Preferences, income, and life satisfaction: An equivalence result

Mathematical Social Sciences 75 (2015) 20–26

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Preferences, income, and life satisfaction: An equivalence result Holger Strulik University of Göttingen, Department of Economics, Platz der Göttinger Sieben 3, 37073 Göttingen, Germany

highlights • Theoretical foundation of a log-linear association between income and life satisfaction. • Empirical slope of the curve is supported with or without status concerns and habits. • Results hold true for inward and outward orientation of consumption comparisons.

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Article history: Received 4 April 2014 Received in revised form 22 December 2014 Accepted 6 February 2015 Available online 20 February 2015

abstract In this paper I investigate the nexus between life time utility (life satisfaction) and income predicted by the standard model of endogenous economic growth under different behavioral assumptions. The solution rationalizes why the empirical association between income and life satisfaction is approximately loglinear. I show that the solution is observationally equivalent when individuals compare their consumption (i) with others, (ii) with their own past consumption achievements, and (iii) not at all (ordinary preferences). This finding suggests that the observed slope of the income–life satisfaction curve is uninformative about the presence and strength of habits or reference-dependent utility. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Early research on the nexus between income and life satisfaction (or happiness) focused on a piece-wise linear relationship between both variables and concluded that at sufficiently high income levels, additional income fails to buy more happiness (e.g. Easterlin, 1974; Frey and Stutzer, 2002; Layard, 2003). Inspired by these results economists introduced reference-dependent preferences into economic models. Recent research on the nexus between income and life satisfaction, in contrast, suggests that the relationship is log-linear (Deaton, 2008; Stevenson and Wolfers, 2008) and that there is no point of satiation (Stevenson and Wolfers, 2013). From these observations it has been concluded that relative income comparisons are of lesser importance for individual life satisfaction than previously thought (Stevenson and Wolfers, 2008).1

E-mail address: [email protected]. 1 I follow the happiness literature and regard the empirical observation of life satisfaction from surveys as a reasonable approximation of life time utility (Clark et al., 2008; Stevenson and Wolfers, 2008). Most of the economics literature uses happiness and life satisfaction interchangeably but it has been argued that happiness describes better the instantaneous component of subjective well-being while life satisfaction is the more appropriate measure of its evaluative, long-term component (Deaton, 2008; Stevenson and Wolfers, 2008). http://dx.doi.org/10.1016/j.mathsocsci.2015.02.003 0165-4896/© 2015 Elsevier B.V. All rights reserved.

The flattening of the income–happiness curve, however, could as well be a simple consequence of ordinary preferences, represented by a concave utility function. In fact the empirical association of income and life satisfaction may be uninformative about the presence and strength of reference-dependent preferences. In order to derive this result I follow Carroll et al. (1997, 2000) who introduced comparison utility as formalized by Abel (1990) into a standard endogenous growth model.2 I show that the model can be solved to provide an explicit solution for life time utility (life satisfaction) as a function of current income. The solution fits the cross-country data on average income and average life satisfaction reasonably well and supports the assumption that the association between these variables is approximately log-linear. Reported life satisfaction can be conceptualized as a function R that maps the true association between income and life satisfaction

2 The model has become a benchmark for theoretical and quantitative analyses of the impact of consumer reference stocks on growth and has been developed further in several other papers, for example, Futagami and Shibata (1998), AlvarezCuadrado et al. (2004), and Alonso-Carrera et al. (2005). The issue of the present paper, i.e. the provision of a theoretical motivation for the observable log-linear association between income and life satisfaction, remained unexplored. There exists, however, a growth literature aiming to explain why there could be no or a negative association between income and life satisfaction, see De la Croix (1998) and Cooper et al. (2001).

