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Nom original: spadaro.pdfTitre: Social Preferences Revealed through Effective Marginal Tax RatesAuteur: Amedeo Spadaro

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Social Preferences Revealed through Effective Marginal Tax Rates

François Bourguignon and Amedeo Spadaro1

Preliminary version
October 2000

1

Respectively, The World Bank and Delta, Paris, Universitat de les Illes Balears and Delta.

1

Abstract
This paper inverts the usual logic of the applied optimal income taxation literature.
Standard practice analyzes the shape of the optimal tax schedule that is consistent with a
given social welfare function, a statistical distribution of individual productivities that fits
available data on labor incomes and given preferences between consumption and leisure. In
this paper, we go in the opposite direction. We start from the observed distribution of gross
and disposable income within a population and from the observed marginal tax rates as
computed in standard tax-benefit models. We then show that, under a set of simplifying
assumptions,

it is possible to identify the social welfare function that would make the

observed marginal tax rate schedule optimal under some assumption about consumptionleisure preferences. This provides an alternative way of reading marginal tax rates
calculations routinely provided by tax-benefit models. In that framework, the issue of the
optimality of an existing tax-benefit system may be analyzed by considering whether the
social welfare function associated with that system satisfies elementary properties. Likewise,
the reform of an existing system may be seen as a change in the underlying social welfare
function which may prove to be less consensual than the reform itself. A detailed application
is given in the case of France, and of a basic income/flat tax reform of the tax-benefit system
in that country. For comparability, an application is also made to several other EU countries.

Introduction
Several attempts were recently made at analyzing existing redistribution systems in
several countries within the framework of optimal taxation theory. The basic question asked
in that literature is whether it is possible to justify the most salient features of existing
systems by some optimal tax argument. For instance, under what condition would it be
optimal for the marginal tax rate curve to be U-shaped - see Diamond (1998) and Saez (1999)
for the US and Salanié (1998) for France? Or could it be optimal to have 100 per cent
marginal tax rates at the bottom of the distribution through some kind of guaranteed income
program - see d'Autume (2000) and Bourguignon and Spadaro (2000) in the case of France
and other European countries. Such questions were already addressed in the early optimal
taxation literature and in particular in Mirrlees (1971, 1986) but the exercise is now more
relevant because of the possibility of relying on large and well documented micro data sets
rather than on hypothetical distributions.
The results obtained when applying the standard optimal taxation framework to actual
data depend very much on several key ingredients of the model. The shape of the social
welfare function may be the most important one. As already pointed out by Atkinson and
Stiglitz (1980) in their comments of Mirrlees' original work, using a Rawlsian social objective
or a utilitarian framework make a big difference. The first would lead to very high effective
marginal rates for low individual abilities, whereas the second would be closer to a linear tax
system with a constant marginal tax rate. It is not unexpected to find the same sensitivity with
actual distribution of abilities rather than hypothetical. But then, what should be done? Should
actual redistribution systems be closer to the 'optimal' schedule using a Rawlsian objective or
on the contrary should one pick up a much less demanding social welfare function?
The point of view taken in this paper is to some extent the opposite of the previous
one. Instead of taking the social welfare function as given and deriving the optimal schedule
of effective marginal tax rates along the income or ability scale, we do the reverse. Namely,
we consider the effective marginal tax rates schedule that corresponds to an actual
redistribution system and we look for the social welfare function according to which that
schedule would be optimal. This approach is the dual of the previous one. In the first case,
wondering about the optimality of an actual redistribution system consists of comparing an
optimal effective marginal tax rate schedule derived from some 'reasonable' social welfare
function with the actual one. In the second case, it consists of checking whether the social
3

welfare function implied by the marginal tax rate schedule is in some sense 'reasonable', that
is whether the marginal social welfare is positive and decreasing along the horizontal axis and
possibly the rate at which it decreases - the absolute aversion of society toward inequality. 2
In effect, the approach that is proposed here is simply a way of 'reading' the average
and marginal net tax schedules that may be obtained from simulating a tax-benefit system on
a population of households. This reading simply translates the observed outcome of the
redistribution process into social welfare language. Because of this, comparing two
redistribution systems or analyzing the reform of an existing system can be made directly in
terms of social welfare. Instead of determining who is getting more out of redistribution and
who is getting less - or paying more net taxes - this reading of the marginal tax rate schedule
informs directly on the differential implicit marginal social welfare weight given to one part
of the distribution versus another.
Of course, this revelation of social preferences must rely on several auxiliary
assumptions about labor supply behavior and about the distribution of abilities of individuals
or households who are not observed working. The optimal tax schedule depends crucially on
these assumptions. The same is true of the social preferences revealed by a given marginal tax
schedule. It is even conceivable that apparent anomalies in these preferences may be due to
these assumptions being not satisfied. The observation of the marginal tax rate schedule may
thus reveal more than social preferences. In some cases, it may suggest either that the tax
schedule is inconsistent with optimality or that some common assumptions on labor supply
behavior or on the distribution of abilities are unsatisfactory. This would seem equally useful
information.
The paper is organized as follows. Section 1 recalls the optimal taxation model and
derives the duality relationship between the effective marginal tax rate schedule and the
marginal social welfare function in the simple case where individual preferences between
consumption and leisure are assumed to be quasi-linear. The second section discusses the
empirical application of the preceding principle. In section 3, we characterize the social
welfare function under a set of simple alternative assumptions about the labor supply
elasticity and some characteristics of the distribution of abilities in the case of France. In this
section we also use that framework to discuss a possible reform of the tax-benefit systems in
terms of changes in social preferences or in terms of alternative views on labor supply

