Download Revisiting Asset Pricing under Habit Formation in an Overlapping

Revisiting Asset Pricing under Habit Formation in an
Overlapping-Generations Economy
Sei-Wan Kim
Department of Economics
Ewha Womans University
Seoul, S. Korea
Joshua Krausz
Syms School of Business
Yeshiva University
500 W. 185th Street
New York, NY 10033
Kiseok Nam
Syms School of Business
Yeshiva University
500 W. 185th Street
New York, NY 10033
[email protected]

Corresponding Author
1
Revisiting Asset Pricing under Habit Formation in an
Overlapping-Generations Economy
Abstract
Incorporating habit formation into an overlapping-generations economy, we show that the
middle-aged consumers’ savings decision has a substantial impact on the high equity premium
under the borrowing-constrained economy. The higher incentive for savings for the middle-aged,
resulting from the habit formation preference, causes an even higher demand for bonds and a
lesser demand for equity, which eventually generates a lower risk-free rate and a higher required
return for holding equity than does the framework of non-habit forming models. the effect of
habit formation on the demand for equity and bonds is more profound under the borrowingconstraint economy than under the borrowing-unconstraint economy. Calibration results verify
that (a) the habit formation setting together with an OLG framework is capable of yielding lower
bond returns and higher equity returns than the standard CRRA utility models, and (b) the
borrowing constraint imposed to the young-aged consumers amplifies the positive effect of habit
formation preference on equity premium. Therefore, we argue that the habit formation
preferences in the OLG framework, with the borrowing constraint imposed on the young
generation, can provide a more satisfactory explanation of the equity premium puzzle.
JEL Classification: E21; G10
Key words: Equity premium; habit formation preference; overlapping-generations economies;
consumption asset pricing model; calibration
2
1. Introduction
In their seminal work, Mehra and Prescott (1985) identify the phenomenon that the
historical real returns of stock over government bonds are anomalously high. They show that the
historical equity premium, which is defined as equity returns less government bond returns,
exhibits an abnormally high level not only in the United States but also in many other
industrialized countries, over long time periods. 1
Since the equity premium is supposed to reflect the relative risk of stocks compared to
risk-free government bonds, the unexpectedly large percentage of the risk premium for equity
implies an implausibly high level of risk aversion among consumers. 2 The problem, known as
the equity premium puzzle, is that the magnitude of the equity premium is too large to reflect a
reasonable level of compensation justified under the standard neoclassical equilibrium asset
pricing model.
Due to the importance of its economic implications, the equity premium puzzle has
spawned an extensive research effort to resolve the puzzle in both macroeconomics and finance.3
In general, most of the literature explaining the puzzle takes the approach of either finding
factors requiring adjustment to the empirical side of the puzzle, or exploring alternative
1
They demonstrate that it is difficult to reconcile the empirical fact of a suspiciously high level of equity
premium and the process of consumption growth with a reasonable assumption about the relative rate of risk
aversion and the pure rate of time preference, in a conventional infinite-horizon model with an additively separable,
constant relative rate of risk aversion (CRRA) utility function.
2
By looking at the disparity from a different perspective, Weil (1989) raises an issue, known as the risk-
free rate puzzle, on why bond returns are lower than equity returns. Ebrahim and Mathur (2001) suggest an
equilibrium model reflecting investor heterogeneity, market segmentation and leverage to resolve the two puzzles.
3
The excessive magnitude of the equity premium has many important economic implications, such as those
for resource allocation, social welfare, and economic policy, other than financial market implications. See Grant and
Quiggin (2005) for more details.
3
theoretical frameworks. The studies focusing on the empirical side of the puzzle include the
question of sample time periods and mean reversion or aversion by Siegel (1992a, 1992b).
On the other hand, the studies attempting to modify the theoretical features of the Mehra
and Prescott (1985) model propose alternative assumptions about preference (Constantinides
1990; Abel 1990; Epstein and Zin 1991; Meyer and Meyer 2005; Giordani and Söderlind 2006),
disaster states and survivorship bias (Reitz 1988; Brown, Goetzmann and Ross 1995; Barro
2006), incomplete markets (Constantinides and Duffie 1996; Heaton and Lucas 1997;
Storesletten, Telmer and Yaron 1999), and market imperfection (Bansal and Coleman 1996;
Alvarez and Jermann 2000; Constantinides, Donaldson and Mehra 2002).
Also, more recent studies of the puzzle attempt to provide different rationales for
explaining the equity premium, such as investor prospects (Benartzi and Thaler 1995; Durand,
Lloyd and Tee 2004; Fielding and Stracca 2007), macroeconomic influences (Campbell and
Cochrane 1999), and changes in tax rates (McGrattan and Prescott 2001 & 2006) 4. Despite a
great deal of literature suggesting a wide range of useful theoretical and empirical tools, the
puzzle has not been completely resolved.
In this paper, we extend the framework of Constantinides, Donaldson and Mehra (2002).
Incorporating habit formation into an overlapping-generations (hereafter OLG) economy with
the borrowing constraint imposed to the young generation, we verify that there is a positive
impact of habit formation on the savings levels of middle-aged consumers. The higher incentive
4
McGrattan and Prescott (2001) suggest that the changes in tax rates can explain the equity premium
puzzle. They show that the large reduction in individual income tax rates and the increased income from tax shelter
opportunity have led to a dramatic increase in equity prices between income 1960 and 2000. In turn, this increased
equity prices generate much higher ex post returns on equity than on debt, such that they argue that at least for the
post-WWII period, the equity premium is not puzzling.
4
for savings for the middle-aged, resulting from the habit formation preference, causes an even
higher demand for bonds and a lesser demand for equity, which eventually generates a lower
risk-free rate and a higher required return for holding equity than does the framework of nonhabit forming models. Calibrating our model, we confirm that our model yields a lower risk-free
rate and a higher equity return than do other general non-habit forming models. We thus argue
that habit formation preferences within the overlapping-generations framework under the
borrowing-constrained economy can provide a more improved explanation of the equity
premium puzzle.
The rest of the paper is organized as follows. In section 2, we discuss the related works
on habit formation in the equity premium puzzle. In section 3, we derive the optimum savings of
the habit-forming middle-aged consumers under a borrowing-constrained economy, and confirm
a positive effect of the habit formation preference on middle-aged consumers’ savings. In section
4, we discuss the calibration of our model and its results. Section 5 concludes the paper.
2. Related Works
Habit formation has been widely used in recent studies of financial economics as an
important assumption in explaining the dynamic equilibrium path of consumption. For example,
Constantinides (1990) and Abel (1990) show that habit-forming consumption, with its flexibility
in modeling risk aversion and consumption paths, can partially resolve the equity premium
puzzle posed by Mehra and Prescott (1985). 5 This finding has indeed motivated a line of habit
