Effects of Varying Tax Receipts when Timing of Tax Payments is Indeterminate: The Case of a Tax on Corporate Distributions

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

EFFECTS OF VARYING TAX RECEIPTS WHEN THE TIMING OF TAX PAYMENTS IS

INDETERMINATE: THE CASE OF A TAX ON CORPORATE DISTRIBUTIONS

David F. Bradford

October 1983 CP-83-53

CoZZaborative Papers r e p o r t work which h a s n o t been p e r f o r m e d s o l e l y a t t h e I n t e r n a t i o n a l I n s t i t u t e f o r A p p l i e d Systems A n a l y s i s and which h a s r e c e i v e d o n l y l i m i t e d r e v i e w . V i e w s

o r

o p i n i o n s e x p r e s s e d h e r e i n do n o t n e c e s s a r i l y r e p r e s e n t t h o s e o f t h e I n s t i t u t e , i t s N a t i o n a l - M e m b e r O r g a n i z a t i o n s ,

o r

o t h e r o r q a n i - z a t i o n s s u p p o r t i n g t h e work.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, A u s t r i a

FOREWORD

This Collaborative Paper is one of a series embodying the outcome of a workshop and conference on Economic Structural Change: Analytical Issues, held at IIASA in July and August

1983. The conference and workshop formed part of the con-

tinuing IIASA program on Patterns of Economic Structural Change and Industrial Adjustment.

Structural change was interpreted very broadly: the topics covered included the nature and causes of changes in different sectors of the world economy, the relationship between inter- national markets and national economies, and issues of organi- zation and incentives in large' economic systems.

There is a general consensus that important economic structural changes are occurring in the world economy. There are, however, several alternative approaches to measuring these changes, to modeling the process, and to devising appropriate responses in terms of policy measures and institutional re- design. Other interesting questions concern the role of the international economic system in transmitting such changes, and the merits of alternative modes of economic organization in responding to structural change. All of these issues were addressed by participants in the workshop and conference, and will be the focus of the continuation of the research program's work.

Geoffrey Heal

Anatoli Smyshlyaev

Erne) Zalai

E f f e c t s of Varying Tax Receipts When t h e Timing of Tax Payments i s Indeterminate: The Case of

A Tax on Corporate D i s t r i b u t i o n s

David F. Bradford*

1. I n t r o d u c t i o n .

This paper concerns a n a s p e c t of t h e q u e s t i o n : When do govern- ment d e f i c i t s m a t t e r ? My o r i g i n a l i n t e n t w a s t o s k e t c h o u t as a s t a r t i n g p o i n t t h e e x t e n s i o n of p r e v i o u s work (1981) showing how endogenously generated d e f i c i t s might have no real e f f e c t , a result obtained under a n assumption of p e r f e c t s u b s t i t u t a b i l i t y between

government and p r i v a t e d e b t , t o a world of r i s k y d e b t where t h e l a t t e r assumption would no l o n g e r hold. l f y i n t u i t i o n w a s t h a t t h e e x t e n s i o n would go through in a s t r a i g h t f o r w a r d way. A s i t t u r n s o u t , t h e exten- s i o n is n o t q u i t e a s d i r e c t as I had expected, and I t h i n k t h e d i f - f e r e n c e may be of some g e n e r a l interest. I n p a r t i c u l a r , t h e condi- t i o n s f o r n e u t r a l i t y seem less l i k e l y t o be f u l f i l l e d in a p r a c t i c a l c o n t e x t .

The underlying i d e a i s t h a t it should n o t matter when t a x e s are paid, provided t h e r e i s an a p p r o p r i a t e compensating interest element in t h e postponed l i a b i l i t y . This n o t i o n c o n f l i c t s w i t h t h e assump- t i o n o f t e n employed t h a t i t is t h e gwerrrment's cash flow balance

*

P r i n c e t o n U n i v e r s i t y and National Bureau of Economic Research. I would l i k e t o e x p r e s s my a p p r e c i a t i o n f o r t h e h e l p f u l d i s c u s s i o n s w i t h c o l l e a g u e s a t t h e Workshop on Economic S t r u c t u r a l Change:

A n a l y t i c a l I s s u e s , h e l d a t IIASA during August, 1983.

t h a t counts, w e n though c u r r e n t d e f i c i t s may be o f f s e t by corres- pondingly l a r g e r l i a b i l i t y f o r f u t u r e tax payments, and s u r p l u s e s may r e f l e c t drawing down l i a b i l i t i e s f o r f u t u r e taxes. This i s s u e a r i s e s e x p e c i a l l y s t r o n g l y in t h e c o n t e x t of a n a l y s i s of proposals f o r consumption-type t a x e s , where t h e r e i s a choice between a l i t e r a l consumption t a x and a t a x on wage and t r a n s f e r r e c e i p t s . Typically t h e two approaches g e n e r a t e t h e same l i a b i l i t i e s in a p r e s e n t v a l u e sense, b u t v e r y d i f f e r e n t cash flows.