2

See Fleurbaey, Hagneré, Martinez and Trannoy (1999).
4

behavior. Finally, section 4 extends the analysis to a few other EU countries taking advantage
of preliminary results obtained with the EUROMOD model.3

1. The duality between optimal marginal tax rates and the social welfare function
The basic optimal taxation framework is well-known.4 Agents are supposed to choose
the consumption (y) /labor (L) combination that maximizes their preferences, U(y, L) given
the budget constraint imposed by the government :
productivity of the agent

y = wL - T(wL), where w is the

and T( ) the net tax schedule. If the distribution of agents'

productivity is given by the density function f(w) defined on the support [w0, A], the optimal
taxation problem may be written as :
A

Max T ()

∫ G[V [w, T ( )]]. f (w).dw

(1.1)

w0

under the constraints :

V [ w, T ( )] = U ( y*, L*)
( y*, L*) = Argmax [ U ( y, L); y = wL − T ( wL), L ≥ 0]

(1.3)
(1.2)

A

∫ T (wL*). f (w).dw ≥ B

(1.4)

w0

where G[ ] is the social welfare function that transforms individual indirect utility V( ) into
social welfare and B is the budget constraint of the government.
We shall not try to deal with this general specification of the model. Instead we shall
be focusing on the special case where the function U(y, L) representing agents' preferences is
quasi-linear with respect to y and iso-elastic with respect to L:
1+

U ( y , L) = y − k .L

1
ε

(2)

In that case, the labor supply function that is solution of (1.2) above is given by :
L * = A.wε .[1 − T ' ( wL*)]ε

3
4

(3)

See Immervoll, O'Donoghue and Sutherland (2000).
See for instance Atkinson and Stiglitz (1980) or Tuomala( 1990).
5

where A is some constant. In that expression, ε, appears as the elasticity of labor supply , L,
with respect to the marginal return to the labor of the agent, the latter being his/her
productivity corrected by the marginal rate of taxation T'( ).
This particular case has been studied in some detail in the literature5. It leads to the
following simple characterization of the optimal tax schedule :

t ( w)
1 1 − F ( w)
= (1 + ).
. (1 − S ( w) / S ( w0 )]
ε w. f ( w)
1 − t ( w)

(4)

In that expression, t(w) is the marginal tax rate faced by an agent with productivity, w, and
therefore with earnings w.L*, where L* is given by (3), F(w) is the cumulative distribution
function, whereas S(w) stands for the average marginal social utility of all agents with
productivity above w :

A

1
S ( w) =
G' [V ( w, T ()]. f ( w).dw
1 − F ( w) ∫w

(5)

As the duality between the marginal rate of taxation and the social welfare function
that we want to exploit in the rest of this paper lies in the two preceding relationships, it is
important to have a good intuition of what they actually mean. Consider the following thought
experiment. Starting from an existing tax system, the government decides to increase the tax
payment by a small increment dT for people whose labor income is Y and labor productivity
W, leaving the rest of the tax schedule unchanged. Such a measure essentially has three
effects : a) it reduces the labor supply of people at Y and just below that level whose marginal
labor income is reduced by dT ; b) it increases the tax payment by dT for all people whose
earnings is above Y ; c) it increases total tax receipts by the difference between effects b and a.
At an optimum, we want the total effect of these changes in terms of social welfare to be
equal to zero.
The tax reduction effect a) depends on the marginal rate of taxation, t(W), the elasticity
of labor supply, ε, the productivity itself, W, and the density of people around that level of
productivity, W. This tax reduction effect (TR) may be shown to be equal to :

5

See for instance Atkinson (1995) or Diamond (1998).
6

TR =

t (W ) W . f (W )
.
.dT
1 − t (W ) 1+1 / ε

The tax increase effect (TI) is simply equal to the proportion of people above the productivity
level W times the infra-marginal increase in their tax payment, dT :
TI = [1 − F (W )].dT

In order for the government’s budget constraint to keep holding, the resulting net increment in
tax receipts, TI – TR, is to be redistributed. Since net effective marginal tax rates are not to be
changed, except at Y, this requires redistributing a lump sum TI – TR to all individuals in the
population. The marginal gain in social welfare of doing so is given by (TI – TR).S(w0). The
loss of social welfare comes from people above W whose disposable income is reduced by dT.
Indeed, people whose marginal tax rate is actually modified – i.e. people at W and just below
– are not affected because they compensate the drop in the effective price of their labor and its
negative effect on consumption by a reduction in the quantity they supply and an increase in
their leasure. This is the familiar enveloppe theorem. Under these conditions the loss of social
welfare is simply equal to the proportion of people above W times their average social
marginal welfare, S(W). The optimality condition may thus be written as :

[1 − F (W )].S (W ).dT = (TI − TR ).S (w0 )
and after dividing through by S(w0) and dT:

[1 − F (W )]. S (W ) . = TI − TR
S ( w0 )

dT

(6)

What is attractive in that expression is that the right-hand side is essentially positive whereas
the left-hand side is essentially normative. The right hand side measures the net tax gain by
Euro confiscated from people above W. The left hand side measures the relative marginal
social loss of doing so.
The preceding expression illustrates the duality that we exploit in the rest of this paper.
For a given distribution of productivities, f( ), the right-hand side may be easily evaluated by
observing the tax-benefit system in a given economy and its implied effective marginal tax
7

rate schedule, provided that some estimate of the labor supply elasticity is available. Then (6)
yeilds information on the social welfare function that is consistent with the tas-benefit system.
Viewed in the other direction, (6) shows the tax-benefit system that is optimal for a given
social welfare function. The latter is the usual approach in the applied optimal taxation
literature. The former approach that ‘reveals’ the social welfare function consistent with an
existing tax-system, under the assumption that this system is indeed optimal in the sense of
model (1) is less conventional.
Characterizing precisely the social welfare function optimally implied by a tax-benefit
system requires some additional step. Define the right-hand side of (6) as the following
‘positive’ function :

