + β)]
1
(19)
Applying the properties of stochastic dominance, we show that this equation is negative,
i.e. e∗1β ≤ e∗1 , if and only if (−1)k+1 u(k+1) ≤ 0 ∀k = 1, ..., n.
If the background risk is introduced in the second period, the agent maximisation
problem is defined by eq. (6). As done previously, if we replace the risk ˜ by another
background risk βe that is riskier than ˜ in terms of nth-order stochastic dominance. The
agent problem is the following:
e + (1 − p(e))E[v(w2 + β)].
e
V2β (e) = u(w1 − e) + p(e)E[v(w2 − l + β)]
(20)
Let’s denote e∗2β its solution. Evaluating this condition at e∗2 , we obtain:
0
e − E[v(w2 + β)]
e − E[v 0 (w2 − l + ˜)] + E[v 0 (w2 + ˜)]). (21)
V2β
(e∗2 ) = p0 (e∗2 )(E[u0 (w2 − l + β)]
0
V2β
(e∗2 ) ≥ 0 is equivalent to e∗2 ≤ e∗2β if and only if (−1)k+1 v (k+1) ≤ 0 ∀k = 1, ..., n. Indeed,
0
e Results
using the function g, V2β
(e∗2 ) ≥ 0 is equivalent to E[g(w2 + ˜)] ≤ E[g(w2 + β)].
are summarised in the following proposition.
Proposition 3
An increase in the first period background risk in terms of nth-order stochastic dominance decreases the optimal level of prevention (e∗1β ≤ e∗1 ) if and only if (−1)k+1 u(k+1) ≤ 0
∀k = 1, ..., n.
An increase in the second period background risk in terms of nth-order stochastic dominance increases the optimal level of prevention (e∗2β ≥ e∗2 ) if and only if (−1)k+1 v (k+1) ≤ 0
∀k = 1, ..., n.
Note that if we restrict change in background risk to the special case of nth-degree
increase in risk defined by Ekern (1980), we then have that an nth-degree increase in the
first (second)-period background risk decreases (increases) the optimal level of prevention if and only if (−1)n+1 u(n+1) ≤ 0 ((−1)n+1 v (n+1) ≤ 0). Note also that proposition 1
corresponds to the special case of second-order increase in risk since E(˜) = 0.
9
As an illustration, consider the special case of third-order increase in risk where ˜ and
β̃ write as ˜ = [−k, ˜l; 12 , 21 ] and β̃ = [0, −k + ˜l; 12 , 12 ] with k > 0 and E(˜l) = 0.We then have
e∗1β ≤ e∗1 if and only if the individual is temperant4 in the first period (e.g. iff u(4) ≤ 0).
In the case where β̃ is an 4th-degree increase in risk i.e. in the case where ˜ = [k̃, ˜l; 12 , 12 ]
and β̃ = [0, k̃ + ˜l; 12 , 12 ] with E(˜l) = E(k̃) = 0 (and ˜l and k̃ independent), then e∗1β ≤ e∗1 if
and only if the individual is “edgy5 ” in the first period (e.g. iff u(5) ≥ 0).
4.2. State-dependent background risk increase
Assume that the agent faces a background risk ˜ in the loss state of nature. The
maximisation problem is defined by eq. (9). Assume that the risk ˜ is replaced by βe such
that βe nSD ˜. The problem becomes:
V2bβ (e) = u(w1 − e) + p(e)E[v(w2 − l + β̃)] + (1 − p(e))v(w2 ).
(22)
0
The optimal level of prevention e∗2bβ is given by V2bβ
(e∗2bβ ) = 0. Comparing this optimal
level of prevention to the optimal one with the background risk ˜, we have:
0
e
V2bβ
(e∗2b ) = p0 (e∗2b )(−E[v(w2 − l + ˜)] + E[v(w2 − l + β)])
(23)
This equation is positive for all utility function v such that (−1)k+1 v k ≥ 0 ∀k = 1, .., n.
Hence, the introduction of an even riskier background risk in term of nth-order stochastic
dominance in the loss state of nature increases the optimal level of prevention (e∗2bβ ≥ e∗2b ).
In the case where the background risk appears in the good state of nature, we compare
the solution e∗2g given by eq. (12) to the solution of the following problem (denoted by
e∗2gβ ).
e
V2gβ (e) = u(w1 − e) + p(e)v(w2 − l) + (1 − p(e))E[v(w2 + β)].
(24)
If βe is such that βe nSD ˜. then e∗ ≤ e∗ for all utility function v such that
2gβ
2g
(−1)k+1 v k ≥ 0 ∀k = 1, .., n. This gives the following proposition.
Proposition 4
An increase in the second period background risk in terms of nth-order stochastic dominance in the loss state increases the optimal level of prevention (e∗2bβ ≥ e∗2b ) if and only
if (−1)k+1 v (k) ≥ 0 ∀k = 1, ..., n.
An increase in the second period background risk in terms of nth-order stochastic dominance in the no-loss state decreases the optimal level of prevention (e∗2gβ ≤ e∗2g ) if and
only if (−1)k+1 v (k) ≥ 0 ∀k = 1, ..., n.
Note as in the previous case that if we restrict change in background risk to the special
case of nth-degree increase in risk defined by Ekern (1980), we then have that a nthdegree increase in the second-period background risk loss state (no-loss state) increases
4
Temperance (defined by u(4) ≤ 0) was introduced by Kimball (1992) in the context of risk management in the presence of background risk. A decision maker is temperant when “an unavoidable
(background) risk leads him to reduce exposure to another risk even if the two risks are statistically
independent”.
5
Edginess (defined by u(5) ≥ 0) was introduced by Lajeri-Chaherli (2004) in the context of multiple
risks in a two-period model. Specifically, edginess captures the reactivity to multiple risks on precautionary motives. It is a necessary condition to have preferences exhibiting standard prudence or precautionary
vulnerability.
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(decreases) the optimal level of prevention if and only if (−1)n+1 v (n) ≥ 0 . Note also that
proposition 2 is a special case of proposition 4.
5. Conclusion
The presence of a background risk can modify individual behaviour with respect to
other risks. In this paper we looked at how the introduction of a background risk affects
optimal prevention with respect to another risk. We carried out our analysis in a twoperiod framework and used three configurations of background risks. These background
risks have a different impact on prevention since they affect differently the marginal cost
of prevention or its marginal benefit. We found that the introduction of a background risk
in the first period reduces the level of prevention if and only if the individual is prudent
in the first period, while it increases prevention if the background risk is introduced in
the second period and the individual is prudent in the second period. Yet, when the
background risk is state-dependent, prudence is not required any more and results are
only driven by risk aversion. A risk-averse individual increases prevention in the face a
background risk in the loss state but reduces prevention in the face of a background risk
in the no-loss state. In a second step, we investigated how an increase in background
risk affects prevention activities using the concept of stochastic dominance. This makes it
possible to generalise our results to higher orders increase in risk. We show that depending
on the type of increase in risk considered, more restrictive conditions than prudence are
required, such as temperance or edginess.
Two natural extensions of this work would be to consider the case of either multiplicative background risks or multivariate utility functions where individual preferences
depend on other arguments than final wealth. While it will add complexity to the model,
it will not provide additional insights to the analysis.
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