THE STRUCTURE OF INCOME TAXATION
THE STRUCTURE OF INCOME TAXATION
1.
INTRODUCTION
The
implications of the equal marginal sacrifice doctrine may be seen from the
following simple model. Suppose that individuals differ in their earning
ability, denoted by w. The before-tax earnings of a person of type w are
denoted by Z (w) and the tax paid, by T (w). The utility derived from after-tax
income is given by Uw(Z(w) – T(w)), so that if F(w) is the
cumulative distribution of the people of type w, the integral of individual
utilities is denoted by
(13-1)
The
government determines the taxes paid (where T(w) may be positive or negative)
so as to maximize total utility subject to raising the required revenue R0
:
(13-2)
If the marginal utility of income
schedule is assumed to be identical for everyone, the tax structure is such
that after-tax incomes are equalized, “A system of equimarginal sacrifice fully
carried out would involve lopping off the tops of all incomes above [a certain]
income and leaving everybody, after taxation, with equal incomes’ (Pigou, 1947,
pp. 57-8). Possibly because of its radical implications, the least sacrifice
theory came under a great deal of attack. Three main lines of criticism may be
distinguished :
1.that the minimim sacrifice theory
takes no account of the possible disincetive effect of taxation (that Z(w) may
be influenced by the tax structure) ;
2.that the underlying utilitarian
framework is inadequate ;
that account must be taken of the
restrictions on the types of taxes that may be levied. The first of these was
clearly recognized by those writing in the utilitarian tradition.
SIMPLE
MODEL
The
Model
The individual’s earnings are
assumed to depend only on earnings ability (w) and on the number of years of
education recevied (D); i. e., hours of work (effort) are assumed to be fixed.
While undergoing education the individual has zero earnings. At work he earns a
constant amout Z(w,D) and he retires income, discounted at interest rate r back to the start of his aducation.
Since Z is
constant over time, this is proportional to
The
tax schedule is assumed to be linear, so that, as in earlier lectures, there is
a guaranteed income G and a constant marginal tax rate, denoted by t :
If we assume that there is an equal
number of people, with the same distribution of w, in eachage group, ans that
revenue R0 is required per cohort, the government budget constraint
is
Choice
Open to Government
If we now consider the implications
of individual behaviour for the government’s choice of t dan G, we can see that
the revenue constraint may take one of two forms. In the first (case A), the
level of w0 is set such that all individuals have D>0 (i.e.
w>w0). We can then subtitute from (13-8a) into the revenue
constraint. If we normalize such that
,
then
Writinr
for the mean value of w, and re-arranging,
In
the second case (B), some individuals choose D=0 (w≤w0). The revenue
constraint is then :
. Introducing te notation for the incomplete
mean,
This expression may be rewritten as :
(13-14)
From (13-9) we can see that, where
the individual has D > 0, he would rank combinations of G and t according to : 3
(13-16)
This
gives indifference curve of the type illustrated in Fig. 13-1. In particular,
the slope of the curve
=
constant is given by
(13-17)
The preferred tax is that:
G
Indifferencecurve of individual with
Revenue constrait
0
1
-R0
Where the difference curve is tangent to the constraint ;
i.e., from (13-15) and (13-17)
(13-18)
Social
Welfare Maximization
For ease of exposition, we take the
isoelastic form :
.
.
In the Pareto case
LINEAR INCOME TAX
This means that we can examine the
degree of progression in terms of the behaviour of the averge rate tax
implies that the tax is progressive in this
sense), but that we can throw no light on the way in which the marginal rate
should vary with income.
The
Government’s Problem
The individual maximizes utility
subject to
since
pre-tax income Z=wL where w is the wage rate. The first-order
conditions give
(13-26a) or
In the population as a whole,
individuals are assumed to be identical in all respects except their wage rate
(earning ability). (The implications of other differences are discussed below).
It can then be shown that there is a critical wage w0 such that:
If
on the production side we assumed constant producer prices and no profits, and
if the revenue requirement is R0,
then the production constraint is
We
normalize again by setting
= 1 ; and, using the individual budget
constraint (13-25), this may be rewritten as a revenue constraint
.