The s p e c i f i c c a s e I analyze involves a t a x on d i s t r i b u t i o n s by c o r p o r a t i o n s t o e q u i t y h o l d e r s , in essence, a dividend tax. I f such a tax i s a s s e s s e d a t a f l a t r a t e which i s n o t expected t o change, and if n e g a t i v e d i s t r i b u t i o n s ( s a l e s of new e q u i t y ) a r e included

( i . e . , s u b s i d i z e d ) , t h e case i s q u i t e compelling in a p a r t i a l equi- l i b r i u m s e t t i n g , t h a t t h e l e v e l of t h e t a x should have no i n f l u e n c e on real o r f i n a n c i a l t r a n s a c t i o n s of a c o r p o r a t i o n a c t i n g in t h e interest of i t s stockholders. The reason i s simply t h a t t h e f l a t t a x changes p r o p o r t i o n a t e l y t h e consequences of a l l d e c i s i o n s a s f a r as stockholder outcomes are concerned. I n p a r t i c u l a r , t h e trade-off in a f t e r - t a x d o l l a r s f o r t h e s h a r e h o l d e r s between a l a r g e r d i s t r i b u t i o n today and t h e consequently s m a l l e r d i s t r i b u t i o n a t some f u t u r e time is u n a f f e c t e d by t h e rate of tax.

In t h e absence of all t a x e s (and t r a n s a c t i o n s c o s t s ) t h e v a r i o u s v e r s i o n s of t h e Modigliani-Miller (1958) Theorem t e l l us t h e corporation.

w i l l be i n d i f f e r e n t between debt and e q u i t y finance. An i m p l i c a t i o n i s t h a t t h e timing of dividend payments i s a matter of i n d i f f e r e n c e .

Since, a s I have j u s t argued informally, a f l a t t a x on dividends has

no e f f e c t on t h e optimal f i n a n c i a l p o l i c y , something l i k e t h e Modigliani-Miller Theorem should continue t o hold. Hence, govern- ment r e c e i p t s from such a tax will be determined by t h e whims of c o r p o r a t e managers and n e e d f o l L m n o p a r t i c u l a r p a t h as f a r as wealth-maximizing c a l c u l a t i o n s a r e involved. Will f l u c t u a t i o n s in t a x r e c e i p t s , a s might a r i s e in such a n economy, have real e f f e c t s ?

I n Bradford (1981) I s p e l l e d o u t an overlapping-generation

model in which government d e b t does g e n e r a l l y i n f l u e n c e t h e r a t i o n a l e x p e c t a t i o n s e q u i l i b r i u m p a t h , b u t in which v a r i a t i o n s in government debt a t t r i b u t a b l e t o v a r i a t i o n s in d i s t r i b u t i o n t a x r e c e i p t s do n o t matter. The i n d i f f e r e n c e about f i n a n c i a l p o l i c y a t t h e level of t h e f i r m , in s p i t e of t h e tax on d i s t r i b u t i o n s , carries over t o n e u t r a l i t y of t h e economy's p a t h t o t h e choice of f i n a n c i a l p o l i c y ,

even though t h e flow of tax r e c e i p t s is a f f e c t e d .

The b a s i s f o r this conclusion i s f a i r l y simple: The c a p i t a l investment level chosen by t h e f i r m is governed by t h e going i n t e r e s t rate. Therefore, a d e c i s i o n t o i s s u e an extra d o l l a r of debt i m p l i e s a d e c i s i o n t o d i s t r i b u t e an e x t r a d o l l a r t o shareholders. T h i s i n t u r n Implies extra t a x r e c e i p t s of t d o l l a r s (where t is t h e r a t e of t a x ) . I f real government spending i s f i x e d , t h e e x t r a t d o l l a r s a r e devoted t o reducing t h e p u b l i c debt. The result t h u s f a r i s a n e t a d d i t i o n of 1-t d o l l a r s t o t h e supply of f i n a n c i a l assets.

