Happiness and Loss Aversion: When Social Participation Dominates Comparison. Maarten Vendrik Geert Woltjer 1. Preliminary Version

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1 Happiness and Loss Avesion: When Social Paticipation Dominates Compaison Maaten Vendik Geet Woltje 1 Peliminay Vesion Abstact A cental finding in happiness eseach is that a peson s income elative to the aveage income in he social efeence goup is much moe impotant fo he life satisfaction than the absolute level of he income. This dependence of life satisfaction on elative income can be elated to the efeence dependence of the value function in Kahneman and Tvesky s (1979) pospect theoy. In this pape we investigate whethe the chaacteistics of the value function like concavity fo gains, convexity fo losses, and loss avesion apply to the dependence of life satisfaction on elative income. This is tested with a new measue fo the efeence income on a lage Geman panel fo the yeas We find indications fo significant concavity of life satisfaction in positive elative income, but supisingly stongly significant concavity of life satisfaction in negative elative income as well. The latte esult is shown to be obust to exteme distotions of the epotedlifesatisfaction scale. It implies a ising maginal sensitivity of life satisfaction to moe negative values of elative income, and hence loss avesion. We explain this in tems of inceasing financial obstacles to social paticipation. 1 Both authos, Depatment of Economics, Maasticht Univesity, P.O. Box 616, 6200 MD Maasticht, The Nethelands, coesponding autho: Tel ( ). We thank Andew Clak, Eik de Regt, Alois Stutze, Bob Sugden, Pete Wakke, and (othe) paticipants to the 2 nd Wokshop on Capabilities and Happiness in MilanoBicocca, June 1618, 2005, fo helpful comments on an ealie vesion of this pape. We ae also indebted to Denis de Combugghe and Syband Schim van de Loeff fo econometic advice.
2 1. Intoduction A basic postulate of utility theoy is that utility ises with income. Howeve, empiical eseach on subjective wellbeing (see Fey and Stutze, 2002a, fo an oveview) has shown that in most developed counties aveage happiness has not o only slightly inceased ove the last fifty yeas despite high economic gowth. This paadox has been explained by Eastelin (1974, 2001) and Fank (1997) in tems of ising aspiations and social compaison. The level of one s income elative to that of othe people is much moe impotant than its absolute level, and if absolute income ises fo eveybody at the same pace, elative income does not change. This impotance of elative income was aleady found by Duesenbey (1949) in his seminal study on individual consumption and savings behaviou. In addition, Duesenbey postulated that social compaisons of income ae not symmetic. In the context of happiness this means that the happiness of pooe people is negatively affected by the income of thei iche pees, wheeas iche people do not get happie fom compaing thei income with that of pooe membes of thei efeence goup. Thity yeas late, Kahneman and Tvesky (1979) intoduced thei pospect theoy as an altenative to the standad micoeconomic theoy of choice. It includes a value function that is defined ove gains and losses with espect to some natual efeence point and that is concave fo gains and convex fo losses. It is also assumed to be steepe fo losses than fo gains, which is efeed to as loss avesion (Tvesky and Kahneman, 1991). Finally, the value function is hypothesized to be as convex fo losses as it is concave fo gains (eflection effect; see Kahneman, 2003, and see Köbbeling et al., 2004, fo a diffeent hypothesis). These chaacteistics of the value function have been tested quite extensively fo decision utility and psychological peceptions (see Köbbeling et al. fo an oveview), but, as Kahneman (1999, p. 19) states, the extent to which loss avesion is also found in expeience is not yet known (see, howeve, Galante, 1990). In this context thee is an inteesting elationship with the dependence of happiness on the aveage income in a social efeence goup as postulated by Eastelin, Fank and othes. Such a social efeence income povides a natual efeence point fo people to compae thei own income with. Income elative to this social efeence income then coesponds to gains fo positive values and to losses fo negative values. Moeove, the assumed asymmety in the income compaison amounts to loss avesion in Kahneman and Tvesky s theoy. This then also aises the question whethe the othe chaacteistics of the value function like concavity fo gains and convexity fo losses may apply to this dependence of happiness on elative income as well. In this pape we test the chaacteistics of the value function fo individual life satisfaction functions fo income levels above and below social efeence incomes. A significant positive dependence of life satisfaction on elative income has aleady been found in the econometic cosssection and panel studies of McBide (2001), Stutze (2004) and FeeiCabonell (2005). 2
3 Futhemoe, the last study finds suppot fo the asymmety postulate of Duesenbey. Howeve, the measues fo the efeence income that wee used in the thee studies have some dawbacks that we ty to impove on by intoducing a new measue. With this new measue we then test all the chaacteistics of the value function mentioned fo a lage Geman panel (GSOEP) spanning the yeas 1984 to To test concavity fo positive elative income and convexity fo negative elative income, we need a moe flexible specification of life satisfaction in tems of elative income than the usual logaithmic one since the latte is concave fo negative as well as positive elative income. Fo that pupose a powefunction specification is used, which includes the logaithmic specification as a limiting case. We then find indications fo significant concavity of life satisfaction in positive elative income, but supisingly vey significant concavity of life satisfaction in negative elative income as well. Moeove, unde plausible assumptions on the cadinal popeties of the tuelifesatisfaction scale, the latte esult tuns out to be obust to exteme distotions of the epotedlifesatisfaction scale (such as suggested by Oswald, 2005). It means a stong ejection of the hypothesized convexity and implies a ising maginal sensitivity of life satisfaction to moe negative values of elative income. Ou explanation fo this esult is that as a peson s income falls inceasingly shot of that in the efeence goup, it becomes