Bounded Rationality and Lifecycle Consumption
Hyeon Park and James Feigenbaumy
October 2015
Abstract
This paper develops a lifecycle model of bounded rationality for those
consumers who foresee the future only to a degree and optimize periodby-period via maximizing their utilities over only a subset of their entire
lives. It then explores the general equilibrium characteristics of the model,
focusing in particular on whether the model can produce a hump-shaped
lifecycle consumption pro…le similar to the existing data on the average
U.S. consumer. Using an analytic solution, this paper shows that an
increasing income pro…le along with exogenously imposed retirement is
su¢ cient to induce a hump for a simple model. Then, with no mechanism that can account for a hump other than bounded rationality, the
model produces a consumption hump with a location and magnitude that
are consistent with the data in a well-calibrated general equilibrium. Unlike a partial equilibrium, wherein matching the data is trivial given any
parameter set, there exists, in general equilibrium, a stylized relationship
among the parameters for model calibration and standard macroeconomic
targets. Moreover, this paper demonstrates that a model with a planning
horizon of approximately 20 years provides the best …t for the salient features of the consumption data, which is well-supported by the behavioral
evidence found in surveys on retirement planning. Finally, this paper incorporates social security and performs a welfare analysis, con…rming that
the model is consistent with the data in an even more realistic environment
that includes social security and mortality risk.
JEL Classi…cation: E21, D91
Keywords: bounded rationality, re-optimization, lifecycle model, general equilibrium, consumption hump, social security
Contact: Hyeon Park, Email: [email protected], Department of Economics
and Finance, Manhattan College. 4513 Manhattan College Pkwy., Riverdale, NY 10471. Tel:
(718) 862–7462.
y Contact: James Feigenbaum, Email: [email protected], Department of Economics and Finance, Utah State University, Logan, UT 84322 Tel: (435) 797–2316.
1
1
Introduction
A well-known feature of lifecycle consumption data is that mean consumption increases while the consumer is young, peaks while the consumer reaches middle
age, and then decreases afterward. This prominent characteristic of the consumption pro…le is known as the consumption hump.1 The consumption hump
has been the central interest of many researchers who study lifecycle consumption because it cannot be predicted by standard economic theory. The standard
lifecycle model, in which the agent is fully rational, predicts a monotone consumption over time if the model does not assume any friction (such as borrowing
constraints) or uncertainty (such as stochastic income processes). This prediction has a strong theoretical implication in lifecycle models because monotonicity is achieved regardless of the functional speci…cation for periodic utility as
long as it satis…es strict concavity.2 In the standard model, optimal consumption is expected to be monotonically increasing, decreasing, or constant over the
agent’s life. Therefore, there is no way to break the monotonicity and achieve
the consumption hump.3 In fact, many explanations that exhibit full rationality in the standard model have di¢ culty …tting the data in general equilibrium,
especially when social security is included. In this study, we contribute to the
literature by examining a model that solves the lifecycle consumption puzzle in
general equilibrium, even with social security taken into account.
The discrepancy between the standard model’s prediction and the existing
data on the consumption hump has not gone unnoticed. Among researchers
who explore the consumption hump, there has recently been a growing level of
interest in explaining the consumer’s optimization behavior based on bounded
rationality, deviating from the traditional assumption of full rationality without
breaking the general optimization rule. In bounded rationality, agents experience
limits in formulating and solving complex problems. The term also implies that
agents experience limits in processing information, i.e. in receiving, storing,
retrieving, and transmitting it.4 Therefore, bounded rationality indicates that
the agent’s decision-making may be incompletely based on rationality if the
agent either does not have full information regarding his options or if costs
exist, physical or mental, in relation to the decision. The incompleteness of
decision-making in bounded rationality models is thought to add other possible
solutions to the lifecycle consumption puzzle.
In this paper, however, we …nd that a bounded rationality model in which
1 The earliest hump shown in the literature is from Thurow (1969): it is shown that both the
lifecycle consumption and income streams are hump shaped, but the age of the consumption
peak comes slightly earlier than the age of the income peak, which is similar to the data
estimated Gourinchas and Parker (2002).
2 When the periodic utility is strictly increasing and strictly concave, the Euler equation
of the standard model indicates that u0 (ct ) = Ru0 (ct+1 ): Because marginal utility is strictly
decreasing, it must be satis…ed that ct < ct+1 or ct > ct+1 or ct = ct+1 ; whenever R > 1
or R < 1 or R = 1.
3 This is often called the “consumption puzzle.”
4 The term is attributable to Herbert Simon. The description here is from the citation in
Williamson, O., (1988), “The Economies of Organization: The Transaction Cost Approach.”
2
decision-making adapts to changes in the economic environment, such as new
income information, can solve the discrepancy between the lifecycle consumption
prediction and its data by generating a consumption hump in a well-calibrated
general equilibrium. Speci…cally, we …nd that, in a short-term planning horizon
model wherein consumers optimize period-by-period through maximizing their
utilities over only a subset of their entire lives, the general equilibrium result
can be consistent with the known characteristics of the lifecycle consumption
data. In fact, the quantitative result of the simulation exercise shows that the
model with a planning horizon of approximately 20 years provides the best
…t for the consumption pro…le supplied by U.S. data. In this paper, we will
demonstrate that my results in the calibrated general equilibrium are robust to
alternative sets of parameters within a reasonable range. The implications of
the general equilibrium results are important because, in partial equilibrium,
matching the data is trivial given any parameter set. One can match the data
simply by changing consumer parameters, such as the discount factor and risk
aversion coe¢ cient, for any target variable, such as the interest rate. In a
general equilibrium, however, the interest rate and therefore the ratio of the
interest rate to the wage rate are no longer free values. Because both the
labor market and the bond market must clear simultaneously, the equilibrium
condition of one market directly a¤ects the other. Therefore, many partial
equilibrium results may not be supported by general equilibrium.5 Moreover,
because of the interdependence of the macroeconomic variables, the calibration
result may be able to be interpreted more properly in a general equilibrium
model.
This paper also explores the best mechanism for simulating the consumption hump within a more realistic model economy by incorporating PAYG social
security into the baseline model. We …nd that the model that includes social
security preserves the study’s main result, with a planning horizon of approximately 20 years still providing the best …t to the data. This is true even within
the extended model that incorporates social security and mortality risk. An
important issue related to social security is welfare direction in relation to timeinconsistent preferences, i.e. whether or not social security can improve the
welfare of agents who exhibit myopic or hyperbolic discounting. In fact, many
studies6 have found that welfare improvement is not supported for these agents.
However, when agents have irrational or boundedly rational expectations about
their future incomes, this paper suggests that there is a possibility that social
security does improve welfare. In fact, the short-term horizon mechanism can
explain why the present social security system of the U.S. is optimal, something
that other explanations for the hump cannot do. In this paper, furthermore,
we quantify the welfare gain under social security for the bounded rationality
model; this is a model that is closely related with time-inconsistent preferences,
though it is clearly di¤erent from these in terms of expectations regarding future
5 In the case of the consumption hump, all of the various mechanisms proposed to explain this work in partial equilibrium when the interest rate and preferences are calibrated
independently. But very few of these mechanisms work in a general equilibrium.
6 This will be discussed in Section 5.
3
income realization.7
We set up the model with the assumption that the agent is not fully rational
and foresees the future only to a degree. Because of this, the agent plans for
only a fraction of his life; the planning horizon is shorter than the usual lifecycle
length. It is inferred that, the shorter the planning horizon, the greater the
degree of bounded rationality. Because the agent does not foresee perfectly, he
needs to re-optimize as further information or new resolutions regarding future
plans are revealed over time. One may imagine that the agent will want to
reset his consumption plan as new information about future income becomes
available.8 By the mechanism of re-optimization, it is shown that actual consumption is a series involving the initial consumption of each planned path,
being an envelope. Moreover, the realized consumption is more closely tied
with income stream through this rebalancing. The standard lifecycle model
with perfect foresight is a special case of this, one in which the planning horizon
is exactly equal to the lifecycle length. Likewise, the hand-to-mouth consumer
is another special case, one with a planning horizon of zero.
We consider a general equilibrium model with (T + 1)-period overlapping
generations and (T + 1) types of identical cohorts in a stationary economy.
To calibrate the model, we propose the three standard macroeconomic targets.
Also, the model has four standard parameters. The risk aversion coe¢ cient
and the discount factor are the parameters related to the consumer, and the
capital share of the production function and depreciation rate are the parameters related to the producer. Within a model of bounded rationality, however,
there is one …nal parameter to consider because the model assumes a shorter
than full-term planning horizon: the planning horizon parameter. Among these
…ve parameters, the four scalar parameters are jointly set to match the targets,
taking the planning horizon as given, in order to see how closely the general
equilibrium model replicates the data as the planning horizon changes. After
this has been calculated, we then look for the best planning horizon for allowing the consumption pro…le of the model to …t the mean consumption pro…le
estimated by Gourinchas and Parker (2002).
1.1
Related Literature
There are many related papers in both the bounded rationality and the standard full rationality frameworks. Regarding bounded rationality, the hyperbolic
discounting model9 of Laibson (1997) should be addressed. This model is based
on evidence that people tend to value immediate utilities di¤erently from future
ones. That is, delayed outcomes are heavily devalued compared to immediate
ones, especially with respect to discounting. Unlike standard exponential discounting, the hyperbolic discounting on which Laibson’s model is formulated
captures this type of time inconsistency. Laibson posits the dynamically incon7I
call this time inconsistent expectations.
this may be the simplest logic behind re-optimization, the bounded rationality
model admits other psychological reasoning as well.
9 Quasi-hyperbolic discounting, precisely; also
model or present-biased preference.
8 Although
4
sistent agent at a certain time as a unique self and sets a T -period consumption
and saving problem in a T -period dynamic game among T -type selves facing
asset constraints. He …nds a unique subgame perfect equilibrium (SPE) strategy
under certain assumptions. The empirical implication of his work is that the
existence of a commitment device such as illiquid assets plays a role that can
produce a consumption pro…le that tracks the income ‡ows. That is, consumption and income can co-move.
A similar result from a di¤erent mechanism can be found in Park (2014).
Park analyzes the behavioral implications of myopic consumers who have a
time-inconsistent taste for immediate grati…cation. Through an analytic solution, Park shows the possibility of a consumption hump without resorting to
any other constraint when the myopic agents re-optimize, negating an earlier
resolution for current consumption. The key mechanism in the paper inducing
the consumption hump is time preferences,10 which incorporate with the interest rate through re-optimization. This implies that a high interest rate may be
required to induce a hump, and thus the hump may not be supported under certain parameter values in general equilibrium. If there is a borrowing constraint,
this will generally reinforce the hump mechanism. The current paper’s bounded
rationality (or short-term planning) model is di¤erent in that it does not rely
on time-inconsistent tastes, while it does rely on time-inconsistent expectations
regarding future income. Therefore, the income stream induces a hump in the
bounded rationality model but not in time-inconsistent taste models.