There remains, however, t h e e q u i t y interest in t h e c o r p o r a t i o n , t h e v a l u e of which i s p r e d i c t e d t o f a l l , n o t by one d o l l a r , as a result of t h e e x t r a debt cum d i s t r i b u t i o n , but by 1-t d o l l a r s , in reflec- t i o n of t h e government's claim t o a f r a c t i o n t of all cash flows

t o s h a r e h o l d e r s . As a consequence, a l l markets c o n t i n u e t o c l e a r . The f i r m ' s d e c i s i o n has no real e f f e c t , even though tax r e c e i p t s are i n c r e a s e d .

The a n a l y s i s j u s t sketched s u f f i c e d t o demonstrate t h e p o i n t

t h a t a t a x on c o r p o r a t i o n d i s t r i b u t i o n s may n o t have t h e often-assumed i n c e n t i v e e f f e c t s w i t h r e s p e c t t o real and f i n a n c i a l a l l o c a t i o n s .

The n e u t r a l i t y of t h e t a x as f a r as t h e a g g r e g a t e economy i s concerned was something of a by-product, necessary f o r t h e rkgorous v a l i d i t y of t h e f i r m - l w e l p r o p o s i t i o n , b u t n o t t h e main p o i n t . I n t h e p r e s e n t i n v e s t i g a t i o n t h e emphasis i s r e v e r s e d , in t h a t I am p r i n c i p a l l y con- cerned w i t h how t h e government cash-flow n e u t r a l i t y result i s a f f e c t e d if t h e p e r f e c t s u b s t i t u t a b i l i t y among f i n a n c i a l assets, used in t h e argument above and a t t r i b u t a b l e t o t h e assumption of c e r t a i n t y , i s r e p l a c e d i n t h e c o n t e x t of an e x p l i c i t t r e a t m e n t of u n c e r t a i n t y . Here t h e indeterminacy of t h e f i r m ' s f i n a n c i a l p o l i c y i s needed t o a l l o w governnent r e c e i p t s t o v a r y .

As i t t u r n s o u t , t h e earlier results c a r r y

over

w i t h o u t s i g n i f i - c a n t c o m p l i c a t i o n when o n l y e q u i t y i s r i s k y w h i l e government and c o r p o r a t e d e b t are r i s k f r e e , and t h e r e f o r e p e r f e c t s u b s t i t u t e s . When, however, both government and c o r p o r a t e d e b t are r i s k y , t h e n e u t r a l i t y c o n c l u s i o n s r e q u i r e two r e s t r i c t i o n s . F i r s t , t h e r i s k p r o p e r t i e s of gwernment d e b t m u s t be pre-specified. Second, t h e gwernment must b e conmcitted t o m a i n t a i n i n g a p o r t f o l i o of c o r p o r a t e bonds amounting t o a f r e m t of t h e t o t a l supply. While t h e p r a c t i c a l requirements of implementing t h e f i r s t r e s t r i c t i o n .do n o t seem p a r t i c u l a r l y demanding, t h e second r e s t r i c t i o n would

i n v o l v e a major change i n government f i n a n c i a l behavior.

I n s e c t i o n 2 below I review t h e c e r t a i n t y results a s a way of i n t r o d u c i n g t h e b a s i c model. The e x t e n s i o n t o a model w i t h uncer- t a i n t y i s p r e s e n t e d in S e c t i o n 3.

2. R w i e w of t h e C e r t a i n t y Model.

The model underlying t h e a n a l y s i s i s i n t h e Samuelson (1959) consumption l o a n t r a d i t i o n . I n d i v i d u a l s l i v e f o r two p e r i o d s in an i n f i n i t e tlme h o r i z o n world. I n t h e f i r s t l i f e - p e r i o d each i n d i v i d u a l works ( o f f e r i n g one u n i t of l a b o r i n e l a s t i c a l l y ) , consumes, and saves f o r r e t i r e m e n t . "Retirement" d e s c r i b e s t h e second l i f e-period, when each i n d i v i d u a l d i s s a v e s and consumes, l e a v i n g n o t h i n g t o h i s h e i r s . A l l production t a k e s p l a c e in t h e consolidated c o r p o r a t e s e c t o r which is modeled as a s i n g l e price-taking firm. Production c o n d i t i o n s are d e s c r i b e d by a n e o c l a s s i c a l p r o d u c t i o n f u n c t i o n of c a p i t a l and l a b o r employed, w i t h c o n s t a n t r e t u r n s t o scale. The c a p i t a l a v a i l a b l e t o t h e c o r p o r a t i o n in any p e r i o d i s i n h e r i t e d from t h e previous period and is t h u s f i x e d in amount b e f o r e t h e time of a c t u a l production. The o u t p u t of a p e r i o d may e i t h e r be consumed o r f r o z e n i n t o i n f i n i t e l y d u r a b l e c a p i t a l . Investment i s regarded as r e v e r s i b l e .