formation approaches in dynamic modeling of optimal consumption, savings and portfolio
5
See Cochrane and Hansen (1992) and Kocherlakota (1996) for surveys on the equity premium puzzle.
5
decisions (Sundaresan, 1989; Jermann, 1998; Campbell and Cochrane, 1999; Uhlig, 2000; and
Guvenen, 2009). 6
On the other hand, Constantinides, Donaldson, and Mehra (2002) (hereafter CDM)
propose an overlapping-generations (hereafter OLG) model that explicitly captures the saving
and dissaving behavior of consumers subject to a borrowing constraint. CDM show that with a
simple time separable utility function and a borrowing constraint, consumers in a three-period
overlapping-generations economy have an incentive to hold a diversified portfolio for different
stages, over their life cycle. That is, a borrowing constraint prevents the young
-aged generation from holding equity, and that equity prices are assumed to be exclusively
determined by the middle-aged consumers. Knowing that their future retirement income is either
zero or deterministic, and that their future consumption is highly correlated with equity income,
the middle-aged consumers will save more by holding more bonds and less equity. Therefore, the
middle-aged consumers’ savings decision has a dominant impact on the level of the equity and
the bond return.
In this paper, we extend CDM’s work by incorporating habit formation into the OLG
economy, such that the habit-forming consumers’ optimal savings decision is derived from an
overlapping-generations framework. Under the habit formation utility and the OLG economy,
the impact of the middle-aged consumers’ savings decision on the demand for equity and bonds
is affected not only by the presence of a borrowing constraint, but also by the habit formation
process. A habit-forming middle-aged consumer will have a much higher incentive to smooth
6
Guvenen (2009) proposes an asset pricing model focusing on the limited stock market participation and
heterogeneity in the elasticity of intertemporal substitution in consumption. His model is partially successful in
calibrating major features of asset pricing such as high equity premiums, smooth interest rates, procyclical stock
prices, and countercyclical variations in the equity premium.
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consumption over time, and he/she will have a lower incentive to bear risk in order to guarantee
a stable consumption for the next period, by demanding more bonds and less equity than a
nonhabit-forming consumer. Thus, the habit formation utility causes an even higher demand for
bonds (yielding a lower risk-free rate) and a lower demand for equity (yielding a higher required
return for holding equity), and thereby yielding a higher equity premium, than does a non-habit
formation utility such as the CRRA utility suggested by CDM (2002). Also, we show that the
effect of habit formation on the demand for equity and bonds is more profound under the
borrowing-constraint economy than under the borrowing-unconstraint economy. In sum, the
combination of the habit formation utility, and an OLG economy with a borrowing constraint,
yields better results which can be used in explaining the equity premium.
3. Habit formation and Optimum Savings under Borrowing Constraint
In this section, we present a habit formation exchange economy in the OLG framework
and derive the optimum savings of the middle-aged generation with the borrowing constraint
imposed on the young generation. Under the borrowing-unconstrained economy, the young will
borrow to purchase equity, thereby raising bond returns. The increase in bond returns induces the
middle-aged investors to shift their portfolio holdings from equity to bonds, thereby reducing
equity return. The increase in the demand for equity by the young will be overweighed by the
decrease in equity demand by the middle-aged, such that the net effect is an increase in equity
returns. The increase in equity returns and the increase in bond returns together shrink the equity
premium. Under a borrowing-constrained economy, however, the young are prevented from
borrowing for equity investments, so that both the equity and bond returns are exclusively
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determined by the habit-forming middle-aged investors. Due to the inability of the young to hold
equity, resulting from the borrowing constraint together with high fluctuations in equity income,
the demand for equity is reduced, and consequently, the net demand for bonds by middle-aged
consumers is raised. Thus, the middle-aged consumers’ savings decision has a substantial impact
on the high equity premium (i.e., a high risk premium and a lower risk-free rate) under the
borrowing-constrained economy.
To derive the optimum savings decision under the borrowing-constrained economy, we
consider a utility-maximizing representative agent in an OLG economy, where each generation
lives within three discrete generational periods: as members of the young, middle, and old aged
generations. The representative consumer born at t  0 with no endowment assets receives labor
income w0 in period t  0 , w1 in period t  1 , and zero labor income in period t  2 . In the first
period, the consumer receives a relatively low labor income sufficient only to satisfy his or her
first period consumption requirements. In the second period, the consumer receives increased
wage income, and seeks to accumulate sufficient assets for third period consumption. The
consumer retires in the third and last period, and consumes the assets accumulated during the
second period. Savings for smoothing lifetime consumption is done by holding a diversified
portfolio of equity and bonds.
Following Sundaresan (1989) and Constantinides (1990), we assume that the
representative consumer’s utility exhibits habit formation preferences, such that the habit level of
consumption at time t, X t , is a positive fraction of the consumer’s own previous consumption
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level, i.e., X t  Ct 1 . 7 The parameter  is the constant habit persistence parameter and it is
assumed to have a value between 0 and 1, which characterizes the consumption of non-durable
goods and services. 8 Since the representative consumer in the first period does not have the
previous period consumption for habit formation, the consumer is assumed to have a habit
formation utility function from the second period on. Consequently, the consumer has the
following sum of discounted utility flows over three periods:
U
[C0 ]1
[C  X 1 ]1
[C  X 2 ]1
 1
2 2
,
1 
1 
1 
(1)
where C0 , C1 and C2 are consumption at t  0 , t  1 and t  2 , respectively, and all are
assumed to be positive. Habit level at time t is determined by X t  Ct 1 .  is the constant
subjective discount factor, and   0 is the constant RRA coefficient.
The representative consumer faces the following budget constraints over his life cycle:
C0  w 0  S 0 ,
(2)
C1  w1  R1 S 0  S1 , and
(3)
C2  R2 S1 ,
(4)
7
Abel (1990) proposes another type of habit formation, i.e., Catching up with the Joneses, where the habit-
forming behavior is based upon the consumption of other consumers. Comparing his or her own consumption to that
of others, a consumer could get utility from knowing that he or she is consuming more than others.
8
The standard CRRA utility is a special case of the habit formation utility with   0 .   0 implies
negative habit formation, which is applied to the consumption of durable goods.
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where S 0 and S1 are savings of young and middle generations, R1 and R2 are the gross rates of
return for the middle and old generations, and w0 and w1 represent the labor income of the
young and middle generations. With Eq. (2), (3), and (4), the objective function U becomes the
following value function:
V
[w0  S0 ]1
[( w1  R1S0  S1 )   (w0  S0 )]1
[ R S   (w1  R1S0  S1 )]1
.