H. Strulik / Mathematical Social Sciences 75 (2015) 20–26

V (y) to reported happiness R(V (y)). It could then be that the observed log-linearity results from the curvature of the reporting function R, rather than from the true association V (y). Since cardinal utility is defined up to an affine-linear transformation this would imply that reported life satisfaction is not a good proxy of actual life satisfaction, a conclusion that could threaten the meaningfulness of the empirical happiness literature. So far this issue has been rarely addressed. Oswald (2008) uses actual and reported height and cannot reject that the reporting function is linear. Here, I provide a theoretical argument that the true association between income and life satisfaction is log-linear and not an artifact of the reporting function. Log-linearity of the income–utility curve, however, is equally well supported by reference-dependent preferences and ordinary preferences. Since the cardinal utility function is only defined up to an affine-linear transformation, the result suggests that it is not feasible to derive conclusions about the strength of status concerns or consumption habits from the empirical association of income and life satisfaction. 2. Setup of the model and steady-state solution



1−γ

(ci /hi )γ

1−σ

/(1 − σ ), where σ denotes the

coefficient of relative risk aversion, σ ̸= 1, and γ ∈ (0, 1) measures the strength of reference-based consumption.3 The implied intertemporal (life-time-) utility is given by V (t ) =



∞

γ 1−σ

ci /hi



t

1−σ

·e

−θ(τ −t )

dτ ,

The case of inward looking preferences implies the law of motion h˙ i = ρ(ci − hi ).

gc ≡

(1)

(2)

(3)

The speed at which the reference stock adjusts to current consumption is given by ρ . The larger ρ the more important is consumption of the recent past. The important difference between (2) and (3) is that there is strategic interaction between consumption and the reference stock in the case of inward-looking preferences but not in the case of outward-looking preferences. When making their utility maximizing consumption plans, individuals with inward-looking preferences take the feedback of their consumption ci on the evolution of their habit stock into account. Individuals with outward-oriented preferences take consumption of others, which equals average consumption c, as given.

3 If there were no long-run growth, then c = h at the steady state. In this sense γ is the share of consumption which matters only in relative terms, either compared to the consumption of others or compared to own achievements. 4 In order to obtain closed form solutions, I focus on linear functions for the evolution of the reference stock, following Carroll et al. (1997, 2000) and most of the related literature. There are, of course, other functions conceivable (see e.g. De la Croix, 1998), which may lead to different results.

(4)

It is the conventional condition for bounded utility (e.g. Barro and Sala-i-Martin, 2004, Chapter 4.1), scaled by factor (1 − γ ). Carroll et al. show that the steady-state solution of the optimization problem is the same irrespective of whether habits are formed outward-looking or inward-looking and is given by

x≡

in which θ denotes the rate of pure time preference. The maximized life time utility will be associated with life satisfaction. In order to square this approach with the fact that individual life is finite, one needs, as usually, an altruistic head of a dynasty. Consumption comparisons may be formed with respect to household i’s own past consumption (inward-looking preferences) or with respect to consumption of others (outward-looking preferences).4 In the latter case the reference stock evolves according to h˙ i = ρ(c − hi ).

The equation of motions (2) and (3) can also be interpreted differently, taking into account the debate about whether individuals are aware of the adjustment of reference stocks (Clark et al., 2008). Eq. (3) would then capture the case of outward oriented preferences when individuals are realizing the fact that the Joneses are just like them and will thus follow the same consumption strategy. Eq. (2) would represent inward looking consumers which are not aware of the adjustment of their consumption habits. Output is produced using capital and a linear production function such that income of household i is yi = Aki where ki is capital (wealth) of household i and A is capital productivity. Capital depreciates at rate δ , implying that the capital stock of household i evolves according to k˙ i = (A − δ)ki − ci . In case of inward looking preferences we impose σ > (1 + γ )/γ in order to ensure that the utility function is concave in both elements such that the Hamiltonian is concave in control and states. In case of outward looking preferences h is exogenous and σ > 0 is sufficient for concavity of the Hamiltonian. Moreover we assume that the time preference rate is sufficiently large in order to get finite life-time utility. As shown below, this requires

θ > (A − δ)(1 − σ )(1 − γ ).

