Gomes, F. A. R.

English

Research Papers in Economics

The Effect of Future Income
Uncertainty in Savings
Decision
Fábio Augusto Reis Gomes
Insper Working Paper
WPE: 082/2007Copyright Insper. Todos os direitos reservados.
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citação completa do documentoThe Effect of Future Income Uncertainty in Savings Decision
∗ 
 
Fábio Augusto Reis Gomes
♣ 
Ibmec São Paulo 
 
Abstract 
This paper analyzes consumption and savings decisions in a two-period consumption 
setting, supposing that future income is uncertain in the sense of Knight (1921). The 
results imply that uncertainty-averse agents save more than risk-averse agents. 
 
Key-words: consumption, savings, uncertainty aversion. 
Jel Classification: D81, D91. 
 
                                                       
∗ I am grateful to Sergio Werlang, Heleno Pioner, Paulo Coimbra, Guilherme Hamdan and Cleomar 
Gomes for their comments. The usual disclaims apply. 
♣  Address: Rua Quatá, 300, 4º Andar, Sala 422, Vila Olímpia. São Paulo, SP, Brasil - CEP: 04546-042. 
Phone: (55) 11 4504-2776. E-mail: FabioARG@isp.edu.br. 1 Introduction 
Consumption decision is a central subject for economists and, since the 
development of Friedman’s (1957) Permanent Income Hypothesis (PIH), economists 
have agreed that consumption decisions depend on random variables such as future 
income, existing risk or uncertainty
1. According to Knight (1921), risk is a situation in 
which a single additive probability measure on the states of nature is available to 
conduct choice. However, under uncertainty, information is too imprecise to be 
summarized by any additive probability measure. In general, economic models assume 
the rational expectations hypothesis, which means that individuals know the objective 
probability law, or the Bayesian approach in which they have a (single) prior subjective 
probability distribution. In both cases uncertainty is doffed. 
Recently, Manski (2004, p.1330) has extensively argued that “observed choice 
may be consistent with many alternative specifications of preferences and 
expectations”.  However, the prevailing practice is to assume the rational expectations 
hypothesis, which implies that other possible explanations for economics phenomena 
are discarded  and applied works are reduced to inference about preferences alone. 
Manski advocates that researchers must use data on expectations formation to test 
assumptions about expectations. Following the advice, the first difficulty emerges 
immediately; in general, survey respondents have only the option to report expectations 
in probabilistic form because the questions asked are about point prediction of random 
events or verbal assessments of likelihood.
2 However, Das and Soest’s (1997, 1999) 
results generate indirect evidence in favor of uncertainty. Studying the Dutch Socio-
Economic Panel from 1984 to 1985, Das and Soest (1997) found that a large fraction of 
households (34.9%) underestimated their future income growth while only a small 
                                                       
1 Some authors use ambiguity instead of uncertainty. 
2 To enable persons to express uncertainty, survey researchers could elicit ranges of probabilities rather 
than precise probabilities for events of interest (Manski, 2004). proportion (15,5%) overestimated their income change. According to the authors, a 
possible explanation is that some people are simply too pessimistic, on average. Das and 
Soest (1999) extended the sample until 1989, finding a similar pattern. Additionally, 
using a formal test, they rejected the rational expectations hypothesis.  
A generalization of these results seems to be very premature. However, it is 
worth investigating what effects income uncertainty has on consumer behavior. In order 
to implement this analysis, this paper employs a two-period consumption model using 
the Choquet Expected Utility (CEU) approach, an axiomatic treatment of uncertainty, 
developed by Schmeidler (1989), in which the agent’s belief is represented by a convex 
non-additive probability function. Uncertainty aversion is introduced by means of a 
uniform contraction of any additive probability measure.  
The paper proceeds as follows. Section 2 presents useful results of the CEU 
model. Section 3 develops and solves the model. Final section summarizes the 
conclusions in light of other departures from PIH.   
 