 2 2 1
1 
1 
1 
(5)
Assuming a constant RRA in the overlapping generation economy, CDM (2002) show
that, imposing the borrowing constraint on the young-aged generation reduces the risk-free rate
and increases the equity return. Under the borrowing constraint, the young generation bears a
restriction on borrowing against future labor income, which is realistic in that human capital
alone does not collateralize major loans in modern economies for reasons of moral hazard and
adverse selection. In addition to this constraint, the young-aged generation’s labor income ( w0 )
is assumed to be at a lower level than the middle-aged generation’s labor income ( w1 ), so that it
is enough only for the consumption at t  0 . These two assumptions together rationalize the zero
savings of the young-aged generation, i.e., S 0  0 . Since the young-aged generation is excluded
from participating in the equity markets, the equity price (and, thus, the equity premium) is
exclusively determined by the middle-aged consumers’ savings decision.
Using comparative statics on the optimum savings decision of middle-aged consumers,
we examine the effect of habit-formation preferences on the optimum savings level. First, we
derive the optimum savings level of the middle-aged generation by solving the maximization
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problem of the discounted utility over the life cycle. Then, we show the positive impact of the
habit formation preferences on the optimum savings level of the middle-aged.
Differentiating the value function V (with S 0  0 ) of Eq. (5) with respect to S1 yields the
first order condition Q as follows:
Q :  2 ( R2   )[ R2 S1*   (w1  S1* )]   [( w1  S1* )  w0 ]  0 ,
(6)
where S1* is the optimal savings level of the middle-aged generation. The second order condition
for the maximization problem is also satisfied as follows:
  [( w1  S1 )  w0 ] 1   2 ( R2   ) 2 [ R2 S1   ( w1  S1 )] 1  0 .
(7)
Given that the first and second order conditions are satisfied, the effect of habit formation on the
dS1*
optimum savings level of the middle-aged generation can be examined by the sign of
.
d
dS1*
Q Q
( 
) can be expressed as follows:
d
 S1*
 2 [ R2 S1*   (w1  S1* )]   2 ( R2   )( w1  S1* )[ R2 S1*   ( w1  S1* )] 1  w0 [w1  S1*  w0 ] 1
.
[w1  S1*  w0 ] 1   2 ( R2   ) 2 [ R2 S1*   (w1  S1* )] 1
Using the budget constraints, Eq. (8) can be rewritten as follows.
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(8)
dS1*  2 [C2  C1 ]   2 ( R2   )C1[C2  C1 ] 1  C0 [C1  C0 ] 1

