If the consumers risk aversion behavior varies intertemporally and if the risk aversion coefficient on future consumption becomes very large, the consumer tends to aim at a fixed future consumption target. A by-product is a reinterpretation of subsistence theories of consumption.Intertemporal Consumer Choice, theories of consumption, risk aversion coefficient

Ashok Guha

Brishti Guha

English

Research Papers in Economics

 
 
ANY OPINIONS EXPRESSED ARE THOSE OF THE AUTHOR(S) AND NOT NECESSARILY THOSE OF 
THE SCHOOL OF ECONOMICS & SOCIAL SCIENCES, SMU 
 
 
     
 
 
 
 
 
 
 
 
Utility functions, future consumption targets and 
subsistence thresholds 
 
 
 
 
 
 
Brishti Guha,  Ashok Guha  
October 2008 
 
 
 
 
 
 
 
   Paper No. 13-2008 Utility functions, future consumption targets and subsistence
thresholds 
Ashok Guha Brishti Guha
Indian Statistical Institute, New Delhi Singapore Management University
Abstract
If the consumer’s risk aversion behavior varies intertemporally and if the risk aversion
coefficient on future consumption becomes very large, the consumer tends to aim at a fixed
future consumption target. A by-product is a reinterpretation of subsistence theories of
consumption.
Citation: Guha, Ashok and Brishti Guha, (2008) "Utility functions, future consumption targets and subsistence thresholds."
Economics Bulletin, Vol. 4, No. 30 pp. 1-4
Submitted: October 4, 2008.  Accepted: November 2, 2008.
URL: http://economicsbulletin.vanderbilt.edu/2008/volume4/EB-08D90009A.pdf1.  Introduction 
 
Empirical evidence from consumer surveys and the like suggests that consumers often 
save to meet a fixed future consumption target.
1  If this were so, it would have important 
consequences for saving behavior and capital markets.  Saving would be negatively interest-
elastic, the gross substitutability property would not obtain between current and future 
consumptions, resulting in instability and multiplicity of equilibria, including some with high 
saving at very low, possibly zero, interest rates
2. Under certain assumptions, such behavior can 
generate a rich dynamics, including historical traps, implosions involving dwindling capital stock 
and increasing interest rates, and optimal one-time interventions [we refer readers to another 
paper of ours for a detailed treatment of the implications for equilibrium and dynamics].  Further, 
with future consumption targets fixed, income shocks would be absorbed essentially by current 
consumption, yielding an alternative explanation of the excess-sensitivity-of-consumption-to-
income puzzle.   
However, theoretical work has essentially ignored fixed consumption targeting behavior.  
Perhaps this is because such behavior has been generally regarded as an oddity, irreconcilable 
with standard utility maximization. The present paper seeks to dispel this notion and show how 
such behavior can be generated from conventional utility functions.  
Carroll (2004) has explored a somewhat similar theme – that of buffer stock saving. 
However he is concerned with consumers who aim at a fixed ratio of wealth to permanent 
income, which is quite different from the phenomenon we seek to explore.  Our consumer targets 
a fixed consumption level rather than a saving ratio and this target, unlike Carroll’s, is 
independent of wealth, interest rates and permanent income. 
The mathematical analysis that we apply, both to this theme and to a second application 
(subsistence thresholds) is very simple. But it has never been done before and does shed light on 
empirically observed phenomena (like future consumption targets and subsistence thresholds) 
that have never quite been explained within the framework of standard utility functions. 
 
2.  The Model 
 
 Consider  an  additive  two-period utility function 
1
2
12 1
1
(, ) ()
(1 )(1 )
c
Ucc uc
θ
θ ρ
− −
=+
−+
                                 (1)               
Here second period felicity is CRRA in c2 with risk-aversion coefficient θ and time preference 
rate ρ. First period felicity is any function of first period consumption which does not share the 
same risk aversion coefficient as second period felicity, and such that u’( ) > 0, u’’ < 0.  Given a 
wealth constraint and a fixed interest rate r, an interior maximum of U(c1,c2) implies 
2
1
1
(1 ) '( ) 1
c
uc r
θ
ρ
−
=
+ +
                                            (2) 
or  
1/ 1
2
(1 ) '( )
[
1
uc
c
r
]
θ ρ − +
=
+
                                        (3) 
                                                 
