in Economics and Finance
Prova finale di Laurea
The impact of temporal
util-ity evaluation and changes
of loss distribution on
opti-mal prevention
Relatrice
Prof.ssa Paola Ferretti
Felix Peter Kalkert
854837@stud.unive.it
Matricola 854837
Anno Accademico
2015/2016
Contents i
List of Figures ii
1 Introduction 1
1.1 Overview . . . 1
1.2 Optimal Prevention . . . 4
1.3 Risk Aversion and Impatience . . . 8
1.4 Risk Aversion and Prudence . . . 11
1.4.1 Risk Aversion . . . 11
1.4.2 Prudence . . . 13
2 Optimal Prevention in Two States Model and Impatience 15 2.1 Framework and Assumptions . . . 15
2.2 Common Optimum under No-Risk Condition . . . 17
2.3 Differences in the Optimum under Risk . . . 19
2.4 Changes in levels of Impatience . . . 23
2.5 Summary . . . 28
3 Optimal Prevention in Three States Model and Impatience 29 3.1 New Framework and related Assumptions . . . 29
3.2 Results from the Common Optimum under No-risk condition . . . 31
3.3 Resulting Differences in the Optimum under Risk . . . 32
3.4 Summary . . . 38 4 Conclusion 39 5 Acknowledgement 43 A Appendix 44 A.1 . . . 44 A.2 . . . 45 Bibliography 47 i
2.1 Convexity in comparison with Linearity of the Marginal Utility . . . . 21
2.2 Concavity in comparison with Linearity of the Marginal Utility . . . . 22
2.3 Convexity in comparison with Linearity of the Marginal Utility with change in Impatience . . . 25
2.4 Concavity in comparison with Linearity of the Marginal Utility with change in Impatience . . . 26
3.1 Graphical representation of (3.18) . . . 35
3.2 Graphical representation of (3.19) . . . 36
4.1 Loss distributions . . . 40
I
NTRODUCTION
1.1
Overview
There are many kinds of risks that one has to face. May it be health risk or fi-nancial risk. However, there are also many ways to deal with these risks, either by insuring oneself against the outcomes or by reducing the probability of a haz-ardous event. This last method is called prevention. However, these prevention activities come at a cost. For example the price of the safe-driving class or the vaccine, but also the free-time one has to spend in order to undertake the activity. Thus, the question arises of an optimal amount of effort one is willing to spend on prevention.
Examples for preventive activities would be
• Defensive-Driving Class (Menegatti, 2009, [11]): The effort the agent would spend is the cost of the class and the free-time that needs to be spend. The upside would be, the agent is trained to follows rules, which reduce the risk of an accident, and also, how to react in order to avoid an accident in a dangerous situation.
• Vaccine (Hofmann, Peter, 2011, [8]): A vaccine can reduce the probability of becoming sick, however it comes at the cost, of maybe the price for the
medical care, the free time one has to spend, or even the pain due to side effects.
• Advanced Driver Assistance Systems or Electronic Stability Control: One has to spend a higher price, if one wishes these systems to be included in one’s car, however they clearly reduce the probability of an accident (Ellero, Ferretti, 2014, [6]).
• Measuring and Control Systems for Industrial Processes: Systems that help measuring and controlling processes, such that critical points won’t be reached, which could cause damages to the facility and in the worse case harm working people, could be implemented.
• Safety-Induction: Similar, to the above example. Safety-Inductions are performed, in order to sensitise people with dangerous task to proper be-haviour, which reduces the probability of an accident.
Examples for preventive activities, where the models do not apply would be • Air Traffic Safety
• Safety of Nuclear Power Plants
Due to the severe consequences of an accident in those two fields, the question of optimal prevention does not apply, as the only goal must be the maximum re-duction of probability of such an event.
In the previous literature one has generally modelled this problem as a situation of two possible states. One with probability p that the hazardous event occurs and the other state with probability (1 − p) that it does not. By spending a cost e, the agent then is able to influence p. However, this does not consider situations where there are different grades of possible damages. For example, a car accident could lead to a total loss or just to a partial damage, which can be repaired. Thus,
one needs a more general approach to model the probability of the hazardous event.
Since Menegatti pointed out (Menegatti, 2009, [11]), there exists also a time di-mension, meaning that one has to be more precise in when the effort is spent and when the loss may occurs. Many papers have been published which take this into consideration ([8],[9],[11]). However, none of these papers put much focus on the time point of evaluation. In general, all these papers assume, at least in parts, that effort is spent today and influences the probability of the hazardous event tomorrow. So, the agent has to evaluate his utility for both periods today, in order to be able to find his optimal decision. The consequence is that one has to move the future expected utility through time. However, most papers neglect this issue or just impose fixed discount rates ([11], [8], [9]), which do not seem to be an appropriate solution for this problem.
Therefore, this thesis will focus the attention on explicit time dependence of rep-resentative variables, in order to present a better method for the above stated problems with the aim of finding new insight to the problem of optimal preven-tion.
1.2
Optimal Prevention
Prevention or "[...] self-protection [...]"(Ehrlich, Becker, 1972, [7]) is a cost spent in order to reduce the probability of a hazardous event. It was first studied by Ehrlich and Becker as its own method to counter an uncertain loss. They mod-elled the world as one-period with two states, one, with probability p, were the loss occurs, and one state, with probability (1 − p), were it does not. With this framework they then analysed the impact of income and insurance (both market-and self-insurance) on prevention (Ehrlich, Becker, 1972, [7]).
It was Dionne and Eeckhoudt, who first studied the effect of risk-aversion to-wards prevention (Dionne, Eeckhoudt, 1985, [4]).They used the same one-period two states setting of Ehrlich and Becker and focused on the optimum of preven-tion an individual would choose, and they studied how the optimal level changes if one would increase the level of risk aversion. They showed that the effect of increasing risk-aversion towards prevention is ambiguous and so they argued that "[...] increased risk aversion reduces risky activities is shown not to apply to self-protection [...]" (Dionne, Eeckhoudt, 1985, [4]). The ambiguity of prevention towards increasing risk aversion was confirmed by Briys and Schlesinger (Briys, Schlesinger, 1990, [2]). However, they argued that this is not a counter-intuitive result with respect to risk aversion, as "[...] lowering the level of self-protection cannot be viewed as a risky activity [...]"(Briys, Schlesinger, 1990, [2]). Their argu-ment is that prevention reduces the probability of a loss but does not increase the predictability for the decision maker. Thus, a situation were the decision maker exerts more prevention is not dominant towards a situation where he exerts less in the second-order stochastic dominance sense (Briys, Schlesinger, 1990, [2]). Since the results for the impact of risk-aversion to prevention could not be used to evaluate changes in the optimum, because of differences in the utility function, Eeckhoudt and Gollier proposed prudence as the main determinant in exertion of
prevention activities (Eeckhoudt, Gollier, 2005, [5]). The concepts of risk aversion and prudence are similar, as both rely on the curvature of the utility function. However, the main idea is that prudence implies linearity of the first deriva-tive of the utility function for an agent who is neither prudent nor imprudent, and convexity for a prudent agent, and so it allows a qualitative comparison. Thus, in order to be able to distinguish the two effects, Eeckhoudt and Gollier first showed that if a risk-neutral agent chooses his optimal level of prevention, such that p = 12, then also a risk-averse agent would choose the same level, as long as he is both prudent and imprudent. Therefore, they analyse the case where the risk-neutral agent would find his optimum such that p = 12, as in this case the effect of risk-aversion is eliminated, while studying non-linear utility functions (Eeckhoudt, Gollier, 2005,[5]). They were able to proof the following results
• If the risk-neutral agent chooses his optimal effort en such that p(en) = 12,
then a prudent agent selects his optimal effort ep, as ep < en, whereas an
imprudent agent would choose an optimal effort ei, such that ei > en.
