Happiness Maintenance and Asset Prices

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Happiness Maintenance and Asset Prices Antonio Falato 2008-19 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Happiness Maintenance and Asset Prices Antonio Falato1 This draft: January, 2008 1Federal Reserve Board - Division of Research and Statistics, Washington DC. Phone: (202) 452-2861. Email: antonio.falato@frb.gov. This paper is a revised version of Chapter 2 of my Ph.D. dissertation at Columbia University. John Donaldson provided much needed encouragement and guidance and invaluably contributed to deepening my understanding and clarifying the exposition oftheconceptspresentedinthepaper. IalsoacknowledgetheadviceandsupportofThomasCooley, TanoSantos,andPaoloSiconol(cid:133). IthankYacineAit-Sahalia,GurdipBakshi,AlbertoBisin,Andrew Caplin, George Constantinides, Jon Elster, Ido Erev, John Geanakoplos, Gur Huberman, Dalida Kadyrzhanova, Rajnish Mehra, Janet Metcalfe, Edmund Phelps, Steven Zeldes; participants at the 2004 Midwest Finance Association Meeting and the "Equity Premium" Conference (University of Exeter); and seminar participants at Columbia University, and the University of Montreal (HEC) for helpful comments. All remaining errors are mine. The views expressed in the paper do not necessarily re(cid:135)ect the views of the Federal Reserve Board or its sta⁄.

Abstract This paper constructs a simple dynamic asset pricing model which incorporates recent evidence on the in(cid:135)uence of immediate emotions on risk preferences. Investors derive direct utility from both consumption and (cid:133)nancial wealth and, consistent with the happiness maintenance feature documented by Isen (1999) and others, become more cautious toward theirwealthingoodtimes. Mildpro-cyclicalchangesinriskaversionoverwealthcauselarge pro-cyclical(cid:135)uctuationsinthecurrentprice-dividendratiowhich,duetogeneralequilibrium restrictions,translateintocounter-cyclicalvariationinthecurrentconsumption-wealthratio and, in turn, in expected future returns. With a realistic consumption growth process and reasonable preference parameters, the model generates a sizable equity premium, a low and stable risk-free rate, volatile and predictable stock returns, and price-dividend and Sharpe ratios in line with the data. Keywords: state-dependent utility, a⁄ect and decision making, equity premium puzzle. JEL Codes: D81, D91, E44, G12.

I. INTRODUCTION Empirical research in (cid:133)nance has extensively documented the failure of traditional asset pricingmodelstoaccountforthehistoricallyobservedlevel, volatility, andcyclicalbehavior of asset returns. The last decade has witnessed some progress in bringing theory closer to the data (see, for example, Campbell and Cochrane (1999) and Bansal and Yaron (2004)). Nevertheless, the identi(cid:133)cation of the fundamental sources of aggregate risk that drive expected returns remains an open challenge. This paper proposes a new analytical framework based on the link between immediate emotionsandriskpreferences. Thestartingpointofmyanalysisisthewellreplicated(cid:133)nding that individuals who feel good are more risk averse than individuals who feel neutral (see Isen (1999) for a thorough review of the evidence). This (cid:133)nding is consistent with the notion that individuals have a preference for happiness maintenance: accepting a gamble when happy puts their happiness, in addition to any tangible stake, at risk. I embed happiness maintenance preferences into an otherwise standard equilibrium asset pricing model along the lines of Lucas (1978) and Mehra and Prescott (1985). In my model, investorsderivedirectutilityfrombothconsumptionand(cid:133)nancialwealth. Thecurrentstate of the economy changes their attitude toward wealth gambles, which I refer to as "hedonic" risk aversion: in good times investors prefer not to (cid:148)push their luck(cid:148)- i.e., they become more conservative toward their portfolio risk. I show, both analytically and numerically, that happiness maintenance goes a long way toward accounting for stock market facts. With a realistic consumption growth process and reasonable risk aversion and time preference parameters, my model delivers a sizable equity premium, a low and stable risk-free rate, volatile and predictable stock returns, and price-dividend and Sharpe ratios in line with the data. Whatistheeconomicmechanismbehindtheseresults? Inessence,itisa"leveragee⁄ect" that makes equity riskier for an investor with happiness maintenance preferences. To see this point, consider that changes in hedonic risk aversion induce changes in investors(cid:146)views about how valuable wealth is relative to consumption. This is the reason why the level of 1

the wealth to consumption ratio, rather than its volatility, emerges as an extra source of risk: even if wealth were to stay constant, investors in my model would consider equity risky, due to the need to hedge swings in their hedonic risk aversion. Thus, changes in hedonic risk aversion increase the standard deviation of investors(cid:146)intertemporal marginal rate of substitution. The increase in perceived risk leads investors to require a higherreturn on risky assets for any given level of consumption and wealth risk. It is worth stressing the two main features that distinguish happiness maintenance from previous models. First, in contrast to previous models that add a second term to the utility function, such as (cid:133)nancial or housing wealth (e.g. Bakshi and Chen (1996), Piazzesi et al (2005), Barberis et al (2001)), happiness maintenance generates a large equity premium without the need for (cid:133)nancial wealth to be too volatile. This is the case since the leverage e⁄ect depends on the level, rather than the volatility, of the wealth-consumption ratio. Second, in contrast with habit persistence models that rely on high and strongly countercyclical e⁄ective risk aversion (e.g. Campbell and Cochrane (1999), but also Barberis, Huang and Santos (2001) who need strongly counter-cyclical loss aversion), low and mildly pro-cyclical hedonic risk aversion is su¢ cient for the leverage e⁄ect to be operative. Thus, my results highlight that counter-cyclical risk aversion is not a necessary condition for equilibrium asset pricing models to replicate stock market facts. Happiness maintenance di⁄ers from the standard consumption-based approach of CampbellandCochrane(1999)alsoin termsofthepredictionsofthemodel. Intheconsumptionbased model, stock return volatility is generated through changes in risk aversion that are drivenbyconsumption. Thus, incontrastwiththedata, stockreturnsandconsumptionare strongly correlated. My wealth-based framework weakens the correlation between returns and consumption since stock return volatility is driven not only by consumption, but also by wealth. Thus, the multi-factor structure of my model distinguishes it from standard consumption-based approaches. My study makes the following four additional contributions to the literature. First, I identifyafundamentallydi⁄erentsourceofequityrisk, whichiscomplementarytostandard consumption and wealth volatility (see Kocherlakota (1996) for a comprehensive survey). 2

Second,mymodelcontributestotheliteratureonpredictability(forasurveyseeCochrane (2000)). In fact, in my model the consumption-wealth ratio predicts expected returns. Lettau and Ludvigson (2001) provide empirical evidence of a common component in expected returns and the consumption-wealth ratio. With happiness maintenance preferences, such commoncomponentemergesinequilibrium, astheleveloftheconsumption-wealthratiodirectlya⁄ectstheriskinessofstocks. Thus,mymodelo⁄ersageneralequilibriumperspective over the earlier empirical (cid:133)ndings. Third, my work contributes to the wealth-based asset pricing literature (e.g. Bakshi and Chen (1996), Epstein and Zin (1989, 1991)) by addressing the Campbell (1993) critique. This class of models identi(cid:133)es the volatility of wealth as a risk factor. However, these models do not generate volatile wealth with low risk aversion since they imply a constant price/dividend ratio. In other words, the fact that consumption is smooth and wealth is volatile is itself a puzzle that must be explained. My full-(cid:135)edged general equilibrium setting directly address this issue: even with low risk aversion, the leverage e⁄ect allows me to generate endogenously a realistic volatility of wealth. Finally, my paper contributes to the growing literature on emotions and investor behavior (e.g. Lo and Repin (2001), Mehra and Sah (2001); see Loewenstein (2000) for a survey). Saunders (1993) and Kamstra, Kramer and Levi (2000) document an empirical relationship between weather or length of the day and asset returns, which they interpret as evidence of an impact of moods on asset prices. Lo and Repin (2001) o⁄er direct evidence of an emotional reaction of investors to risk by documenting signi(cid:133)cant correlation between changes in stock market traders(cid:146)cardiovascular variables and market volatility. Mehra and Sah (2001) formally explore the role of small (cid:135)uctuations in investors(cid:146)preferences within a non-rational expectations framework. To the best of my knowledge, my paper represents the (cid:133)rst comprehensive equilibrium treatment of emotions and asset prices. Outline of the paper The (cid:133)rst section develops a formal representation of happinessmaintenance preferences which are then embedded into an otherwise standard equilibrium asset pricing model. The second section characterizes equilibrium returns in an economy 3

populated by investors with happiness-maintenance preferences. The results of a simple calibration exercise are in the third section, which investigates the quantitative implications of happiness maintenance. The fourth section concludes. Algebraic derivations and proofs are in Appendix A. Tables and Figures are in Appendix B. II. A WEALTH-BASED CAPITAL ASSET PRICING MODEL In this section I develop a formal representation of investors(cid:146)happiness maintenance preferencesandembedintoanotherwisestandarddynamicequilibriumassetpricingmodel. II.1 Setup Mine is a standard (cid:148)endowment economy(cid:148)(Lucas (1978), Mehra and Prescott (1985)) populated by a large number of in(cid:133)nitely-lived investors, who are identical with respect to their preferences, endowments and expectations and face a standard consumption/saving problem. Given these assumptions, it is customary to aggregate investors into a representative agent. There is one consumption good. The only source of income is a large number of identical and in(cid:133)nitely-lived fruit trees, each in (cid:133)xed supply. Without loss of generality, the supply of trees is normalized to unity and it is assumed that there exists one tree per individual, sothattheamountoffruitproducedbyatreeinperiodt, denotedy , represents t the output or dividend per capita. Fruits are non-storable, cannot be used to increase the number of trees and can only be used for consumption. They are uncertain and evolve according to y = x y , where x (cid:21) ; ... , (cid:21) is the growth rate of output which t+1 t+1 t t+1 1 n 2 f g follows a given stationary stochastic process to be detailed on later. Each tree has a single perfectly divisible equity claim outstanding on it. In each period there is a spot market for the consumption good and a (cid:133)nancial market in which equity shares are exchanged at a price p : Consequently, the gross rate of return on equity holdings from period t to period t t+1 is de(cid:133)ned as R = pt+1+yt+1: A one-period risk-free asset in zero net supply at a t+1 pt f price p completes the description of the (cid:148)technology(cid:148)side of the economy. It pays a gross t interest rate R f = 1 . t pf t 4

Investor preferences.(cid:151) Investors derive utility from a composite good, g ; which includes both current (pert capita) consumption, c ; and current (per-capita) (cid:133)nancial wealth, w . They rank random t t sequences of the composite good according to U = E 1 (cid:12)tu(g ); u(g ) = g t 1 (cid:0) (cid:11) 0 0 t t 1 (cid:11) t=0 (cid:0) X where (cid:12) (0;1) is the subjective discount factor, E [ ] is the expectations operator con- 0 2 (cid:1) ditional on the information available at time zero, and (cid:11) > 0 has the conventional interpretation of the parameter of relative risk aversion. The composite good, g ; represents my t main departure from standard assumptions and the remaining part of this section details its connection with happiness maintenance. In contrast to standard models, investors(cid:146)(cid:133)nancial wealth, w ; enters their preferences t directly over and above the indirect utility of the consumption services it provides;1 that is g = g(c ;w ;(cid:18) ) = c1 (cid:18)tw(cid:18)t t t t t t(cid:0) t The parameter (cid:18) [0;1] controls the (relative) demand for (cid:133)nancial happiness: values of t 2 (cid:18) close to the lower (upper) bound of the [0;1] interval correspond to a low (high) demand t for happiness relative to consumption. The level of (cid:133)nancial wealth measures (cid:133)nancial performance. It is introduced directly into the utility function to capture the wide range of non-consumption related pleasures associated with ownership of (cid:133)nancial assets, such as, for example, power and social status, but also sense of security and control from having resources. Total(cid:133)nancialwealth, w ;isde(cid:133)nedbythevalueofthebeginning-of-periodasset t holding, s ; and dividends, y ; at the current prices, p ; i.e. w = (p +y )s : t t t t t t t A⁄ect-maintenancepreferencesaremodelledasaninstanceofstate-dependentpreferences by postulating a state-dependent demand for (cid:133)nancial happiness. Appendix A gives a standard set of axioms and a representation theorem for these preferences. To facilitate intuition on the connection between (cid:18) and happiness maintenance, it is useful to rewrite t the utility function as U = E 1 (cid:12)t c t 1 (cid:0) (cid:11) w t (1 (cid:0) (cid:11))(cid:18)t (1) 0 0 1 (cid:11) c t t=0 (cid:0) (cid:18) (cid:19) X 5