The short-term planning approach of Caliendo and Aadland (2007) belongs
to this type of time-inconsistent preference (in terms of expectations, to be
precise) as well. In fact, a short planning horizon involves a special case of
time-inconsistent discounting as explained in Feigenbaum (2014). The partial
equilibrium model of Caliendo and Aadland (2007) aims to explain behavioral
phenomena noted in several surveys regarding retirement plans among workers.
They use a continuous time-control model to explain why many people do not
adequately prepare for retirement. Because many workers are not far-sighted,
they do not think seriously about retirement until it is imminent and in view.
Related to the time inconsistency, but taking a di¤erent perspective, is procrastination. As argued in O’Donoghue and Rabin (1999), the self of an agent
may be divided into the naive and the sophisticated selves, who have beliefs
about future selves. Only the sophisticated agent has correct beliefs and does
not procrastinate. This model implies that when there are many errors in retirement planning for the majority of workers due to present biased preferences,
policies with cautious paternalism may help resolve this problem. Thus, a policy
such as tax incentives designed to increase savings may increase the cost of procrastinating regarding saving and can boost saving. Also, infrequent transaction
dates incur great costs for procrastinators but small costs for time-consistent
selves. Another line of bounded rationality is the deviation from full information,11 known as inattentiveness. Based on information friction, this approach
1 0 In
the model of Park (2014), this implies
Sims (2003) and Moscarini (2004).
and :
1 1 See
5
explicitly assumes costs related to information; due to these costs, planning
takes place infrequently. Reis (2006) proposes a partial-equilibrium model with
an inattentive consumer who incurs costs acquiring, absorbing, and processing
information to form expectations and make decisions. His result shows that
in a partial equilibrium with …xed interest rates, inattentive consumers face
more uncertainty and save more for precautionary reasons.12 Other than these,
habit formations (Fuhrer, 2000), dual-self (Levine and Fudenberg, 2006), rational inattentiveness (Luo, 2005), and overcon…dence (Caliendo and Huang,
2008) are considered for bounded rationality as well as near-rationality (Caballero, 1995).
Regarding the lifecycle consumption pro…le implied by the standard assumption of a fully rational agent, several studies should be mentioned. Borrowing
constraints, mortality risk, consumption and leisure substitutability, income uncertainty, and precautionary savings are the main issues because they can induce
a consumption hump. First, the relative importance of precautionary savings
related to the borrowing frictions of the model is well studied in the general
equilibrium model of Feigenbaum (2009). Feigenbaum shows that, along with
the consumption hump, in a general equilibrium lifecycle model with an exogenous borrowing constraint, observable macroeconomic variables are insensitive
to simultaneous changes in the discount factor and risk aversion coe¢ cient that
preserve the equilibrium interest rate. This calibration implication shares a
common feature with my work in that the two parameters are not de…ned in
terms of each other. His work also demonstrates that the unobservable fraction
of aggregate savings due to precautionary motives increases with consumers’
risk aversion, and the e¤ect of the parameter on observable macro variables
di¤ers depending on the assumption about borrowing constraints in each proposed model. When we turn to the other studies that induce a hump, we
address the following: (1) family-size e¤ect (Attanasio et al., 1999), (2) borrowing constraints (Deaton, 1991), (3) mortality risk (Feigenbaum, 2008a; Hansen
and I·mrohoro¼
glu, 2008), (4) choice between consumption and leisure (Heckman,
1974; Bullard and Feigenbaum, 2007), (5) income uncertainty and precautionary
savings (Hubbard et al., 1995; Carroll, 1997; Gourinchas and Parker, 2002; Aiyagari, 1994; Feigenbaum, 2008b), (6) consumer durables (Fernández-Villaverde
and Krueger, 2011), and (7) home production (Dotsey et al., 2014). A couple of
caveats need to be mentioned. Fernández-Villaverde and Krueger (2011) show
how the interaction between durable and nondurable consumption may work
to explain the hump in a model in which durable goods serve as collateral for
loans. Bullard and Feigenbaum’s (2007) calibration work with choice between
consumption and leisure as well as Heckman’s (1974) model produce similar results. Mortality risk (Feigenbaum, 2008a; Hansen and I·mrohoro¼
glu, 2008) also
needs to be noted as an explanation for the hump.
1 2 One can easily show that inattentiveness induces a consumption hump in a lifecycle model
through a mechanism similar to that for precautionary saving in a standard rational agent
model, although Reis does not explicitly work on this topic.
6
2
An Analytic Model
Consider a boundedly rational agent who lives for four13 periods, t = 0; 1; 2;
3; but maximizes his periodic utility for only two periods, = fcurrent; nextg.
Assume that the agent has a nonnegative income stream of fy0 ; y1 ; y2 ; y3 g and
1
his utility is speci…ed by a CRRA utility form; i.e: u(c)14 = c1 , where
denotes the degree of risk aversion or the inverse of the intertemporal elasticity
of substitution. The agent discounts future utility by : Also, assume that
there is no borrowing constraint and that the agent can borrow or lend freely at
the market interest rate R: Then, the optimization problem of the boundedly
rational agent at t = 0 is
c10
1
U0 (c0 ; c1 ) = M axfc0 ; c1 g
c11
1
+
(1)
subject to
c0 + b1 = y0
c1 = y1 + Rb1
where b1 is the bond holdings for the next period. Solving the maximization
problem yields the optimal consumption pro…le for = f0; 1g; which is
c0 =
c1 =
where15
1
=
( R)1=
R
to consume c1 =
R
R
y0 +
1+
y1
R
1
y0 +
1+
y1
R
1
(2)
!
(3)
y
: Thus, he consumes c0 =
y0 + R1
1+ 1
at t = 0 and intends
y
y0 + R1
1+ 1
the next period. At t = 1; however, the agent
now foresees his future income for the next period t = 2 and realizes that the
consumption he planned for the period is no longer optimal. Thus, instead of
y
y0 + R1
, he wants to adjust his consumption according to
consuming c1 = R 1+
1
this future income realization. Therefore at t = 1; the boundedly rational agent
again solves the following maximization problem:
U1 (c1 ; c2 ) = M axfc1 ; c2 g
c11
1
+
c12
1
(4)
subject to
c1 + b2 = y1
1 3 Four
is the smallest period to generate a consumption hump for a short-term planner.
utility is de…ned by u(c) = ln(c) when = 1:
15
= ( R) 1= R: This parameter was originally introduced by Feigenbaum (2005). A
value of contains a combined e¤ect from three parameters, i.e., , R, and :
1 4 CRRA
7
c2 = y2 + Rb2
where y1 = y1 +Rb1 .16 The solution to this problem is the optimal consumption
pro…le for = f1; 2g; starting from t = 1: Notice that the optimal consumption from this maximization is di¤erent from the agent’s calculation from last
period. To emphasize the di¤erence between the new consumption and the
planned consumption from last period, it is better to call the new consumption c11 by denoting the adjustment for the new consumption plan starting from
t = 1: Likewise, c12 is the planned consumption for the next period, i.e., t = 2;
calculated from t = 1: Thus,
y + y2
(5)
c11 = 1 1R
1+
!
y
R y1 + R2
1
c2 =
:
(6)
1+ 1
It is noticeable that the adjusted consumption c11 is di¤erent from the planned
one c01 in that the new consumption is a function of future income realization y2 :
Similarly, the optimal consumption pro…le for = f2; 3g; starting from t = 2;
is
y + y3
(7)
c22 = 2 1R
1+
!
y
R y2 + R3
2
c3 =
:
(8)
1+ 1
Altogether, the adjusted consumption pro…le for the entire four periods, t =
0; 1; 2; 3, is
(
!)
y0 + yR1
y1 + yR2
y2 + yR3 R y2 + yR3
1 2 2
fc0 ; c1 ; c2 ; c3 g =
;
;
;
: (9)
1+ 1
1+ 1
1+ 1
1+ 1
The last term in the bracket follows from the fact that the agent no longer
needs to adjust his consumption in the last period because there is no more
income process to be realized. Solving recursively for the bond demand of each
period and substituting into yt = yt + Rbt for t = 0; 1; 2 pins down the entire
consumption pro…le. The bond demand is
1
b1 =
R
b2 =
2
y0 +
y0
1+
1
1
R y1
1
2
Ry1
R(1 + 1 )2
(10)
1
+ 1 y2
:
(11)
1 6 In fact, this term corresponds to the ‘cash on hand’ by Deaton (1990), but here, it is
indexed by t = 1:
8
Then the realized consumption fc0 ; c1 ; c2 ; c3 g
fc00 ; c11 ; c22 ; c23 g is obtained
from the above consumption equations by substituting the bond demands. The
consumption is17
(Ry0 + y1 )
(12)
c0 =
R 1+ 1
R
c1 =
(Ry0 + y1 ) + (1 + 1 )y2
R 1+
R
2
(Ry0 + y1 ) +
R
1
(1 + 1 )y2 + 1 +
c2 =
R 1+
c3 =
0
RB
@
R
2
(Ry0 + y1 ) +
R
1
2
y3
(14)
1
1
1
3
(1 + )y2 + 1 +
R 1+
(13)
2
3
1
2
1
y3 C
A:
(15)
It is worthwhile to see how consumption is related to income. Rewrite the
above to obtain
fc0 ; c1 ; c2 ; c3 gBoundedly Rational =
c0 ;
R
1+
where c0 =
c0 +
Ry0 +y1
:
R(1+ 1 )
1+
y2
;
R
R
1+
c1 +
1+
y3
;
R
R
c2
Notice that consumption at each period, except for the
initial and the last, is directly related to income for the next period together
with previous consumption. In fact, this is the core property of the bounded
rationality model: re-optimization ties consumption more closely to income.