Savings may be h e l d in t h r e e forms, bonds i s s u e d by t h e corpora- t i o n , s h a r e s of i t s common s t o c k , o r bonds i s s u e d by t h e government.

A given g e n e r a t i o n of i n d i v i d u a l s a c q u i r e s t h e s e f i n a n c i a l assets a t t h e end of i t s f i r s t l i f e - p e r i o d , a f t e r p r o d u c t i o n f o r t h a t p e r i o d has been completed, and a f t e r t h e investment and f i n a n c i a l p l a n s of t h e c o r p o r a t i o n have been r e a l i z e d .

The h o l d e r s of i t s conrmon s t o c k "own" t h e c o r p o r a t i o n . The owners a t t h e beginning of a p e r i o d c o n t r o l t h a t p e r i o d ' s production and t h e r e a l investment which determines t h e amount of c a p i t a l which w i l l b e a v a i l a b l e f o r u s e in production a t t h e beginning of t h e n e x t p e r i o d . The owners of t h e f i r m a t t h e beginning of a period s p e c i f y a s w e l l t h e f i n a n c i a l p o l i c y f o r t h a t p e r i o d , which means t h e y set t h e amount of f u n d s t o be d i s t r i b u t e d t o themselves a s d i v i d e n d s and t h e amount of c o r p o r a t e borrowing.

The government's real spending program i s assumed f i x e d , f o r s i m p l i c i t y a t a z e r o l e v e l . The government i s t h u s modeled a s a mechanical c a s h f l o w manager: in each p e r i o d t h e i n h e r i t e d d e b t

o b l i g a t i o n must b e p a i d o f f , w i t h any s h o r t f a l l of tax r e c e i p t s covered by t h e i s s u e of new d e b t .

No t a t i o n .

To d e s c r i b e t h e results of t h e a n a l y s i s f o r m a l l y , I u s e t h e f o l l a w i n g n o t a t i o n ( i n v o l v i n g minor changes from t h a t of t h e 1981 paper t o f a c i l i t a t e e x t e n s i o n t o i n c o r p o r a t e u n c e r t a i n t y ) :

L:

t o t a l number of l a b o r u n i t s a v a i l a b l e f o r a p p l i c a - t i o n d u r i n g t h e p e r i o d ( e q u a l s t h e number of

i n d i v i d u a l s born in t h e p e r i o d , exogenously given)

.

K: s t o c k of c o r p o r a t e c a p i t a l a t t h e beginning of t h e p e r i o d (used f o r p r o d u c t i o n d u r i n g t h e p e r i o d ) . B: t o t a l s t o c k of c o r p o r a t e indebtedness a t t h e

beginning of t h e p e r i o d , which must b e r e p a i d d u r i n g t h e p e r i o d .

F(K,L): t h e production f u n c t i o n , c h a r a c t e r i z e d by c o n s t a n t r e t u r n s t o scale.

D: t o t a l d i s t r i b u t i o n made by t h e c o r p o r a t i o n d u r i n g t h e p e r i o d .

gg: t o t a l s t o c k of government indebtedness a t t h e beginning of t h e p e r i o d , which must be r e p a i d d u r i n g t h e p e r i o d .

t : r a t e of t a x on c o r p o r a t e d i s t r i b u t i o n s t o stock- h o l d e r s .

There are two o r d i n a r y p r i c e s in t h e model ( c u r r e n t output i s n u n e r a i r e ) , and one p r i c e - l i k e " v a l u a t i o n function":

w : wage.

d: t h e discotmt rate; a bond payable next period sells f o r d in t h e c u r r e n t period.

V(K+,B+): a f u n c t i o n r e l a t i n g t h e "ex dividend" v a l u e of c o r p o r a t e e q u i t y , t h a t is, t h e v a l u e a t t h e end of a p e r i o d , a f t e r production i s complete, t o t h e f i n a n c i a l and investment d e c i s i o n s of t h e c u r r e n t owners.

