JOL~,NAL OF PUBLIC ECONOMICS

ELSEVIER Journal of Public Economics 62 (1996) 327-338

Taxes , redistribution, and growth

Randall Wright Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia,

PA 19104, USA

Received December 1994~ revised version received August 1995

Abstract

This paper develops a simple model to study the eff:cts of growth on economic policy, and, in particular, on redistributive taxation. I do not explain economic growth; I assume it is exogenous, and ask how changes in the growth rate (and in several other variables) affect the outcome of a political process that determines taxes and transfers. It is shown that faster-growing economies will choose more or less redistribution, depe~ding on risk aversion. In the empirically relevant case, the model predicts lower taxes in faster-growing economies.

Keywords: Growth; Taxation, Redistribution; Voting

J E L classification: H5

1. Introduction

Thi s p a p e r d e v e l o p s a s imple m o d e l wi th in which o n e can s t udy the ef fec ts of g r o w t h on e c o n o m i c pol icy , and , in pa r t i cu l a r , on r ed i s t r ibu t ive t axa t i on . I d o no t p u r p o r t to exp la in g rowth h e r e ; r a t h e r , I a s s u m e tha t it is e x o g e n o u s , a nd ask how ch an g es in the g rowth r a t e (and in seve ra l o t h e r va r i ab l e s ) affect the o u t c o m e of a pol i t ical p rocess t ha t d e t e r m i n e s t axes a n d t r ans fe r s . B e c a u s e of its s impl ic i ty , the m o d e l de l ivers c lean ana ly t i c resul ts .

M u c h r e c e n t r e sea rch has e m p h a s i z e d the impac t o f r ed i s t r ibu t iv¢ t a x a t i o n o n g r o w t h (see, for e x a m p l e , Pe r s son and Tabe l l i n i , 1994; A l e s i n a and R o d e r i c k , 1993; Pe ro t t i , 1993; a n d S t o k e y and R e b e l o , 1993). In the real w o r l d , no t on ly does fiscal pol icy affect g rowth , bu t g rowth also affects

004%2727/96/$15.00 ~ 1996 Elsevier Science S.A. All rights reserved S S D I 0047-2727(95)01570-I

328 R Wright / Journal o f Public Economics 62 (1996) 3 2 7 - 3 3 8

po l i cy , and the net r e l a t i o n s h i p can be qu i t e c o m p l e x . F o r e x a m p l e , Kruse l l e t al. (1994) p r e s e n t o n e m o d e l wi th f e e d b a c k f r o m g r o w t h to pol icy and vice ve r sa , bu t it is so c o m p l i c a t e d it can only be a n a l y z e d us ing n u m e r i c a l t e c h n i q u e s . I t h ink tha t it is useful to cons ide r a s imp le r m o d e l , wi th f e e d b a c k in on ly o n e d i r ec t i on , w h e r e we can find ana ly t ic so lu t ions tha t can be u s e d to d e v e l o p o u r u n d e r s t a n d i n g o f ce r t a in k e y effects . E v e n w h e n c a u s a t i o n runs on ly f r o m g r o w t h to pol icy , it will be seen tha t t h e r e a re s eve ra l f ac to r s t h a t in f luence e q u i l i b r i u m taxes and t r ans f e r s , inc lud ing i n e q u a l i t y , r isk ave r s ion , t ime p r e f e r e n c e , a l t ru i sm, social mob i l i t y , pol i t ical s t ab i l i ty , a n d the d i s t r i b u t i o n o f pol i t ical p o w e r , and the m o d e l g e n e r a t e s c l ea r p r e d i c t i o n s a b o u t all o f these t h i n g s )

T h e first t h ing o n e n e e d s to do in such an exerc i se is to t a k e a s t a n d as to w h y r e d i s t r i b u t i o n occurs . T a x e s on the rich and t r ans f e r s to the p o o r a re m o d e l e d h e r e as the o u t c o m e o f a pol i t ical p rocess in which the rich a re wi l l ing to p a r t i c i p a t e b e c a u s e t h e y real ize t ha t t h e y , o r p e r h a p s t he i r o f f s p r i n g , m a y be p o o r t h e m s e l v e s o n e day . T h a t is, t a x a t i o n and r ed i s t r ibu - t ion c o n s t i t u t e social i n su rance . S o m e sa l ien t aspec t o f social i n su rance c a p t u r e d by the m o d e l in this p a p e r inc lude the fact t h a t the s y s t e m is l ong - l i ved , so t ha t r ich a g e n t s have to t a k e in to a c c o u n t t h a t t h e y m a y o n e d a y b e c o m e p o o r a n d vice ve r sa , and tha t t axes m u s t be pa id t o d a y e v e n t h o u g h benef i t s m a y acc rue o n l y in the fu tu re .

