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SOEPpapers
on Multidisciplinary Panel Data Research
50
NN
Richard Layard
Guy Mayraz
Stephen J. Nickell
The marginal utility of income
- revised version -
–
Berlin, April 2008
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Contact: Uta Rahmann | urahmann@diw.de
The marginal utility of income∗
R. Layard†,G. Mayraz‡, S.Nickell§
Revised: 14 January 2008
Abstract
In normative public economics it is crucial to know how fast the
marginal utility of income declines as income increases. One needs this pa-
rameter for cost-benefit analysis, for optimal taxation and for the (Atkin-
son) measurement of inequality. We estimate this parameter using four
large cross-sectional surveys of subjective happiness and two panel sur-
veys. Altogether, the data cover over 50 countries and time periods be-
tween 1972 and 2005. In each of the six very different surveys, using a
number of assumptions, we are able to estimate the elasticity of marginal
utility with respect to income. We obtain very similar results from each
survey. The highest (absolute) value is 1.34 and the lowest is 1.19, with a
combined estimate of 1.26. The results are also very similar for subgroups
in the population. We also examine whether these estimates (which are
based directly on the scale of reported happiness) could be biased upwards
if true utility is convex with respect to reported happiness. We find some
evidence of such bias, but it is small—yielding a new estimated elasticity
of 1.24 for the combined sample.
JEL classification: I31, H00, D1, D61, H21.
Keywords: Marginal utility, income, life satisfaction, happiness, public
economic, welfare, inequality, optimal taxation, reference-dependent pref-
erences.
∗ We thank the Esmee Fairbairn Foundation for financial support to the CEP well-being
programme.
† Centre for Economic Performance, London School of Economics.
‡ Department of Economics and Centre for Economic Performance, London School of Eco-
nomics.
§ Nuffield College, Oxford, and Centre for Economic Performance, London School of Eco-
nomics.
1
1 Introduction
In normative public economics it is crucial to know how fast the marginal
utility of income declines as income increases. One needs this parame-
ter for cost-benefit analysis, for optimal taxation and for the (Atkinson)
measurement of inequality. For example, in cost benefit analysis a central
problem is how to aggregate the costs and benefits that accrue to differ-
ent people. A natural way to do this is to weight each person’s change in
income by his or her marginal utility of income1 .
The key issue for public economics is not how strongly income affects
utility but how this effect changes with income. To focus on this question
we assume that the elasticity, ρ, of marginal utility with respect to income
is constant. Thus, utility, u, is given by
( 1−ρ
y −1
1−ρ
ρ 6= 1
u= (1)
log y ρ=1
where y is income. It follows that if we take two people, A and B, the
ratio of their marginal utilities is given by
∂uB /∂y yA
A
= ( B )ρ (2)
∂u /∂y y
Thus, for example, if ρ = 1 (and so u = log y) the marginal utilities
are inversely proportional to income: someone with an income of $10,000
has ten times the marginal utility of someone getting $100,000. Both
Bernoulli (1738, 1954), who invented the mathematical idea of utility,
and Dalton (1920), who pioneered the measurement of inequality, made
just this logarithmic assumption2 .
What is needed, in this context, is a cardinal measure of utility where
unit intervals have the same meaning at all points along the scale. As is
well-known, such a measure is assumed whenever decisions are explained
on the basis of a decision function involving the weighted addition of utility
in different states (choice under uncertainty) or in different time periods
(intertemporal choice). Thus, studies of such behaviour have been used
to obtain estimates of ρ. However, these estimates have two key prob-
lems for our present purpose: (1) We are interested in utility as actually
experienced ex-post rather than in utility as evaluated ex-ante for pur-
poses of decision-making. As Kahneman (Kahneman, 1999) points out,
the forecasts of utility that are used for decision making often turn out to
be inaccurate. (2) Moreover, the estimates of ρ obtained from choice be-
haviour are very indirect, involving many extraneous assumptions, and it
is not therefore surprising that they yield a wide range of estimates. Those
1 This corresponds to a utilitarian social welfare function, which is a simple average of the
utility of all individuals. If the social welfare function is concave in the individual utilities,
a further weighting is needed for the diminishing marginal social value of an increment to
individual utility (Atkinson and Stiglitz, 1980, Part 2).
2 Dalton measured inequality by W /W , where W is the welfare that would be obtained
e e
if everybody received the average income. This measure is only invariant with respect to
equi-proportional changes in all incomes if ρ = 1. By contrast, Atkinson measured equality
by ye /¯y where ye is the equal income which would yield W if everybody had it. This measure
is invariant to equi-proportional changes in all incomes, even if ρ 6= 1.
2
based on choice under uncertainty range from 0 to 10 or even higher3 . Es-
timates based on intertemporal choice vary less4 , but in any case the
assumption of intertemporal additivity is problematic5 . For both these
reasons a new estimate could be helpful, especially if it yields consistent
estimates from a wide variety of data sources.
Our approach is based not on inferences from behaviour, but on the
direct measurement of experienced happiness in six major surveys which
also asked questions about income and other variables. Most of the surveys
have been analysed before to examine the impact of income on happiness6 ,
but as far as we are aware, none of these studies address the crucial ques-
tion for public economics of the curvature of the relationship—in other
words the value of ρ.
That is the focus of this paper. In Section 2 we discuss some key issues
which arise in the measurement of utility. In Section 3 we describe the
survey data in more detail. In Sections 4 and 5 we give our results, and
conclude in Section 6.
2 Measuring utility
In the surveys that we analyse, a typical happiness question is ”Taking
all things into account, how happy are you these days?”. The respondent
then chooses one of a number of values, for example:
0 1 2 3 4 5 6 7 8 9 10
extremely unhappy extremely happy
In most surveys, individuals are surveyed only once, but in the German
Socio-Economic Panel (GSOEP) and the British Household Panel Study
(BHPS) they are surveyed in a number of successive years. The datasets
we use are described in detail in Table 1 and the happiness questions in
Table 2. Histograms of the replies are in Figure 1.