Investors(cid:146)expected utility has a decision component, which depends on current consumption, and a hedonic component, which depends on the performance of their portfolios. Financial income relative to consumption is assumed to provide a (cid:133)rst approximation indicator of this performance and, hence, a direct source of happiness with respect to (cid:133)nancial wealth. Thus, the term wt (1 (cid:0) (cid:11))(cid:18)t formalizes the intuition that investors(cid:146)utility depends ct on their experienced hap(cid:16)pin(cid:17)ess. If (cid:18) = (cid:18) = 0; the model reduced to a standard wealtht 6 based setup (Bakshi and Chen (1996), Epstein and Zin (1989)). When (cid:18) = (cid:18) = 0; the t model reduces to the standard consumption-based asset pricing framework. Thus, these extreme parametrizations provide useful benchmarks to gauge the marginal contribution of happiness maintenance. One important implication of (1) is that a mean preserving spread of (cid:133)nancial wealth (relative to consumption) directly reduces investors(cid:146)utility. If (cid:133)nancial wealth is a source of happiness, the desire to maintain happiness should determine the size of the reduction of investors(cid:146)utility. In fact, while investors(cid:146)relative risk aversion over g gambles, (cid:11); is t constant, investors(cid:146)hedonic risk aversion, a ; that is their risk aversion2 over portfolio risk, t is a simple function of their demand for (cid:133)nancial happiness. By de(cid:133)nition,3 hedonic risk aversion is a = ((cid:11) 1)(cid:18) +1 (1;(cid:11)) t t (cid:0) 2 As far as (cid:11) > 1 - a restriction maintained throughout the paper and necessary to satisfy the standard transversality condition for the in(cid:133)nite horizon problem4 - a is increasing in t (cid:18) . In this sense, for any given (cid:11); (cid:18) determines by how much a mean preserving spread t t of (cid:133)nancial wealth (relative to consumption) reduces investors(cid:146)utility. I use the following simple speci(cid:133)cation: (cid:18) = (cid:18)x (n); (cid:18) > 0 (2) t t n n 1 1 y t (cid:28) x t (n) = x t (cid:28) = (cid:0) n+1 (cid:0) n+1 y t (cid:28) 1 X (cid:28)=0 X (cid:28)=0 (cid:0) (cid:0) where x (n) is the average of the recent n states of the economy and (cid:18) is an increasing t t functionofthestateoftheeconomy. Noticethat,foranygiven(cid:11);a ishigheringoodtimes, t 6

i.e., risk aversion is pro-cyclical: in good times investors become more risk averse toward (cid:133)nancial wealth (relative to consumption) in an attempt to maintain their happiness. The parameter n controls how far back in the past investors look to determine whether times are good or bad. When n = 0; de(cid:133)nition (2) simpli(cid:133)es to (cid:18) = (cid:18)x = (cid:18) yt : In this t t yt 1 (cid:0) case good times are measured simply by the current state of the economy. If n 1; a mean (cid:21) of the recent past states of the economy measures investors(cid:146)hedonic risk aversion. This is broadly consistent with the psychological evidence on incidental emotions, which documents the existence of a durability or projection e⁄ect (see Loewenstein et al. (2001) for a survey): investors(cid:146)current moods are a⁄ected by the recent economic trend. An alternative interpretation is thatinvestors(cid:146)views aboutcurrent times are formed byextrapolatingfrom the recent past. In summary, the choices of an investor with a⁄ect-dependent preferences are fully characterized by the triple ((cid:12);(cid:11);(cid:18) ); i.e., respectively, by his subjective rate of time preference, t (cid:12); his relative risk aversion, (cid:11); and his relative demand for (cid:133)nancial happiness, (cid:18) : Variables t (s ;y ;(cid:18) ) are su¢ cient relative to the entire history of shocks up to, and including, time t t t t for predicting the subsequent evolution of the economy. They thus constitute legitimate state variables for the model. Further discussion of assumptions Happiness maintenance is a well replicated (cid:133)nding (for a detailed overview of the experimental psychology (cid:133)ndings see Isen (1999) and Appendix A). While earlier studies indicate a tendency toward conservatism for individuals in a good mood, the study by Isen et al. (1988) is most relevant since it focuses directly on the notion of risk aversion typically employed in economics and (cid:133)nance. The study examined the slope of the utility associated with various outcomes, as a function of positive a⁄ect induced by means of a small bag of candy. Participants were asked to make choices between pairs of simple 50-50 gambles in such a way that a set of indi⁄erence points could be found and individual utility functions constructed. The average utility curves were computed for the two groups and people in whom positive a⁄ect had been induced displayed steeper utility function than controls. 7

Finally, notice that in my speci(cid:133)cation u( ) is iso-elastic, which insures stationarity of (cid:1) returnsandisbroadlyconsistentwithanestablishedstylizedfactoftherelationshipbetween individualemotionalwell-beingandaggregateeconomicconditions(seeEasterlin(2000)and FreyandStutzer(2002)forrecentsurveys): thereisnoclearcuttrend, positiveornegative, in self-reported subjective well-being over periods of 20 to 30 years in rich countries. In particular, in the United States between 1946 and 1991, per capita real income has risen by a factor of 2.5, but happiness, on average, remained constant. II.2 The consumption-saving problem Given the asset price function, p = p(s ;y ;(cid:18) ); initial asset holdings, s ; the initial state t t t t 0 of the economy, y ; and initial a⁄ective state, (cid:18) ; the problem of the (cid:148)stand-in(cid:148)investor is 0 0 to choose a sequence of plans for consumption, c ; and end-of-period asset holdings, s ; t t+1 that maximizes her present discounted expected utility subject to the budget constraint. Formally, the investor chooses consumption and asset holdings that solve the following 1 max E (cid:12)tu(g(c ;w ;(cid:18) )) (3) t t t f ct;st+1 g t=0 X c +p s = (p +y )s = w t t t+1 t t t t c > 0; s (0;1]; and s ;y (cid:18) given t t+1 0 0 0 2 where u(g ) is de(cid:133)ned in (1) and (cid:18) is de(cid:133)ned in (2): t t This problem admits a recursive formulation which, through standard perturbation arguments, delivers5 the following Euler equation: u (g(c ;w ;(cid:18) ))p c t t t t = (cid:12)E [(u (g(c ;w ;(cid:18) ))+u (g(c ;w ;(cid:18) )))(p +y )] (4) t c t+1 t+1 t+1 w t+1 t+1 t+1 t+1 t+1 where u and u denote the partial derivative of the utility function with respect to conc w sumption and wealth, respectively. The basic intuition is common to a broad class of wealth-based asset pricing models: by reducing consumption by p units in the (cid:133)rst pet riod, the agent can purchase one unit of the asset, thereby raising consumption by s t+1 8

units in the second period. Importantly, this decision entails a portfolio adjustment which has a direct e⁄ect on investor(cid:146)s utility, as indicated by the second term on the right-hand side of equation (4). The distinctive feature of happiness maintenance is that the extent to whichportfolioadjustmentschangeinvestors(cid:146)utilitydependsonhedonicriskaversion, a . t+1 Moreover,investorsfullyanticipatechangesintheirhedonicriskaversionand,consequently, can fully hedge this extra source of uncertainty. The Euler equation is derived using only the preferences and budget constraint of the investor. Before exploiting the restrictions imposed by equilibrium on asset returns, it is useful to consider the consequences of the individual budget constraint and preferences for asset returns. The risk-free interest rate is given by 1 R = (5) f (cid:12)E ct+1 (cid:0) (cid:11) wt+1=ct+1 1 (cid:0) at+1 k wt at (cid:0) at+1 ct wt=ct t+1 ct (cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) where a t = ((cid:11) (cid:0) 1)(cid:18) t +1 is the hedonic risk aversion, k t+1 (cid:17) 1 (cid:0) 1 (cid:18)t (cid:18) + t 1 1+ 1 (cid:18)t (cid:18) + t+ 1 1w ct t + + 1 1 ; (cid:0) (cid:0) and (cid:18) and (cid:18) are de(cid:133)ned in (2). How does happiness mainte(cid:16)nance h(cid:17)el(cid:16)p to match a rel(cid:17)at t+1 tively low rate of returns on riskless securities? To see the intuition, consider the following two parametric choices. First, when (cid:18) = (cid:18) = (cid:18) = 0 the risk-free rate reduces to the t t+1 standard consumption-based asset pricing framework of Mehra and Prescott (1985): 1 R = (6) f (cid:11) (cid:12)E ct+1 (cid:0) ct (cid:20) (cid:21) (cid:16) (cid:17) Under (6); it is di¢ cult to generate a low risk-free rate since the very feature that helps to explaintheequitypremium-i.e., ahigh curvature(cid:11) ofutilityoverconsumption -alsoleads to a strong desire to smooth consumption intertemporally, generating high interest rates. As standard in wealth-based models (e.g. Bakshi and Chen (1996)) and in models that introduce a second variable into the utility function (e.g. Piazzesi et al (2005)), happiness maintenance adds a second factor, wealth growth, to the determination of returns. To see this, consider the case when (cid:18) = (cid:18) = (cid:18) = 0 : t t+1 6 1 R = f (1 (cid:11))(1 (cid:18)) 1 (1 (cid:11))(cid:18) (cid:12)E ct+1 (cid:0) (cid:0) (cid:0) wt+1 (cid:0) k~ ct wt t+1 (cid:20) (cid:21) (cid:16) (cid:17) (cid:16) (cid:17) 9

where k~ 1+ (cid:18) ct+1 : Adding a second risk factor could in principle help, since it t+1 (cid:17) 1 (cid:18)wt+1 (cid:0) allows to expla(cid:16)in the equity(cid:17)premium with a lower curvature (cid:11) of utility, thus contributing to a lower risk-free rate relative to Mehra and Prescott (1985): However, it is well known that adding wealth as a second factor does not really help to resolve the puzzles, since high riskaversion is stillneededtogenerate enoughvolatilityof wealthinequilibrium(Campbell (1993)). Happiness maintenance adds an extra source of volatility, a ; which is fundamentally t di⁄erent from standard consumption and wealth uncertainty. To see this, consider the last termin the denominatorof (5); wt at (cid:0) at+1 : since hedonic riskaversion changes, investors(cid:146) ct marginal utility varies even if th(cid:16)ere(cid:17)is no consumption or wealth uncertainty. Importantly, large consumption or wealth swings are not needed to generate enough volatility since happiness maintenance introduces a "leverage e⁄ect," which depends on the level, rather than the growth rate, of the wealth/consumption ratio. It is this leverage e⁄ect that gives to mild variation in hedonic risk aversion potency for asset pricing. The intuition for how the leverage e⁄ect works is straightforward: changes in hedonic risk aversionmakea a stochasticallynegativeorpositive. Thus,incontrasttoothermodels t+1 t (cid:0) withstate-dependentpreferencessuchasCampbellandCochrane(1999),evensmallchanges in hedonic risk aversion can generate substantial volatility as the wealth to consumption ratio operates as either a discount or a compound factor depending on whether a a t+1 t (cid:0) is negative or positive. This e⁄ect magni(cid:133)es the contribution of consumption and wealth volatilitytothestandarddeviationofinvestors(cid:146)intertemporalmarginalrateofsubstitution, thus increasing investors(cid:146)perceived risk. As aresult, investors seekingsafetyin the risk-free asset bid its price up, which lowers the risk-free return. Insummary, sincethemainmechanisminmymodeloperatesthroughchangesinhedonic risk aversion, I do not need high curvature of the utility function to match the equity premium. This is the way my model contributes to the resolution of the risk-free rate puzzle. Moreover, since I only need small changes in hedonic risk aversion, I can maintain a relatively stable interest rate. The leverage e⁄ect implied by happiness maintenance depresses the price of risky assets. 10

In fact, the expected return on equity is c t+1 (cid:0) (cid:11) w t+1 =c t+1 1 (cid:0) at+1 w t at (cid:0) at+1 1 = (cid:12)E k R ; (7) t+1 t+1 c w =c c " t t t t # (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) and the implied (conditional) expected premium demanded by the investor to hold her wealth in equities, E(cid:5) = ER R ; is t+1 f (cid:0) c t+1 (cid:0) (cid:11) w t+1 =c t+1 1 (cid:0) at+1 w t at (cid:0) at+1 R cov k ;R (8) f t+1 t+1 (cid:0) c w =c c t t t t ! (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The expected premium, as stated in (8), depends upon the familiar covariation of the intertemporal marginal rate of substitution and the return on equities. In line with the discussion of the risk-free rate, the leverage e⁄ect that operates under happiness maintenancerepresentsanovelsourceofriskasthecovariancein(8)isnonzeroevenwithconstant consumptionandwealthgrowth. Inotherwords,aninvestorwitha⁄ect-maintenancepreferences would still require a premium to hold equities even if she were to face no consumption or wealth uncertainty. To see the intuition for this last point, consider the term wt at (cid:0) at+1 in (8) : equity is ct riskier for an investor with happiness maintenance preference(cid:16)s sin(cid:17)ce she fears that changes in hedonic risk aversion will change her views about how valuable wealth is relative to consumption. This is the reason why the level of the wealth to consumption ratio matters for risk:asa a stochasticallychangesfromnegativetopositive,thewealthtoconsumption t+1 t (cid:0) ratio induces lower discounting. Thus, happiness maintenance introduces a leverage e⁄ect that increases the standard deviation of investors(cid:146)intertemporal marginal rate of substitution. The increase in perceived risk leads investors to require a higher return on risky assets for any given level of consumption and wealth risk. Itisworthstressingthreeimportantfeaturesthatdistinguishhappinessmaintenancefrom previous resolutions of the equity premium puzzle. First, the wealth to consumption ratio is bounded even in a growing economy, since the budget constraint of the investor holds. Thus, in contrast to previous consumption-based models with state-dependent preferences (e.g. Danthine et al. (2003) and Gordon and St-Amour (2000)), happiness maintenance preferences preserve stationarity of returns. Second, in contrast to previous models that 11

add a second term to the utility function, such as (cid:133)nancial or housing wealth (e.g. Bakshi and Chen (1996) and Piazzesi et al (2005)), happiness maintenance does not need (cid:133)nancial wealth to be too volatile to be able to generate large enough premiums. This is the case since the leverage e⁄ect increases the impact of wealth volatility on the volatility of the pricing kernel. Finally, in contrast with most habit persistence models that rely on a strongly counter-cyclical risk aversion,6 as the next section will make clearer, I only need mild variation in hedonic risk aversion for the leverage e⁄ect to be operative. Thus, happiness maintenance does not need to make interest rates counterfactually volatile. III. AGGREGATE ASSET PRICING IMPLICATIONS OF HAPPINESS MAINTENANCE Can a⁄ect maintenance provide a satisfactory analytical account of the main stylized facts of (cid:133)nancial markets? To address this question, I characterize equilibrium asset prices and returns in an economy populated by investors with happiness maintenance preferences. I then study the quantitative properties of the model vis-a-vis the historical record of US stock returns in the post-war period. III.1 Equilibrium characterization of returns Given the (cid:133)xed supplies of goods and assets, it is trivial to determine quantity choices in a competitive equilibrium: all fruits are consumed during the period in which they are produced, i.e., c = y , and the representative investor holds all her wealth in the risky t t asset, s = s = 1. Since consumer-investors are assumed to have identical preferences, t t+1 per-capita consumption of the representative investor equals aggregate consumption, which then equals aggregate output. Equilibrium is characterized by the asset price function that supports this allocation, that is by the function p = p(y ;s ;(cid:18) ) that solves (4): Loosely speaking, the optimality t t t t conditions that correspond to the solution of the investor(cid:146)s problem de(cid:133)ned in (3) and the requirement of market clearing in the aggregate provide the equilibrium pricing equation 12