T h e m a i n q u e s t i o n is this: will f a s t e r - g r o w i n g e c o n o m i e s c h o o s e m o r e o r less r e d i s t r i b u t i o n , a n d w h y ? A l t h o u g h the e q u i l i b r i u m t a x - t r a n s f e r po l icy d e p e n d s on the n a t u r e o f the pol i t ical p rocess and on the va r ious o t h e r f a c t o r s d e s c r i b e d a b o v e , as well as the g r o w t h r a t e , the a n s w e r to this q u e s t i o n t u rn s cr i t ica l ly on o n e p a r a m e t e r , a , the coef f ic ien t o f re la t ive r isk a v e r s i o n (or , e q u i v a l e n t l y , t he i n t e r t e m p o r a l e las t ic i ty of subs t i t u t i on ) . N a m e l y , f a s t e r g r o w t h impl ies h i g h e r taxes on the rich and m o r e g e n e r o u s t r a n s f e r s to the p o o r if and on ly if a < 1.

T h i s has n o t h i n g to do wi th w h e t h e r f a s t e r - g r o w i n g e c o n o m i e s are r i c h e r - t h e spec i f i ca t ion u t i l ized fo r m o s t o f the ana lys is impl ies t h a t an inc rease in

t h e l e v e l of i n c o m e d o e s no t af fect the tax r a t e at all ( a l t h o u g h I also br ief ly c o n s i d e r a m o r e g e n e r a l spec i f ica t ion to s h o w tha t , in p r inc ip le , the level c o u l d m a t t e r ) . R a t h e r , t he e f fec t resu l t s f r o m the fact t ha t the r e t u r n to p a y i n g t axes , which is the poss ib i l i ty of rece iv ing t r ans f e r s in the f u t u r e , is h i g h e r w h e n the g r o w t h ra t e is g r ea t e r . T h e impac t o f a c h a n g e in this r a t e

~ These considerations may help to explain why recent empirical studies have failed to find robust significant relationships between policy and 7.xowth- Among the large number of papers that regress cross-country growth rates on government policy variables (and many other things). Kormendi and Meguire (1985) find government spending affects growth positively although not significantly: Grier and Tullock (1989) and Barro (1991) find a significant negative effect: and Levine and Renelt (1972) conclude that "'none of the broad array of fiscal indicators.., studied is robustly correlated with growth" (p. 959).

R. Wright / Journal o f Public Economics 62 (1996) 327-338 329

o f r e t u r n d e p e n d s on h o w ind iv idua l s we igh the t r a d e - o f f b e t w e e n p r e s e n t t axes a n d t h e poss ib i l i ty o f f u tu r e t r ans fe r s , wh ich is c a p t u r e d by a .

P r e s u m a b l y t h e empi r i ca l l y r e l e v a n t case is a > 1 (see the d i scuss ion in P r e s c o t t , 1986, fo r e x a m p l e ) ; in this case , the m o d e l p r ed i c t s l ower t axes a n d less r e d i s t r i b u t i o n in f a s t e r - g r o w i n g e c o n o m i e s . O f c o u r s e , th is d o e s n o t m e a n t h a t l o w e r t axes c:r',~e h i g h e r g r o w t h , s ince g r o w t h is e x o g e n o u s in t h e m o d e l . O n e c o n t r i b u t i o n of t he p a p e r is to po in t ou t t ha t e v e n if o n e can f ind a n e g a t i v e r e l a t i o n s h i p b e t w e e n t a x a t i o n a n d g r o w t h in the d a t a , by n o m e a n s d o e s this imp ly t h a t r e d u c i n g taxes will l ead to i m p r o v e d e c o n o m i c g r o w t h .

T h e res t o f t h e p a p e r is o r g a n i z e d as fo l lows. Sec t ion 2 d e s c r i b e s t h e bas ic m o d e l . Sec t ion 3 d e r i v e s t h e m a i a resu l t c o n c e r n i n g the r e l a t i o n s h i p b e t w e e n t axes a n d g r o w t h . SectiorJ 4 uses a s impl i f ied t w o - p e r i o d ve r s ion o f t h e m o d e l t o i l lus t ra te h o w , in g e n e r a l , the level o f i n c o m e as well as t h e g r o w t h r a t e m a y m a t t e r . Sec t ion 5 p r e s e n t s s o m e b r i e f c o n c l u d i n g r e m a r k s .

2. The basic model

T h e bas ic s e t -up is a v a r i a n t o f the pol i t ica l e c o n o m y m o d e l in W r i g h t ( 1 9 8 6 ) , e x t e n d e d to a c c o m m o d a t e g r o w t h . T h e r e a r e a l a rge n u m b e r o f i n d i v i d u a l s wi th iden t i ca l p r e f e r e n c e s d e s c r i b e d by

U = E ~ ~ 'u (c , ) , ( 1 ) / = 0

w h e r e c, is c o n s u m p t i o n at da t e t, E d e n o t e s the e x p e c t a t i o n , a n d #1 • (0 , 1)- It d o e s n o t m a t t e r fo r c u r r e n t p u r p o s e s if i nd iv idua l s a re a s s u m e d to live f o r e v e r o r if t he se p r e f e r e n c e s a r e o v e r the c o n s u m p t i o n s t r e a m s o f d y n a s t i c fami l ies . In the l a t t e r case , o n e s h o u l d i n t e r p r e t fl as c a p t u r i n g t h e d e g r e e t o w h i c h ind iv idua l s ca re a b o u t t he f u t u r e ut i l i ty o f t he i r o f f sp r ing ( i n t e r g e n e r a - t i ona l a l t ru i sm) .