• If p(en) ≥ 12, then ep < en,
• If p(en) ≤ 12, then ei > en(Eeckhoudt, Gollier, 2005, [5]).
Courbage and Rey (Courbage, Rey, 2006, [3]) added to the results of Eeckhoudt and Gollier the concept of fear of sickness. "[...] Fear of sickness measures the ’de-gree of future pain’ induced by the occurrence of the illness, where pain is mea-sured via a decrease in utility [...]" (Courbage, Rey, 2006, [3]). Thus, they used a bivariate utility function to include health as a second objective. They confirmed Eeckhoudt’s and Gollier’s result that prudence is the main determinant, which drives prevention also in this more general case (Courbage, Rey, 2006, [3]). However, Menegatti (Menegatti, 2009, [11]) pointed out that many preventive activities are undertaken in advance and reduce the probability of an hazardous
event in the future. Therefore, he argued that one should consider prevention not as a one period decision, where prevention activities is exerted simultaneously with its effect, but rather that the preventive measure is conducted anticipating the event. Thus, he modelled the problem as a two time-periods model, where prevention is exerted in the first period and the expected utility in the second one (Menegatti, 2009, [11]). Using this framework he proved the following results
• If the risk-neutral agent chooses his optimal effort en such that p(en) = 12,
then a prudent agent selects his optimal effort ep, as ep > en, whereas an
imprudent agent would choose an optimal effort ei, such that ei < en.
• If p(en) ≥ 12, then ep > en,
• If p(en) ≤ 12, then ei < en(Menegatti, 2009, [11]).
These results are the complete opposite of those Eeckhoudt and Gollier found. So, Menegatti showed that time plays an important role in the decision problem of optimal prevention with respect to prudence, this is because "[...] a prudent agent desires a larger wealth in the period where he bears the risk[...]"(Menegatti, 2009, [11]). Thus, in a one time period world the agent has to spent the prevention cost in the same period where he bears the risk and so he spends less. How-ever, if the cost of prevention anticipates the risk, during a period of no risk, it increases the expected wealth for the risky period and so a prudent agent exerts more risk (Menegatti, 2009, [11]). Afterwards, Hofmann and Peter (Hofmann, Peter, 2011, [8]) extended Menegatti’s two time period model by introducing two preventive activities, an anticipatory effort in period one as Menegatti and an-other effort is conducted contemporaneously during the second period similar to the effort in the one period models that have been proposed before Menegatti. Moreover, they add a discount factor to the second period. Furthermore, they analyse the relation of the anticipatory to the contemporaneous preventive
ac-tivity, and changes in the discounting factor, for a risk-neutral, a risk-averse, a prudent and an imprudent agent. Although, they do not compare the optima of these agents (Hofmann, Peter, 2011, [8]).
However, in a second paper Hofmann and Peter (Hofmann, Peter, 2013, [9]) in-vestigate the changes of increased risk aversion on the optimal level of preven-tion and obtain some qualitative results. They are able to state that the effect of in-creased risk aversion in a two period model depends on the possibility of savings. The point is that in case of saving possibilities the agent can obtain consump-tion smoothing through savings, while in a model without savings the agent has to smooth his consumption through the prevention method. However, they ar-gue that individuals often optimize there "[...]mind accounts[...]" (Hofmann, Pe-ter, 2013, [9]) and thus the optimization problem of prevention is independent from savings. Furthermore, they show that a more risk averse agent exerts more preventive effort if current consumption exceeds an endogenously determined threshold, if savings is not available. In case of savings the risk averse agent ex-erts more effort if the probability of the loss state is low enough (Hofmann, Peter, 2013, [9]).
1.3
Risk Aversion and Impatience
In the above section it was said that in the previous literature it is not taken into consideration that the agent is not just facing uncertainty in his decision but that there also exists a time dimension which has to be taken into account. In his paper Nachman (Nachman, 1975, [13]) developed a measure for risk aversion with time varying preferences. His underlying idea is described by the following situation. An engineer develops a new product and can either sell his idea now for a fixed certain amount or gain an uncertain profit in the future. Nachman then develops and analyses a method where the agent would be indifferent between the two choices. Thus, in the end the agent is indifferent about an uncertain future expected utility and a present deterministic utility. So, Nachman’s works can be adopted to the optimal prevention problem and enable us to move the future uncertain period to present and to transform the uncertain problem in a deterministic one. Thus, the agent is able to value its future expected utility in the present, when he also makes his choice of prevention activities. For his modelisation, Nachman first defines a bivariate utility function function U (t, w) where t describes time and w wealth. So,U is a function U : F → R with the following assumptions
• t ∈ R+= [0, ∞)
• w ∈ R = (−∞, ∞) • F = {(t, w) ∈ R+× R}
The set F is assumed to be a topological space or a measurable space for either case in which there is a random variable part of w or not. Furthermore, there are three postulates which describe the utility function
• Monotonicity. For each t, t ∈ R+, the function Ut(w) = U (t, w) is strictly
increasing in w.
• Comparability. For each t0
, t ∈ R+ with t0 < tand for each w ∈ R, there is a
w0 ∈ R, such that U(t0, w0) = U (t, w).
The first postulate is necessary to obtain a smooth function to avoid critical points. The second mainly states that for a fixed point in time utility is increasing with wealth. So, it is just the usual assumption that an agent always prefers more to less. The last postulate says that it is possible to achieve the same utility at dif-ferent times with corresponding wealth or vice versa. Meaning, it allows us to compare the utility at different time points by comparison of the corresponding wealth. Together with the first two postulates this leads to a conclusive descrip-tion. With the above described utility function Nachman then is able to introduce a so called temporal risk premium with which one can transform a future ex-pected utility of a random wealth into a present utility of a deterministic wealth status. Namely, the temporal risk premium τs,t(w, ψ)is implicitly defined by the
following equation
U (s, E(ψ) + w − τs,t(w, ψ)) = EU (t, w + ψ)
where s < t and ψ is a random variable. Furthermore, he states that τs,t(w, ψ)
is well defined. Thus, it should be emphasized, that the temporal risk premium τs,t(w, ψ)is depending on ψ, however it is not a function of it. Moreover,
Nach-man argues that the temporal risk premium incorporates two effects. First, it describes the willingness of the agent to transform an uncertain outcome into a deterministic one at a fixed time. Secondly, it evaluates, how much the agent is willing to pay to move this, now certain wealth, through time.