This section contains a brief recap of Carroll et al.’s model. Consider an economy populated by identical households (of measure one) maximizing an infinite stream of utility derived from consumption ci relative to a reference stock hi . Instantaneous utility is given by ui = ci

21

p≡

c˙ c c h k h

A−δ−θ , γ (1 − σ ) + σ   1 A−δ−θ gc =1+ =1+ , ρ γ (1 − σ ) + σ ρ   1 ρ(γ (1 − σ ) + σ ) + (A − δ − θ ) = , ρ (A − δ) [(1 − σ )γ + σ − 1] + θ

=

(5a) (5b)

(5c)

in which gc denotes the balanced growth rate. Individual indices have been dropped since all households are identical. Carroll et al. (1997) show that the steady state is a saddlepoint for in- and outward looking preferences. In contrast to the conventional Ak growth case, the model displays adjustment dynamics, which are uniquely obtained as movement along the stable manifold towards the steady state. Compared with the neoclassical growth model, the speed of adjustment is very fast. In the calibration of Carroll et al. (1997) an initial gap is closed with a half-time of 5.4 or 4.9 years, depending on the orientation of preferences. These fast movements towards the steady state may justify that the following discussion focuses on the steady state, assuming that economies are sufficiently close to it. The steady state focus allows to derive results analytically. In the Appendix I show, for some numerically specified example economies, that the main result is also obtained off the steady state when adjustment dynamics are taken into account. 3. Income and life-time utility (life satisfaction) While the results so far are well known, the implied association between income and life satisfaction remained unexplored. To solve this problem we begin with noting that at the steady state consumption grows at rate gc , such that c (τ ) = c (t )egc (τ −t ) . Inserting this information and (5a) and (5b) in (1) provides (6). V (t ) =

x−γ (1−σ ) c (t )(1−γ )(1−σ ) 1−σ

∞



e[gc (1−γ )(1−σ )−θ ](τ −t ) dτ .

(6)

t

Inserting gc from (5a) into the exponent of the integrand confirms that condition (4) is needed for bounded utility. Using the fact that

22

H. Strulik / Mathematical Social Sciences 75 (2015) 20–26

c (t ) = (x/p) · k(t ) and inserting (5b) and (5c) provides utility as a function of the current capital stock per capita. Finally, substituting y(t ) = Ak(t ) as implied by the production function we get maximized intertemporal utility (life satisfaction) as a unique function of current income per capita.

β0 · y(t )β1 β1  γ (1−σ ) gc + ρ β0 ≡ (1 − γ ) · ρ  (1−γ )(1−σ )−1 [θ − (A − δ)(1 − σ )(1 − γ )] · σ (1 − γ ) + γ · A−(1−γ )(1−σ ) β1 ≡ (1 − γ )(1 − σ ).

V (t ) =

(7)

The term in square brackets is positive when life-time utility is finite, as initially assumed. This means that β0 is positive whereas the sign of β1 is uniquely pinned down by σ . The main observation is that the curvature parameter of the function, β1 , is a compound of the coefficient of relative risk aversion σ and the strength of habits or status concerns γ . Quantitative comparisons of life satisfaction, like estimating its association with income, require a cardinal concept of utility. Since a cardinal utility function is only defined up to an affine linear transformation (Strotz, 1953), no inferences about the underlying preferences can be made from the observation of β0 . The curvature parameter β1 , however, is uninformative about the strength of habits or status concerns unless the coefficient of relative risk aversion is known. Proposition 1 (Equivalence Result). The association between income and life satisfaction is uniquely pinned down by the coefficient of relative risk aversion σ and the strength of habits or status concerns γ . For any combination (σ , γ ) with habits or status concerns (i.e. γ > 0), there exists another combination (σ , 0) that motivates the same association between income and life satisfaction without relying on habits or status concerns. This means that inferences on status concerns or consumption habits cannot be derived from an estimate of the empirical association of income and life satisfaction if the workhorse model of endogenous growth augmented by status concerns or consumptions habits is a reasonable approximation of reality. The conjecture of the earlier literature that people are subject to status concerns and consumption habits because the observed empirical association of income and life satisfaction is flat at high income levels is not supported by theory. Likewise the conclusion that status concerns or habits play a relatively minor role, as recently derived from the observation that the empirical association between income and life satisfaction is not as flat as previously thought, is not supported by theory, at least not by general equilibrium theory that puts consumption habits or status concerns in the framework of endogenous growth. Notice furthermore that the growth rate gc shows up only in the constant β0 . Since the cardinal utility function is defined only up to an affine linear transformation, the constant β0 cannot be identified with data, which implies the following result. Corollary 1 (Economic Growth). The impact of economic growth on life satisfaction (according to the present endogenous model) cannot be inferred from data on income, income growth, and life satisfaction. This result offers an explanation for the seemingly puzzling finding that empirical studies provide a strongly significant association between income levels and life-satisfaction and simultaneously fail to identify a positive impact of long-run growth on life satisfaction (see, for example, Easterlin et al., 2010). The present