inceasingly and moe than popotionally had to aise the funds to paticipate in the social activities of the efeence goup. This effect of ising maginal social paticipation costs of falls in elative income appaently dominates a possible social compaison effect of diminishing maginal sensitivity of life satisfaction to such falls, which is suggested by the theoy of Kahneman and Tvesky. Moeove, it is shown to imply significant loss avesion (at least fo lage losses). The pape is stuctued as follows. Section 2 fomulates the hypotheses to be tested, and Section 3 discusses the poblems of testing cadinal popeties of tue life satisfaction and ou way to tackle these poblems. Section 4 pesents the econometic specifications, Section 5 intoduces the new measue of efeence goup, and Section 6 discusses the data and estimation pocedue. Section 7 pesents and analyses the estimation esults, Section 8 deives thei implications fo cadinal popeties of tue life satisfaction, and Section 9 tests the obustness of the esults fo othe specifications and populations. Section 10 discusses ou explanation fo the esults and discusses possible implications fo behaviou. Finally, Section 11 concludes. 2. Hypotheses While the influence of social compaison on individual pefeences in decisionmaking and on income and job satisfactions has aleady been investigated fo quite some time (see FeeiCabonell fo a ecent oveview), econometic studies that estimate the impact of social compaison on individual life satisfaction ae athe ecent. In cosssection and panel studies fo Ameican, Swiss and Geman data 3
4 McBide (2001), Stutze (2004) and FeeiCabonell (2005) find that the oveall life satisfaction of individuals negatively depends on the aveage (eal) incomes in thei social efeence goups. Moeove, the diffeence between thei own income and such social efeence income (elative income) tuns out to be a moe impotant deteminant of individual life satisfaction than the absolute level of income. The empiical findings on the impact of social compaison on individual pefeences, income and job satisfactions, and life satisfaction ae patially consistent with the moe geneal valuefunction theoy of Kahneman and Tvesky (1979) fo decisions unde isk and of Tvesky and Kahneman (1991) fo iskless choice (see the latte pape fo cossefeences). The social efeence income povides a natual efeence point fo people, to which they compae thei own income, and the asymmety esult of FeeiCabonell suggests loss avesion. 2 On the othe hand, most studies employ a utility function that is concave fo negative as well as positive elative income. 3 chaacteistics of the value function have been tested quite extensively fo decision utility and psychological peceptions (see Köbbeling et al. fo an oveview), but, as Kahneman (1999, p. 19) indicates and as fa as we know, chaacteistics like loss avesion have not been tested fo expeienced utility as measued ex post. Some indications on possible popeties of such expeienced utility ae given by psychophysical expeiments epoted by Galante (1990). In one such expeiment subjects wee asked to imagine that they lost $5 (o $50 esp $500), and to name the amount of money they would have to lose to make them exactly twice as upset as losing $5 (o $50 esp $500). The geometic means of the subjects judgments was $21 (o $160 esp $1850), and fitting this to a powe function of the fom b U = am yielded an exponent b equal to A vaiety of cossmodality matching expeiments yielded simila values of b, always a bit lage than 0.5, fo losses, but fo gains values of b always a bit smalle than 0.5. In geneal, loss avesion was found, and this finding as well as those of concavity fo gains and convexity fo losses is consistent with the chaacteistics of Kahneman and Tvesky s value function. 4 Howeve, the utility that was measued in these expeiments is not expeienced utility as measued ex post, but athe expeienced utility as imagined ex ante. In the teminology of Kahneman et al. (1997) this may be called pedicted utility as distinct fom expeienced utility pe se. It might be consideed as an intemediate case between expeienced utility and decision utility and may deviate The 2 Howeve, the tstatistic values computed by FeeiCabonell ae not conclusive since they neglect possible covaiances between the two compaed paamete estimates. 3 Clak and Oswald (1998) show that while compaisonconcave utility leads to following behaviou, compaisonconvex utility implies deviant behaviou. 4 Kahneman (1999, p. 19) claims that the diffeence in the estimates of exponent b fo gains and losses would not account fo the extensive loss avesion obseved in choice expeiments. Howeve, we did not find evidence fo this claim in the analysis of Gallante (see his Fig. 4). 4
5 fom both. In this pape, we focus on expeienced utility as measued ex post in suveys. Moe specifically, we test the chaacteistics of the value function fo the shapes of life satisfaction functions fo income levels above and below the aveage incomes in social efeence goups. The hypotheses tested ae: H1: Individual life satisfaction (S) depends significantly negatively on the aveage (eal) family income in a social efeence goup, contolling fo (eal) family income and othe vaiables. This hypothesis is based on what one would expect on the basis of psychological and sociological theoies of social efeence pocesses (e.g., Festinge, 1954), and has been confimed by the empiical studies of McBide, Stutze, and FeeiCabonell cited above. On the othe hand, Senik (2004) finds a significantly positive sign fo Russian panel data, which she explains by the use of the infomational content of changes in othe people s incomes in foming one s own income expectations (see also Schyns, 2002). We conside family income instead of individual income fo two easons. Fist, on a conceptual level, family income may be a moe adequate deteminant of S since the impact of income on S uns pimaily via individual consumption and this is elated to family athe than individual income. Second, the dataset that we use (see Sec. 5) contains bette data fo family than fo individual income. Howeve, since the consumption out of the family income is divided ove all household membes, we contol fo numbe of adults and numbe of childen in the household. H2: S depends significantly positively on the diffeence between family income and the aveage family income in a social efeence goup (elative income), contolling fo family income (absolute income) and othe vaiables. This hypothesis coesponds to the efeence dependence of Kahneman and Tvesky s value function, and is explicitly confimed by FeeiCabonell. H3: S depends significantly moe positively on elative income than on absolute income. This hypothesis investigates the dominance of efeence dependence ove efeenceindependent effects. H4: S is concave in positive values of elative family income and convex in negative values of elative family income. 5