Θ( w) = [1 − F ( w)] −

t ( w) w. f ( w)
.
1 − t ( w) 1 + 1 / ε

(7)

Normalize the welfare function G( ) in such a way that the mean marginal social welfare is
equal to unity, S(w0) = 1. Finally take derivatives on both sides of (5). It comes that the
marginal social welfare, H(w), associated with the productivity level, w, is given by:

H ( w) = G ' [V ( w, T ()]= − Θ' ( w) / f ( w)

(8)

On that expression, it may readily be seen that the revealed marginal social welfare function
depends not only on the effective marginal tax rate and the density of the distribution of
productivities, but also on the derivatives of these two functions with respect to productivity.
Clearly, eyeballing a marginal tax rate schedule and the distribution of income is not
sufficient to say much on the social welfare function that may optimally be consistent with
them.
It is the function H( ) that may be associated with the tax-benefit systems observed in
several countries and with some basic assumptions on ε, and possibly on f( ) for extreme
ranges of productivity that we want to study in the rest of this paper. Before we see how this
may be done practically, however, it is worth reflecting on the restrictions that permit such a
simple characterization.
The preceding methodology relies on several restrictions about the preferences of
agents. A slightly less restrictive form consists of assuming separability of preferences with

8

respect to consumption and labor without assuming quasi-linearity. In other words,
preferences are represented by the following function :
U(y, L) = A(y) – B(L)
where A( ) is not supposed to be linear anymore. In that case, it may be shown that the
optimal taxation formula (4) becomes :

t ( w)
1 1 − F ( w)
a[ y ( w)] = (1 +
.[a[ y ( w)] − S ( w) / S ( w0 )]
).
ε ( w) w. f ( w)
1 − t ( w)
where a( ) is the inverse of the derivative of the utility of income A( ), a () is the mean value
of that inverted marginal utility for people with productivity above w and ε(w) is the elasticity
of labor supply for somebody whose productivity is w.
If an estimation of the labor-supply elasticity function ε(w) were available, then the
inversion technique discussed above could still be used. But, clearly, it would permit to
identify some combination of the marginal social utility and private marginal utility of
income. Some additional assumption on the functions A( ) and B( ) would be necessary to go
beyond this. This is quite natural. Marginal social utility may be revealed by actual marginal
tax rates only if individual utility U( ) has been normalized in one way or another. The quasilinear case considered above does it in the simplest way by setting A(y) = y. Other
normalizations are theoretically possible. In particular, when A( ) and B( ) are both iso-elastic
functions, the elasticity of labor supply becomes constant again and normalization essentially
consists of setting the elasticity of one of the two functions A( ) or B( ). However, we have
not yet experimented with this possibility.

2. Basic principles for empirical implementation
To implement the previous methodology, estimations of the elasticity of labor supply,
ε, the distribution f(w) and the marginal rate of taxation, t(w), are necessary. Practically, what
is observed in a typical household survey? Essentially total labor income, wL, and disposable
income, y , or equivalently, total taxes and benefits, T(wL)6. When the household survey is
6

To keep with the logic of the optimal taxation model, we ignore non-labor taxable income in all what follows.
9

connected with a full tax-benefit model, it is possible to compute the latter on the basis of the
observed characteristics of the household and the official rule for the calculation of taxes and
benefits. With such a tax-benefit model, it is also possible to evaluate the marginal tax rate by
modifying observed labor income by a small amount. To be in the situation to apply the
formulae above for the inversion of the marginal tax rates into marginal social welfare, it is
thus necessary to impute a value of the productivity parameter, w, to the economic unit of
analysis, usually the household, and to deduce from that imputation the statistical distribution
of productivities, f( ).
A rather natural approach would consist of assimilating the productivity terms with
observed hourly wage rates, and then to estimate econometrically the labor-supply elasticity,
ε, which, without loss of generality, may even be specified as a function of the productivity
characteristics, w. In the present case, we refrain from following this approach for several
rather fundamental reasons. First, labor supply may differ quite significantly from working
hours when unobserved efforts are taken into account. Second, the econometric estimation of
a labor supply model requires taking into account the non-linearity introduced by the taxbenefit system actually faced by individuals, and in particular the endogeneity of marginal tax
rates. Econometric estimations of this type are now known to be little robust 7. However,
relying on simpler alternative estimates, based on simple linear specifications, is like
introducing some arbitrariness in the analysis. Third, it is well known that the elasticity of
labor supply differs substantially across various types of individuals. In particular, it is small
for household heads and larger for spouses, young people and people close to retirement age.
Under these conditions, what value should be chosen? Fourth, and more fundamentally, it
seems natural to chose the household as the economic unit in a welfare analysis of taxes and
benefits. But then, the problem arises of aggregating at the household level concepts or
measures that are valid essentially at the individual level. In particular, how should individual
productivities be aggregated so as to define “household productivity” ? Likewise, if the
elasticity of labor supply has been estimated at the individual level and is different across
various types of individuals, how should it be averaged within the household?
Somehow, the approach that we propose here goes in a direction that is opposite to the
previous one. Instead of starting from observed productivity and deriving an estimate of labor
supply from observed labor incomes, we start from an arbitrary assumption about the
elasticity of labor supply and derive from observed labor incomes the implicit productivity

