Now contrast an income tax, at rate xy, with a consumption tax imposed at rate Xc. Under the income tax, individual m consumes an amount (l-xy)(l-s)wlin and i ...
John Kay
Consumption and Income Taxation: Horizontal Equity and Life Cycle Issues· I. Introduction
"What reason is there, that he which laboureth much, and sparing the fruits of his labour, consumeth little, should be more charged, than he that liveth idlely, getteth little, and spendeth all he gets: Seeing the one hath no more protection from the commonwealth than the other?" Hobbes, Leviathan, Ch. XXX
H Hobbes were alive today, he would be required to express himself somewhat differently. Consider two individuals, one of whom laboureth much, supplying labour 1m, while the other, living idlely, supplies Ii. Both face the same wage, w, while m but not i spares a fraction s of the fruits of his labour. Now contrast an income tax, at rate xy, with a consumption tax imposed at rate Xc. Under the income tax, individual m consumes an amount (l-xy)(l-s)wlin and i an amount (l-xy)wlj. H the two individuals have identical utility functions in consumption and labour u( c,l) with Uc > 0, UI
S
0, they enjoy respective utilities
• I am grateful to Syed Ahsan and Wilhelm Pfahler for discussion of an earlier version. M. Rose (ed.), Heidelberg Congress on Taxing Consumption © Springer-Verlag Berlin · Heidelberg 1990
86
JobnKay
m: u( (l-xy) (l-s)wlm ,1m)
i: u( (l-xy )wli ,Ii)
and pay respective taxes
Since 1m > lj, m makes a greater tax payment although if, for example, s > 1 then i enjoys higher utility.
+, m
Under a consumption tax at rate c, however, utilities achieved by the two individuals are now (l-~
m: u(
1 +
)wlm
wli ,1m)
i: u ( - - , l d 1+xc
Xc
and respective tax payments Xc
Xc
m: - - ( 1- s )wlm
1+xc If ul
= 0, then um >
i: - - wli
1+xc
Uj
if tax payments by m are greater than tax payments by i, and the
inequities of the income tax are avoided.
This formulation is at once longer and less elegant than that of Hobbes, but it does have the advantage of focussing attention immediately on a number of issues which have been central to the subsequent literature on equity questions in the choice between income and consumption taxes. First, note that Hobbes anticipates not only a number of fiscal policy arguments, but also the Keynesian consumption function. He which laboureth much spareth the fruits of his labour, while he that liveth idly chooses, or is compelled to, spend all he gets. Thus the average propensity to consume diminishes with income, at least in crosssection data. The distribution of income is more unequal than the distribution of consumption: correspondingly, the distribution of tax payments based on income is more egalitarian than the distribution of tax payments based on consumption, if the two tax schedules are the same.
Consumption and Income Tsxation: Horizontal Equity and life Cyr:le Issues
87
Next, observe that, as presented, sparing the fruits of one's labour is simply irrational. He which laboureth much would enjoy unambiguously higher utility by ceasing to spare the fruits of his labour. This problem can be dealt with in one of two ways. We might assume that the spared fruits of labour generate utility just as do the consumed fruits. Thus the utility function becomes u(c,s,l) with Us > 0 and in this case respective utilities under the income tax may be written m: u( (l-xy) (l-s )wIm • (l-xy) sw1m .1m)
i: u((I-xy )w1 i .O.1 i ) with tax payments as before. Although the inequity which was the subject of earlier concern is not now impossible, the case in which the arguments of m's utility function are simply dominated by those of i can now be excluded. This "spot" view of intertemporal choice - in which current utility depends on potential as well as consumption in the current period as well as actual consumption in that period - underlies much discussion of the comprehensive income tax. An alternative means of resolution is to observe that m may well be sparing the fruits of his labour with a view to enjoying retirement in which he spares also the labour itself. Here consumption is C! in period 1 and C2 in period 2 with utility function u(c1.c2,1). Fruit spared in period 1 is enhanced at a rate r by period 2 but this enhancement is, of course, subject to income tax. Now we have under the income tax respective utilities of m: u( (l-xy ) (1- s )w1m • (1+ (1-xy ) r) s (l-xy )w1m .1m )
