REDISTRIBUTION AND LABOUR SUPPLY

REDISTRIBUTION AND LABOUR SUPPLY
Autores: Jorge Onrubia(a)
Rafael Salas(b)
José Félix Sanz(c)
P. T. N.o 17/01
(a) Instituto de Estudios Fiscales y Universidad Complutense de Madrid.
(b) Instituto de Estudios Fiscales y Universidad Complutense de Madrid. Address for correspon­
dence: Rafael Salas, Departamento de Análisis Económico I, Universidad Complutense de Madrid,
Campus de Somosaguas, 28223 Madrid (Spain). Phone: 34 91 3942512, Fax: 34 91 3942561,
E-mail: R.Salas@ccee.ucm.es.
(c) Instituto de Estudios Fiscales y Universidad Complutense de Madrid.
N.B.: Las opiniones expresadas en este trabajo son de la exlusiva responsabilidad de los auto­
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INDEX
1. INTRODUCTION
2. THE MODEL
3. APPLICATIONS TO ALTERNATIVE TAXES AND LABOUR SUPPLY
3. SPECIFICATIONS
3. 3.1. Proportional Tax Case
3.
3.
3.
3.
3.
3.2.
3.2.
3.2.
3.2.
3.2.
The linear tax under different labour supply specifications
3.2.1. CES
3.2.2. The linear labour supply (Hausman, 1980, 1981)
3.2.3. The iso-elastic labour supply specification (Burtless and
3.2.3. Hausman, 1978)
4. CONCLUDING REMARKS
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ABSTRACT
This paper explores the effect of personal income taxes on redistribution
when labour supply reactions are taken into consideration. We find that classical
non-behavioural results on redistribution are not necessarily satisfied in a more
general behavioural framework. In this respect, we show that the relevant tran­
sition for measuring redistribution is not the transition from the initial post-tax
to the final post-tax income distribution but the one from the more precise ini­
tial pre-tax to the final post-tax income distribution. In addition, we postulate
necessary and sufficient conditions to ensure redistribution in this wider setting.
We also find that the functional specification of the labour supply may condition
final results.
JEL Classification: H31, H23, D31 and D63
Key Words: Redistribution, inequality, taxation and labour supply
Acknowledgements: This paper has benefited from support from the Eu­
ropean Commission Project #ERBCHRXCT980248 and the Spanish Ministry of
Education Project #PB98-0546-C0202 and comments by Miguel Ángel LópezGarcía. Errors are our own.
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Instituto de Estudios Fiscales
1. INTRODUCTION
This note explores the effect of taxes on redistribution when labour supply
behavioural reactions are taken into account. In this respect, redistribution is
measured in the classical way, defined in terms of the transition from pre-tax to
post-tax income distributions. We prove that non-behavioural (static) standard
results on redistribution are not necessarily fulfilled in a more general behaviou­
ral (dynamic) framework1 To this extent, correct redistribution analysis requires
incorporation of behavioural effects induced by taxes.
In this new setting, we have to distinguish among three different income dis­
tributions:
(i) the initial pre-tax income distribution, which corresponds to gross incomes in the absence of taxes,
(ii) the initial post-tax income distribution, which is the gross income dis­
tribution once changes in labour supply have been taken into conside­
ration as a result of tax change,
(iii) the final post-tax income distribution, which reflects the distribution of
initial post-tax incomes net of taxes.
In exploring redistribution, this conceptual distinction is theoretically relevant
as it implies that net income and tax liability are endogenously determined. With
this framework, the relevant transition to measure redistribution is the one
from the initial pre-tax to the final post-tax income distribution, and not the one
from the initial post-tax to the final post-tax income distribution, as the static
setting assumes.
In this wider context, we postulate necessary and sufficient conditions to
ensure redistribution. Moreover, standard Jakobsson-Fellman-Kakwani (JFK)
results can be preserved under restricted conditions on the structure of the
tax system and on the specification of the labour supply. As a result, we also
conclude that labour supply specification matters and may influence the final
redistributive results of a given tax reform. Therefore, testing the correct
functional form for the labour supply would be a compulsory requirement in
applied work.
The plan of the paper is as follows. Section 2 sets the model and quantifies
progressivity in terms of wage elasticities. Section 3 offers some applications
to alternative tax structures and labour supply functions. Finally, section 4
concludes.
1
The seminal papers in static literature are Fellman (1976), Jakobsson (1976) and Kakwani
(1977). Extensions to personal income taxes with positive thresholds can be found in Keen et
al. (1996). As far as we know, Preston (1987, 1989) is the only reference using a dynamic
approach.
— 7 —
2. THE MODEL
Assume a distribution of homogeneous individuals i=1,...,H with differences
only in the exogenous gross wage rates W=(w1,...,wH), with the same non­
labour income mi (assumed initially to be zero). Assume the initial pre-tax income vector Y=(y1,...,yH) generated by yi=wiLi, where Li is the pre-tax labour
supply and the distribution Y is confined to positive pre-tax income levels
yi∈R++ ≡ (0,∞), as usual.
Labour supply
The labour supply adopts this general form:

∂L
∂L
≥ 0,
≥0
L(wi,mi ),
∂m
∂w
i
i


Li = 

 L,
wi ≥ w


where mi is non-labour income, initially assumed to be zero, and L is the maxi­
mum labour supply, obtained for w . This general form includes typical cases in
the literature such as the CES (Stern, 1976 and Zabalza, 1983), linear (Hausman,
1980, 1981) and log-linear (Burtless and Hausman, 1978) specifications. These
specific functional forms will be observed below.
Tax structure
The tax structure adopts the general form, T: R+→R, such that T(u) is conti­
nuous, non-decreasing and differentiable on u and
standard assumptions in the literature.
∂T
< 1, ∀u . These are the
∂u
Redistribution
Our aim is to generalise the main standard results on redistribution of JFK. In
doing so, we use the concept of a local residual progression. According to these
authors, a necessary and sufficient condition for the existence of non-negative
redistribution is that local residual progression is always less than or equal to
one (and higher than or equal to zero), for every pre-tax income distribution.
When comparing two tax systems, the necessary and sufficient condition for
non-lower redistribution is that the residual progression should be reduced (and
non-negative).
— 8 —
Instituto de Estudios Fiscales
In order to allow for labour supply effects, we distinguish between the initial
post-tax income vector2 Y'∈R++, generated by y'i=wiL'i, and the final post-tax income vector X' ∈R++, generated by x'i=yi'-T(yi'). L'i is the post-tax labour supply.
With this setting, redistribution focuses on the transition from the initial pre-tax
income distribution (Y) to X'.
The redistribution effect is consistently defined with the Lorenz curve crite­
rion of second-order relative inequality dominance as proposed by Atkinson
(1970).
Definition
Given any initial pre-tax and final post-tax distributions, Y and X' ∈RH++, a
tax system is redistributive (progressive) if and only if X' ≥L Y; that is, if and only
if X' weakly dominates Y:
X'≥L Y
⇔
k
'
x(i)
∑ µ(X')
i=1
≥
k
y(i)
∑ µ(Y)
,∀k = 1,...,H
i=1
where µ(X') and µ(Y) denote the mean of X' and Y, respectively. The terms x'(i)
and y(i) are the ith smallest elements of the corresponding distributions.
Making use of this definition we can state the following proposition, which is
a natural extension of JFK:
Proposition
Given any initial pre-tax and final post-tax distributions Y and X' ∈RH++, a
necessary and sufficient condition for a tax system to be non-negative redistri­
butive (progressive) is
0 ≤ ηx',y = ηy',y ηx',y' ≤ 1
That is, the local elasticity of x' with respect to y (local residual progression)
is always not greater than one (and not lower than zero). As can be noticed,
JFK's condition (0≤ηx',y'≤1) is a particular case when labour supply behaviour is
discarded, so that ηy',y=1, which implies ηx',y=ηx',y'. It is worth noting that η y ', y is
the key concept here, which captures labour supply behaviour changes.
This key concept η y' , y can be expressed in terms of wage-income elastici­
ties as:
η y',y =
2
η y',w
η y,w
Note that the initial post-tax income Y' corresponds to the actual tax base.
— 9 —
Since ηy',w =ηL',w +1, it can be also written as a function of wage-labour su­
pply elasticities:
ηy',y =
ηL',w + 1
ηL,w + 1
Note that these elasticities are expressed in terms of the exogenous pre-tax
wage rate as it identifies individuals.
This can be extended to characterise redistribution for two alternative tax
structures. Given any initial pre-tax income distribution Y, assume two alternati­
ve taxes T' and T''. These tax structures generate, respectively, two initial post­
tax distributions, Y' and Y'', and two final post-tax distributions, X' and X''. Then,
T' is more redistributive than T'' if and only if local residual progression from Y
to X' is always not greater than the one from Y to X'':
0 ≤ ηx',y ≤ ηx'',y
which is equivalent to:
0 ≤ ηy',yηx',y' ≤ ηy'',yηx'',y''
3. APPLICATIONS TO ALTERNATIVE TAXES AND LABOUR
3. SUPPLY SPECIFICATIONS
In the following, let us analyse the labour supply effects and the particular
value of ηy',y for different taxes under different labour supply specifications. In
general, we prove that the condition mentioned above for positive redistribu­
tion is difficult to satisfy, even in the homogeneous case. As an illustration of this,
firstly we analyse the case of the proportional tax, for which standard zero­
redistribution is only guaranteed under restricted conditions. Secondly, we stu­
dy linear progressive taxation applied to alternative labour supply specifications
in order to search for conditions that guarantee positive redistribution.
3.1. Proportional Tax Case
We see that under very restricted conditions a proportional tax achieves
non-negative redistribution with this tax behavioural framework.
Initial pre-tax income yi is:
w L(w , 0),
yi =  i i
wi L,
0 ≤ wi ≤ w
wi ≥ w
— 10 —
Instituto de Estudios Fiscales
Post-tax labour supply Li' is:
L'i