for the risk-free and risky assets. For the probability structure speci(cid:133)ed in the next section, Appendix A contains a proof of the existence of equilibrium. The following proposition o⁄ers a characterization of equilibrium asset returns. Proposition 1 Given the preferences in (3) the equilibrium risk-free interest rate and equilibrium expected return on equity satisfy, respectively, 1 R = (9) f (cid:12)E x (cid:11) ft+1+1 (1 (cid:0) (cid:11))(cid:18)t k(f )(f +1)(1 (cid:11))((cid:18)t+1 (cid:18)t) (cid:0)t+1 ft+1 t+1 t+1 (cid:0) (cid:0) (cid:20) (cid:21) (cid:16) (cid:17) and 1 = (cid:12)E x (cid:11) f t+1 +1 (1 (cid:0) (cid:11))(cid:18)t k(f )(f +1)(1 (cid:11))((cid:18)t+1 (cid:18)t)R (10) (cid:0)t+1 f +1 t+1 t+1 (cid:0) (cid:0) t+1 " t # (cid:18) (cid:19) where k(f t+1 ) = 1 (cid:0) 1 (cid:18)t (cid:18) + t 1 1+ 1 (cid:18)t (cid:18) + t+ 1 1 (f t+1 +1) (cid:0) 1 ; (cid:18) t and (cid:18) t+1 are de(cid:133)ned in (2) and (cid:0) (cid:0) f = pt. (cid:16) (cid:17)(cid:16) (cid:17) t yt Proof. see Appendix A. An important quali(cid:133)cation of my results transpires from this characterization of equilibrium returns. Since all the investor(cid:146)s wealth is (cid:133)nancial, in equilibrium there is a tight mapping between the properties of the wealth to consumption ratio and the price-dividend ratio. However, recent empirical work by Menzly, Santos and Veronesi (2004) and Lustig, Van Nieuwerburgh and Verdelhan (2007) has shown that these two ratios behave quite differently in the presence of human capital, government transfers or, in general, non (cid:133)nancial income. While beyond the scope of this paper, integrating non-(cid:133)nancial wealth into the model is an important question for future research. A Note on Aggregation.(cid:151) The equilibrium pricing equations in (9) and (10) are derived under the assumption that investors are homogeneous. This is certainly a strong assumption. Investors may be heterogeneous along a variety of dimensions, which raises the question of aggregation. Since in my economy (cid:133)nancial markets are competitive and complete, and investors(cid:146)preferencessatisfytheaxiomsofexpectedutility(seeAppendixAforaformalproof), existence 13

of a representative (single agent) economy with the same aggregate consumption series as the heterogeneous agent economy and the same asset price functions is guaranteed by construction (Prescott and Mehra (1980)). However, it is well known that these properties do not guarantee strict-aggregation, i.e. that the representative agent can be constructed independent of the underlying heterogeneous agent economy(cid:146)s initial wealth distribution. The question of whether happiness maintenance admits strict-aggregation is important but beyond the scope of this paper. However, there are reasons to believe that the intuition ofmymodelislikelytogothroughunderseveraltypesofinvestorheterogeneity. Theresults should generalize to any particular form of heterogeneity such that aggregate hedonic risk aversion still varies with the aggregate state of the economy. Another form of heterogeneity that does aggregate is when investors have di⁄erent wealth levels, but identical wealth to income ratios, a case that can be modelled by having several cohorts of investors, each with a continuum of equally wealthy investors. While this is speculative, there are good reasons to believe that my intuition will survive more extreme forms of heterogeneity, such as, for example, when investors di⁄er in their hedonic risk aversion, a. Even if a varies across investors, as far as each individual investor has pro-cyclical hedonic risk aversion there is no reason to believe that this property will be lost in the aggregate. III.2 Quantitative assessment In order to gain insight into the quantitative e⁄ect of a⁄ect maintenance on aggregate assetreturns, Icomputenumericalsolutionstotheproblemoftheinvestorde(cid:133)nedin(3)for various parameter choices and use these solutions to compute the associated time averaged risk-free rate, market rate, and risk premium as implied by equations (9) and (10). The choice of speci(cid:133)c values for the behavioral parameters and the aggregate growth rate of output, x , is crucial for the empirical evaluation of the model. t Calibration of the (cid:148)technology(cid:148)side of the model is standard. I consider two alternative stochastic processes for the aggregate output growth, x . In the (cid:133)rst case, as in Mehra and t 14

Prescott (1985), x follows a Markov chain and the number of states n is limited to two t ((cid:21) ;(cid:21) );with transition probabilities given by (cid:5) and de(cid:133)ned as 1 2 (cid:21) = 1+(cid:22)+(cid:14); (cid:21) = 1+(cid:22) (cid:14) (11) 1 2 (cid:0) (cid:25) 1 (cid:25) (cid:5) = (cid:0) 2 3 1 (cid:25) (cid:25) (cid:0) 4 5 Theparameters(cid:22);(cid:14) and(cid:25) arechosentomatchrespectivelythemean, standarddeviation and(cid:133)rstorderautocorrelationofaggregateconsumptiongrowthintheUSeconomybetween 1889 and 1985. The values required are (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: In the second case, aggregate output growth follows an iid lognormal process (as in Campbell and Cochrane (1999)), i.e. logx = (cid:22)+(cid:14)(cid:15) (12) t+1 t+1 Investor preference parameters, ((cid:12);(cid:11);(cid:18) ); are chosen based on evidence from (cid:133)eld studies t andtheconsensusviewinthepreviousliterature. Itiscustomarytochooseatimepreference coe¢ cient, (cid:12); close to and lower than one. A negative rate of time preference has been shown to be e⁄ective in (cid:148)solving(cid:148)the risk-free rate puzzle, but introspection provides a strong argument in support of a positive rate of time preference. Consequently, I choose the (cid:12) lower than one that optimizes the performance of the model with respect to the risk-free rate. TheliteratureafterMehraandPrescott(1985)deemsreasonableriskaversioncoe¢ cients within the (0;10) interval. Field studies support this choice, since an (cid:11) beyond 10 would implyrejectionsofconsumption(andwealth)betsthatmostsubjectsinexperimentsdonot turn down. I choose risk aversion close to the lower bound of the interval (0;10) to explore the full extent to which happiness maintenance can account for stock market facts without resorting to high risk aversion. Given that (cid:11) > 1 is needed to satisfy the transversality condition of the in(cid:133)nite horizon problem, I take (cid:11) to be equal to 3. Given the simple speci(cid:133)cation of (cid:18) in (2); I need to determine reasonable ranges of t two parameters, (cid:18) and n. Recall that, for the composite good g to be well de(cid:133)ned, (cid:18) t t 15

has to lie at every point of time within the [0;1] interval. This suggests a (cid:133)rst restriction for (cid:18) 0; 1 : Neither psychological experiments nor introspection provide guidance 2 xt(n) on how tho furthier restrict (cid:18): I take a pragmatic stand and consider the interval [0;0:5]. This is a conservative choice, since it constrains the demand for happiness to be lower than that for consumption, and hedonic risk aversion to be always lower than (cid:11) and less volatilethantheunderlyingstateoftheeconomy. Infact,with(cid:11) = 3hedonicriskaversionis a = 2(cid:18) +1 = 2(cid:18)x (n)+1;anditsstandarddeviationis(cid:27)(a ) = (cid:27)((cid:18) ) = 2(cid:18)(cid:27)(x (n)) 2(cid:18)(cid:14): t t t t t t (cid:20) Thus, (cid:18) [0;0:5] implies a (1;2) and (cid:27)(a ) (0;0:036): Finally, since the model is t t t 2 2 2 calibrated on yearly frequency, I constrain n within the [0;5] interval so as to roughly span theaveragelengthofabusinesscycle. Thisrestrictionimpliesthatinvestorsspanatmostan entire cycle in their assessment of recent economic conditions. In summary, my preference parameter values are (cid:12) (0:97;0:99); (cid:11) = 3; and (cid:18) [0;0:5]: t 2 2 Two features of my calibration are worth emphasizing. First, my parametrization implies low and mildly volatile hedonic risk aversion, which is consistent with the emphasis of happinessmaintenanceexperimentsonmild everydayemotions. Second, mytwo-parameter speci(cid:133)cation of (cid:18) is relatively parsimonious in that it allows for fewer free parameters than t previouswealth-basedmodels(see, forexample, Barberisetal. (2001)). Finally, incontrast to Campbell and Cochrane (1999), I am not forcing the (cid:18) process to match the stochastic t properties of returns. Computing returns Expectedreturnscannotbesolvedforinclosedformandneedto be computed using numerical methods. As in Mehra and Prescott (1985), the de(cid:133)nition of returns can be used to rewrite equation (10) in terms of the price-dividend ratio. Since the pricing kernel does not depend on the level of consumption, I do not expect asset prices to dependonconsumption levelseither. Thus, itisnaturaltoassume thatthe priceofequities is pe(c ;(cid:21) ;(cid:18) ) = f c ; where f = pe(ct;(cid:21)i;(cid:18)i) is a price-dividend ratio function related to (cid:21) , t i i i t i ct i the growth rate of output, both directly and through the dependence of (cid:18) on (cid:21) . Under t i probability structure (11); the Euler equation de(cid:133)nes a system of (nonlinear) (cid:133)rst-order di⁄erence equations in unknown price-dividend ratios (see Appendix A for details): Using 16

these price relationships, I can compute the conditional and unconditional expectation of asset returns.7 The number of (cid:133)rst-order di⁄erence equations and price-dividend ratios de(cid:133)ned by the Euler and the methods adopted to actually solve these equations depend on the value of n: In particular, when n = 0; (cid:18) = (cid:18)(cid:21) ; the price-dividend ratio, f = pe(ct;(cid:21)i) ; depends only i i i ct on the growth rate of output, and the Euler equation de(cid:133)nes a system of two equations in two unknowns which can be solved as in Mehra and Prescott (1985). When n [1;5]; the 2 Euler in general de(cid:133)nes more than two equations. This is the case since (cid:18) depends on past i realizations of aggregate dividend growth, which induces time-variation in expected returns and makes the price-dividend ratio, f = pe(ct;(cid:21)i;(cid:18)i) ; depend e⁄ectively on two states, (cid:21) i ct i and (cid:18) . Since the traditional solution methodology is not applicable to this case, I employ i standard simulation methods: In particular, I employ a simple parametrized expectation algorithm (see Marcet and Marshall (2002) for a detailed description of the algorithm) and then compute summary statistics by simulating the solved system to generate a long times series of 50,000 data points.8 III.2.a The risk-free rate, the equity premium, and the volatility puzzles.(cid:151) In the US, the mean excess return of equities over relatively riskless securities such as bonds - i.e., the equity premium - has been historically about six percent, with riskless securities paying an average return of about one percent. Riskless securities have displayed signi(cid:133)cantly lower volatility of returns than equities, with a di⁄erence of about ten percentage points. Consequently, the so called Sharpe ratio, which is de(cid:133)ned as the mean of the equity premium divided by its standard deviation, has been in the neighborhood of .32. The (cid:133)rst three columns of Table 1 show that these facts hold robustly across di⁄erent time periods. The (cid:133)rst column is especially important since it is an updated version of the sample used in Mehra and Prescott (1985), which constitutes the classical benchmark to evaluate the empirical performance of asset pricing models. To facilitate comparison, the fourth column of Table 2 reports the predictions of the traditional consumption-based asset pricing model (to which my model reduces if I shut down 17

demand for happiness by setting (cid:18) = (cid:18) = 0) under the parameters chosen and Markov t t+1 consumptiongrowth. Thiso⁄ersaquantitativecounterparttomydiscussionofassetpricing puzzles in the previous section: the predicted premium is one order of magnitude smaller than the actual one (the (cid:148)equity premium puzzle(cid:148)of Mehra and Prescott (1985)), and the risk free rate is too high (the (cid:148)risk-free rate puzzle(cid:148)of Weil (1989)). Moreover, both returns display excessively low volatilities and the implied Sharpe ratio is too low. Adding wealth as a second term in the utility function provides limited help toward the resolution of the puzzles. In fact, contrary to the partial equilibrium results of Bakshi and Chen (1996), the predictions of a basic wealth-based model - to which the present model reduces when happiness maintenance is shut down by setting (cid:18) = (cid:18) = (cid:18) = t t+1 6 0 - are virtually indistinguishable from those of the consumption-based model. In fact, general equilibrium implies that the restrictions imposed by the budget constraint on the relationship between wealth and consumption need to be taken into account. In this case, the (cid:133)fth column of Table 2 shows that introducing a demand for wealth increases prices and lowers both returns, but has virtually no e⁄ect on the premium and on the volatility of returns. This is consistent with Campbell (1993): if risk aversion is kept relatively low as in my calibration, standard wealth-based models fail to generate volatile wealth since the implied price/dividend ratio is essentially constant. The fourth and (cid:133)fth columns of Table 1 contrast these results with the predictions of happiness maintenance. With relatively low (on average about 1.5) and mildly volatile (standard deviation of about 0.01) hedonic risk aversion, the implied premium is more than ten times bigger than either the wealth-based or the consumption-based benchmarks. Moreover, stock market volatility ceases to be a puzzle. A low level of risk aversion and a reasonable rate of time preference need not be inconsistent with the basic facts of asset markets, i.e. a fairly stable and low average return on riskless securities and a sizable and fairly volatile premium of equities over bonds. While these results obtain regardless of the stochastic process chosen for consumption growth, with iid consumption growth the mean and volatility of the price-dividend ratio are closer to the data. GiventhattheimplicationsofthemodelareanalogousunderiidandMarkovconsumption 18