In order to make use of these data, we need to investigate how they are
generated. First, we assume that individual i has a level of experienced
utility ui which is cardinal and is a function of a set of observable variables
3 Lanot et al. (2006) includes a survey of measures showing a wide variety of estimates.
For example, Metrick (1995) studied game shows and estimated that risk behaviour is close
to risk neutrality, while Cohen and Einav (2005) investigated insurance purchases, and es-
timated a coefficient of risk aversion of more than 10. As is well known, explaining the
equity premium within expected utility theory requires a coefficient of the order of 30 or even
more (Mehra and Prescott, 1985, 2003).
4 See Blundell et al. (1994) and Chetty (2006) for examples of estimates of ρ based on
intertemporal choices.
5 Many economists believe that habit formation plays an important role in intertemporal
substitution decisions (Attanasio, 1999).
6 See, for example, Clark and Oswald (1996), Easterlin (1995), Winkelmann and
Winkelmann (Winkelmann and Winkelmann, 1998), Di Tella et al. (2003), Helliwell (2003),
Van Praag (2004), and Blanchflower and Oswald (2004). For a recent survey of the happiness
and income literature see Clark et al. (2008).
3
(e.g. income, work, etc. ). We suppose ui is comparable across individuals.
In order to generate an answer to a happiness question, individual i applies
an idiosyncratic, strictly increasing, function fi to ui . Thus, reported
happiness for individual i, hi , is given by
hi = fi (ui ) (3)
Under this assumption, reported happiness is an ordinal non-comparable
measure of true utility, because each individual applies their own mono-
tonic transformation to ui . It is our intention to make a further restrictive
assumption, namely that fi is common to all individuals up to a random
additive term, that is
hi = fi (ui ) = f (ui ) + vi (4)
with vi representing an error term that is independent of the circumstances
affecting true utility. There is a considerable body of evidence to justify
this assumption. First, evidence suggests that respondents report their
level of happiness in a way that is consistent with other meaningful mea-
sures of their utility. For example we can ask a person’s friend to rate the
subject’s happiness, or even ask an independent observer who has never
met the person before. The reports of these ”third parties” turn out to be
correlated across people with the subject’s own report (Diener and Suh,
1999). Second, neuropsychologists can now measure the level of activity
in the areas of the brain where positive and negative affect are expe-
rienced. These levels of activity are well correlated with the subject’s
self-report (Davidson, 1992, 2000; Davidson et al., 2000) both across peo-
ple and over time. Third, another important exercise in validation is to
ask whether the resulting measure relates to external factors in an ex-
pected manner. All the evidence that we have is that it does. For exam-
ple, income increases reported happiness, separation and divorce decrease
it, etc. (Kahneman et al., 1999). Likewise, measured happiness predicts
quitting, marital break-up, and other behaviours7 . Overall, in light of this
evidence, we feel quite justified in making the assumption embedded in
Equation 4, namely that individuals’ reported happiness reflects this ex-
perienced utility in a manner which is at least partially comparable across
individuals and, in particular, is independent of income.
Nevertheless, there remains a problem with Equation 4 arising from the
possible non-linearity of the function, f (·). If f is non-linear, then equal
intervals on the reported happiness scale do not reflect equal intervals of
true utility. For example, given our interest in estimating the decline of
marginal utility, a significant concern is that happiness reports may be
concave in true utility. We investigate this potential problem in Section 5.
We cannot reject the possibility of such a non-linear relationship, but show
that it would only have a small effect on our conclusions.
7 A good survey of papers studying the predictive value of satisfaction measures can be
found in Clark et al. (2008).
4
Table 1: Description of surveys as used. Details relate only to the subset of observa-
tions used in the regressions. The happiness variable was categorical when levels are
indicated, and numerical when a range is indicated.
Survey Name Type Countries Years Obs. Happiness Household
variable income
variable
General Social Sur- Cross- United 1972-2004 17,603 Happiness (3 Yearly
vey section States (25 waves) levels) gross
World Values Sur- Cross- Worldwide 1981-2003 37,288 Life satisfac- Varies
vey section (50) (4 waves) tion (1-10)
European Social Cross- Europe (22 2002 and 26,687 Both happiness Monthly
Survey section in 1st wave, 2004 (2 and life satis- net
26 in 2nd waves) faction (0-10)
wave)
European Quality Cross- Europe 2003 8,175 Both happiness Monthly
of Life Survey section (28) and life satis- net
faction (1-10)
German Socio- Panel Germany 1984-2005 78,877 Life satisfac- Monthly
Economic Panel (22 waves) tion (0-10) net
Survey
British Household Panel Britain 1996-2004 43,484 Life satisfac- Monthly
Panel Survey (7 waves) tion (1-7) net
3 Data and strategy
Our aim is to estimate a direct utility function. The dependent variable
is happiness, h, and the independent variables are income, y (acting as a
proxy for consumption), hours of work (measured in various ways), sex,
age, education, marital status, and employment status. A few comments
are needed on these variables.
3.1 The reported happiness variable
The happiness questions are listed in Table 2. As Figure 1 shows, allow-
ing for the different ranges the histograms look similar. For our subse-
quent statistical analysis we want to compare surveys, so we normalise all
the happiness data to a 0 to 10 scale using the appropriate affine trans-
formation8 . When both happiness and satisfaction with life as a whole
8 For example, 1-7 was shifted and stretched to run from 0 to 10. The qualitative scale in
the General Social Survey (not very happy, fairly happy, very happy) was first turned into a
numerical scale: 0, 1, 2 and then stretched.
5
General Social Survey World Values Survey European Social Survey European Social Survey
Happiness Life satisfaction Happiness Life satisfaction
.6
.6
.6
.6
.4
.4
.4
.4
.2
.2
.2
.2
0
0
0
0
0 5 10 0 5 10 0 5 10 0 5 10
European Quality of Life Survey European Quality of Life Survey German Socio−Economic Panel British Household Panel Survey
Happiness Life satisfaction Life satisfaction Life satisfaction
.6
.6
.6
.6
.4
.4
.4
.4
.2
.2
.2
.2
0
0
0
0
0 5 10 0 5 10 0 5 10 0 5 10
Fig. 1: Histograms of answers to happiness and life satisfaction questions.
were recorded, the responses were averaged to produce a single depen-
dent variable. The averaged variable was typically more closely related to
external circumstances than either happiness or life satisfaction on their
own, consistent with the possibility that each of these variables includes
an independent measurement error.