6 This hypothesis coesponds to the diminishing sensitivity of the maginal value of both gains and losses, which is implied by Kahneman and Tvesky s value function and suggested by the esults of Galante epoted above. H5: S is as concave in positive values of elative family income as it is convex in negative values of elative family income (fo equal absolute magnitudes of negative and positive elative family income). This hypothesis coesponds to the eflection effect with espect to gains and losses as hypothesized by Kahneman and Tvesky (1979, pp ) except fo the asymmety implied by loss avesion (see below). Seveal studies on decision utility suggest that the value function fo gains and losses is faily well appoximated by powe functions with simila exponents, both less than unity (Swalm, 1966; Tvesky and Kahneman, 1992; Kahneman, 2003). These exponents eflect the degees of concavity of the value function fo gains and of convexity fo losses. On the othe hand, fo the case of pedicted utility Galante epots highe estimated exponents fo losses than fo gains (see above). This suggests that expeienced utility of money may be less convex fo losses than it is concave fo gains. Fo decision utility this is also hypothesized by Köbbeling et al. (2004). They ague that the psychological peceptionofquantity effect as decibed by Kahneman and Tvesky s value function is then confounded with the economic intinsicutilityofmoney effect, which is assumed to exhibit diminishing maginal utility. This agument can explain Galante et al. s findings fo pedicted utility as well. Howeve, in the pesent lifesatisfaction context the intinsicutilityofmoney effect can be associated with the effect of absolute income on S, while the effect of elative income may be fully consistent with the chaacteistics of the value function, including the eflection effect. We theefoe maintain hypothesis H6. Accodingly, with espect to the effect of absolute income we hypothesize: H6: S is concave in (positive values of) absolute family income. Finally, H7: S as a function of elative income is significantly steepe fo negative values of elative income than fo positive values (at equal absolute levels of negative and positive elative income). This hypothesis coesponds to the loss avesion implied by Kahneman and Tvesky s value function and suggested by the esults of Galante. 6
7 3. Testing cadinal popeties of tue life satisfaction To test ou hypotheses, fist of all a eliable measue of life satisfaction is needed. Usually, such a measue involves a peson s answe to a suvey question about he/his oveall life satisfaction. Fo example, in the GSOEP data set we use in this pape the satisfaction question is: Please, answe accoding to the following scale: 0 means completely dissatisfied, 10 means completely satisfied. How satisfied ae you with you life, all things consideed? 5 The esulting scoe of epoted life satisfaction (R) can be consideed as a easonably eliable measue of a peson s tue life satisfaction (S), but the eliability of the scoe is limited by its disceteness and nonandom measuement eos (see, e.g., Fey and Stutze, 2002a, 2002b; FeeiCabonell and Fijtes, 2004). In paticula, thee may be cultual noms that lead people to oveestimate o undeestimate thei happiness (see, e.g., JohanssonStenman and Matinsson, 2005). This may especially affect cadinal popeties of the life satisfaction function like concavity and convexity, as addessed in ou hypotheses H4H6. To see this, assume that a peson s epoted life satisfaction R is a stepwiseinceasing function of he tue life satisfaction o wellbeing S plus a measuement eo tem (Blanchflowe and Oswald, 2004). If the measuement eo would be only andom, an imaginay OLS egession of R on S would yield a linea elation with slope 1. Howeve, conside a cultue (like pehaps the Bitish) whee people ae eluctant to give a high assessment of thei happiness (Oswald, 2005). 6 This will lead to an undeestimation of high values of S, implying a systematic downwad bias of R and a concave shape of the egession elation between R and S. Estimated concavity of R in elative o absolute income Y could then be due to a compaable degee of concavity of the estimated Rˆ in S, leaving no (significant) concavity of tue life satisfaction S in Y. This can be seen by witing R ˆ( Y ) as R ˆ ( S ( Y )) (see Oswald, 2005, fo a fomal poof, and see Sec. 8 fo moe details). On the basis of this possibility Oswald concludes that the usual finding of diminishing etuns of income in happiness eseach does not pove diminishing maginal utility. Moeove, in the context of ou study, it seems to dismiss egessions of epoted life satisfaction R on elative and absolute income as a eliable means to test hypotheses H4H6. At fist sight, a bette way to test these hypotheses seems to be estimation of the elation of S to elative o absolute income by an odeed pobit o logit, whee the latent vaiable Z can be consideed as a poxy fo the tue wellbeing S (Blanchflowe and Oswald; Fey and Stutze, 2002b, p. 406). Howeve, the question is whethe this intepetation of Z is valid not only in an odinal, but also 5 This is the English tanslation in the GSOEP of 2001 of the Geman question. Supisingly, in pevious yeas the same question was tanslated in tems of unhappy and happy instead of dissatisfied and satisfied (see, e.g., FeeiCabonell, 2005). We assume the moe ecent tanslation is the moe coect one. 6 It is inteesting to note that a simila eluctance to give high gades is found in the gading system at Bitish univesities, whee an 8 tends to be the vitual maximum (Sugden, pesonal communication). 7
