Connecting possibilistic prudence and optimal saving

— In this paper we study the optimal saving problem in the framework of possibility theory. The notion of possibilistic precautionary saving is introduced as a measure of the way the presence of possibilistic risk (represented by a fuzzy number) influences a consumer in establishing the level of optimal saving. The notion of prudence of an agent in the face of possibilistic risk is defined and the equivalence between the prudence condition and a positive possibilistic precautionary saving is proved. Some relations between possibilistic risk aversion, prudence and possibilistic precautionary saving were established.


I. INTRODUCTION
HE effect of risk on saving was studied for the first time by Leland [1], Sandmo [2] and Drèze and Modigliani [3].They showed that if the third derivative of the utility function is positive, then the precautionary saving is positive.Kimball introduced in [4] the notion of prudence and established its relation with optimal saving.This paper aims to approach optimal saving and prudence in the context of Zadeh's possibility theory [5].The first contribution of this paper is a model of optimal saving, similar to the one in [4] or [6], p. 95.The notion of possibilistic precautionary saving (associated with a weighting function f, a fuzzy number A representing the risk and a utility function representing the consumer) is introduced and necessary and sufficient conditions for its positivity are established.The second contribution is the definition of the notion of prudence in possibilistic sense and its characterization in terms of possibilistic optimal saving.The third contribution refers to some relations between the degree of absolute prudence [4], possibilistic risk aversion [7] and possibilistic precautionary saving.Among others, the possibilistic precautionary premium is defined as a possibilistic measure of precautionary motive.
This notion is analogous to (probabilistic) precautionary premium of [4].
We will survey the content of the paper.In Section 2 are recalled, according to [8], [9], [10] the definition of fuzzy numbers and some associated indicators: possibilistic expected utility, possibilistic expected value and possibilistic variance.The equivalence between the concavity (resp.convexity) of a continuous utility function and a possibilistic Jensen-type inequality is proved.
In Section 3 the possibilistic two-period model of precautionary saving is studied.The consumer is represented by two utility functions u and v and the risk, present in the second period, is described by a fuzzy number.The expected lifetime utility of the model is defined with the help of the notion of possibilistic expected utility.The main introduced notion is possibilistic precautionary saving.It measures the changes on optimal saving produced by the presence of risk in the second period.If this indicator has a positive value then by adding the risk the consumer will choose a greater level of optimal saving.The main result of the section characterizes the positivity of possibilistic precautionary saving by the condition . One also proves an approximate calculation formula of possibilistic precautionary saving.
In Section 4 the notion of prudence of an agent in the face of risk situation is described by a fuzzy number.The definition of this notion follows the line of [11], [12], where we find a formal presentation of probabilistic prudence.The main result of the section is a theorem which characterizes possibilistic prudence in terms of the previously studied optimal saving model.
Section 5 begins by recalling the Arrow-Pratt index [13], [14], the degree of absolute prudence [4] and posibilistic risk premium [7].A result of the section characterizes the property of possibilistic risk premium to be decreasing in wealth by the comparison between prudence and absolute risk aversion (prudence is larger than absolute risk aversion).Then the notion of possibilistic precautionary premium is introduced and some of its properties which establish relations between prudence, possibilistic risk aversion and possibilistic precautionary saving are proved.
The paper ends with a section of concluding remarks.