3.2 The income variable
We elected to use total household income, rather than the respondent’s
own income. Household income was adjusted to units of constant purchas-
ing power with the exception of the World Values Survey9 . It was not,
however, normalised to income per equivalent adult as we see children
as a choice variable, so that income normalisation is misleading for our
purposes. Exact income data are available for the panel surveys (German
Socio-Economic Panel and British Household Panel Survey). In the cross-
section surveys only income bands are available, and these we converted
into numerical values using the mid point of each band. For respondents
9 The period of time and the currency varied by country and wave, and we did not have
sufficient documentation available to ensure correct normalisation. This does not matter for
the estimation of ρ in Section 4.3 because we used country/wave and income interaction terms.
6
Table 2: The survey questions for the reported happiness variables we used.
Survey Variable Question
General Social Sur- Happiness Taken all together, how would you say things are these days
vey would you say that you are very happy, pretty happy, or not
too happy?
World Values Sur- Life sat. All things considered, how satisfied are you with your life as
vey a whole these days? Please use this card to help with your
answer. [range of 1-10 with 1 labelled ”Very Dissatisfied” and
10 labelled ”Very Satisfied”]
European Social Happiness Taking all things together, how happy would you say you are?
Survey Please use this card. [range of 0-10 with 0 labelled ”Extremely
unhappy”, and 10 labelled ”Extremely happy”]
European Social Life sat. All things considered, how satisfied are you with your life as
Survey a whole nowadays? Please answer using this card, where 0
means extremely dissatisfied and 10 means extremely satisfied.
[range of 0-10 with 0 labelled ”Extremely dissatisfied” and 10
labelled ”Extremely satisfied”]
European Quality Happiness Taking all things together on a scale of 1 to 10, how happy
of Life Survey would you say you are? Here 1 means you are very unhappy
and 10 means you are very happy.
European Quality Life sat. All things considered, how satisfied would you say you are
of Life Survey with your life these days? Please tell me on a scale of 1 to 10,
where 1 means very dissatisfied and 10 means very satisfied.
German Socio- Life sat. In conclusion, we would like to ask you about your satisfac-
Economic Panel tion with your life in general. Please answer according to the
following scale: 0 means ‘completely dissatisfied’, 10 means
‘completely satisfied’. How satisfied are you with your life, all
things considered?
British Household Life sat. How dissatisfied or satisfied are you with your life overall?
Panel Survey [range of 1-7 with 1 labelled ”Not satisfied at all” and 7 labelled
”Completely satisfied”.]
7
in the lowest income band we assumed an income of two thirds of the
upper limit of the band, and for respondents in the highest income band
we assumed an income of 1.5 of the lower income limit of the band.
3.3 Other causal variables
We also have to control for other variables affecting happiness—especially
hours of work. Unfortunately, hours data can in many cases be unreliable,
and we proceed in two ways.
In our main analysis we measure employment by the following set of
dummies: inactive (the omitted variable), unemployed job seeker, full-
time work, and part-time work (the distinction between full-time and
part-time work is not available in all the surveys). In a separate analysis
we combine the last two categories into a single dummy, and also interact
this dummy with a cubic polynomial in hours of work.
We also include the following control variables in all our analysis:
• Sex
• Age (including a quadratic)
• Education (dummies for degrees of achievement, or years of educa-
tion and years of education squared)
• Marital status (dummies)
In addition to these standard controls we include country or wave dum-
mies, and country/wave interaction terms when there is more than one
wave per country. The inclusion of these dummy variables enables us to
control for common fixed effects to do with country/wave. Most impor-
tantly, it allows us to control for systematic additive reporting biases. For
example, in the European Social Survey a given level of income in Den-
mark is associated with a higher reported happiness than a similar level of
income in Poland. This may have to do with the higher quality of life in
Denmark, but it may also be the case that people in Denmark will report
any given level of utility more positively than would people in Poland. By
using country/time dummies we avoid this problem altogether and obtain
ceteris paribus results that are valid across different countries and time
periods10 .
3.4 Sample
In order to have a relatively homogeneous population, we confined our-
selves to people aged 30-55, for whom annual income tends to be well
correlated with permanent income. For the same reason we excluded the
5% at either extreme of the distribution of fitted residuals from a linear
regression of log y on a set of standard regressors. We believe that many
of these observations are the result of measurement error, or else reflect
transitory deviations from usual income. Taking such observations at face
10 The flip side is that our evidence can shed no light on the source of inter-country variabil-
ity in reported happiness, or the stability of reported happiness across time (The Easterlin
Paradox).
8
value could lead to misleading conclusions about the long-run relation-
ship between utility and income (consistent with this the plots of Figure 3
show that no clear functional relationship with happiness exists for these
extreme income observations, while an excellent fit exists for all the other
observations).
3.5 Strategy of analysis
For our analysis we make the following assumptions:
1. Reported happiness, h, is linked to true utility, u, via a fixed trans-
formation f , as in Equation 4.
2. The experienced utility of person i at time t in country c can be
written as follows:11
1−ρ
yit −1 X
uit = αct + βj xjit + γi + γct + ǫ′it (5)
1−ρ j
where yit is person’s i’s income at time t, which enters the equation
via a constant elasticity of marginal utility function with parame-
ter ρ. In this formulation ρ corresponds to minus the elasticity of
marginal utility with respect to income. ρ = 0 corresponds to a
linear relationship between utility and income, with higher values
of ρ denoting increasing concavity. Marginal utility is constant for
ρ = 0, and is diminishing for ρ > 0. The coefficient on income, αct ,
is assumed to be the same for all people within a given country at a
given point in time, but it can vary between countries or at different
time points. xjit are other controls, γi is a person fixed effect, and
γct is a country/wave dummy. Finally, ǫ′it is the error.