.
d
 [C1  C0 ] 1   2 ( R2   ) 2 [C2  C1 ] 1
We calibrate Eq. (9) to determine the sign of
(9)
dS1*
under a plausible range of parameters and the
d
consumption set. Since it is always the case that C1  C0 and C2  C1 (in order to have
dS1*
positive utility levels), the denominator of Eq. (9) is positive. Thus, the sign of
is
d
determined by the sign of the numerator of Eq. (9).
Given permissible parameter values for , , , and R2 under several combinations of
consumption paths over the life cycle, calibrating the model readily confirms the following
inequality:
9
 2 [C2  C1 ]   2 ( R2   )C1[C2  C1 ] 1  C0 [C1  C0 ] 1 .
(10)
dS1*
The above inequality indicates that
 0 , which implies that habit formation has a
d
positive impact on the optimum savings level, thereby showing a higher incentive to save more
than with the CDM framework. Solving Eqn. (6) for the equilibrium savings decision, we also
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Boundaries of parameters and variables are determined as follows. For the value of discount factor ( ) we
assign around 0.955 per year. By recalculating  in terms of 20 years (one generation period), we get 0.3982. For the
habit persistence parameter () we use the value 0.615 following Otrok et al. (2002). The value of the relative risk
aversion parameter  is set between 1 and 10 following Mehra and Prescott (1985). For the consumption over
different age cohorts, we use values between 20,000 and 40,000, which are consistent with the Consumer
Expenditure Survey (conducted by the Bureau of Labor Statistics) from 1984 to 1996.
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calculate the optimum savings for the middle-aged agents for various levels of habit persistence
under three different levels of risk aversion (γ = 2, 4, 6), with and without borrowing constraints.
There are two notable findings. (a) The average growth rate in aggregate savings increases
monotonically as habit persistence increases, for both the borrowing-constrained and the
borrowing-unconstrained economies. (b) For a given level of habit persistence, the growth rate in
aggregate savings is greater under the borrowing-constrained economy than under the
borrowing-unconstrained economy. Indeed, our result is consistent with Lahiri (1998, JET) and
Carroll et al. (2000, AER), who also show a positive impact of habit formation on the aggregate
savings – though they do not consider borrowing constraints in their models.
4. Calibration Results for Equity Premium
In this section we calibrate the habit formation preference in the OLG framework for both
the borrowing-constrained and the borrowing-unconstrained economies. The only difference
between the two economies is that while the young agents are able to save under the borrowingunconstrained economy, they are not allowed to save under the borrowing-constrained economy.
Under the borrowing-constrained economy the young agents are prevented from issuing bonds to
buy equity, such that their future income is determined by their forthcoming wages in their
middle age, while the future income of the middle-aged agents is derived from their savings in
bonds and equity.
In our calibration, a bond represents a risk free asset and is supplied at the beginning of
the period. The supply of bonds is fixed at b units in perpetuity. We consider the bond to be a
representative asset for long-term government bond, i.e., the 20 year US Treasury bond. Each
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bond pays a fixed amount of coupon ( b  0 ) every period permanently, which is financed out of
the economy’s capital income payments. The bond price after the coupon payment in period t is
denoted by q tb . This bond price can be interpreted as the value of a claim to the coupon b paid in
perpetuity from period t  1 . The representative consumer born in period t  0 has a zero
endowment of this bond. This consumer purchases z 0b of the bond in period t  0 (when young).
In period t  1 (when middle aged), the consumer adjusts the bond holding to z1b . Then, with a
no bequests assumption, the consumer liquidates all of his/her bonds in period t  2 (when old).
Thus, the bond market clearing condition is that the total demand for bonds by the young and
middle-aged consumers must equal the fixed supply of bonds:
z0b  z1b  b
(10)
Equity is a claim on the dividend stream and pays net dividends d t in period t. Like
bonds, equity is supplied at the beginning of each period and its supply is fixed at one unit in
perpetuity. The equity value after the dividend payment in period t is denoted by q te , which is the
claim to the net dividend stream in perpetuity in period t  1 . With no initial endowment of
equity, the representative consumer purchases z0e equity in period t  0 (when young), adjusts
the equity holding to z1e in period t  1 (when middle aged), and sells his/her entire portfolio in
period t  2 (when old). The equity market clearing condition is that the demand for equity by
the young and middle-aged consumers must equal the fixed supply of equity:
z 0e  z1e  1
(11)
14
The representative consumer receives fixed wage income ( w0 ) in period t  0 , stochastic
wage income ( w1 ) in period t  1 , and zero wage income ( w 2  0 ) in period t  2 . As
mentioned by CDM (2002), the assumptions on the income process reflect three important