For any v a r i a b l e , X, l e t X+, X*, e t c . , r e p r e s e n t i t s v a l u e in succeeding periods; X

-

its v a l u e i n t h e preceding p e r i o d . Let X+ e denote t h e v a l u e of X expected t o o b t a i n in t h e next p e r i o d .

There are t h r e e c l a s s e s of a g e n t s in t h e model, two d i s p l a y i n g 1 2

maximizing behavior. The "young" t r y t o maximize U(c ,c+)

( l a b o r i s s u p p l i e d i n e l a s t i c a l l y ) , where c i i s consumption d u r i n g t h e ith p e r i o d of t h e l i f e c y c l e . The "old" manage t h e c o r p o r a t i o n t o maximize c 2

,

which m e a n s maximize (1-t) D+V(K+,B+)

.

The t h i r d a g e n t i s t h e government which, as has been mentioned, a c t s m e c h a n i c a l l l y in managing t h e cash flow a c c o r d i n g t o dB+ g = B g - t ~ .

Evolution of t h e Economy.

The s i t u a t i o n i n h e r i t e d from the p a s t is d e s c r i b e d by ( K , B , B ~ , L )

,

w i t h L e v o l v i n g exogenously along a known path. The requirements placed on t h e model world are t h a t t h e e v o l u t i o n t o (K+,B+,B: ,L+) be determined by c l e a r i n g of c o m p e t i t i v e "spot" markets f o r l a b o r , c o r p o r a t e bonds, government bonds, c o r p o r a t e e q u i t y and goods, and

that the e x p e c t a t i o n s on which demands and s u p p l i e s depend be " r a t i o n a l . "

The g e n e r a l n o t i o n of r a t i o n a l e x p e c t a t i o n s , a t t r i b u t e d t o Muth (1961), h e r e encompasses two p r o p e r t i e s : E x p e c t a t i o n s are c o r r e c t , and they are determinate, in t h e sense that they a r e governed by knowledge of t h e economic s t r u c t u r e and t h e c u r r e n t state of t h e economy. For

r a t i o n a l e x p e c t a t i o n s t o make sense, t h e r e must be an a p p r o p r i a t e degree of determinacy of t h e model as a whole, i n c l u d i n g i t s e x p e c t a t i o n s - formation. Solvfng t h e model Involves showing t h a t t h e endogenous v a r i a b l e s , i n c l u d i n g p r i c e s and e x p e c t a t i o n s , can b e expressed as s t a t i o n a r y f u n c t i o n s of t h e state v a r i a b l e s , K, B , and Bg, given t h e known p a t h o f L. P r o p e r l y , t h e r e should a s w e l l b e a demonstration t h a t t h e proposed e q u i l i b r i u m p a t h i s a t least l o c a l l y unique-other- wise, why s h o u l d t h e economic a g e n t s p i c k t h e r e q u i r e d q e c t a t i o n s - forming r u l e ?

I n my previous work I d e s c r i b e d a n e q u i l i b r i u m path of t h e economy sketched o u t above. ( I d i d n o t a c t u a l l y demonstrate l o c a l uniqueness, although I t h i n k t h e s o l u t i o n i s probably g l o b a l l y

unique.) The n e u t r a l i t y r e s u l t concerning t h e r a t e of t a x on c o r p o r a t e d i s t r i b u t i o n s followed from t h e conclusion t h a t t h e e q u i l i b r i u m

v a l u a t i o n f u n c t i o n i s given by (1)

.

The o b j e c t i v e of t h e owners of t h e f i r m is t o set employment t o g e t h e r with K+, B+ and D t o n m x i d z e (I-t)D

+

V(K+,B+)

.

^{I f w e sub-}

s t i t u t e f o r D i n t h e o b j e c t i v e f u n c t i o n , u s i n g accounting r e l a t i o n - s h i p (2) among t h e o u t l a y s by t h e f i r m , B+ drops o u t .

Maximizing v a l u e s of the o t h e r v a r i a b l e s are independent of t , w h i l e the f i n a n c i a l s t r u c t u r e of t h e f i r m is i n d e t e r m i n a t e .