i: u((I-xy )w1 i .O.1 i ) and tax payments
while under the consumption tax utilities are
m:
(l-s)wIm u(----
(1+r) swIm 1m) (l+xc )
i:
W1i u ( - - . O. 1i) l+xc
88
John Kay
and tax payments
Xc
m: ---l+xc
Xc
wlm (l+rs)
We may note here that if Xy =
i: ---l+xc
W1i
t:Xc ' then the tax payments by m and i are identical
under both income and consumption taxes. Since 1m >
~,
individual m pays more in
both cases. In the life cycle model, the potential inequity is that he which laboureth much pays more tax than he that liveth idly, and this is true whether he which laboureth much chooses to spare the fruits of his labour or not. Whether there is indeed inequity may depend on the role played by Ii and 1m. If Ul
=
0, then m enjoys unambiguously higher utility than i, and it is difficult to disagree
with an outcome in which this is reflected in higher tax payments. Our concern is that labour involves disutility and hence that i may achieve higher utility levels. The key problem is our inability to tax leisure and that is common to both income and consumption taxes. But this should alert us to a broader problem in the treatment of equity issues. If m and i indeed have identical utility functions, why does one labour much while the other liveth idlely? If, on the other hand, they are different, what is the basis of the comparisons under which horizontal equity - the equal treatment of similar individuals - is to be assesed? Hobbes' model therefore introduces us to the principal issues of equity in the choice between income and consumption taxes. Is equity to be judged from a "spot" or "life cycle" perspective? Do we rank individuals by reference to the consumption possibilities available to them over their lifetime, or over some shorter interval? How should an equitable tax system respond to differences between individuals in the tradeoff between goods and leisure? How, if at all, should the principle of horizontal equity be interpreted? And what are the distributional implications of consumption or income based taxation? Subsequent sections of this paper consider these issues. Sections II and ill develop a framework for analysing equity issues over the life cycle. Section IV explores the implications for horizontal equity.
Consumption and Income Taxation: Horizontal Equity and Life Cycle Issues
89
II. Income and Consumption Taxes in a Simple Life Cycle Model The life cycle framework is a useful means of exploring the choice between income and consumption taxation. Auerbach and Kotlikoff (1987) and Zodrow (1989) have developed simulation models by employing specific assumptions about income patterns and utility functions. The approach adopted here is closer to that of Bradford (1980), Graetz (1980) and Bradford (1986) who consider the relations between alternative tax structures under general assumptions about earnings profiles and preferences. Although their arguments are now well enough known to be described by Zodrow as "the standard paradigm", no rigorous or comprehensive explanation of these relationships appears to exist. In this section I seek to develop a framework for that purpose. Over his lifetime (O,L) an individual receives labour income wet) and spends at a rate z(t). He accumulates assets A(t) and, in the absence of gifts and bequests, A(O) A(L)
=
= o. In any period, incomings are therefore (w+rA) and outgoings Z, so that in
the absence of taxes
(1)
A = w+rA-z
and hence
f L
(2)
A(t)
J
t
(z_w)e-r(y-t)dy
t
(w_z)er(t-Y)dy •
0
Proposition 1 In the absence of gifts and bequests, and given a fixed and certain return on capital, a consumption tax Xc and a tax on labour income XL are identical if XL = Xc / (1 + Xc).
With a tax on labour income at rate XL, (1) becomes
(3)
A=
(l-xL)w+rA-z
and with a tax on consumption at rate Xc, (1) becomes
90
John Ray
(4)
A
= w+rA-(I+xc)z .
With a tax on labour income, a consumption plan z(t) is feasible if
I
{(I-XL)w+rA-z-A}e-rtdt
I
(I-XL )we-rtdt
L
(5)
~
0
~
0
o
ie. if L
(6)
o
since
A(O) ... A(L)
=
0
Under a consumption tax z(t) is feasible if
I
(w+rA-(I+xc)z-A)e-rtdt
I
I
L
(7)
o
as above if
L
(8)
L
we-rtdt
o
~
(I+xc)z(t)e-rtdt .
0
Thus the sets of feasible consumption plans are identical under the two taxes if and only
if (1-xL)
= (1 : Xc)
so that XL
=~
as in section I.
