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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. 6 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 7 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 8 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. 9 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 10 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. 11 (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 9 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. 12 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 13 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. 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Weil, P., 1989. The equity premium puzzle and the risk free rate puzzle, Journal of Monetary Economics 24, 401-421. 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