growth, I present the next results under the Markovian assumption. The results are not meaningfully di⁄erent under iid consumption growth.9 Volatility bounds and the market price of risk. To fully explore the quantitative e⁄ectsofhappinessmaintenanceonreturns,itisusefultoconsiderHansenandJagannathan (1991)(cid:146)s statement of the equity premium puzzle, according to which the largest possible Sharpe ratio is given by the conditional standard deviation of the log stochastic discount factor. More formally, E Ri Rf (cid:27) (m ) t t+1 (cid:0) (cid:12) (cid:0) (cid:27)( (cid:1) Ri) (cid:12) (cid:20) E t m t+1 (cid:12) (cid:12) (cid:12) (cid:12) where E( (cid:27) R ( i R ) (cid:0) i) Rf is the Sharpe r (cid:12) (cid:12)atio, (cid:27) E t( t m m t t + + 1 1 )(cid:12) (cid:12)is the market price of risk and m t+1 is the investors(cid:146)intertemporal marginal rate of substitution. In this formulation, the (cid:148)equity premiumpuzzle(cid:148)liesinthefactthatstandardmodelsmaketheconditionalstandarddeviation of the pricing kernel too small. I have argued in the previous section that happiness maintenance increases the volatility of the pricing kernel through a leverage e⁄ect, wt at (cid:0) at+1 : Panels B-D of Figure 1 show ct that, even with low and moderately procyclica(cid:16)l he(cid:17)donic risk aversion, the leverage e⁄ect is strong enough to bring the equilibrium pricing kernel well within Hansen-Jagannathan bounds. This is in sharp contrast to Panel A, which shows that, even with a risk aversion of 10, the market prices of risk implied by power utility falls short from satisfying the bounds. Inspecting the mechanism. In contrast to the habit-persistence preferences of Campbell and Cochrane (1999), happiness maintenance matches the equity premium without high implied risk aversion. In fact, risk aversion in my model is constant and equal to (cid:11): Moreover, while the elasticity of intertemporal substitution is di⁄erent from 1 and is a (cid:11) complicated function of the other parameters of the model, in my benchmark calibration it is equal to 0.4, a value which is well within the range commonly assumed in the literature.10 The comparative dynamics of the coe¢ cient of relative risk aversion, (cid:11); and the rate of time preference, (cid:12); under happiness maintenance are standard and, thus, omitted for 19

brevity:Not surprisingly, the value of hedonic risk aversion is key for the empirical performance of the model. Table 3 illustrates this point by experimenting with several di⁄erent combinations of values for (cid:11) and (cid:18) that, however, imply the same mean and volatility of hedonic risk aversion as in my benchmark calibration. The message is that, as far as hedonic risk aversion preserves the features of my baseline calibration, the model can match the Sharpe ratio for a wide range of combinations of values of (cid:18) and (cid:11). Even with risk aversion as low as 2, the model can deliver a premium in line with the data. III.2.b Cyclicality and the correlation puzzle.(cid:151) The (cid:133)rst column of Table 4 reports stylized facts about the price-dividend ratio over the business cycle: the price-dividend ratio is procyclical and displays positive auto-correlation. In contrast to standard consumption-based models that imply a counter-cyclical pricedividend ratio, the cyclicality and autocorrelations of the price-dividend ratio implied by mymodelareinlinewiththedata. Inthenextsection, Idiscusstheintuitionforthisresult in the context of predictability. Before moving on, though, it is worth contrasting another dimension along which happinessmaintenanceimprovesthe(cid:133)twiththedata. Itiswellknownthatstandardconsumptionbased models imply a correlation between stock returns and consumption growth equal to one, which is much higher than in the data. This is the (cid:148)correlation puzzle(cid:148)of Cochrane and Hansen (1992). Happiness maintenance alleviates the puzzle since it predicts a correlation well below 1. To understand this, note that in my model stock returns are made up of two standard components, news about consumption and wealth, and a novel component, changes in hedonic risk aversion. Although changes in hedonic risk aversion are ultimately caused by changes in consumption, the rich multi-factor structure of the model lowers the correlation between returns and consumption. III.2.c Long-horizon predictability and volatility.(cid:151) Can pro-cyclical changes in hedonic risk aversion reproduce the observed patterns of predictabilityofassetreturns? Todevelopintuitionaboutthisquestion,itishelpfultostart 20

by considering the implications of happiness maintenance for the volatility of equity prices. Tothisend,Table5presentstheresultsofregressionsoflong-horizonlogstockpricechanges on long-horizon log consumption changes. The (cid:133)rst column shows that in the data these regressions yield coe¢ cients invariably greater than one, and as high as 1.61 at a 20-year horizon. Barsky and DeLong (1993) characterize this (cid:133)nding as the "excess volatility" of stock prices. In contrast to traditional consumption-based asset pricing models, happiness maintenance generates stock prices that are more volatile than underlying consumption fundamentals (as evidenced by the coe¢ cients in the third and forth columns which are consistently greater than one). To see how the model generates prices that are more volatile than the underlying dividends, suppose that there is a positive consumption innovation this period. This shock increases current hedonic risk aversion, a ; which is pro-cyclical. However, as indicated by t the term wt at (cid:0) at+1 in (7);11 what matters for a forward-looking investor is the anticict pated chan(cid:16)ge(cid:17)in hedonic risk aversion, a +1 a : Thus, a positive consumption innovation t t (cid:0) makes the future look relatively brighter, i.e. it actually implies a lower expected change in hedonic risk aversion. This increases the value of wealth relative to consumption for the investor (recall from (1) that (cid:18); and, thus, a; control the relative demand for wealth). As a result, as it is clear from (7);the investor discounts the future dividend stream at a lower rate, giving stock prices an extra jolt upward. A similar mechanism works for a negative shock, which generates low current hedonic risk aversion and pushes prices lower. The ultimate e⁄ect is that prices are more volatile than consumption growth. Thismechanismisalsounderlyinglonghorizonpredictability: sincetheinvestor(cid:146)shedonic riskaversionvariesovertimedependingonthestateoftheeconomy,expectedreturnsonthe risky asset also vary. To understand this in more detail, suppose once again that there is a positive shock to consumption, which increases the investor(cid:146)s current hedonic risk aversion. This makes wealth relatively more valuable with respect to consumption, since the investor expect a lower change in hedonic risk aversion. The higher demand for wealth pushes the stock price still higher, leading to a higher price-dividend ratio. At the same time, higher demand for wealth implies that subsequent stock returns will be lower on average, 21

since the investor demands a lower premium to bear equity risk. Price-dividend ratios are therefore inversely related to future returns. Notice that, although my model generates predictability through state-dependent preference as Campbell and Cochrane (1999), the economic mechanism is distinct from theirs. One advantage of my leverage e⁄ect is its non-linearity in risk aversion changes - i.e. small switches in risk aversion are su¢ cient for the term wt at (cid:0) at+1 to operate as either a discount or a compound factor. Thus, I only ct need mild(cid:16), rat(cid:17)her than strong, cyclicality of risk aversion. The results reported in Table 6 con(cid:133)rm this intuition about predictability. In particular, I report the slope coe¢ cients, (cid:12) ; and R2 obtained from running the following regression of k k cumulative log returns over a k-year horizon on lagged price-dividend ratio: r +r +:::+r = (cid:11) +(cid:12) (p y )+(cid:15) t+1 t+2 t+k k k t t k;t (cid:0) where r is log return and k = 1;2;3;5 and 10. For ease of comparison, the corresponding t+k values in the data and in the standard consumption-based model are reported in the (cid:133)rst andsecondcolumnsofTable6,respectively. ThestylizedpatternsdocumentedbyCampbell and Shiller (1988) are well replicated by the model: the coe¢ cients are negative; they start low and then increase. Finally, the R2 increases with the return horizon. III.3 Robustness checks The results in Table 7 show that the model is robust to alternative speci(cid:133)cations of the happiness demand process which preserve the pro-cyclicality of hedonic risk aversion. In particular, the third column of Table 7 shows that replacing speci(cid:133)cation (2) with a di⁄erent pro-cyclical process for (cid:18) produces returns that are virtually indistinguishable t from the benchmark calibration. Finally, the second column of Table 7 contrasts the implication of happiness maintenance (reported, for ease of comparison, in the (cid:133)rst column) with those of a model where hedonic risk aversion is counter-cyclical. I report summary statistics of asset returns when the happiness demand process (cid:18) is perfectly negatively correlated with consumption growth. t This departure from the baseline model makes the premium shrink to about a quarter of 22

its benchmark value, mostly due to the fact that the risk-free rate is four times as high. IV. CONCLUSION Drawing on ingredients from outside the usual domain of (cid:133)nance and economics can help to understand several otherwise puzzling features of (cid:133)nancial markets. In particular, I use an equilibrium model simple to the extreme and show that mild everyday feelings have rich implications for aggregate asset returns. Happiness maintenance, a well documented feature of the immediate emotional perception of risk, increases the risk associated with equity, thus contributing to resolve some of the most prominent documented asset pricing puzzles. Further, it provides a novel perspective over a broad set of important stylized features of (cid:133)nancial markets, such as, the predictability of asset returns and the volatility of asset prices at long horizons. An important feature of my model is that it does not depart from conventional asset pricing wisdom along any dimensions other than investors(cid:146)preferences. Moreover, my twoparameterspeci(cid:133)cationofhappinessmaintenancepreferencesisrelativelyparsimoniouswith respect to other wealth-based models. Overall, my results strongly suggest that pro-cyclical risk aversion provides a potentially useful construct to understand stock market facts. 23

Notes 1Pigou(1947)elaboratedonthenotionofamenityutilityprovidedbywealth,intheformofpower,sense ofsecurity, and controlfrom having resources. Carroll(2000) shows a modelwith directutility from wealth might help to explain the high saving rates of the rich. Zou (1994) and Bakshi and Chen (1996) study preferences based on an interpretation of Max Weber(cid:146)s spirit of capitalism as the pursuit of wealth for its own sake. The lossaversion motive studied in Barberis,Huang and Santos(2001)isyetanotherinstance of preferences that depend on changes in (cid:133)nancial wealth. 2a ismorepreciselythelocalcurvatureoftheutilityfunctionwithrespecttowealth,i.e. uwww :With t (cid:0) uw t a slight abuse of terminology we refer to it as (cid:148)hedonic risk aversion.(cid:148) 3See Appendix A for details. 4The transversality condition is that E 0 1t=0 (cid:12)tu(g t )< 1 or, equivalently, (cid:12)(1+(cid:21))1 (cid:0) (cid:11)k<1; where (cid:21) is the expected growth rate of output and kP>1. Clearly, since (cid:21)>0; (cid:11)>1 guarantees that the inequality holds. This argument is analogous to Bakshi and Chen (1996). For a detailed formal development see Kocherlakota (1990). 5For the details on the derivation of the Euler equation see Appendix A. 6Campbell and Cochrane (1999) is perhaps the only habit persistence model that avoids problems with the volatility of the risk-free rate through a clever choice of functional form that enables them to use precautionary saving to counterbalance the strong counter-cyclical variation in risk aversion. 7Details on the remaining part of the computation are given in Appendix A. 8The fortran codes to solve and simulate the model are available upon request from the author. 9The results for the iid case are available upon request. 10Appendix A contains a detailed derivation. 11Or, equivalently, by the term (f t+1 +1)(1 (cid:0) (cid:11))((cid:18)t+1(cid:0) (cid:18)t ) in (10): 24

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[35] Melino, Angelo, and Alan X., Yang, 2003, (cid:148)State Dependent Preferences can explain the Equity Premium Puzzle,(cid:148)Review of Economic Dynamics, 6(4), pp.806-30. [36] Menzly, Lior, Tano Santos and Pietro Veronesi, 2004, "Understanding predictability," Journal of Political Economy, 112, pp. 1(cid:150)47. [37] MilnorJ.,1997,TopologyfromtheDi⁄erentiableViewpoint,PrincetonUniversityPress,Princeton, 2nd edition. [38] Piazzesi, Monica, Martin Schneider, and Selale Tuzel, 2005, "Housing, Consumption, and Asset PRicing," forthcoming, Journal of FInancial Economics. [39] Pigou, Arthur C., 1947, (cid:148)Economic Progress in a Stable Environment,(cid:148)Economica, 14, pp.180-88. [40] Prescott, Edward C., and Rajnish Mehra. 1980, (cid:147)Recursive Competitive Equilibrium: The Case of Homogeneous Households.(cid:148)Econometrica, 48, pp. 1365 (cid:150)1379. [41] Weil, Philippe, 1989, (cid:148)The Equity Premium Puzzle and the Riskfree Rate Puzzle,(cid:148)Journal of Monetary Economics, 24, pp.401-21. [42] Zhou, Heng-fu, 1994, (cid:148)The Spirit of Capitalism and Long-Run Growth,(cid:148)European Journal of Political Economy, 10, pp.279-93. [43] Zin, Stanley E., 2001, (cid:148)Are Behavioral Asset-Pricing Models Structural?(cid:148)Journal of Monetary Economics, 49, pp.215-28. 28