Thus, he defines two underlying premia. The instantaneous risk premium πt(w, ψ),
which transforms the random event into a non-random status at a fixed time, and the time preference premium νst(w, y), where y is a deterministic increment of w
U (s, w + y − νst(w, y)) = U (t, w + y)
1.4
Risk Aversion and Prudence
1.4.1
Risk Aversion
"[...] A risk averter is defined as one who, starting from a position of certainty, is unwilling to take a bet which is actuarially fair[...]" (Arrow, 1970, [1]) Arrow wrote this definition in mathematical form by defining the following inequality
u(w0) >
1
2u(w0 − h) + 1
2u(w0− h)
where w0 is a certain wealth and h is a fixed amount one can loose or win with
probability 12. Furthermore, u() is a utility function with the following properties
• u0(w) > 0, the agent prefers always to have more wealth.
• limw→0u(w)and limw→∞u(w)exist and are finite. Thus, u() is bounded in
w.
From the above definition of a risk-averter it follows then u(w0) >
1
2u(w0− h) + 1
2u(w0+ h) ⇔ u(w0) − u(w0− h) > u(w0+ h) − u(w0)
From which we have the third property of u() • u00(w) < 0. So, u0()is strictly decreasing in w.
Thus, u() is a strictly concave function if the agent is a risk averter (Arrow, 1970, [1]).
Pratt (Pratt, 1964, [14]) came to the same conclusion using a different approach. He defines an equality equation, which describes the indifference curve of an agent, who decides between a certain event and a risky one. So,
where w is a certain wealth and z is a random variable. π(w, z) is called risk premium, which describes an amount the agent is willing to pay in order to face the certain event rather than the risky one. If we interpret the random variable as a Bernoulli random variable, it is easy to see from the condition of strict concavity for a function f, namely
f ((1 − α)x + αy) > (1 − α)f (x) + αf (y) , α ∈ [0, 1]
that if π(w, z) ≥ 0 it follows that u() is concave, which so results in u00() < 0. For π(w, z) > 0, u() is strictly concave. More precisely, if π(w, z) = 0, then the agent is indifferent between taking risk or certainty. Thus, he is risk-neutral and so u00() = 0 and u() is a linear function. Furthermore, if π(w, z) < 0, the agent would want to be paid in order to prefer certainty over risk and so he would be a risk taker and u00() > 0(Pratt, 1964, [14]).
We can sum up, that we are able to conclude based on u00(), if an agent is
risk-averse, risk-neutral or risk-affine. Based on this, Arrow and Pratt developed quantitative measures for risk-aversion, namely
r(w) = −u
00(w)
u0(w)
which is called absolute risk-aversion (Arrow, 1970, [1]) or local risk aversion (Pratt, 1964, [14]) and
r(w) = −wu
00(w)
u0(w)
which is called the relative risk aversion (Arrow, 1970, [1]) or local proportional risk aversion (Pratt, 1964, [14]).
1.4.2
Prudence
Rothschild and Stiglitz analysed the effect of increased risk. First, they defined different concepts for increased risk of a random variable compared to another (Rothschild, Stiglitz, 1970, [15]). They mainly identified three definitions
• a random variable X is preferred to a random variable Y by all risk- averters, meaning EU (X) ≥ EU (Y ) and that X is less risky than Y.
• Y = X + Z, where Z is any from X independent noise. • There is more weight in the tails of Y than of X
Based on these definitions, they prove, in their second paper (Rothschild, Stiglitz, 1971, [16]) that, given an utility function, which is a function of a parameter α and a random variable θ, an individual will maximize his expected utility by
max
α
Z
u(θ, α)dF (θ)
which yields the first order condition Z ∂u(θ, α)dF (θ)
∂α = E
∂u(θ, α)
∂α = 0 (Rothschild, Stiglitz, 1971, [16]). Consider the two random variables θ1 and θ2, where θ1is considered to be riskier
than θ2. So, according to the first definition, a risk averter, thus an agent with
a concave utility function, will prefer the less risky situation to the riskier one, which leads to Z ∂u(θ 1, α)dF (θ1) ∂α ≤ Z ∂u(θ 2, α)dF (θ2) ∂α .
If α∗is the α that satisfiesR ∂u(θ2,α)dF (θ2)
∂α = 0, it is
R ∂u(θ1,α∗)dF (θ1)
∂α∗ ≤ 0. Now,
assum-ing that α∗is the unique optimizer and in its neighbourhood, u() is monotonically
decreasing in α. Then, it follows that in case of concavity of u() in θ an increase of riskiness will decrease α∗ and respectively if u() is convex in θ, then an increase
Following this proposal, Kimball introduced a new concept called prudence and he proposed its measure. "[...]The term prudence is meant to suggest the propen-sity to prepare and forearm oneself in the face of uncertainty, in contrast to risk aversion, which is how much one dislikes uncertainty and would turn away from uncertainty if possible [...] "(Kimball, 1990, [10]). Thus, this "propensity to pre-pare and forearm oneself in the face of uncertainty " is given in case of the con-vexity of ∂u(θ,α)∂α , as risk aversion is given by concavity of u().
Then, Kimball develops a similar measure for prudence, as Pratt and Arrow de-veloped for risk aversion. At first, he defines, in the setting of Rothschild and Stiglitz, that θ = θ0+ θ∗, where θ0 is a deterministic quantity and θ∗ is a random
variable. With this he defines the so called precautionary premium ψ E∂u(θ0+ θ ∗, α 1) ∂α = ∂u(θ0− ψ, α1) ∂α
for any α1. Kimball then shows that the precautionary premium is given similarly
as Pratt’s risk premium by
ψ(θ∗, θ0, α) = η(θ, α) σ2 θ 2 + o(σ 2 θ)
where η(θ, α) is the measure for prudence and is defined as η(θ, α) = −
∂3u(θ,α)
∂θ2∂α
∂2u(θ,α)
∂θ∂α
Given the assumption that ∂2∂θ∂αu(θ,α) is either uniformly positive or negative (Kim-ball, 1990, [10]), so an effect of a change in α, which effects θ, would always either monotonically increase or decrease the utility. Note, that the problem of optimal prevention represents exactly this framework.
O
PTIMAL
P
REVENTION IN
T
WO
S
TATES
M
ODEL AND
I
MPATIENCE
2.1
Framework and Assumptions
In this chapter we want to show that by combining Menegatti’s (Menegatti, 2009, [11]) and Nachman’s (Nachman, 1975, [13]) proposals it is possible to obtain the same results as Menegatti, while evaluating everything at the time point when the decision has to be made. The setting follows basically Menegatti’s proposal from 2009 (Menegatti, 2009, [11]). Thus, we have two time points t0and t1, where
t1 is later than t0. Furthermore, an agent has a fixed wealth at each time point,
meaning a fixed wealth w0 at t0 and w1 at t1. Moreover, a hazardous event can
occur with probability p at t1 causing a loss L to the agent. The agent in turn can
spend an effort e to reduce the probability p, which so is a function of e, namely p(e). 1 P P P PPq w0− e t0 t1 w1− L w1 p(e) 1 − p(e) 15
Introducing the time dimension following Nachman (Nachman, 1975, [13]), we have for the bivariate Utility function the same assumptions
• t ∈ R+= [0, ∞)
• w ∈ R = (−∞, ∞) • F = {(t, w) ∈ R+× R}
The set F is assumed to be a topological space or a measurable space for either case in which there is a random variable part of w or not. Furthermore, there are three postulates which describe the utility function
• Continuity. U is continuous on F.
