Estimating Eta: Revisions, Critique and Extensions - WebMeets.com

ESTIMATING THE ELASTICITY OF MARGINAL UTILITY: REVISIONS, EXTENSIONS AND
PROBLEMS
Abstract The elasticity of marginal utility is an important parameter for the
purposes of calculating the Social Time Preference Rate as well as
determining appropriate distribution weights in Cost Benefit Analysis. This
paper conducts a review of the empirical evidence on the value of this
parameter for the United Kingdom. Four different empirical methodologies
are identified namely the Euler equation approach, the wants-independence
approach, the equal sacrifice approach and the subjective wellbeing
approach. The paper then presents a new set of estimates obtained using
both contemporaneous and historical data, alternative estimation procedures
and where possible, testing the underlying assumptions. The theoretical
restrictions underpinning wants-independence, one of the most popular
approaches, are rejected. These estimates are then combined using meta-
analytical techniques in order to produce a single preferred estimate of
the elasticity of marginal utility of 1.5. Critically, the confidence
intervals for this estimate do not include unity which is the current
official estimate of the elasticity of marginal utility. Keywords: Elasticity of Marginal Utility, Social Time Preference Rate, Cost
Benefit Analysis, Inequality Aversion 1. Introduction Numerous techniques have been developed to appraise public sector projects
whose costs and benefits span multiple time periods. In recent years
however there has been a growing consensus that the appropriate discount
rate is the Social Time Preference Rate (HM Treasury, 2003). The formula
for the STPR is given by (Ramsey, 1928): [pic] Where ? is the rate of pure time preference, g is the growth rate and ? is
the elasticity of marginal utility.[1] The elasticity of marginal utility
is therefore a key component within the STPR calculation. In fact two approaches are normally considered when determining social
discount rates: the STPR and the social opportunity cost (SOC). The latter
is normally associated with the rate of return to the marginal project in
the private sector. In an economy without any distortions or imperfections
these should be the same but in practice they might differ substantially.
According to one well-respected approach the appropriate procedure is to
weight the funds necessary for a particular project according to their
source: consumption or investment (Bradford, 1975). An obvious problem with
this approach however is determining for each project the proportion of
investment funds from displaced consumption and displaced investment. Until 2003 the basis of the discount rate used in the United Kingdom (6
percent for most purposes) was the rate of return to marginal private
investment projects. [2]The current rate of discount however, is 3.5
percent and is based explicitly on the Ramsey formula. The decision by the
United Kingdom Government to use the STPR in policy appraisal and
evaluation (HMT, 2003) has brought to the fore the question of what is an
appropriate value for the elasticity of marginal utility. The purpose of
our paper is to review the empirical evidence on the magnitude of this
parameter. And in a number of instances we not only review the empirical
evidence but also develop our own estimates using more recent data and / or
using what we believe are more appropriate techniques. In one case we test
the validity of the key assumption underlying a particular revealed
preference approach.

The current assumption is that the elasticity of marginal utility is unity
although previously the Treasury had assumed a higher value of 1.5. Yet an
incorrect assumption regarding the magnitude of this parameter would result
in the wrong value for the STPR and potentially, a serious misallocation of
public funds. Nuclear power offering short term benefits in the form of
cheap energy but high decommissioning costs in the long term provides a
perfect illustration of a project of national significance whose
desirability is highly sensitive to changes in the STPR. Apart from discounting knowledge of the elasticity of marginal utility is
also essential for the derivation of welfare weights necessary for policies
that redistribute income such as might occur in the context of regional
policy. An alternative perspective of course, is that welfare weights
should not be used but instead should a project result in a burden on one
part of society then compensation should be paid to those who are adversely
impacted (the Hicks-Kaldor compensation criterion). Our paper confines itself to estimating only the elasticity of marginal
utility and not with any other component such as the rate of pure time
preference. We likewise limit ourselves to considering estimates developed
specifically for the United Kingdom. The theoretical discounting literature
now extends beyond the basic Ramsey Rule. We however do not concern
ourselves with the literature on discounting over the long term beyond
noting that an estimate of the elasticity of marginal utility is still
invariably required (e.g. Gollier, 2009). We also sidestep the recent
literature attempting to separate risk aversion, intertemporal substitution
and inequality aversion (e.g. Atkinson et al. 2009). We consider the theoretical basis for and the empirical evidence generated
by four different revealed preference approaches: the intertemporal
allocation of consumption (Euler equations) approach, the 'wants
independent' goods approach, the subjective wellbeing approach and the
'equal sacrifice' approach. Each of these it transpires, poses significant
empirical challenges such that no single technique is likely to produce a
convincing answer to the question what is the elasticity of marginal
utility. We do not however review the stated preference approach to
estimating the elasticity of marginal utility generated by controlled
experiments examining aversion to inequality or risk aversion. In our
opinion such exercises produce results that may be specific to the nature
of the experiment or to the characteristics of the respondents (e.g.