1 Samwick (1998) cites evidence from the Survey of Consumer Finance (1992).  
2 On the aggregative plane, such behavior has been observed, for instance, in China, which saves 50% of its GDP 
while the Bank of China’s deposit rate, after two recent increases, has just reached 2.52% with CPI inflation 
currently running at 5%.  There have of course been other attempts at explaining the Chinese paradox. 
  1In the limit as θ → ∞, c2 → 1 – provided u’(c1) is independent of  θ.  Asymmetrically infinite 
second-period risk aversion suffices to ensure target saving behavior.  There is no loss of 
generality due to the unit value of c2 since we can choose units as we please. 
  Alternatively, in the total utility function (1), consider the second-period felicity  
component.  As θ →   ∞, (1 – θ) → –  ∞,  and 
1
2 1
lim
1
c
θ
θ θ
−
→∞
−
=
−
0 for c2 ≥ 1                                      (4)  
= – ∞ for c2 < 1                                 (4a) 
(1), at this limit, reduces to 
U(c1,c2) = u(c1) for c2 ≥ 1                                                   (5) 
= – ∞ for c2 < 1                                                   (5a) 
This explains why consumers target one and only one unit of consumption in the second period.  
Given the wealth constraint, any additional second period consumption would reduce first period 
consumption and felicity and, therefore according to (5), total utility as well. Any shortfall in 
second period consumption below the target level results on the other hand in infinite misery and 
must therefore be avoided if this is at all possible within the wealth constraint. 
  Future consumption targeting is thus associated with intense future risk aversion – so 
intense, in fact, that interest rate, time preference and wealth level cease to matter. Such intense 
risk aversion may not be uncommon among consumers in economies that have a long history of 
turbulence and volatility. 
 
3.  A by-product: a reinterpretation of subsistence theories of consumption 
 
While the consumption targets considered above admit of neither upward nor downward 
variation, the concept of a subsistence level of consumption (as used for instance in Steger 
(2000), Ben-David (1998), Sharif (1986), Stigler (1945)) implies only downward rigidity.  A by-
product of our results however is such a subsistence model. Consider utility function (1) with c1 
= c2 = c and ρ = 0: 
U(c) = u(c) + (c
1 – θ – 1)/(1 – θ)                                       (1a) 
We have now switched from a two-period model of consumption choice to a static utility 
function with two components: an arbitrary one and another of CRRA form.  These two 
components correspond to two possible elements of the total utility derived from consumption:  a 
sociological element (based for instance on concepts like status value) and a biological one.   We 
assume that biological utility is additively separable from sociological utility and that one of the 
two component functions, possibly the biological,  is CRRA.  θ is the risk aversion coefficient 
associated with biological utility and a high value of θ reflects the fact that people are reluctant to 
gamble with the biological aspects of consumption. 
Repeating the argument in equations (4) to (5a) above, this reduces, in the limit θ → ∞, to 
U(c) = u(c) for c ≥ 1                                                   (6) 
= – ∞ for c < 1                                                   (6a) 
A downward-rigid subsistence consumption level c = 1 is implied (without loss of generality due 
to the unit value of c).  This is of some interest because there is no evidence of a rigid 
physiological minimum level of consumption required to sustain life.
3  What we suggest here is 
                                                 
3 See Sukhatme and Margen (1981), Edmundson and Sukhatme (1990). 
  2a model of subsistence based on risk aversion.
4  Needless to say, it is easy to recast our argument 
in the framework of multi-period choice. 
 
 
References 
Ben-David, D. (1998) “Convergence Clubs and Subsistence Economies” Journal of 
Development Economics 55, 155-171. 
Carroll, C. (2004) “Theoretical Foundations of Buffer Stock Saving” NBER Working Paper 
10867. 
Dalgaard, C.J and H. Strulik (2007) “A Bioeconomic Foundation of the Malthusian Equilibrium : 
Body Size and Population Size in the Long Run” Diskussionspapiere der 
Wirtschaftswissenschaftlichen FakultAOt der UniversitAOt Hannover dp-396, Universitat 
Hannover, Wirtschaftswissenschaftliche Fakultat. 
Dasgupta, P. (1997) “Nutritional Status, the Capacity for Work and Poverty Traps” Journal of 
Econometrics 77, 5-38. 
Edmundson, W.C and P.V Sukhatme (1990) “Food and Work : Poverty and Hunger?” Economic 
Development and Cultural Change 38, 263-280. 
Guha, B. and A.S. Guha (2008) “Target Saving in an Overlapping Generations Model” B.E 
Journal of Macroeconomics 8, 1-24.  
Samwick, A.(1998) “Tax Reform and Target Saving” National Tax Journal  51: 621-635. 
Sharif, M. (1986) “The Concept and Measurement of Subsistence : A Survey of the Literature” 
World Development 14, 555-577. 
Steger, T. (2000) “Economic Growth with Subsistence Consumption” Journal of Development 
Economics 62, 343-361. 
Stigler, G. (1945) “The Cost of Subsistence” Journal of Farm Economics 27 : 303-314. 
Sukhatme, P.V and S. Margen (1981) “Relationship Between Under-Nutrition and Poverty” 
Indian Economic Review 16, 13-39. 
 
 
  
                                                 
4 The subsistence concept can be endogenized either (1) in terms of a relationship between nutrition and working 
efficiency – as in Dasgupta (1997) – which yields a given subsistence level for any specific level of technology and 
complementary inputs – or (2) in terms of a relationship between nutrition and fertility which yields a given 
subsistence level for any specific mortality rate – as in Dalgaard and Strulik (2007). Such an endogenous subsistence 
concept is not our focus in this paper. 
  3

Utility functions, future consumption targets and subsistence thresholds

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