• Monotonicity. For each t ∈ R+, the function Ut(w) = U (t, w) is strictly
increasing in w.
• Comparability. For each t0
, t ∈ R+ with t0 < tand for each w ∈ R, there is a
w0 ∈ R, such that U(t0, w0) = U (t, w).
Moreover, we have the usual assumptions in an optimal prevention context, namely • p0(e) < 0
, so the preventive activity reduces the probability of the hazardous state.
• p00(e) > 0, meaning that the effect of the preventive activity looses its effect,
when more is exerted. This assumption also ensures that the second order condition is satisfied.
2.2
Common Optimum under No-Risk Condition
In order to be able to compare the optimal choices of a risk neutral agent to a risk averse or prudent agent, it is necessary to find a common point. Following Menegatti[11] and Eckhoudt, Gollier[5] this is given for a situation with no risk, as it is assumed that both agents would choose the same level of prevention in this case. The no risk situation is modelled by introducing a certain loss of p(e)L at t1. This results in the objective function
V (e) = U (t0, w0 − e) + U (t1, w1− p(e)L) (2.1)
= U (t0, w0 − e) + U (t0, w1− p(e)L − νt0,t1(w1, −p(e)L)) (2.2)
where we have introduced the time preference premium νt0,t1(w1, −p(e)L)in
or-der to move the utility at t1 to t0. Recalling Nachman’s definition of the time
preference premium
U (s, w + y − νst(w, y)) = U (t, w + y) (Nachman, 1975, [13])
and giving the first order condition,
d[U (w0− e) + U (w1− p(e)L − νt0,t1(w1, −p(e)L))]
de = 0 (2.3)
where the time parameter is now neglected (as it will from now on always be t = t0), we obtain −U0(w0−e)+ ∂νt0,t1(w1, −p(e)L) ∂x2 −1
p0(e)L U0(w1−p(e)L−νt0,t1(w1, −p(e)L)) = 0
(2.4) For a risk neutral agent we have that,
U0(w0− e) = U0(w1− p(e)L − νt0,t1(w1, −p(e)L)) = const.
It follows that the optimal level of the risk neutral agent enmust satisfy
−p0(en) =
1
L [1 −∂νt0,t1(w1,−p(en)L)
∂x2 ]
where νt0,t1()is seen as a function of two variables x1and x2. Thus
∂νt0,t1(w1,−p(en)L)
∂x2
denotes the first partial derivative with respect to the second variable at the point (w1, −p(en)L). By assumption we have that both agents will exert the same level
of preventive activity. Thus, it is true also for the non risk-neutral agent, that is U0(w0− en) = U0(w1− p(en)L − νt0,t1(w1, −p(en)L). (2.6)
Although, it should be emphasized that in case of a non risk-neutral agent, this is only true for this specific point and thus U00() 6= 0, U0()is not constant.
For the second order condition we have that V00(en) = U00(w0− en)
+ ∂νt0,t1(w1, −p(e)L)
∂x2
− 1
p00(e)L U0(w1− p(e)L − νt0,t1(w1, −p(e)L)
− ∂
2ν
t0,t1(w1, −p(e)L)
∂x2 2
(p0(e)L)2 U0(w1− p(e)L − νt0,t1(w1, −p(e)L)
+ ∂νt0,t1(w1, −p(e)L)
∂x2
− 1 2
(p0(e)L)2 U00(w1− p(e)L − νt0,t1(w1, −p(e)L).
From both first order (based on sign equivalence in (2.5)) and second order con-dition (in order to ensure V00(en) < 0) we can conduct that
∂νt0,t1(w1, −p(e)L)
∂x2
− 1
< 0 which can be rewritten to
∂νt0,t1(w1, −p(e)L)
∂x2
< 1.
Furthermore, it is reasonable to assume that 0 < ∂νt0,t1(w1,−p(e)L)
∂x2 , as this would
mean that one is always willing to pay more in order to move more through the same time. Thus, we obtain
0 < ∂νt0,t1(w1, −p(e)L)
∂x2
< 1. (2.7)
Moreover, it is safe to assume with the same reasoning that the above condition should hold more generally for
0 < ∂νt0,t1(x1, x2)
∂x2
2.3
Differences in the Optimum under Risk
In the section above, we were able to obtain two main results. • −p0(e n) = 1 L[1−∂νt0,t1 (w1,−p(en)L) ∂x2 ] • 0 < ∂νt0,t1(x1,x2) ∂x2 < 1.
With these new results we can now proceed to define the original problem with uncertainty, represented by the two states in the second time-point. Thus, the objective function writes as
V (e) = U (t0, w0− e) + p(e)U (t1, w1− L) + [1 − p(e)]U (t1, w1) (2.9)
Evaluating everything at t0, it follows that
V (e) = U (t0, w0−e)+p(e)U (t0, w1−L−νt0,t1(w1, −L))+[1−p(e)]U (t0, w1−νt0,t1(w1, 0))
For notation purposes, lets define ν−L = νt0,t1(w1, −L), ν0 = νt0,t1(w1, 0).
More-over, the time variable will be neglected from now on, as everything will be at time t0. So the above function will be written as
V (e) = U (w0− e) + p(e)U (w1 − L − ν−L) + [1 − p(e)]u(w1− ν0) (2.10)
and the first order condition writes as
V0(e) = −U0(w0− e) − p0(e)[U (w1− ν0) − U (w1 − L − ν−L)] = 0. (2.11)
For the second order condition we have that V00(e) = U00(w0− e) | {z } <0 − p00(e) | {z } >0 [U (w1− ν0) − U (w1− L − ν−L)] | {z } >0 < 0 ∀ e ∈ (0, w0) (2.12) where [U (w1− ν0) − U (w1 − L − ν−L)] < 0is given because L + ν−L > ν0, which
is confirmed by assumption 0 < ∂νt0,t1(x1,x2)
∂x2 < 1, as from this we obtain that
Now, lets assume the risk-neutral agent would choose an optimal preventive level en, such that p(en) = 12 and lets define ν−1
2L= νt0,t1(w1, −p(en)L) = νt0,t1(w1, −
1 2L).