Indeterminacy of f i n a n c i a l s t r u c t u r e corresponds t o i n d e t e r - minacy of government t a x r e c e i p t s . The r e a s o n t h e e q u i l i b r i u m p a t h of t h e economy's real v a r i a b l e s and p r i c e s i s n o n e t h e l e s s d e t e r m i n a t e i s suggested by t h e expression (3) of e q u a l i t y between t h e v a l u e of demanded and s u p p l i e d claims t o f u t u r e consumption:

where c+(d,w) 2 i s t h e r e t i r e m e n t - p e r i o d consumption (which w i l l t a k e . p l a c e n e x t p e r i o d ) demanded by a r e p r e s e n t a t i v e young person, where t h e v a l u e s of K+ and B+ have been set by t h e c u r r e n t owners of t h e

f i r m , and where I have taken f o r g r a n t e d w+ e = w+. The e x p r e s s i o n on t h e l e f t i s t h e v a l u e of claims demanded on t h e b a s i s of l i f e - c y c l e o p t i m i z a t i o n by members of t h e young g e n e r a t i o n . The f i r s t two terms on t h e r i g h t are t h e v a l u e s of bonds s u p p l i e d , and t h e t h i r d i s t h e v a l u e of c o r p o r a t e e q u i t y . E x p l o i t i n g t h e government's budget c o n s t r a i n t , dB: = B g - t ~ , and t h e a l r e a d y mentioned accounting rela- t i o n s h i p (2) between D and t h e o t h e r v a r i a b l e s , a l l o w s us t o w r i t e c o n d i t i o n (3) as

Once a g a i n , B+ has been e l i m i n a t e d , s o t h e c o n d i t i o n is independent of c o r p o r a t e f i n a n c i a l p o l i c y .

Equation (5) e x p r e s s e s a f u r t h e r reformulation of t h e same c o n d i t i o n , t a k i n g advantage of E u l e r ' s theorem (an open q u e s t i o n : Can one dispense with linear homogeneity of

F?)

:

The left-hand s i d e of (5) is, as b e f o r e , t h e demand by t h e young genera- t i o n f o r assets. The right-hand s i d e , t h e economy's n e t supply of assets, i s t h e d i f f e r e n c e between gwernment bonds c a r r i e d over from t h e p a s t and t h e tax r e c e i p t s t h a t would be generated if t h e c o r p o r a t i o n were t o b e l i q u i d a t e d in t h e c u r r e n t period. The l a s t term, in o t h e r words, a f f e c t s t h e real e q u i l i b r i u m p a t h like an a n t i c i p a t e d t a x r e c e i p t "asset"

of the government, o f f s e t t i n g e x p l i c i t government d e b t . Through t h i s r e l a t i o n s h i p one o b t a i n s a clear sense of why d e f i c i t s o r s u r p l u s e s due

t o v a r i a t i o n s in c o r p o r a t e d i s t r i b u t i o n s have no e f f e c t on t h e real path of t h e economy, even though government debt does matter. An increase in d i s t r i b u t i o n s simultaneously reduces government debt and t h e p o t e n t i a l tax r e c e i p t asset by equal amounts.

3. I n t r o d u c i n g Uncertainty.

There a r e v a r i o u s ways one might i n t r o d u c e u n c e r t a i n t y t o t h i s model. I have chosen t o make t h e f u t u r e production c o n d i t i o n s de- pendent upon t h e p a r t i c u l a r state ( f o r example, weather c o n d i t i o n s ) occurring. S p e c i f i c a l l y , I assume t h a t t h e investment d e c i s i o n is f i x e d in t h e c u r r e n t p e r i o d , b u t t h e a c t u a l production f u n c t i o n is determined i n t h e next period.

L e t S s t a n d f o r the set of p o s s i b l e states t h a t might o b t a i n in t h e c ~ l r e n t p e r i o d , S+ t h e set of p o s s i b l e states in t h e next p e r i o d , and s o on. The l a r g e r dimensionality of t h e problem r e q u i r e s some new n o t a t i o n . The following d e s c r i b e s my compromise between comprehensiveness and mnemonics. Ln g e n e r a l , v a r i a b l e s a r e now understood t o have a s u b s c r i p t t o d e s i g n a t e t h e s t a t e and t h e w i t h which they are a s s o c i a t e d . Thus, w i&S+, r e f e r s t o t h e wage rate

i

'

r e d l i z e d i n state i next period. Since t h e s t o r y starts with a knowa c u r r e n t s t a t e , we can l e t v a r i a b l e s with no s u b s c r i p t r e f e r t o t h e v a l u e s c u r r e n t l y r e a l i z e d .