Writing the model in this way allows for the possibility that income
is not measured in comparable units in different countries or at dif-
ferent points in time. This is a particular problem in the World
Values Survey in which the documentation of the measurement of
income is incomplete, but is also a potential concern in other sur-
veys due to the limitations of purchasing power parity calculations.
In the model of Equation 5 yit can be multiplied by an arbitrary
constant µct with no change to the fit of the model, and hence no
bias in the maximum likelihood estimate of ρ12 .
Note, the function explaining uit is additive in form. This is a crucial
assumption. Without imposing some such functional form on the
relationship between u and y, we cannot identify the curvature of
this relationship. We have more to say on this issue in Section 5.
11 Changes in income (and not only the level) are believed to affect utility, but in our two
panel surveys income changes come out as insignificant in regressions. The same is true if
additional lags are included.
12 If the coefficient on income had not been allowed to vary between countries or different
points in time, the model would have a special status for ρ = 1. This is because when
ρ = 1 (log y) any normalisation errors are absorbed by the country/wave dummies and do
not contribute to the error. Allowing α to vary by country/wave eliminates the effect of
normalisation errors for any ρ.
9
3. ρ is the same for different people. This assumption is not essen-
tial for our estimates to be useful, but it allows a particularly clear
interpretation of the results. We relax this assumption by estimat-
ing ρ separately in different surveys, and also in selected population
subgroups.
Our goal is to obtain an estimate of ρ. In the main part of our analysis
we assume that the transformation f in Equation 4 is linear. That is,
hit = uit + vit , cov(vi , yi ) = 0 (6)
Combining this assumption with Equation 5 we obtain
1−ρ
yit −1 X
hit = αct + βj xjit + γi + γct + ǫit (7)
1−ρ j
where ǫit = ǫ′it + vit . The main analysis includes the following three steps:
1. We investigate the logarithmic hypothesis (ρ = 1) using cross-sectional
analysis.
2. Panel fixed effects analysis: we test whether including person fixed
effects in the regressions substantially changes the estimated rela-
tionship between income and utility.
3. Maximum likelihood estimates of the curvature parameter, ρ, based
on Equation 7.
Finally, in Section 5 we relax the linearity assumption in Equation 6.
We investigate whether the function relating reported happiness to true
utility is concave, and compute the implied correction to the estimate of
ρ obtained in our main analysis.
4 Main results
Figure 2 shows the simple cross-sectional relationship between happiness
and income in one of the surveys. Each of the 20 points corresponds to
5% of the people arrayed according to income. The body of our results is
an attempt to analyse the curvature of this relationship in greater detail.
4.1 The logarithmic model: cross-section
We begin by exploring the hypothesis that ρ = 1. In other words, that
happiness depends linearly on log income. We therefore estimate
X
hit = α log yit + βj xjit + γct + ǫit (8)
j
The fit is quite good and is plotted in Figure 3 where both income and
happiness have been adjusted for the effect of all the control variables.
As can be seen, if we ignore the 5% at each end of the income distribu-
tion, the plots lie quite well along a straight line. For the moment we
are focusing on the cross-sectional fit, i.e. on the six upper panels in the
10
Reported happiness as a function of income
7
7
6.5
6.5
6
6
5.5
5.5
5
5
0 50000 100000 150000 8 9 10 11 12
Household income (2004 dollars) Log household income (2004 dollars)
Fig. 2: The simple cross-sectional relationship between reported happiness and income
in the U.S. General Social Survey.
figure. Considering the variety of countries, the similarity of the slopes is
interesting.
However, the obvious question is whether the relationship with log
income is truly linear. We therefore add a term in log income squared.
The estimated equation is as follows:
X
yct ) + α2 log2 (yit /¯
hit = α1 log(yit /¯ yct ) + βj xjit + γct + ǫit (9)
j
As can be seen, we have now normalised income by the average income
in that country and time period. This is because log income and log in-
come squared are highly correlated across the whole sample, which makes
the interpretation of the estimated coefficients quite difficult. The coun-
try dummy does not eliminate this problem, which is why we normalise
income by average income in the country and year concerned.
The results are shown in Table 3. As can be seen, in all the surveys the
coefficient on the squared term is negative—meaning that the relationship
between happiness and income is more concave than implied by the log
function alone, i.e. ρ is greater than 1. To estimate ρ we need to proceed
to maximum likelihood estimation.
4.2 The logarithmic model: panel fixed-effects
However, first we can exploit further the longitudinal data from the Ger-
man Socio-Economic Panel and British Household Panel Survey by in-
cluding person fixed effects. In this way, we can eliminate any bias caused
by correlations between income and unmeasured personal characteristics
which also directly affect happiness. Such correlations are the main source
of endogeneity in the income variable. We therefore estimate Equations 8
11
General Social Survey World Values Survey
1
1
.5
.5
0
0
−1 −.5
−1 −.5
−2 −1 0 1 2 −2 −1 0 1 2
European Social Survey European Quality of Life Survey
1
1
.5
.5
0
0
−1 −.5
−1 −.5
−2 −1 0 1 2 −2 −1 0 1 2
German Socio−Economic Panel British Household Panel Survey
1
1
.5
.5
0
0
−1 −.5
−1 −.5
−2 −1 0 1 2 −2 −1 0 1 2
German Socio−Economic Panel (FE) British Household Panel Survey (FE)
1
1
.5
.5
0
0
−1 −.5
−1 −.5
−2 −1 0 1 2 −2 −1 0 1 2
Fig. 3: The partial relationship between reported happiness (y-axis) and log income
(x-axis). FE indicates person fixed-effects were included in the regression. The graphs
show a consistent near-linear relationship, with some variation in the slopes.
Table 3: Partial regression coefficients of happiness on log relative income and log
relative income squared (t-scores in parentheses). Relative income is relative to mean
income in country/year. Country/year dummies and other personal controls are in-
cluded but not shown.