Thus, we obtain with (2.5) −p0(en) = 1
L [1−∂νt0,t1 (w1,−p(en)L) ∂x2 ] and (2.6) U0(w0 − en) = U0(w1− p(en)L − νt0,t1(w1, −p(en)L) V0(en) = − U0(w0− en) | {z } =U0(w 1−p(en)L−ν− 1 2L ) + 1 L(1 − ∂ν− 12L ∂x2 ) [U (w1− ν0) − U (w1− L − ν−L)] = 0 (2.13) ⇔ w1−ν0 Z w1−L−ν−L U0(x)dx = U0(w1− 1 2L − ν−12L) L (1 − ∂ν−1 2L ∂x2 ) (2.14)
For the risk-neutral agent we have that U0(x) = U0(w1−12L − ν−1
2L) = const. ⇒ L + ν−L− ν0 = L(1 − ∂ν−1 2L ∂x2 ) ⇔ ∂ν− 1 2L ∂x2 = ν0− ν−L L (2.15)
So, we are able to rewrite the first order condition to
V0(en) = 1 L + ν−L− ν0 w1−ν0 Z w1−L−ν−L U0(x)dx − U0(w1− 1 2L − ν−12L) = 0 (2.16)
This condition should hold also for the risk-averse agent, if he would exert the same effort en, as the risk-neutral agent. However, following Menegatti’s
ap-proach (Menegatti, 2009, [11]) and considering the function T (x) = U0(w1−12L −
ν−1 2L)+U 00(w 1−12L−ν−1 2L)[x−(w1− 1 2L− ν0+ν−L
2 )], which is the tangent line of U 0(x) at the point ((w1−12L − ν0+ν−L 2 ), U 0(w 1−12L − ν−1 2L)), if we assume ν− 1 2L = ν0+ν−L 2 ,
we have that (see Appendix A.1)
w1−ν0 Z w1−L−ν−L T (x)dx = U0(w1− 1 2L − ν−12L) (L + ν−L− ν0). (2.17)
Figure 2.1: Convexity in comparison with Linearity of the Marginal Utility
Thus, substituting T (x) in the first order condition (2.16), the equation is verified. However, we know due to the fact that a prudent agent shows a convex marginal utility function, that the following is given
w1−ν0 Z w1−L−ν−L T (x)dx < w1−ν0 Z w1−L−ν−L U0(x)dx, (2.18)
which can also be seen in Figure (2.1). So it follows, that evaluating the first order condition for a prudent agent with the optimizer enof the risk-neutral agent, we
have that
V0(en) > 0.
As it is V00(.) < 0 we obtain that prudent agent will choose an optimal level higher than the risk-neutral agent. Analogously for an imprudent agent, whose
Figure 2.2: Concavity in comparison with Linearity of the Marginal Utility
marginal utility function is concave, it follows that
w1−ν0 Z w1−L−ν−L T (x)dx > w1−ν0 Z w1−L−ν−L U0(x)dx, (2.19)
which is graphically represented in Figure (2.2) and thus, an imprudent agent will exert an level lower than the risk-neutral agent. Furthermore, an agent who is both prudent and imprudent (i.e. U0()is linear), however not necessarily risk-neutral, would exert the same preventive effort en. This is clear from Figures (2.1)
2.4
Changes in levels of Impatience
Up until, we have assumed that all agents have the same level of impatience and we compared the differences in optimal prevention for different levels of pru-dence. Now, we will compare the optimal prevention point for an agent with dif-ferent levels of impatience, but the same level of prudence and risk-preference. Consider two agents a and b, who show the same preference towards risk and prudence, however with different levels of impatience, which is described as fol-lows
νta0,t1(x1, x2) = νtb0,t1(x1, x2) + β (2.20)
where β > 0 and constant. So, agent a is always willing to pay β more than agent b to have x1 + x2 at t0 instead of at t1, meaning agent a is more impatient than
agent b. Now, lets say that agent b had the same level of impatient, as the agents had until now in this chapter, so we can drop the superscript for b
νta0,t1(x1, x2) = νtb0,t1(x1, x2) + β = νt0,t1(x1, x2) + β (2.21) Moreover, it is ∂νa t0,t1(x1, x2) ∂x2 = ∂νt0,t1(x1, x2) ∂x2 (2.22) If we rewrite than the objective function in the situation of a certain loss p(e)L with the impatience of agent a, we have
V (e) = U (w0 − e) + U (w1− p(e)L − νta0,t1(w1, −p(e)L)
| {z }
=νt0,t1(w1,−p(e)L)+β
) (2.23)
and the first order condition is written as −U0(w0−e)+
∂νt0,t1(w1, −p(e)L)
∂x2
−1
p0(e)L U0(w1−p(e)L−νt0,t1(w1, −p(e)L)−β) = 0
(2.24) Again for a risk neutral agent we have that,
−p0(en) =
1
L [1 −∂νt0,t1(w1,−p(en)L)
∂x2 ]
(2.25) And so we get that also for the non-risk-neutral agent we have
U0(w0− en) = U0(w1− p(en)L − νt0,t1(w1, −p(en)L) − β)
From before, we had for the agent with less impatience that U0(w0− en) = U0(w1− p(en)L − νt0,t1(w1, −p(en)L)
As we assumed that both agents are only different in their impatience but not in their risk-aversion or prudence, it follows that this is true, if U0(x)is shifted to the left by β. Now, we consider the situation with risk for the impatience of agent a, meaning the two states model, where everything is evaluated at t0. The objective
function is then as follows
V (e) = U (w0− e) + p(e)U (w1− L − ν−La |{z} =ν0+β ) + [1 − p(e)]u(w1− ν0a |{z} =ν0+β ) (2.26)
where we used the same subscripts as before. Consequently, the first order con-dition is
V0(e) = −U0(w0− e) − p0(e)[U (w1− ν0− β) − U (w1− L − ν−L− β)] = 0 (2.27)
As before we assume, that the risk-neutral agent chooses an optimal level of pre-vention ensuch that p(en) = 12 and so we obtain for the first order condition
−U0(w1− 1 2L − ν−12L− β) + 1 L[1 −∂ν− 12L ∂x2 ] [U (w1− ν0− β) − U (w1− L − ν−L− β)] = 0 (2.28) Which again can be rewritten as
−U0(w1− 1 2L − ν−12L− β) + 1 L[1 −∂ν− 12L ∂x2 ] w1−ν0−β Z w1−L−ν−L−β U0(x)dx = 0 (2.29)
Figure 2.3: Convexity in comparison with Linearity of the Marginal Utility with change in Impatience
As before, for the risk neutral agent U0(x)is constant and we obtain ∂ν−1
2L
∂x2
= ν0− ν−L
L (2.30)
Now, lets consider the function T (x) = −U0(w
1−12L − ν−1 2L− β) + −U 00(w 1−12L − ν−1 2L− β)[x − (w1− 1 2L − ν0+ν−L
2 − β)] and it is (see Appendix A.2) w1−ν0−β Z w1−L−ν−L−β T (x)dx = U0(w1− 1 2L − ν−12L− β)(L + ν−L− ν0) (2.31)
And so, we have again the same situation as before. For, a risk averse and pru-dent agent, we have due to the fact of convexity of U0(x), that it is
w1−ν0−β Z w1−L−ν−L−β T (x)dx < w1−ν0−β Z w1−L−ν−L−β U0(x)dx (2.32)
Figure 2.4: Concavity in comparison with Linearity of the Marginal Utility with change in Impatience
and for a risk averse agent, who is imprudent, we have, because of the concavity of U0(x), that w1−ν0−β Z w1−L−ν−L−β T (x)dx > w1−ν0−β Z w1−L−ν−L−β U0(x)dx (2.33)
and finally for a risk averse agent, who is both prudent and imprudent, we have, because of the linearity of U0(x)
w1−ν0−β Z w1−L−ν−L−β T (x)dx = w1−ν0−β Z w1−L−ν−L−β U0(x)dx. (2.34)
So, we see that the agents with higher impatience exert the same level of preven-tion with respect to the agents with a lower impatience, but the same preference towards risk and prudence, as it is U0(w
0−en) = U0(w1−p(en)L−νt0,t1(w1, −p(en)L)−
β) = U0(w1−p(en)L − νt0,t1(w1, −p(en)L)). It follows as can be seen in Figures (2.3)
fact that ∂νa t0,t1(x1, x2) ∂x2 = ∂ν b t0,t1(x1, x2) ∂x2
the factors of the integrals stay the same. So, we have that a change in impatience has no impact on the optimal level of prevention.