I w a h l t h e m o d e l to be c o n s i s t e n t wi th a "ba lanced growth" e q u i l i b r i u m , w h e r e o u t p u t a n d c o n s u m p t i o n g r o w at the s a m e ra te . T h i s m e a n s t h a t t h e m o m e n t a r y ut i l i ty f u n c t i o n , u(c), m u s t be o f the fo l lowing class: e i t h e r

C l-¢~t ~ 1

u ( c ) = 1 - , , , " ( z )

f o r ~x >~ 0 a n d c~ # 1; o r

u(c) = l o g ( c ) , ( 3 )

w h i c h is t h e l imi t ing case as a ~ I . T h e p a r a m e t e r a is t he coef f ic ien t o f

330 R. Wright / Journa l o f Publ ic E c o n o m i c s 62 (1996) 3 2 7 - 3 3 8

r e l a t i v e r isk a v e r s i o n ; e q u i v a l e n t l y , 1/c~ is t h e i n t e r t e m p o r a l e l a s t i c i ty o f subs t i tu t ion . -"

T h e m o d e l ha s a v e r y s i m p l e p r o d u c t i o n t e c h n o l o g y : at e a c h d a t e t, i n d i v i d u a l i c an t r a n s f o r m o n e un i t o f l a b o r in to y , un i t s o f n o n - s t o r a b l e o u t p u t . L a b o r will be s u p p l i e d ine l a s t i ca l l y , as l o n g as t h e a f t e r - t a x i n c o m e f r o m w o r k i n g e x c e e d s t h e i n c o m e f r o m n o t w o r k i n g . F o r e v e r y i a n d t , Yi, E { w , , 0}. T h i s , i n d i v i d u a l i e i t h e r has acces s to t h e ab i l i ty to p r o d u c e w, un i t s o r z e r o un i t s o f o u t p u t . 3 F o r s imp l i c i t y , w, d o e s n o t d i f f e r a c ro s s a g e n t s , b u t it d o e s g~ow o v e r t i m e . W e a s s u m e t h a t w, = W T ' , w h e r e W is t h e base w a g e a n d 3t is t h e g ros s g r o w t h rate ( i . e . 7 - 1 is t h e p e r c e n t a g e r a t e o f g r o w t h ) . We t a k e 7 t o be e x o g e n o u s , a n d to sa t i s fy /3~/1-~ < 1 ( s ince o t h e r w i s e U will n o t b e b o u n d e d a l o n g t h e b a l a n c e d g r o w t h p a t h .

A n i n d i v i d u a l ' s y~, s w i t c h e s b e t w e e n w, a n d 0 a c c o r d i n g to a i a r k o v p r o c e s s , w i t h

p r ( y , + ~ = O ] y , = w , ) = A a n d p r ( Y t + l - - W t + t l y t = O ) = p . (4 )

F o r s i m p l i c i t y , A a n d p a r e t h e s a m e fo r all a g e n t s ( a l t h o u g h th is is ea s i ly g e n e r a l i z e d ) , a n d s t r i c t ly b e t w e e n 0 a n d 1. T h e r e is n o a g g r e g a t e u n c e r t a i n - t y , in t h e s e n s e t h a t t h e s a m e n u m b e r o f a g e n t s h a v e access to w t at e a c h d a t e . L e t ] d e n o t e t h e s t a t e o f an a g e n t , w h e r e j = 1 if he ha s access t o w, a n d ] = 0 o t h e r w i s e , a n d le t nj d e n o t e t h e n u m b e r o f a g e n t s in s t a t e ]. T h e n nj will n o t c h a n g e o v e r t i m e if a n d o n l y if

n o = A / ( O + A) a n d n~ = p / ( p + A) . (5 )

W e c a n t h i n k o f t h e a g e n t s in s t a t e 0 as low i n c o m e , p o o r , o r u n e m p l o y e d , a n d t h e a g e n t s in s t a t e 1 as h igh i n c o m e , r i ch , o r e m p l o y e d . If we i n t e r p r e t t h e s t a t e ] = 0 as u n e m p l o y m e n t , t h e n A is t h e l a y o f f r a t e , p is t h e reca l l r a t e , a n d no is t h e a g g r e g a t e u n e m p l o y m e n t r a t e . M o r e g e n e r a l l y , A a n d p m e a s u r e ( d o w n w a r d a n d u p w a r d ) s o c i o - e c o n o m i c m o b i l i t y o f i n d i v i d u a l s o r f a m i l i e s .