L(wi (1 − t), 0), 0 ≤ wi ≤ w
=
L,
wi ≥ w

Initial post-tax income yi' is:

w L(wi (1 − t), 0), 0 ≤ wi ≤ w
y'i =  i
wi L,
wi ≥ w

Final post-tax income xi' is:
x i'

wi (1− t)L(wi (1− t), 0), 0 ≤ wi ≤ w
=

wi ≥ w
wi (1− t)L
As ηx',y' = 1, the necessary and sufficient condition to guarantee redistribu­
tion neutrality is ηx',y = ηy',y = 1. Hence, ηL',w = ηL,w . This stringent condition is
satisfied in cases such as the log-linear labour supply specification and, when
mi=0, in the cases of the Cobb-Douglas and the linear labour supply func­
tions.
3.2. The linear tax under different labour supply specifications
Now we study the affine tax system as analysed by Atkinson and Stiglitz
(1980), Atkinson (1995) and Hall and Rabushka (1995):
T(y) = −Z + ty, Z ≥ 0 and1> t ≥ 0
This tax scheme will be examined below under three alternative labour su­
pply specifications: the CES, linear and log-linear.
3.2.1. CES
The CES utility function is defined as follows:
U(y,L) = y ρ + α(L − L)ρ
α > 0, ρ < 1
from which the pre-tax labour supply L can be recovered as:
L(wi, 0) =
L (wi / α)σ
wi + (wi / α)σ
,
wi ≥ 0
where L represents the maximum labour and σ = 1/(1-ρ) is the elasticity of
substitution.
— 11 —
Post-tax labour supply L' is:
L'(wi, 0) =
L (wi(1− t) / α)σ − Z
wi (1− t) + (wi (1− t) / α)σ
,
0 ≤ wr ≤ wi
where wr is the reservation wage.
It can be proved that for any α>0, σ>0 (or ρ<1), Z ≥ 0 and 1 > t > 0, and
wi ≥ wr,
ηL',w > ηL,w
So
ηy',w > ηy,w
Hence
η y',y > 1
Positive redistribution is not guaranteed, as ηx',y'<1. Note also that under the
proportional tax case, Z=0 and t>0 case, negative redistribution arises, unless
σ converges to zero, which is the Cobb-Douglas case.
3.2.2. The linear labour supply (Hausman, 1980, 1981)
Pre-tax labour supply L is:
aw , 0 ≤ w ≤ w
i
Li (wi, 0) =  i
wi ≥ w
L,
where a>0
Post-tax labour supply L' is:
aw (1− t) − bZ, 0 ≤ w ≤ w ≤ w'
r
i
L'i (wi, 0) =  i
wi ≥ w'
L,
where b>0 and
'
w < w' .
It can be proved that for any a, b>0, Z > 0 and 1 > t > 0, and w ≥ wi ≥ wr,
ηL',w > ηL,w = 1
So
η y',y > 1
Nevertheless, positive redistribution is produced -between wr and the lowest
of w and w*=2/b(1-t)- as:
ηx',y = ηx',y'ηy',y < 1
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Instituto de Estudios Fiscales
Note also that under the proportional tax case (Z=0 and t>0), the polar ca­
se of zero redistribution arises. In addition, in this case Z<0 implies negative
redistribution (between wr and the lowest of w and w*). In general, for mi=m,
the zero redistribution case is reached at Z=-m.
3.2.3. The iso-elastic labour supply specification (Burtless and Hausman 1978)
Pre-tax labour supply L is:
Awαmβ , 0 ≤ w ≤ w
i
i
i
Li (wi, mi ) = 
wi ≥ w
L,
where A, α>0 and β≤0 and mi>0.
Post-tax labour supply L' is:
A(w (1− t))α (m + Z)β , 0 ≤ w ≤ w'
i
i
L'i (wi, mi ) = 
i
wi ≥ w'
L,
'
where w < w' .
It can be proved that for any A,α>0, β≤0, Z >0 and 1 > t > 0, and w ≥ wi ≥ 0,
ηL',w = ηL,w = α
So
ηy',w = ηy,w
Hence
η y',y = 1
Positive redistribution is produced for Z>0 (between 0 and w ). Note also
that zero redistribution arises under the proportional tax case Z=0 and that
negative redistribution is induced under Z<0. Moreover, for any tax system
(with non-negative virtual incomes), η y',y = 1 so ηx',y = ηx',y' (between 0 and w )
which is the JFK result under no-tax behaviour.
4. CONCLUDING REMARKS
Making use of the concept of local residual progression, this paper decompo­
ses redistribution into two components:
— 13 —
(i)
the contribution to progression due to the impact on labour supply
behaviour induced by the tax change, captured by the transition from
the initial pre-tax to the final post-tax income distribution,
(ii) the contribution to progression as a consequence of the actual tax lia­
bility, quantified by the move from the initial to the final post-tax income distribution.