To further characterize the consumption property of the boundedly rational
agent, …rst look for the standard lifecycle consumption pro…le of a fully rational
agent who is looking forward up to T . The standard optimization predicts
that the marginal utility of consumption between any two periods conforms
to the Euler rule, and thus the consumption pro…le over all periods exhibits
monotonic movement. By monotonicity, the consumption pro…le of the rational
agent is increasing, decreasing or constant over the entire lifetime. Thus, the
fully rational agent’s consumption
pro…le with T = 3 is characterized by
2
3
fc0 ; c0 ; c0 ; c0 g
if R = 1
1
2
3
5
fc0 ; c1 ; c2 ; c3 gRational = 4
c0 ; R c0 ; R c0 ; R c0
if R 6= 1
y1
y2
y3
y0 + R
1 + R2 + R3
: It is important to notice that the monotonic1 + 11 + 12 + 13
ity of these consumption pro…les is preserved across any choice of income processes.
where c0 =
1 7 One
c11 =
may …nd how realized consumption is related to intended consumption. For example,
R (Ry +y )+
0
1
R 1+
1
1+ 1 y2
2
=
1+
c01 +
y2
R
:
9
By comparing the consumption of the boundedly rational agent with this monotone
pro…le, it is easily seen that for the boundedly rational agent, this monotonicity
no longer holds except at the last stage.18 The breakage of monotonicity can be
obtained for a boundedly rational consumer even with a shorter lifetime period,
and a proof with a three-period model is presented in the Appendix. Another
property to consider is whether the consumption pro…le of the boundedly rational agent could generate a consumption hump. To analytically derive the
conditions for a hump, we want to …rst de…ne the consumption hump for any
T -period consumption pro…le in a strong sense:19
De…nition A consumption hump for a T-period model is a consumption
pro…le fct gTt=0 that satis…es the following:
i) There is a consumption peak at time t 2 (0; T ):
ii) Consumption is monotonically increasing up to t.
iii) Consumption is monotonically decreasing beyond t.
For example, if there are four periods t = 0; 1; 2; 3, the hump condition
requires either fc0 < c1 < c2 > c3 g or fc0 < c1 > c2 > c3 g depending on
whether the peak occurs at c1 or c2 : Can the hump be obtained with the fourperiod model solved above? To explore this question, assume that the gross
interest rate is greater than zero, i.e., 1 + r = R > 0, and that the net interest
rate satis…es > r:20 Furthermore, assume that > 0 to ensure CRRA:
Proposition 1 If the boundedly rational agent’s income stream fy0 ; y1 ; y2 ; y3 g
(1 R + 1 )
(Ry0 + y1 ); then the optimal consumption is increasing
satis…es y2 >
1+ 1
initially, i.e., c0 < c1 , regardless of the agent’s choice of time preference. For the
agent with > 1=R; the increasing property is achieved with a weaker condition
1
for y2 >
1+ 1
(Ry0 + y1 ):
Proof. First, consider the case of = 1=R. Because R = 1; the condition
reduces to y2 + yR2 > y0 + yR1 ; and the consumption of t = 0 and t = 1 is
y
(Ry0 +y1 )+(y2 + R2 )
0 +y1
: Thus, if the condition is satis…ed, then
c0 = RyR+1
and c1 =
1
(R+1)(1+ R )
y2
y1
(y2 + R ) + (Ry0 + y1 ) > (y0 + R ) + (Ry0 + y1 ); the RHS of which is equal to
(1 + R1 )(Ry0 + y1 ): Therefore, c0 < c1 : Second, consider the case of > 1=R:
Because
R
> 1; it is satis…ed that y2 + y2 > 1 (Ry0 +y1 ) > 1
y1 ): Therefore, c0 < c1 : Third, consider the case of
it is satis…ed that either y2 +
y2
> 1
R
+
1
R
+
1
< 1=R: Because
(Ry0 + y1 ) >
1
(Ry0 +
R
< 1;
(Ry0 + y1 ) or
1 8 This comes from the assumption that the boundedly rational agent plans for two periods.
If he plans for three periods, then the monotonicity holds for the last two periods.
1 9 A weak sense would allow a wiggle over the horizon with several local peaks.
2 0 These assumptions are just for computational purposes. In fact, these assumptions are
not restrictive at all and are easily satis…ed in general.
10
1 R + 1 (Ry0 + y1 ) > y2 + y2 > 1 (Ry0 + y1 ). It is clear that only the …rst
induces c0 < c1 : Therefore, c0 < c1 regardless of :
To see the meaning of the condition, rewrite it to obtain
y2 +
y2
> 1
R
+
1
(Ry0 + y1 ):
(16)
If
= 1=R; then the condition yields y2 + yR2 > y0 + yR1 :21 In a line of roughly
hump-shaped income data, this inequality implies that if y2 is greater than y0
and if either y2 is as good as y1 or, when y2 is big enough relative to y0 ,
is not too small compared to y1 ; then the increasing property of consumption
at an early stage of life is obtained, i.e., c0 < c1 . Notice that this condition is
always satis…ed with an increasing income pro…le up to t = 2: But this condition
may not be satis…ed if y2 is much smaller than y1 : Now turn to the other two
cases. If < 1=R then because R= < 1; the condition says that y2 + y2 >
1 R + 1 (Ry0 + y1 ) > 1 (Ry0 + y1 ) > y0 + y1 : This inequality implies that,
given and R; the increasing property of consumption requires a higher income
level of y2 than that in the case where = 1=R: Conversely, if
> 1=R; then
the condition implies that given and R; the increasing property of consumption
can be achieved with lower y2 : Intuitively, when > 1=R; the initial savings is
greater, and this helps to keep consumption growing even with a lower income
level at later periods. Therefore, given any combination of and R; the desired
>1=R
level of y2 for the increasing consumption pro…le has the ordering of y2
<
=1=R
<1=R
y2
< y2
while keeping both y0 and y1 the same for all three cases.
Proposition 2 If the boundedly rational agent’s income stream fy0 ; y1 ; y2 ; y3 g
satis…es that there is at least one period with non-zero income except for the last
period, and the last income is su¢ ciently small, i.e., yT = y3 = "; then c1 > c2
regardless of the agent’s choice of time preference. The su¢ cient condition of
the last income is yT = y3 = 0.
Proof. From the consumption pro…le obtained above, the condition for c1 > c2
R
is 1
+
1
R
(Ry0 + y1 ) + 1 +
1
y2
> 1+
1
2
y3 : When
= 1=R;
2
the inequality condition reduces to R1 Ry0 + y1 + 1 + R1 y2 > 1 + R1 y3 :
Therefore, if the last period income y3 is su¢ ciently small so that y3 = " <
(1 + R1 ) 2 y0 + R1 y1 + R1 (1 + R1 )y2 and at least one income among fy0 ; y1 ; y2 g
is non-zero, then c1 > c2 : Because with a non-negative income stream, it is
always true that y0 + R1 y1 + R1 (1 + R1 )y2 > 0, thus " = 0 is the su¢ cient condition for c1 > c2 : If
< 1=R; it is true that r < R <
because r < : Likewise, if
> 1=R; it is true that r <
< R: In either
case, it is satis…ed that 1 R + 1 > 0: Therefore, if y3 is su¢ ciently
small, then c1 > c2 for all choices of : The su¢ cient condition is y3 = 0 <
1+
1
2
2 1 Note that
inequality.
1
R
+
1
R
(Ry0 + y1 ) + 1 +
1
y2 :
is no longer in the expression, implying that
11
does not play any role in this
It is easily seen that if the last period income is zero such that yT = y3 = 0;
or very small22 such as y3 = ", then the inequality holds for general classes of
parameter values. Therefore, if retirement is exogenously imposed, then this
condition is seldom violated. Finally, combining both conditions yields the
consumption hump.
Proposition 3 If the boundedly rational agent’s income stream fy0 ; y1 ; y2 ; y3 g
(1 R + 1 )
(Ry0 + y1 ) and (B): y3 = 0, then the
satis…es both (A): y2 >
1+ 1
consumption pro…le of the agent produces a hump regardless of the choice of
time preference of the agent.
Proof. For the agent with 6 1=R; (A) is a necessary and su¢ cient condition
for c0 < c1 and (B) is a su¢ cient condition for c1 > c2 : Similarly, for the
agent with > 1=R; (A) is a su¢ cient condition for c0 < c1 and (B) is also
a su¢ cient condition for c1 > c2 : Therefore, combining both conditions yields
c0 < c1 > c2 regardless of time preferences: Thus the consumption hump is
achieved for fc0 ; c1 ; c2 g:23
Example Suppose fy0 ; y1 ; y2 ; y3 g = f1; 3; 2; 0g:24 Then the consumption
pro…le of a boundedly rational agent who is forward looking only for two periods
Bounded Rational
and has 1=R for his time preference is characterized
=
n
o by fc0 ; c1 ; c2 ; c3 g
2
2
2
5+R+
5+R+
5+R+
3+R
R
R
R
1+R ; (1+R)(1+ 1 ) ; (1+R)(1+ 1 )2 ; (1+R)(1+ 1 )2 : Using the standard annual inR
R
R
terest rate Rann = 1:035 and per length
= 15 year, this yields fc0 ; c1 ; c2 ;
c3 gBounded Rational = f1:748; 1:842; 1:153; 1:153g: Clearly, c0 < c1 > c2 and a
hump is achieved.
The following two …gures explain these propositions. The second …gure
(B) replicates the example above, and the …rst …gure (A) is obtained with
fy0 ; y1 ; y2 ; y3 g = f1; 2; 3; 0g. The consumption of the last period is residual,
and because = 1=R, it is equal to the consumption of the previous period.
That is, c3 = c2 : In both …gures, it may be noticed that the consumption peak
comes no later than the income peak, which is related to the length of planning
horizon relative to the lifecycle horizon in the short term planning model. In
fact, the consumption and income data (Section 4) show that the age for the
consumption peak comes slightly earlier than that for the income peak. In this
four-period model, the overall period is too small to show this detailed characteristic. Also, one cannot miss the observation that the initial consumption is
higher than the initial income, which is also supported by the data. We revisit
these properties in the quantitative analysis with the full model.
2 2 The common application for the case of zero income in the last period is retirement and
the case of small income is social security.
2 3 The last consumption c follows the standard path starting from c , increasing, constant,
3
2
or decreasing depending on >=< 1=R and may not be the main interest of the analysis.
2 4 This income stream is the case where y is greater than y and y is not too small relative
2
0
2
to y1 . But with an increasing income pro…le f1; 2; 3; 0g; the hump is always achieved with
= 1=R:
12
13
3
A Lifecycle Model
In this section, the full model with a boundedly rational consumer is presented,
…rst in partial equilibrium and then in general equilibrium, by including technology in an overlapping generations economy.
3.1
Consumer
3.1.1
Environment
Time is discrete and denoted by : At each time, a generation of identical
cohorts is born. The population is constant over time and each agent who is
indexed by age t lives for T +1 periods in a (T +1)-period overlapping generations
economy. During working periods, agents are endowed with one unit of labor
productivity, measured in e¢ ciency units, which is supplied inelastically. There
is no borrowing constraint so that agents can borrow and lend freely under
market determined interest rate R. There is no government, and there exists a
single good, which can be either consumed or saved, in which case it is called
capital. There is no uncertainty in this model. However, the agents are not
fully rational to perfectly foresee all the way through their lifetime T . Instead,
they care and plan only up until S; which is sooner than T . Finally, retirement
occurs exogenously at t = Tw +1, where Tw < T .