M a r k e t s a r e a s s u m e d to be i n c o m p l e t e , in t he s e n s e t h a t in t h e a b s e n c e o f r e d i s t r i b u t i o n a l po l i c i e s , i n d i v i d u a l s s i m p l y c o n s u m e t h e i r i n c o m e . L e s s

-" G i v e n a f ixed i n t e r e s t r a t e r . it is e a s y t o d e r i v e t he resu l t tha t u(c) m u s t d i s p l a y c o n s t a n t r e l a t i v e r isk a v e r s i o n . If t he g r o w t h r a t e o f i n c o m e is ~ t h e n b a l a n c e d g r o w t h m e a n s t h a t c,+ ~ = ~/c,. a n d t he s t a n d a r d m a r g i n a l c o n d i t i o n r e l a t i ng c o n s u m p t i o n t o d a y a n d c o n s u m p t i o n t o m o r r o w c a n b e w r i t t e n u ' ( c ) = / 3 ( 1 + r)u'(~,c). D i f f e r e n t i a t i n g wi th r e s p e c t t o c . w e ge t u " ( c ) = / 3 ( 1 +r)u"(Tc) 'g . M u l t i p l y i n g t he s e c o n d c o n d i t i o n b y c a n d d iv id ing b y t he first c o n d i t i o n , w e ge t u " ( c ) c / u ' ( c ) = u"(~,c)~,c/u'(yc). T h e r e f o r e . t he coe f f i c i en t o f r e l a t ive r i sk a v e r s i o n is c o n s t a n t .

3 W e c o u l d a s s u m e m o r e g e n e r a l l y t ha t y,, E {wt , . wo, }. w h e r e wo, = Xw, , wi th X E [0. 1) ( t h e m o d e l in t h e t ex t is the. spec ia l c a s e w h e r e )¢ -- 0) . T h e r e su l t s a re bas i ca l ly u n c h a n g e d , as will b e s k e t c h e d b e l o w .

R. Wright / Journal o f Public Economics 62 (1996) 327-338 33I

res t r ic t ive ly , we could a s sume tha t agen ts can b o r r o w and lend bu t do no t h a v e access to insu rance m a r k e t s , as H a n s e n and I m r o h o r o g l u (1992) a s s u m e in a s imilar e n v i r o n m e n t ; this would not change the basic message , bu t wou ld compl ica te the analysis cons iderab ly . Even if t he re were c o m p l e t e i n s u r a n c e m a r k e t s , the policies desc r ibed be low could still be ins t i tu ted as a pol i t ical equ i l ib r ium (at least in the gene ra l i zed vers ion of the mode l wi th ind iv idua l h e t e r o g e n e i t y - see Wright , 1986).

T h e pol icy s c h e m e is desc r ibed as follows. T h e income of r ich agen t s at da t e t is t axed at ra te ~',, and the r e v e n u e is d i s t r ibu ted to p o o r agen t s , w h o each get b,. Fo r the mos t par t , the in te res t is in s t a t iona ry tax policies , w h e r e T is c o n s t a n t wi th r e spec t to t ime and the re fo re b, g rows at the s ame ra t e as w,. N o t e tha t we are impos ing tha t ~r is cons t an t for all t. H o w e v e r , we do no t w a n t ~- to d e p e n d on the level of i ncome at the da te at which the pol icy is d e t e r m i n e d , which is why we adop t the class of p r e f e r e n c e s tha t l ead to b a l a n c e d g rowth . W i t h these p r e f e r ences , agents have the s ame e v a l u a t i o n of the costs and benef i t s of a policy at every da te t, i r respec t ive of the c u r r e n t level of i ncome ( b u t see Sect ion 4).

W i t h a c o n s t a n t tax ra te ~-, the g o v e r n m e n t ' s ba l anced t~udget cons t r a in t is n ob ̀ = n ~ r w , . Subs t i tu t ion of n o and n~ f rom (5) into this condi t ion yields

b, = w , z p l A . (6)

So at each da t e t the c o n s u m p t i o n of a r ich individual is cl, = (1 - ~ ' ) W T ' and the c o n s u m p t i o n of a p o o r individual is co, = ( r p / A ) W T ' . Final ly , we add the c o n s t r a i n t cx, >t co,. This says tha t those wi th the i n c o m e - e a r n i n g o p p o r t u n i t y y~, = w, c o n s u m e m o r e t han those with y,, = 0, which is obvious ly necessa ry if agen t s a re go ing to w o r k voluntar i ly . T h e cons t ra in t holds if and only if T ~ ? - - - - A / ( A + p ) . A t 7 = ?, t h e r e is f u l l i n s u r a n c e , in the sense tha t c~, = Cot = W T t p / ( p + A).

3. Equilibrium

W e d e n o t e by V, the e x p e c t e d l i fet ime utility at da t e t of an agen t in s ta te j , j = 0, 1, for a given tax policy ~-. T h e Vj,'s will be cal led the va lue f u n c t i o n s . T h e y d e p e n d on t ime only because w, = W y ' is g rowing ( tha t is, if 3, = 1, t h e n Vj, does not vary wi th t) . T h e va lue func t ions satisfy the fo l lowing recurs ive re la t ionsh ips :

E , = u[(1 - "r)w,] + fl[AVo,+, + (1 -- A ) E , + , ] , (7)

E,, = u(.,-,, , ,p/A) + ,Bit, v , , + , + (1 - o ) v o , + , l . ( 8 )

Eq. (7) says tha t the value to be ing rich today is the utili ty of c o n s u m i n g o n e ' s a f te r - tax income this pe r iod , plus the d i scoun ted va lue of the next

3 3 2 R. Wright / Journa l o f Publ ic E c o n o m i c s 62 (1996) 3 2 7 - 3 3 8

p e r i o d b e i n g p o o r w i th p r o b a b i l i t y A a n d r ich w i th p r o b a b i l i t y 1 - A. E q . (8 ) h a s a s i m i l a r i n t e r p r e t a t i o n in t e r m s o f t h e v a l u e o f b e i n g p o o r t o d a y .

