Multiple Pareto-rankable equilibria may obtain in an overlapping generations model where consumers save to reach a fixed target. Existence and uniqueness conditions are discussed. The model displays excess consumption sensitivity to current income and perfect old-age insurance.Multiple equilibria, saving, overlapping generations, excess sensitivity.

Ashok S 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 
 
 
 
 
 
 
 
  
Future Targets and Multiple Equilibria  
  
   
 
  
 
 
 
  
 
Ashok S Guha, Brishti Guha 
October 2005 
 
 
 
 
 
 
 
 
   Paper No. 21-2005 
S S SM M MU U U    E E EC C CO O ON N NO O OM M MI I IC C CS S S    & & &    S S ST T TA A AT T TI I IS S ST T TI I IC C CS S S    
W W WO O OR R RK K KI I IN N NG G G    P P PA A AP P PE E ER R R    S S SE E ER R RI I IE E ES S S FUTURE TARGETS AND MULTIPLE EQUILIBRIA 
Ashok S. Guha
*                                                                                     Brishti Guha
†
Jawaharlal Nehru University                                  Singapore Management University 
 
Abstract 
  Multiple Pareto-rankable equilibria may obtain in an overlapping generations 
model where consumers save to reach a fixed target. Existence and uniqueness conditions 
are discussed. The model displays excess consumption sensitivity to current income and 
perfect old-age insurance. 
 
JEL Classification: E21, D91. 
Keywords: Multiple equilibria, saving, overlapping generations, excess sensitivity. 
  
 
1. Introduction 
 
Consumers often aim at target consumption levels of specific goods.  Consumer 
durables (houses, cars) frequently come in indivisible units and the consumer is often 
satiated with a single unit :  he desires just one piece, no more and no less.  Where such a 
good bulks large in the consumer’s budget, changes in its price may have unusual 
repercussions on demand for other goods.  The gross substitutability property may not 
obtain:  a rise in the price of the target good may reduce the budget available for other 
goods and the consumer’s demand for them.  This affects the existence and uniqueness of 
market equilibrium. 
An important example occurs when the consumer aims at a future consumption 
target – say a house to retire to or a fixed payment to an old age home that will support 
him.  This affects his saving behavior, particularly his reactions to interest rate changes.  
If such behavior occurs on a large scale, equilibrium in the capital market, the interest 
rate and the level of output are affected in curious ways.   
We focus on the possibility of multiple equilibria in such environments.  We 
suggest that an economy may persist in a high-saving equilibrium even at very low 
interest rates, offering one possible explanation for syndromes such as Japan’s over the 
last fifteen years.  We discuss conditions for uniqueness and demonstrate a multiplicity of 
Pareto-rankable equilibria for some parameters.  An implication is excess response of 
consumption to current income.  While a violation of the permanent income hypothesis,  
this is in line with the fact that there is an “excess sensitivity” puzzle, particularly for 
developing countries where consumers often violate the permanent income hypothesis 
(see Rao [2005], Lavi [2003], Chakrabarty and Schmalenbach [2002]).   
 
 
                                                 
* School of International Studies, Jawaharlal Nehru University, New Delhi 110067, India. E-mail: 
ashoksanjayguha@hotmail.com. 
 
†Corresponding author. Department of Economics, School of Economics and Social Sciences, Singapore 
Management University, 90 Stamford Road, Singapore 178903. E-mail : bguha@smu.edu.sg The authors 
acknowledge input from Kaushik Basu. 
 