2 Uncertainty Aversion
3 
First of all, define Ω as a set of the states of nature and Λ as an algebra from its 
subsets. Thus, i)  Λ ∈ Ω ; ii)  Λ ∈ ∪ ⇒ Λ ∈ B A B A, , and iii)  Λ ∈ ⇒ Λ ∈
c A A , where 
c A  is the set of elements of Ω not in A. The elements of Λ are the events. A function 
[] 1 , 0 : → Λ P  is a non-additive probability if i)  ( ) 0 = φ P , where φ  is the empty set; ii) 
()1 = Ω P , and iii)  () ( ) B A B P A P B A ⊂ ≤ Λ ∈   if   , , . Imposing the additional restriction:  
iv) () () ( ) ( ) B P A P B A P B A P B A + ≥ ∩ + ∪ Λ ∈ , , , we obtain a convex non-additive 
probability function P.    
                                                       
3 Proofs are omitted and the reader is referred to Schmeidler (1986, 1989), Gilboa (1987), Gilboa and 
Schmeidler (1989) and Dow and Werlang (1992). In the presence of uncertainty, information is too scarce for the agent to discard 
the additive probabilities until there is only one left.  For each action, an uncertainty 
averse agent considers the additive measure that accentuated the probability weights 
associated with the least favorable outcome (Mukerji and Tallon, 2001). The core of P, 
() P C , identified these additive probability:  
( )( ) ( ) ( ) {} Ω ⊂ ≥ Ω ∆ ∈ = X X P X Q Q P C   all for  , | , (1) 
where () Ω ∆  is the set of all additive probability measure on Ω and X is a random 
variable.    
The Choquet expected value of a random variable X is defined as: 
[] () [] () α α α α d X P d X P dP X X EP ∫ ∫ ∫
∞
∞ − Ω
≥ + − ≥ = =
0
0
1  (2) 
whenever these integrals exist (in the improper Riemann sense) and are finite. 
Alternatively, the expected value of X  can be defined based on  ( ) P C :   
[]
() [ ] { } X E X E Q
P C Q
P
∈
= min    (3) 
Thus, the uncertainty averse agent evaluates an uncertain act by the minimum 
expected value that may be associated with it. Two useful results are: 
[ ] [ ] b X aE b aX E b a P P + = + ℜ ∈ ≥ ∀ , , 0  and if   ℜ → ℜ : u  is a concave function, then 
() [] X u EP  is concave. 
Definition [Dow and Werlang (1992)]: Let P be a probability and  Ω ⊂ A  an event. The 
uncertainty aversion of P at A is defined by 
() ( ) ( )
c A P A P A P − − =1 , θ    (4) 
Thus, the degree of uncertainty aversion is proportional to the amount of 
probability “lost”. As Dow and Werlang (1992) showed, P can be generated by 
increasing the uncertainty aversion of an additive probability Q: fix  [] 1 , 0 ∈ θ , let ()1 = Ω P  and  () ( )() A Q A P θ − = 1  for  Ω ≠ A . Then, for all  Ω ≠ A ,  () θ θ = A P, , the 
uncertainty aversion is constant and identical to the probability “lost” by the uniform 
contraction. Although this result seems to be very restrictive, it is not the case. The 
degree of uncertainty aversion may be interpreted as an individual parameter and, in 
general, economists assume that the agents’ parameters are constant for all events. 
Lastly, assuming that  ( ) 0 inf ≥ =
Ω ∈ w X X
w
L , it is possible to show that 
[] () [] X E X X E Q
L
P θ θ − + = 1 . As anticipated, a CEU model gives an extra weight to the 
worst event, leading to a pessimistic decision criterion.  
 
 3 Consumption Model 
  Consider a consumption model in which individuals live for two periods. In the 
first period, the consumer has an income  1 w  and chooses consumption, 1 c , and 
savings,s. In the second period, the consumer picks consumption, 2 c , taking into 
account income, 2 w , and financial wealth, Rs, where R  is the gross rate of return.  
Suppose that  2 w  is uncertain.  
Assume that utility function u is 
2 C  and  0 '> u ,  0 ' ' < u . Monotonic preferences 
imply that budget constraints are binding and the consumer’s problem becomes
4  
( ) ( ) [ ] 2 1
, , 1 2 1
max c u c u EP
s c c
β +  
st.  s w c − = 1 1  and  sR w c + = 2 2  
where 1 0 < < β  is the intertemporal discount factor and  () ⋅ P E  is the expected value on 
the convex non-additive probability P. Substituting the restrictions on the objective 
function,   () ( ) ( ) sR w u E s w u s U P + + − = 2 1 β  is obtained and the consumer chooses only 
                                                       