paper suggests that the weak empirical results on growth and lifesatisfaction do not necessarily mean that the association is not there. In (7) it is clearly present and positive but it is not identifiable because of measurement problems resulting from the notion of cardinal utility. Notice that this holds true irrespective of the presence, nature, and strength of habits or reference-dependent utility. The result thus advises caution concerning the interpretation of the finding of a non-significant association between long-run growth and life-satisfaction. Notice that a similar argument applies to the speed of adaptation ρ , which also shows up only in the constant β0 . Again, this means that no inferences on the actual speed of adaptation can be made from the observed association between income and life satisfaction. In order to visualize the result I take data for life-satisfaction across countries from the most recent Gallup (2013) poll and data on income per capita (real PPP GDP per capita) from the most recent Penn World Tables (Feenstra et al., 2013). Most of the Gallup data and all of the income data is for the year 2011. Altogether we have data points for 148 countries, represented by dots in Fig. 1. In order to normalize data and model predictions I take up the suggestion of the happiness literature and compute z-scores. In case of the model I feed income data into (7) and compute from the resulting life time utility V , V˜ =

V − µ(V ) s( V )

+ b,

in which µ(V ) is the mean of V and s(V ) its standard deviation. Notice that this constitutes an affine linear transformation of V . The life satisfaction data is normalized in a similar fashion. The parameter b is another shifter needed in order to get the origin right. It is set to 1.3 for the model and to zero for the data. The panels on the left and right hand side of Fig. 1 show the same information. The only difference is the scale of the income axis. I took up the idea from Stevenson and Wolfers (2008) and present results for income measured in absolute terms as well as in logs. The blue line is the prediction of the model for β1 = 0.01 that is, for example, for γ = 0 and σ = 0.99 (no status concerns) or γ = 0.7 and σ = 0.96 (strong status concerns). The case of β1 → 0 is interesting because the recent empirical happiness literature has imposed (rather than estimated) a log association between income and life satisfaction (Deaton, 2008; Stevenson and Wolfers, 2008). We see that the log assumption, here approximated by β = 0.01, fits the data quite well irrespective of whether there are strong status concerns or non at all. Imposing strong status concerns modifies the estimated coefficient of relative risk aversion only marginally; σ declines from 0.99 to 0.96. Dashed red lines in Fig. 1 show the model prediction for another important case, with reference to Layard et al. (2008). This study uses results from six different social surveys (mostly on life satisfaction but also on happiness) and estimate the associated ‘‘income elasticity of marginal utility’’ here denoted by ϵ . The point estimates are remarkably consistent across surveys and vary between 1.19 and 1.34 with quite narrow 95% confidence intervals. The ‘‘combined estimate’’ is 1.26. In the present context this means that β1 = 1 − ϵ = −0.26. Apparently, the model provides theoretical support of the iso-elastic functional form imposed by Layard et al. In Fig. 1, Layard et al.’s combined estimate is reflected by red dashed lines, which represent, for example, values of γ = 0 and σ = 1.26 (no status concerns) or γ = 0.7 and σ = 1.87 (strong status concerns). The model predictions fit the income and life satisfaction data somewhat less well than the ‘‘quasi log’’–prediction, a detail which, however, becomes only visible when the data is inspected in log-linear scale. For all cases considered the implied range of values for σ is remarkably narrow and compatible with values imposed in

H. Strulik / Mathematical Social Sciences 75 (2015) 20–26

23

Fig. 1. Life satisfaction across Countries: data and model prediction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Life satisfaction equivalence.