  12.The Model 
 
  Assume a world of overlapping generations in which all individuals are identical.  
Population is constant with the number in each generation being normalized to one. Each 
individual lives for two periods.  At the outset of the first, he borrows capital from the 
previous generation, now retired, and promises to repay principal and prenegotiated 
interest after the production process.  He then produces according to a production 
function yt = f(kt) + εt, where εt is a production shock with zero mean.  Thereafter, he 
makes the contractual repayment and allocates the remaining output between   
consumption ct and saving in the light of his interest rate expectations and preferences.  
At the beginning of the next period, he lends the savings to the new generation of 
producers at the interest rate expected in that period and lives in retirement thereafter on 
his interest and capital.  Agents are all risk-neutral.   
  As in all models in which population is constant and individuals exhaust their 
income over their life-cycle, there is no aggregate saving:  however, the young save to 
finance their retirement.  
Competition for capital between producers drives the contractual interest rate to 
the expected marginal product of this capital 
'() t f k . This represents the non-random 
component of marginal product of capital, which can be fully forecast given the capital 
stock at the time of contracting (which is just the savings carried over from the previous 
period by the then working generation). Given rational expectations, savers form correct 
expectations of this contractual interest rate and base their savings decisions on it. Thus 
the contract is signed before the production shock hits. As savers are guaranteed the 
interest rate 
'() t f k , they are fully insured against the production shock – whether 
positive or negative - which is borne accordingly by the current producing generation. 
Since all agents are risk-neutral, the current generation is willing nonetheless to guarantee 
the older generation an interest rate of 
'() t f k - the expected value of the production shock 
being zero.  
Suppose now that the second period consumption of each individual is fixed at a.   
Let the younger generation’s consumption in period t – the first period of its life – be 
denoted by   while the older’s consumption  is   (the older generation’s consumption 
in its youth was  ). Because individuals are saving to meet fixed targets, we always 
have  
1,t c 2,t c
2, 1 t c −
2,t c = a = 
'
2, 1 (1 ( )) tt f ks − +                                    (1) 
 
  After paying the older generation, the younger one also wants to save just enough 
to ensure a second period consumption of a.  Accordingly the production shock is entirely 
absorbed in  , the current consumption of the working generation.  This excess 
sensitivity of current consumption to transitory income goes against the permanent 
income hypothesis, but is a characteristic of target-savers.  The following equation 
describes the younger generation’s behavior in period t+1, when it gets old:  
1,t c
2, 1 t c + = a =
'
12 , (1 ( ) tt f ks + +                                          (2) 
Moreover, we have 
  212 , tt ks + =        t ∀                                                   (3) 
From (1), (2) and (3) we can easily check that  
2, 2, 1 tt ss − == s ,   1 tt kk k + == 1 tt kk k + = = ,     s = k                                        (4) 
  Thus the game is stationary, despite the uncertain production shock.  If in any 
period, the shock is so adverse as to reduce the total income of the younger generation – 
after interest payments on the older generation’s loan – below the minimum required to 
reach the second period target,  the young will not save, but will instead consume 
everything left over after making interest payments.  No capital will be available next 
period to sustain production and the economy will disappear.   An economy can be wiped 
out by a sufficiently unfavorable shock if 
'
' () ()
1(
a
fk k f k
) f k
ε −− <
+
 
for any ε .  As long as production shocks are bounded, there is always a parameter space 
where this will never happen.  For simplicity, we assume that production shocks are 
uniformly distributed in the interval [-ε *,ε *], and for the rest of this paper concentrate 
on the case where even the worst shocks leave the young with enough to reach their 
second period target – so that the economy is in no danger of disappearing. 
 
3. Equilibrium 
 
 The  supply of savings from the old is given by 
* 1
a
sk
r
==
+
                                                     (5) 
where r* is the pre-negotiated interest rate that savers will face in the beginning of the 
next period.  The demand of the young for savings, or equivalently for capital, is given by 
' ()
* f kr =                                                  (6) 
Thus equilibrium capital stock solves the equation 
' (1 ( )) kf ka + =                                                      (7) 
The supply curve is negatively interest-elastic and  representable as a rectangular 
hyperbola in (s,r) space.  It is asymptotic to the vertical axis and to  the horizontal 
through r = -1;  it intersects the horizontal axis at s = a .. The shape of the demand curve 
depends on the production function, and determines the number of intersections.  Given 
diminishing marginal productivity of capital, the demand curve is also down-sloping so 
that multiple intersections are possible.  We now prove some properties of the supply and 
demand curves and give examples of specific production functions which give rise to (a) 
a unique equilibrium, and (b) multiple equilibria. 
Proposition 1: With neo-classical production functions, the demand curve always lies 
below the supply curve for sufficiently small s: and if there is no capital saturation (f’(k) 
> 0 for all k),  there is at least one intersection. 
Proof : If, as k → 0, the limiting elasticity of capital-labour substitution is less than one,  
r converges to a finite ceiling
‡, so that the demand curve intercepts the vertical axis at a 
finite r.  In this case clearly the demand curve lies below the supply curve (which is 
asymptotic to the vertical axis) for sufficiently small k (or s).  If the elasticity of 
                                                 