8 in a cadinal sense. The answe seems to be no since odeed pobit o logit analysis only uses the odinal infomation in the epoted R scoes (Van Paag and FeeiCabonell, 2004, p. 32; Feei Cabonell and Fijtes, 2004, Sec. 3.2), and hence can neve contain cadinal infomation on R, and so S. Moe specifically, pobit (o logit) analysis depats fom the assumption that epoted R is a stepwiseinceasing (o deceasing) function of an undelying latent vaiable Z. Combined with the above assumption that R is a stepwiseinceasing function of tue wellbeing S, this implies that Z is a monotonously nondeceasing function of S. We can make this elation somewhat stonge by plausibly assuming that Z is a (continuously diffeentiable) monotonously inceasing function of S. Odinal popeties of S ae then the same as the odinal popeties of Z, but cadinal popeties of S and Z may diffe fom each othe. A moe adequate appoach then is cadinal pobit analysis, which makes use of the cadinal infomation in the epoted R scoes as well (Van Paag and FeeiCabonell, 2004, p. 32). S is then assumed to be well appoximated by the standad nomaldistibution function N as a function of Z plus an eo tem ε, i.e. S = N(Zε). Howeve, just as egession analysis, cadinal pobit analysis suffes fom the possibility that estimated Rˆ is concave in S. This may then account fo estimated concavity of S in elative o absolute income. Then the question aises how stong concavity of estimated Rˆ in S might be. This question is investigated in Section 8. To make this investigation tactable, we make the following cucial assumptions: (i) Tue life satisfaction S has, just as R, values on a scale fom 0 to 10 (up to a linea, i.e. affine, tansfomation 7 ). (ii) The scale of S is continuous and cadinal (cf. Van Paag and FeeiCabonell, 2004, Sec. 2.7). (iii) Scoes of epoted life satisfaction R deviate fom tue life satisfaction S within cetain exteme limits (to be specified in Sec. 8). Cadinality of the scale of S means that a diffeence in S between, say, 7.5 and 8.5 epesents the same diffeence in satisfaction intensity as the diffeence between 5.5 and 6.5. This as well as the othe assumptions ae metaphysical assumptions since tue life satisfaction is a metaphysical concept. Hence, they cannot be veified empiically. We can only obseve epoted life satisfaction R. 8 Still, the 7 This excludes scales with values anging fom  and/o to. Thus, the basic assumption is that the tue life satisfaction scale is bounded fom above and below. Thee ae a maximum level and a minimum level of tue life satisfaction, coesponding to complete satisfaction and dissatisfaction, espectively. 8 In the futue, cadinal infomation on tue life satisfaction might come available fom bain measuements, but even then it will be unknown if such measues epesent linea tansfomations of the subjective scale of tue life satisfaction! 8
9 assumptions seem plausible (to us). One exteme limit to the deviation of R fom S is suggested by a possibility aised by Oswald: people who expeience a maximal tue life satisfaction S = 10 may epot this by a scoe of R = 8 (in line with the Bitish gading system, see footnote 3). Section 8 shows that such distotions imply degees of concavity that ae still significantly lowe than estimated degees of concavity of R in elative income accoding to least squaes estimation (LS). This esult implies that estimated significant concavity of R in elative income accoding to LS cannot be (fully) explained by possible concavity of Rˆ in S, and hence implies significant concavity of S in elative income. Thus, epoted life satisfaction R is not assumed to be cadinal, but is nevetheless supposed to contain cadinal infomation on tue life satisfaction S, conditional on assumptions (i)(iii). In the eseach epoted hee this infomation tuned out to be sufficient to test cadinal popeties of S like concavity and convexity in elative and absolute income. Hence, we might say that R is assumed to be cadinal in sufficiently good appoximation (fo ou puposes). Mutatis mutandis, simila aguments will hold with espect to estimated concavity of S in elative o absolute income accoding to a cadinal pobit o logit analysus. Howeve, an LS appoach is much simple and intuitively easie to intepet. Still, a geneal dawback of the usual LS of a loglinea specification is that it in pinciple equies estimation on the eal axis, wheeas the ange of R scoes is bounded between 0 and 10. This may be emedied by egessing a logit o lognomal specification, but then significance poofs become tedious. Moeove, a loglinea specification may give a good appoximation since the vaiation in R that is explained by the explanatoy vaiables is small elative to the [0,10] scale. In the case of LS with individual fixed effects (which we apply in this pape), the loglinea specification may, fo each individual, be consideed as a fistode Taylo expansion in the ln explanatoy vaiables (X) aound an individualspecific baseline level of wellbeing as detemined by the value of the individual fixed effect (cf. Fey and Stutze, 2002b, p. 407). Regessing the loglinea specification ove all individuals in the sample then amounts to aveaging the coefficients of X ove all these individual fistode Taylo expansions. We include individual fixed effects to account fo diffeences in pesonality chaacteistics, which explain up to 80% of the total vaiation in happiness (see...?). Applying LS with individual fixed effects implies that only the vaiation of R with X ove time is estimated (within effects). This has cetain disadvantages (see Sec. 6), but a geat advantage in the pesent context is that it seems to equie only intetempoal (appoximately) cadinal compaability of R scoes (as in intetempoal utilitymaximizing models), but no intepesonal cadinal o odinal compaability of R. 9 This is because the individual fixed effects also coect fo diffeences in the levels at which individuals 9 Thus, ou assumptions coespond to geneal assumption A1 in FeeiCabonell and Fijtes (2004) with the addition that we make some plausible assumptions about the scale of tue wellbeing S and maximal concavity of the positive monotonic tansfomation of S into epoted life satisfaction R. 9