Survey log(y/¯
y) log2 (y/¯
y)
General Social Survey 0.70 (14.0) -0.07 (1.7)
World Values Survey 0.62 (26.1) -0.09 (3.7)
European Social Survey 0.59 (24.9) -0.15 (5.8)
European Quality of Life Survey 0.81 (17.4) -0.08 (1.5)
German Socio-Economic Panel (no fixed effects) 0.55 (27.2) -0.10 (2.7)
British Household Panel Survey (no fixed effects) 0.35 (14.0) -0.06 (2.6)
German Socio-Economic Panel (panel fixed effects) 0.36 (11.0) -0.08 (0.1)
British Household Panel Survey (panel fixed effects) 0.25 (6.7) -0.10 (3.5)
12
and 9 with a person fixed effect added in, in order to get a better handle
on causality.
The results, reported in the bottom two panels of Figure 3 and Table 3,
are similar to those of the cross-sectional analysis of the previous section, a
result is consistent with the causal interpretation13 . As compared with the
cross-section, the slopes in the panel fixed effects regressions are somewhat
lower. The downward bias resulting from measurement error in income
is exacerbated in fixed effects models (Griliches and Hausman, 1986), so
some reduction in the slope coefficient is to be expected14
4.3 Maximum likelihood estimation of ρ
We now come to the central analysis of this paper, where we directly es-
timate the curvature of the happiness-income relationship. We use maxi-
mum likelihood to estimate the parameters of Equation 7, that is
1−ρ
yit −1 X
hit = αct + βj xjit + γct + ǫit (10)
1−ρ j
To find the maximum likelihood estimate we compute the log likelihood
for values of ρ between 0 and 3 in steps of 0.1, and then use a quadratic
approximation in the vicinity of the maximum. As before observations
with extreme income values in the top or bottom 5% are excluded.
The results are shown in the first column of Table 4. In each of these
six surveys, using data on different countries and time periods, the maxi-
mum likelihood estimate of ρ falls in the narrow range of 1.19-1.34. This
similarity is in spite of the great differences between the countries, cul-
tures, and languages in the surveys we used, as well as the differences
between the questionnaires in the different surveys.
We can now ask what single value of ρ best explains the data in the six
surveys we have. To obtain the best overall estimate we add up for each
value of ρ the log likelihood terms from each of the six surveys, enabling
us to find how good each value of ρ is at explaining the data from all the
surveys put together. The overall maximum likelihood estimate we obtain
is ρ = 1.26, with a 95% confidence interval of 1.16-1.37.
The critical question, however, is whether this overall estimate is in
fact consistent with all the separate sets of data. The test for that is how
ρ = 1.26 compares with the confidence intervals in the different surveys,
and the answer is that it falls well within the 95% confidence interval
of all the individual surveys. Thus, unlike the case with choice under
13 In an annual time series, variations in happiness are unlikely to be a major cause of
variation in income. Another potential concern is time-dependent common causes, such as a
promotion, which may have a direct effect on reported happiness, as well as on income. To
demonstrate conclusively that these effects are secondary it is necessary to utilise exogenous
events which affect income, but have no direct effect on utility. A start in this direction
has been made by Gardner and Oswald (2007), who studied winners of medium sized lottery
prizes.
14 Income is measured with some error not only because of simple misreporting, but also
because income at the time of the survey is not always representative of the person’s typical
income.
13
Table 4: Maximum likelihood estimates of ρ in different surveys using first the lin-
ear dependent variable model, and then ordered logit. 95% confidence intervals in
parentheses.
Survey Standard estimate Ordered logit estimate
General Social Survey 1.20 (0.91-1.48) 1.26 (0.96-1.55)
World Values Survey 1.25 (1.05-1.45) 1.26 (1.06-1.46)
European Social Survey 1.34 (1.12-1.55) 1.25 (1.02-1.49)
European Quality of Life Survey 1.19 (0.87-1.52) 1.05 (0.71-1.38)
German Socio-Economic Panel 1.26 (0.90-1.63) 1.15 (0.81-1.49)
British Household Panel Survey 1.30 (0.97-1.62) 1.32 (0.99-1.65)
Combined estimate 1.26 (1.16-1.37) 1.23 (1.12-1.34)
uncertainty, we find that estimates of ρ based on happiness surveys yield
a single estimate that is consistent with very different data sets.
4.4 The robustness of the estimate of ρ
Even if a single value of ρ is consistent with data from different cultures,
it can still be the case that significantly different values of ρ may be
found in different subgroups of the same population. To investigate this
possibility we looked at subgroups defined by sex, age, education level, and
by marital or cohabiting status15 . After defining the different subgroups
we proceeded to find the estimate of ρ that is combined across the different
surveys, but computed for each subgroup separately.
The results are reported in Table 5. As in the case of the different
surveys, we find high degree of consistency between the estimates for dif-
ferent subgroups. The formal test for this is that the overall maximum
likelihood value of 1.26 lies not only within the 95% confidence interval
of the different surveys, but also well within the 95% confidence interval
for all the different subgroups. Taken together, these results suggest that
ρ does not vary systematically between different subgroups of the popu-
lation defined by sex, age, education, or marital status. Our results, of
course, say nothing about the possibility of non-systematic variation in
ρ between different individuals. Furthermore, these results are consistent
with systematic differences in choice behaviour16 , but imply that such dif-
ferences are likely to be due to factors other than underlying differences
in experienced utility.
15 In the case of education we defined a low, medium and high education subgroups in each
survey, so that roughly one third of the sample is in each of the three categories. In the
case of relationship status we split the sample between couples, singles, and other (separated,
divorced, or widowed). In four of the surveys the marital status variable contained a living
with partner category which we combined with the married category to create the couples
subgroup. In the two surveys in which living with partner was not part of the marital status
variable the couples category equals the married category.
16 There is some evidence for systematic group differences in risk attitudes (Dohmen et al.,
2006) and in intertemporal choice (Blundell et al., 1994).
14
Table 5: Maximum likelihood estimates of ρ in different subgroups (using a standard
linear dependent variable model and with 95% confidence intervals in parentheses).