‡ For proof see Guha (1963). 
  3substitution in the limit exceeds one, as k → 0, the relative share of capital in total output 
goes to zero while output tends to a finite limit
§; if the elasticity equals one, the relative 
capital share remains constant while output goes to zero.  In either event, the absolute 
share of capital sinks to zero  This implies that the area below the demand curve would 
go to zero, while that below the supply curve goes to a positive limit, a,  as r → ∞.  Thus, 
the demand curve must lie below the supply curve as r → ∞(or as k → 0) whatever the 
elasticity of substitution..  Now suppose there is no capital saturation, so that the demand 
curve never intercepts the horizontal axis.  In this case, clearly the demand curve lies 
above the supply curve for large enough values of k, as the supply curve intersects the 
horizontal axis at k =  a.  Since the demand curve lies below the supply curve for small k, 
and above it for large k, then given the continuity of both curves, there must be at least 
one intersection. Q.E.D. 
 
Corollary: If there is no capital saturation, there is an odd number of intersections : 
capital saturation is necessary for an even number of intersections. 
  
However, not every intersection between the supply and demand curves 
constitutes an equilibrium in which the economy is sure to survive unfavorable 
production shocks.  To insure the economy against vanishing due to lack of savings,  
capital stock in equilibrium must be such that 
* () f ka ε < −        (using (7))                                           (8) 
Thus equilibria where the capital stock is not large enough to satisfy (8) are not “safe”: 
there is some danger of the economy disappearing. 
   
Examples 
1. Consider the standard Cobb-Douglas production function f(k) = k
α .  Using (7), in 
equilibrium,  
kk a
α α + =                                                     (9) 
While the right hand side of (9) is invariant with k, the left hand side is monotonically 
increasing : its derivative being 
2( 1 ) 1 k
α α
−− +  > 1 > 0. 
Therefore, only one k satisfies (9).  Moreover, there is no capital saturation : 
only as k .  Therefore, by Proposition 1, a unique equilibrium 
exists (Figure 1).  Consider parameter values 
'( 1 ) () 0 fk k
α α
−− =→ →∞
α  = 1/3, a = 7/24, ε  = 1/8. Then the 
equilibrium k satisfying (9) is k = 1/8.  Moreover, (8) is satisfied, as its left hand side is 
1/8 while its right hand side is 5/24 > 1/8. Thus in this equilibrium, there is no danger of 
not being able to save enough to meet the target. 
 
2. Consider the piecewise quadratic production function 
f(k) = ak – bk
2 for k ≤ a/2b 
= a
2/4b for k ≥ a/2b. 
This production function displays capital saturation, a necessary condition for an even 
number of intersections.  Here the demand curve is linear with a horizontal intercept at 
                                                 
§ Op.cit. 
  4a/2b and a vertical intercept at a (Figure 2).  Since the horizontal intercept of the supply 
curve is a, we see that if a/2b <  a, then provided the demand curve crosses the supply 
curve at least once, it will do so an even number of times – specifically, twice for this 
production function.  Then, again using (7), any equilibria must satisfy 
2 2 ka k b k a + −=  
or 
2 2( 1 ) bk a k a 0 − ++ =                                    (10) 
The solutions to (10) are given by 
2
12
1( 1 ) 8
,
4
aa
kk
b
+± + −
=
b a
                                      (11) 
These roots are both real provided 
2 (1 ) 8 ab +> a                                                       (12) 
Moreover, for any a, b > 0, the sum and product of roots is positive, so that both roots are 
positive.  For both roots to be less than the saturation level of capital (so that the interest 
rate is non-negative), a sufficient condition is that the larger root be less than a/2b. This 
reduces to 
2
a
a
b
<                                                           (13) 
– the condition mentioned in the text for the horizontal intercept of the demand curve to 
be less than that of the supply curve (permitting two crossings). Consider parameter 
values of a = 8, b = 1, a=7.5,  *. 0 8 ε = .  Then we have  1 3.4 k = ,  .  These 
parameters satisfy both (12) and (13), which is why we have two real positive roots.  
Both roots are less than the value at which capital saturation sets in, which is 4 for these 
parameter values.  Output at the first of these equilibria is 
2 1.1 k =
1 ( ) 15.6 fk =  while output in 
the second equilibrium is  2 ( ) 7.59 fk = .  Moreover, (8) is satisfied at both these outputs 
so that even in the most unfavorable case, there is no danger of the economy 
disappearing.  However, these two equilibria are clearly Pareto-rankable, as output in the 
first exceeds that in the second.   
 