W e c a n t h i n k o f ( 7 ) a n d ( 8 ) as a s y s t e m o f d i f f e r e n c e e q u a t i o n s in (V~,, I,"2, ) a n d , g iven t h e f u n c t i o n a l f o r m f o r u(c) d e s c r i b e d a b o v e , t h e i r s o l u t i o n h a s a v e r y c o n v e n i e n t r e p r e s e n t a t i o n . L e t us c o n s i d e r t h e ca se w h e r e u(c) is g iven b y (2 ) . T h e n , V# = vjT (1-~)t, w h e r e

[W(1 -- ~ ' ) ] ' -~ (W~'p/A) '-~ = -- + AA8 + B , ( 9 ) v~ A ( 1 c5+p¢5) 1 - - a 1 - - a

[ w ( 1 - z ) ] ' - ~ ( w z p / a ) ~ - ~ = + A ( 1 - A + 3,8) + C , ( 1 0 ) va Ape5 1 -- a 1 -- a

w h e r e ¢5----/3T t - , , a n d w h e r e A , B a n d C a r e p o s i t i v e c o n s t a n t s t h a t n e e d n o t c o n c e r n us h e r e . T h e ca se w h e r e u(c) is g i ven b y (3 ) c an be u n d e r s t o o d b y first a n a l y z i n g th i s ca se a n d t a k i n g t h e l imi t as a ~ 1.

T h e i m p o r t a n t t h i n g a b o u t ( 9 ) a n d ( I 0 ) is t h a t t h e y a r e i d e n t i c a l t o t h e v a l u e f u n c t i o n s t h a t o n e ge t s in an e c o n o m y wi th n o g r o w t h b u t w i t h a d i f f e r e n t d i s c o u n t f a c t o r , 6 i n s t e a d o f ft. T h a t is, if y = 1, w e can se t V# = Vj a n d so lve (7 ) a n d (8 ) f o r Vt a n d V,_, a n d t he s o l u t i o n will be o f t h e f o r m g i v e n in (9 ) a n d ( 1 0 ) w i t h /3 r e p l a c i n g 8. T h i s is b e c a u s e t h e s e u t i l i ty f u n c t i o n s i m p l y c h a n g e s in T a f fec t t h e w a y i n d i v i d u a l s e v a l u a t e t a x - t r a n s f e r p o l i c i e s o n l y b y c h a n g i n g 8. In w h a t fo l lows we call 6 t h e effective discount factor. A l s o , to m a k e t h e d e p e n d e n c e o f v l a n d v0 o n t he t ax r a t e exp l i c i t , t h e y will b e w r i t t e n as v lO ' ) a n d v0(~').

T o d e f i n e a n d ~olve f o r a po l i t i ca l e q u i l i b r i u m r e q u i r e s d e c i d i n g h o w t h e p r e f e r e n c e s o f t h e r ich a n d p o o r m a p i n t o p o l i c y d e c i s i o n s . W i t h s i m p l e m a j o r i t y v o t i n g , t h e d e c i s i v e p r e f e r e n c e s a r e t h o s e o f t h e m e d i a n v o t e r , w h o wil l b e e i t h e r a r ich o r a p o o r i n d i v i d u a l d e p e n d i n g o n l y o n w h e t h e r n t > n o o r n o > n t. I f we i n t e r p r e t t h e low i n c o m e s t a t e as u n e m p l o y m e n t , t h e n it s e e m s r e a s o n a b l e to a s s u m e t h a t n t > no; if we i n t e r p r e t it d i f f e r e n t l y , t h e n t h e o p p o s i t e m a y s e e m m o r e r e a s o n a b l e . In m a n y s i t u a t i o n s w e w o u l d a r g u e t h a t t h e r ich a r e dec i s i ve e v e n if t h e r e a r e f e w e r o f t h e m , b e c a u s e po l i t i ca l p o w e r d o e s n o t a l w a y s t r a n s l a t e l i t e r a l ly i n t o o n e m a n , o n e v o t e . In a n y e v e n t , b o t h ca ses ( r ich d . . 2sive a n d p o o r dec i s ive ) will be d i s c u s s e d b e l o w ?

W e c o n s i d e r f irst a c u r r e n t l y r ich p e r s o n f ac ing a s t a t i o n a r y p o l i c y g i v e n b y % =-~ f o r all t. S u p p o s e he ha s to d e c i d e on his p r e f e r r e d t ax r a t e ~- a t a s ing le p o i n t in t i m e , u n d e r t h e a s s u m p t i o n t h a t t h e po l i cy will r e v e r t t o -~ a t e a c h f u t u r e p o i n t in t i m e . A s t a t i o n a r y e q u i l i b r i u m r e q u i r e s ~-= ~. H e e v a l u a t e s his p r e f e r r e d t ax r a t e at t h e s ingle p o i n t in t i m e , t, a c c o r d i n g to t h e c r i t e r i o n :

I n t e r m e d i a t e c a s e s , w h e r e 7 is b e t w e e n t h e p r e f e r r e d t a x r a t e s o f r i c h a n d p o o r a g e n t s , c a n a l s o b e h a n d l e d .

