Progressive Taxation and the Equal Sacrifice Principle

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

Progressive Taxatipn and the Equal Sacrifice Principle

HP. Young

June 1986 CP-86-18

Collaborative P a p e r s r e p o r t work which has not been performed solely at t h e International Institute f o r Applied Systems Analysis and which h a s received only limited review. Views or opinions expressed herein do not necessarily represent those of t h e Institute, its National Member Organizations, or o t h e r organizations supporting the work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

Foreword

A traditional justification f o r progressive taxation is that it imposes equal sa- crifice on all taxpayers in loss of utility. Nevertheless, many plausible utility functions yield strictly regressive taxes when all taxpayers sacrifice equally. If tax rates are required t o be progressive, nonnegative, and independent of scale, then t h e r e is a unique family of positive, increasing, continuous utility functions that is consistent with equal rate of sacrifice. They determine a unique family of tax schedules that seem not to have been studied before except in isolated cases.

Alexander B. Kurzhanski Chairman

System and Decision Sciences Program

-

iii

-

Progressive

Taxation

and t@e Equal Sacrifice Principle

HP. Young

School of Public Affairs University of Maryland College P a r k , Md. 20742

1. Introduction

'? have not been able t o discover who w a s t h e first t o proclaim t h e thesis that because t h e degree of utility of income decreases when income increases, it follows that equality of sacrifice entails progressive taxation". ([2], p. 48). So begins a learned paper by A.J. Cohen Stuart (1889) on t h e utilitarian foundations of pro- gressive taxation. Cohen Stuart points out that 'equal sacrifice' may be interpret- ed in two ways. Equal absolute sacrifice means that in paying taxes everyone gives up t h e same amount of utility relative t o his initial position. Equal r a t e of sa- crifice means that everyone gives up the s a m e percentage in utility. Assuming t h a t t h e marginal utility of income f a l l s as income rises, it is clear that the r i c h e r must pay more in tax than the poorer if all are t o sacrifice equally by either criterion.

It is not s o clear, however, that the r i c h e r must pay a higher percentage of their incomes in taxes t o achieve equality of sacrifice. In fact, f o r many plausible utili- t y functions this is not t h e case. By itself, then, t h e equal sacrifice doctrine does not imply progressive taxation.

Instead of attempting to justify progressive taxes by equal sacrifice, w e may turn t h e argument around: if a progressive (or flat) tax is deemed t o be the ap- propriate outcome of an equal sacrifice model, what implications does this have f o r t h e form of the utility function? More generally, might it be possible to deduae pertinent infomation about t h e utility function by examining t h e properties of the tax schedules that result from t h e equal sacrifice principle? The answer is affir- mative. We shall show t h a t t h r e e simple properties of the tax system-

*

This work was supported by the National Science Foundation under Grant SES-8319530 a t the University of Maryland.

nonnegativity, progressivity, and scale-invariance-are enough to completely determine t h e form of t h e utility functions, whether equal absolute o r equal rate of sacrifice is t h e criterion. In both cases, the implied family of tax schedules is t h e same.

The study of formal properties of tax schedules h a s a long and distinguished tradition going back t o t h e work of Cohen S t u a r t [2], Edgeworth [3]. and o t h e r s around t h e t u r n of t h e last century. This axiomatic point of view has recently been revived in t h e work of W.F. Richter [9],

[lo],

Eichhorn and Funke [4], Eichhorn, Funke and Richter [5], and Pfaehler [?I, among others. It differs substantially from t h e optimal tax l i t e r a t u r e (e.g., Mirrlees [ 6 ] ) in t h a t tax systems a r e derived from elementary principles of equity and consistency, r a t h e r than from c r i t e r i a of economic optimality

.

2. The equal sncrifice principle

Let z r e p r e s e n t an individual's taxable income, and u (z) t h e utility of income level z. For purposes of this discussion, every individual is assumed t o have t h e s a m e utility function f o r income. W e make t h e following regularity assumptions:

u (z) is defined f o r all z

>

0. u (z) may o r may not be bounded above o r below.

A t a z schedule is a function t = J (z) defined f o r all z

>

0, where t i s t h e tax paid on income z and 0 S J (z )

<

^z

.

That is, taxes are nonnegative and nonconfis- catory. J is progressive if, f o r all z , J (z ) / z is monotone nondecreasing in z

.

J is s t r i c t l y progressive if J (z) / z is strictly monotone increasing in z

.

Thus t h e flat tax P(z) = rz where 0 S r

< 11

is progressive but not strictly so.