aspects of reality; first, the condition that w1  w0  w 2 reflects the incentive that the middleaged are willing to save; second, due to future wage uncertainty, the young would like to borrow
against future income and invest in equity. Under the borrowing constraint, however, the young
cannot borrow for equity investment. Third, due to the no wage uncertainty, the saving middleaged consumers have more flexibility to invest in a diversified portfolio of equities and bonds
With the consumption in period t denoted as C t ( t  0, 1, 2 ), we specify the dynamic
budget constraints for the representative consumer, as follows:
When young : C0  z0b  q0b  z0e  q0e  w0 ,
(12)
When middle aged : C1  z1b  q1b  zte,1  q1e  w1  z0b (q1b  b)  z0e (q1e  d1 ) ,
(13)
When old : C2  z1b (q2b  b)  z1e (q2e  d 2 ) ,
(14)
where C0 , C1 , and C2 are the consumption levels for the young, middle-aged, and old
consumers, respectively. z0e and z1e are the equity levels held by the young and middle-aged
consumers, respectively. z 0b and z1b are the demand levels for bonds held by the young and
middle-aged consumers, respectively. q 0e , q1e , and q 2e are the ex-dividend prices for equity at
each period, q 0b , q1b , and q 2b are the ex-coupon prices of bonds at each period. d1 and d 2 are the
15
dividends paid at period t  1 and t  2 , respectively. Note that under the borrowing constraint,
the young cannot issue a bond for equity investment, and hence z0b  z 0e  0 .
To assess the effect of habit formation in the three-generation OLG framework, a total of
fifty-four different versions of the economy are considered, with different ranges of parameters
set for both the borrowing-constraint and borrowing-unconstraint economies. These economies
range from the conventional ‘constant relative risk averse (CRRA) utility economy’, i.e.,   0 ,
to the strong ‘habit formation utility economy’, i.e.,   0.8 , with an increment of 0.1. We also
set three different level of the risk aversion parameter, i.e.,   2, 4, 6 . In each economy, a
representative consumer maximizes his/her utility by choosing the optimal consumption and
investment policies under the constraints of non-negativity consumption and the budget
constraints specified in equations (12) through (14). To obtain dynamic equilibrium asset returns,
we define the equilibrium condition of each economy, with the following first order conditions,
as specified in Definition-1.
Definition-1
A stationary rational expectation equilibrium in a three-generation economy is a pair of price
functions, q e (k ) and q b (k ) that satisfy:
4
i)
U 0 (C0 )q e    U1 (C1 )  (qke  d ke ) jk
ii)
U 0 (C0 )q b    U1 (C1 )  (qkb  b) jk
iii)
U1 (C1 )q e    U 2 (C2 )  (qke  d ke ) jk
k 1
4
k 1
4
k 1
16
4
iv)
U1 (C1 )q b    U 2 (C2 )  (qkb  b) jk
v)
z1e  1  z0e
vi)
z1b  b  z0b ,
k 1
Where U t (Ct ) is the t period’s marginal utility against consumption, k represents four different
states of the economy (k = 1, 2, 3, 4), and  jk is the Markov transition matrix. Substituting the
dynamic budget constraints from equation (12) through (14) into consumption, the above
equilibrium conditions can be expressed as follows:
vii)
4
U 0 ( w 0  q e z 0e  q b z 0b )q e   U 1 ([ q ke  d ke ]z 0e  [q kb  b]z 0b  w1k  [q ke z1e  q kb z1b ])  (q ke  d ke ) jk
k 1
viii)
4
U 0 ( w 0  q e z 0e  q b z 0b )q b   U 1 ([ q ke  d ke ]z 0e  [q kb  b]z 0b  w1k  [q ke z1e  q kb z1b ])  (q kb  b) jk
k 1
ix)
4
U 1 ([ q e  d ke ]z 0e  [q e  b]z 0b  w1  q e z1e  q b z1b )q e    U 2 ([ qke  d ke ]z1e  [qkb  b]z1b )( qke  d ke ) jk
k 1
x)
4
U 1 ([ q e  d ke ]z 0e  [q e  b]z 0b  w1  q e z1e  q b z1b )q b    U 2 ([ qke  d ke ]z1e  [qkb  b]z1b )( qkb  b) jk
k 1
Instead of specifying the joint process of the wage income of the middle-aged generation
and the dividends, ( w1 , d t ), we specify the joint process of the aggregate income and the wages
of the middle-aged generation, ( yt , w1 ).As in CDM (2002), the aggregate income in period t to
17
all generations is specified as yt  w0  wt1  b  d t . In the calibration, yt and wt1 have two
values for the good and bad states for each variable, such that four possible realizations of the
pair ( yt , wt1 ) are represented by the state variable st  k , where k = 1, 2, 3, 4.
10
After log
linearization, the equilibrium conditions in the Definition-1 can be rewritten as the following
eight equations:
i)
1e q1e   2e q2e   3e q3e   4e q4e  1b q1b   2b q2b   3b q3b   4b q4b  0
ii)
1e q1e  2e q2e  3e q3e  4e q4e  1b q1b  2b q2b  3b q3b  4b q4b  0
iii)
 1e q1e   2e q2e   3e q3e   4e q4e   1b q1b   2b q2b   3b q3b   4b q4b  0
iv)
1e q1e  e2 q2e  e3 q3e  e4 q4e  1b q1b  b2 q2b  b3 q3b  b4 q4b  0
v)
 1e q1e   2e q2e   3e q3e   4e q4e   1b q1b   2b q2b   3b q3b   4b q4b  0
vi)
 1e q1e   2e q2e   3e q3e   4e q4e   1b q1b   2b q2b   3b q3b   4b q4b  0
vii)
1e q1e   2e q2e   3e q3e   4e q4e  1b q1b   2b q2b   3b q3b   4b q4b  0
viii)
 1e q1e   2e q2e   3e q3e   4e q4e   1b q1b   2b q2b   3b q3b   4b q4b  0 ,
10
Following CDM (2002), we set the correlation between yt and wt1 as 0.1. We also confirmed that the
main results are robust with respect to the different levels of the correlation. The joint process of
modeled as a Markov chain with the following transition matrix.
(Y1 , w ) 
1 
 (Y1 , w2 ) 
(Y2 , w11 ) 