γ 0.00 0.30 0.60 0.90

Income elasticity of marginal utility (ϵ ) 3.00

2.50

2.00

1.50

1.25

0.99

0.75

0.50

3.00 3.86 6.00 21.0

2.50 3.14 4.75 16.0

2.00 2.43 3.50 11.0

1.50 1.71 2.25 6.00

1.25 1.36 1.63 3.50

0.99 0.99 0.98 0.90

0.75 0.64 0.37 −1.50

0.50 0.29 −0.25 −4.00

The entries in the table report the coefficient of relative risk aversion (σ ) that provides a given income elasticity of marginal utility (ϵ = 1 − β1 ) for alternative strengths of habits and status concerns (γ ).

other calibration studies as well as with recent empirical estimates (e.g. Chetty, 2006). This observation holds irrespective of the presence and strength of status concerns or consumption habits. Table 1 shows results over the whole range of feasible γ and conceivable curvatures of the utility function. For that purpose I take up the terminology of Layard et al. and represent the slope of the curve by the income elasticity of marginal utility ϵ ≡ 1 − β1 . This number shows by how many percent marginal utility (marginal life satisfaction) declines when income increases by 1%. The entries in the Table show the coefficient of relative risk aversion (σ ) that supports a given curvature ϵ for alternative values γ . For an understanding of the results it is helpful to recall that individuals compare consumption with the reference stock now and in the future. The curvature of the utility function, summarized in σ determines whether present or future consumption comparisons are more important. If σ is small, the intertemporal elasticity of substitution is high, and individuals give up future consumption in order to improve present relative consumption (current status). If σ is large, individual are induced to smooth heavily their consumption comparisons with the Joneses or own achievements. This means that individuals subjected to status concerns or habits give up consumption today in order to better compete with the reference stock in the future. The threshold is where σ equals unity such that both effects cancel each other and the presence of status concerns or habits has no impact on observed behavior. Table 1 shows that the observation of an income elasticity of marginal utility close to one, allows to make an inference about σ , which is in this case predicted to be close to one irrespective of the strength of status concerns γ . The strength of status concerns γ , in contrast, cannot be generally inferred from the empirically observed income elasticity of utility. Only some specific (γ , ϵ ) combinations can be excluded by imposing plausible constraints on σ . For example, it is impossible for individuals to display a low income elasticity of marginal utility and to be simultaneously risk averse (σ > 1) and strongly subjected to status concerns or consumption habits (γ ≥ 0.6). Likewise the combination of a high income elasticity and strong status concerns or habits can be

excluded because it would require an implausibly large coefficient of risk aversion. For income elasticities in the empirically plausible range between 1.0 and 1.5 (Layard et al., 2008), however, it is impossible to draw conclusions about status concerns or habits because there is (yet) to little precision in the estimate of σ such that all possible values of γ are compatible with empirically supported values of σ . 4. Subtractive reference-dependent preferences Comparisons of consumption to a reference stock can be made in multiplicative way, as so far in this paper, or in a subtractive way. Both ways are used in the literature and a priori it is hard to come up with a reason to prefer one over the other. However, in the literature of endogenous growth to which this paper relates, the multiplicative form of habit formation or status concerns dominates. The reason is, presumably, that status concerns or habits in subtractive form have no impact at all on behavior, economic growth, and the consumption function at the steady state of an endogenously growing economy. In the following I provide the proof for this claim. The subtractive analogon to the multiplicative utility function of Section 2 is given by ui = (ci − γ hi )1−σ /(1 −σ ). The implied life time utility is V (t ) =