Say t h a t J (z) i n d u c e s equal sactipice if

J (z ) i n d u c e s an equal r a t e of sactipice if

Note that, since 0 S J (z)

<

^z and u (z) i s strictly increasing, w e must have c ² 0 in (1). Moreover, c

>

0 unless J(z) is identically zero. For (2) t o be well- defined, u (z ) # 0, and hence by continuity either u (z )

>

0 f o r all z

>

0, o r u(z)

<

⁰ f o r all z

>

0. The former case is the more natural, and implies t h a t 0 S r

<

1 i n (2).

O u r goal is to gain more information about t h e form of t h e utility functions in (1) and (2) by studying t h e p r o p e r t i e s of t h e tax schedules t h a t they engender. W e have already required t h a t t a x e s b e nonnegative, i.e., t h a t t h e r e b e no minimum in- come guarantee o r o t h e r form of d i r e c t subsidization through t h e tax system. This ^, is a significant assumption, but accords with much c u r r e n t practice. Second, i t seems intuitively reasonable t h a t a tax schedule which metes o u t equal sacrifice will be e i t h e r progressive o r flat, but not strictly regressive. In o t h e r words, if a proposed measure of utility yields nonprogressive taxation, t h e n w e may suspect t h a t it a t t r i b u t e s too g r e a t a marginal value of income to t h e rich.

Third, i t seems reasonable t h a t tax r a t e s should depend only on t h e relative distribution of incomes, not on t h e absolute monetary units in which incomes are measured. In o t h e r words, if incomes all increase by a fixed percentage, so t h a t t h e relative distribution of incomes remains unchanged, and if t h e total tax burden increases by t h e same percentage, then t h e relative distribution of taxes should remain unchanged. Such a tax system is s c a l e i n u a t i a n t . Scale-invariance can be interpreted as a simple principle of distributive equity: if taxes a n d incomes all increase or d e c r e a s e in some fixed proportion, then equity i s preserved. I t can also b e r e g a r d e d as merely a pragmatic principle t h a t allows t h e tax system t o b e re-indexed without changing t h e relative distribution of t h e tax burden.

3. Utility functions and tax myatema consintent with absolute equality of - d i c e

Let f ( z ) b e a tax schedule t h a t induces equal absolute sacrifice under t h e utility function u (z). Thus f and u satisfy (I), o r equivalently

Note t h a t t h e inverse of u exists because u i s assumed to b e strictly increas- ing. Expression (3) shows t h a t f can b e regarded as a function of two variables:

t h e income level z and t h e amount of sacrifice c . As c varies, f ( z , c ) defines a family of tax schedules t h a t allows different amounts of tax to be raised relative to any fixed distribution of incomes. f ( z , c ) is called a parametric t a z schedule.

Parametric families c r o p up in a wide variety of f a i r division problems

[Ill.

Scale-invariance of t h e tax system based on f ( z , c ) amounts t o saying t h a t , f o r e v e r y scale f a c t o r A

>

0, and f o r e v e r y constant c Z 0 , t h e r e exists a constant c' (depending on A) such t h a t

In o t h e r words, t h e t a x schedule c a n b e r e - i n d e r a d by changing c t o c', and t h e proportion of t a x paid to income received will remain fixed f o r a l l persons.

If j imposes absolute s a c r i f i c e and i s scale-invariant t h e n s e v e r a l important p r o p e r t i e s of t h e utility function may b e deduced. In p a r t i c u l a r , differences in utility levels will b e p r e s e r v e d (in an ordinal sense) under a uniform change of scale. Consider two individuals, one with initial income r and a f t e r - t a x income y , 0

<

y S r , and a n o t h e r with initial income r ' and a f t e r tax income y ', 0

<

y ' S r

'.

Equal absolute s a c r i f i c e implies t h a t u ( 2 )

-

^{u ( y )} = u (2')

-

^{u ( y ') = c 2 0. By}

definition of J , y = r

-

^j ( r , c ) and y ' = r ' - j ( r ' , c ) . Since j i s scale-invariant, f o r e v e r y A

>

0 t h e r e i s a c ' 2 0 such t h a t

Ay

=

Xz - j ( X z , c f ) and Ay'

=

Xz' - j ( X z ' , c f )

.

Hence by (3)

That is, f o r e v e r y z , r ', y , y '

>

0 and e v e r y A

>

^{0 ,}

Such a function u is said to b e homogeneous i n d m r e n c e s .

Lemma. If u ( r ) is continuous, nondecreasing, and homogeneous in differences, t h e n u ( r ) i s of form

(i) u ( r ) = b , o r

(ii) u ( z ) = a . Z n z + b , a > O , o r (iii) u ( z ) = a z P + b , ap > O

.