3.1.2
Consumer Optimization
Let us …rst de…ne the consumption notation. A consumer who is in age t with
a planning horizon S has consumption denoted by cSs (t). The subscript s represents consumption time. The age t also represents planning time in the model.25
For convenience, the planning horizon is suppressed whenever the notation is
unambiguous. Then, the representative consumer who plans for S+1 periods at
each planning time maximizes for t = 0; 1; :::; T;
U (t) =
t+S
X
1
s t cs
s=t
(t)
1
(17)
subject to
cs (t) + bs+1 (t) = wes + Rbs (t)
b0 (0) = 0; bt (t) = given; bt+S+1 (t) = 026
where cs (t) is consumption planned at t for time s, and bs+1 (t) is bond
demand purchased at s for the next period, indexed by planning time t. The
consumer has a stream of productivity pro…le over his lifetime so that he supplies
es e¢ ciency units of labor at s into production and earns labor income of wes
each time, where w is the market determined real wage rate, which is assumed to
2 5 The
physical age is t + 25 if consumers start working when they are 25 years old.
implies that the agent plans to have no debt or savings by the end of a planning
term [t; t + S].
2 6 This
14
be stationary over time. Because the consumer can save or borrow freely under
market determined interest rates, he will earn …nancial income of Rbs (t) or incur
…nancial cost of Rbs (t) if he carries the bond to the next period. However, the
model does not restrain to less than one.27 This implies that a consumer with
a short-term planning horizon may have a higher evaluation of future than of
current consumption. Solving the consumer optimization problem implies that
for s = t; :::; t+S–1;
cs (t) = (R )cs+1 (t)
(18)
Let ct (t) be the optimal initial consumption starting from the planning date (or
age) t. Using the budget constraint, one can pin down ct (t), which is
Pt+S wes
s t + Rbt (t)
(19)
ct (t) = Ps=t R
h
is t
1=
S
s=t
(R )
R
where bt (t) is the initial bond holding at each planning date t and b0 (0) = 0: Let
us de…ne the total wealth at any time over the planning horizon [t; t + S] as
the sum of human wealth and …nancial wealth: W (t) = h (t) + Rb (t); where
Xt+S wes
h (t) =
: This produces a compact form for initial consumption at
s= Rs
each planning time t, i.e., ct (t), which is
Wt (t)
ct (t) = P
h is
t+S 1
(20)
t
s=t
1
where
= (R )1= R 1 : Therefore, for any time
[t; t + S], it is satis…ed that
W (t)
c (t) = P
h is
t+S 1
over the planning horizon
:
(21)
s=
3.1.3
Planned and Realized Consumption
Assume that the consumer has a productivity schedule over the working periods
that shows an inverse U shape. This agrees with common sense as well as
lifecycle income data. If a consumer at age or time t perfectly foresees only
S periods forward, then he maximizes his utility from time t to time t + S
following any income stream available for this planning horizon. The solution
to this maximization procedure is a vector of a consumption schedule for a
planning horizon S starting from t,
C P (t) = fct (t); ct+1 (t); ct+2 (t); :::; ct+S (t)g:
(22)
Let us call this planned consumption determined at t for the subsequent S periods. Initially, the consumer intends to follow the planned consumption stream
2 7 Because the model assumes a …nite time horizon, the usual restriction on
time horizon is not necessary.
15
for an in…nite
and consumes ct (t) at t. However, at t+1, he realizes that the planned consumption for t+1; i.e., ct+1 (t), is no longer optimal because a new income for
time (t+1)+S is in view. He has to re-plan to incorporate this and solves again
for the consumption stream starting from t+1. He keeps doing this as long as
new income comes into view: as long as there is a drift between the planned
consumption and the optimal consumption with a new planning horizon. Because each time, the consumer follows a planned path only at the initial time of
the planning horizon, the actual consumption will be an envelope of the entire
planned consumption path over the entire life. The realized consumption from
age t is
C P (t) = fct (t); ct+1 (t + 1); ct+2 (t + 2); :::; cT
S (T
S)g:
(23)
Therefore the lifecycle consumption pro…le of the representative consumer with
any planning horizon S is
fct gTt=0 = fcS0 (0); cS1 (1); cS2 (2); :::; cST (T )g28
(24)
where the consumption is indexed by a planning time, a consumption time, and
a planning horizon. Likewise, the pro…le of lifecycle asset demand is
bS0 (0)
fbt+1 gTt=01 = fbS1 (0); bS2 (1); :::; bST (T
1)g
(25)
and
= 0: Figure 1 shows a planned and realized consumption pro…le. The
realized consumption is the envelope of all of the planned consumption series.29
3.2
Technology and General Equilibrium
To explore a general equilibrium model, let us add production technology to the
economy. Assume that there is a continuum of identical perfectly competitive
…rms. Speci…cally, this model introduces the Cobb-Douglas production function F (K; N ) = K N 1
for the representative …rm. To de…ne a competitive
equilibrium for the model, …rst obtain the marginal productivity:
FK =
K
N
FN = (1
)
1
(26)
K
N
(27)
De…nition A competitive equilibrium in this economy is an allocation fct gTt=0 ,
a set of bond demands fbt+1 gTt=0 , an interest rate R; and a wage rate w such
that given R and w, the following are satis…ed:
T 1
solve the consumer’s problem.
i) fct gTt=0 and fbt+1 gt=0
ii) Factors are paid out of their marginal productivity:
2 8 The consumer follows a standard consumption path for the residual periods in the last
phase of life. Therefore, the consumption beyond time T S is the same as the planned path.
2 9 This result is obtained with S=10, =3, R=1.045, and =0.98.
16
Figure 1: Planned and Realized Consumption Pro…le for the model with S=10.
The realized consumption is the envelope of the planned consumption series.
w = FN and R 1 = FK
iii) The labor and bond markets clear:
PT
PT
K = t=0 bt and N = t=0 et
The market clearing condition in the last line speci…es that consumption
loans cancel out in the aggregate so that the excess demand for bonds should be
equal to the capital stock. Also, the aggregate labor supply that sums up over
all cohorts should be equal to the aggregate labor demand. By the equilibrium
1
K
K
and R 1 =
:
condition ii), it is obvious that w = (1
)
N
N
1
K
R 1+
Rewrite the last equation to obtain
=
: The capital-to-labor
N
demand ratio is written as a function of the interest rate and other parameters.
Rearrange the capital as a function of the interest rate to obtain
K(R) = N
R
1+
1
1
:
(28)
By the equilibrium condition iii), the market equilibrium condition to determine
R is
1
T
T
1 X
X
R 1+
et :
(29)
bt (R) =
t=0
t=0
17
Once the equilibrium interest rate R is obtained, the wage rate w is determined by
R 1+
:
(30)
w(R) = (1
)
4
Quantitative Analysis
The goal of this quantitative analysis is to assess how well a calibrated, general equilibrium model of bounded rationality can account for the stylized facts
regarding lifecycle consumption data. For the following simulation exercises,
we utilize the mean pro…le of lifecycle consumption and income estimated by
Gourinchas and Parker (2002). Because the equilibrium series of consumption,
income, bond demand, and labor supply in the model of an overlapping generation economy can be interpreted as economy-wide cohort averages, the mean
pro…les should work well. To quantify the lifecycle model, we set the period of
the model to a year. The agents are born to be 25 years old. The economy is
stationary and there is no population growth. Let the agents live for sure from
t = 0 to T = 55. This corresponds to a physical life from 25 to 80 years old.
The agents work until 65 years old and because there is no other income sources
than earnings from labor, their income is zero after retirement.30
4.1
Targets in US Data
We propose three targets in standard macroeconomic variables representing US
data: interest rate (R), capital-output ratio (K=Y ), and consumption-output
ratio (C=Y ). Following Rios-Rull (1996), we set 2.9431 as a target value for the
capital-output ratio and 0.748 for the target ratio of consumption to output.
The third macroeconomic target is the real interest rate, which is determined
by the equilibrium condition of the model. Following McGrattan and Prescott
(2000), we set the rate at 3.5%. Regarding lifecycle consumption data that the
model aims to accomplish, we use the mean consumption in Gourinchas and
Parker’s estimation mentioned above. Feigenbaum (2008a) interpolates their
estimation into a septic polynomial function of age:
cGP
t
= 1:062588 + 0:015871t 0:00184t2 + 0:000109t3 + 0:00000413t4(31)
0:00000056t5 + 0:0000000163t6 0:0000000001475t7
According to this pro…le, the consumption peak occurs at 45 years old and the
ratio of peak consumption to initial consumption is 1.1476. These values, as
well as the mean squared error between the data and the model, are used to
assess di¤erent consumption pro…les produced by models with di¤erent planning
3 0 Thus
Tw = 40 and et = 0 if t > 40:
authors suggest 2.5 for this ratio. When I simulate the model with this target
ratio, I …nd that the main result of the model is not altered in terms of the best planning
horizon. The alternative target ratio produces a very similar consumption pro…le for each ;
but with a slightly lower than the baseline model.
3 1 Other
18
Figure 2: Consumption and Productivity pro…le adapted from US consumption
and income data estimated by Gourinchas and Parker (2002). With an inelastic
labor supply, the mean income schedule serves as a productivity pro…le of the
representative consumer.
horizons. Also, regarding the lifecycle income schedule, Feigenbaum (2008a)
suggests a quadratic …t for the US data. Because labor is supplied inelastically
in the model, income is proportional to productivity. Therefore, Feigenbaum’s
quadratic …t to the income data of Gourinchas and Parker (2002) can serve for
the productivity pro…le as well. The pro…le is
et = 1 + 0:0181t + 0:000817t2
0:000051t3 + 0:000000536t4
(32)
According to this income stream, the peak occurs around 48 years old. Figure
2 shows both the consumption and income (productivity) pro…le from US data.
From the …gure, it is clear that both lifecycle consumption and income streams
are hump shaped, but the peak age of consumption comes slightly earlier than
the peak age of income.
4.2
Methodology
The model has four scalar parameters f ; ; ; g32 and one planning horizon
parameter fSg. Unlike other parameters, S allows only a small set of distinct
integers, from one to …fty-…ve, at most.33 Therefore, instead of calibrating all
…ve parameters together, we set the planning horizon as given and have the
3 2 The risk aversion coe¢ cient
and the discount factor are from the consumer optimization, and the capital share and the depreciation rate are from technology.
3 3 The maximum length S = 55 corresponds to a plan from 25 years old to 80 years old.
19
other four parameters jointly set to match the targets. [Table 1] summarizes
the description of parameters and targets.
[Table 1] Parameters and Targets
Variable
Description
Target
Risk Aversion
Free
Discount Factor
Free
Capital Share
Free
Depreciation Rate
Free
Planning Horizon
Given
S
R
Interest Rate
3:5 %
Capital-Output
2:94
K=Y
C=Y
Consumption-Output 0:748
Then, we evaluate the planning horizon itself in terms of the best model
…tted to the consumption data. For this purpose, we employ both quantitative
and qualitative comparisons. For the quantitative …t, we calculate the model’s
deviation from the consumption pro…le implied by the data using mean squared
error (MSE) and the least absolute deviation (LAD). For the qualitative …t, we
compute the consumption ratio: the ratio of the consumption at its maximum to
the consumption at the starting age. This is a common measure to quantify the
consumption hump, proposed by Hansen and I·mrohoro¼
glu (2006) and Bullard
and Feigenbaum (2008a). Another measure for the qualitative …t is the age of
the consumption peak.