R. Wright / Journal o f Public Economics 62 (1996) 327-338 333

/ , 0 ) = u I W ( 1 - ~)1 + @ , ( ÷ ) , (11)

where @t(÷) is the (expected, d iscounted) cont inuat ion value under ÷ one per iod hence, and thus depends on ? but not on ~'. Since maximization of ft (~,) yields ~" = O, obviously ~ = 0 is the only equil ibrium. Why would anyone choose to pay taxes when the con temporaneous benefits are nil and future policy is independen t of his current choice?

But this misses the point entirely. It misses the point because it complete- ly severs the relat ionship be tween con temporaneous taxation and the potent ia l future benefits to the individual. Suppose, for example, that the political process, for whatever reason, constrains the tax rate chosen at any date to be in force for two periods. T h e n a rich agent evaluates ~- according to the cri terion:

A O ) = u [ w ( 1 - r ) l + , ~ t 3 u ( W ~ p / a ) + (1 - a ) t 3 u [ W ~ ( 1 - , ) l + @_-(÷) •

( 1 2 )

where @2(÷) is the cont inuat ion value under ÷ two periods hence. Because u ( c ) = c ~ - ~ / ( 1 - a ) , the growth rate can bc factored out of the utility funct ion and into ~ = /3y ' -~ so that (12) can be writ ten:

/-_b') = [ w ( 1 - " , ) l ' - " + , ~ a ( w ' , p / a ) "- '~ + ( 1 - ,~)a [ w ( 1 - -,-)]'-~'

1 - a

+ @,(÷). (~3)

The maximizat ion of f20") does not general ly yield r = 0. In genera l , if the policy were in place for k periods, then the following

recursive represen ta t ion describes the cri teria of the rich and the poor:

L ( ~ ) = [W(l - ~-)1"-~ l-a +8(1--A)fk_,('r)+SAgk_t(~')+@k(f'), (14)

g ~ ( ¢ ) - (WTp/A) ~ -'~

1 - a + apL_ , ( - , - ) + 8 ( i - . ) g , _ , ( - , . ) + % ( ÷ ) . (15)

where @~(÷) and ~ ( ÷ ) are their cont inuat ion values under ~. Maximizat ion of (14) yields ~r > 0, because taxation today has a potential benefit in the future . The point is that as long as elections or political decisions are less f requen t than possible changes in individual status, non-trivial dynamic considera t ions arise.

A l though not crucial for the basic message, for simplicity we focus on the limiting case where the political process constrains the tax rate chosen to be in force forever: k = ~. In this case, fk('r) = vl0") and gk0") = v00"), and .rich and poor ind:.viduals evaluate taxes according to the expressions in (9) and

334 R. Wright / Journal o f Public Economics 62 (1996) 327-338

(10) . Since vj(~-) is strictly concave , there will be a un ique p re fe r red tax rate for an agent in state j in the feasible set [0, ~], where ~ = A/(A + p) is the m a x i m u m (ful l - insurance) tax rate. For rich agents , v ~ ( 0 ) > 0 > v ~ ( ~ ) , so the i r p re fe r r ed tax rate is g rea te r than 0 but less than -~. Hence , if the rich are decisive, then there is a unique equi l ibr ium ~-* ~ ( 0 , - ~ ) satisfying v~(T*) = 0. For poo r agents , v0(-Y)>0, so if they are decisive, then the un ique equ i l ib r ium is 7 = ~. 5

G iven the funct ional fo rm of u(c), v~(~'*)= 0 can be solved explicitly for the o u t c o m e when the rich are decisive. The answer is r * = ~p/(1 + tp), w h e r e ~p is given by

~A-,p l--. t P ~ = 1--c$ + p S " (16)

Reca l l the a s sumpt ion tha t t$ = / ~ y ~-~ < 1 (which is needed for the individual ob jec t ive funct ion to be wel l -def ined) . This implies 1 - 8 •gt$ > 0 , and the re fo re tha t 0~-*/08 :> 0, and ~---->-~ as ~$--> 1. The effect of the growth ra te o n ~- d e p e n d s exclusively on its effect on the effective discount factor , ~, : - /3T l -~ . H e n c e , w h e n the rich are decisive, we conclude the fol lowing: 0T*/0T > 0 if and only if t~ < 1. In the empir ical ly reasonab le case of a > 1, the m o d e l predicts r* is decreas ing in y. Also , regardless of a , r* is i n d e p e n d e n t of W, so it is the growth rate o f income that ma t t e r s , and not the level o f income.