Consumption and Income Taxation: Horizontal Equity and life Cycle Issues
91
Hence if this condition holds, the same consumption plan z(t) will be chosen under both tax regimes. Moreover, under the labour income tax the present value of revenue is
while under the consumption tax it is
J L
(w+rA-A)e-rtdt from (4)
(9)
o
(10)
and hence the two taxes yield identical revenue. With a comprehensive income tax at rate:Ky, (1) becomes
(11)
A=
(l-xy )(w + rA)-z .
Consider first an individual with a fixed pattern of lifetime labour income w(t). Then under the labour income tax the present value of lifetime tax payment is
J L
Xa
XLwe- rt dt
o
and under the comprehensive income tax
92
JobnKay
L
I
X =
xy(w+rA)e-rtdt
o
Proposition 2 follows immediately.
Proposition 2
A comprehensive income tax raises more or less revenue than a labour income tax at the same rate, or an equivalent rate consumption tax, as the average discounted value of lifetime assets
The value of
S~
Ae - r t dt is greater or less than zero.
S~
Ae - r t dt reflects, in a loose sense, how late in a lifetime an individual
chooses to consume. We might seek to make this more precise. Figure 1 illustrates two alternative consumption paths, each with the same present value.
Zl
is, in an obvious sense, later than
Z2
and this is reflected in Figure 2, which
compares cumulative values. The analogy with Lorenz dominance is obvious [see Atkinson (1970)]. The relationships in Figure 1 and Figure 2 are straightforward but, as Figures 3 and 4 show, the dominance pattern shown in F'IgUfe 2 is consistent even with a much more ambiguous pattern than that displayed in Figure 1. Thus a consumption plan
Z
is unambiguously later over the lifetime than an
alternative Z2 with the same present value if L
I
zle+r(T-t)dt S
V
T
E (O,L)
•
o We are also, however, interested in comparing profiles with different present values. Consider the patterns Z2' and Z2" in Figure 5. z' is, we might suggest, later than Z2' (although it is mainly smaller than Z2') and it might also be seen as later than Z2". If we adopt this definition then we have a more general concept.
93
Consumption BIld Income TllXJltion: Horizontal Equity lind life CYCle Issues
Definition 1
A consumption planZl(t) is later than an alternative Z2(t) if
J T
J T
zle+r(T-t)dt
~
o
z2
er (T-T)dtforsome
To
and V
T
~
To·
o
We can now look at the effect of earlier or later consumption plans on tax liabilities and on the value of lifetime consumption.
Proposition 3
For a fixed time pattern of labour income w, an individual with a later consumption plan pays more tax under a comprehensive income tax, and enjoys a lower present value of lifetime consumption.
With a comprehensive income tax, we have (12)
A ..
(l-xy) (w+rA)-z
and with fixed w (13)
AI-A2 = (1-xy )r(A1 -A2 }-(Zl-Z2)
•
Hence:
(14)
(l-xy}r
J
(Al-A, )e-rtdt
Tl
and so
94
JobnKJJy
(15)
[(AI-A2
I
T2
)e- r t ] T2 Tl
- -Xyr
-I
T2
(AI-A2)e- r t dt
Tl
(Zl-Z2)e- r t dt •
Tl
We show first that the single crossing case (as Z2" in FJgUI'e 5) cannot arise where w is the same on both consumption paths. Suppose there is some ~ such that
o. o Then by setting T2 =Tand Tl =0 in (15). we have T
and also by setting T2 = L and T1 = T
(17)
'"T Hence
I L
(AI-A2)e- r t dt S 0
o
Now set T2 = Land Tl = 0 in (15) and find
95
Consumption and Income Taxation: Horizontal Equity and life Cyr:Je Issues
J L
xyr
(18)
J
T2
(A 1 -A2 )e- r t dt =
o
(Z2- Z1)e- r t dt
Tl
which makes
(19)
Thus we need only consider cases like Z2 in Figure 5 where
J T
(zl-z2)e- rt dt < 0 V T in (O,L) .
o
Then set T 1 = 0 and T2 = L in (15) and it is immediate that Z2' involves a higher present value of lifetime consumption and a lower present value of lifetime tax payment. Now relax the assumption of fixed w and consider individuals who differ in their lifetime earnings profile. We can define early and late earnings patterns in essentially the same way as we have defined early and late consumption patterns. Thus: Definition 2 An earnings profile Wl(t) is later than an alternative W2(t) if
J T
o
J
T
wle-r(T-t)dt
~
w2 e - r (T-t)dt for some
T
and V
T