∞

 t

(ci − γ hi )1−σ −θ (τ −t ) ·e dτ . 1−σ

(8)

Individuals maximize (8) subject to the budget constraint k˙ i = (A − δ)ki − ci . In case of inward looking preferences they take the evolution of habits h˙ i = ρ(ci − hi ) into account. In case of outward looking preferences the evolution of habits h˙ i = ρ(c − hi ) is exogenous at the individual level. In both cases the steady state is characterized by a constant consumption-habit stock ratio and constant ratio between habit stock and capital stock given by: x=1+

A−δ−θ

ρσ

,

p=

A − δ − θ + ρσ . ρ [(A − δ)(σ − 1) + θ ]

(9)

In contrast to the multiplicative case, I cannot refer to the literature for the derivation of the solution. I thus provide the derivation of (9) in the Appendix. From (9) we obtain the steady state growth rate gc ≡ c˙ /c = h˙ /h = ρ(c − h)/h = (A − δ − θ )/σ and the consumption function c = (x/p)k = [(A − δ)(σ − 1) + θ ]/σ k. Notice that neither economic growth nor the consumption function c (k) dependents on the strength of status concerns and habits γ or the speed of adaptation ρ . The solution coincides with that for the conventional

24

H. Strulik / Mathematical Social Sciences 75 (2015) 20–26

Ak-growth model without consumption habits (Barro and Sala-iMartin, 2004, Chapter 4). Using c = kx/p, we can rewrite life time utility (8) in terms of capital,

(1 − γ /x)1−σ V (t ) = 1−σ



xk(t )

1−σ 

p

∞

e[(1−σ )gc −θ ](τ −t ) dτ .

(10)

t

As for the associated conventional model without habits we need to impose that θ > (1 − σ )(A − δ) for the utility integral to be finite. Then, using the fact that k = y/A, the solution is V (t ) =

[(x − γ )/(pA)]1−σ

(1 − σ ) [θ − (1 − σ )(A − δ)]

· y(t )1−σ .

(11)

At the steady state, life time utility is a unique function of income. The curvature of this function is – in contrast to the multiplicative utility case – independent from the parameters of habit formation. Status concerns γ enter as a shifter of V (t ) but since cardinal utility is defined only up to an affine linear transformation, their impact on life satisfaction cannot be identified. The following proposition summarizes these insights. Proposition 2 (Equivalence Result). For reference comparisons of consumption in a subtractive way, i.e. for utility ui = (ci − γ hi )1−σ / (1 − σ ), status concerns have no impact on consumption behavior, on economic growth, and on the income gradient of life time utility at the steady state. This in turn means that for the case of subtractive utility comparisons with a reference stock, the empirically observable loglinear association between income and life satisfaction is entirely motivated by the elasticity of intertemporal substitution and independent from habits and status concerns. 5. Discussion This paper has calculated the implied life satisfaction (life time utility) when the basic model of endogenous growth is extended by status concerns or consumption habits. This way it has been shown that the estimated iso-elastic or log-linear association between and income and life satisfaction has a theoretical foundation. The slope of the estimated relationship however is found to be equally well supported by ordinary preferences and (strong) status concerns or consumption habits. In case of subtractive utility comparisons, the slope is entirely motivated by the elasticity of intertemporal substitution and independent from habits or status concerns. Inferences from the observed association of income and life-satisfaction on the presence and strength of reference-based utility, made by the earlier (Easterlin, 1974) and recent (Stevenson and Wolfers, 2008) happiness literature, are not supported by dynamic general equilibrium theory. In order to arrive at an explicit solution for life satisfaction, the economic framework was chosen to be deliberately simple, built upon the Ak growth model. Based on Rebelo (1991), it has been argued by Carroll et al. (2000) in a similar context that the linear Ak production function is the ultimate structure of all endogenous growth models. The model has been modified to allow for wealth heterogeneity and status derived from wealth or education (e.g. Futagami and Shibata, 1998; Kawamoto, 2009) with little effect on the qualitative results discussed in the present paper (as long as σ > 1). Likewise the empirical estimate of ϵ has been shown to vary little across socioeconomic strata (Layard et al., 2008). These facts indicate that the equivalence result is more general than shown in the present paper. In fact, any model employing utility function (1) would display the result at a steady-state along which consumption grows at a constant rate.