The government is assumed to maximize the social welfare function :
where
different assumptions about
yield, for example, the utilitarian
and Rawlsian abjectives. Forming the
Lagrangean :
we
may derive the first-order conditions with respect to G and t (Sheshinski, 1972) :
where
it should be noted that L = 0 and
.
The Optimum Linear Income Tax
The problem is parallel in several
respects to the Ramsey tax model. In particular, there is no reason to expect
the problem to be well-behaved, and we have to be careful in employing the
first-order conditions. Where more than one solution exist, a global comparison
must be made fo the levels of welfare. Bearing these qualifications in mind, we
may examine the implicatons. Our earlier experience with the Ramsey problem
suggests that it may be illuminating to use the Slutsky relationship
where
SLL is the subtitutions
term (compensated response of labour to the marginal net wage) and is
non-negative. Using this, and the fact that
,
where a is the private marginal
utility of income, we can rearrange :
If
we now define, as in the previous Lecture, the net social marginalvaluation of
income,
If we assume that the distribution
has strictly positive density at all non-negative w (so that there are always some individuals not working), then the
problem reduces
The
first-order condition with respect to t yields
by
nothing that
for which
).
Using the government revenue constraint.
. In
the Cobb-Douglas case,
, Hence :
(13-43)
numerical
calculations
the model
just described is no more than illustrative. among other things, the
cobb-douglas utility function may give a misleading impression of the
elasticity of labour suplly. this aspect has been investigated by stern (1976),
Stern
formulates the problem in terms of the goverment choosing G and t maximize
With y=0, we
have the benthamite utilitarian objective. For heiger values of y the function
is more concave, and the optimum tax rate rises markedly for all values of
. for
. The rawlsian case, the tax rates are all excess of
2. GENERAL INCOME TAX
The
assumption of a linear tax schedule has precluded any discussion of whether it
is desirable for marginal tax rates to rise or to fall whith income. We turn
now to this question, considering a general income tax schedule T(Z). The
results at a general level are rather limited ( although they yield interesting
counter – examples to certain beliefs), and in the last part of the section we
descibe numerical results obtained in the special cobb-douglass case.
Geometric
Exposition
which we now
modify by introducing-net income households with discretely differnt wage rates
( in increasing
order), and by measuring along the horizontal axis gross eaarnings Z=wL ( for
any households this is proportional to L), rather tahn labour ( as in mirrlees.
1977). To set the scence, let us suppose that the tax schedule has been fixed
up to the point P ( gross income
) chosen by households (i-1). And that we are deciding
how to extend it beyond P. The indifference curve for (i-1) is shown( for P to
be chosen by him, the schedule at heigher.
Indifferent Curve for i
Indifferent
Revenue
curve for (i-1)
Q
Z
Gross
income
(a)
Indifferent
curve for (i-1)
Gross income
(b)
13-2 determination of optimum tax rates
Heuristic
Solution
The basic
formulation is parallel to that iin earlier sections. A person of type w
maximize
where Y(w)=Z-T(Z) and Z = wL(w). There is again
a value
such that
. Where it is
assumed that the tax function is differentiable and that
(the government
chooses the function T(Z) to maximize
the social welfare function ( 13-30) subject to production constraint (
13-28);i.e., we have retained the simplifying assumptions of constant producer
prices and no profits. We introduce the multiplier
associated with
the differential equation (13-48) and writing the hamiltonian as (where f is the density of the distribution of w)
Where
satisfies
The last
term in this expression is obtained differentiating dU/dw with respect to U,
golding L constant but allowing for
the dependences of Y on U. The control variable is then chosen
to maximize
, and the first-order condition is
Where this
is only necessary where Z is strictly
increasing this expression may be –re arranged as
In order to
simplify the ensuing discussion, we assume thatt
i.e.p, with
this representation
is indepedent
of Y. This allows us to integrate (13-50) to obtain
Where we
have used the transversality consition that
=0.
The
key conditions(13-52) and (13-53) may be combined to yield (where we have
exploited the fact that
in
differentiating within the square bracket on the right-hand sideof the former
and the individual first-order condition has again been used)
Where
In the
rawlsian case (with
, the government maximizes U (
) and
for w
, so that the integral declines throughout the range (
broken lines in fig. 13-3). In this case, we are interested solely, as far as
the heiger groups are concerned, in the revenue raised-we have the situation
discussed diagrammatically. The earlier exposition suggested that these
considerations need to be balanced against the effect on the labour supplied by
those at w. This letter effect is
represented by
. From this we can see that the effect depends on:
1.