10 ancho thei R scale (Winkelmann and Winkelmann, 1998). These levels may not be andom but coelated with explanatoy vaiables like income and educational level (e.g. via pesonal chaacteistics like optimism and intelligence). This causes the egession coefficients to be biased in the absence of individual fixed effects. Thus, contolling fo such effects implies a stong unbiased test of popeties like concavity of R in X, which seems to avoid the poblematic assumption of intepesonal compaability of R scoes. Because of this advantage, we adopt LS with individual fixed effects as ou baseline estimation. 4. Econometic specifications The usual specification used in econometic wok on life satisfaction and social efeence income (e.g., FeeiCabonell, 2004) is: ~ R = α β lny γ lny δ X e, (1) whee Y is family income, Y is the aveage family income in a social efeence goup, X is a vecto of (ln) contol vaiables, e is an eo tem, and the Geek symbols stand fo paametes. Paametes ~ β and γ ae supposed to be nonnegative and δ is a vecto of paametes. Equation (1) can be ewitten as ~ R = α ( β γ )lny γ (lny lny ) δ X e, (2) which sepaates the elative income effect of the diffeence between ln income and ln efeence income fom the absolute income effect of the ln income level. This elative income vaiable can also ~ be witten as ln( Y / Y ), and we denote the absoluteincome coefficient β γ as β. In the context of Kahneman and Tvesky s valuefunction theoy Y Y seems the elevant vaiable, but ln( Y / Y ) can be ewitten as ln[1 ( Y / Y 1)] = ln[1 ( Y Y ) / Y ], i.e. as a monotonously inceasing function of (pecentual) factions of Y Y. We denote this elative income gap ( Y Y ) / Y. This ( Y Y ) / Y indicates gains and losses as Y, which seem even moe elevant than the absolute gains and losses ( Y Y ) / Y as G. Hence, specifications (1) and (2) can be used to test hypotheses H1H3 and H7. Howeve, specification (2) cannot be used to test hypotheses H4 and H5 since it is concave in G fo losses as well as gains. Fo that pupose, we nest specification (2) into a moe flexible powefunction specification that is given by ρ R = α β ln Y γ[( Y / Y ) 1] / ρ δ X e, ρ 0. (3) Hee the powefunction tem appoaches ln( Y / Y ) fo ρ 0. Rewiting the powe function tem as ρ γ[( 1 ( Y Y ) / Y ) 1] / ρ, it is easily seen that it is concave in G / ( Y Y ) Y fo < 1 ρ and convex in G fo ρ > 1. Concavity of S in positive G and convexity of S in negative G can then be tested as follows. Fist, we define positive elative income G, espectively negative elative income G as equal to G fo positive, espectively negative values of G, but as equal to zeo fo negative, espectively positive values of G (cf. FeeiCabonell). Then, we modify specification (3) as 10
11 ρ ρ R = α β ln Y γ [(1 G ) 1]/ ρ γ [(1 G ) 1]/ ρ δ X e, (4) whee we allow fo diffeent paametes S in G and convexity of S in γ and γ (see below) as well as G then coespond to ρ < 1 and ρ > 1. ρ and ρ. Concavity of Assuming that epoted life satisfaction R is a good poxy fo tue life satisfaction S (see the pevious section), the hypotheses in Section 2 imply the following inequalities fo the paametes in specifications (1)(3). Hypothesis H1 coesponds to a significantly positive value of γ in specification (1), and H2 to significantly positive values of γ in specifications (2) and (3) and of and ~ γ β 2γ β in specification (2). γ in specification (4). Hypothesis H3 coesponds to a significantly positive values of Hypothesis H4 (concavity in gains and convexity in losses) coesponds to ρ being significantly lowe than one and ρ being significantly highe than one in specification (4). Fo hypothesis H5 (equal degees of concavity in gains and convexity in losses) we need a measue fo concavity and convexity. Fo ou powe function specification a convenient (though somewhat ± abitay) measue is ρ 1, which measues the degee of deviation fom a linea elationship ± ( ρ = 1 ) and which is positive fo a convex cuvatue and negative fo a concave one. 10 Hypothesis H5 then implies that ρ 1 should not diffe significantly fom concavity of S in absolute family income (hypothesis H6), we eplace the tem σ γ ( ρ 1) = 1 ρ. To test the β lny in specification (4) by the moe flexible tem β ( Y 1) / σ, σ 0 (which appoaches β lny fo σ 0 ), and estimate whethe σ is significantly lowe than one while β is significantly positive. Hypothesis H7 (loss avesion) is consistent with a significantly positive value of γ in specifications (2) and (3) fo ρ < 1 (concavity) since the maginal utility of elative income G / ( Y Y ) Y then ises as G falls fom positive to negative values. In specification (4) we have a ± ± ± ± 1 ρ ± ± S / G = γ (1 G ) = γ fo G = 0 if γ γ, but not if ρ ρ discontinuity in the slope and γ = γ. It is then easily seen that thee is loss avesion fo all positive values of G = G if γ γ, ρ < 1 and ρ < 1 (concavity fo gains as well as losses). Howeve, if hypothesis H4 (concavity in gains and convexity in losses) holds, this condition cannot be fulfilled since then be geate than one. Fulfilment of hypothesis H7 then depends on the values of elative to γ and ρ and of = ρ will γ and G G G in a complicated way accoding to the slope condition S / G ( G) > S / G ( G). On the othe hand, if hypothesis H4 is ejected, mattes may become less complicated (see Sec. 7). A stonge vesion of hypothesis H7 is that this slope condition also holds in the limit fo G appoaching zeo. This implies a kink in the satisfaction function fo G = 0 ρ 10 Fo concave elations this measue equals minus Patt s measue of elative isk avesion 2 2 ( Y / Y ) ( Y / Y )( R / ( Y / Y ) ) /( R / ( Y / Y )) = 1 ρ, which is a commonly used index to descibe cuvatue of utility. 11
12 (see Kahneman, Knetsch and Thale, 1991), and so a significantly highe value of γ. γ than the value of It is inteesting to note that the powefunction specification used hee diffes fom that employed by Galante (1990) and Tvesky and Kahneman (1992, p. 309). In the pesent context the ρ latte specification would amount to tems γ G and ρ γ ( G ) fo the elative income effects in specification (4), whee all paametes ae supposed to be positive. Fo decision utility Tvesky and Kahneman obtained paamete estimates that in the pesent context would imply that both ae smalle than one (about 0.9) and equal to each othe, and that ρ and ρ γ is moe than twice as high as γ. This would mean simila concavity in gains and convexity in losses (H4, H5) and loss avesion (H7). Fo pedicted utility Galante calculated paamete values that in ou context would imply that ρ = 0.45 and ρ = (the elative size of diffeent scaling in combination with γ and γ cannot meaningfully be infeed because of ρ ρ ). Hence, we would again have concavity in gains and convexity in losses (H4), but not simila (H5). Futhemoe, in the context of Galante s utility function loss avesion can easily be shown to hold except fo vey small values of money. Howeve, in contast to specification (4), the Galante and Tvesky and Kahneman (GTK) specification has the dawback of a lack of flexibility: Fo G and G  appoaching zeo the slopes = 1 ρ S / G γ ρ G and fo ρ > 1 and ρ > 1, while the slopes G = G = 0 fo any values of 1 ρ S / G = γ ρ ( G ) go to fo ρ < 1 and ρ < 1, but to 0 ρ and ± S / G