7

See Blundell, Duncan and Meghir (1998).
10

characteristic of households. This operation is a simple inversion of the labor supply equation
(3) . Multiply both sides of that equation so that the gross labor income, Y, appears on the left
hand side :
Y = wL * = B.w1+ε .[1 − T ' ( wL*)]ε

(9)

After inversion, one gets for a given value of ε :
1
+
1
w = C.Y ε

−ε
1
.[1 − T ' ( wL*)] +ε

(10)

where C is a constant. So, implicit productivity appears as an increasing function of observed
gross labor income and the marginal tax rate. The second relationship is easily understood.
For a given gross labor income, the higher the marginal tax rate, the lower is labor supply as
given by (3), and therefore the higher the implicit productivity.
The preceding inversion equation allows for a satisfactory definition of all functions
necessary for recovering the social welfare function from the optimal taxation formula. For
household i, observed with gross labor income Yi, and marginal tax rate ti, a value of the
implicit productivity characteristic wi, may be imputed through (10). Then all households may
be ranked by increasing value of that productivity

and it is possible to identify the

distribution function F(w), the marginal tax rate function, t(w), and finally the function Θ(w)
from which the social marginal welfare function H(w) may be inferred - see (7) and (8) above.
Equation (10) yields the basic principle of the methodology. However, its actual
implementation raises additional complications. We list them below, indicating in each case
the choices made to go over them.
a) Continuity and differentiability
The application of the inverted optimal taxation formula, (7)-(8), requires the
knowledge of the continuous functions f(w), Θ(w) and the derivative of the latter. As just
discussed, above, however, what may be obtained from households data bases is a set of
discrete observations of the imputed productivity characteristic, wi, the associated cumulative
distribution function, F(wi) and the marginal tax rate function, t(wi). In order to get an

11

estimate of the derivative of the function Θ( ) in (8), and therefore of the marginal social
welfare function, we proceeded as follows.
(i) For each observation obtain an estimate of the density function f(wi) by standard kernel
techniques - using a Gaussian kernel with the so-called 'optimal' window. 8
(ii) Compute the function Θ() given by (7) for each observation, i.
(iii) Estimate the derivative of Θ() at any point w, using again a kernel approximation
computed over the whole sample.9
Dividing then by the kernel approximation of f(w) obtained in (i) yields the marginal social
welfare function defined by (8) for any value of w.
b) Household size
It was assumed in the preceding section that all households had identical preferences
and indirect utility functions. Practically, however, actual tax-benefit systems discriminate
households according to various characteristics. Size and household composition are the main
dimensions along which this discrimination is taking place. Under these conditions, the issue
arises of the way in which these characteristics might be implicitly or explicitly incorporated
in the imputation of the social welfare function.
The results shown in the next sections are based on two extreme views. In the first
one, the size of households is simply ignored in both the imputation of productivity and in tax
optimization. The implications of this choice are somewhat ambiguous. It may be seen in (10)
that size will affect productivity through two channels. On the one hand, a larger family - in
terms of the number of potentially active adults - will generally have a greater gross labor
income, which will contribute to a larger estimate of productivity. On the other hand, it will
also face a different marginal tax rate. At the middle or the top of the distribution, it will
probably be smaller, thus contributing to a smaller productivity estimate, which compensates
the preceding effect. At the bottom of the distribution, it will be the opposite. Overall, it is
thus difficult to say very much on the effect of simply ignoring household size or
composition.
8

See Härdle (1990).
The function Θ( ) itself may be approximated by Kernel techniques and then differentiated numerically. It is
simpler to compute a kernel approximation of the derivative of Θ( ). At a point w, the first is given by

9

Θ( w) =

∑ Λ(w ).e
i

− h ( w − wi ) 2

where Λ(wi) is the value of the RHS of (7) for observation i, the summation is over

i

12

The other extreme assumption consists of considering groups of households with the
same size or the same composition as independent populations of which the redistribution
authority seek to maximize the social welfare. In other words, the optimal taxation problem
involves finding an optimal tax-benefit schedule separately for each household group. This is
implicitly done under some exogenous budget constraint, which makes the aggregate
redistribution of income across the various groups of households exogenous. In other words,
the analysis does not seek to recover any information about the social preferences of the
redistribution authority among various household compositions.
Other choices would probably be possible, which would precisely lead to inferring
social preferences about household composition. However there are two issues involved here.
One has to do with the autonomous labor supply behavior of households with different
composition - i.e. how the preferences U(y, L ) should depend on size. The other is concerned
with the weight of households with different composition in social preferences. Some
arbitrary assumption is necessary on the former to be able to recover some information on the
latter. The simple attempts we made in that direction consisted of defining the preferences
U(y, L) on a per capita basis. However, this had for consequence that the optimal tax-benefit
system had to satisfy a homogeneity property close to the 'quotient familial' principle in the
French income tax. According to that property, taxes and benefits per capita should depend
only on gross income per capita.10 But as this property is seldom satisfied by the whole taxbenefit system neither in France itself11 nor in other EU countries, we did not follow that
route.
c) Households with zero income and households with apparently irrational behavior
In presence of a guaranteed minimum income in a tax-benefit system, some
households may find it optimal not to work at all. In the simple labor supply model above,
this would correspond to a situation where the marginal tax rate is 100 per cent. However,
there is some ambiguity about these situations. It arises because it is not totally clear whether
actual marginal tax rates are indeed 100 per cent and because transitory situations are
observed where households may not be at their preferred consumption-labor combination.
the entire sample and h represents the inverse of the square of the window’s size. The derivative may then be
directly evaluated by :
10

Θ' ( w) = ∑ − 2.h( w − wi ).Λ( wi ).e − h ( w− wi )