(Y2 , w12 ) 
1
1
 jk
(Y1 , w11 )

(Y1 , w12 )

(Y2 , w11 )

 

 
H
H
H

 

(Y2 , w12 )
H

 

18






( y t , wt1 ) is
where each of , , , , , , , and  is the vector consisting of the combination of eight
1
parameters (, , , w1gs , wbs
, y gs , ybs , w0 ). Note that y gs and ybs are the aggregate incomes for
1
the good and bad states, respectively. Also, w1gs and wbs
are the wage incomes of the middle-
aged agent for the good and bad states, respectively. The above dynamic equilibrium equations
can be solved for the eight target price variables ( q1e , q 2e , q 3e , q 4e , q1b , q 2b , q 3b , q 4b ) by employing the
Method of Undetermined Coefficients. For robustness of calibration results, we calibrate the
model over fifty-four different economy sets, for both the borrowing-constrained and the
borrowing-unconstrained economies respectively, by changing the coefficient of relative risk
aversion () to 2, 4, and 6, and the habit forming parameter () from 0.0 to 0.8 with an increment
of 0.1. The set of parameters employed for calibration in this study is reported in Table 1. Note
that, since one-period implies one-generation in our OLG model, all parameters are converted to
twenty-year values, such that the annualized return is defined as the geometric average over a 20year holding period return, i.e., (1+ 20-year hold period return)1/20 – 1.
[Insert Table 1 here]
Table 2 presents the annualized mean equity return, risk-free bond return, and equity
premium under each economy. For comparison, we test fifty-four different economies defined by
three different values of the CRRA parameter ( = 2, 4, and 6), and eight different values of the
habit formation parameter ( = 0.0 through 0.8 with an increment of 0.1) with and without the
borrowing constraint. Economies with   0.0 represent the simple (non-habit forming) CRRA
utility and are used as benchmark economies for comparison to the other habit formation
economies in each group of the CRRA parameters. Likewise, each of the economies with
19
  0.8 is characterized by the highest level of habit formation preference in each CRRA group.
Note that we calibrate the model both with and without borrowing constraints.
There are several notable observations on bond returns ( r f ) from the calibration results.
First, regardless of the borrowing constraint, incorporation of habit formation into the OLG
framework, consistently reduces bond returns ( r f ). Except for the economy with   0.2 and
  0.1 , for all three values of the risk averse parameter , bond returns under a habit formation
utility (   0.1 through 0.8) are all less than those under a non-habit formation utility (   0.0 ).
Second, for a given value of the CRRA parameter, bond returns on average decline as the degree
of habit formation increases. For example, for   0.2 with no borrowing constraint imposed,
bond returns decrease from 7.76% under   0.0 to 7.72% under   0.2 , 6.75% under   0.4 ,
and 5.35% under   0.8 (7.46% under   0.0 to 6.67% under   0.2 , 6.20% under   0.4 ,
and 5.00% under   0.8 , with the borrowing constraint imposed). Also, for   0.6 with no
borrowing constraints imposed, bond returns decrease from 8.93% under   0.0 to 7.88% under
  0.2 , 6.67% under   0.4 , and 5.83% under   0.8 (7.00% under   0.0 to 5.96% under
  0.2 , 4.99% under   0.4 , and 4.16% under   0.8 , with the borrowing constraint
imposed). The results imply that as habit persistence increases, bonds are relatively more
attractive than equity, so that bond returns decrease as the demand for bonds increases. Third, for
a given coefficient level of  and , the magnitude of bond returns is relatively smaller under the
borrowing-constrained economy than under the borrowing-unconstrained economy.
Calibration results on equity returns ( r e ) also show consistent patterns. First, habit
formation on average raises equity returns. For a given level of the CRRA parameter, equity
returns under a habit-forming utility are all greater than the equity returns under a non-habit
CRRA utility for both the borrowing-constraint and the borrowing-unconstraint economies. Also,
20
the level of equity return increases as habit persistence increases. For example, with no