Proof.* If u (z ) is homogeneous in differences, then t h e r e is a single-valued func- tion F such that f o r all 0

<

y r z and all X

>

0,

Since u is continuous, F is too. _Let D = [ u ( z ) -u(y):O

<

y r z j . By con- tinuity, D is an interval of R ⁺ of form [O,m ) o r [O,m ], where m

>

0 unless u (z ) is constant, which is case (i) of t h e lemma. Suppose then that m

>

0. If z ,z' 2 0 and z + z ' < m , then z + z t = u ( z ) - u ( y ) f o r some z and y . Since 0 S z S u ( z ) - u ( y ) , wehave

Thus by t h e continuity of u ( z ) t h e r e exists some w between z and y such that u ( w ) = u ( z ) - 2 ; t h a t is z = u ( z ) - u ( w ) . Hence also z '

=

u ( w ) - u ( y ) . This argument shows t h a t f o r every X

>

0,

F(z +z',A)=F(z,X)+F(z',X) provided z , z ' 2 O and z + z t < m

.

(6) Since F is continuous, F may be continuously extended s o t h a t (6) also holds when- e v e r z ,z' r 0 and z

+

z ' = m . It follows from [I] (Section 2.1.4, Theorem 3) that

F(z,X) = c(X)z f o r all 0 S z S m , all X > O

.

Since u is nondecreasing,

u ( & )

-

^u(Xy)

⁼

^c(X)[u(z)

-

u ( y ) I f o r all =,'I/

>

0

.

Fix y*

>

0, and let d(X)

=

u (Ay*)

-

c(X)u (y*). Then

u ( & ) = c(A)u(z)

+

d(X) f o r all z,X

>

0.

.-

For all real numbers r define u * ( r ) = u ( e f ) , c*(r) = c ( e f ) , and d * ( r ) = d ( e f ) . Making t h e substitutions s = In X and t

=

In z , t h e preceding becomes

By [I] (Section 3.1.3, Theorem 1) this, together with t h e continuity and monotonici- ty of u* , implies t h a t

*The proof o f t h i s r e s u l t originated in collaboration w i t h Janos Aceel during s pleasant d r i v e t o - g e t h e r a c r o s s southern Germany in July, 1985. His contribution and helpful comments on t h e manuscript a r e g r a t e f u l l y acknowledged.

Therefore.

I t should b e remarked t h a t , if u ( z ) is assumed t o be continuous, homogeneous in differences, and n o n c o n s t a n t (instead of nondecreasing), then t h e above argu- ment shows t h a t u must b e of form (i) o r (ii) with a # 0, o r (iii) with ap f 0.

Theorem. Let t h e utility of income u ( z ) b e continuous and strictly increasing f o r all incomes z

>

0, and let f ( z ,c ) be a scale-invariant, progressive family of tax schedules t h a t induces a n equal level of sacrifice c 2 0 f o r all positive income lev- els. Then f o r some fixed a , p

>

0 and all z

>

0,

f ( z r c ) = ( l - e - I a ) z and u ( z ) = a Lnz + b , a >O; (0

f ( 2 . c )

=

z

-

Z a n d u ( z ) = - c z z * + b , a , p > O

.

(ii) [I +;ZP / a ll/P

Proof: By t h e lemma, u ( z ) must b e of t h e form aln z

+

b f o r some a

>

0 o r a x P

+

^{b f o r} ap

>

0. The case of constant u ( z ) i s ruled out since it is assumed h e r e t h a t u ( z ) i s s t r i c t l y increasing. f ( z ) yields a constant level of sacrifice c

r

0 if and only if

If u ( z ) = aln a

+

^b , (7) implies t h a t

which is case (i) of the theorem. If u ( z )

=

^{a z P}

+

b w e have two possibilities:

C-1: u ( z ) = a z P + b , a > O , p > O

.

Then u - l ( y ) = [ ( y - b ) / a l i / p , and ( 7 ) becomes f ( z , c ) = z - [ z P - c / a l l / P . This i s not progressive f o r c > 0 , since f ( z , c ) / z = 1 - [ 1

-

^{c /} z P a ] l / P is strictly decreasing in z .

Came2: U ( Z ) = a z P + b , a < O , p < O .