More generally, the equilibrium with the larger capital stock will involve larger 
output as marginal productivity of capital is positive (marginal productivity is equated to 
the prenegotiated interest rate, which can never be negative – or else people would prefer 
to store their savings instead of investing them in the production technology).  Because 
savings are negatively interest-elastic, the equilibrium with the higher capital stock and 
higher output is associated with a low interest rate, while the other equilibrium is a high 
interest-low output equilibrium. The former type of equilibrium appears to describe  the 
Japanese economy, where high savings have persistently coexisted with very low interest 
rates, though the levels of output and employment have been high.  If indeed much of the 
population comprises target savers, a low-interest equilibrium is better, in terms of output 
than one with a higher interest rate – accordingly, attempts to move from one to the other 
may be misguided. 
 
  It is also possible to have multiple equilibria where the equilibrium with the 
smaller capital stock yields an output too low to satisfy (8).  In this case, although the low 
  5output-high interest equilibrium can be maintained, given a run of good luck, a strong 
adverse shock can destroy the economy.  In contrast, the high-output low-interest 
equilibrium could have been robust to strong unfavorable shocks.  Therefore, if strong 
shocks are possible, it is even more desirable to co-ordinate on the high-capital stock 
equilibrium than on the Pareto inferior one.  Whether government or any other institution 
can play such a co-ordinating role, aiding equilibrium selection, remains an open 
question. 
 
 
4. Conclusion 
 
  We have analyzed an overlapping generations model of “target savers”.  With 
rational expectations, the older generation can insure themselves against all production 
shocks by setting a pre-negotiated interest rate on the savings they loan to the productive 
younger generation.  The model exhibits excess sensitivity of consumption to transitory 
income.  We have identified some conditions governing the existence of equilibrium, and 
the number of equilibria.  We have given examples of a unique equilibrium and have also 
shown that multiple Pareto-rankable equilibria are possible.  In the latter case we may 
have high-interest low-saving equilibria as well as low-interest high-saving equilibria : 
the latter seem to describe the situation of the Japanese economy. Co-ordinating on the 
Pareto superior equilibrium is even more important if the Pareto-inferior one is such that 
a particularly unfavorable shock may destroy the economy.  
 
References 
Chakrabarty, M and A. Schmalenbach (2002) : “ The Effect of Current Income on 
Aggregate Consumption”, Economic and Social Review, 33: 297-313. 
Diamond, Peter (1965) : “ National Debt in a Neoclassical Growth Model”, American 
Economic Review, 55: 1126-1150. 
Guha, Ashok S(1963) : “ Scarcity of Specific Resources as a Limit to Output”, Review of 
Economic Studies, 30: 37-42. 
Lavi, Yaacov (2003) : “Do Changes in Current Income Help to Explain Changes in 
Consumption in Israel?”, Israel Economic Review, 1: 113-135. 
Rao, Bhaskara (2005) : “Testing Hall’s Permanent Income Hypothesis for a Developing 
Country : the Case of Fiji”, Applied Economics Letters, 12: 245-248. 
 
 
 
 
  6S
D
k,s
r
r*
k*
a
Figure 1
-1k,s
r
k1 k2
S
D
r1
r2
a
2
a
b
a
Figure 2

Ashok S(1963) : “ Scarcity of Specific Resources as a Limit to Output”,

Do Changes in Current Income Help to Explain Changes

National Debt in a Neoclassical Growth Model”,

Testing Hall’s Permanent Income Hypothesis for a Developing Country : the Case of Fiji”,

Future Targets and Multiple Equilibria

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