2

i

For a discussion of this property see Moyes and Shorrocks (1998). See also the discussion in Bourguignon
and Spadaro (2000).
11
Indeed, the way in which the RMI benefit or housing benefits in France depend on family size is different
from the quotient familial used in the income tax.
13

Taking the French minimum income program as a references, two different situations
may be envisaged. The fact that some additional income reduces a given benefit one for one
only after some adjustment period means that the short-run marginal tax rate may be actually
considered to be lower than 100 per cent. For instance, people receiving the minimum income
RMI lose only 50 per cent of any additional income from activity during some interessement
period. At the end of that period, however, they would lose all of it if they kept receiving the
RMI. Averaging over time, consider that the marginal tax rate for a 'RMIste' is 75 per cent,
then his or her budget constraint writes :
y = RMI + .25* wL
Ignoring the other benefits going to non-RMI low income households, their budget constraint
is simply:
y = wL if wL > RMI
On that basis, we should never observe a person or a household not receiving the RMI
and working for a total income between the RMI and 1.333 times the RMI - i.e. the point at
which the two preceding budget constraints cross. But, of course, there are households in the
survey used for France whose labor income falls in that range, which is inconsistent with the
model being used and/or the assumption made on the marginal tax rate associated with the
RMI.
One way of dealing with this inconsistency is to assumed that all gross labor incomes
are observed with some measurement error drawn from some arbitrary distribution. These
may the be drawn repeatedly until no household in the sample finds itself between the two
preceding budget constraints. This is a treatment that is analogous to the original econometric
model of a household facing a non-linear and possibly discontinuous budget constraint by
Hausman (1985).
An alternative to this treatment is indeed to consider that households receiving the
guaranteed minimum income face a 100 per cent marginal tax rate.12 In the framework of the
simple labor supply model used in the present paper, all households whose productivity is
below some minimum threshold , w0, should rationally be receiving this minimum income
and supplying zero labor. This threshold may be computed as the level of gross labor income
12

We take 100 per cent as being an absolute upper bound. Practically, some situations may lead to marginal tax
rates higher than 100 per cent but we take them as unwanted anomalies.
14

yielding a disposable income just equal to the guaranteed minimum income. But, here again,
we face the problem of apparently irrational households whose gross labor income is strictly
positive but below the RMI. As before, a way of resolving that contradiction is to assume
some measurement or optimization error, possibly due to the fact that the household is not
observed at 'equilibrium'. Drawing repeatedly in an arbitrary distribution of errors and putting
a zero lower limit on 'true' gross incomes, one may get rid of these supposedly inconsistent
observations.
The inversion formula (8) does not work anymore for those households at the
minimum income and facing 100 per cent marginal tax rate. The marginal social welfare of all
households receiving the minimum income, S0, may easily be recovered. With the same
condition as before that the mean marginal social welfare is normalized to unity, we must
have that :

S 0 .F ( w0 ) + [1 − F ( w0 )].S ( w0 ) = 1
where the threshold, w0, may be recovered as indicated above. Of course, for such a taxbenefit system to be optimal some conditions must be met below w0.13 We do not insist on
them here because it it not really clear that the quasi-linear preferences for consumption and
labor used here can offer the best justification of an optimal minimum income with 100 per
cent marginal tax rate scheme.
3. Application to the case of France
We now apply the methodology which has just been discussed to French data. This
application draws on a recent version of Sysiff based on the 1994 French household budget
survey and the 1994 tax-benefit system.14 The sample comprises 10.214 households. To keep
with the logic of the optimal taxation model, all households for which non-labor income,
including pension and unemployment benefits, represented more than 10 per cent of total
income were eliminated from the sample. Non-labor income itself is ignored in the analysis.
Net income is computed through official rules for taxes and benefits instead of being taken
directly from the data. This guarantees that effective net marginal tax rates, which are
calculated through the same rules, are consistent with average net tax rates.
13

From (3) above it would seem necessary that [1-F(w)].[1+1/ε(w)]/[w.f(w)] be uniformly very high below some
threshold w0.

15

Following the last set of remarks in the preceding section, several applications have
been run. They differ with respect to the value selected for the elasticity of labor supply and
the choices made for handling household size on the one hand and the issue of the 100 per
cent marginal tax rate and apparently irrational households on the other. Results are
summarized by curves showing the marginal social welfare of the various quantiles of the
population of households ranked by level of productivity.
There are three panels in figure 1. The left hand panel shows the effective net marginal
tax rates faced by the various quantiles of the population as computed on the basis of official
rules modeled in Sysiff. On the same panel, with a different scale, is represented the
distribution of gross labor incomes, the population mean being normalized to unity. These
two curves are the two inputs of the whole analysis. The center panel shows the distribution of
productivity consistent with the two preceding curves, under two alternative assumptions
about the elasticity of labor-supply, a 'very low' elasticity case with ε =.1 and a 'moderate'
elasticity case with ε=.5. In both cases, the mean productivity is normalized to unity. Finally,
the right hand panel shows the marginal social welfare consistent with the previous curves for
various quantiles of the population. Here again, the mean marginal social welfare is
normalized to unity.
This first figure calls for several remarks. The marginal tax rate curve has a U-shape,
being extremely high at the bottom of the distribution - recall that some proportion of
households face marginal tax rates equal to 100 per cent in this scenario - falling until a little
after the median and then increasing very slowly with the progressivity of the income tax. It is
only in the last one or two centiles that the slope of that the curve becomes steep, but this
something that is difficult to see with the scale used in figure 1a. The center figures show the
density functions associated with the distribution of gross incomes and marginal tax rates in
the preceding panel. It can be seen that less inequality is obtained for productivities when the
elasticity of

labor supply is moderate than when it is very low.