Sub-group ML estimate
Men 1.22 (1.06-1.39)
Women 1.26 (1.11-1.40)
Age 30-42 1.27 (1.12-1.42)
Age 43-55 1.26 (1.10-1.41)
Low education 1.13 (0.85-1.40)
Mid education 1.21 (1.01-1.42)
High education 1.26 (1.16-1.37)
Couples 1.27 (1.11-1.43)
Singles (never married) 1.44 (1.13-1.77)
Widowed/Separated/Divorced 1.34 (0.85-1.83)
All subjects 1.26 (1.16-1.37)
As a further test of robustness we tested the alternative specification
for hours that we described in Section 3.3, namely using a single dummy
for being at work, and adding its interaction with hours, hours2 , and
hours3 . In so doing we also kept a dummy for being unemployed, as
distinct from being outside the labour force. The result of this alternative
specification is a maximum likelihood estimate of ρ = 1.24 with a 95%
confidence interval of 1.14-1.35, as compared with ρ = 1.26 with a 95%
confidence interval of 1.15-1.37 when using our standard specification. The
estimate of ρ thus appears robust to different treatments of working hours.
Finally, some readers may be interested in the estimated coefficient when,
in addition to controlling for hours worked, only gender and age controls
are used. The resulting estimate in this case is ρˆ = 1.32, which lies within
the 95% confidence interval for the standard specification.
5 Compression of the reported happiness
scale
Finally we need to raise the following question: Is the scale which re-
spondents use in reporting their utility the true scale (up to an arbitrary
positive affine transformation), or some non-linear transformation of it?
In the terms of Equation 4, is f a linear function?17
In particular, there comes from psycho-physics the suspicion that re-
ports using bounded scales (e.g. 0-10) may be a concave function of the
true scale of utility. In reports of physical sensations this often seems to
17 Loosely speaking, of course. Since utility reports are discrete, whereas true utility is
continuous, the function f cannot strictly be linear. But, if we think of the reports as the
rounding off to the nearest point on the scale of a continuous function of utility, we can give
precise meaning to the notion of f being linear or else a non-linear concave function.
15
happen, since replies on bounded scales are generally concave with respect
to the use of unbounded scales. For example respondents have been asked
to report the duration of noises of differing length (Stevens, 1975, p. 229).
In one experiment, a bounded scale of 1 to 7 was offered and respondents
were exposed to different durations (the shortest and longest durations
being exhibited first). Respondents assigned a category to each duration.
These values turned out to be a concave function of the true durations. By
contrast when respondents were offered an unbounded scale, their replies
were roughly proportional to the true durations.
One could argue that findings relating to the sensation of external
physical stimuli are irrelevant to the reporting of utility. When people
report their utility they are reporting not on their experience of an exter-
nal stimulus, but directly on their internal state. Thus, there may well
be a non-linear relationship between external circumstances and utility
(as we suggest for the case of income), but there is no obvious reason
why the relationship between utility and reported happiness should be
non-linear. In fact, there is evidence from the neuroscience of physical
sensation (Johnson et al., 2002) that the relationship between neural ac-
tivity and verbal reports is linear. If we are willing to assume that sub-
jective experience is linear in neural activity this evidence implies that
verbal reports are linear in subjective experience. Finally, while many
people would agree on the meaning of intervals in reported happiness,
there is less sense in which we can talk of the concept of zero happiness.
So presenting people with an interval scale seems to be a natural way to
proceed.
Our strategy to explore this issue is as follows. Consider first a general
model of the relationships between reported happiness, hi , true experi-
enced utility, ui , and income, yi . Thus we might have
hi = f (ui ), f ′ > 0 (11)
yi1−ρ−1
ui = g( , zi , ǫi ) (12)
1−ρ
where zi reflects other variables and ǫi is an unobserved error term.
The only investigation we can undertake is of the function
yi1−ρ − 1
hi = f (g( , zi , ǫi )) (13)
1−ρ
We are interested in the curvature of ui , and hence that of the function
g with respect to yi . Any analysis of Equation 13 can tell us nothing
about this without making a prior assumption about the function g. So
we continue to maintain the assumption of linearity, that is
yi1−ρ − 1
ui = α + zi + ǫi (14)
1−ρ
Conditioning on this assumption, our strategy is then to investigate
the function f in the equation
yi1−ρ − 1
hi = f (α + zi + ǫi ) (15)
1−ρ
16
In particular, we see if we can find any evidence that f ′′ < 0. We use
three approaches based on this framework:
1. Ordered logit: if f is concave and if the error term ǫi is symmet-
ric, then the estimated cut points of the ordered logit will tend to
be convex with respect to reported happiness (implying a concave
transformation from true utility to reported happiness).
2. Variance of the residuals: if we estimate a linear version of the hap-
piness regression (Equation 15), then if f is concave, the fitted resid-
uals will tend to have a lower variance at higher values of predicted
happiness, assuming ǫi is homoscedastic. This is essentially because
at high values of predicted happiness, variations in actual happiness
generated by variation in ǫi tend to be lower because of the concavity
of f in the true model.
3. Spline: assuming f is concave, then if we combine the regressors into
a single prediction variable by running a linear regression, and then
run separate regressions of reported happiness on each half of the
sample, the slope should be higher for low values of ˆ
hi and lower for
ˆi.
high values of h
In the next three sections we pursue these three approaches and com-
pute a corrected estimate of ρ. This turns out to be quite close to the
estimate generated in Section 4.3. We then give the intuition for this
outcome.
5.1 Ordered logit
Ordered logit makes it possible to allow for a non-linear f in the estimate
of ρ. The ordered logit cut points are displayed in the top left panel
of Figure 4, and are consistent with the proposed concave relationship
between reported happiness and true utility. The maximum likelihood
estimate of ρ obtained using ordered logit is reported in the right column
of Table 4. The resulting estimate of ρˆ = 1.23 is only marginally lower
than the linear regression estimate of 1.26 (Section 4.3).
This calculation relies on the assumption of a symmetric error dis-
tribution. A negatively skewed error distribution could also account for
convex cut points even if f is linear.