4.3
Results
The simulation exercise reveals that the short-term planning model does produce
a consumption hump with the correct size and location for the consumption
peak in a well-calibrated general equilibrium. The location of the peak falls
between 40 and 54 years old for all planning horizons from 5 to 26, and the
location is reversely related to the length of the planning horizon. By this, it
is inferred that there is a planning horizon that generates the peak location
seen from the data. Likewise, there is a planning horizon that produces a
ratio of peak consumption to initial consumption closest to that in data. We
summarize the result by an approximately 20-year planning horizon. When
the planning horizon is approximately 18 to 22 years, the model …ts very well
with the consumption data by all means generally considered. Finally, the
robustness report con…rms the positive contribution of the short-term planning
general equilibrium model.
4.3.1
Simulation
Initially, let us set the planning horizon to 18. This number is chosen because
the general equilibrium consumption pro…le from an 18-year planning horizon
best …ts the consumption data in terms of the minimum deviation from the
20
objective. Figures 3 and 4 show the optimal lifecycle consumption and the
corresponding asset demand from the model with an 18-year planning horizon.
In the …gure, the simulated consumption C(t) shows a hump-shaped pro…le, in
the strong sense of the term de…ned earlier, together with an increasing section
at the tail. This is the residual,34 explained in Section 2, which comes from the
assumption that agents follow the planned path once they reach the …nal stage
of planning when death is in view.35
Figure 5 shows a series of consumption pro…les that follow a gradual increase
of the planning horizon from 10 years to 26 years.36 To compare them together,
all of the pro…les are generated by models with the same value for = 0:5.
This is because 0.5 belongs to the small range of that can produce a general
equilibrium with an acceptable range for the entire planning horizon from 10
to 55. In the case of a high ; the equilibrium requires a very high
if the
planning horizon is relatively short. In fact, among the few discrete values of
= f0; 0:5; 1; 1:5; 2; 2:5; 3; 3:5g;
= 0:5 is the maximum at which
the model generates a consumption pro…le that agrees with the targets without
invoking
> 1; for all of the planning horizons from 10 to 55. At = 1, the
general equilibrium is obtained with = 0:5967 for S = 10. Note that this is the
shortest planning horizon in this simulation. The next subsection shows that
becomes lower as the planning horizon grows longer. Thus, it follows that when
is higher than 0.5967, the discount factor exceeds one. Therefore, it may be
logical to set = 0.5 to compare all of the planning horizons.37 Needless to say,
a planning horizon admits many di¤erent combinations of and set to match
the targets.38
If is allowed to exceed one, then for any choice of , it is possible to obtain
an optimal lifecycle consumption pro…le for every planning horizon. This is
because
adjusts accordingly to any change in the risk aversion parameter,
keeping the model in equilibrium. From this, we highlight two interesting facts
with respect to lifecycle consumption. First, given a value for the risk aversion
parameter, as the planning horizon grows longer, the agents tend to consume
more initially because they are aware of the income increase in later years. Also,
the consumption peak arrives earlier but at a lower level.39 This means that the
lifecycle consumption pro…le looks ‡atter when the planning horizon is longer.
This agrees with the common notion that as agents see farther into the future,
they tend to plan for a longer period. Thus, smoother consumption becomes
3 4 Because
R > 1 in this …gure, the consumption is increasing during the residual years.
mortality risk (together with bequests) is introduced to the model, then the tail would
be smoother. But as described in the introduction, the main objective of the paper is to
analyze the short-term planning model to induce a hump without any other mechanism that
might account for it. Mortality risk is one such factor.
3 6 A planning horizon beyond this may not be interesting because consumption becomes
closer to the standard lifecycle model as the planning horizon increases.
3 7 This …nding is also supported by Gourinchas and Parker (2002), in which they estimate
= 0.5.
3 8 This comes from the fact that the parameters are under-identi…ed: in the model, a composition of the two parameters is calibrated, but not or , separately.
3 9 Again, this is because the agents are aware that they will have zero income in later years
after retirement.
3 5 If
21
Figure 3: Optimal Consumption Pro…le C(t) for the model with S=18, =1,
=0.986 and R=1.035. C(GP) represents the mean consumption pro…le from
Gourinchas and Parker (2002).
Figure 4: Optimal Bond Pro…le b(t) corresponding to the consumption pro…le
for the model with S=18, =1, =0.986 and R=1.035.
22
Figure 5: Optimal Consumption Pro…les for the models with di¤erent planning
horizons from 10 to 26. The farthest (darkest) one is for the shortest planning
horizon S=10, and the nearest (brightest) one is for the longest horizon S=26.
All models are set to = 0.5 given the three targets.
23
more likely.
Notice that the analysis addressing initial consumption and the consumption
peak are related to the attribute of the consumption-income data in Section
2. If we revisit them, (A) the initial consumption is higher than the initial
income and (B) the age of the consumption peak comes slightly earlier than
that of the income peak. If we compare the model’s behavior with the data
for the two terms, we …nd the following: for all models with S > 18,40 the
initial consumption is higher than the initial income. For all models with S <
23, the age of the consumption peak comes earlier than that of the income
peak.41 Therefore, the planning horizons between the two inequalities satisfy
both properties.
Another issue to discuss is the limiting behavior of the model. What would
happen to the general equilibrium if the planning horizon lengthens to become
the full length? Given a target interest rate value R = 1:035, the value of R
gets smaller as the planning horizon increases because for a …xed ; the value of
becomes smaller with a longer horizon. This implies that R may converge to
a value near one, so that in the extreme, one gets very a smooth consumption
pro…le similar to the one predicted by the standard theory. To see this, we
compute the value of R for the full horizon model, i.e., S = T = 55: It turns
out that R = 0.9956 when = 2 and R = 0.9989 when = 0:5: Both yield a
very smooth but slightly decreasing consumption pro…le because R < 1: This
implies that in the well-calibrated general equilibrium, the standard lifecycle
consumption pro…le with 6 2 is not constant if the full length is 55 years,
although it is nearly ‡at and monotonic.42 This suggests that if the model
assumes a life span longer than 55, then there is a planning horizon slightly
higher than 55 that yields R = 1. Both should generate a completely ‡at
consumption pro…le over the entire life.
4.3.2
and
Among the four scalar parameters, we …nd that the model is well calibrated
jointly with a set of three parameters. Those three parameters are ; ; and :
Among the three, represents a combined value of the two consumer parameters
f ; g because = ( R) 1= R: The joint set f ( ; ); ; g that minimizes the
deviation from the targets provides a unique value of for each planning horizon
along with the other two production parameters. Unlike ; the two production
parameters are found to be independent of the planning horizon S. This leaves
the model with two free parameters, and : Therefore, for a choice of , it
is reasonable to have set to match the target for each planning horizon. The
three-dimensional graph of the parameters f ; ; Sg can be observed in the
following. In the …gure, a value of is determined by the joint position of
4 0 This
result is obtained with K/Y=2.94. When I use K/Y=2.5, the horizon is S=17.
…ts exactly with a three-year di¤erence between the consumption peak (45 years old)
and the income peak (48 years old).
4 2 This implies that unless
is very high, completely ‡at consumption cannot be expected
with S = T = 55 years.
4 1 S-20
24
and S:
[Table 2] shows the optimal values for di¤erent over the major horizon
grids. Figure 6 extends the result to all values of 22 planning horizons from
S = 5 to S = 26:
[Table 2] Optimal Values over Planning Horizons
S = 5 S = 10 S = 15 S = 20 S = 25
0:5 1:0580 0:9944 0:9800 0:9742
0:9710
1
1:1586 1:0235 0:9940 0:9822
0:9759
1:5 1:2688 1:0517 1:0081 0:9903
0:9809
2
1:3894 1:0818 1:0225 0:9985
0:9858
2:5 1:5215 1:1128 1:0371 1:0068
0:9908
3
1:6661 1:1447 1:0519 1:0151
0:9957
Figure 6 demonstrates the following facts. First, is inversely related with
planning horizon S for all values of , and it is more so with a higher .
Second, when is relatively low,
does not change a lot over the planning
horizon. Third, with a longer planning horizon, values are relatively stable
for all choices of : Because both and belong to the reasonable range of
parameter values around the optimal planning horizon, i.e., S = 20; this result
may justify the claim that the general equilibrium model of short-term planning
would be a good approach to explain the data. Figure 6 shows the evolution of
for each planning horizon over all risk aversion parameters from 0.5 to 3.
If is restricted to be less than one, then for some planning horizons, not
every value of the designated is admissible. [Table 3] shows the results ad25
Figure 6: Optimal Beta Values for the models with di¤erent planning horizons
from 5 to 26. The darkest graph is from the lowest gamma, =0.5 and the
lightest one is from the highest gamma, =3.
dressing this fact. From Figure 7 and [Table 3], it is easy to note the following
summary facts. First, is positively related with for all choices of the planning horizon. Second, is inversely related with the planning horizon for any
: Third, for any planning horizon longer than 10, is good enough to be over
0.5 even with the restriction of < 1: Fourth, the longer the horizon is, the
wider the range of admissible for < 1.
[Table 3] Optimal < 1
S = 5 S = 10
0:01 0:9679 0:9667
0:05 0:9759 0:9689
0:1
0:9839 0:9718
0:15 0:9929 0:9746
0:2
0:9774
0:5
0:9944
1
1:2
2
2:5
3
3:42
Value over Planning
S = 15 S = 20
0:9665 0:9664
0:9676 0:9670
0:9689 0:9678
0:9703 0:9686
0:9717 0:9694
0:9800 0:9742
0:9940 0:9822
0:9996 0:9855
0:9985
0:9994
Horizon
S = 25
0:9663
0:9667
0:9672
0:9676
0:9681
0:9710
0:9759
0:9779
0:9858
0:9908
0:9957
0:9999
The logic behind the smaller range of with a shorter planning horizon is
this: when the planning horizon is very short, it is less likely to accumulate
enough wealth to be transformed into capital, clearing the bond market. Therefore, to compensate for this shortage and achieve equilibrium, it is necessary
26
Figure 7: Optimal Beta Values for the models with risk aversion parameters,
from 0.5 to 3. The darkest graph is for the shortest planning horizon S=5 and
the lightest one is for the longest horizon S=25.
to let the consumer have very low risk aversion or, in other words, very high
intertemporal substitution.43 This implies that the agent should yield enough
saving to be matched with the necessary capital in equilibrium. [Table 4] shows
the highest values of for some major grids in the planning horizon.