accoding to specification (4) equal ± γ at ρ. Hence, in the fome as opposed to the latte case the slopes at G = G = 0 do not vay continuously and independently fom the concavity and convexity paametes ρ and ρ. Anothe dawback of the GTK specification in the pesent context is that, in contast to specification (4), it does not include the asymmetic vaiant of the usual loglinea specification (2) (cf. FeeiCabonell) as a special o limiting case. 5. Measues fo social efeence income An impotant question is how to define the (social) efeence goup(s) of a peson, i.e. who belongs to his/he efeence goup(s). The efeence goup that has the stongest influence on the peson is likely to be the social goup to which the individual belongs and which consists of people of simila age, education, income, egion of esidence, etc. (FeeiCabonell, 2002; cf. Senik, 2004). In addition, thee will be a less stong influence fom wide goups like the peson s community o egion of esidence (Diene et al., 1993; Stutze, 2004), the peson s cohot (McBide, 2001), o the peson s county (Eastelin, 1995). Thus, in geneal thee will be moe than one efeence goup the aveage income of which can be assumed to affect a peson s life satisfaction (see Kapteyn and Wansbeek, 1985; Vendik and Hiata, 2005). Howeve, in econometic eseach it is usually assumed that thee is 12
13 only one efeence goup which is eithe identified as the social goup of the peson (Feei Cabonell) o appoximated by one of the wide goups mentioned above. FeeiCabonell uses education, age and egion as efeence goup categoies. He efeence goup contains all individuals with a simila education level, inside the same age backet, and living in the same egion (West o East Gemany fo FeeiCabonell s Geman data). The five diffeent education categoies ae: less than 10, 10, 11, 12, and moe than 12 yeas of education, and the age backets ae: younge than 25, 2534, 3544, 4565, and 66 yeas o olde. This leads to 50 diffeent (exogenous) efeence goups. Howeve, this measue has the dawback that the age backets ae fixed. This implies, fo instance, that when a peson becomes 35 yeas old, his/he efeence goup suddenly changes fom to yeas old. This is, of couse, implausible, and in this espect the measue used by McBide (2002) is bette. Fo USA data, he appoximates the influence fom the efeence goup by the aveage income in a peson s cohot consisting of eveyone fom 5 yeas younge than the peson to 5 yeas olde. Thus, this efeence goup moves along with the age of the peson. On the othe hand, a sevee downside of this measue is that it does not distinguish between diffeent education categoies, as FeeiCabonell s measue does. Theefoe, in this pape we intoduce a new measue which combines the best of both measues, i.e. a moving age backet as McBide s in combination with the education and egion categoies of FeeiCabonell and sex. We added sex since especially men may pimaily be influenced by the family income of othe men. 6. Data, vaiables and estimation pocedue The database used fo all estimations is the Geman SocioEconomic Panel (GSOEP; see Wagne et al., 1993; Koh and Spiess, 2005). The panel coves the yeas fo West Gemany and fo East Gemany. The sample includes about individuals who stayed on aveage moe than eight yeas in the panel. Fo most households all individuals ae in the sample. The numbe of missing obsevations fo the vaiables we use is less than 3%. The dependent vaiable used in all estimations is epoted individual life satisfaction (see Sec. 3). The main independent vaiable is eal family income, i.e. family income coected fo the consume pice index. Fo social efeence income we calculated the diffeent measues discussed in the pevious section. As contol vaiables we use moe o less the standad vaiables: Fist, the numbe of adults and childen in the households (see Sec. 2); second, the numbe of yeas of education, since people with a highe educational level tend to have moe skills to become happy; thid, unemployment, since unemployed people will have a lowe level of happiness; fouth, ln age since happiness vaies with age 11 ; fifth, a numbe of dummies fo family situation (maied without childen, single paent, 11 We did not include ln age squaed to account fo a possible Ushape of happiness as a function of age since we estimate with fixed individual and time effects that catch the effect of the vaiation of age between individuals 13
14 maied with childen up to o highe than sixteen yeas old); and finally, woking hous with a dummy fo missing obsevations. Timeinvaiant vaiables like gende ae not included, because we estimate with fixed individual effects that catch the effect of those vaiables. Fo ou baseline estimations we use nonlinea least squaes with fixed time and fixed individual effects. We use fixed time effects to account fo peiod effects on happiness. Fo example, people may be influenced by the phase of the business cycle, events in histoy like the eunion of Gemany, and the pice index may not always be adequate to calculate eal income. We use fixed individual effects fo two elated easons: Fist, since then only the vaiation of life satisfaction with explanatoy vaiables ove time is estimated (within effects), we seem to avoid the poblematic assumption of intepesonal compaability of life satisfaction scoes ove time (see Sec. 2). Second and moe geneally, we pevent biases due to unobseved pesonality taits like optimism and intelligence affecting at the same time life satisfaction and explanatoy vaiables like income and educational level, giving ise to spuious coelations between life satisfaction and these vaiables. Howeve, estimating only within effects of couse means that the infomation fom between effects of inteindividual vaiation is not used. As noted above, this eliminates the impact of timeinvaiant explanatoy vaiables like gende and geatly educes that of age. In a dynamic context it means that only shotun shock effects ae estimated, implying a potential undeestimation of longun level effects of explanatoy vaiables (Van Paag et al., 2003; Van Paag and FeeiCabonell, 2004, p. 50). Fo that eason, we also un estimations with andom individual effects (but fixed time effects) as a obustness check of the esults fom the baseline estimations with fixed individual effects (see Sec. 9). 