We can also ask: H o w does T* depends on o the r p a r a m e t e r s in the m o d e l ? Le t us cons ider , for example , the r isk-aversion p a r a m e t e r , t~. It is useful to d e c o m p o s e the net effect as follows:

07 ( 0 ~ - ) 0~-&5 - - + - - - - ( 1 7 )

T h e first t e rm is the effect of risk avers ion on ~-* for a fixed effective d i scount ra te ; it can be shown to be unambiguous ly posit ive. The second t e r m is the effect of a change in the effective discount rate; it is negat ive for all T > 1, and can actual ly d o m i n a t e the first t e rm if T is large. If T is not too la rge , howeve r , then the first t e rm domina te s and more r i sk-average agents will necessar i ly chose m o r e taxes and more social insurance.

Since ~-* is increasing in /3, the more individuals care abou t their fu ture c o n s u m p t i o n , or the more they altruistically care abou t fu ture generat ions"

N o t e tha t the p o o r a lways c h o o s e the m a x i m u m feas ib le tax ra te desp i t e the fact tha t t hey k n o w t h e y will o n e day be rich. In genera l , the incent ive to s m o o t h c o n s u m p t i o n t ends to m a k e a g e n t s des i re full insurance , and agen t s on ly c h o o s e o t h e r than full insurance b e c a u s e the pr ice is no t ac tuar ia l ly fair in the m o d e l due to the ' pay n o w . rece ive benef i t s later" na tu re o f the s y s t e m . This w o u l d normal ly cause the rich to p re fe r less than full insurance and the p o o r to c h o o s e m o r e than full insurance ; bu t since full insurance is the m a x i m u m feasible pol icy , the p o o r e n d up at a c o r n e r so lu t ion .

R. Wright / Journal o f Public Economics 62 (1996) 327-338 335

consumpt ion , the higher is the tax rate when the rich are decisive. Also, ~-* is always increasing in A, and it is decreasing in p, at least in the case where a > 1. It is also true that ~ is increasing in A and decreasing in p. Hence , w h e t h e r the rich or the poor are decisive, the model predicts that taxes increase with the transit ion rate from rich to poor , A, and decrease with the t ransi t ion rate from poor to rich, p. Since these pa ramete r s measure mobility, the model predicts that the degree of both downward and upward mobility affects the political equil ibrium.

Finally, we can also show that • increases with inequali ty, in the following sense. Suppose we generalize the model so that income is Yi, E {wtl, w,~}, where wot = Xw~t with X E (0, 1) ( the above model is the limiting case where X = 0). The smaller is X, the grea ter is (pre-tax) income inequality. A similar analysis to the one used above implies that if the rich are decisive, then ~'* will be strictly positive if and only if ~, is not too large, and that as long as 7 * > 0 we have 0~-*/08 > 0 exactly as in the model with • = O. Moreover , it is easy to check that more inequali ty (a reduct ion in ,~) increases ~-*. Note tha t this is t rue whe the r or not we adjust the base wage to mainta in the same average income when we increase inequali ty (i.e. whe the r we consider a mean~preserving or mean-al ter ing spread in income), because the prefer- ences we are using to genera te a balanced growth path imply that the ou tcome is independen t of average income.

These findings bear on the ambiguous cross-country evidence on the re la t ionship be tween growth and taxation ment ioned in footnote 1. The model predicts that there are many reasons for taxes to vary for a given growth rate, including risk aversion, time preference, al truism, mobility and inequali ty. Moreover , the degree of political stability or commi tmen t mat ters : e.g. if taxes are only in place for only one period, then the rich always prefer 7 = O. Finally, the model predicts that the grea ter the political power in the hands of the poor the higher is T. 6 This suggests that the empir ical evidence on the relation be tween growth and taxation is compli- ca ted not only because countr ies may differ in terms of preference pa rame- ters or social mobility, but also because they may differ in terms of the political power held by different income groups or in te rms of political stability.

4. A two-period e c o n o m y

The model described above intentionally ruled out weal th effects. In o rde r to discuss the interact ion be tween taxes and the level (in addit ion to

6 This can be easily seen in the above analysis by not ing that the tax ra te chosen by the p o o r is -~. wh ich always exceeds that p r e f e r r ed by the rich; but in a gene ra l i zed vers ion of the m o d e l . w h e r e t he re is a m o r e compl i ca t ed d is t r ibut ion of i ncome , s imilar resul ts apply.

336 R. Wright / Journa l o f Publ ic E c o n o m i c s 62 (1996) 327-3 .38

t he g r o w t h ra te ) o f i n c o m e , it is ins t ruc t ive t , c o n s i d e r a ve r s ion o f the e c o n o m y t h a t lasts on ly two per iods . T h e ut i l i ty func t ion is

U(c t , c - , ) = u (c l ) + Ef lu (c2 ) . (18)

N o f u n c t i o n a l f o r m a s s u m p t i o n s are i m p o s e d on u(c) . In each p e r i o d , a f r a c t i o n n I = p / ( p + A) o f the agen t s have access to w,, w h e r e w~ = W a n d w2 = T W. A g e n t s swi tch s t a tus a c c o r d i n g to the p robab i l i t i e s A and p, so t ha t n~ d o e s no t c h a n g e , as in the m o d e l a n a l y z e d above . T h e s a m e tax r a t e r is i m p o s e d on rich agen t s in b o t h p e r i o d s , and the p r o c e e d s are d i s t r i b u t e d to t he p o o r , w h o each rece ive b, = Tw, p/A, exac t ly as above .