4 Inada’s conditions are assumed to guarantee interior solution. current savings.  Furthermore, the objective function is concave and, consequently, i) 
0 > s  if the right side derivative of  ( ) s U , evaluated in  0 = s , is greater than zero, 
() 0 0
' > + U ; ii)  0 < s  if the left side derivative of  ( ) s U , evaluated in  0 = s , is lesser than 
zero,  () 0 0
' < − U  and, iii)  0 = s  if  ( ) ( ) 0 0 0
' '
− + ≤ ≤ U U .    
Suppose that P is generated from a uniform contraction of an additive 
probability, Q, using an uncertainty aversion measure [ ] 1 , 0 ∈ θ , thereby  
() ( ) ( ) ( ) ( ) sR w u E sR w u s w u s U Q
w
+ − + + + − = 2 2 1 1 min
2
θ β βθ    (5) 
In practice, the lower bound of income is zero, so define 0 inf ≥ =
Ω ∈
w w
w
L . Thus, 
() ( ) sR w u sR w u
L
w + = + 2
2
min  and, accordingly 
i)  0 > s  if  () ( ) ( ) ( )R w u E R w u w u Q
L
2 1 ' 1 ' ' θ β βθ − + < ;  
ii)  0 < s  if  () ( ) ()( )R w u E R w u w u Q
L
2 1 ' 1 ' ' θ β βθ − + > ; 
iii)  0 = s  if  () ( ) ( ) ( )R w u E R w u w u Q
L
2 1 ' 1 ' ' θ β βθ − + = .  
In the first case, in order to increase the current marginal utility, consumer decreases 
consumption. In the second case, the opposite occurs. And the third case does not 
require lending or borrowing.  Notice that, if the agent is risk averse ( 0 = θ ), savings 
have a similar pattern, except that future marginal utility is replaced by () R w u EQ 2 '.  
Graph 1 shows hypothetical savings functions for  0 > θ  and  0 = θ , where 
() R w u E a Q 2 ' β =  and ( ) ( ) ( )R w u E R w u b Q
L
2 ' 1 ' θ β βθ − + = . Because  2 w w
L ≤  and u is 
concave,  b a < . Once the uncertainty averse agents are more pessimist to forecast future 
income than risk averse agents, when the latter group starts to borrow, the former is still 
lending in order to smooth consumption path.  
< Insert Graph 1 > 
The following proposition discusses the impact of θ  on savings. Proposition: If future income  2 w  is uncertain and its probability function is generated 
by a uniform contraction of any additive probability measure, then the consumer’s 
savings increases with the uncertainty aversion measure. 
Proof: Objective function 
( )( ) ( ) ( ) ( ) sR w u E sR w u s w u s U Q
L + − + + + − = 2 1 1 θ β βθ  
FOC: 
() ( ) ( ) ( ) 0 ' 1 ' ' 2 1 = + − + + + − − R sR w u E R sR w u s w u Q
L θ β βθ  
Using the implicit function theorem,  
( ) ( ) [ ]
() () () ( ) []
0
' ' 1 ' ' ' '
' '
2
2
1
2 >
+ − + + + −
+ − + −
=
sR w u E sR w u R s w u
sR w u E sR w u R
d
ds
Q
L
Q
L
θ θ β
β
θ
 