7. Estimation esults This section pesents estimation esults fo ou new efeence goup measue. We estict ouselves to the WestGeman sample, because the tansfomation pocess in Easten Gemany may have geneated less stable behaviou ove time when the fome county was adapting to its new situation. We focus on the coefficients that ae elevant fo testing the hypotheses. The othe coefficients will be discussed as fa as they yield unexpected esults. To test hypothesis H1 we estimated equation (1) with the logaithmic specification of the impact of socialefeence income. The estimated coefficients ae pesented in the fist column of Table 1. The coefficient γ of ln Y (social efeence income) is indeed negative and vey significant. Thus, hypothesis H1 is confimed. Moeove, γ is stikingly simila in absolute magnitude to the and pat of the intetempoal effect, espectively (see below), and since adding ln age squaed gave ise to a significantly negative coefficient of absolute income, which is implausible. In the latte vaiant the coefficients and exponents of elative income wee not essentially diffeent fom those without ln age squaed, but they wee moe stongly significant while the coefficients wee highe (pehaps as compensation fo the negative coefficient of absolute income). 14
15 coefficient β of ln Y (family income). The size 0.31 of β indicates that a ceteis paibus incease in family income by 10 % would aise life satisfaction R of the aveage West Geman individual by On the othe hand, if at the same time the family income of people in the social efeence goup ises by the same 10%, implying a ise in net effect of these ises in Y and Y by 10%, this would lowe R by the same amount. The Y on R would then be zeo! In a nutshell this explains the empiical finding of no effect of income gowth on aveage happiness in Gemany (as in most othe developed counties) ove time (Glatze, 1991; Eastelin, 1974, 1991). Note that the sepaate effects of such ises in Y and Y on life satisfaction ae small, but vey significant. Hee one should keep in mind that income is only one of the many deteminants of life satisfaction (see also FeeiCabonell). Futhe note that the coefficients of the othe deteminants in Table 1 have plausible and mostly significant values. Hypothesis H2 was tested using equation (2) with the logaithmic specification of the effect of elative income as well as equations (3) and (4) with the powefunction specification. The estimates fo equation (2), as pesented in the second column of Table 1, show that the coefficient γ of ln (elative family income) is indeed positive and vey significant. In fact, this and the othe OLS estimates except that fo absolute income ae identical to the coesponding estimates fo equation (1). Note that the coefficient of absolute income in equation (2) ( β = ~ β γ ) diffes fom the income coefficient in equation (1) ( β ~ ), and is insignificant. In equation (3) (see the fist column of Table 2) the estimated powe coefficient ρ is small and not significantly diffeent fom zeo, implying that the specification does not significantly diffe fom the logaithmic specification (2) (suppoting the eliability of a logaithmic specification). Accodingly, the coefficient γ of the elative income tem deviates only little fom the estimate fo γ in equation (2) and is again significantly positive. Fo the asymmetic specification (4) (second column of Table 2) we estimated γ = and γ = 0. 40, which ae significantly positive as well. The estimated exponents ρ = and ρ = ae not significantly diffeent fom zeo, again implying that the specification does not significantly diffe fom a logaithmic one. We theefoe also estimated the coesponding asymmetic vaiant of equation (2), yielding simila and significant estimates of γ and γ (thid column of Table 1; cf. Feei Cabonell, 2005, fouth specification). Thus, hypothesis H2 was confimed fo all specifications. The elative and absoluteincome coefficients in equation (2) stongly suggest that they ae significantly diffeent (hypothesis H3). A Wald test indeed showed this maginally to hold (p = 0.06), thus giving suppot to hypothesis H3. Supisingly, hypothesis H4 (concavity in gains and convexity in losses) is only patially suppoted by the estimates of ρ and ρ in equation (4): ρ = is significantly lowe than one ( x standad eo = 0.585), implying significant concavity of life satisfaction in negative elative income, and ρ = is consideably lowe than one, but not significantly due to 15
16 the lage standad eo 1.22 ( x 1.22 = 1.51). The nonlinea estimation pocedue equied stating values of ρ and To obtain suitable stating values of values of ρ and ρ close to thei final values due to the closeness to 0 (causing oveflow). ρ and ρ, we fist estimated equation (4) fo a gid of fixed ρ between 4 and 4 with a diffeence 0.1 between successive values. The log likelihoods fo all these fixed values of ρ and besides the shape absolute maximum fo both ρ evealed a lowe and vey flat local maximum ρ and ρ (see Fig. 1a and b). Fo ρ this did not pose a poblem since the local maximum is located at about ρ = 3, i.e fa below 1. Applying a likelihoodatio test to the nullhypothesis H 0 : ρ = 1. 0 yielded 2 2 ln L( 0. 22) ln L( 1. 0) = > ( χ (1) theshold value at p =.05), and so ejection of H 0. Thus, just as with the ttest (o, equivalently, Wald test), we can conclude that ρ = is significantly lowe than one. Fo ρ mattes ae moe complicated since the local maximum is now located at ρ = 2. 0, i.e. above 1. This seems to explain the lage standad eo in the estimate fo ρ. Howeve, applying a likelihood atio test to the nullhypothesis H 0 : ρ = 2. 0 (i.e. at the localmaximum) yielded 2 ln L( 0. 93) ln L( 2. 0) = > 3. 86, and so ejection of H 0. Since ln L ( ρ ) is lowe than the local maximum ln L ( 2. 0) fo any othe value of to one, it then follows that a likelihoodatio test ejects the nullhypothesis H 0 : 0 1 ρ highe than o equal = ρ 0 ρ fo any ρ. Thus, in contadiction to the ttest esult, this suggests that ρ = is significantly lowe than one! Such contadictions ae wellknown in nonlinea estimation, and it does not seem to be justified to tust one test esult moe than the othe. Nevetheless, in this case of bimodality of the loglikelihood function, the likelihoodatio test seems to make moe sense. This would imply that life satisfaction is significantly concave in positive elative income as well Inset Fig. 1 about hee Hypothesis H5 (equal degees of concavity in gains and convexity in losses) is obviously ejected since R is significantly concave in negative elative income. On the othe hand, the estimated exponents ρ = and ρ = imply ρ 1 = and ρ 1 = 0. 78, and hence suggest that R is moe concave in positive elative income than it is in negative elative income. Howeve, due to the lage standad eo 1.22 in ρ, the tvalue of the diffeence in degee of concavity ρ ρ is 1.02, implying no significant diffeence. On the othe hand, a likelihoodatio test of H 0 : ρ = 0 ρ yields 2 ln L( 0. 93, 0. 22) ln L( 0. 2, 0. 2) = > 3. 86, and so a significant diffeence in degee of concavity. Thus, accoding to this test, R is significantly moe concave in positive elative income than it is in negative elative income. Hypothesis H6 (concavity in absolute income) could not be tested because of oveflow poblems that ae elated to the insignificance of the effect of absolute income. 16