This decomposition allows a generalisation of the standard JFK conditions on
redistribution when labour supply reactions to taxes are taken into account. In
this richer framework, we also found that the labour supply specification is rele­
vant in evaluating redistribution of taxation. Further research may explore the
extension of the concept of redistribution to incorporate the notion of equality
of opportunities.
— 14 —
REFERENCES
ATKINSON, A. B. (1970): “On the measurement of inequality”, Journal of Econo­
mic Theory, 2, 244-263.
– (1995): Public Economics in action: Basic income-flat tax proposal, Clarendon
Press, Oxford.
ATKINSON, A. B., y STIGLITZ, J. E. (1980): Lectures on Public Economics, McGrawHill, Maidenhead, Berkshire, UK.
BURTLESS, G., y HAUSMAN, J. A. (1978): “The effect of taxation on labour supply:
Evaluating the Gary negative income tax experiment”, Journal of Political Eco­
nomy, 86, 1103-1130.
FELLMAN, J. (1976): “The effects of transformations on Lorenz curves, Econome­
trica, 44, 869-881.
HALL, R. E., y RABUSHKA, A. (1995): The Flat Tax, 2nd ed., Hoover Institution
Press, Stanford, CA.
HAUSMAN, J. A. (1980): “The effects on wages, taxes and fixed costs on wo­
men’s labour force participation”, Journal of Public Economics, 14, 161-194.
– (1981): “Labour supply”, in H. J. Aaron and J. A. Pechman (eds.), How taxes
affect economic behaviour, 27-72, Brookings, Washington, DC.
JAKOBSSON, U. (1976): “On the measurement of the degree of progression,
Journal of Public Economics, 5, 161-168.
KAKWANI, N. C. (1977): “Applications of Lorenz curves in economic analysis”,
Econometrica, 45, 719-727.
KEEN, M.; PAPAPANAGOS, H., y SHORROCKS, A. (1996): “Progressivity effects of
structural income tax reforms”, Studies in Economics, University of Kent at
Canterbury No. 96/12.
PRESTON, I. P. (1987): “The redistributive effect of progressive taxation, with
endogenous labor decisions”, M. Phil. Thesis, Oxford University, unpublished.
– (1989): “The redistributive effect of progressive taxation”, D. Phil. Thesis,
Oxford University, unpublished.
STERN, N. H. (1976): “On the specification of models of optimum income taxa­
tion”, Journal of Public Economics, 6, 123-162.
ZABALZA, A. (1983): “The CES utility function, non-linear budget constraints and
labour supply. Results on female participation and hours”, Economic Journal,
93, 312-330.
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19/01 Imposición marginal efectiva sobre el factor trabajo: Breve nota metodológica y
comparación internacional.
Autores: Desiderio Romero Jordán y José Félix Sanz Sanz
Páginas 40.
10/01 A non-parametric decomposition of redistribution into vertical and horizontal components.
Autores: Irene Perrote, Juan Gabriel Rodríguez y Rafael Salas.
Páginas 28.
11/01 Efectos sobre la renta disponible y el bienestar de la deducción por rentas ganadas en el IRPF.
Autora: Nuria Badenes Plá.
Páginas 28.
12/01 Seguros sanitarios y gasto público en España. Un modelo de microsimulación para las
políticas de gastos fiscales en sanidad.
Autora: Ángel López Nicolás.
Páginas 40.
13/01 A complete parametrical class of redistribution and progressivity measures
Autores: Isabel Rabadán y Rafael Salas.
Páginas 20.
14/01 La medición de la desigualdad económica.
Autor: Rafael Salas.
Páginas 40.
— 22 —
15/01 Crecimiento económico y dinámica de distribución de la renta en las regiones de la UE:
un análisis no paramétrico.
Autores: Julián Ramajo Hernández y María del Mar Salinas Jiménez.
Páginas 32.
16/01 La descentralización territorial de las prestaciones asistenciales: efectos osbre la igualdad.
Autores: Luis Ayala Cañón, Rosa Martínez López y Jesus Ruiz-Huerta.
Páginas 48.
17/01 Redistribution and labour supply.
Autores: Jorge Onrubia, Rafael Salas y José Félix Sanz.
Páginas 24.
— 23 —
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