, which is a measure of the elasticity of labour
supply. It is not however the conventional defenition of the elasticity,
-1 being in fact the response of L to a change in w, holding
constant. 16
the value of
is Equal to 1
if there is a constant marginal disutility of effeort, and
rises as the
supply becomes less elatic.
2.
The number of work units affected (wf). If the potential labour affected is
low, then the marginal tax rate can be heigh. This suggets that the rate would
be heigh at the lower and ( to a lesser extent) the upper tails of the
distribution.
Numerical
Calculations
The most important for policy purpose include:
1.
The optimal tax structure is appromasimately linear ,
i.e. a constant marginal tax rate , with an exemption level below which
negative tax supplements are payable:
2.
The marginal tax rates are rather low (“I must confess
that I had expected the rigorous analysis of income – taxation in the
utilitarian manner to provide arguments for high tax rates. It has not done
so”. (Mirlees, 1971, p.207));
3.
“ The income tax is a much less effective tool for
reducing inequalities than has ofen been thought” ( Mirlles, 1971,p. 208)
In case i
the revenue requirement is positive, in case II it is negative (the government
sposing of the profits of public sector production).
Indeference
the level of utility for a person with ability w is given by
The
government chooses T(w) subject to the revenue constaint. Since T(w) can be
varied independently, the first –order condition for the utilitarian objective
is that
=
where
is the multiplier associated with the revenue
constraint. Since
=-1/Y, it follows that incomes are equalized , but not
utilities:
for
CONCLUDING
COMMENTS
If the
social wealfare funcction takes account of the posibility that some individuals
are more “productive” in consumption, then tthe results may be quite different.
There are a number of developments that remain to be carried out. We have
referred in passing to uncertainty
surrounding the elasticity of labour supply , and to the interpretation of F as a probability distribution , but
this needs to be built in explicitly. Finally, administrative costs are of
crucial importance when considering alternative schedules – and different tax
bases-and must be incorporated. deciding on the direction of such research ,
one must bear firmly in.
FOURTEEN
A MORE
GENERAL TREATMENT OF THE OPTIMAL TAX PROBLEM
1. A More General Specification of the
Tax Structure
for the number of hours of labour supplied by
individual h ( we assume for simplicy that he has only one job),
for his wage rate ,
for his consumption vector (of n goods), and
for a vector of observable characteristicd,
such as age, or number of children, which may be relevant for tax purposes.
Where, as assumed below, only
is
observed, then those variables must enter in this form.
Direct Versus Indirect Taxation
If
we turn to definitions based on the
economics effects of different types of taxation,that most commonly found in
the public finance literature is based on same presumption about the ultimate
incidate of the tax.
ARGUMENTS FOR DIRECT OR INDIRECT
TAXATION
One
can,in the literature,distinguish two main lines of argument.The first is that
there should be a broad balance between direct and indirect taxation,a view
that appears to find favour with many politicians,as is illustrated by the
following piece of glandstonian rhetoric. His reason for holding such a view
are not apparent , but in much popular discussion one can discern a form of
assignment of instruments to largets. Directtaxation is assigned to the equity
objective, and indirect taxation is assignedto the goal of raising
revenueefficiently.
INDIRECT
TAXES AND LINEAR DIRECT TAXATION
In
this section we examine the role of
commodity taxation in a model where there is an (optimal) linear direct
tax.We use the framework of thenprevious lecture,letting G be the uniform
lump-sum subsidy and
the consumer price of good i.Normalizing all
producer prices to unity.We have
.
As explained earlier,a uniform commodity tax is equivalent in this model to a
tax on wage income.We therefore normalize by setting the tax rate on wages to
be zero.This means that,if it turns out
that the optimal tax structure involves
for all i,then we have established that no
commodity taxes need be employed (the optimum can be achieved by a tax on wage
income alone).
IDENTICAL INDIVIDUALS
This
straightforeard, and intuitive, observation has two important implications.