This is because the

distribution of gross income tends to amplify more the inequality of productivity in the first
case. Thus, for a given distribution of gross incomes and marginal tax rates, there is less
inequality in productivities when labor supply is more elastic.
The last panel shows the marginal social welfare for various quantiles of productivity.
As marginal tax rates at the bottom are assumed to be equal to 100 per cent, the curves start at
the quantile for which gross incomes are above the RMI, that is approximately the 4th
14

The same version of the model was used in Bourguignon and Spadaro (2000). An version updated to 2000 is
about to be ready.
16

percentile. This is the explanation of the initial gap. A second technical observation raised by
these curves has to do with the kernel approximation. If both curves are rather smooth
throughout almost all the productivity range, they become somewhat agitated when entering
the last 7 or 8 centiles of the population. Clearly, the kernel window seems to small in that
part of the distribution and the smoothened curve is still very much influenced by a few
outliers. This is natural because individual incomes or productivities are much more distant
from each other in that part of the distribution. A solution could have been to increase the size
of the window. This indeed tends to smoothen more the right-hand part of the curve. But it
does so in the left hand side too, eliminating very much of the changes in curvature and
possibly in the slope of the curve. On the other hand, adopting a log-normal kernel simply
shifts the irregularity from the extreme right to the extreme left. The window size finally used
reflects all these tradeoffs. In all cases, we tried to obtain the maximum smoothness for high
productivity households while keeping the main accidents in the rest of the curve. For the ease
of the presentation, we also approximate the resulting curves in the remaining of this paper by
a 6th order polynomial. It turns out that this approximation has the advantage of fitting well
the kernel estimate in the first part of the curve. And to smoothen further the upper part, the
details of which are simply ignored.
Overall, revealed marginal social welfare is declining with the level of household
productivity. This is indeed reassuring since it suggests that the redistribution system in
France exhibits some minimum features of optimality in the sense of maximizing a standard
concave social welfare function of individual utility levels. This is an interesting result
because it is certainly not guaranteed by the inversion methodology used in this paper.
A second feature readily apparent in figure 1 is that the slope of the marginal social
welfare curve increases quite substantially with the elasticity of labor supply. This result was
to be expected, though. It was seen above that the observed distribution of gross incomes was
leading to more inequality in the distribution of productivity when the elasticity of labor
supply was low - equivalently, the elasticity of labor supply contributes to amplifying the
inequality of productivity. But because the observed marginal rates of taxation remain the
same, it must be the case that revealed social preferences are less equalizing when the
elasticity of labor supply is low. The difference shown in figure 1 is indeed quite sensible.
Marginal social welfare falls from 110 to 90 per cent of the mean when ε=.1. It goes from 150
to 50 per cent when ε=.5. As indirect private utility in the present case has the dimension of
income, the social marginal rate of substitution between the income of the very rich and the
very poor is only slightly above unity in the first case. It is equal to 3 in the second case. This
17

is an interesting way of evaluating the implicit importance of the assumptions being made on
the elasticity of labor supply in applied taxation work.
A third feature, that is more surprising, is the fact that the progressivity of the marginal
social welfare function in the case ε=.5 seems to concentrate in the portion of the distribution
between the 55th and the 90th centiles. The curve is almost flat before this lower limit - it
might even be increasing around the median. It then declines quite abruptly. Thus, everything
is as if social preferences behind the French redistribution system were opposing the 6 bottom
deciles or so to the rest of the population. Although this requires a little more theoretical
analysis, a possible justification for such a shape could rely on some kind of median voter
argument or more generally some political economy determination of the tax system.
The next figures show the effect of changing some of the assumptions behind the
preceding result. Figure 2 refers to the whole population of households but is based on the
assumption that marginal tax rates for RMI recipients is not 100 per cent and on the
methodology described above for correcting observations inconsistent with that assumption.
Unlike in the preceding case, it is then possible to recover the marginal social welfare of the
bottom of the productivity distribution. Figure 2 shows 6th order polynomial approximation
of the kernel estimates of the marginal social welfare curves in that case. The features present
in the preceding figure can still be observed. In addition, it turns out that the marginal social
welfare function appears to be decreasing at the very bottom of the productivity distribution,
whereas it remains flat over a long interval extending from slightly below the first decile to
almost the 6th decile. Under the present set of assumptions, it would thus seem that the main
source of progressivity in social preferences at the bottom of the productivity scale is
concerned with the very poorest. This conclusion is to be taken with some care, however,
because it may depend on the arbitrary assumptions made on the marginal tax rates of
individuals who receive the minimum income, RMI, and the way apparently inconsistent
observations were corrected.
We now turn to the case where the size and the composition of households is taken
into account by applying the preceding methodology separately to various groups of
households which are homogeneous in terms of demographic composition. As mentioned
before, this is equivalent with considering as exogenous the redistribution that takes place
across these groups independently of productivity and income. Thus, figure 3 shows the
results of the inversion of the marginal rate curve into the marginal social welfare curve for
singles, couples, couples with one child and couples with two children. These curves are