I s h a l l a l s o assume t h a t S is f i n i t e in each p e r i o d , and t h a t an o r d e r i n g has been agreed upon f o r t h e states in each p e r i o d , s o we can u s e the' s u b s c r i p t "+" t o r e f e r t o t h e v e c t o r of v a l u e s of a v a r i a b l e o b t a i n i n g in t h e n e x t period. (I make an exception f o r

==+

and I+, which keep t h e i r previous scalar i n t e r p r e t a t i o n . ) Thus,

w

+

e

'

r e f e r s t o t h e v e c t o r of wage r a t e s p r e s e n t l y a n t i c i p a t e d f o r next p e r i o d , w i t h components wi, i&S+, and h a s dimensionality equal t o t h e number of states in S+.

The f i r m and t h e government must now s p e c i f y , i n s t e a d of t h e single-dimensional bonds, v e c t o r s of s t a t e - s p e c i f i c claims, where, f o r example, Bi, iES+, pays one u n i t if state i i s r e a l i z e d n e x t period, and z e r o ff a n o t h e r state i s r e a l i z e d . To i s s u e a r i s k l e s s bond is e q u i v a l e n t t o s e l l i n g one u n i t each of claims

Bi, iES+.

I n s t e a d of t h e s i n g l e discount f a c t o r d, we now have a v e c t o r of p r i c e s of u n i t claims c o n t i n g e n t on t h e n e x t p e r i o d ' s state. I f e

+

i s t h e v e c t o r of ones of a p p r o p r i a t e dimension, e+-d is t h e p r i c e of a riskless bond, Since w e s h a l l want t o continue t o use t h e o r d i n a r y s u b s c r i p t t o refer t o t h e state and time in which a p a r t i c u - l a r v a r i a b l e is r e a l i z e d , f o r example, di, i G + is t h e v e c t o r of discount rates ( a p p l i c a b l e t o claims on o u t p u t two p e r i o d s

hence) r e a l i z e d if s t a t e i occurs n e x t p e r i o d , I s h a l l use parentheses when I wish t o i d e n t i f y a p a r t i c u l a r element of d. Thus, (d) i, i&S+,

is a scalar, m e l y , t h e p r e s e n t p r i c e of a claim t o one unit i f state i occurs next period. The 8-01 d+ r e f e r s t o t h e natrir of d i s - count f a c t o r v e c t o r s , one f o r each p o s s i b l e state t h a t might be r e a l i z e d n e x t period.

We are now in a p o s i t i o n t o study t h e analogues in t h e world of r i s k t o t h e temporary e q u i l i b r i u m r e l a t i o n s h i p s discussed earlier.

We may guess, f o r example, t h a t t h e v a l u a t i o n f u n c t i o n f o r c o r p o r a t e e q u i t y is given by (6).

This is simply t h e v a l u e of t h e v e c t o r of p a y o f f s , c o n t i n g e n t on t h e state r e a l i z e d , t h a t t h e purchaser of t h e e q u i t y e x p e c t s t o o b t a i n in t h e form of d i s t r i b u t i o n p l u s proceeds from s a l e of t h e e q u i t y interest.

To a n a l y z e asset market c l e a r i n g in t h i s c a s e , i t i s n o t s u f f i c i e n t t o l o o k a t t h e a g g r e g a t e v a l u e of p r e s e n t c l a i m s , as w e d i d above. We must now l o o k f o r t h e s t a t e - b y - s t a t e e q u a l i t y of s u p p l i e s and demands of c o n t i n g e n t claims. Recall t h a t c+(v,d) 2 is now t o be i n t e r p r e t e d

a s the v e c t o r of consumption planned by t h e r e p r e s e n t a t i v e young person f o r t h e retirement p e r i o d , c o n t i n g e n t on t h e state r e a l i z e d . A s s e t market c l e a r i n g now r e q u i r e s t h e v e c t o r e q u a t i o n (7)

where, as b e f o r e , w e a r e t a k i n g f o r g r a n t e d t h e d e t e r m i n a t i o n of K+

and w+ via o t h e r e q u i l i b r i u m r e l a t i o n s h i p s , given w and d. I n t h e c e r t a i n t y c a s e w e were a b l e t o u s e t h e government budget r e l a t i o n - s h i p , d.B: = Bg

-

^{t D ,} t o g e t h e r w i t h t h e r e l a t i o n s h i p between D and B+, ^{t o} eliminate both B+ and B! from t h e asset market c l e a r i n g c o n d i t i o n . Under u n c e r t a i n t y , t h e combination of t h e s e two r e l a t i o n - s h i p s is no l o n g e r s u f f i c i e n t . Whereas b e f o r e , c o n s t r a i n i n g t h e v a l u e of t h e govermnent's bond sales o r purchases determined t h e q u a n t i t y

(given d)