First, it casts doubt on the standard formulation of the Ramsey problem with
identical individuals. In effect this problem arises only because the
constraint of no poll taxation is imposed, and there seems no clear rationale
for ruling out this simple lump-sum tax.
Distributional
Objectives
The
distributional characteristic
depens on how the net social marginal
valution of income
changes with w. In general there is a
presumption that it is a declining fuction, and hence that the compensated
reduction in demand (onthe samuelson interpretation) is greater for goods whose
consumption increases more with w. (note that it is variation with w, and not
M, that is relevant, and that the changes in
may reflect substitutability for leisure or,
in a more general model, variation in tastes.) where there are no income
effects, the behavior of
depends solely on the social marginal utility
of income. But where
≠ 0, it is possible (as noted in Lecture
13) that
rises with w, if the pattern of demand is
such taht it is goods with a high income elasticity that are consumed by the
well off, then it might turn out that it is goods consumed by people with a low
wage rate thet tend to be taxed more ( in the sense of having a larger
reduction in compensated demand). The important point brought out by this is
that the relevant characteristic is not the social marginal utility of income (
), but the net social marginal valuation
(
), which allows for the effect on revenue
Optimal
Tax Stucture and the Properties of Demand Fuctions
The
optimal tax structure is then tax rate is equal to the variance of the social
marginal valuation of income ( it is assumed that the variation in w is not
such that some demand are zero). This special case is not in itself of great
interest:it does however bring out the fact that the case for differential
taxation depends on second-and higher-order derivatives of the demand functions
(and on the higher moments of the distribution). The functional forms typically
assumed (such as costant elasticity) impose strong restrictions on these
derivatives, but we can have little confidence that in an unconstrained
estimation procedure the available data would allow us to determine them with
great drecision.
NONLINEAR
TAX SCHEDULES AND TAX EXEMPTIONS
The
direct tax considered in the previuos section has a particularly simple form.
Although there may be strong administrative reasons for governments resricting
themselves to a linear tax schedule, we should also consider the more general
case of a schedule with variable
marginal rates of tax. In this section, we examine the choice between direct
and indirect taxes in this context, and discuss a further variation in the
possible tax instruments.
Implications For Direct And Indirect
Taxation
terms of our earlier discussion of
differing views about the direct-indirect tax problem ,the weak separability
result provides some limited support for she second broad view-that direct
taxes are superior on both efficiencyand equity counts,not just in the case of
the linear expenditure systems,there is no need to employ terentiated indirect
taxation to achieve an optimum.this does not quire separabality between
goods,just weak separability between labour all goods.at the same time,it does
not provide a blanket justification or the view that direct taxes are
superior,and it is quite possible that this parability requirement may not in
practice be met,for exalmple,in the of leisure goods.for this reason no such
catagorical assertion can be as that by from and taubman quoted earlier.
Nonlinear
Indirect Taxes and Tax Deductions
The
distributional characteristic
depens on how the net social marginal
valution of income
changes with w. In general there is a
presumption that it is a declining fuction, and hence that the compensated
reduction in demand (onthe samuelson interpretation) is greater for goods whose
consumption increases more with w. (note that it is variation with w, and not
M, that is relevant, and that the changes in
may reflect substitutability for leisure or,
in a more general model, variation in tastes.) where there are no income
effects, the behavior of
depends solely on the social marginal utility
of income. But where
≠ 0, it is possible (as noted in Lecture
13) that
rises with w, if the pattern of demand is
such taht it is goods with a high income elasticity that are consumed by the
well off, then it might turn out that it is goods consumed by people with a low
wage rate thet tend to be taxed more ( in the sense of having a larger
reduction in compensated demand). The important point brought out by this is
that the relevant characteristic is not the social marginal utility of income (
), but the net social marginal valuation
(
), which allows for the effect on
revenue.
Optimal
Tax Stucture and the Properties of Demand Fuctions
The
optimal tax structure is then tax rate is equal to the variance of the social
marginal valuation of income ( it is assumed that the variation in w is not
such that some demand are zero). This special case is not in itself of great
interest:it does however bring out the fact that the case for differential
taxation depends on second-and higher-order derivatives of the demand functions
(and on the higher moments of the distribution).
NONLINEAR
TAX SCHEDULES AND TAX EXEMPTIONS
The
direct tax considered in the previuos section has a particularly simple form.