borrowing constraint imposed, the average equity returns increase from 8.89 under   0.0 to
9.17% under   0.2 , 9.24% under   0.4 , and 9.53% under   0.8 (8.23% under   0.0 to
8.38% under   0.2 , 8.53% under   0.4 , and 9.03% under   0.8 , with the borrowing
constraint imposed). Second, the calibration results show that imposing the borrowing constraint
on average reduces the level of equity returns for all levels of habit formation for a given CRRA
coefficient level of . Third, the equity returns increases as the CRRA parameter increases.
The above patterns in the equity and bond returns provide two important observations
about
equity premiums, namely: (a) habit formation raises equity premiums, (b) equity
premiums increases as the degree of habit formation increases, (c) imposing the borrowing
constraint raises the level of equity premiums, for all levels of habit formation, for a given ., and
(d) for a given level of the CRRA parameter, the highest level of equity premium occurs under
the borrowing-constrained economy with the highest level of habit persistence. This confirms
that habit formation preferences within the overlapping-generations framework under the
borrowing-constrained economy can provide the most improvement in resolving the equity
premium puzzle. Figure 1 also shows the above patterns in the calibration results.
The habit-forming middle-aged agent becomes more risk averse to consumption variation
than those with a simple non-habit preference, thereby having a stronger incentive to choose a
safer asset. Thus, bonds become relatively more attractive than equity to the habit-forming
middle-aged agent. Consequently, the net increase in the demand for bonds and the decrease in
equity demand reduce the bond returns and increase the equity returns, thereby raising the equity
21
premium.
11
This pattern is more profound under a no borrowing constraint imposed on the
young. In sum, our calibration results show that (a) the habit formation setting together with an
OLG framework is capable of yielding lower bond returns and higher equity returns than the
standard CRRA utility models, and (b) the borrowing constraint imposed to the young-aged
consumers amplifies the positive effect of habit formation preference on equity premium.
[Insert Table 2 here]
[Insert Figure 1 here]
5. Summary and Conclusions
In this paper, as an attempt to explain the equity premium puzzle, we incorporate habit
formation into an overlapping-generations economy. Using comparative static analyses and
calibrations, we show that incorporating habit formation preferences into the three-period OLG
model has a positive impact on the savings level of the middle-aged consumers. When compared
to the non-habit formation preferences, the explicit inclusion of habit formation within an
overlapping-generations model results in a stronger incentive for agents to secure their future
11
Habit formation preference results in the income effect and the substitution effect on the demand for
equity and bonds. Due to the stronger incentive to save more, habit formation increases the portion of wealth to be
invested in assets as savings. Thus, under the income effect, habit formation increases the demand for equity and
bonds, thereby decreasing both equity and bond returns. In contrast, habit formation causes the representative agent
to have a stronger incentive to invest in a safer asset, in that the agent prefers bonds to equity. Under the substitution
effect, habit formation reduces the bond returns, but increases the equity returns. Our calibrations show a decreasing
pattern of bond returns and an increasing pattern in the equity returns, under habit formation, which implies that the
substitution effect outweighs the income effect in the portfolio choice between equity and bonds for the middle-aged
consumers.
22
consumption, so that the habit-forming middle-aged consumers will save even more than the
middle-aged consumers in the non-habit formation case. Our calibration results verify that the
higher incentive to save, causes a higher demand for bonds, and a lower demand for equity,
