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Precautionary saving and fuzzy information
*
Nils Hauenschild, Peter Stahlecker
University of Hamburg, Institute for Statistics and Econometrics, Von-Melle-Park 5, 20146 Hamburg, Germany
Received 4 January 2000; accepted 15 June 2000
Abstract
We consider a two-period life-cycle model where uncertainty about future labour income is modelled by a fuzzy set. Applying a defuzzification strategy that explicitly takes the individual’s behaviour towards risk into account, we show that pessimistic individuals engage in precautionary savings even if marginal utility is not convex, e.g. in case of a quadratic utility function. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Intertemporal life-cycle models; Fuzzy sets; Defuzzification strategy; Precautionary saving
JEL classification: D81; D91
1. Introduction
In the analysis of intertemporal life-cycle models it has early been recognized that risk aversion (i.e. a strictly concave utility function u) is not the only relevant aspect of preferences in determining optimal consumption and savings in the presence of stochastic incomes (Leland, 1968; Sandmo, 1970;
`
Dreze and Modigliani, 1972). If the utility function is quadratic, individual behaviour exhibits certainty equivalence, i.e. consumption and savings are just the same as if income were certain and equal to the stochastic income’s expectation. In case of a convex marginal utility function (u- .0), on the other hand, individuals save more than under certainty equivalence, i.e. they build up
precautionary savings to protect themselves against low incomes and hence low consumption levels
in future periods. This kind of behaviour is generally characterized as prudence (Kimball, 1990). Recently, the problem of precautionary savings has been studied intensively in the literature both
1
theoretical and applied, because it turned out that several empirical patterns of consumption and especially some so-called consumption puzzles (see Caballero, 1990) could be explained by prudence.
*Corresponding author. Tel.: 149-404-2838-3539; fax: 149-404-2838-6326. E-mail address: [email protected] (P. Stahlecker).
1
An (incomplete) list of the most important papers contains Skinner (1988), Zeldes (1989), Caballero (1990, 1991), Kimball (1990), Hubbard et al. (1994), and Carroll and Samwick (1997, 1998).
In this paper we show that the analysis of precautionary saving need not be restricted to preferences with convex marginal utility nor to stochastic models but can just as well be carried out with a quadratic utility function and a different notion of uncertainty, for example. For this purpose we leave the stochastic model framework and simply assume that the individual has some vague ideas about his future labour income instead of knowing its stochastic distribution exactly. This vagueness is modelled by a fuzzy set with a corresponding membership function that represents the individual’s subjective beliefs about the uncertain income. After defining an adequate defuzzification strategy to capture the individual’s degree of pessimism (respectively optimism) we show that sufficiently pessimistic individuals engage in precautionary savings, no matter whether marginal utility is convex or not. In particular, we will derive an explicit (crisp) solution to the utility maximization problem for the case of a quadratic utility function. Formally, the resulting consumption function is a straight-forward extension of the corresponding ones obtained in deterministic and stochastic models, but it exhibits precautionary savings if the individual is ‘pessimistic’. Moreover, the precautionary savings term enters the consumption function linearly and only depends on the individual’s subjective beliefs and his degree of pessimism.
To some extent, our analysis parallels the studies of van der Ploeg (1992, 1993), who also demonstrates the existence of precautionary savings in case of quadratic utility functions. On the one hand, his approach is fundamentally different since he sticks to the stochastic framework and introduces prudent behaviour via non-expected utility maximization. On the other hand, the individual’s optimization problem in his study can be characterized as a max–min or a max–max problem, respectively, which is a noteworthy analogue to the objective function resulting from our
2
defuzzification strategy. In our approach, however, the individual need not be perfectly informed about the income generating process, namely its probability distribution, in order to engage in precautionary savings. He only has to be aware of the fact that his future income may deviate from the one he considers most possible and have some aversion against ‘negative surprises’, i.e. a certain degree of pessimism.
2. The model
Consider an individual who lives for only two periods. In both periods the individual works and receives the real wage w , tt 51,2. He is assumed to have a time-additive utility function with an instantaneous utility function u:(a,b)→R, (a,b),R1, that is twice continuously differentiable, strictly increasing and strictly concave. The individual has to decide upon his consumption c , tt 51,2, in his first period of life, where savings s can be invested on a perfect capital market, earning the1 deterministic and known interest rate r. Hence, the individual has to solve the optimization problem:
1
]]
u(c )1 1 u(c )2 →max ! (1)
c ,c ,s 11u 1 2 1
subject to:
c15w12s1 (2)
2
c25(11r)s11w2 (3)
where u$0 denotes the rate of time preference.