II. POSSIBILISTIC EXPECTED UTILITY
Fuzzy numbers are the most studied class of possibility distributions [10].Their indicatorsthe expected value and variance represent the main instrument in the possibilistic study of risk phenomena [7], [9].
In this section we will define the fuzzy numbers and their indicators and we will prove a characterization theorem of convex (resp.concave) functions by possibilistic Jensen-type inequalities.
International Journal of Artificial Intelligence and Interactive Multimedia, Vol. 2, Nº 4.
-39-Let X be a non-empty set.A fuzzy subset of X (shortly, fuzzy set) is a function A:X[0,1].A fuzzy set A is normal if A(x)=1 for some xX.The support of A is defined by supp(A)={xR|A(x)>0}.The possibilistic expected utility E(f,u(A)) is defined by: If u is the identity function of R then E(f, u(A)) is the possibilistic expected value [9]: E(f,A) and Var(f,A) are the notions introduced by Carlsson and Fullér in [8].
Lemma 1. [15] Let u:RR be a continuous utility function.The following are equivalent: a) u is concave; b) For any a, bR, ) Taking into account that f0 and applying Jensen inequality it follows: . By hypothesis, we will have This inequality holds for any a, bR and u is continuous.By Lemma 1, it follows that u is concave. Corollary 1.If u is a continuous utility function then the following are equivalent: a) u is convex; b) u(E(f,A))E(f,u(A)) for any fuzzy number A.
The following result appears implicitly in the proof of Proposition 4.4.2 of [7].
In this section we define a notion of precautionary saving in the framework of an optimal saving possibilistic model.The positivity of precautionary saving shows that the presence of risk increases the level of optimal saving.Intuitively this points out that the agent is prudent in the face of possibilistic risk.The main result of the section characterizes this prudence in an intuitive sense by the positivity of the third derivative of one of consumer's utility functions.
The probabilistic two-period model of precautionary saving from [6], p. 65 is characterized by the following data:  u(y) and v(y) are the utility functions of the consumer for period 0, resp. 1  for period 0 there exists a sure income 0 y and for period 1 an uncertain income given by a random variable y ~  x is the level of saving for period 0 Assume that u, v have the class 2 . The expected lifetime utility of the model is: ) where r is the rate of interest for saving.The consumer's problem is to choose that value of s for which the maximum of V(s) is attained.
The possibilistic model of optimal saving that we are going to build further starts from the same data, except for the fact that y ~will be replaced by a fuzzy number.
We fix a weighting function f and a fuzzy number A whose The (possibilistic) expected lifetime utility W(s) of our model will be defined using the notions of possibilistic expected utility from the previous section.
)) ) The relation ( 5) can be written: By derivation, from (6) one obtains: which can be written: Deriving it one more time it follows One considers the following optimization problem: (ii) follows from (i).  By Proposition 4 (ii) and ( 8), it follows that the optimal solution  s is determined by the following equality: ) The optimal solution  s of problem (10) has the approximate value: Applying the first order Taylor formula one has: Applying again the first order Taylor formula it follows )) , ( Replacing in (13) it follows: From ( 11), ( 13), ( 14) we obtain From where the following approximate value of We consider now the optimal saving model in which in period 1 we don't have uncertainty any more: the uncertain income A is replaced by the sure income E(f,A).The lifetime utility of the model is: and the optimization problem becomes: (16) In this case one has The optimal solution )) , ( ( , which, by (17), is written: The difference    1 s s will be called possibilistic precautionary saving (associated with 0 y , r and A).This indicator measures the way the presence of the possibilistic risk A causes changes in consumer's decision to establish the optimal saving.
The following proposition is the main result on our optimal saving model.The key-element of its proof is the application of Proposition 2. Proposition 6.The following assertions are equivalent: Proof.Let A be a fuzzy number.From ( 17) and ( 11) one obtains, by denoting The previous inequality holds for any fuzzy number A, thus, by Corollary 1, the following equivalences follow: Condition (i) of Proposition 6 (=the positivity of possibilistic precautionary saving) expresses the fact that the presence of risk leads to the increase of optimal saving, and condition (ii) is the well-known property of prudence introduced by Kimball in [4].Since condition (ii) is present both in Kimball's result and in Proposition 6, we conclude that the positivity of possibilistic precautionary saving is equivalent with the positivity of probabilistic precautionary saving.
Example 1.We consider the possibilistic optimal saving model with the following utility functions: for any yR.
Let A be a fuzzy number and f a weighting function.Then By Proposition 5, the optimal solution of problem (10) will have the approximate value: The approximate value of  s can be written: )) , ( In this section we will define the meaning that an agent is prudent in the face of risk modeled by a fuzzy number.This definition is inspired by the concept of prudence in possibilistic sense as it has been defined in [12], [11].Using the results from the previous section we will find an equivalent formulation of possibilistic prudence in terms of precautionary saving. Remark 2. The above theorem provides a more intuitive meaning to the notion of possibilistic prudence formally introduced by Definition 1. Indeed, by the equivalence (a) (c) it follows that the agent v is possibilistically prudent iff in the presence of risk he chooses a higher level of optimal saving.Remark 3. In the optimal saving model of Section 3, the consumer is represented by the pair of utility functions (u,v).As the risk may appear only in period 1 (when the consumer 's behavior is described by v), the prudence of consumer (u,v) in the face of risk coincides with v's prudence in the face of risk.Therefore, under condition (c ) of Theorem 1, we deal with the prudence of consumer (u, v).

V. PRUDENCE AND POSSIBILISTIC RISK AVERSION
Following the line of Kimball from [4], in this section we will investigate the relation between prudence and possibilistic risk aversion, issue treated in [7].Both topics describe two attitudes of an agent in the face of risk.By defining possibilistic precautionary premium as a case of possibilistic

Proposition 4 .
(i) W is a strictly concave function.(ii)The optimal solution )

.
If we replace A with the fuzzy point E(f,A) it follows: In this case we obtain an approximate value of the optimal solution of problem (10):

Remark 1 . 7 .
f a weighting function.If X is a random variable then M(X) is its expected value and M(u(X)) is the expected utility associated with u and X.for any triple (x,k,A) with the above significance.According to (24), the agent is possibilistically prudent iff the possibilistic utility premium w(x,A,u) is decreasing in x.Proposition Assume that the utility function u .Proof.Deriving (21) w.r.t.x we obtain

From 1 .
the previous inequality and taking into account Remark 1 and Corollary 1 the equivalence of the following assertions follows:the agent u is possibilistically prudent  We go back now to the possibilistic precautionary saving model from Section 3 (u(y) and v(y) are the utility functions of the consumer for period 0, resp.1).Theorem Under the conditions of Section 3 the following assertions are equivalent: The agent v is possibilistically prudent.Proof.(a)(b) By Proposition 6. (b)(c ) By Proposition 7.