A potential limitation of the study is that the income gradient of life-satisfaction is entirely motivated by the direct impact of income on consumption expenditure and life time utility, controlling for habit formation. Income, however, has potentially also an indirect effect on life satisfaction, most obviously perhaps through health expenditure and longevity (Jones, forthcoming; Dalgaard and Strulik, 2014) but also through more sublime positive and negative channels (e.g. quality of institutions, pollution). The empirical estimates of an income elasticity of marginal utility above unity imply a coefficient of risk aversion σ greater than unity. For σ > 1 the model suggests that the rate of economic growth is a positive function of the strength of habits or status concerns γ (as shown in (5a), replicating the earlier literature, Carroll et al., 1997, 2000). Put into general equilibrium context, the empirical evidence thus suggests that strong status concerns or consumption habits are good for economic growth. In a related paper Strulik (in press) I show that status concerns may also, through the growth channel, be conducive to the experience of a happy life. Acknowledgments I would like to thank Carl-Johan Dalgaard, Christian Groth, Timo Trimborn, and an anonymous referee for helpful comments. Appendix A. Derivation of the solution for additive utility In the case of outward-looking preferences, individuals maximize (8) subject to the budget constraint k˙ i = (A − δ)ki − ci .

(A.1)

The associated Hamiltonian reads H =

(ci − γ hi )1−σ + λ [(A − δ)ki − ci ] . 1−σ

(A.2)

The first order conditions are:

(ci − γ hi )−σ − λ = 0 ˙ λ(A − δ) = λθ − λ.

(A.3) (A.4)

Differentiating (A.3) with respect to time, using the result in (A.4) and eliminating λ provides:

  σ c˙i − γ h˙ i σ [c˙i − γ ρ(ci − hi )] A−δ−θ = = ci − γ hi ci − γ hi

(A.5)

where the last equality follows from applying symmetry, ci = c, and inserting the evolution of outward looking preferences h˙ i = ρ(c − hi ).

(A.6)

In order to solve the model suppose that consumption, habits, and capital stock grow at equal rates and set c /h ≡ x and k/h ≡ p. Then (A.5) can be rewritten as

(A − δ − θ ) =σ





x p

−

A−δ−

γ



p x p



x p

− γρ



x p

−

1 p



.

(A.7)

Using (A.1) and (A.6) we can rewrite the steady-state condition h˙ i /hi = k˙ i /ki as A−δ−

x p

= ρ(x − 1).

(A.8)

H. Strulik / Mathematical Social Sciences 75 (2015) 20–26 Table A.1 Equivalence results taking adjustment dynamics into account.

ρ = 0.5 ρ = 0.25

(γ = 0.5, σ = 1.5)

(γ = 0.5, σ = 2.5)

(γ = 0.7, σ = 2.5)

1.385 1.378

1.759 1.717

1.713 1.665

The economy starts 10% below its steady state (in terms of k). The entries in the table show the value of σ that implies the same life time utility without habits as the model with habits and γ , σ , and ρ as specified in the header and the first column. Other parameters: δ = 0.08, θ = 0.015, A = 0.12.

Solving (A.7) and (A.8) for x and p provides the optimal solution5 : x=1+

A−δ−θ

ρσ

,

p=

A − δ − θ + ρσ , ρ [(A − δ)(σ − 1) + θ ]

which is (9) from the main text. In case of inward-looking preferences, the maximization of utility takes the evolution of habits, h˙ i = ρ(ci − hi ),

(A.9)

into account. The Hamiltonian reads H =

(ci − γ hi )1−σ + λ [(A − δ)ki − ci ] + µρ(ci − hi ). 1−σ

(A.10)

The first order conditions are:

(ci − γ hi )−σ − λ + µρ = 0 ˙ λ(A − δ) = λθ − λ.