In case of uncertain incomes it is usually assumed that the individual decides upon consumption and saving after observing w . In a stochastic setting (only) w has to be interpreted as a random1 2
˜
variable (w ) then, and the objective function contains an expected utility term E [u(c )], where E [2 1 2 1 ?] denotes the expectation operator conditional on the information available at time t51.
In this paper we will retain the assumption that the first period income has already been observed 3 but that future income w is uncertain. Unlike all previous studies of intertemporal life-cycle models,2 we suppose that the individual does not regard w as a random variable with a perfectly known2 probability distribution. Instead, the individual only has some vague ideas about w2 that can be represented by a fuzzy set@5hsw ,ms ddw :w [R j, wherem:R →[0,1] is the membership function.
2 2 2 1 1
Here, for every w2[R1, the valuem(w ) denotes the degree of membership to which w belongs to2 2
@, where m(w )50 represents nonmembership (e.g. an extraordinarily high income) andm(w )51
2 2
4
refers to the most possible income values. In this context, note that degrees of membership must not 5
be identified with probabilities. In particular, the membership values do not add up to one.
Since w is fuzzy, the budget constraint (3) is fuzzy as well, and the optimization problem (1–3)2 cannot be solved directly. In order to obtain crisp decisions c and c , the multivaluedness in w has1 2 2 to be resolved by some adequate modifications of the objective function. Therefore we will construct a suitable defuzzification strategy that makes explicit use of the individual’s attitude towards deviations of income from its most possible value, i.e. his ‘behaviour towards risk’. In this way we obtain a crisp optimization problem whose objective function has a clear economic interpretation.
3. A defuzzification strategy
Given a fuzzy set@ with corresponding membership functionm, one can define the family of crisp sets:
3 [hw [R :ms dw $aj (4)
a 2 1 2
for all a[(0,1], which are calleda-cuts of @. We now propose the following procedure to obtain a
6
crisp decision: According to (1–3), second period income only has a direct impact on second period utility. Given his consumption level c , the individual will thus be interested in those incomes that1 provide him with the highest and the lowest second period utility. Of course, those utility levels will also depend on the degree of membership a to which the respective incomes belong to @. It is thus reasonable to calculate the incomes that maximize or minimize second period utility with respect to all
3
See, for example, the references cited in the Introduction. As far as we know, income uncertainty in life-cycle models has not been modelled via fuzzy sets yet.
4
A crisp set B,R1 is included as the special case where m is a mapping into the set h0,1j. Thus, m reduces to the well-known characteristic function.
5
For more details on fuzzy sets as well as their relation to probability theory, see, for example, Zimmermann (1996) and Klir and Wierman (1998).
6
incomes with degree of membership m(w )2 $a for given c and1 a[(0,1] at first, and to aggregate over all a-cuts afterwards, i.e. to consider:
1 aggregates all utility minima (maxima) when w runs over all a-cuts 3 , the function h giving
2 a
appropriate weights to the different degrees of membership. Typically, h will be increasing (e.g.
h(a)52a), such that high degrees of membership are given higher weight.
]
Obviously, H (H ) represents a worst (best) case scenario in that it only takes the lowest (highest)] possible utility levels (given a) into account. Hence, an extremely pessimistic individual will choose
c by maximizing u(c )1 1 11 / 11uH(c ) so as to avoid any negative surprises, i.e. lower utility levels] 1 ] than expected. An extremely optimistic individual, on the other hand, will take u(c )1 11 / 11uH(c )1
as his objective function, thereby accepting the possibility of negative surprises. A more ‘inter-mediate’ (e.g. neutral) individual will certainly take both scenarios into account. We therefore introduce the well-known Hurwicz optimism–pessimism index q, 0#q#1, to obtain a weighted average of the worst and the best case scenario. Taking everything together we finally arrive at the individual’s new objective function:
4. Optimal consumption and precautionary saving
We will now show that an individual maximizing the objective function (7) builds up precautionary
*
savings if q is sufficiently large. For this purpose, consider an income level w2 for which certainty equivalence prevails. In order to maintain the analogy to the stochastic notion of certainty
7
*
equivalence, w2 can be viewed as some expected value E[w ], but might as well be given by an2
7
*
It is in fact possible to take w2 as the expected value of a (possibly subjective) probability distribution. The possibility
*
that w may deviate from w , however, is not incorporated into the decision problem via a density function or the standard2 2
*
Let c1 denote the optimal consumption in case of certainty equivalence, i.e. when using
*
*
w2 5E[w ] in the corresponding stochastic model. In view of (1)–(3), c2 1 is determined by the (necessary and sufficient) optimality condition:
*
*
*
u9(c )1 5r?u9((11r)(w12c )1 1w )2 (8)
*
where r[(11r) /(11u). We suppose that there exists a solution c1 of (8). For an individual maximizing the objective function (7), we obtain the analogous condition:
1
where we have made use of the fact that u is strictly increasing.