[Table 4] The highest value of when
1
Planning Horizon (S)
Risk Aversion ( )
5
(0;
0:189401]
10
(0;
0:596736]
15
(0;
1:214161]
20
(0;
2:010267]
25
(0;
3:424499]
*The second number in the brackets is the value when
4.3.3
=1
The Best Planning Horizon
Regarding the quantitative …t, both methods of MSE and LAD yield the same
result in terms of the smallest deviation. A planning horizon of 18 years best
…ts the consumption data by both methods: the errors are M SE = 0:005045
and LAD = 0:05308: Although the two quantitative measures do not produce
identical rankings over all of the planning horizons, they do unambiguously
produce the same ranking near the planning horizon of minimum deviation
(S = 18) for all values of between zero to three. One fact to mention is that
any planning horizon less than 10 is not considered to be a good …t in terms of
MSE or LAD. Figure 8 summarizes this.
4 3 With
CRRA, risk aversion is equal to the inverse of intertemporal substitution.
27
Figure 8: The Best Fitting Planning Horizon based on the minimum deviation
error from the baseline consumption (GP). The minimum is achieved at the
planning horizon of 18, and the MSE is approximately 0.005.
For the age of the consumption peak, S = 20 …ts better than any other
planning horizon to the data where the peak age is 45. Also, the peak age
increases constantly (proportionally) as the planning horizon becomes shorter,
but only up to the point of S = 11; beyond which it stalls at 54 and stays the
same. Finally, the data displays that the ratio of mean consumption at the peak
of the hump to the one at the beginning is 1.1476. In terms of this measure,
S = 22 …ts best. One remarkable feature is that the consumption ratio decreases
as the planning horizon gets longer, suggesting a smoother consumption pro…le
with a longer horizon.
4.3.4
Robustness
In this section, we check the sensitivity of the model to alternative calibrations of
the baseline parameters. Notice that is going to adjust accordingly to a speci…c
for a predetermined planning horizon S: Therefore, S is not considered to be
a parameter for this analysis. For the three targets of macroeconomic variables
we report several sensitivity checks around the best planning horizons found
above. [Table 5] shows two sets of sensitivity reports, one with a baseline = 2
and the other one with = 0:5; for S = 15; 20 and S = 18; 22; respectively.
The second column, Model, of the table shows di¤erent alternative calibrations
to the baseline model, keeping the other parameters intact. Thus, for example,
the …rst group of the table represents the alternative calibration to the baseline
model fS = 15; = 0:5; = 0:979; = 0:289; = 0:063g by changing only one
parameter, except S; and keeping the others constant.
28
S
Baseline
S=15
Baseline
S=18
Baseline
S=20
Baseline
S=22
5
[Table 5]44 Sensitivity to Alternative Calibration
Model
r
C=Y
K=Y Age Cm =Co
=0:5; =0:979 3.50% 0.748 2.94
50
1.369
= 0:40
3.60% 0.753 2.91
50
1.365
= 0:30
3.93% 0.747 2.92
51
1.380
= 0:07
3.80% 0.751 2.77
50
1.351
= 0:98
3.61% 0.732 3.02
50
1.395
=2; =1:006
3.50% 0.748 2.94
47
1.259
= 1:90
3.39% 0.743 2.97
47
1.262
= 0:33
4.43% 0.712 3.07
47
1.328
= 0:07
3.20% 0.746 2.83
47
1.239
= 1:00
3.83% 0.763 2.84
47
1.250
=0:5; =0:974 3.50% 0.748 2.94
45
1.202
= 0:60
3.64% 0.754 2.90
45
1.199
= 0:30
3.58% 0.730 3.01
45
1.210
= 0:08
3.34% 0.747 2.66
45
1.169
= 0:98
3.08% 0.707 3.17
45
1.232
=2; =0:993
3.50% 0.748 2.94
43
1.139
= 2:10
3.58% 0.752 2.92
43
1.153
= 0:30
3.73% 0.737 2.98
43
1.168
= 0:05
4.09% 0.755 3.18
43
1.189
= 0:98
4.32% 0.784 2.71
43
1.139
M SE
0.0063
0.0061
0.0054
0.0058
0.0060
0.0051
0.0054
0.0019
0.0071
0.0041
0.0058
0.0053
0.0048
0.0083
0.0072
0.0074
0.0071
0.0058
0.0039
0.0049
Social Security
Social security is often cited as providing intergenerational insurance and preventing high poverty rates among older individuals, such as in the case of Great
Recession (Peterman and Sommer, 2014). At the same time, however, it is said
that social security does not improve the welfare of agents with time-inconsistent
preferences alone (Imrohoroglu et al., 2003).45 By examining a richer class
of time-inconsistent preferences, Caliendo (2011) shows that for naive timeinconsistent individuals, social security cannot improve welfare regardless of the
functional form of time discount when it has a negative net present value. Moreover, in Feigenbaum (2014), social security does not improve welfare for myopic
agents who have rational expectations about future income.46 In fact, all of
these agents follow consumption rules that are not di¤erent from the standard
4 4 Age represents the peak consumption age and C =C represents the ratio of peak conm
o
sumption to initial consumption.
4 5 Imrohoroglu et al. (2003) show this in partial equilibrium for agents with hyperbolic
discounting.
4 6 With CRRA in …nite-horizon models, Feigenbaum (2014) …nds that the consumption
rule represented by being proportional to lifetime wealth nests the standard exponential discounting model, the naive hyperbolic discounting model, and the short-horizon myopic model.
Thus, in a steady state with constant interest rates, it can be inferred that any proportional
consumption rule may be derived either from one with time-consistent preferences or one with
time-inconsistent preferences that follows re-optimization.
29
one: consumption is proportional to lifetime wealth. As long as individuals
have rational expectations of future income, their consumption is not di¤erent from the Lifecycle-Permanent Income (LCPI) prediction, and thus, the IIJ
Impossibility Theorem (Caliendo, 2011) may be applied.
However, as seen in Findley and Caliendo (2009), when agents have irrational
or boundedly rational expectations about their future income such as Caliendo
and Aadland (2007) and this paper suggest, then there is the possibility that
social security does raise welfare. In fact, in Findley and Caliendo (2009), it is
the limited expectation about future income beyond the current time horizon
that puts social security into play to raise welfare. Therefore, in this section,
we explore further to …nd if the bounded rationality model can improve welfare under social security. Together with this, we also want to analyze if the
model still produces the hump-shaped consumption pro…le in calibrated general
equilibrium under the new environment of social security.
5.1
The Model with PAYG
In this section, we examine two di¤erent environments for social security models.
First, we include the social security system into the baseline model in section 3
and analyze the e¤ect of social security on boundedly rational agents’behavior
in a calibrated general equilibrium.47 We introduce government in the model
and assume that the government implements a pay as you go (P AY G) social
security system …nanced by a payroll tax ; which is determined exogenously.
The government is assumed to have a balanced budget each time. The agent
receives the social security bene…t t from the point of retirement TRET , where
TRET is set by government policy. Thus, the consumer’s optimization problem
in this extended model is to maximize the lifetime utility, Eq.(17), over short
term s = t; :::; t + S 1; for t = 0; 1; :::; T; subject to the following budget
constraint:
cs (t) + bs+1 (t) = (1
) wes + Rbs (t) +
s 1 (t
> TRET )
(33)
where the index function 1 (t > TRET ) represents the condition by which the
social security bene…t is included in the consumer’s income. Inclusion of social
security imposes an extra mechanism to the bond demand (private saving) that
transfers income from the young to the old. The consumer’s optimization rule
stays the same as before, but with social security, the size of bene…t is determined
in equilibrium and given by
PTRET 1
PT
wet = t=TRET t
t=0
For calibration purposes, we assume that consumers pay payroll tax =
7:65%48 before they retire, and the social security income is …xed during retirement years. Figure 9 shows the general equilibrium result. It clearly demon-
4 7 Following this subsection, a more realistic model of social security with mortality risk and
bequests is analyzed in 5.2.
4 8 If the self-employed are included, the overall tax rate is approximately 10%.
30
Figure 9: Optimal Consumption Pro…les C(t) for the model with S = 17, one
with social security and the other without it. The tax rate is 7.65% and T = 80.
strates a hump-shaped consumption pro…le for the bounded rationality model
with social security. The pro…le is obtained from the planning horizon of 17
years, which is found to be the best …tting planning horizon in terms of mean
squared error. We add a consumption pro…le of the model with the same planning horizon but without social security for comparison. Although both pro…les
show a similar pattern regarding the hump and the peak consumption age (47
years old), it is noticeable that social security reduces consumption over younger
years. Imposing a payroll tax implies a lower wealth level for consumption during working years and a higher consumption possibility in retirement years.
With social security, the best …tting planning horizon in terms of peak consumption age (45 years old) is S = 20, which is not di¤erent from the result of
the baseline model without social security. In terms of the smallest deviation
from the target consumption pro…le (GP), S = 17 …ts best, and in terms of
Cmax =C0 ; S = 24 …ts best. Figure 10 summarizes this. In summary, the model
with social security still preserves the main …nding of Section 4 that a planning
horizon of approximately 20 years provides the best …t to the data.
The fact that the model with social security still accounts for the consumption hump has important implications in the lifecycle literature. In a standard
model with social security, higher interest rates induced by lower private saving
tend to increase the consumption projection49 toward the later years, and the
consumption peak would occur after retirement. Therefore, the hump-shaped
pro…le with a peak around 45 may not be achieved in the standard world.
But with bounded rationality models, the shorter than full-planning horizon
4 9 With
CRRA, the consumption growth rate is determined by ( R)1= :
31
2
0.025
1.6
0.02
1.2
0.015
0.8
0.4
C-ratio
0.01
1.14756
0.005
0
10
15
20
MSE
0
25
10
15
20
25
Planning Horizon
Planning Horizon
Figure 10: The Best Fitting Planning Horizon with social security. In terms of
MSE, S = 17 …ts best and in terms of CM ax =C0 ; S = 24 …ts best.
generates time-inconsistent expectations regarding income, and the usual consumption rule of proportion to lifetime wealth of the standard model does not
apply.