APPENDIX A: EVIDENCE OF HAPPINESS MAINTENANCE, DERIVATIONS, AND PROOFS Happiness maintenance is a well replicated (cid:133)nding. For example, Isen and Patrick (1983) conducted an experiment to study the in(cid:135)uence of positive a⁄ect on choices under uncertainty. Participants, a large sample of college students, were randomly assigned to two groups: positive a⁄ect was induced only in participants in one group by receipt of a small gift, a McDonald(cid:146)s gift certi(cid:133)cate worth $.50. Subjects were given ten poker chips and told thatthesechipsrepresentedtheircreditforparticipatinginthestudy. Riskpreferenceswere measured in terms of the amount of chips actually bet by the two groups of participants in a game of roulette. They found that individuals in a positive mood bet signi(cid:133)cantly less than controls on gambles with a meaningful probability of losing (about 20% chance of winning). In particular, individuals in a neutral state bet on average about six times as many chips as individuals in a positive mood12. Isen and Geva (1987) used the level of the probability of winning before accepting a bet of (cid:133)xed amount and found again that, when a meaningful amount was at stake, namely their whole endowment of chips, individuals in a positive mood, in contrast to those in a control group, set a level for the probability of winning as a cuto⁄point for accepting a given gamble on average about 30% higher than controls. Isen et al. (1984) documented that individuals in whom a positive mood had been induced by receipt of a small gift expressed greater preference in a lottery choice for a $1 ticket rather than a $10 ticket relative to a control group. Nygren et al. (1996) provided stronger support for an in(cid:135)uence of a⁄ect on risk taking: they asked participants in whom positivea⁄ecthadbeeninduced,aswellasnomanipulationcontrols,tomakeactualbetting decisions in twelve di⁄erent three-outcome gambles. The mean bet value of a⁄ect condition participants was found to be consistently lower of about 30% than controls, regardless of the riskiness of the gambles, i.e. the ratio of the probability of winning and loosing or of the amounts. While these early results indicated a tendency toward conservatism in risk preferences, Fong and McCabe (1999) replicate their essence within a very careful experimental setup 29

that through the adoption of auction theoretic techniques (see Kagel (1995)) enables them to avoid potential di¢ culties with the studies mentioned so far, especially associated with the possible role of uncontrolled variables, the lack of monetary incentives and the lack of mechanismtoensurethattruthfulrevelationofprivatevaluesofthelotterywasadominant strategy. They endowed their subjects with lottery tickets and let them bid for the tickets in both a sealed-bid and an English auction. Subjects could earn up to $10 in each lottery or as little as zero in each. They found that average exit price is signi(cid:133)cantly lower for subjects whose mood had been improved by a minor manipulation, indicated a higher risk aversion in a⁄ect subjects. The perspective suggested by these (cid:133)ndings is well described by the idiom: don(cid:146)t push your luck. It is worth contrasting it with the (cid:133)ndings of illusion of control or (cid:148)gambling with the house money(cid:148)of Thaler and Johnson (1990), which motivate the work of Barberis et al. (2001). As suggested in Arkes et al. (1988), the presence of a meaningful loss might be the crucial determinant of the discrepancy between the (cid:133)ndings of the two classes of experiments. In one experiment, where a meaningful loss was nonexistent, a⁄ect participants exhibited relatively more risk-prone behavior compared to controls. In a second experiment dealingwithinsurancebuyingbehaviorwhereparticipantswereforcedtofocusonpotential loss, positive a⁄ect participants displayed a greater risk aversion than did controls. Nygren et al. (1996) further illustrates this point: positive a⁄ect participants signi(cid:133)cantly overestimated the probability of winning while participants in the control group did not, in accord with the (cid:133)ndings of studies such as Johnson and Tversky (1983). Nevertheless, in actual gambling situations, a⁄ect condition participants were much less likely to gamble than were controls. Axioms and representation theorems for state-dependent utility Technically, the speci(cid:133)cation chosen for the a⁄ect-dependent utility belongs to the wider classofstate-dependentutilityfunctions. Thestructureofthepreferencesunderlyingstatedependent utility functions is relatively well understood. Karni (1985) and more recently 30

Dreze and Rustichini (2001) present a thorough analysis of alternative axiomatizations. I follow Myerson (1991) and give a list of axioms and a representation theorem for statedependent preferences of the type informally illustrated in the text. Notation.(cid:151) For any (cid:133)nite set Z; let (cid:1)(Z) denote the set of probability distributions over Z: That is, de(cid:133)ne: (cid:1)(Z) = q : Z R q(y) = 1 and q(z) 0; z Z 8 ! j (cid:21) 8 2 9 < y X2 Z = Let X denote the set of possible prizes that the decision maker could ultimately get, (cid:10) : ; denote the set of possible states of the world, and assume both X and (cid:10) are (cid:133)nite. De(cid:133)ne a lottery to be any function f that speci(cid:133)es a nonnegative real number f(x t); for every j prize x in X and every state t in (cid:10); such that f(x t) = 1 for every t in (cid:10): x X j 2 Let L denote the set of all such lotteries. ThPat is, L = f : (cid:10) (cid:1)(X) f ! g For any state t in (cid:10) and any lottery f in L; f( t) denotes the probability distribution (cid:1) j over X designated by f in state t. That is, f( t) = f(x t) (cid:1)(X) (cid:1) j f j gx X 2 2 Let (cid:4) denote the set of all events, S, so that (cid:4) = S S (cid:10) and S = f j (cid:18) 6 ;g For any two lotteries f and g in L and any event S in (cid:4); I write f %S g if and only if (i⁄) the lottery f would be at least as desirable as g; in the opinion of the decision-maker, if he knew that the true state of the world was in the set S: In other words, f %S g i⁄the decision-maker would be willing to choose the lottery f when he has to choose between f and g and he knows only that the event S has occurred. Given the relation %S ;I can de(cid:133)ne f S g i⁄f %S g and g %S f (cid:31) f S g i⁄f %S g and g (cid:28)S f (cid:24) 31

where f g and f g have the customary meanings of (conditional) indi⁄erence and S S (cid:31) (cid:24) (conditional) strict preference. Naturally, %(cid:10) ; (cid:10) and (cid:10) correspond to the familiar %; (cid:31) (cid:24) (cid:31) and ; that is when no conditioning event is considered, I refer to prior preferences. (cid:24) For any number (cid:11) such that 0 (cid:11) 1; and for any two lotteries f and g in L; (cid:11)f + (cid:20) (cid:20) (1 (cid:11))g denotes the lottery in L such that (cid:0) ((cid:11)f +(1 (cid:11))g)(x t) = (cid:11)f(x t)+(1 (cid:11))g(x t) (cid:0) j j (cid:0) j for all x X and t (cid:10): 2 2 Finally, a conditional-probability function on (cid:10) is any function p : (cid:4) (cid:1)((cid:10)) that ! speci(cid:133)es nonnegative conditional probabilities p(t S) for every state t in (cid:10) and every j event S, such that p(t S) = 0 if t = S and p(r S) = 1: j 2 r S j 2 P Axioms.(cid:151) The axioms are to hold for all lotteries e;f;g and h in L, for all events S and T in (cid:4); and for all numbers (cid:11) and (cid:12) between 0 and 1: Axiom 2 (Completeness) f %S g or g %S f: Axiom 3 (Transitivity) If f %S g and g %S h; then f %S h: Axiom 4 (Relevance) If f( t) = g( t); t S; then f g. S (cid:1) j (cid:1) j 8 2 (cid:24) Axiom 5 (Monotonicity) If f h and 0 (cid:12) < (cid:11) 1; then (cid:11)f + (1 (cid:11))h S S (cid:31) (cid:20) (cid:20) (cid:0) (cid:31) (cid:12)f +(1 (cid:12))h: (cid:0) Axiom 6 (Continuity) If f %S g and g %S h; then there exists some number (cid:13) such that 0 (cid:13) 1 and g (cid:13)f +(1 (cid:13))h: S (cid:20) (cid:20) (cid:24) (cid:0) Axiom 7 ((Strict) objective substitution) If e( S ) %S f and g %S h and 0(<) (cid:31) (cid:20) (cid:11) 1; then (cid:11)e+(1 (cid:11))g( S ) %S (cid:11)f +(1 (cid:11))h: (cid:20) (cid:0) (cid:31) (cid:0) Axiom 8 ((Strict) subjective substitution) Iff( S ) %S g andf %T g andS T = ?; (cid:31) \ then f( S T ) %S T g: (cid:31) [ [ 32

Axiom 9 (Interest) For every state t in (cid:10); there exist prizes y and z in X such that [y] [x]; where [ ] denotes the lottery that always gives the prize for sure. t (cid:31)f g (cid:1) A representation theorem.(cid:151) A utility function can be any function from X (cid:10) into the real numbers, : A utility (cid:2) < function is state-independent i⁄there exists some function U : X , such that u(x;t) = ! < U (x); for all x and t: Theorem 10 The eight axioms are jointly satis(cid:133)ed if and only if there exists a utility function u : X (cid:10) and a conditional-probability function p : (cid:4) (cid:1)((cid:10)) such that: (cid:2) ! < ! maxu(x;t) = 1 and minu(x;t) = 0; t (cid:10); x X x X 8 2 2 2 p(R T) = p(R S)p(S T); j j j R; S; T : R S T (cid:10);S = ; 8 8 8 (cid:18) (cid:18) (cid:18) 6 ; f %S g i⁄ E p [u(f) S] E p [u(g) S]; j (cid:21) j f;g L; S (cid:4); 8 2 8 2 where E [u(f) S] = p(t S) u(x;t)f(x t) is the expected utility value of the p j t S j x X j 2 2 prize determined by f,Pwhen p( S)Pis the probability distribution for the true state of the (cid:1) j world. Proof. see Myerson (1991). Discussion and caveats.(cid:151) Axiom 11 (State neutrality) For any two states r and t in (cid:10); if f( t) = f( t) and (cid:1) j (cid:1) j g( t) = g( t) and f %r g; then f %t g: (cid:1) j (cid:1) j Theorem 12 Given the axioms above, state neutrality is also satis(cid:133)ed if and only if the conditions of the representation theorem can be satis(cid:133)ed with a state-independent utility function. Proof. see Myerson (1991). 33

De(cid:133)nition, existence and uniqueness of the equilibrium solution This Appendix provides a more formal de(cid:133)nition of equilibrium for an exchange economy populated by investors with happiness maintenance preferences. It also contains a proof that such equilibrium exists. De(cid:133)nition of equilibrium.(cid:151) Equilibrium is de(cid:133)ned by a pair of functions, p : + +; the asset pricing function, < ! < and v(s;y;(cid:18);p), a value function, such that: 1. v : + + +;v(s;y;(cid:18);p( )) = maxE (cid:12)tu(g ) ; subject to c +p s < (cid:2)< ! < (cid:14) 0 1t=0 t t t t+1 (cid:20) s t (p t +y t ) given F ( ), s 0 = s 0 < 1;y 0 ;(cid:18) 0 . (cid:2)P (cid:3) (cid:1) 2. s = s = 1; c = y : b t+1 t t t Euler equations and returns.(cid:151) The functional equation associated with the investor(cid:146)s maximization problem is: V (s ;y ;(cid:18) ) = max u(g(c ;w ;(cid:18) ))+(cid:12)E V (s ;y ;(cid:18) ) t t t t t t t t+1 t+1 t+1 ct;st+1 f g f g c +p s = (p +y )s = w t t t+1 t t t t c > 0; s (0;1]; and s ;y (cid:18) given t t+1 0 0 0 2 where E (x ) = x dF (y ;x ;y ;x ) is the expectation operator. The (cid:133)rst order and t t t t+1 t+1 t t envelope conditioRns are respectively: u (g((p +y )s p s ;(p +y )s ;(cid:18) ))p = (cid:12)E V (s ;y ;(cid:18) ) c t t t t t+1 t t t t t t 1 t+1 t+1 t+1 (cid:0) V (s ;y ;(cid:18) ) = u (g((p +y )s p s ;(p +y )s ;(cid:18) ))(p +y ) 1 t t t c t t t t t+1 t t t t t t (cid:0) +u (g((p +y )s p s ;(p +y )s ;(cid:18) ))(p +y ) w t t t t t+1 t t t t t t (cid:0) where V is the derivative of the value function with respect to s . 1 t Substituting back for consumption from the budget constraint and using the de(cid:133)nition of w , the Euler equation (4) can be written as t 34