2.5
Summary
In the above chapter it was shown that including Nachman’s impatience in Menegatti’s two state model for optimal prevention can be used to confirm Menegatti’s re-sults in a more general setting. Furthermore, it seems more intuitive that the agent would evaluate his utility, including the expected utility to be obtained at t1, at t0, when he has to decide his optimal effort on preventive activities. We
were able to obtain the following results • 0 < ∂νt0,t1(x1,x2) ∂x2 < 1 • ∂νt0,t1(w1,−p(en)L) ∂x2 = νt0,t1(w1,0)−νt0,t1(w1,−L) L
Furthermore, we confirmed Menegatti’s results that
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
the optimal effort of a prudent agent ep will be higher than en
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
the optimal effort of an imprudent agent eiwill be lower than en
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
the optimal effort of an agent, who is neither prudent nor imprudent, will be the same as en
Moreover, it was shown that a change in impatience of the kind νta0,t1(x1, x2) = νtb0,t1(x1, x2) + β
has no impact on the optimal level of prevention as long as the preference for risk or prudence is the same.
O
PTIMAL
P
REVENTION IN
T
HREE
S
TATES
M
ODEL AND
I
MPATIENCE
3.1
New Framework and related Assumptions
In the above chapter and in almost all previous literature it is assumed that the agent faces the risk of an hazardous event and the outcome is in case of occur-rence the total loss L or the event does not occur and thus there is no loss. How-ever, it is a more realistic scenario that the in case of a hazardous event the agent does not necessarily face the total loss L, but rather a fraction of L. Thus, we generalise the two state model from chapter 2 by introducing a third state. So, we still have two time-points t0 and t1. At t0 the agent owns a certain wealth
w0 and can spend the prevention activity e in order to reduce the probability of
the hazardous event. Thus, in t1 the agent has a certain wealth w1and faces three
possible states
• with probability q(e), the agent faces a loss of αL, where 0 < α < 1. • with probability r(e), the agent faces a loss of L.
• with probability 1 − r(e) − q(e) the agent does not face any loss. So, to sum up we have the following situation
1 P P P PPq -w0− e t0 t1 w1− L w1− αL w1 r(e) q(e) 1 − r(e) − q(e)
We will keep almost all assumptions, which were used in chapter 2. So, we have the utility function following Nachman’s setting, meaning
• t ∈ R+= [0, ∞)
• w ∈ R = (−∞, ∞) • F = {(t, w) ∈ R+× R}.
The set F is assumed to be a topological space or a measurable space for either case in which there is a random variable part of w or not. Furthermore, there are three postulates which describe the utility function
• Continuity. U is continuous on F.
• Monotonicity. For each t ∈ R+, the function Ut(w) = U (t, w) is strictly
increasing in w.
• Comparability. For each t0
, t ∈ R+ with t0 < tand for each w ∈ R, there is a
w0 ∈ R, such that U(t0, w0) = U (t, w).
Furthermore, we have the same assumptions for r(e) and q(e) as we had for p(e). Thus
• r0(e) < 0
and q0(e) < 0
3.2
Results from the Common Optimum under
No-risk condition
First, we would like to conduct as in Chapter 2 the common optimum of a risk-neutral agent and a non-risk-neutral agent. Again this is given under a situation with no risk, which is so given by a certain loss of (r(e) + αq(e))L at t1. The
corre-sponding objective is thus written as
V (e) = U (t0, w0− e) + U (t1, w1− (r(e) + αq(e))L). (3.1)
Let’s define for notation purposes
ν−(r+αq)L= νt0,t1(w1, −(r(e) + αq(e))L).
Now, we are able to evaluate all terms at t0
V (e) = U (t0, w0− e) + U (t0, w1− (r(e) + αq(e))L − ν−(r+αq)L). (3.2)
From now on everything will be evaluated at t0 so we can neglect the time
vari-able. The first order condition then is as follows V0(e) = −U0(w0−e)+
∂ν−(r+αq)L
∂x2
−1
(r0(e)+αq0(e))LU0(w1−(r(e)+αq(e))L−ν−(r+αq)L) = 0.
(3.3) Due to the fact, that for the risk-neutral agent it is true that U0(x) = cons., it
follows that r0(en) + αq0(en) = 1 L[∂ν−(r+αq)L ∂x2 − 1] (3.4) ⇔ −r0(en) = 1 L[1 −∂ν−(r+αq)L ∂x2 ] + αq0(en) (3.5)
As we assume that both agents would choose the same optimum, (3.4) is also given for a non-risk neutral agent. And so it follows, by substituting (3.4) in the objective function of a non-risk neutral agent, that for a non risk-neutral agent it is given
3.3
Resulting Differences in the Optimum under
Risk
In the section above we identified some relations for both a risk-neutral and a non risk-neutral agent for the special case, where they both don’t face a risk. Using these results as a basis we are able to move to a situation with risk, namely the three states model described in the first section of Chapter 3. Thus, we have the following objective function
V (e) = U (t0, w0−e)+r(e)U (t1, w1−L)+q(e)U (t1, w1−αL)+[1−r(e)−q(e)]U (t1, w1).
(3.7) Again, we want to evaluate everything at time-point t0. So, let’s define our
tem-poral premia as follows
ν−L= νt0,t1(w1, −L)
ν−αL= νt0,t1(w1, −αL)
ν0 = νt0,t1(w1, 0).
This gives us the objective function entirely evaluated at t0, such that
V (e) = U (w0−e)+r(e)U (w1−L−ν−L)+q(e)U (w1−αL−ν−αL)+[1−p(e)−q(e)]U (w1−ν0).
(3.8) Again, the time parameter is now neglected as it is always t0.
This yields the first order condition
−U0(w0−e)−r0(e)[U (w1−ν0)−U (w1−L−ν−L)]−q0(e)[U (w1−ν0)−U (w1−αL−ν−αL)] = 0
(3.9) ⇔ −U0(w 0− e) − r0(e) w1−ν0 Z w1−L−ν−L U0(x)dx − q0(e) w1−ν0 Z w1−αL−ν−αL U0(x)dx = 0. (3.10)
Now, we can evaluate this function for the optimal choice of the risk-neutral agent, which we developed in the section above. Thus, we substitute equation
(3.5) in (3.10) and obtain −U0(w0−en)+ 1 L(1 − ∂ν−(r+αq)L ∂x2 ) +αq0(en) wZ1−ν0 w1−L−ν−L U0(x)dx−q0(en) w1−ν0 Z w1−αL−ν−αL U0(x)dx. (3.11) Due to equation (3.6) we have that U0(w
0 − en) = U0(w1 − (r(en) + αq(en))L −
ν−(r+αq)L)and we can rewrite (3.11)
−U0(w1− (r(en) + αq(en))L − ν−(r+αq)L) +
1 L(1 − ∂ν−(r+αq)L ∂x2 ) w1−ν0 Z w1−L−ν−L U0(x)dx −(1 − α)q0(en) w1−ν0 Z w1−αL−ν−αL U0(x)dx + αq0(en) w1−αL−ν−αL Z w1−L−ν−L U0(x)dx.
