The Marginal Utility of Income
Richard Layard*
Guy Mayraz*
Steve Nickell**
* CEP, London School of Economics
** Nuffield College, Oxford
Goal
• The marginal utility of income is a central concept in public
economics
• The rate at which it declines is a very important number
• Given a CRRA utility function:
the parameter ρ (the coefficient of risk aversion) is a measure
of this rate of decline.
• Our purpose is to estimate ρ
Method
• Our method is very simple.
• First, treat answers (on a 1-10 scale) to the question “Taking
all things into account, how happy are you these day?” as a
measure of utility.
• Second, relate these answers to income in a cross-section or
time-series analysis, and estimate the relevant parameters.
• Of course, the basic problem is how to persuade people that
this procedure generates the parameters of interest.
Alternative Methods
• Our method is based on attempting to measure ex-post
“experienced” utility, which is what is required in this context.
• Alternative methods of estimating are based on studies of
behaviour.
a) choice under uncertainty or b) intertemporal choice.
• Behaviour is assumed to be based on a decision function
involving the weighted addition of ex-ante “decision” utility in
different states or future time periods.
Problems with Alternative Methods
• Ex-ante “decision” utility often turns out to be systematically
different from ex-post “experienced” utility.
• These methods involve dubious extraneous assumptions eg.
Intertemporal additivity or expected utility maximization.
• Not surprisingly, they yield a very wide range of estimates of
ρ - eg. Those based on choice under uncertainty range from 0
to 10.
Measuring utility
• First, the use of overall “judgment” type questions (i.e. how
happy or satisfied are you, all things considered?) may be
questioned.
• The day reconstruction method (DRM) is an alternative (DRM
involves dividing a day into episodes and in each, provide a
rating on happiness, worry, frustration, etc. Then aggregate
these into a combined score).
• Each has advantages and disadvantages.
• The measure we use is consistent with other meaningful
measures of utility.
True utility and reported happiness
• Second, what is the relationship between reported “happiness”,
h, and true utility, u ?
• Normalising u so that 0 is the bottom level (extremely
unhappy) and 10 is the top level (extremely happy), suppose
h=f(u). Assume f’>0.
• Now consider three possibilities:
The relationship between u and h
First possibility
•
Each individual has their own idiosyncratic interpretation of
the scale. Thus the replies are not comparable:
hi= fi (ui)
•
If true, it is hard to see how cross-sections yield rather
precise relationships between h and variables such as
income, employment status etc.
•
Also, it is hard to see how, when person dummies are
introduced into panel data (thus concentrating on time-series
variation for each person), one obtains results which are
similar to those generated by a cross-section.
The relationship between u and h
Second possibility
•
Individuals use the scale in the same way, but potentially it
reflects some non linear transformation of true utility:
hi= f(ui)
• We investigate this, but initially we assume the third
possibility.
Third possibility
•
Same linear scale:
hi= ui
Data
• We use happiness scores or life satisfaction scores. We
renormalise, if necessary, onto a 0-10 scale. If a survey
contains both, we average.
• The income variable is total real household income, not
equivalised, and sample members are restricted to those aged
30-55.
Reported happiness histogram
European Social Survey
0
.1
.2
.3
Happiness
0
2
4
6
8
10
Data (cont.)
• We use multiple years of four cross-section surveys:
–
–
–
–
The US General Social Survey (GSS)
European Social Survey (ESS)
European Quality of Life Survey (EQLS)
World Values Survey (WVS)
• In addition we use two panel surveys:
– German Socio-Economic Panel (GSOEP)
– British Household Panel Survey (BHPS)
Data (cont.)
•
•
Because we are estimating a direct utility function, we must
include an hours of work variable.
In addition we include standard controls:
– Sex
– Age (years + quadratic)
– Education (years + quadratic or attainment level
dummies)
– Marital status dummies
– Employment status dummies
– Country Dummies
– Year dummies
Strategy
• Maintained model
hit  f (uit )
uit   ct g ( yit )    j ijt  i  ct   it
j
i  individual, t  time, c  country
Strategy (cont.)
•
•
Assuming hit= uit
–
Cross-section analysis (assuming ρ=1)
–
Panel analysis (assuming ρ=1)
–
Estimation of ρ
Investigation of the form of f
5
5.5
6
6.5
7
Reported happiness as a function of income
0
50000
100000
Household income (2004 dollars)
Source: General Social survey 1972-2004. All subjects. No controls.
Fig. 2: The simple cross-sectional relationship between reported happiness and income in the US General Social Survey.
150000
h vs. log y in cross-sections
•
The result is that reported happiness is approximately linear in
log income.
•
Subjects with extreme incomes (5% on either side) deviate from
general relationship, but this may reflect yearly blips or
measurement problems.
•
If we exclude observations with sharp change in reported
income, the remaining observations fit the linear relationship
well.
•
However, introducing a quadratic term suggests a further degree
of concavity.
•
Panel analysis with fixed effects yields similar results.
.5
0
-.5
-1
-1
-.5
0
.5
1
World Values Survey
1
General Social Survey
-2
-1
0
1
2
-2
-1
0
1
2
.5
0
-.5
-1
-1
-.5
0
.5
1
European Quality of Life Survey
1
European Social Survey
-2
-1
0
1
2
-2
0
1
2
.5
0
-.5
-1
-1
-.5
0
.5
1
British Household Panel Survey
1
German Socio-Economic Panel
-1
-2
-1
0
1
2
-2
-1
0
1
2
.5
0
-.5
-1
-1
-.5
0
.5
1
British Household Panel Survey
1
German Socio-Economic Panel
-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 included but
not shown.