18

derived under the assumption that people receiving the minimum income face a marginal tax
rates of 100 per cent and are therefore ignored.
Except for singles, the shape of the marginal social welfare curve is comparable in
these various cases to the one found for the whole population. It is decreasing throughout the
whole population of households of a given size, once they are ranked by productivity level.
However, the general shape of the curve is slightly different from what was seen in the
preceding figures. In particular, the flat part at the beginning of the curve is considerably
reduced. The slope of the curve is now negative the second decile, whereas it was practically
zero until the sixth decile for the curves referring to the whole population of households. This
suggests that part of the flatness of the marginal social welfare curve in that framework was
explained by the heterogeneity of the way the French tax-benefit system handles households
with different size and composition.
The marginal social welfare curve for singles shows that there certainly is no reason
why the marginal social welfare imputed from the marginal tax rate should be falling. The
curve shows a hump in the second decile and another, less pronounced, in the 7th decile.
Thus, in this case, it can be said that the existing tax-benefit system in France is inconsistent
with the maximization of a standard concave and increasing social welfare function restricted
to households of single persons.
Identifying what causes these humps in the marginal social welfare curve of singles or,
equivalently, the symptoms of the non-optimality of the tax-benefit system when restricted to
singles is uneasy. It may be seen from (7) and (8) above that marginal social welfare is a
function that depends on the marginal tax rate and the density functions, but also on their
derivatives. In turn the slope of the marginal social welfare curve depends in a rather complex
way on the second derivative of the function that links the marginal tax rate and productivity
and the second derivative of the density function. It should be possible to analyze in some
detail how the slope of the marginal social welfare curve depends on the shape of the
marginal tax rate and density curves, but we did not get into this direction here.
Another feature that is readily apparent on the 'singles' panel of figure 3 is that the
'shape' of the social marginal welfare curves obtained with the two elasticities .1 and .5 of the
labor supply is rather different. Going back to (7) and (8) again, one could believe that
switching from the first to the second value would simply multiply the distance at which the
marginal social welfare if from unity - i.e. the mean - by (1+1/.1)/(1+1/.5) = 11/3. This
clearly is not the case in figure 2, however. The reason for this is simply that modifying the

19

elasticity also modifies the density function which is an important ingredient of the shape of
the marginal social welfare curve.
To finish this analysis of the social welfare function implicit in the French tax-benefit
system, it seemed interesting to evaluate the sensitivity of the results obtained with respect to
the marginal tax rate curve. As a matter of fact, changing the marginal tax rate curve while
maintaining the distribution of productivity constant is an interesting way of looking at
possible reforms of a tax-benefit system.
In the present framework, any tax reform can be analyzed as a kind of instantaneous
change in the social preferences of the redistribution authority. For instance, figure 4 shows
the effect of replacing the existing marginal tax rate curve implied by the French system by a
simple basic income/flat tax rate system. In the new system, the marginal tax rate is flat at a
level arbitrarily set at 40 per cent - which is the mean marginal tax rate with the present
system. If we ignore the top of the distribution for which we know there may be problems
associated with the fact that the density of income and productivity is too low in the available
sample, the reform contributes to rotating the marginal social welfare curve clockwise, and,
therefore, making it more progressive. In other words, advocates of a reform that would move
the French tax-benefit system toward a basic income/flat tax rate system with a marginal tax
rate at 40 per cent, would appear, according to our criterion, more inequality averse than those
who believe the present tax-benefit system is optimal.
Of course, the preceding argument is incomplete because the government's budget
constraint is totally absent from the analysis or, equivalently, because normalizing the mean
marginal social welfare does not make very much sense when simulating a tax reform. To go
further would thus require computing the actual change in disposable income of households
implied by the reform and their behavioral response. Again, this is left for future work. Note
also that instead of interpreting a tax reform as going from an old to a new set of social
preferences, it may be interpreted as moving from a sub-optimal tax-benefit system to an
optimal one. If this is the case, the inversion methodology developed in this paper is clearly
useless.
4. Application to other countries
It is tempting to see whether the preceding results are specific of the French taxbenefit system and the demo-economic structure of French households. Thus, we have
repeated the preceding exercises using tax-benefit models and household data bases available

20

in other countries in relation with the construction of the integrated EU tax-benefit model,
EUROMOD.
We show here results obtained using UK's and Spain's models. In both cases, we could
obtain data sets derived from national tax-benefit models that included gross income by
sources, disposable income, the effective marginal tax rate and household demographic
characteristics. It was thus possible to select the same kind of subsample as in the French
system, eliminating in particular households for whom labor income would represent less than
90 per cent of total gross income. In the case of UK, households with 100 per cent marginal
tax rates because of the minimum income mechanism were ignored since the marginal social
welfare that corresponds to them cannot be evaluated. Results are shown for household size
groups treated separately.
The revealed marginal social welfare curve associated with singles and childless
couples in the UK are decreasing throughout the productivity scale, except for a small initial
hump for singles - much less pronounced, however, than in the case of French singles. For
couples, however, the difference with French results is rather striking. Marginal social welfare
is not downward sloping. Actually, there are two humps more or less at the same place in both
curves, the first one around the 3rd decile and the second one a little before the 8th decile. It is
tempting to interpret this difference with the French case as the result of the way family size
is treated in the two systems, and in particular to the quotient familial in France - and its
equivalent for benefits.
Spain's revealed marginal social welfare curves are downward sloping after a small
initial hump at the first decile for couples with children. For singles and childless couples,
however, a hump is observed around the median of the distribution. Because there is an
initial hump at a higher level in the marginal social curve for singles, its general shape is
downward sloping. For childless couples, however, the hump is unique and the whole curve
has an inverted U-shape.
As already stressed above, it would be interesting to go a little more into what causes
these differences in the shape of the marginal social welfare curve and thus to undertake
comparisons of national tax-benefit systems on the basis of observed marginal tax rate
curves, and of course gross income distribution. But this requires a more detailed analysis of
the conditions under which the marginal social welfare as given by (7) and (8) is increasing
or decreasing.