Let’s consider the following special situation, where α = 1
2, r(en) = 1
4 and q(en) = 1
2. This results in the same expected outcome as in Chapter 2.
We denote for the above fixed values that we have ν−(r+αq)L= ν−αL= ν−1 2L −U0(w1− 1 2L − ν−12L) + 1 L(1 − ∂ν12L ∂x2 ) w1−ν0 Z w1−L−ν−L U0(x)dx (3.12) −1 2q 0 (en) w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx + 1 2q 0 (en) w1−12L−ν− 1 2L Z w1−L−ν−L U0(x)dx (3.13) If we assume that ν0− ν1 2L= ν 1
2L− ν−L, which means that the lengths of the
inte-grals in (3.13) are equal. Then, we obtain, due to the fact that (3.12) is equivalent to equation (2.12) in Chapter 2 and for a risk-neutral agent it is U0(x) = cons., that the risk-neutral agent exerts exactly the same level of prevention as in Chapter 2. Thus, the risk-neutral agent exerts the same level of prevention for a differently distributed outcome, with the same expected value.
Moreover, a risk-averse agent will exert a prevention level lower than that in Chapter 2, as U0(x)is a decreasing function and so (3.13) is negative. Thus, we
optimal level lower than the risk-neutral agent. The question, if a risk-averse and prudent agent obtains a higher or lower level of optimal prevention than the risk-neutral agent, is not so easily answered. In order to see this, we have to rewrite the first order condition again. First, we recall that
U0(w1− 1 2L − ν−12L) = 1 L(1 − ∂ν− 12L ∂x2 ) w1−ν0 Z w1−L−ν−L T (x)dx where T (x) = U0(w1− 12L − ν−1 2L) + U 00(w 1− 12L − ν−1 2L)[x − (w1− 1 2L − ν−12L)].
We can substitute this in (3.12)-(3.13)
− 1 L(1 − ∂ν− 12L ∂x2 ) w1−ν0 Z w1−L−ν−L T (x)dx + 1 L(1 − ∂ν12L ∂x2 ) w1−ν0 Z w1−L−ν−L U0(x)dx −1 2q 0 (en) w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx + 1 2q 0 (en) w1−12L−ν− 1 2L Z w1−L−ν−L U0(x)dx
Now, we add and subtract 12q0(en)
w1−ν0 R w1−L−ν−L T (x)dx, as well as12q0(en) w1−ν0 R w1−12L−ν− 1 2L U0(x)dx. So, we obtain 1 L(1 − ∂ν12L ∂x2 ) wZ1−ν0 w1−L−ν−L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx −1 2q 0 (en) w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx + 1 2q 0 (en) w1−12L−ν− 1 2L Z w1−L−ν−L U0(x)dx +1 2q 0(e n) w1−ν0 Z w1−L−ν−L T (x)dx − 1 2q 0(e n) w1−ν0 Z w1−L−ν−L T (x)dx +1 2q 0 (en) w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx − 1 2q 0 (en) w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx
Figure 3.1: Graphical representation of (3.18)
which can be rewritten 1 L(1 − ∂ν12L ∂x2 ) wZ1−ν0 w1−L−ν−L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx (3.14) +1 2q 0 (en) wZ1−ν0 w1−L−ν−L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx (3.15) −1 2q 0 (en) 2 w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx (3.16) r0(en) + αq0(en) = 1 L[∂ν−(r+αq)L ∂x2 − 1]
For the fixed values of r(en) = 14, q(en) = 12 and α = 12 and some computations we
obtain 1 2q 0 (en) = 1 L[1 −∂ν− 12L ∂x2 ] − r0(en) (3.17)
Figure 3.2: Graphical representation of (3.19)
If we substitute (3.17) in (3.15), we can simplify (3.14-3.16) −r0(en) wZ1−ν0 w1−L−ν−L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx (3.18) −1 2q 0 (en) 2 w1−ν0 Z w1−12L−ν− 1 2L U0(x)dx − w1−ν0 Z w1−L−ν−L T (x)dx (3.19)
Due to the assumptions r0(e) < 0 and q0(e) < 0, it is −r0(e) > 0and −12q0(e) > 0. (3.18) and (3.19) are graphically represented in figure (3.1) respectively (3.2). Note that the integrals are of course the areas underneath the functions. We see in fig-ure (3.1) that quantity (3.18) is positive for a prudent agent, equal to zero for an agent, who is both prudent and imprudent, and negative for an imprudent agent. Moreover, we see in figure (3.2) that (3.19) is negative for an agent who is both prudent and imprudent (shaded area). For the prudent agent (3.19) is also nega-tive, but it is bigger than for the agent, who is both prudent and imprudent. And
again, for the imprudent agent (3.19) is negative and even smaller than for the agent, who is both prudent and imprudent. So, we are able to conclude, that in the three states model a risk-averse and both prudent and imprudent agent will exert an optimal prevention level smaller than the risk-neutral agent. Further-more, a risk-averse and imprudent agent will exert an optimal level even smaller than the risk-averse and both prudent and imprudent agent. And finally, the risk averse and prudent agent will exert more optimal preventive activities than the risk-averse and both prudent and imprudent agent. It depends on the level of prudence and risk-aversion, if the agent will consequently exert an optimum higher than the risk-neutral agent.
3.4
Summary
In the above chapter a new modelling for optimal prevention was introduced. Instead of two states in the second period, with either no loss or total loss, we have three sates in t1 with either no loss, a partial loss or total loss.
Then, we analysed the special case that the risk- neutral agent would choose an optimal level ensuch that the probability in the total loss state would be r(en) = 14,
and in the partial loss state q(en) = 12. Moreover, we assumed the outcome in the
partial loss state would be w1 − 12L. Thus, the expected outcome in t1 was the
same as in the analysed two states case in Chapter 2.
We were able to proof the following result for the above fixed values
• a risk-neutral agent will choose exactly the same prevention activity as in Chapter 2
• a risk-averse and both prudent and imprudent agent will exert an optimal level lower than the risk-neutral agent
• a risk-averse and imprudent agent will choose an optimal level lower than the risk-neutral agent and lower than the risk-averse and both prudent and imprudent agent
• a risk-averse and prudent agent will exert an optimal level higher than the risk-averse and both prudent and imprudent agent. With respect to the risk-neutral agent it depends on the degree of risk-aversion and prudence, if the risk-averse and prudent agent chooses a level lower or higher than the risk-neutral agent.
C
ONCLUSION
Prevention or Self-Protection is one of the main activities a person can pursue in order to deal with risk. However, it was not clear for a long time what the main force is that drives a person to exert prevention, as it was shown by Dionne and Eeckhoudt (Dionne, Eeckhoudt, 1985, [4]), that the effect of risk aversion towards prevention is ambiguous.