5.2 Variance of the residuals
Another approach involves a fundamental assumption about what it is we
want to measure in public economics. We are concerned with the impact
of the outside world upon how people feel. If a person notices no difference
in how he feels, that should count as no difference in happiness. And the
natural units in which to measure changes in happiness is the ”just notice-
able difference” (JND). This has been much used in the study of physical
sensation, and is variously defined — for example, as the difference that in
repeated observations would be noticed 50% of the time (Stevens, 1975).
However, we cannot assume that, when people use a response scale
from 0 to 10, they make the differences between each point on the scale
17
represent a given number of JNDs. A way to investigate this is by repeat-
edly asking an individual about his or her happiness (with no change of
circumstances or mood available to explain any change in the answer). In
such a test-retest situation we would not expect the same answer every
time unless the JND was vanishingly small. But we would want the vari-
ance of the replies to be independent of the person’s position on the scale
of happiness — implying that units of the scale reflect the same number
of JNDs at all points on the scale.
We would therefore like to retest individuals in unchanging circum-
stances. But we have no such data for closely repeated observations. We
do, however, have repeated annual observations on the happiness of the
same individual from two panel surveys. We can extract from these data
a person fixed effect, as well as the effect of changes in all observable inde-
pendent variables, leaving an error that approximates a test-retest error.
In the top right panel of Figure 4 we group individuals into 20 groups
with ascending average predicted reported happiness, and plot for each
group the average value of their root mean squared prediction error σi .
This shows clearly that the errors are smaller for higher values of reported
happiness.
If we assume this error reflects test-retest error and possibly other
homoscedastic error, then the error variance for true utility should be the
same at all points on the scale18 . The negative slope in the top right
panel of Figure 4 would then imply that at the upper end the reported
happiness scale is a compressed version of the true utility scale.
In producing a corrected scale of true utility, our psychological as-
sumptions are:
1. True utility, u, should be measured using a scale with constant stan-
dard errors.
2. Reported happiness, h, has variable units which are proportional to
the measured standard errors.
Thus, to convert intervals on the h scale to units on the u scale we
deflate them by σ(h), or for purposes of normalisation by σ(h)/σ0 , where
σ0 is the error variance at a reference point h0 , where true utility will be
defined as equal to reported happiness:
dh
du = (16)
σ(h)/σ0
We can now use the individual data underlying the top right panel of
Figure 4 to estimate the following relationship:
σ(h)
= 1 − c(h − h0 ) (17)
σ0
Where c > 0. Substituting Equation 17 into Equation 16 and inte-
grating we obtain
18 We use panel fixed effects regression to eliminate stable personality differences from the
error. The error is thus made of test-retest error, and other time-varying factors affecting
utility other than the ones we can control for. We assume here that this error is homoscedastic.
18
Ordered logit cut points RMS prediction error as function of predictions
2
2
1.5
0
RMSE
1
−2
.5
−4
0
2 4 6 8 10
−6
Mean predicted reported happiness for person
0 2 4 6 8 10 Source: German Socio−Economic Panel.
Estimated function linking true utility to happiness reports RMS prediction error as function of predictions
1.4
12
10
1.2
True utility
8
RMSE
6
1
4
2
.8
0 2 4 6 8 10 4 6 8 10 12
Reported happiness Mean predicted true utility for person
Source: German Socio−Economic Panel. Source: German Socio−Economic Panel.
Fig. 4: Evidence for the concavity of h = f (u) using the German Socio-Economic
Panel. (i) The top left panel shows the cut points in an ordered logit model. The
x-axis denotes the number of steps in the reported happiness variable. The y-axis is
arbitrary, so only the curvature is significant. The cut points of ordered probit are
virtually indistinguishable from those of ordered logit. (ii) The top right panel shows
the root mean squared prediction error as a function of mean prediction in a panel fixed
effects regression. (iii) The bottom-left panel shows the function relating true utility
to reported happiness that can be estimated from the information in the top-right
panel by assuming homoscedastic test-retest errors in the true utility domain. (iv)
The bottom-right panel shows the test-retest errors transformed into the true-utility
domain.
19
log(1 − c(h − h0 ))
u − u0 = − (18)
c
This function is plotted in the bottom left panel of Figure 4. The bottom,
right panel repeats the plot of root mean square prediction errors against
predictions, but this time in the u (true utility) domain. The lack of
correlation is consistent with the assumption of homoscedastic test-retest
error in the true utility domain.
To estimate the implied correction for ρ we first differentiate Equa-
tion 18 with respect to income:
∂u 1 ∂h
= (19)
∂y 1 − c(h − h0 ) ∂y
∂2u 1 ∂2h c ∂h
= + ( )2 (20)
∂y 2 1 − c(h − h0 ) ∂y 2 (1 − c(h − h0 ))2 ∂y
Denoting the elasticity of true marginal utility with respect to income
by ρu , and that of marginal reported happiness by ρh we can write
(∂ 2 u/∂y 2 )
ρu = − y
∂u/∂y
(∂ 2 h/∂y 2 ) c ∂h
=− y− y
∂h/∂y 1 − c(h − h0 ) ∂y h=h0
∂h
= ρh − cy (21)
∂y h=h0
Here c has units of 1/h, so the correction term is dimensionless. As ρh
was estimated using Equation 10 we have ∂h/∂y = y −ρh . The correction
is therefore given by the following equation:
ρu − ρh = cy 1−ρh (22)
h=h0
Using the overall maximum likelihood value of ρh = 1.26 we estimated
Equation 17 using a panel fixed effects regression on data from the German
Socio-Economic Panel, finding c = 0.153. The value of y 1−ρh for h = h0
was found by taking the sample mean in the range h0 ± 0.2. Plugging
in the numbers into Equation 22 results in an estimated correction term
of −0.021, or ρu ≈ 1.26 − 0.02 = 1.24. An analogous estimate using the
British Household Panel resulted in a correction term of −0.016. Note
that in principal, if c had been large enough, we could have obtained a
corrected estimate of ρu < 0, that is a convex relationship between u and
income (Oswald, 2005). The evidence, however, implies that in practice
the relationship between true utility and income is concave and is not
much different from that between reported happiness and income.