Although there may be strong administrative reasons for governments resricting
themselves to a linear tax schedule, we should also consider the more general
case of a schedule with variable
marginal rates of tax.
Implications For Direct And Indirect
Taxation
terms of our earlier discussion of
differing views about the direct-indirect tax problem ,the weak separability
result provides some limited support for she second broad view-that direct
taxes are superior on both efficiencyand equity counts,not just in the case of
the linear expenditure systems,there is no need to employ terentiated indirect
taxation to achieve an optimum.
Optimal
Tax Stucture and the Properties of Demand Fuctions
The
functional forms typically assumed (such as costant elasticity) impose strong
restrictions on these derivatives, but we can have little confidence that in an
unconstrained estimation procedure the available data would allow us to
determine them with great drecision. This suggests that considerable
circumspection is necessary in applying the theoritical results: it is likely
that empirically calculated tax notas, based on econometric estimates of
parameters, will be determined in structure, not by the measurements actually
made, but by arbitratary, antested (and even unconscious) hypotheses chosen by
the econometrician or practical convenience
NONLINEAR
TAX SCHEDULES AND TAX EXEMPTIONS
The
direct tax considered in the previuos section has a particularly simple form.
Although there may be strong administrative reasons for governments resricting
themselves to a linear tax schedule, we should also consider the more general
case of a schedule with variable
marginal rates of tax. in practice howefer we are constrained to employ income
taxation,and the problem becomes a second-best one.as a result,the
optimuminvolves in general a wedge between before-and after-tax returns to
labour at the margin,but this leaves open the question whether we want to have
the fist-best conditions elsewhere,as it
was put by davis and whinston.
FAXATION
OF SAVINGS
Here
we simply assume for the purposes of argument that, ignoring differences in
tastes, the government is concerned with individual lifetime welfare derived
from consumption and leisure. This in itself creates-at least in a perfect
capital market-a presumption in favour of consumption as the appropriate base,
and hence the total exemption of savings. However, this presumption needs to be
qualified by the efficiency and redistributive considerations, with which we
are here concerned. A number of authors have suggested that the optimal tax
results derived for a timeless economy may be applied directly to the tax
treatment of savings. Indeed, the original Ramsey article included a brief
section on savings.
Since
the static general equilibrium model used in earlier sections can ,given
a straightforward intertemporal interpretation, with commodities being
distinguished according to their dates, it may seem that the previous result
can be applied directly in this way.
This does not however take count of the
possibility that the government may wish to intervene to range the allocation
of consumption over time-to achieve a socially resired intertemporal
distribution.
Referensi from: GURITNO MANGKOESOEBROTO
Income
Tax
Income
tax can be classified into two categories, namely personal income tax and
corporation taxI (Tax Agency, which is the subject of the income tax of a
corporation). Although administratively both types of tax are classified in
direct tax that is not meant to shifted to the other party, but in reality the
tax may be shifted to another party by the payertax.
Individual Income Tax
Personal
income tax imposed on any person who earn above the taxable income in a given
period. To simplify the analysis we assume that the wage received by an
individual depends on the number of hours the person.
Corporate Income Tax
Agency
charged income tax on profits derived by an entity in a given period, and for
the next tax will be referred to corporation tax or income tax liability.
Whether or not the corporation tax shifted to the consumer depends on the
structure of the market and motivated entrepreneurs.
Income tax
Above
has been stated that the income tax with a tax on capital and labor with the
same rate. Because diasumsikanbahwa supply (supply) factors of production does
not change, then the income tax entirely would be borne by the owner because of
factors of production can not be shifted to the other.
Tax on Employee Wages
Tax
on income subject to the payroll of employees in sector both X and Y in the
sector, so there is no desire of employees to avoid tax by working another
sector. In addition it is assumed that the supply of factors of production
(including labor) does not change, then the tax on wages also be shifted to the
consumer and to the owners of other factor produtions (K).
Structure tax
Economists
and politicians like the progressive tax structure because such a tax structure
that led to the distribution of income (taxable after) become more prevalent.
However, economic theory can not express how well the progression of a tax. In
addition, economists realized that the tax is too progressive will cause bad
influence terhdap investment, desire to work, efficiency, and formation
capital.
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