Once  0 ' ' < u , the denominator is negative. Furthermore,  w sR w sR w
L ∀ + ≤ + , 2 , as a 
result  ( ) () sR w u E sR w u Q
L + > + 2 ' '  and the numerator is negative.  
 When  θ  increases, both the expected value of future 
income, ()2 2 1 w E w w E Q
L
P θ θ − + = , and the uncertainty aversion measure are affected. 
Thereby, perhaps the impact of θ  on s may be driven only by the variation in expected 
income. The following example sheds light on this question. 
Example:  Consider a quadratic utility:  ( ) ( ) c a c b a c u > > − − = b   and 0 , 2 /
2 . Then,  
2
2 1
1 R
w RE w b bR
s
P
β
β β
+
− + −
=  
The uncertainty aversion measure affects s only via the expected income. Thus, as in the 
case of risk aversion ( 0 = θ ), the solution of the model exhibits certainty equivalence 
when the quadratic utility is used (there is no precautionary savings). In addition, 
contrary to Hall’s (1978) results, when  1 = R β  is assumed, instead of the consumption 
random walk hypothesis, the following process is obtained: ( ) ( ) 2 1 2 2 1 ε θ θ − + + − = c w w c
L  
where  []0 2 = ε Q E . The pessimist behavior generates an excess of resource in the 
second period, the difference of  2 w  and 
L w . When  0 = θ  the random walk hypothesis is 
restored. 
 
4 Conclusions 
In the literature there are, at least, two important departures from PIH. Firstly, 
the hyperbolic discounting models, which predict that consumers have both a short-run 
preference for instantaneous gratification and a long-run preference for acting patiently 
and, as a consequence from this self-control problem, consumers are likely to increase 
consumption to obtain a higher instant gratification. This approach is able to explain the 
evidence that many consumers save too little (Laibson, 1997; Angeletos et al., 2001). 
On the other hand, as Gourinchas and Parker (2001, p. 406) stress “one of the basic 
motives for saving is the accumulation of wealth to ensure future welfare.”  More 
precisely, if  () 0 ' ' ' > ⋅ u , then introducing risk in future income made current savings rise 
due to a precautionary motive. Despite this prediction, this model is able to explain a 
lower saving behavior if agents have a high discount rate, once the last factor induces 
current consumption but individuals tend to keep a small amount of savings to use in 
case a large fall in income takes place (Carrol, 1992; Carrol, 1997).  
Motivated by Das and Soest’s (1997, 1999) results, this paper investigated 
another departure from PIH, the effect of future income uncertainty on savings and 
concluded that an uncertainty averse agent saves more than a risk aversion agent and 
this gap increases with the degree of uncertainty aversion. This result is also compatible 
with low savings, if the discount rate is sufficiently high. Indeed, the unique implication 
of the model is: people save more when future is uncertain. References 
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Consumption Model: Calibration, Simulation and Empirical Evaluation, Journal of Economic 
Perspective 15 (3), 47-68.  
Carroll, C., 1992, The Buffer Stock Theory of Saving: Some Macroeconomic Evidence, 
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Carroll, C., 1997, Buffer Stock Saving and the Life Cycle/Permanent Income Hypothesis, The 
Quarterly Journal of Economics 112, 1-57. 
Das, V. and A. v. Soest, 1997, Expected and realized income changes: Evidence from the 
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Hall, R. 1978, Stochastic implications of the life cycle-permanent income hypothesis: theory and 
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57, 571–587. Graph  
 
Graph 1 – Saving Functions 
                                            s 
                                                                                                         
                            
                                                        a        b                                          ( ) 1 ' w u  
 
                       0 = θ            0 > θ  
 
 
 

A panel data model for subjective information on household income growth,

A theory of the consumption function (Princeton,

Ambiguity Aversion and Incompleteness of Financial Markets,

Buffer Stock Saving and the Life Cycle/Permanent Income Hypothesis,

Expected and realized income changes: Evidence from the Dutch socio-economic panel,

Expected utility with purely subjective non-additive probabilities,

Maxmin expected utility with a non-unique prior,

Risk, uncertainty, and profit

Stochastic implications of the life cycle-permanent income hypothesis: theory and evidence,

Subjective probability and expected utility without additivity,

The Buffer Stock Theory of Saving: Some Macroeconomic Evidence,

The Empirical Importance of Precautionary Saving,

The Hyperbolic Consumption Model: Calibration, Simulation and Empirical Evaluation,

Uncertainty aversion, risk aversion, and the optimal choice of portfolio,

The Effect of Future Income Uncertainty in Savings Decision

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The Effect of Future Income Uncertainty in Savings Decision

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