17 Hypothesis H7 (loss avesion) is confimed fo equations (2) as well as (3) due to the concavity in elative income ( ρ = < 1 in eq. (3)). Fo equation (4) we found γ > γ, ρ < 1 and ρ < 1, implying loss avesion. Howeve, this loss avesion is not significant fo all values of G = G since γ is not significantly geate than γ (tvalue of γ γ = 0.6) and ρ may not be significantly smalle than one (see above). Theefoe, we pefomed Wald tests on the slope condition ρ 1 ρ 1 S / G ( G) = γ (1 G) > S / G ( G) = γ (1 G) fo fixed values of G. This indicated that the slope condition is significant fo G 0. 16, so fo incomes at least 16 % highe o lowe than the social efeence income (note that ρ 1 S / G ( G) = γ ( 1 G) appoaches fo G going to one). Since the outcome of the Wald test fo a nonlinea paamete estiction tends to depend on the specification of the estiction, we also did the test fo the log of the slope condition. This indicated significance of the slope condition fo G 0. 12, so fo somewhat lowe values of G than above. Fo ou sample consideably less than 50 % of the individuals ae in these domains. Loss avesion is also implied fo the asymmetic vaiant of logaithmic specification (2), but now it is significant fo all values of G = G because γ is now significantly geate than γ (tvalue of γ γ = 3.1). In view of this diffeence in esults fo the geneal specification (4) and the asymmetic vaiant of logaithmic specification (2), we also estimated equation (4) with fixed optimal values ρ = and ρ = The diffeence between the estimated γ = and γ = then tuned out to be maginally significant (tvalue = 1.71, pvalue = 0.086). This implies (maginally) significant loss avesion fo all values of G = G if we assume that the concavity in positive elative income is significant, as suggested by the petinent likelihoodatio test (see above). This would then confim Duesenbey s asymmety postulate except that the impact of positive elative income is still significant. Howeve, the test esults on loss avesion ae mixed. 8. Implications fo cadinal popeties of tue life satisfaction Do the significant cadinal popeties of epoted life satisfaction R as a function of negative and positive elative income that we have found in ou estimations imply simila cadinal popeties of tue life satisfaction S? In paticula, does the estimated significant concavity of R in negative (and positive) elative income eflect simila concavity of S, o can it be explained by possible concavity of OLSestimated Rˆ in S? To tackle this question, we make the plausible assumptions (i), (ii), and (iii) that have been intoduced in Section 3 and conside what may be maximal plausible concavities of R in S. One exteme distotion of R elative to S is suggested by Oswald: people who expeience a maximal S = 10 may epot this by a scoe of R = 8. Howeve, Figue 2 shows that although most individuals in the West Geman sample epoted R = 8, thee ae also a lot of people who epoted a 9 o 10. This implies that even if a numbe of West Gemans who expeienced S = 10 o 9 epoted R = 17
18 8, othes did not. It might be possible that on aveage West Gemans who expeienced S = 10 epoted aound R = 9, but an aveage scoe of R = 8 fo these people seems implausible. Let us theefoe fist conside the fome possibility and constuct a continuous concave function R ˆ( S ) such that R ˆ (5) = 5, R ˆ (7) = 7 and R ˆ (10) = 9. Fo this we need a flexible function with at least thee paametes and a b suitable one tuns out to be the powe function R ˆ( S) = a( S c) with paametes a > 0, 0 < b < 1 and c > 0. The invese of this function is the convex function S( Rˆ) = Rˆ / a c, whee 1 / b > 1. Since Rˆ = R e, it is given by the ighthand side of equation (4) minus e. Substituting this expession fo Rˆ into the function ˆ S (R) then yields S as a function S[ Rˆ( Y, G, G, X )] of Y, G, G, and X. What we want to know is whethe accoding to ou estimates of equation (4) S is significantly concave in G and G despite the convexity of S (Rˆ ). Fo that we need to investigate whethe the secondode patial deivative S / G is significantly negative fo all values of G and R ˆ 5 ( S (Rˆ ) is 2 m 2 m likely to be less convex o even concave fo R ˆ < 5 ). Fist, the fistode deivative m S / G can be deived as S / G m = ( S / Rˆ)( Rˆ / G m ) = ( γ / a ) Rˆ 1 (1 G m m ρ 1 ). Diffeentiating this to m G and imposing negativity yields the condition m m γ m m m m ρ 1 ρ > R ˆ 0 ( 1 ρ ) γ ( 1 G ) 1 m m ρ ρ (5) m fo all values of G and R Rˆ( Y,0,0, X ) 5. On the lefthand side of this equation we see the estimated degee of concavity ˆ0 m 1 ρ of Rˆ in m G, while on the ighthand side the maximal plausible degee of convexity 1 of S in Rˆ appeas Inset Fig. 2 about hee Ou estimates γ = and ρ = fo equation (4) imply γ / ρ =1. 88, which is smalle than R ˆ0 5, making the lefthand side of condition (5) positive. The thee estictions R ˆ (5) = 5, R ˆ (7) = 7, and R ˆ (10) = 9 fo R ˆ( S ) lead to a powe equation in (see Appendix A) that can only be solved numeically and yielded a value = The facto 1 (1 ρ ) / ρ = on the ighthand side of (5) is then negative, endeing this side negative. Condition (5) is then satisfied fo all m G and R ˆ0 5. Estimates γ = and ρ = imply γ / ρ < 0, making the lefthand side of condition (5) again positive. Howeve, the facto 1 (1 ρ ) / ρ, and hence the ighthand side of (5), is then positive. In this case fulfilment of condition (5) is not tivial, but ρ < 0 ρ ρ implies ( 1 G ) < ( 1 0) = 1 fo all G 0, and hence fulfilment of condition (5) fo all G 0 if and only it holds fo G = 0. Condition (5) then simplifies to 5(1 ρ ) > γ ( 1), (6) whee we substituted 5 fo ˆR 0 since the condition fo R ˆ0 = 5 implies the condition fo R ˆ0 > 5. This condition is satisfied fo values of such that > (1), i.e. fo < 33.24, and hence fo the 18
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