(A.12)

γ (ci − γ hi )

(A.13)

−σ

(A.11)

− µρ = µθ − µ. ˙

The solution in general is substantially more involved than for the outward looking case. However, since we are interested in steadystate behavior, we can apply the steady state definition directly to the first order condition. When c and h grow at equal rates, the growth rate of c − γ h is constant such that λ and µ grow at equal rates. This means that derivation of (A.11) with respect to time provides

− σ (ci − γ hi )−σ −1 (˙ci − γ h˙ i ) − λ˙ +

λ˙ ρµ = 0. λ

(A.14)

Substituting ρµ from (A.11) and inserting (A.12) and (A.13) provides

(A − δ − θ ) =σ





x p

−

A−δ−

γ

p

et al. (2008, p. 111) conclude their discussion of the empirical happiness literature: ‘‘Together, this suggests a utility function in which two-thirds of aggregate income has no effect because it is status related’’. The mean value derived from experiments by Alpizar et al. (2005) is 0.45. In their calibration Carroll et al. (2000) use 0.5 as the benchmark value. Heaton (1995) estimates a value of 0.71. Boldrin et al. (1997) find that a value of 0.58 fits best to explain the risk premium. Below, I report results for γ = 0.5 and γ = 0.7. For the speed of adaptation, di Tella et al. (2007) estimate that 8/23 = 34% of an income shock have still an impact on happiness after 4 years. Solving the back-of-the envelope calculation e−4ρ = 0.34 provides an estimate of ρ = 0.27. This is slightly larger than Carroll et al.’s (2000) benchmark case where ρ = 0.2. Heaton (1995) and Boldrin et al. (1997), however, suggest larger values for ρ of 0.58 and 0.7, respectively. Below, I report results for ρ = 0.5 and ρ = 0.25. In order to determine the remainder of parameters, I follow the mainstream DSGE literature and set the time preference rate θ to 0.015 and the depreciation rate δ to 0.08 and then obtain factor productivity A endogenously such that the economy generates a long-run growth rate of 2% annually in line with the long-run growth experience of the US. This provides the estimate A = 0.12. The implied net return on capital, r = A − δ , is 4% annually. I consider adjustment dynamics after a 10% wealth shock, i.e. after a 10% shock of p. In order to solve the model I employ the method of backward integration (Brunner and Strulik, 2002), which approximates non-linear adjustment dynamics correctly up to a pre-specified, arbitrarily small error of discretization (set to 10−15 ). Results are presented in Table A.1 for the case of inward looking preferences. The table shows for some plausible combinations of (σ , γ ) the respective (σ , 0) combination that implies the same life time utility without habits. It can be seen that the adjustment speed of habits ρ has an impact on the estimated (σ , 0)-equivalent, which is albeit of relatively minor magnitude. As in the case of the steady-state analysis in the paper, all σ values are in an empirically reasonable range such that implausibility cannot be invoked to rule out that habits are (or are not) responsible for the observed association between life satisfaction and income. Similar results, available upon request, are obtained for the model variant with outward looking preferences. References



p x

25



x p

− γρ



x p

−

1 p



,

(A.15)

which coincides with (A.7). Thus inward- and outward looking preferences support the same steady-state solution, as in the case of multiplicative utility (see Carroll et al., 1997). Appendix B. The equivalence result off the steady state In order to quantify off-steady-state dynamics, I refer to an earlier version of this paper (Strulik, 2008) in which I in detail explore adjustment dynamics of the Ak growth model with reference utility (albeit with a different focus). For the strength of habits or status concerns γ , the happiness literature suggests a value between 0.5 and 0.7. For example, Clark

5 There exists another solution for (A.7) and (A.8), at which c = γ h . This i i solution, however, is not optimal. It implies infinite marginal utility and thus an infinite shadow price λ (from (A.3)) such that it violates the transversality condition limT →∞ λ(T )k(T ) exp(−ρ T ) = 0.

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