(1 (cid:18) t )c t (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 w t (1 (cid:0) (cid:11))(cid:18)tp t (cid:0) = (cid:12)E (1 (cid:0) (cid:18) t+1 )c ( t+ 1 (cid:0)1 (cid:11))(1 (cid:0) (cid:18)t+1) (cid:0) 1 w t ( + 1 (cid:0)1 (cid:11))(cid:18)t+1 (p +y ) t t+1 t+1 8 < 2 +(cid:18) t+1 c ( t+ 1 (cid:0)1 (cid:11))(1 (cid:0) (cid:18)t+1) w t ( + 1 (cid:0)1 (cid:11))(cid:18)t+1 (cid:0) 1 3 9 = 4 5 (1 (cid:18) t ):c t (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 w t (1 (cid:0) (cid:11))(cid:18)tp t ; (cid:0) (cid:18) c = (cid:12)E t (1 (cid:0) (cid:18) t+1 )c t ( + 1 (cid:0)1 (cid:11))(1 (cid:0) (cid:18)t+1) (cid:0) 1 w t ( + 1 (cid:0)1 (cid:11))(cid:18)t+1 1+ 1 t+ (cid:18) 1 w t+1 (p t+1 +y t+1 ) t+1 t+1 (cid:26) (cid:18) (cid:0) (cid:19) (cid:27) Simple algebraic manipulations deliver the Euler equation (7) that appears in the text. Premium For the expected premium, using the de(cid:133)nition of the risk-free rate I have c t+1 (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 w t+1 (1 (cid:0) (cid:11))(cid:18)t c t+1 (1 (cid:0) (cid:11))((cid:18)t (cid:0) (cid:18)t+1) 1 = (cid:12)E k ER t+1 t+1 c w w " t t t+1 # (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) c t+1 (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 w t+1 (1 (cid:0) (cid:11))(cid:18)t c t+1 (1 (cid:0) (cid:11))((cid:18)t (cid:0) (cid:18)t+1) +cov k ;R t+1 t+1 c w w t t t+1 ! (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) ER t+1 1 = R f c t+1 (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 w t+1 (1 (cid:0) (cid:11))(cid:18)t c t+1 (1 (cid:0) (cid:11))((cid:18)t (cid:0) (cid:18)t+1) +cov k ;R t+1 t+1 c w w t t t+1 ! (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) R = ER +R cov(m ;R ) f t+1 f t+1 t+1 ER R = R cov( m ;R ) t+1 f f t+1 t+1 (cid:0) (cid:0) where m = ct+1 (1 (cid:0) (cid:11))(1 (cid:0) (cid:18)t) (cid:0) 1 wt+1 (1 (cid:0) (cid:11))(cid:18)t k ct+1 (1 (cid:0) (cid:11))((cid:18)t (cid:0) (cid:18)t+1) : t+1 ct wt t+1 wt+1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) Existence of equilibrium.(cid:151) This section proves the existence of a (bounded and strictly positive) equilibrium pricedividend function for probability structure (11): The main complication in establishing existence derives from the endogeneity of the pricing kernel induced by the dependence of the utility function on wealth and, in equilibrium, on the price-dividend function. Such endogeneity prevents us from characterizing the Euler equation as a non-linear counterpart 35

of the linear Fredholm equations much studied in the consumption-based asset pricing tradition. Under the assumed probability structure, the Euler equation de(cid:133)nes the following system of two non-linear equations in two unknown price-dividend ratio functions: f 1 (f 1 +1) (cid:0) a1 (cid:12) (cid:25) 11 (cid:21) 1 1 (cid:0) (cid:11) 1 (cid:0) 1 a (cid:0) 1 (cid:11)f1 1 +1 (f 1 +1)1 (cid:0) a1 = 0 (13) (cid:0) 2 +(cid:25) 12 (cid:21) 2 1 (cid:0) (cid:11)(cid:16) 1 (cid:0) 1 a2 (cid:11)f2 1 +1 (cid:17) (f 2 +1)1 (cid:0) a2 3 (cid:0) 4 (cid:16) (cid:17) 5 f 2 (f 2 +1) (cid:0) a2 (cid:12) (cid:25) 21 (cid:21) 1 1 (cid:0) (cid:11) 1 (cid:0) 1 a (cid:0) 1 (cid:11)f1 1 +1 (f 1 +1)1 (cid:0) a1 = 0 (cid:0) 2 +(cid:25) 22 (cid:21) 2 1 (cid:0) (cid:11)(cid:16) 1 (cid:0) 1 a2 (cid:11)f2 1 +1 (cid:17) (f 2 +1)1 (cid:0) a2 3 (cid:0) 4 (cid:16) (cid:17) 5 where (cid:25) (cid:25) (cid:25) 1 (cid:25) 11 12 (cid:0) 2 3 (cid:17) 2 3 (cid:25) (cid:25) 1 (cid:25) (cid:25) 21 22 (cid:0) 4 5 4 5 To simplify notation, I notice that (13) can be rewritten as (cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) )(f +1) a1 f (f +1) a1 (cid:12) 11 1(cid:0) 1 1 1 (cid:0) = 0 1 1 (cid:0) (cid:0) 2 +(cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) )(f +1) a2 3 12 2(cid:0) 2 2 2 (cid:0) 4 5 (cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) )(f +1) a1 f (f +1) a2 (cid:12) 21 1(cid:0) 1 1 1 (cid:0) = 0 2 2 (cid:0) (cid:0) 2 +(cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) )(f +1) a2 3 22 2(cid:0) 2 2 2 (cid:0) 4 5 (f +1) a2 f (cid:12) (cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) )+(cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) ) 2 (cid:0) = 0 1 (cid:0) 11 1(cid:0) 1 1 12 2(cid:0) 2 2 (f +1) a1 (cid:20) 1 (cid:0) (cid:21) (f +1) a1 f (cid:12) (cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) ) 1 (cid:0) +(cid:25) (cid:21)1 (cid:11)(f +1+(cid:13) ) = 0 2 (cid:0) 21 1(cid:0) 1 1 (f +1) a2 22 2(cid:0) 2 2 (cid:20) 2 (cid:0) (cid:21) x a2 x (cid:12) (cid:25) (cid:21)1 (cid:11)(x +(cid:13) )+(cid:25) (cid:21)1 (cid:11)(x +(cid:13) ) (cid:0)2 1 = 0 1 (cid:0) 11 1(cid:0) 1 1 12 2(cid:0) 2 2 x a1 (cid:0) (cid:20) (cid:0)1 (cid:21) x a1 x (cid:12) (cid:25) (cid:21)1 (cid:11)(x +(cid:13) ) 1(cid:0) +(cid:25) (cid:21)1 (cid:11)(x +(cid:13) ) 1 = 0 2 (cid:0) 21 1(cid:0) 1 1 x a2 22 2(cid:0) 2 2 (cid:0) (cid:20) (cid:0)2 (cid:21) xa1 x (cid:12) (cid:25) (cid:21)1 (cid:11)(x +(cid:13) )+(1 (cid:25) )(cid:21)1 (cid:11)(x +(cid:13) ) 1 1 = 0 1 (cid:0) 11 1(cid:0) 1 1 (cid:0) 11 2(cid:0) 2 2 xa2 (cid:0) (cid:20) 2 (cid:21) xa2 x (cid:12) (cid:25) (cid:21)1 (cid:11)(x +(cid:13) ) 2 +(1 (cid:25) )(cid:21)1 (cid:11)(x +(cid:13) ) 1 = 0 2 (cid:0) 21 1(cid:0) 1 1 xa1 (cid:0) 21 2(cid:0) 2 2 (cid:0) (cid:20) 1 (cid:21) 36

where x f +1 and x f +1: I can than denote (13) as 1 1 2 2 (cid:17) (cid:17) G (x;(cid:25) ) 1 1 G(x;(cid:30)) = = 0 2 3 G (x;(cid:25) ) 2 2 4 5 where x (x ;x ), (cid:30) ((cid:25) ;(cid:25) ); (cid:25) ((cid:25) ;(cid:25) ); (cid:25) ((cid:25) ;(cid:25) ). 1 2 1 2 1 11 12 2 21 22 (cid:17) (cid:17) (cid:17) (cid:17) I resort to a (cid:133)xed point argument (see Milnor (1997) for a detailed treatment) to show that a solution to G exists. It is understood that all parameters other than the probabilities are taken as given. Let (cid:16) = (x;(cid:30)) G(x;(cid:30)) = 0 R 2 ((cid:1))2 f j g (cid:26) (cid:2) I start by proving the following Lemma 13 (cid:16) is a smooth manifold. Proof. By perturbing G with respect to (cid:25) ; I need to show that, for an arbitrarily (cid:133)xed 1 open and full Lebesgue set of parameter values ((cid:12);(cid:21) ;(cid:21) ;(cid:11);a), the Jacobian of the map G 1 2 with respect to (cid:25) and x; D G; has full rank. To this end I study the Jacobian of the map (cid:25);x G with respect to (cid:25) and x: By de(cid:133)nition, I have (cid:11)((cid:21) ) (cid:11)((cid:21) ) 0 1 2 D G = (cid:0) (cid:25);x 2 3 0 (cid:11)^((cid:21) ) (cid:11)^((cid:21) ) 1 2 (cid:0) 4 5 where I de(cid:133)ne (cid:11)((cid:21) ) (cid:21)1 (cid:11)(x +(cid:13) ); (cid:11)((cid:21) ) (cid:21)1 (cid:11)(x +(cid:13) ) xa 1 1 , 1 (cid:17) 1(cid:0) 1 1 2 (cid:17) 2(cid:0) 2 2 xa2 2 (cid:11)^((cid:21) ) (cid:21)1 (cid:11)(x +(cid:13) ) xa 2 2 ; (cid:11)^((cid:21) ) (cid:21)1 (cid:11)(x +(cid:13) ): 1 (cid:17) 1(cid:0) 1 1 xa1 2 (cid:17) 2(cid:0) 2 2 1 Evidently, D G is onto if (cid:11)((cid:21) ) (cid:11)((cid:21) ) = 0 (or, equivalently, (cid:11)((cid:21) ) = (cid:11)((cid:21) )) and (cid:25);x 1 2 1 2 (cid:0) 6 6 (cid:11)^((cid:21) ) (cid:11)^((cid:21) ) = 0 (or, equivalently, (cid:11)^((cid:21) ) = (cid:11)^((cid:21) )): However, 1 2 1 2 (cid:0) 6 6 (cid:11)((cid:21) ) = (cid:11)((cid:21) ) (cid:11)^((cid:21) ) = (cid:11)^((cid:21) ) 1 2 1 2 , Suppose, then, that these equalities hold. (13) simpli(cid:133)es to x = (cid:12)(cid:21)1 (cid:11)(x +(cid:13) )+1 1 1(cid:0) 1 1 xa2 x = (cid:12)(cid:21)1 (cid:11)(x +(cid:13) ) 2 +1 2 1(cid:0) 1 1 xa1 1 37

By taking the ratio, I obtain x 1 xa1 1 (cid:0) = 1 (14) x 1 xa2 2 (cid:0) 2 I need to verify the existence of an open and full Lebesgue measure set of parameter values such that x 1 xa1 1 (cid:0) = 1 x 1 6 xa2 2 (cid:0) 2 When (cid:25) = 1 and (cid:25) = 0 (13) simpli(cid:133)es to 11 21 x 1 = (cid:12)(cid:21)1 (cid:11)x +(cid:12)(cid:21)1 (cid:11)(cid:13) 1 (cid:0) 1(cid:0) 1 1(cid:0) 1 x 1 = (cid:12)(cid:21)1 (cid:11)x +(cid:12)(cid:21)1 (cid:11)(cid:13) 2 (cid:0) 2(cid:0) 2 2(cid:0) 2 x (cid:12)(cid:21)1 (cid:11)x = (cid:12)(cid:21)1 (cid:11)(cid:13) +1 1 (cid:0) 1(cid:0) 1 1(cid:0) 1 x (cid:12)(cid:21)1 (cid:11)x = (cid:12)(cid:21)1 (cid:11)(cid:13) +1 2 (cid:0) 2(cid:0) 2 2(cid:0) 2 x 1 (cid:12)(cid:21)1 (cid:11) = (cid:12)(cid:21)1 (cid:11)(cid:13) +1 i (cid:0) i(cid:0) i(cid:0) i (cid:0) x (cid:1) = (cid:12)(cid:21) i 1 (cid:0) (cid:11)(cid:13) i +1 i 1 (cid:12)(cid:21)1 (cid:11) (cid:0) i(cid:0) (cid:12)(cid:21)1 (cid:11)(cid:13) +1 x = 1(cid:0) 1 x 1 1 (cid:12)(cid:21)1 (cid:11) (cid:17) (cid:3)1 (cid:0) 1(cid:0) (cid:12)(cid:21)1 (cid:11)(cid:13) +1 x = 2(cid:0) 2 x 2 1 (cid:12)(cid:21)1 (cid:11) (cid:17) (cid:3)2 (cid:0) 2(cid:0) and (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) x 1 = 1(cid:0) 1 (cid:3)1 (cid:0) 1 (cid:12)(cid:21)1 (cid:11) (cid:0) 1(cid:0) (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) x 1 = 2(cid:0) 2 (cid:3)2 (cid:0) 1 (cid:12)(cid:21)1 (cid:11) (cid:0) 2(cid:0) Thus x (cid:3)1(cid:0) 1 = (cid:12)(cid:21)1 1(cid:0) (cid:11)(1+(cid:13) 1 ) 1 (cid:0) (cid:12)(cid:21) 2 1 (cid:0) (cid:11) x 1 (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) 1 (cid:12)(cid:21)1 (cid:11) (cid:3)2(cid:0) 2(cid:0) 2 (cid:0) (cid:0) 1(cid:0) (cid:1) and (cid:0) (cid:1) (x )a1 (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) a1 1 (cid:12)(cid:21)1 (cid:11) a2 (cid:3)1 = 1(cid:0) 1 (cid:0) 2(cid:0) (x )a2 1 (cid:12)(cid:21)1 (cid:11) (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) (cid:3)2 (cid:18) (cid:0) 1(cid:0) (cid:19) (cid:18) 2(cid:0) 2 (cid:19) 38