impecunious undergraduates). We note that several surveys already exist of the value of this parameter
but that these are all now rather dated. Stern (1977) advocates an estimate
of the elasticity of marginal utility of 2 but with a possible range of 1
to 10. Pearce and Ulph (1995) suggest a value of 0.7 to 1.5 and a best-
guess estimate of 0.83 based on Blundell et al (1994). Cowell and Gardiner
(1999) point to a range of 0.5 to 4.0. One is not however required to
accept unquestioningly the entire range of possible values offered by these
reviews. As a check on the plausibility of estimates one can perform a
'leaky bucket' test e.g. Pearce and Ulph (op cit). For example, a value for the elasticity of marginal utility equal to 1
implies that an extra £1 of income to someone whose income is X has twice
the weight of an extra £1 to someone whose income is 2X. Put differently a
loss of £1 to the richer individual is worth a gain of 50p for the poorer
person. For a value of 1.5 the weight would be 2.8 and for a value of 2 the
weight would be 4. But for an elasticity of marginal utility equal to 10
cited in Stern (op cit) a loss of £1 to the richer household would be worth
a gain of 0.1p for the poorer household. Such calculations suggest that
high values of the elasticity of marginal utility e.g. above 5 are
implausible. High values of the elasticity of marginal utility obviously
imply an aversion to inequality. The remainder of the paper is organised as follows. Sections two to five
discuss each of the different techniques in turn. Section six takes
estimates from each technique and combines them using meta-analytical
techniques in order to test the hypothesis that the elasticity of marginal
utility is equal to unity. Using the same set of techniques we also
generate a central estimate along with a 95 percent confidence interval and
use it to develop a preferred estimate of the STPR given conventional
assumptions about the rate of pure time preference and the growth rate of
the economy. The final section concludes.
2. Socially revealed inequality aversion: The equal absolute sacrifice
approach In this section we use 'socially revealed' preferences to infer the
elasticity of marginal utility. More specifically, we analyse information
on the progressivity of the income tax schedule to infer the elasticity of
marginal utility under the assumption of equal absolute sacrifice. 2.1. Theory The approach depends on the twin assumptions that the principle of equal
sacrifice holds and that the utility function takes a known form (almost
invariably isoelastic). The justification for the assumption of equal
sacrifice may be traced back to Mill (1848) who stated: "Equality of
taxation, as a maxim of politics, means equality of sacrifice". In such
exercises the elasticity of marginal utility can be interpreted as the
Government's inequality aversion parameter. Algebraically the principle of equal absolute sacrifice implies that for
all income levels y the following equation must hold: [pic]
Where k is a constant, y is gross income, V is utility and T(y) is the
income tax schedule. Assuming an isolelastic utility function: [pic] Substitution yields: [pic] Differentiating this expression with respect to y and solving for ? yields:
[pic] Where y is gross income, T is total tax liability, T(y)/y is the average
rate of taxation and t is the marginal rate of taxation. Cowell and Gardiner (1999) argue that there is good reason to take
seriously estimates derived from tax schedules: decisions on taxation have
to be defended before an electorate and the values implicit in them ought
therefore to be applicable in other areas where distributional
considerations are important such as discounting or the determination of
welfare weights. At the same time there are concerns about whether a
progressive in