5.2
Mortality Risk
Second, we add mortality risk to the model with social security, making the
model even more realistic, and explore the implication of social security in general equilibrium. Unlike the environment of a …xed lifespan in the previous
section, the agents here are subject to mortality risks and survive until age
s > t, with a maximum age T . The mortality risk is represented by a survivor
function Qt , which is assumed to be cohort-independent. Also, we assume that
people who do not survive up to t leave their assets to o¤spring in the form of
bequest Bt . We further assume that the bequests are spread uniformly over the
surviving population. Under this new environment, the representative consumer
maximizes his lifetime utility for t = 0; 1; :::; T; and s = t; :::; t + S 1 :
U (t) =
t+S
X
s t
s=t
Qt
c1s
1
(t)
(34)
subject to
cs (t) + bs+1 (t) = (1
) wes + Rbs (t) + Bs + s 1 (t > TRET )
b0 (0) = 0; bt (t) = given; bt+S+1 (t) = 0
To accommodate this modi…cation to the model, we adjust the condition
of general equilibrium accordingly. The competitive equilibrium condition in
32
equation (28) in the baseline model should be modi…ed to
T
X
Qt bt (R) = N
R
1+d
1
1
t=0
T
X
Qt et
(35)
t=0
The following equation determines the size of the bequest:
T
X
Bs Qt =
t=0
T
X
(Qt
Qt+1 ) Rbt+1
(36)
t=0
Likewise, equation (35) is modi…ed for mortality risk in determining the size of
social security.50
TRET
T
X 1
X
Qt wet =
Qt t
(37)
t=0
t=TRET
For calibration purposes, we adopt the survivor probabilities from Feigenbaum’s
(2008) sextic polynomial function to the mortality data in Arias (2004). According to Arias, a maximum lifespan is 100, which determines the maximum
model age of T = 75 because t = 0 represents age 25. The survivor function is
Qt
exp( 0:01943 0:00031t + 0:000006t2 0:00000328t3
+0:000000003188t5 0:00000000005199t6 )
=
0:0000000306t4
Other than this, we make the same assumptions as in the above section, such
as payroll tax rate = 7:65% and …xed social security income for retirement
years. Under this condition, we …nd that the bounded rationality model of
social security with mortality risk produces the featured consumption hump
but with a better …t to the data than the …xed lifespan model of social security
without mortality risk. Figure 11 exhibits this result: compared with Figure 9,
consumption in early life is increased and the …t to the baseline is improved.
The pro…le is obtained from the planning horizon S = 20, which generates
the smallest MSE of 0.00564. In terms of the peak consumption age, S = 17
provides a better …t than any other, while in terms of consumption ratio, S = 18
…ts best.
5.3
Welfare
In this section, we analyze the e¤ect of social security on welfare for an economy
populated by boundedly rational individuals. As described at the beginning of
the section, social security can improve welfare when individuals have limited
expectations about future income. As in the case of hyperbolic discounting
models, measuring the welfare e¤ects from social security could be controversial. This is because there is no objective way to measure welfare in models with
5 0 If
t
=
for TRET
t
T , then
PTRET
t=0
33
1
Qt wet =
PT
t=TRET
Qt :
Figure 11: Optimal Consumption Pro…les C(t) for the model with social security
when mortality risk is added. The tax rate is 7.65% and S = 20.
dynamically inconsistent preferences.51 However, unlike in hyperbolic discounting models52 , in which the subsequent selves are fully aware of future income
streams throughout the end of life, the subsequent selves have limited expectations regarding the income in the bounded rationality models. This implies that
it is highly unlikely that the initial plan from the time-zero optimization should
be preferred over subsequent plans. Because of this, evaluating welfare based on
the preferences of the time-zero plan is not necessary with the model.53 Next,
we need to quantify the welfare change. Although the welfare can be measured
by a change in lifetime utility, it is more desirable to use a compensating variation, which measures the expenditure increment due to social security tax to
achieve the same utility level. To do so, we de…ne the -representative- lifetime
utility for boundedly rational individuals who are born at time t.
Ut =
T
X
s=0
where
5 1 With
1
s cs;t
1
hyperbolic discounting models, some may argue that equating welfare with the
preferences of a time-zero self may be reasonable, while others may oppose this method because
there is no particularly bene…cial reason to do so. However, if we consider social security to
be a commitment device, the former may be suggested.
5 2 I call hyperbolic discounting models time inconsistent tastes, while bounded rationality
models time inconsistent expectations.
5 3 The common paternalistic approach to obtain this measurement from social security is to
…rst calculate the expected discounted lifetime utility from the initial point of time for one
with social security and the other without it.
34
fcs;t gTs=0 = fcs (s); cs+1 (s + 1); cs+2 (s + 2); :::; cT (T )g54
(38)
as de…ned in the baseline model. To measure the overall welfare from the entire
population at each point of time, we introduce a social planner who takes care
of the utilities from all cohorts. The social planner evaluates the total welfare of
the model economy by adding up each cohort’s utility weighted by the relative
importance of each group.
SWt =
t
X
i
Ui
(39)
i=t (T +1)
where i > 0 is a weighting function to each group. The change in social welfare
is represented by the following compensating variation:
CV =
SWt (tax > 0)
SWt (tax = 0)
1
1
1
(40)
This value determines the welfare change from social security.55 For the simulation exercise below, we posit that the planner should treat the utility of each age
group as though all groups were equally contributing toward average welfare:
i = 1.
Then we perform the welfare analysis by comparing utility with and without
social security for each environment of …xed lifespan (no mortality risk) and
full lifespan (mortality risk). For the model of …xed lifespan, we …nd that if
the planning horizon is shorter than 10 years, social security would increase
welfare. This result concurs with Findley and Caliendo (2009), who …nd a
welfare increase for those with a horizon that is less than 7 years when the tax
rate is 10.6%. With mortality risk, the result is somewhat di¤erent in terms
of the planning horizon because we …nd that the utility increases when the
planning horizon is around the best …tting horizon, S = 18~21. Because of this
property, the model of mortality risk may be more appropriate to justify the
social security program if the welfare increase is quanti…ed.
Based on the measurement, we further …nd that there is positive welfare
change from social security when the planning horizon is around the best …tting
horizon. Speci…cally, we …nd that the welfare increases among the planning
horizons of S = f18; 19; 20; 21g and the highest welfare increase is obtained
when S = 18: Figure 12 represents the compensating variation from a change in
payroll tax rate from 0% to 10%. Together with this, the following table shows
equilibrium social security income and bequest when the tax rate is given by
0% to 10%.
54 c
s;t is the envelop of an optimization series that solve the short-term maximization problem at each planning time.
55 A
positive value re‡ects a welfare increase from a new policy compared to the initial one.
35
Figure 12: Average Compensating Variation over di¤erent tax rates when the
utility of each age cohort is equally treated. S=18.
Tax
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
6
Social security
0.033453
0.066230
0.098331
0.129757
0.160506
0.190580
0.219978
0.248700
0.276746
0.304117
Bequest
0.109402
0.109252
0.109092
0.108921
0.10874
0.108547
0.108342
0.108126
0.107897
0.107654
CV(%)
0.087827
0.161680
0.221206
0.266600
0.298223
0.316113
0.320130
0.311045
0.288485
0.252641
Modi…cations
The main model discussed so far makes a few assumptions. First, the planning
horizon does not change during the agent’s entire life. Second, the agent is
able to adjust his plan at any point during his lifetime. Third, each time the
agent plans to dissave all his foreseeable wealth within the planning horizon,
leaving zero assets. The …rst assumption is based on the premise that an agent
would not learn from the past and does not improve his planning skills for
the future. In other words, the agent’s limited ability to foresee the future
is persistent throughout his entire life. Thus, the optimization process stays
unvarying in an economy as well. This may not be a good assumption if one
accepts the argument that human cognitive ability suggests learning through
36
past experiences. If this is the case, then a modi…cation to the baseline model
on consumer optimization is necessary, and this is the topic for the section.
Here, we consider two modi…cations: one is the case of gradual learning until
the end of the consumer’s life, while the other is gradual learning only up to a
speci…c horizon S.
The second assumption is related to infrequent planning due to planning cost
and has quite common elements in rationality models.56 The third assumption
implies that agents behave as if there would be no further life beyond the end
of the horizon. In other words, they expect that they do not need to consume
any more at the end of the short-term horizon, even though they are still young
when they plan to consume. For this reason, a modi…ed version of the model
may be considered. Agents are allowed to hold some assets for the future at
the end of the horizon. One way to construct this is to assume that the agents
have positive bond holding at the terminal node of each planning horizon. This
modi…cation is, in fact, related with the social security in Section 5.
6.1
Learning
The boundedly rational agent learns from past experience and plans better as
his experience accumulates. For example, each year he plans for one additional
year to his previous planning horizon. But he is still assumed to plan frequently,
based on each year, and without incurring any cost related with planning. Assume that the planning horizon of each time is just one more year than the
previous one and the initial planning horizon is zero: the consumer at t = 10
(age=35) is able to plan for 10 years and at age 36 for 11 years, etc. Then
the representative agent plans for age + 1 periods each time and maximizes the
following for t = 0; 1; :::; T;
U (t) =
t+S
Xt
s=t
1
s t cs
(t)
1
(41)
subject to
cs (t) + bs+1 (t) = wes + Rbs (t)
b0 (0) = 0; bt (t) = given; bt+St +1 (t) = 0
St+1 = St + 1; S0 = 0:
The optimal consumption at age t is obtained by
Pt+St wes
s t + Rbt (t)
ct (t) = Ps=t Rh
is t :
1=
t+St
s=t
(42)
(R )
R
Figure 9 shows the result in general equilibrium when the agent keeps learning
through his life. In this …gure, the age of peak consumption comes earlier than
5 6 Therefore, it is not necessary to discuss this for the modi…cation in the Bounded Rationality model.
37
Figure 13: The gradual learning result in general equilibrium. Planning horizon
expands each year through consumer’s life time. = 0:5:
the data. This implies that as the agent learns more from the past and plans
further, he is more likely to prepare for his retirement early. Consider another
case where the agent’s learning ability grows only up to L years and then stalls.
Figure 10 summarizes this scenario. From the …gure, it is also clear that learning
for longer periods gives the agent more chances to smooth consumption. The
peak consumption age is closest to the data when the length of the gradual
learning is approximately 20.57
6.2
Positive Asset Holding for the Future
Let us modify the baseline model into one in which the agent holds onto some
assets for the future at the end of the S horizon. This modi…cation would
be justi…ed if boundedly rational agents expect a longer life than the speci…c
planning horizon and hold extra savings instead of dissaving all the wealth
within the planning horizon. For example, an agent at age t who plans for
the upcoming S years would not plan to starve beyond t + S and thus may put
aside extra money for the future. This modi…cation can be modeled by imposing
positive bond demand at the terminal node of each planning horizon. Assume
the agent plans for S + 1 periods and maximizes for t = 0; 1; :::; T S;
U (t) =
t+S
X
1
s t cs
s=t
(t)
1
subject to
cs (t) + bs+1 (t) = wes + Rbs (t)
b0 (0) = 0; bt (t) = given
5 7 In
fact, L = 18 is the best by this criterion in the learning model.