21

Conclusion
This paper has explored an original side of applied optimal taxation. Instead of
deriving the optimal marginal tax rate curve associated with some distribution of individual
productivities, the analysis consists of retrieving the marginal social welfare functions that
makes the observed marginal tax rates optimal. Rather surprisingly, such analysis applied in
the most simple way to the observed performances of tax-benefit systems in several EU
countries does not systematically lead to abnormal revealed social preferences. In particular,
the general aspect of the revealed marginal social welfare curves generally is downward
sloping.

There are several exceptions to that rule, though, especially when one treats

independently households with a given composition or size.
Summarizing tax-benefit systems by the marginal social welfare curve they imply to
be considered as being optimal seems to be an interesting way of characterizing them, and
possibly to compare them in space and in time. For this to be the case, however, several
problems that become apparent in this exploratory paper must be dealt with. Many of them
are practical and have to do with the way in which a discrete set of values may be best
approximated by a function that is continuous and differentiable. From a more conceptual
point of view, it is also important to precisely characterize the way in which the revealed
marginal social welfare curve is related to the density of the implicit distribution of
productivity and the way in which the marginal tax rate depends on productivity. To the
extent that the marginal social welfare function depends on the derivatives of these two
curves, both these empirical and conceptual issues are closely linked.

22

Figure 1. Derivation of the marginal social welfare function: France, all households
Figure 1a. Marginal tax rates by productivity ranking
(6th order polynomial approximation)

1
0.9

Figure 1c. Marginal social welfare by productivity
ranking, mean =1. (Kernel estimates and 6th order
polynomial approximation)

2
1.8

0.8
0.7

1.6

0.6

1.4

0.5
0.4

1.2

e=.5

1

0.3
0.2

e=.1

0.8

0.1

0.6

0
0

0.2

0.4

0.6

0.8

1

productivity quantiles

0.2
0

Figure 1b. Kernel estimates of productivity density

0.001

0.4

0

0.2

0.4

0.6

productivity quantiles

0.0009
0.0008

e=.5

0.0007
0.0006
0.0005

e=.1

0.0004
0.0003
0.0002
0.0001

productivity (mean =1)

4.84

4.61

4.37

3.9

4.14

3.67

3.44

3.2

2.97

2.5

2.73

2.26

2.03

1.79

1.56

1.32

1.09

0.85

0.62

0.39

0.15

0

0.8

1

Figure 2. France, all households ; marginal social welfare curve with less than 100 per cent
marginal tax rate at minimum income (6th order polynomial approximation of kernel
estimates)

2
1.8
Marginal social welfare (mean = 1)

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

Quantiles of population ranked by productivity

24

Figure 3. Marginal social welfare curves : France, household size groups treated separately

Marginal social welfare (mean = 1)

(Kernel curves approximated by 6th order polynomial)
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

2

e = .5

1.8

Singles

ee==.5.5

1.6
1.2
1

e = .1

0.8

e = .1

0.6
0.4
0.2
0

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

Productivity rank

0.5

0.6

0.7

0.8

0.9

1

Couples+2

1.8

e = .5

1.6

0.4

2

Couples+1

1.8

0.3

Productivity rank

2
Marginal social welfare (mean = 1)

Couples

1.4

e = .5

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

e = .1

0.6

0.6

0.4

0.4

0.2

0.2

e = .1

0

0
0

0.2

0.4

Productivity rank

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Productivity rank

25

Figure 4. Change in marginal social welfare associated with a switch
to a 40 per cent flat marginal tax rate

2
1.8
1.6
1.4
1.2
1

Initial curve, e= .5

0.8
40 per cent flat marginal
tax rate, e=.5

0.6
0.4
0.2
0
0

0.1

0.2

0.3

0.4

0.5

0.6

Productivity quantile

0.7

0.8

0.9

1

26

Figure 5. Marginal social welfare by productivity rank : UK, household size groups treated separately

Marginal social welfare (mean = 1)

(Kernel curves approximated by 6th order polynomial)
2

2.5

1.8
1.6

Single

e =.5

1.4
1.2

e = .5

1.5

1
e =.1

0.8

1

0.6
0.4

e = .1

0.5

0.2
0
0

0.2

0.4

0.6

0.8

1

Productivity rank
2
Marginal social welfare (mean = 1)

Couples

2

0
0

0.2

0.4

Productivity rank

2

1.8

Couples+1

e = .5

1.6

1.6
1.4

1.2

1.2

1

1

0.8

0.8

e = .1

0.6

0.8

1

Couples+2

1.8

1.4

0.6

e = .5

e = .1

0.6
0.4

0.4

0.2

0.2

0

0
0

0.2

0.4

Productivity rank

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Productivity rank

27

Figure 6. Marginal social welfare by productivity rank : Spain, household size groups treated separately
(Kernel curves approximated by 6th order polynomial)

Marginal social welfare (mean = 1)

1.6

1.8

Singles

e = .5

1.4

1.6
1.2

1

1

e = .1

0.8

0.8

0.6

e = .1

0.6

0.4

0.4

0.2

0.2
0

0
0

0.2

0.4

Productivity rank

0.6

0.8

0

1

0.2

0.4

Productivity rank

1.6
1.6
Marginal social welfare (mean = 1)

Couples

e = .5

1.4

1.2

Couples+1

0.6

0.8

1

0.8

1

Couples+2

1.4

1.4

1.2

1.2

1

1

e = .1

0.8

e = .1

0.8

0.6

0.6

e = .5

e = .5

0.4

0.4

0.2

0.2

0

0
0

0.2

0.4

Productivity rank

0.6

0.8

1

0

0.2

0.4

0.6

Productivity rank

28

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Moyes P, Shorrocks A., (1998) “The Impossibility of a Progressive Tax Structure”, "Journal
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Saez, E. (1998), Using Elasticities to Derive Optimal Income Tax Rates, Mimeo, MIT
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