5.3 The spline test
The spline test is based on the idea that if the function f linking reported
utility, h, with true utility, u, is linear, then the relationship between h
and the linear regression prediction, h, ˆ should have the same slope of 1
20
Table 6: Spline test results showing the estimated δˆ in Equation 23 (t-score in paren-
theses), the F -score of the joint restriction β = 0 and δ = 0, and the implied correction
to ρˆ.
δˆ F -test Correction to ρˆ
General Social Survey -0.13 (-1.18) 2.08 -0.01
World Values Survey -0.04 (-1.16) 0.67 -0.00
European Social Survey -0.23 (-5.27) 13.97 -0.02
European Quality of Life Survey -0.07 (-1.17) 0.84 -0.01
German Socio-Economic Panel -0.30 (-6.05) 18.42 -0.09
British Household Panel Survey -0.17 (-1.79) 10.46 -0.03
Combined -0.08(-4.71) 13.17 -0.01
independently of the value of ˆh, and a constant term of zero. On the other
hand, if f is concave, the slope estimates based on low values of hˆ should
be higher than the slope estimates based on high values of h. ˆ
In each survey we first estimate on the entire sample Equation 10 with
ρ imposed at the overall maximum likelihood estimate of 1.26. We then
identify the median value of h, ˆ and create a dummy, di , which equals 1
ˆ i exceeds that median value, and 0 otherwise. We then estimate the
if h
following equation:
ˆi
hi = α + βdi + (γ + δdi )h (23)
The values reported in Table 6 show that δ < 0 and β 6= 0, consistent
with f being concave. This result is not statistically significant in all
surveys, but is strongly statistically significant in the combined sample (t−
score = 4.71 for δ < 0). The implied correction to ρˆ is -0.01, bringing the
estimate down from 1.26 to 1.25. The calculations involved are explained
below.
Using Equations 11 and 14 we have the following relationship between
the elasticity of true marginal utility, u, and that of marginal reported
happiness, h:
∂ 2 h/∂y 2 f ′′ (·)
ρh = −y = ρu − αy 1−ρh ′ (24)
∂h/∂y f (·)
where α is the coefficient in Equation 14 and y is income. We estimate the
correction to ρˆ computing the derivatives at the median prediction point,
ˆ 0 , and using the average value of y 1−ρh within a 0.2 distance of h
h ˆ0.
To estimate the required derivatives we perform the following regres-
sion separately below and above median reported happiness:
ˆp
hpi = αp + β p h (25)
i
where p ∈ {l, h} denotes whether h ˆ is in the lower or upper part of the
sample with respect to the median. We use the following linear approxi-
mation for the two halves of the sample:
ˆ p ) ≈ αp + β p h
hpi = f (h ˆp (26)
i i
21
Using this approximation we obtain
¯
ˆh ¯
ˆl
ˆ 0 ) ≈ f (h ) − f (h )
f ′ (h ¯
ˆh − h¯
ˆl
h
ˆ h + βˆh h
(α ˆ h ) − (α
ˆ l + βˆl h
ˆl)
= ¯ ¯ (27)
ˆh − h
h ˆl
¯
ˆ p denotes the median predicted happiness in sample half p.
where h
Similarly,
′ ¯
ˆh ′ ¯
ˆl
ˆ 0 ) ≈ f (h ) − f (h )
f ′′ (h ¯h ¯
ˆ ˆl
h −h
βˆ − βˆ
h l
= ¯ ¯l (28)
ˆh − h
h ˆ
5.4 Why the scale compression matters so little
For the non-linear scale compression to have a big effect two conditions
must hold:
1. Rich and poor people should typically be found at significantly dif-
ferent points of the utility scale.
2. The effect of utility on reported happiness should be substantially
different at these different points.
In reality, however, neither of these conditions hold. First, income is
only moderately correlated with utility. For example, in the GSOEP19
the regression coefficient of reported happiness on log income is 0.7, so
that a quadrupling of income is necessary to lift a person one unit on
the 0-10 utility scale. For it to have a significant effect over common
income differences, the slope of the relationship between real utility and
its reports would thus have to be substantially different at intervals of
less than 1 on the utility scale, but, second, the slopes at these different
points are in fact similar. Thus, although we have found evidence for a
non-linear relationship between real utility and its reports, this non-linear
relationship can only have a modest impact on the results of Section 4.
6 Conclusion
A key parameter for policy analysis is the elasticity (ρ) of the marginal
utility of income with respect to the level of income. To estimate this, we
have used six different surveys, including country studies of USA, Britain
and Germany, and multi-country studies involving most first-world coun-
tries and in one case third-world countries also.
We have found a striking uniformity in the estimates obtained from
these totally different surveys. The lowest estimate was 1.19 and the
19 The simple correlation between reported happiness and log income is about 0.2.
22
highest 1.30. This uniformity is in contrast to the wide range of estimates
obtained from behavioural studies.
Our overall estimate is 1.26 (with a 95% confidence interval of 1.16-
1.37). This estimate falls well-within the 95% confidence interval of each
of the individual surveys. It is also well-within the 95% confidence interval
of each sub-group of the population (sex, age, education, marital status),
so that any behavioural differences between these groups cannot be at-
tributed to differences in the curvature of the experienced utility-income
relationship.
To interpret this value of ρ, it is useful to start from the logarithmic
(ρ = 1) where the marginal utility of income is inversely proportional
to income. Our higher estimate of ρ implies that marginal utility falls
somewhat faster than that.
However, there is the obvious question, Do the answers correspond to
the true level of utility, or do respondents in their replies use a happi-
ness scale which is convex or concave with respect to their true utility?
We investigate this in three ways, with similar results. In each case we
find under certain assumptions that true utility is convex with respect to
reported happiness, so that true marginal utility declines less fast than
reported marginal utility. However, we can also compute the size of the
difference. It turns out that the maximal implied correction would only
reduce our estimate of ρ to 1.24.
We have thus confirmed the (cardinalist) assumption of nineteenth
century economists that marginal utility of income declines with income.
Given a number of assumptions, we have been able to estimate a numer-
ical value for the rate at which this occurs. This value can be used to
inform the distributional weights in cost-benefit analysis and the analysis
of optimal tax.
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