Consider now the function x 1 (x )a1 H(x;a) = (cid:3)1(cid:0) (cid:3)1 x 1 (cid:0) (x )a2 (cid:3)2(cid:0) (cid:3)2 It is straightforward to show that @H = 0: In fact, I have @a 6 @H (x )a1 (x )(cid:21)1 = (cid:3)1 log (cid:3)1 @a (cid:0)(x (cid:3)2 )a2 (x (cid:3)2 )(cid:21)2 (x )a1 = (cid:3)1 ((cid:21) logx (cid:21) logx ) (cid:0)(x )a2 1 (cid:3)1 (cid:0) 2 (cid:3)2 (cid:3)2 Clearly, thereexistsanopenandfullLebesguemeasuresetofparametervalues((cid:12);(cid:21) ;(cid:21) ;(cid:11);a) 1 2 such that @H = 0 or, equivalently, (cid:16) is a smooth manifold. @a 6 Lemma 14 There exists a regular value of the map proj((cid:16)) (cid:1); (cid:25) such that (cid:3) ! # proj 1((cid:25) ) = odd: (cid:0) (cid:3) Pr(cid:2)oof. Fix (cid:25) (cid:3)= 1 and (cid:25) = 0: Then (13) simpli(cid:133)es to a system of two linear equations 11 21 x 1 = (cid:12)(cid:21)1 (cid:11)x +(cid:12)(cid:21)1 (cid:11)(cid:13) 1 (cid:0) 1(cid:0) 1 1(cid:0) 1 x 1 = (cid:12)(cid:21)1 (cid:11)x +(cid:12)(cid:21)1 (cid:11)(cid:13) 2 (cid:0) 2(cid:0) 2 2(cid:0) 2 (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) x = 1(cid:0) 1 +1 1 1 (cid:12)(cid:21)1 (cid:11) (cid:0) 1(cid:0) (cid:12)(cid:21)1 (cid:11)(1+(cid:13) ) x = 2(cid:0) 2 +1 2 1 (cid:12)(cid:21)1 (cid:11) (cid:0) 2(cid:0) Clearly, the solution is unique. Lemma 15 The map proj((cid:16)) (cid:1) is proper, that is proj 1((cid:25)) is compact for each com- (cid:0) ! pact subset of probability. Proof. It su¢ ces to show that 1 < proj 1((cid:25)) < : (cid:0) 1 Suppose that (without loss of generality) x = 1: Then 1 1 (cid:12) (cid:25) (cid:21)1 (cid:11)(1+(cid:13) )+(1 (cid:25) )(cid:21)1 (cid:11)(x +(cid:13) ) = 0 11 1(cid:0) 1 (cid:0) 11 2(cid:0) 2 2 xa2 (cid:20) 2 (cid:21) which is obviously impossible. Suppose, by contradiction, that (cid:25)n (cid:25)h such that xh x (cid:25)h : I distinguish 9 ! (cid:17) ! 1 two cases ( is symmetric to zero and therefore ignor(cid:13)ed):(cid:13) (cid:13) (cid:0) (cid:1)(cid:13) 1 (cid:13) (cid:13) (cid:13) (cid:13) 39

(xh)a1 1. 1 K > 0 (xh)a2 ! 2 The second equation becomes xh (cid:13) xh a2 (cid:13) 1 1 = (cid:12) (cid:25) (cid:21)1 (cid:11) 1 + 1 2 +(1 (cid:25) )(cid:21)1 (cid:11) 1+ 2 + 21 1(cid:0) xh xh xh a1 (cid:0) 21 2(cid:0) xh xh " (cid:18) 2 2(cid:19) (cid:0) 1(cid:1) (cid:18) 2(cid:19) # 2 thus in the limit (cid:0) (cid:1) xh 1 = (cid:12) (cid:25) (cid:21)1 (cid:11) lim 1 K 1+(1 (cid:25) )(cid:21)1 (cid:11) 21 1(cid:0) h xh (cid:0) (cid:0) 21 2(cid:0) (cid:20) !1(cid:18) 2(cid:19) (cid:21) The (cid:133)rst equation (again dividing by x and taking the limit) becomes 1 (cid:13) xh (cid:13) xh a1 1 1 = (cid:12) (cid:25) (cid:21)1 (cid:11) 1+ 1 +(1 (cid:25) )(cid:21)1 (cid:11) 2 + 2 1 + 11 1(cid:0) xh (cid:0) 11 2(cid:0) xh xh xh a2 xh " (cid:18) 1(cid:19) (cid:18) 1 1(cid:19) (cid:0) 2(cid:1) # 1 1 = (cid:12) (cid:25) (cid:21)1 (cid:11)+(1 (cid:25) )(cid:21)1 (cid:11) lim xh 2 K (cid:0) (cid:1) 11 1(cid:0) (cid:0) 11 2(cid:0) h xh (cid:20) !1(cid:18) 1(cid:19) (cid:21) Since xh and (xh 1 )a1 K > 0; then xh and xh : Since a = ! 1 (xh)a2 ! 1 ! 1 2 ! 1 1 6 2 (cid:13) (cid:13) xh xh (xh)a1 a 2 ; eit(cid:13)her(cid:13)lim h !1 x 2 h 1 = 1 or lim h !1 x 1 h 2 = 1 : Hence, (xh 2 1 )a2 ! K > 0 is impossible. (cid:16) (cid:17) (cid:16) (cid:17) (xh)a1 2. 1 0 (xh)a2 ! 2 By repeating the same procedure, I have xh a1 (cid:0) 1 1 = (cid:12) (cid:25) (cid:21)1 (cid:11)+(1 (cid:25) )(cid:21)1 (cid:11) lim 1 " 11 1(cid:0) (cid:0) 11 2(cid:0) h !1 (cid:0) xh 2(cid:1) a2 (cid:0) 1 # xh a2 (cid:0) 1 (cid:0) (cid:1) 1 = (cid:12) (cid:25) (cid:21)1 (cid:11) lim 2 +(1 (cid:25) )(cid:21)1 (cid:11) " 21 1(cid:0) h !1 (cid:0) xh 1(cid:1) a1 (cid:0) 1 (cid:0) 21 2(cid:0) # (cid:0) (cid:1) (xh)a1 (xh)a1(cid:0) 1 (xh)a1(cid:0) 1 If 1 0; then either lim 1 = 0 or lim 1 = . Hence (xh 2 )a2 ! h !1 (xh 2 )a2(cid:0) 1 h !1 (xh 2 )a2(cid:0) 1 1 (xh)a1 1 0 is impossible. (xh)a2 ! 2 I are now in a position to state the following 40

Theorem 16 There exists a bounded and strictly positive equilibrium price-dividend function for probability structure (11) Proof. The statement follows directly from Lemmas 13-15. Computation of returns.(cid:151) To write the Euler as a (nonlinear) (cid:133)rst order di⁄erence equation in the price-dividend ratio, recall 1 (cid:18) t+1 c t+1 (cid:0) (cid:11) (w t+1 =c t+1 ) (cid:0) at+1 (cid:18) t+1 c t+1 1 = (cid:12)E (cid:0) 1+ R " 1 (cid:0) (cid:18) t (cid:18) c t (cid:19) (w t =c t ) (cid:0) at (cid:18) 1 (cid:0) (cid:18) t+1 w t+1 (cid:19) t+1 # De(cid:133)ne f = pt to be the price-dividend ratio and observe that in equilibrium I can write t yt w = (f +1)y : The Euler, then, becomes t t t 1 (cid:18) t+1 c t+1 (cid:0) (cid:11) (f t+1 +1) (cid:0) at+1 (cid:18) t+1 c t+1 1 = (cid:12)E (cid:0) 1+ R " 1 (cid:0) (cid:18) t (cid:18) c t (cid:19) (f t +1) (cid:0) at (cid:18) 1 (cid:0) (cid:18) t+1 (f t+1 +1)y t+1 (cid:19) t+1 # 1 (cid:18) t+1 c t+1 (cid:0) (cid:11) (f t+1 +1) (cid:0) at+1 1 (cid:18) t+1 1 1 = (cid:12)E (cid:0) 1+ (cid:0) R " 1 (cid:0) (cid:18) t (cid:18) c t (cid:19) (f t +1) (cid:0) at (cid:18) 1 (cid:0) (cid:18) t f t+1 +1 (cid:19) t+1 # since by de(cid:133)nition returns are R = pt+1+yt+1 = p yt t + + 1 1+1 yt+1 = ft+1+1yt+1; I have t+1 pt (cid:16) p yt tyt (cid:17) ft yt 1 (cid:18) t+1 c t+1 (cid:0) (cid:11) (f t+1 +1) (cid:0) at+1 1 (cid:18) t+1 1 f t+1 +1y t+1 1 = (cid:12)E (cid:0) 1+ (cid:0) " 1 (cid:0) (cid:18) t (cid:18) c t (cid:19) (f t +1) (cid:0) at (cid:18) 1 (cid:0) (cid:18) t f t+1 +1 (cid:19) f t y t # 1 (cid:18) t+1 c t+1 1 (cid:0) (cid:11) (f t+1 +1) (cid:0) at+1 1 (cid:18) t+1 1 f t+1 +1 1 = (cid:12)E (cid:0) 1+ (cid:0) " 1 (cid:0) (cid:18) t (cid:18) c t (cid:19) (f t +1) (cid:0) at (cid:18) 1 (cid:0) (cid:18) t f t+1 +1 (cid:19) f t # Hence, the Euler can be rewritten as f t (f t +1) (cid:0) at = (cid:12)E 1 (cid:0) (cid:18) t+1 c t+1 1 (cid:0) (cid:11) 1+ 1 (cid:0) (cid:18) t+1 1 (f t+1 +1)1 (cid:0) at+1 1 (cid:18) c 1 (cid:18) f +1 " t t t t+1 # (cid:0) (cid:18) (cid:19) (cid:18) (cid:0) (cid:19) Details of the derivations in the text Hedonic Relative Risk Aversion.(cid:151) Consider the a-temporal case where the outcome l L is independent of the preference 2 states S;withprobabilitiesgivenbyP andP respectively. Itisstraightforwardtoderive l s 2 41

the hedonic risk aversion of the investor by using the de(cid:133)nition of relative risk aversion: In fact, u(g) = EU (C;W=C;S) = P P C1 (cid:11) W=C l (1 (cid:0) (cid:11))(cid:18)s l s l(cid:0) 1 (cid:11) l Ls S (cid:0) X2 X2 = P (W=C ) l l V l L X2 where (W=C ) = P C1 (cid:11)W=C l (1 (cid:0) (cid:11))(cid:18)s is the state independent utility function, that V l s S s l(cid:0) 1 (cid:11) 2 (cid:0) is a linear combinatPion with positive weights of conditionally isoelastic concave functions, and thus concave. Moreover, given that S and L are orthogonal, the curvature of (W=C ) l V captures the investors(cid:146)attitude toward atemporal risk. Hence, the Arrow-Pratt coe¢ cient of relative risk aversion with respect to (wealth relative to consumption) lotteries on L is WW RRA = W=CV W=C (cid:0) w V V W = P s C l 1 (cid:0) (cid:11)(cid:18) s W=C l (1 (cid:0) (cid:11))(cid:18)s (cid:0) 1 s S X2 V WW = P s C l 1 (cid:0) (cid:11)(cid:18) s ((1 (cid:0) (cid:11))(cid:18) s (cid:0) 1)W=C l (1 (cid:0) (cid:11))(cid:18)s (cid:0) 2 s S X2 RRA = W=C s S P s C l 1 (cid:0) (cid:11)(cid:18) s ((1 (cid:0) (cid:11))(cid:18) s (cid:0) 1)W=C l (1 (cid:0) (cid:11))(cid:18)s (cid:0) 2 W 2 (cid:0) P s S P s C l 1 (cid:0) (cid:11)(cid:18) s W=C l (1 (cid:0) (cid:11))(cid:18)s (cid:0) 1 2 = ((1 (cid:11))(cid:18)P 1) s (cid:0) (cid:0) (cid:0) s S X2 = (((cid:11) 1)(cid:18) +1) s (cid:0) s S X2 If preferences are state-independent, i.e. (cid:18) = (cid:18) s; then the coe¢ cient of relative risk s 8 aversion is constant and equal to ((cid:11) 1)(cid:18)+1: The coe¢ cient of relative risk aversion for (cid:0) lotteries that are conditional on the realization of a given state s is ((cid:11) 1)(cid:18) +1: Since s (cid:0) each period is associated with a single preference state, ((cid:11) 1)(cid:18) +1 can be interpreted as t (cid:0) the coe¢ cient of relative risk aversion for static lotteries. 42

Intertemporal elasticity of substitution.(cid:151) Recall the Euler 1 = (cid:12)E g(1 (cid:11))(1 (cid:18)t) 1g(1 (cid:11))(cid:18)tk c t+1 (1 (cid:0) (cid:11))((cid:18)t (cid:0) (cid:18)t+1) R c (cid:0) (cid:0) (cid:0) w(cid:0) t+1 w t+1 " t+1 # (cid:18) (cid:19) where g c = ct c + t 1 and g w = w w t+ t 1: and k t+1 (cid:17) 1 (cid:0) 1 (cid:18)t (cid:18) + t 1 1+ 1 (cid:18)t (cid:18) + t+ 1 1w ct t + + 1 1 Along a balanced (cid:0) (cid:0) growth path with constant interest rates I hav(cid:16)e g =(cid:17)g(cid:16)= g and the Eu(cid:17)ler becomes c w (cid:18) c 1 = (cid:12) g (cid:11)+ g (cid:11) (1+r) (cid:0) (cid:0) 1 (cid:18) w (cid:18) (cid:0) (cid:19) (cid:18) c 1 = (cid:12)g (cid:11) 1+ (1+r) (cid:0) 1 (cid:18)w (cid:18) (cid:0) (cid:19) It is straightforward to observe that if I take the term c as exogenous and ignore the w dependence of wealth on returns, then dg 1 = dr (cid:11) Nevertheless, using the investors(cid:146)budget constraint w c t+1 t = R 1 t+1 w (cid:0) w t t (cid:18) (cid:19) and the balanced growth path assumption I can write c g = (1+r) 1 (cid:0) w (cid:16) (cid:17) which provides c as the following function of r w c g = 1 w (cid:0) 1+r substituting for c into the Euler I have w (cid:18) g 1 = (cid:12)g (cid:11) 1+ 1 (1+r) (cid:0) 1 (cid:18) (cid:0) 1+r (cid:18) (cid:0) (cid:18) (cid:19)(cid:19) Taking logs I have (cid:18) g 0 = log(cid:12) (cid:11)logg+log 1+ 1 +log(1+r) (cid:0) 1 (cid:18) (cid:0) 1+r (cid:18) (cid:0) (cid:18) (cid:19)(cid:19) (cid:18) g 0 ((cid:12) 1) (cid:11)(g 1)+ 1 +r (cid:25) (cid:0) (cid:0) (cid:0) 1 (cid:18) (cid:0) 1+r (cid:0) (cid:18) (cid:19) (cid:12) 1+(cid:11)+ (cid:18) +r g = (cid:0) 1 (cid:18) (cid:0) (cid:11)+ (cid:18) 1 1 (cid:18)1+r (cid:0) 43