,

t h e government now may choose among v a r i o u s combinations of s t a t e - c o n t i n g e n t c l a i m s , ( i . e . , d e a l in bonds of d i f f e r e n t r i s k c h a r a c t e r i s t i c s ) . Moreover, t h e same can b e s a i d of t h e f i r m . Thus, i f w e t a k e as a s t a r t i n g p o i n t t h a t t h e government only i s s u e s r i s k l e s s

bonds (buys riskless bonds in t h e c a s e of n e g a t i v e government d e b t ) , we have s t i l l n o t pinned down temporary e q u i l i b r i u m because t h e r i s k c h a r a c t e r i s t i c s o f t h e f i r m ' s d e b t have n o t been determined. Temporary e q u i l i b r i u m t h e r e f o r e depends on t h e f i r m ' s f i n a n c i a l p o l i c y , b u t , s i n c e t h e f i r m i s a price t a k e r , t h a t p o l i c y is s t i l l a m a t t e r of i n d i f f e r e n c e . The economy may a p p a r e n t l y wander a i m l e s s l y .

I f b o t h t h e govermnent and t h e c o r p o r a t i o n a r e r e s t r i c t e d t o riskless d e b t , t h e argument goes through much a s i t d i d in t h e r i s k - f r e e a n a l y s i s . In t h a t c a s e , extra bonds i s s u e d by t h e c o r p o r a t i o n g e n e r a t e n o t o n l y t h e exact withdrawal in v&ue of government bonds

( a s a consequence of e x t r a taxes p a i d ) r e q u i r e d t o m a i n t a i n f i n a n c i a l market e q u i l i b r i u m in view of t h e a s s o c i a t e d d e c l i n e in t h e v a l u e of

c o r p o r a t e e q u i t y , b u t &so t h e matching change in t h e s t a t e - b y - s t a t e c o n t i n g e n t c l a i m s .

N e i t h e r assumption, r i s k - f r e e government bonds o r r i s k - f r e e c o r p o r a t e bonds is & t o g e t h e r a t t r a c t i v e . I n a time of s t o c h a s t i c i n f l a t i o n rates, government bonds are a n y t h i n g b u t r i s k f r e e . More o b v i o u s l y , c o r p o r a t e bonds do carry d e f a u l t r i s k .

What is r e q u i r e d i s a mechanism t o a s s u r e t h a t , in any p e r i o d , tB+

+

^{:B is} independent of t h e f i r m ' s f i n a n c i a l p o l i c y and pre- determined as f a r as government p o l i c y is concerned. Here is one mechanism t o b r i n g about t h e d e s i r e d result: Make i t government p o l i c y t o purchase a f r a c t i o n t of every c o r p o r a t e bond i s s u e , w h i l e f i x i n g i n advance t h e r i s k c h a r a c t e r i s t i c s of government d e b t . I s h a l l d e s c r i b e t h i s f o r m a l l y n o t as t h e purchase of c o r p o r a t e d e b t , b u t as t h e purchase ( i . e . , reduced supply) of t h e e q u i v a l e n t package

of s t a t e - s p e c i f i c c l a i m s on t h e government. Taken l i t e r a l l y , such a p o l i c y would change t h e e x p l i c i t r i s k c h a r a c t e r i s t i c s of government d e b t . Actual implementation, however, should be imagined a s simul- taneous i s s u e of e x t r a government d e b t and purchase of c o r p o r a t e d e b t . The e f f e c t on t h e government's p o r t f o l i o of s t a t e - s p e c i f i c c l a i m s i s t h e same.

The s u g g e s t e d p o l i c y t h e n is d e s c r i b e d by (8)

where a

+

is a p r e - s p e c i f i e d v e c t o r w i t h dimension e q u a l t o t h e s i z e of S+. (For example, a+ might e q u a l e+, i n which case t h e

government i s s u e s riskless bonds.)

Using t h e a c c o u n t i n g r e l a t i o n s h i p s d e s c r i b i n g D

,

c o n d i t i o n (8) i m p l i e s ( a )

p r o v i d i n g t h e d e s i r e d independence of B:

+

^{tB+ from} B+, w h i l e t h e c h a r a c t e r i s t i a s of government d e b t a r e determined by a.

References

Bradford, David F . (1981), "The Lncidence and Allocation Effects of a Tax on Corporate Distributions," Journal of Public Economics, XV, pp. 1-22.

Muth, J. (1961), "Rational Expectations and the Theory of Price Move- ments," Econometrics, XXIX, pp. 315-335.