Eeckhoudt, Gollier (Eeckhoudt, Gollier, 2005, [5]) and Menegatti (Menegatti, 2009, [11]) stated that prudence could be this force that drives prevention. Although, their opposite results show that changes in the modelisation of prevention can have a huge impact. Therefore, it seems to be necessary to generalize the mod-elling.
In this thesis, it was tried to generalize Menegatti’s approach (Menegatti, 2009, [11]) in two ways. First, in Chapter 2 the concept of impatience was introduced, as the agent will evaluate the total utility in the first period, so it is necessary to evaluate in the first period the expected utility of the second period. It was shown, that with this generalization we obtain the same results as Menegatti, namely, it was shown that
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
the optimal effort of a prudent agent ep will be higher than en
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
Figure 4.1: Loss distributions
the optimal effort of an imprudent agent eiwill be lower than en
• if the risk-neutral agent exerts an optimal effort ensuch that p(en) = 12, than
the optimal effort of an agent, who is neither prudent nor imprudent, will be the same as en
Moreover, changes in the level of impatience do not have an impact.
The second generalization refers to the loss distribution. Instead of two states in the second period, three states were introduced: one with no loss, one with total loss and the new third state with only a partial loss. Thus, we modelled the loss distribution in a more general way. In the two states model in Chapter 2 and also in the paper by Menegatti (Menegatti, 2009, [11]) it was analysed the special case, where under the optimal prevention activity of the risk-neutral agent en
the probability that the loss occurs is 1
2. In order to be able to compare the two
modelisations, we considered as a reference case a situation where the partial loss is 12 of the total loss and the probabilities under the optimal prevention activity of the risk-neutral agent are such that the partial loss occurs with probability
1
2 and the total loss with probability 1
4. Thus, the loss distributions in the two
modelisations have the same first moment, however they differ in their higher moments, as it can be seen in Figure (4.1).
• a risk-neutral agent will choose exactly the same prevention activity (en) as
in Chapter 2
• a risk-averse and both prudent and imprudent agent will exert an optimal level (ep,i) lower than the risk-neutral agent (ep,i < en)
• a risk-averse and imprudent agent will choose an optimal level (ei) lower
than the risk-neutral agent and lower than the risk-averse and both prudent and imprudent agent (ei < ep,i < en)
• a risk-averse and prudent agent will exert an optimal level (ep) higher than
the risk-averse and both prudent and imprudent agent. With respect to the risk-neutral agent it depends on the degree of risk-aversion and prudence, if the risk-averse and prudent agent chooses a level lower or higher than the risk-neutral agent (ep,i < ep and ep≤≥en).
This thesis shows that generalizations of the modelisation can lead to new re-sults. However, due to the sheer variety of prevention methods and possible risks, it seems necessary to generalize even further than what has been accom-plished in this thesis.
For example, as Ellero and Ferretti (Ellero, Ferretti, 2014, [6]) point out in their multi-period study of the optimal prevention of a risk neutral agent analysing interior and boundary solution and using static comparative results, the impor-tance of time and the more realistic assumption of several epochs in which a hazardous event may take place.
Another example is the effect of prudence on the optimal level of advanced (be-fore the hazardous event) and contemporaneous (simultaneous with the haz-ardous event) prevention, as recently studied by Menegatti (Menegatti, 2016, [12]). He shows that prudence tends to increase advanced prevention and re-duces contemporaneous prevention, which again highlights the importance of
time in this problem.
To sum up, this thesis brought new insights in the field of optimal prevention and underlines the necessity of further studies.
A
CKNOWLEDGEMENT
Throughout the completion of this thesis Prof.ssa Ferretti supported me tremen-dously in all stages. She was always available for any kind of questions or doubts I had during my work.
For her guidance and scientific advice I am immensely grateful.
A
PPENDIX
A.1
w1−ν0 Z w1−L−ν−L T (x)dx = U0(w1− 1 2L − ν−12L) w1−ν0 Z w1−L−ν−L 1dx | {z } A(x) − U00(w1− 1 2L − ν−12L)(w1− 1 2L − ν−12L) w1−ν0 Z w1−L−ν−L 1dx | {z } B(x) + U00(w1− 1 2L − ν−12L) w1−ν0 Z w1−L−ν−L xdx | {z } C(x) A(x) = U0(w1− 1 2L − ν−12L)(L + ν−L− ν0) B(x) = U00(w1− 1 2L − ν−12L)(w1− 1 2L − 1 2ν−L− 1 2ν0)(L + ν−L− ν0) = U00(w1− 1 2L − ν−12L)[w1L + w1ν−L− w1ν0− 1 2L 2− Lν −L− 1 2ν 2 −L+ 1 2ν 2 0] C(x) = U00(w1− 1 2L − ν−12L) 1 2[(w1− ν0) 2− (w 1− L − ν−L)2] = U00(w1− 1 2L − ν−12L) 1 2[w 2 1 − 2w1ν0+ ν02− (w 2 1+ L 2 + ν−L2 − 2w1L − 2w1ν−L+ 2Lν−L)] = U00(w1− 1 2L − ν−12L)[w1L + w1ν−L− w1ν0− 1 2L 2− Lν −L− 1 2ν 2 −L+ 1 2ν 2 0] 44Thus, B(x) and C(x) cancel out and it is w1−ν0 Z w1−L−ν−L T (x)dx = A(x) = U0(w1− 1 2L − ν−12L)(L + ν−L− ν0)
A.2
w1−ν0−β Z w1−L−ν−L−β T (x)dx = U0(w1 − 1 2L − ν−12L− β) w1−ν0−β Z w1−L−ν−L−β 1dx | {z } A(x) − U00(w1− 1 2L − ν−12L)(w1− 1 2L − ν−12L− β) w1−ν0−β Z w1−L−ν−L−β 1dx | {z } B(x) + U00(w1− 1 2L − ν−12L− β) w1−ν0−β Z w1−L−ν−L−β xdx | {z } C(x) A(x) = U0(w1− 1 2L − ν−12L− β)(L + ν−L− ν0) B(x) = U00(w1− 1 2L − ν−12L− β)(w1− 1 2L − 1 2ν−L− 1 2ν0)(L + ν−L− ν0− β) = U00(w1− 1 2L − ν−12L− β)[w1L + w1ν−L− w1ν0− 1 2L 2− Lν −L− 1 2ν 2 −L+ 1 2ν 2 0 − βL − βν−L+ βν0] C(x) = U00(w1− 1 2L − ν−12L− β) 1 2[(w1− ν0− β) 2− (w 1− L − ν−L− β)2] = U00(w1− 1 2L − ν−12L) 1 2[w 2 1 − 2w1ν0+ ν02+ β 2− 2βw 1+ 2βν0 − (w2 1 + L2+ ν−L2 − 2w1L − 2w1ν−L+ 2Lν−L+ β2 − 2w1β + 2Lβ + 2ν−Lβ)] = U00(w1− 1 2L − ν−12L)[w1L + w1ν−L− w1ν0− 1 2L 2− Lν −L− 1 2ν 2 −L+ 1 2ν 2 0 − βL − βν−L+ βν0]Thus, B(x) and C(x) cancel out and it is w1−ν0−β Z w1−L−ν−L−β T (x)dx = A(x) = U0(w1− 1 2L − ν−12L− β)(L + ν−L− ν0)
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