Replace a by -a and p by p and write u ( z ) = --azv

+

b , where a

>

^0,

p

>

0, and b i s unrestricted. Then u - l ( y ) = [ a / ( b

-

^y ) l l / p and ( 7 ) becomes

These schedules are progressive f o r all c 2 0, and strictly s o f o r c

>

0. 0

S e t a = I and f o r every p > O define f p ( z , c ) = z - z / ( l + C Z P ) ~ / P . Also, define f o ( ~ ^,C ) = ( 1

-

e * ) z f o r all z

>

0. These tax schedules form a single fami- ly with two parameters: p essentially determines t h e progressivity of t h e schedule and f o r each fixed p , c determines t h e amount of tax t o be levied.

4. Utility fundions and tax systems consistent with an equal rate of acrif ice

Consider now a utility function that induces an equal r a t e of sacrifice over all income levels. For this notion t o be well-defined, w e must have u ( z ) # 0 f o r all

z

>

0. That is, t h e utility of income must be everywhere positive o r everywhere

negative. The more natural case is t h e former, and t h e r e is no r e a l loss of gen- erality in assuming it, f o r if u ( z ) is negative, increasing, and an equal rate of sa- crifice regime prevails, then i t also prevails under I / lu ( z ) l , which i s positive and increasing.

Let u ( 2 ) be positive, continuous, and strictly increasing, and let f (2) induce a c o n s h n t rate of sacrifice r , 0 r r

<

1 . By definition,

u ( z -f ( 2 ) )

1

-

^{= r} f o r a l l z > O

.

24 ( 2 ) Thus

Therefore t h e arguments of t h e preceding section apply, since f' ( z ) induces equal absolute sacrifice relative t o t h e transformed utility function In u ( 2 ) . I t follows t h a t t h e only progressive and scale-invariant tax schedules t h a t are consistent with an equal rate of sacrifice are essentially t h e same as those t h a t are con- sistent with equal absolute sacrifice. The difference between t h e two cases is t h e required form of the utility function. There are two possibilities. One is t h a t In u ( z ) = a In z + b , a >0, in which case U ( Z ) = @ z a where a > O and B = e b

>

^0. Then an equal rate of sacrifice at rate r results from imposing t h e flat tax t = y ( z , r )

=

[ I

-

( l - r ) l / a ] z . ^\

The second possibility is t h a t In u ( z )

=

^--az

+

6 , where a , p

>

^0. Then u ( z ) is of t h e form

An equal rate of sacrifice at r a t e t = f ' ( z , r ) = z

-

- z / ( l - [ I n ( l - r ) ] z P / l / P . r results from imposing t h e t a x These results a r e summarized below.

Corollary. Let utility of income u ( z ) be positive, continuous, and strictly in- creasing f o r all incomes z

>

^0, and let

y(z

^{, r ) b e} a scale-invariant, progressive family of tax schedules t h a t induces a n equal rate of sacrifice ⁰ S r

<

1 on all po- sitive incomes. Then

z )= [ I - - r a z and u ( z ) = @ z a , a , 6 > O , ( 9

f ( z , r ) = z

-

Z ^{and u ( z ) = @e-* , a , B. P}

^>

⁰

^.

⁽ⁱⁱ⁾

[ 1 -[In (1 -r ) ] z P / a l l / P

Cohen S t u a r t , who originated t h e formal approach t o equal sacrifice in taxa- tion. advocated t h e use of a utility function of form u ( z )

=

aln z

+

b , a

>

0, to- g e t h e r with equal rate of sacrifice as t h e criterion. This gives r i s e t o tax schedules of t h e form

where r , 0 r r

<

1 is t h e rate of sacrifice. These functions are known as Cohen Stuart tazes. They were also mentioned as a possibility by Edgeworth, who pro-

posed t h a t they might b e a p p r o p r i a t e as a s u r h x yoked onto some o t h e r h x [3].

Let z* = e + l a , which may b e interpreted as t h e "minimum subsistence" level of income. Then Cohen Stuart's utility and h x functions may b e written as

U ( Z ) = aln(z/zL) and t = z - z * ( z / z * ) l 9 .

Cohen S t u a r t taxes do not satisfy t h e conditions of t h e foregoing theorem on several counts. First, t h e rate of sacrifice is undefined when z

=

^{x * ,} i.e., when t h e z e r o level of utility i s reached. Second, when z

<

z*, t a x e s become negative.

Third, and perhaps m o s t disturbing, t h e tax rates paid by individuals depend not only on t h e relative distribution of incomes in society, but also on t h e absolute lev- e l z* t h a t i s supposed to have z e r o utility. In o t h e r words, a p r e c i s e definition of what constitutes t h e subsistence level i s crucial to t h e validity of t h e scheme in equal sacrifice terms.