A r ich ind iv idua l in the first p e r i o d e v a l u a t e s ~- a c c o r d i n g to

v(r ) = u[W( 1 -- r ) ] + Aflu(W3rrp/A) + ( 1 -- A)flu[WT(1 -- r ) ] . (19)

A s s u m i n g r > 0 . which is t r u e if u ' ( 0 ) = ~ , and ~ - < ? , wh ich is t rue if f l T ( P + A - - 1 ) < 1, m a x i m i z a t i o n wi th r e spec t to ~" impl ies

v ' ( r ) = - -Wu ' ( c t l ) + WTflpu'(Coz ) -- WT/3 ( I -- A)u'(cl2 ) = 0 , (20)

w h e r e cj, d e n o t e s c o n s u m p t i o n for an ind iv idua l in s ta te j in p e r i o d t, a n d j = 0 o r 1 c o r r e s p o n d s to b e i n g p o o r or rich. Th i s impl ies

0 r -- A/3(1 -- A)u ' (c ,2)[1 -- R(c ,2) ] - At3pu'(c,,2)[1 - R(c, ,2)] . (21)

0T

w h e r e A----- - - W / o " ( r ) > 0 and R ( c ) = - - c u " ( c ) / u ' ( c ) d e n o t e s the coeff ic ient o f r e l a t i v e r isk ave r s ion . A f t e r s o m e a lgeb ra , this can be r e w r i t t e n

Or A W 0r - - - - u ' (c~t ) [1 -- g(c11)] + - - - - (22) OT T T OW"

H e n c e , in g e n e r a l , t he e f fec t o f g r o w t h on r d e p e n d s no t on ly on w h e t h e r R(c) is a b o v e o r b e l o w un i ty , bu t also on the w e a l t h e f fec t OT/OW. In the spec ia l case w h e r e u ( c ) = c ~ - ~ / ( l - a ) , we can eas i ly s h o w OT/OW=O, a n d 0~-/0), d e p e n d s on ly on w h e t h e r a is a b o v e o r b e l o w un i ty , exac t ly as in the i n f in i t e -ho r i zon m o d e l .

5 . C o n c l u s i o n

T o the e x t e n t t h a t r ed i s t r i bu t ive t a x a t i o n can be m o d e l e d as p r o v i d i n g socia l i n s u r a n c e , a n d to the e x t e n t tha t c o n t e m p o r a n e o u s pol icy dec i s ions h a v e s o m e long- l ived c o n s e q u e n c e s , t he r e is an incen t ive for those wi th pol i t i ca l p o w e r to m a k e the i r cho ices wi th an eye to the fu tu re . G i v e n this , t h e i r pol~.cy dec i s ions will d e p e n d on the i r e f fec t ive d i scoun t f ac to r , which i t se l f d e p e n d s on the g r o w t h r a t e o f the e c o n o m y . It has b e e n shown t h a t a

R. Wright / Journal o f Public Economics 62 (1996) 327-33~ 337

rise in the growth rate can increase or decrease the effective discount factor, and therefore can lead to more or less taxation, depending on the r/sk- aversion pa rame te r a . In the empirically relevant case, a > 1, the model predicts that taxes and growth are negatively related. This does not imply that lower taxes lead to higher growth, since in this model growth is exogenous . Hence , even if we can find a negative relat ionship between taxes and growth in the data , we cannot take this as evidence for the pcsit ion that lowering taxes will increase growth.

An impor tan t quest ion concerns the quant i ta t ive relevance of the above analysis. It would be potential ly useful to think carefully about what types of policies can be captured by the redistr/butional tax-transfer system modeled in this paper . Some actual policies seem to fit directly, such as unemploy- men t insurance. Addi t ional , any government spending on things like public parks or educat ion has impor tant redistr ibutional consequences to the extent that their use is not perfectly connected to the tax revenue that an individual contr ibutes; but careful measuremen t of the costs and benefits to different income groups of such public spending is well beyond the scope of this paper . Finally, it should be emphasized that the introduct ion of assets (l ike physical capital or flat money) would mitigate, but presumably not elimi- na te , the effects derived here, because then r :distributive policy would be less impor tan t in terms of smoothing consumpt ion; such assets were not included in the model because they considerably increase the complexity of the analysis. The goal here was to describe the feedback from growth to fiscal policy in a simple model , and to point out some of the o ther / a c t o r s - like inequali ty, risk aversion or mobility, for e x a m p l e - that may lead different countr ies to choose different policies even if they have the same growth rates.

Acknowledgements

I wish to thank And y Postlewaite, Frank Diebold and seminar particip- ants at the Universi ty of Essex, the Federal Reserve Bank of Minneapol is , the N B E R S u m m e r Institute, and the 1994 Society for Economic Dynamics and Contro l meet ings in Los Angeles . Two referees also provided some part icularly insightful comments . The NSF provided financial support . The usual disclaimer applies.

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