Survey
General Social Survey
World Values Survey
European Social Survey
European Quality of Life Survey
German Socio-Economic Panel (no fixed effects)
British Household Panel Survey (no fixed effects)
Log (y/ym)
0.70 (14.0)
0.62 (26.1)
0.59 (24.9)
0.81 (17.4)
0.55 (27.2)
0.35 (14.0)
Log2 (y/ym)
-0.07 (1.7)
-0.09 (3.7)
-0.15 (5.8)
-0.08 (1.5)
-0.10 (2.7)
-0.06 (2.6)
German Socio-Economic Panel (panel fixed effects)
British Household Panel Survey (panel fixed effects)
0.36 (11.0)
0.25 (6.7)
-0.08 (0.1)
-0.10 (3.5)
Estimating ρ
• Reported happiness is modelled as linear in a CRRA function
with parameter ρ (see eq. 7)
• We plot the log likelihood of the observations as a function of
ρ.
• We combine the datasets to produce the overall maximum
likelihood estimate of ρ.
• MLE is ρ = 1.26.
Table 4:
Maximum likelihood estimates of ρ in different surveys using first the linear
dependent variable model, and then ordered logit. 95% confidence intervals in
parentheses.
Survey
General Social Survey
World Values Survey
European Social Survey
European Quality of Life Survey
German Socio-Economic Panel
British Household Panel Survey
Combined estimate
Standard estimate
Ordered logit estimate
1.20 (0.91-1.48)
1.25 (1.05-1.45)
1.34 (1.12-1.55)
1.19 (0.87-1.52)
1.26 (0.90-1.63)
1.30 (0.97-1.62)
1.26 (1.16-1.37)
1.26 (0.96-1.55)
1.26 (1.06-1.46)
1.25 (1.02-1.49)
1.05 (0.71-1.38)
1.15 (0.81-1.49)
1.32 (0.99-1.65)
1.23 (1.12-1.34)
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 uncertainty, we find that
estimates of ρ based on happiness surveys yield a single estimate that is consistent
with very different data sets.
Table 5:
Maximum likelihood estimate of p in different subgroups (using a standard linear
dependent variable model and with 95% confidence intervals in parentheses).
Sub-group
Men
Women
ML estimate
1.22 (1.06-1.39)
1.26 (1.11-1.40)
Age 30-42
Age 43-55
1.27 (1.12-1.42)
1.26 (1.10-1.41)
Low education
Mid education
High education
1.13 (0.85-1.40)
1.21 (1.01-1.42)
1.26 (1.16-1.37)
Couples
Singles (never married)
Widowed/Separated/Divorced
1.27 (1.11-1.43)
1.44 (1.13-1.77)
1.34 (0.85-1.83)
All subjects
1.26 (1.16-1.37)
Non-linearity of the h-u relationship
• Our results indicate that, under the assumption that h=u, the
estimate of ρ is 1.26.
• This will be an over-estimate if hi  f (ui )
and f’>0, f’’<0, because true utility is then a convex function
of reported utility. So, true utility will be a less concave
function of income than reported utility.
Non-linearity of the h-u relationship
• Indeed, in theory this effect could be so large that true utility
could even be convex in income.
• How can we investigate the curvature of f ?
• We focus on the equation
yi1   1
hi  f (
 zi   i )
1 
zi  other variables. Note additive function is maintained assumption.
The shape of ƒ(u), I
• We use three methods.
(i) Ordered Logit. The ordered logit estimate of ρ is 1.23.
The cut points are slightly curved, consistent with slight
curvature of ƒ.
This method relies on the assumption that ε has a symmetric
distribution.
RMS prediction error as function of predictions
1
0
-4
.5
-2
RMSE
0
1.5
2
2
Ordered logit cut points
-6
2
2
4
6
8
10
RMS prediction error as function of predictions
.8
2
4
1
6
RMSE
8
1.2
10
12
Estimated function linking true utility to happiness reports
10
Sourc e: German Socio-Economic Panel.
1.4
0
4
6
8
Mean predicted reported happiness for person
0
2
4
6
Reported happiness
Sourc e: German Socio-Economic Panel.
8
10
4
6
8
10
Mean predicted true utility for person
Sourc e: 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
12
The shape of ƒ(u), II
(ii)
Variance of the Residuals.
Assume true utility, u, has a scale with a constant standard
error for test-retest mistakes. Reported happiness, һ, has
variable units proportional to measured standard errors
based on repeated observations on same person in
GSOEP.
The shape of ƒ(u), III
dh
 ( h) /  o
•
du 
•
Estimate  (h)  1  c(h  ho )
•
Subst. (2) into (1) and integrate
o
log(1  c( h  ho ))
u  uo  
c
Plotted in bottom left panel in Fig. 4.
• Use this to correct ρ. After correction, ρ = 1.24.
(1)
(2)
The shape of ƒ(u), IV
(iii) The Spline Test.
If ƒ is concave, if we combine the regressors into a single
prediction variable, h, by running a linear regression and then
run separate regressions of reported happiness on the
prediction variable in each half of the sample, the slope should
be higher for low values of h than for high values of h.
This can be used to compute a measure of the curvature of ƒ.
The corrected measure of ρ is 1.25.
Conclusions
1.
We have estimated the elasticity of marginal utility of
income with respect to income (ρ) based on measure of
experienced utility.
2.
We have used six different surveys.
3.
We have attempted to investigate the extent to which the
results are distorted because reported happiness is non-linear
function of true utility.
4.
Under our assumptions, ρ is around 1.24.