thereby yielding a lower risk-free rate and a higher required return for holding equity, than do
any other non-habit forming models. Therefore, we argue that the habit formation preferences in
the OLG framework, with the borrowing constraint imposed on the young generation, can
provide a more satisfactory explanation of the equity premium puzzle.
23
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27
Table 1. Parameters Set on 20-Year Basis
coefficient of relative risk averseness: 
subjective discount rate: 
habit formation parameter: 
average share of income going to labor: E[w0  w1 ] / E[ y]
average share of income going to the labor of the young: w0 / E[ y ]
average share of income going to interest on government debt: b / E[ y ]
coefficient of variation of the 20-year wage income of the middle aged:
 (w1 ) / E[w1 ]
coefficient of variation of the 20-year aggregate income:  ( y ) / E[ y ]
Note: * implies  = 0.96/year.
28
2, 4, and 6
0.44*
0.0, 0.1, 0.2, …, 0.8
0.6 - 0.75
0.10-0.25
0.03
0.20-0.70
0.10-0.30
Table 2. Calibration Results of the Equity Premium with the Equity and Bond Returns
 = 2.0
 = 4.0
 = 6.0
Average
 = 2.0
 = 4.0
 = 6.0
Average
 = 0.0
 = 0.1
 = 0.2
r e : 8.01%
r f : 7.76%
r p : 0.25%
r e : 8.60%
r f : 7.77%
r p : 0.63%
r e : 10.05%
r f : 8.93%
r p : 1.12%
r e : 8.89%
r f : 8.15%
r p : 0.67%
r e : 8.31%
r f : 7.85%
r p : 0.46%
r e : 8.79%
r f : 7.26%
r p : 1.53%
r e : 10.21%
r f : 8.21%
r p : 2.00%
r e : 9.10%
r f : 7.77%
r p : 1.33%
r e : 8.44%
r f : 7.22%
r p : 1.22%
r e : 8.83%
r f : 6.90%
r p : 1.93%
r e : 10.23%
r f : 7.88%
r p : 2.35%
r e : 9.17%
r f : 7.33%
r p : 1.83%
r e : 8.00%
r f : 7.46%
r p : 0.54%
r e : 8.21%
r f : 7.44%
r p : 0.77%
r e : 8.47%
r f : 7.00%
r p : 1.47%
r e : 8.23%
r f : 7.30%
r p : 0.93%
r e : 8.22%
r f : 7.35%
r p : 0.87%
r e : 8.23%
r f : 6.59%
r p : 1.64%
r e : 8.55%
r f : 6.42%
r p : 2.13%
r e : 8.33%
r f : 6.79%
r p : 1.55%
r e : 8.23%
r f : 6.67%
r p : 1.56%
r e : 8.32%
r f : 6.18%
r p : 2.14%
r e : 8.58%
r f : 5.96%
r p : 2.62%
r e : 8.38%
r f : 6.27%
r p : 2.11%
Habit Formation Parameter
 = 0.4
 = 0.5
Without Borrowing Constraint
e
:
8.44%
r
r e : 8.51%
r e : 8.68%
r f : 7.21%
r f : 6.75%
r f : 6.05%
r p : 1.23%
r p : 1.76%
r p : 2.63%
r e : 8.88%
r e : 8.91%
r e : 8.93%
r f : 6.16%
r f : 6.19%
r f : 6.02%
r p : 2.72%
r p : 2.72%
r p : 2.91%
e
e
r : 10.26% r : 10.30% r e : 10.42%
r f : 7.06%
r f : 6.67%
r f : 6.62%
r p : 3.20%
r p : 3.63%
r p : 3.80%
r e : 9.19%
r e : 9.24%
r e : 9.34%
r f : 6.81%
r f : 6.54%
r f : 6.23%
r p : 2.38%
r p : 2.70%
r p : 3.11%
With Borrowing Constraint
r e : 8.30%
r e : 8.52%
r e : 8.58%
r f : 6.63%
r f : 6.20%
r f : 5.56%
r p : 1.67%
r p : 2.32%
r p : 3.09%
r e : 8.33%
r e : 8.38%
r e : 8.61%
r f : 5.56%
r f : 5.62%
r f : 5.29%
p
p
r : 2.77%
r : 2.74%
r p : 3.32%
r e : 8.65%
r e : 8.70%
r e : 8.78%
r f : 5.36%
r f : 4.99%
r f : 4.91%
r p : 3.29%
r p : 3.71%
r p : 3.87%
r e : 8.43%
r e : 8.53%
r e : 8.66%
r f : 5.85%
r f : 5.60%
r f : 5.25%
r p : 2.58%
r p : 2.92%
r p : 3.43%
 = 0.3
29
 = 0.6
 = 0.7
 = 0.8
r e : 8.78%
r f : 5.64%
r p : 3.14%
r e : 8.99%
r f : 5.67%
r p : 3.32%
r e : 10.48%
r f : 6.57%
r p : 3.91%
r e : 9.42%
r f : 5.96%
r p : 3.46%
r e : 8.78%
r f : 5.63%
r p : 3.15%
r e : 9.10%
r f : 5.49%
r p : 3.61%
r e : 10.56%
r f : 5.78%
r p : 4.72%
r e : 9.48%
r f : 5.63%
r p : 3.83%
r e : 8.79%
r f : 5.35%
r p : 3.44%
r e : 9.25%
r f : 5.53%
r p : 3.97%
r e : 10.56%
r f : 5.83%
r p : 4.73%
r e : 9.53%
r f : 5.57%
r p : 4.05%
r e : 8.79%
r f : 5.19%
r p : 3.60%
r e : 8.71%
r f : 5.11%
r p : 3.60%
r e : 8.93%
r f : 4.76%
r p : 4.17%
r e : 8.81%
r f : 5.02%
r p : 3.79%
r e : 9.02%
r f : 5.05%
r p : 3.97%
r e : 8.98%
r f : 4.86%
r p : 4.12%
r e : 9.03%
r f : 4.17%
r p : 4.86%
r e : 9.01%
r f : 4.69%
r p : 4.32%
r e : 9.02%
r f : 5.00%
r p : 4.02%
r e : 9.01%
r f : 4.72%
r p : 4.29%
r e : 9.07%
r f : 4.16%
r p : 4.91%
r e : 9.03%
r f : 4.63%
r p : 4.41%
Figure 1. Equity Premium in different economies
0.05
0.04
rp
0.03
0.8
0.7
0.6
0.5
0.4

0.3
0.2
0.1
0.02
0.01
0
2

0
4
6
(A) Borrowing-Unconstrained Economies
0.05
0.04
0.03
0.8
0.7
0.6
0.5
0.4
0.3 
0.2
0.1
rp
0.02
0.01
0
2

0
4
6
(B) Borrowing-Constrained Economies
30