Proposition.
*
*
(i ) There is a unique q [(0,1) for which c solves (9), i.e. an individual maximizing (7) for
1
*
*
q5q consumes the certainty equivalent c .
9
*
where the sign of Z (c ) follows from the assumptions made at the beginning of this section and from0 1 the strict concavity of u. Analogously we obtain for q51:
1
9
*
*
*
*
Z (c )1 1 5r
E
s
u9((11r)(w12c )1 1w )2 2u9s
(11r)(w12c )1 1]w (2 a) h(dd
a) da,0 09
Since Z is continuous in q and:q
1
9
*
≠Z (c )q 1 ]
]]]≠q 5 r
E
s
u9((11r)(w12c )*
1 1w (2 a))2u9((11r)(w12c )1*
1w (]2 a)) h(d
a) da(10) 0
, 0
*
9
*
the intermediate value theorem yields the existence of a unique q [(0,1) with Zq*(c )1 50. Furthermore, the implicit function theorem implies the existence of solutions to (9) in some
*
neighbourhood %,[0,1] of q as well as:
9
9
≠Z ≠Z
≠c1 q q
]≠ 5 2]]
Y
]],0 (11)q ≠q ≠c1
because of the strict concavity of u, which establishes (ii). Part (iii) of the proposition immediately follows from (11). h
*
In view of the proposition, an individual can be considered ‘risk neutral’ in case of q5q because he
*
behaves as if second period income was certain and equal to w , and thus exhibits some kind of2
*
certainty equivalence. A pessimistic individual is characterized by q.q . He will consume less than
a neutral individual and build up precautionary savings. Moreover, we have ≠s /1 ≠q5 2 ≠c /1 ≠q.0, i.e. the more pessimistic the individual the higher his precautionary savings.
It is important to emphasize that the individual’s prudent behaviour only results of his awareness that second period income may deviate from the most possible (or ‘expected’) value and his aversion against negative surprises. The individual does not necessarily possess information about the income generating process or even its exact probability distribution. Hence, the existence of precautionary savings can be viewed as a most natural consequence of information deficiencies and vagueness.
5. The quadratic utility case
8 As an example consider the quadratic utility function:
1 2
]
u(c)5c22c , c[(0,1) (12)
8 2
If both incomes w and w are known with certainty, the optimization problem (1–3) is rather trivial.1 2
If second period income is stochastic, the objective function (1) contains the expected utility term
E [u(c )] instead of u(c ). This yields the well-known consumption function (with r1 2 2 5u):
1
sto ]] ˜
c1 5
s
(11r)w11E w1f gd
2 (14)21r
which obviously exhibits certainty equivalence. For an individual maximizing the objective function 9
where g:(0,1]→R1 is a decreasing function. This gives us the possibility to derive an explicit
]] ]
*
*
solution. For a-cuts of the form (15) we obviously have w (]2 a)5w2 2
œ
g(a) and w (2 a)5w2 1 ]]9
g(a). Taking this as well as (12) into account, the optimality condition Z (c )50 given by (9)
œ
q 1q 51 / 2 and will exhibit certainty equivalence behaviour. For q.1 / 2, on the other hand, the individual will engage in precautionary savings despite of his quadratic utility function.
9 2 2
* *
The most frequently used membership functions likem1(w)51 /(11(w2w ) ) or2 m2(w)5exp(2(w2w ) ) allow for2
a representation of thea-cuts as in (15).
10
Note that the deterministic model with certain income w is also included in our model. In that case, we have g(2 a);0
*
References
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