5. Conclusion

The equal sacrifice approach to progressive taxation is appealing in princi- ple. But i t i s far f r o m clear a pFiori what form t h e utility function should t a k e , or whether equal absolute or equal rate of sacrifice is t h e m o s t a p p r o p r i a t e cri- terion. One can gain insight into these questions by imposing elementary condi- tions on t h e tax functions t h a t a r i s e f r o m a n equal sacrifice model. A s w e have seen, equal sacrifice plus nonnegativity, progressivity, and scale-invariance is consistent with only a very special class of utility functions.

The f l a t tax implies equal absolute sacrifice if and only if utility is represent- ed by u (z ) = aln z

+

b for some a

>

0, and implies equal rate of sacrifice if and only if utility is represented by u (z) = @za, a , @

>

^0. A s t r i c t l y progressive, nonnegative, and scale-invariant tax implies equal absolute sacrifice if and only if utility is r e p r e s e n t e d by u (z) = -ax*

+

b , where a , p

>

0. I t implies equal rate of sacrifice if and only if utility is represented by u ( z ) =bee', where a , b , ^p

>

0. The class -uz*

+

b i s bounded above and unbounded below, whereas t h e class be-' i s positive, bounded both above and below, and u ( z ) -, 0 as z ^-, 0. Since a positive utility of income to b e more sensible intuitively than one t h a t is unbounded below, i t a p p e a r s t h a t equal r a t e of sacrifice seems to b e t h e more natural criterion when w e examine both t h e form of t h e utility functions and of t h e tax schedules t h a t are consistent with it.

-

¹⁰

-

Finally, t h e r e is only one family of tax schedules t h a t is consistent with equal sacrifice u n d e r e i t h e r d e n i t i o n , and is nonnegative, progressive, and scale- invariant. This s a m e family has another desirable feature: namely, a tax can be levied in successive installments with t h e same outcome as if i t had been levied all at once [12].

Schedules of t h e above type can b e f i t quite closely to r e c e n t tax schedules in t h e U.S. and in West Germany [12]. This fact suggests t h a t one could use actual tax

d a t a to estimate t h e value of p in formula (ii) of t h e Theorem (or t h e Corollary).

For r e c e n t US and West German data, t h e estimated values of p are in t h e vicinity of .65 to .75. If one hypothesizes t h a t tax schedules to some d e g r e e reflect what a given society perceives to b e equal sacrifice, then such estimates might b e t r e a t e d in t u r n as estimates of t h e parameters in t h e utility functions themselves.

[I] Aczel, J. Lectures o n f i n c t i o n a l Equations a n d their Applications, N e w York, Academic P r e s s , 1966.

[2] Cohen S t u a r t , A.J. On progressive taxation, in R.A Musgrave and P.B.

Peacock (Eds.), Classics in the Theory o f h b l i c finance, New York: McGraw Hill, 1958.

[3] Edgeworth, F.Y. The p u r e theory of taxation, in R.A. Musgrave and P.B.

Peacock (Eds.), Classics in the Theory of h b l i c Ftnance. New York:

McGraw Hill, 1958.

[4] Eichhorn, W. and H. Funke, A c h a r a c t e r i z a t i o n of the proportional income t a z , P r e p r i n t , University of Karlsruhe, West Germany, 1984.

[5] Eichhorn, W., Funke, H. and W.F. Richter, Tax progression and inequality of income distribution, Journal ofMathematical Economics 3 3 (1984), 127-131.

[6] Mirrlees, J.A. An exploration in t h e theory of optimum taxation, Review of Economic S t u d i e s 38 (1971), 175-208.

[7] Pfaehler, W. Normative Theorie d e r f i s k a l i s c h m Bestsuerung, Frankfurt, West Germany: P e t e r Lang Verlag, 1978.

[8] Pfingsten, A. The Measurement of Trrz Progression, Ph.D. Dissertation, University of Karlsruhe, West Germany, 1985.

[9] Richter, W.F. "A normative justification of progressive taxation: How to compromise on Nash and Kalai-Smorodinsky", in 0. Moeschlin and D. Pal- laschke (eds), Came m e o w a n d Mathematicad Economics, Amsterdam:

North-Holland Publishing Company, 1981.

[lo]

^{Richter, W.F.} From ability t o pay t o concepts of equal sacrifice, Journad of Public Economics 20 (1983). 211-229.

[Ill

Young, H.P. "On dividing a n amount according t o individual claims o r liabili- ties", Mathematics of m e r a t i o n s Research, t o a p p e a r .

[12] Young. H.P. 'The design of progressive tax schedules1', College P a r k , Md.:

The University of Maryland School of Public Affairs, 1986.