38
(43)
Figure 14: The gradual learning result in general equilibrium. Planning horizon
expands each year only up to L periods and stalls. = 0:5: The interest rate is
…xed at the target rate.
bt+S (t) = bbS (t) > 0
b
where bS (t) is bond holding at the end of planning horizon S, indexed by
planning time t. The agent, who intends to hold onto extra wealth at the
terminal node, may achieve this by reducing the consumption stream over the
planning periods S + 1. This formulation gives positive asset holding even at
the very last period of life (i.e., bT +1 > 0), although it may be fairly plausible
to assume that at the terminal planning horizon, agents fully dissave all wealth
they have accumulated: bT +1 = 0. The former assumes that there is accidental,
unanticipated death among the population at age 80, while the latter implies
that when death is in view, agents will plan accordingly.58 When bT +1 > 0; it is
necessary to impose an extra condition for equilibrium: the wealth accumulation
at the …nal stage of life would spread evenly among the remaining population
in terms of bequest Bt . Solving the optimization problem returns the following
consumption pro…le for t = 0; 1; :::; T; with planning horizon S.
hP
i b
S
bS (t)
Bs +wes
s
s=t
R
Rs + Rbt (t)
ct (t) =
(44)
h
is t
P
1=
S
s=t
(R )
R
where Bs is the bequest from the older generation. Bs > 0 speci…es the model
of accidental bequest, while Bs = 0 speci…es the model of terminal adjustment.
Figure 15 shows the two types of bond demand in the general equilibrium. With
5 8 In fact, this is the same assumption as in the baseline model regarding optimization over
the residual years.
39
Figure 15: Optimal Bond Pro…le over lifecycle. Asset holding is 7% of the initial
income. S = 18.
Figure 16: Optimal Consumption Pro…le over lifecycle. Asset holdings are
f10%; 30%g for the ‘Bequest’ and f10%g for the ‘No Bequest’ of the initial
income. S = 18.
40
the accidental bequest, the agent does not fully dissave toward the end of life,
and bT +1 is positive.
The result regarding consumption is presented in the next …gure. Figure 16
shows the two types of consumption pro…le relative to the data (GP). In the …gure, each model is simulated with the assumption that the positive saving should
be a certain portion, such as 10% of the agent’s normalized initial income. The
…gure plots two levels of asset holding for the model of bequest to demonstrate
the change in consumption: bbS = f10%; 30%g.59 From this, it is clear that a
strong intention for private saving tends to increase overall consumption, which
in turn decreases interest rates, and the slope of consumption curve grows lower
during later years of life, indicating the opposite case to social security.
7
Conclusion
The prominent feature of lifecycle consumption data known as the consumption
hump has been a central issue in the lifecycle consumption literature because
standard lifecycle theory cannot produce consumption with such property. The
standard theory implies that consumption is not directly a¤ected by income
‡uctuation: it is the average of income that matters, not the individual ups and
downs. Therefore, hump-shaped lifecycle consumption cannot be expected in a
model of standard assumptions, even though the income, the resources that the
consumption relies on, is roughly hump-shaped over life. In this paper, we study
a bounded rationality model that is based on time inconsistent preferences, and
that solves the discrepancy between the monotonic prediction and the humpshaped data of lifecycle consumption. The standard lifecycle models usually
require extra assumptions about the structure of the model beyond preferences
to reconcile the discrepancy. Motivated by this we explore a model with a novel
framework that deviates from the main core of the traditional theory: the full
rationality assumption. The rational agent is now allowed to be boundedly rational and maximizes his utility over a certain period of time, which is shorter
than a full life span. In this way we …nd that it is possible to construct a model
in which the optimal consumption pro…le clearly links to the income stream. We
speci…cally highlight the general equilibrium approach through this paper and
characterize a model of a boundedly rational agent under a (T + 1)-period overlapping generations general equilibrium with production. Quantitative analysis
shows that the general equilibrium result is consistent with the known characteristics of the lifecycle consumption pro…le. We also …nd that the result is quite
robust with the extended model of social security and mortality risk. Finally, we
introduce two modi…cations of the model incorporating (1) learning and (2) positive asset holding at the end of the planning horizon. These modi…cations help
understand how the model with a short horizon works to induce a consumption
hump.
The simulation exercise demonstrates that the general equilibrium model
with a planning horizon of approximately 20 years provides the best …t to the
5 9 In
the case of no bequest, the consumption pro…le does not change signi…cantly.
41
salient features of the lifecycle consumption data. One remarkable thing is
that this result is very consistent with people’s actual behavior, as reported
in many surveys, speci…cally the survey on retirement planning (Retirement
Con…dence Survey, 2007). The …ndings from the survey can be summarized
as the following: the average US worker tends to start to save for retirement
around age 45, planning to retire at 65, with an expectation of being retired
for 20 years. By this statement, it is expected that the representative consumer
who has a planning horizon of 20 years and an exogenous retirement age of 65,
does not realize that he needs to accumulate assets for retirement until 45 years
old. At age 45 he can see clearly that there is no more income after 20 years and
must establish an retirement account by dramatically reducing the consumption
he has enjoyed to date. This fact exactly coincides with the model’s prediction
if a planning horizon of 20 years is used, which this paper demonstrates is the
optimal planning horizon for the model.
References
[1] Attanasio, O., J. Banks, C. Meghir and G. Weber, (1999), “Humps and
bumps in lifetime consumption,”Journal of Business and Economic Statistics 17: 22-35.
[2] Bullard, J. and J. Feigenbaum, (2007), “A Leisurely Reading of the Lifecycle Consumption Data,” Journal of Monetary Economics 54: 2305-2320.
[3] Bütler, M., (2001), “Neoclassical life-cycle consumption: a textbook example,” Economic Theory 17: 209-221.
[4] Caballero, R. J., (1990), “Consumption Puzzles and Precautionary Savings,” Journal of Monetary Economics 25: 113-136.
[5] Caballero, R. J., (1995), “Near-Rationality, Heterogeneity and Aggregate
Consumption,” Journal of Money, Credit and Banking 27: 29-48.
[6] Caliendo, F., (2011), “Time-Inconsistent Preferences and Social Security:
Revisited in Continuous Time,”Journal of Economic Dynamics and Control
35: 668-675.
[7] Caliendo, F. and D. Aadland, (2007), “Short-Term Planning and the LifeCycle Consumption Puzzle,” Journal of Economic Dynamics and Control
31: 1392-1415.
[8] Caliendo, F. and K. Huang, (2008), “Overcon…dence and Consumption over
the Life Cycle,” Journal of Macroeconomics 30: 1347-1369.
[9] Carroll, C. D., (1997), “Bu¤er-Stock Saving and the Life-Cycle / Permanent Income Hypothesis,” Quarterly Journal of Economics 112: 1-55.
42
[10] Deaton, A., (1991), “Saving and Liquidity Constraints,” Econometrica 59:
1221-1248.
[11] Dotsey, M., W. Li and F. Yang, (2014), “Consumption and TIme Use Over
the Life Cycle,” International Economic Review 55:665-692.
[12] Fehrer, J. C., (2000), “Habit Formation in Consumption and Its Implications,” American Economic Review 90: 367-390.
[13] Feigenbaum, J., (2005),“Second-, Third-, and Higher-Order Consumption
Functions: A Precautionary Tale,” Journal of Economic Dynamics and
Control 29: 1385-1425.
[14] Feigenbaum, J., (2008a), “Can Mortality Risk Explain the Consumption
Hump?” Journal of Macroeconomics 30: 844-872.
[15] Feigenbaum, J., (2008b), “Information Shocks and Precautionary Saving,”
Journal of Economic Dynamics and Control 32: 3917-3938.
[16] Feigenbaum, J., (2009),“ Precautionary Saving Unfettered,” Working Paper, University of Pittsburgh.
[17] Feigenbaum, J., (2014), “Equivalent Representations of Non-Exponential
Discounting Models,” Working Paper.
[18] Fernández-Villaverde, J. and D. Krueger, (2011), “Consumption and Saving
over the Life Cycle: How Important are Consumer Durables?,” Macroeconomic Dynamics 15: 725-770.
[19] Findley, S. and F. Caliendo, (2009), “Short horizons, time inconsistency,
and optimal social security,”International Tax and Public Finance 16, 487513.
[20] Gourinchas, P. and J. A. Parker, (2002), “Consumption over the Life Cycle,” Econometrica 70: 47-89.
[21] Hansen, G. D. and S. I·mrohoro¼
glu, (2008), “Consumption over the Life
Cycle: The Role of Annuities,” Review of Economic Dynamics 11: 566583.
[22] Heckman, J., (1974), “Life Cycle Consumption and Labor Supply: An
Explanation of the Relationship Between Income and Consumption Over
the Life Cycle,” American Economic Review 64: 188-194.
[23] I·mrohoro¼
glu, Ayşe, Selahattin I·mrohoro¼
glu, and Douglas H. Joines, (2003),
“Time-Inconsistent Preferences and Social Security,” Quarterly Journal of
Economics 118: 745-784.
[24] Laibson, D., (1997), “Golden Eggs and Hyperbolic Discounting,”Quarterly
Journal of Economics 112: 443-477.
43
[25] Levine, D. and D. Fudenberg, (2006), “A Dual Self Model of Impulse Control,” American Economic Review 96: 1449-1476.
[26] Luo, Y.,(2005), “Consumption Dynamics under Information Processing
Constraints,” Working Paper.
[27] Lusardi, A., (2003), “Planning and saving for retirement,”Working paper,
Dartmouth College.
[28] McGrattan, E. R. and E. C. Prescott, (2000), “Is the Stock Market Overvalued?”Federal Reserve Bank of Minneapolis Quarterly Review 24: 20-40.
[29] Moscarini, G., (2004) “Limited information capacity as a source of inertia,”
Journal of Economic Dynamics and Control 28: 2003 –2035.
[30] O’Donoghue, T. and M. Rabin, (1999), “Doing It Now or Later,”American
Economic Review 89: 103-124.
[31] Park, H., (2014), “ Present-Biased Preferences and the Constrained Consumer,” Working Paper.
[32] Peterman, W. and Sommer, K. (2014), “How Well Did Social Security Mitigate the E¤ects of the Great Recession,” Federal Reserve Board. Finance
and Economics Discussion Series 2014-13.
[33] Reis, R., (2006), “Inattentive Consumers,”Journal of Monetary Economics
53: 1761-1800.
[34] Rios-Rull, J., (1996), “Life-Cycle Economies and Aggregate Fluctuations,”
Review of Economic Studies 63: 465-489.
[35] Sims, C., (2003), “Implications of Rational Inattention,”Journal of Monetary Economics 50: 665-690.
[36] Skinner, J., (1988), “Risky Income, Life Cycle Consumption, and Precautionary Savings,” Journal of Monetary Economics 22: 237-255.
[37] Thurow, L. C., (1969), “The Optimum Lifetime Distribution of Consumption Expenditures,” American Economic Review 59: 324-330.
44