Hence, the elasticity of intertemporal substitution is (cid:11)+ (cid:18) 1 + (cid:12) 1+(cid:11)+ (cid:18) +r (cid:18) 1 dg 1 (cid:18)1+r (cid:0) 1 (cid:18) 1 (cid:18)(1+r)2 = (cid:0) (cid:0) (cid:0) dr (cid:16) 2 (cid:17) (cid:11)+ (cid:18) 1 1 (cid:18)1+r (cid:0) (cid:16) (cid:17) 1 (cid:18) (cid:12) 1+(cid:11)+ (cid:18) +r = 1+ (cid:0) 1 (cid:18) (cid:0) (cid:11)+ (cid:18) 1 2 1 (cid:18) (cid:11)+ (cid:18) 1 (1+r)23 1 (cid:18)1+r (cid:0) 1 (cid:18)1+r (cid:0) (cid:0) 4 (cid:16) (cid:17) 5 44

APPENDIX B: TABLES AND FIGURES Table 1 - Summary of unconditional (cid:133)rst and second moments of returns in the benchmark calibration US data US data US data HM, iid HM, Markov (MP sample) 1891-1998 1947-1998 (cid:18) = :26;(cid:12) = 0:97 (cid:18) = :25;(cid:12) = 0:99 E Rf 0.80 1.91 0.90 1.77 0.84 E(cid:0)(Re)(cid:1) 6.98 7.91 8.08 7.81 6.95 E(Rep) 6.18 6.00 7.18 6.03 6.11 (cid:27) Rf 5.44 5.44 1.75 2.22 5.55 (cid:27)(cid:0)(Re)(cid:1) 19.02 18.60 15.65 22.82 23.17 (cid:27)(Rep) 18.53 18.50 15.27 22.61 22.57 E(P=D) 23.75 23.75 28.31 28.34 38.12 (cid:27)(P=D) 7.6 7.6 11.5 7.16 6.4 E(Rep) 0.33 0.32 0.47 0.34 0.27 (cid:27)(Rep) Note: This table reports historical data and model-implied moments of (cid:133)nancial variables for various assets. All statistics are annualized and in percent terms. The (cid:133)nancial variables are the riskfree rate, Rf, the equity return, Re; the excess return of equity over the riskfree rate, Rep, and the price to dividend ratio, P=D: The moments are mean, E, and standard deviation, (cid:27). The historicaldatafortheMPsampleisfromMehraandPrescott(1985)andcoversthe1889-1985period. Otherwise,historicaldataisfromCampbell(1999). Theonlydi⁄erenceinthecalibrationofcolumns "HM,iid"and"HM,Markov"isthespeci(cid:133)cationoftheprocessfor(log)consumptiongrowthunder whichthehappinessmaintenancemodelissolved. Theprocessisspeci(cid:133)edasiidlognormalfor"HM, iid" (speci(cid:133)cation (12) in the text) and Markov for "HM, Markov" (speci(cid:133)cation (11) in the text). Parameter values: (cid:11) = 3; n = 2;(cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: Implied hedonic risk aversion: E(a ) = 1:5; (cid:27)(a ) = 0:009; a = 1:52; a = 1:48: t+1 t+1 tmax tmin 45

Table 2 - Summary of unconditional (cid:133)rst and second moments of returns when (cid:18) = (cid:18) t t+1 US data US data US data No Happiness No HM (MP sample) 1891-1998 1947-1998 ((cid:18) = (cid:18) = 0) ((cid:18) = (cid:18) = 0:25) t t+1 t t+1 E Rf 0.80 1.91 0.90 5.74 4.64 E(cid:0)(Re)(cid:1) 6.98 7.91 8.08 6.22 5.24 E(Rep) 6.18 6.00 7.18 0.48 0.60 (cid:27) Rf 5.44 5.44 1.75 1.57 2.20 (cid:27)(cid:0)(Re)(cid:1) 19.02 18.60 15.65 4.87 5.65 (cid:27)(Rep) 18.53 18.50 15.27 4.60 5.20 E(Rep) 0.33 0.32 0.47 0.10 0.11 (cid:27)(Rep) Note: This table reports historical data and model-implied moments of (cid:133)nancial variables for various assets. All statistics are annualized and in percent terms. The (cid:133)nancial variables are the riskfree rate, Rf, the equity return, Re; the excess return of equity over the riskfree rate, Rep, and the price to dividend ratio, P=D: The moments are mean, E, and standard deviation, (cid:27). The historical data for the MP sample is from Mehra and Prescott (1985) and covers the 1889-1985 period. Otherwise, historical data is from Campbell (1999). The di⁄erence between columns "No Happiness" and "No HM" is in the calibration of the hedonic risk aversion parameter, (cid:18) ;under t whichthehappinessmaintenancemodelissolved. Theparameterissetequaltozeroincolumn"No Happiness" and as a constant equal to 0.25 in column "No HM". Parameter values: (cid:11) = 3; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 46

Table 3 - Inspecting the mechanism: a t E Rf E(Rep) E(Rep) (cid:27)(Rep) US data 0(cid:0).80(cid:1) 6.18 0.32 (cid:11) = 2; (cid:18) = 0:42 0.21 4.27 0.23 (cid:11) = 3; (cid:18) = 0:24 0.84 6.11 0.27 (cid:11) = 4; (cid:18) = 0:16 2.36 6.12 0.28 (cid:11) = 5; (cid:18) = 0:12 3.71 6.19 0.30 (cid:11) = 6; (cid:18) = 0:10 5.05 6.16 0.32 (cid:11) = 7; (cid:18) = 0:08 6.22 6.23 0.33 (cid:11) = 8; (cid:18) = 0:06 7.41 6.18 0.35 (cid:11) = 9; (cid:18) = 0:05 8.44 6.24 0.37 (cid:11) = 10; (cid:18) = 0:04 9.54 6.11 0.39 Note: This table reports historical data and model-implied moments of (cid:133)nancial variables for various assets. All statistics are annualized and in percent terms. The (cid:133)nancial variables are the riskfreerate,Rf,andtheexcessreturnofequityovertheriskfreerate,Rep:Themomentsaremean, E,andstandarddeviation,(cid:27). ThehistoricaldataisfromMehraandPrescott(1985)andcoversthe 1889-1985 period. Model-implied moments are from the Happiness Maintenance model for various values of hedonic risk aversion, a . t Parameter values: (cid:12) = 0:99; n = 2; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 47

Table 4 - Cyclicality and Correlations US data No Happiness HM-1 HM-2 (cid:18) = (cid:18) = 0 n = 2 n = 5 t t+1 P =D p-cyclical -0.15 0.08 0.09 t t (cid:26)(P =D ;P =D ) 0.78 -0.15 0.55 0.63 t t t 1 t 1 (cid:0) (cid:0) (cid:26)(P =D ;P =D ) 0.59 0.01 0.33 0.54 t t t 2 t 2 (cid:0) (cid:0) (cid:26)(P =D ;P =D ) 0.54 0.00 -0.04 0.43 t t t 3 t 3 (cid:0) (cid:0) (cid:26)(P =D ;P =D ) 0.36 0.00 0.00 0.21 t t t 5 t 5 (cid:0) (cid:0) Note: This table reports historical data and model-implied moments of (cid:133)nancial variables for variousassets. Allstatisticsareannualizedandinpercentterms. The(cid:133)nancialvariablesaretheprice todividendratios, P=D;anditslagsupto5years:Themomentsarecorrelations,(cid:26). Thehistorical data is from Campbell (1999) and covers the 1889-1985 period. The only di⁄erence among columns "No Happiness," "HM-1," and "HM-2" is the calibration of the hedonic risk aversion parameter, (cid:18) ;under which the happiness maintenance model is solved. The parameter is set equal to zero in t column "No Happiness" and as in the benchmark parametrization (speci(cid:133)cation (2) in the text) for columns "HM-1" and "HM-2" with 2 and 5 lags, respectively. Parameter values: (cid:11) = 3; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 48

Table 5 - Long-horizon volatility US Data No Happiness HM-1 HM-2 (cid:18) = (cid:18) = 0 n = 2 n = 5 t t+1 (cid:12) 1.14 1.0 4.9 2.0 5 (cid:12) 1.5 1.0 4.5 2.0 10 (cid:12) 1.61 1.0 3.9 1.7 20 (cid:12) 1.42 1.0 2.8 1.4 40 R2 60% 98% 61% 80% 5 R2 76% 98% 52% 80% 10 R2 70% 98% 40% 75% 20 R2 61% 98% 36% 75% 40 Note: This table reports regression results of the change in (log) prices, p p ; on a cont t k (cid:0) (cid:0) stant and the change in (log) consumption, c c ;for k = 5;10;20;40 years using historical t t k (cid:0) (cid:0) and model-implied data. The reported statistics are the estimated slope coe¢ cient, (cid:12); and R2 of these regressions. The "US data" column runs regressions with annual NYSE data for the 1889- 1985 period. The only di⁄erence among columns "No Happiness," "HM-1," and "HM-2" is that the regressions are run on model-generated data under alternative calibrations of the hedonic risk aversionparameter, (cid:18) . Theparameterissetequaltozeroincolumn"NoHappiness,"andasinthe t benchmark parametrization (speci(cid:133)cation (2) in the text) for columns "HM-1" and "HM-2" with 2 and 5 lags, respectively. Parameter values: (cid:12) = 0:99; (cid:11) = 2; (cid:18) = 0:24; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 49

Table 6 - Long-horizon predictability US data No Happiness HM-1 HM-2 (cid:18) = (cid:18) = 0 n = 2 n = 5 t t+1 (cid:12) -1.5 -15.6 -6.9 -5.2 1 (cid:12) -3.0 -15.6 -10.8 -6.5 2 (cid:12) -3.7 -15.7 -10.2 -7.7 3 (cid:12) -6.6 -15.1 -10.4 -11.1 5 (cid:12) -12.1 -14.7 -10.2 -10.5 10 R2 4% 7% 28% 13% 1 R2 8% 5% 44% 15% 2 R2 10% 4% 41% 18% 3 R2 19% 3% 42% 25% 5 R2 39% 2% 40% 21% 10 Note: This table reports regression results of log stock returns, n r ; on a constant and the k=1 t log price-dividend ratio for n = 1;2;3;5;10 years using historicaPl and model-implied data. The reportedstatisticsaretheestimatedslopecoe¢ cient,(cid:12);andR2 oftheseregressions. The"USdata" columnrunsregressionswithannualNYSEdataforthe1889-1985period. Theonlydi⁄erenceamong columns "No Happiness," "HM-1," and "HM-2" is that the regressions are run on model-generated data under alternative calibrations of the hedonic risk aversion parameter, (cid:18) . The parameter is set t equal to zero in column "No Happiness," and as in the benchmark parametrization (speci(cid:133)cation (2) in the text) for columns "HM-1" and "HM-2" with 2 and 5 lags, respectively. Parameter values: (cid:12) = 0:99; (cid:11) = 2; (cid:18) = 0:24; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 50

Table 7 - Summary of unconditional (cid:133)rst and second moments of returns under alternative speci(cid:133)cations of the happiness demand process HM Countercyclical HM-1 ((cid:18) = 0:24;n = 2) ((cid:18) = 0:24;n = 2) ((cid:18) = 0:25 > (cid:18) = 0:23) h l E Rf 0.84 3.85 0.84 E((cid:0)Re)(cid:1) 6.95 5.41 6.95 E(Rep) 6.11 1.55 6.11 (cid:27) Rf 5.55 2.89 5.55 (cid:27)(cid:0)(Re)(cid:1) 23.17 13.84 23.17 (cid:27)(Rep) 22.57 13.50 22.57 E(Rep) 0.27 0.11 0.27 (cid:27)(Rep) Note: This table reports model-implied moments of (cid:133)nancial variables for various assets. All statistics are annualized and in percent terms. The (cid:133)nancial variables are the riskfree rate, Rf, the equity return, Re; the excess return of equity over the riskfree rate, Rep, and the price to dividend ratio,P=D:Themomentsaremean,E,andstandarddeviation,(cid:27). Thedi⁄erencebetweencolumns "HM," "Countercyclical," and "HM-1" is in the calibration of the hedonic risk aversion parameter, (cid:18) ;under which the happiness maintenance model is solved. The parameter is set as in the standard t parametrization (speci(cid:133)cation (2) in the text) for column "HM," as perfectly negatively correlated with consumption growth for column "Countercyclical," and as a two-state Markov-process that takes values 0.25 and 0.23 in expansions and recessions, respectively. Parameter values: (cid:11) = 3; (cid:22) = 0:018; (cid:14) = 0:036; (cid:25) = 0:43: 51

Figure 1 - Hansen-Jagannathan Bounds 1.2 1.2 1 1 0.8 0.8 )y(0.6 )y(0.6 s s 0.4 0.4 0.2 0.2 0 0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 E(y) E(y) Panel A: The equity premium puzzle, (cid:18) = 0 Panel B: Happiness maintenance, (cid:18) = :05 1.2 1.2 1 1 0.8 0.8 )y(0.6 )y(0.6 s s 0.4 0.4 0.2 0.2 0 0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 E(y) E(y) Panel C: Happiness maintenance, (cid:18) = :15 Panel D: Happiness maintenance, (cid:18) = :24 Note: The (cid:133)gure plots Hansen and Jagannathan (1991)(cid:146)s bounds and the model implied market price of risk for di⁄erent values hedonic risk aversion. The market price of risk is the ratio of conditional standard deviation to mean of the model implied pricing kernel. X axis displays the conditional mean, Y axis displays conditional standard deviation. 52