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Working Paper 8801

TOBIN'S Q, INVESTMENT, AND THE ENDOGENOUS
ADJUSTMENT OF FINANCIAL STRUCTURE

b y W i l l i a m P. Osterberg

W i l l i a m P. Osterberg i s an economist
a t t h e Federal Reserve Bank o f Cleveland.
T h i s paper i s based on t h e f i r s t c h a p t e r
o f t h e a u t h o r ' s Ph.D. d i s s e r t a t i o n .
The a u t h o r i s g r a t e f u l t o W i l l i a m Brock,
C h r i s F l i n n , Mark G e r t l e r , and Donald Hester
for numerous he1p f u l comments and suggestions.
Working papers o f t h e Federal Reserve Bank
o f Cleveland are p r e l i m i n a r y m a t e r i a l s c i r c u l a t e d
t o s t i m u l a t e d i s c u s s i o n and c r i t i c a l comment.
The views s t a t e d h e r e i n a r e those o f t h e a u t h o r
and n o t n e c e s s a r i l y those o f t h e Federal Reserve Bank
of C l e v e l a n d o r o f t h e Board o f Governors
of t h e Federal Reserve System.
A p r i l 1988

ABSTRACT

This paper asks whether q t h e o r y must be modified t o take account of
f i n a n c i a l s t r u c t u r e by analyzing a q model o f investment i n which f i n a n c i a l
s t r u c t u r e a f f e c t s f i r m value.

The model i s a p e r f e c t f o r e s i g h t model o f

general e q u i l i b r i u m s i m i l a r t o t h a t o f Brock and Turnovsky (1981), except t h a t
there i s a debt- related agency c o s t .

The combination o f the agency c o s t and

taxes t h a t favor debt y i e l d s an i n t e r i o r s o l u t i o n f o r the debt- to- equity
r a t i o , which v a r i e s endogenously.

We f i n d t h a t although q i s i n f l u e n c e d by

f i n a n c i a l s t r u c t u r e , i t i s s t i l l a " s u f f i c i e n t s t a t i s t i c " f o r investment.
However, the model i m p l i e s t h a t analyses which i g n o r e the endogenous
adjustment o f f i n a n c i a l s t r u c t u r e w i l l s y s t e m a t i c a l l y e r r i n p r e d i c t i n g
investment.
To i l l u s t r a t e these p o i n t s , we examine the comparative s t a t i c s and
dynamics o f changing the corporate tax r a t e .
corporate tax r a t e lowers the c o s t o f c a p i t a l .

An u n a n t i c i p a t e d increase i n t h e
The presence o f r e a l agency

costs, however, may cause the c a p i t a l stock t o d e c l i n e .

TOBIN'S Q, INVESTMENT, AND THE ENDOGENOUS
ADJUSTMENT OF FINANCIAL STRUCTURE

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

Analyses o f the l i n k between f i n a n c i a l markets and t h e investment
decisions o f f i r m s have g e n e r a l l y d i f f e r e d between t h e f i e l d s o f
macroeconomics and finance.

Macroeconomic analyses o f investment now focus on

the r e l a t i o n between T o b i n ' s q and the r a t e o f investment.
finance,

I n the f i e l d of

analyses tend t o focus on the r e l a t i o n s between t h e cost of c a p i t a l ,

f i r m s ' value, and f i r m s ' f i n a n c i n g decisions i n t h e presence o f c a p i t a l market
imperfections.

These l a t t e r analyses t y p i c a l l y describe f i n a n c i n g d e c i s i o n s

by the 1everage or debt- to- equi t y r a t i o .

'

Most q models ( f o r example, see Abel and Blanchard C19831, Hayashi C19821,
o r Von Furstenberg C19771) assume t h a t n e i t h e r the market value o f a f i r m nor
i t s cost o f c a p i t a l i s a f f e c t e d by the d e c i s i o n o f how investment i s
financed.

Analyses o f c a p i t a l market i m p e r f e c t i o n s (see Bradley,' J a r r e l l , and

Kim C198411, on t h e o t h e r hand, f a i l t o p r o v i d e i n s i g h t about the dynamic
r e l a t i o n between f i n a n c i a l s t r u c t u r e and investment.

I n t h i s paper, we

i n v e s t i g a t e whether q theory needs t o be m o d i f i e d t o take account o f f i n a n c i a l
structure.
The f a i l u r e o f q t o perform w e l l e m p i r i c a l l y i s another source o f
m o t i v a t i o n t o modify q theory.

Although q theory i m p l i e s t h a t past values o f

q should n o t m a t t e r i n a r e g r e s s i o n o f investment r a t e s on c u r r e n t and p a s t q,
most e m p i r i c a l t e s t s (see Abel and Blanchard C19831, Hayashi C19821, o r Von
Furstenberg C19771) f i n d t h a t c u r r e n t q has low explanatory power and t h a t
r e s i d u a l s are h i g h l y c o r r e l a t e d .

These e m p i r i c a l r e s u l t s suggest t h a t we

examine 1i n k s between f i n a n c i a l markets and investment o t h e r than q.

I n t h i s paper, the debt- to- equity r a t i o a f f e c t s f i r m value and i s
determinate, l y i n g between 0 and i n f i n i t y .

The optimal debt- to- equi t y r a t i o ,

chosen by f i r m s , i s determined by agency costs o f debt together w i t h t a x r a t e s
f a v o r i n g debt.

The agency c o s t o f debt a r i s e s from the presence of bond

covenants and o t h e r l e g a l o r i n s t i t u t i o n a l r e s t r i c t i o n s on f i r m s .

Jensen and

Meckl i n g (1976) and o t h e r s have shown how agency problems may be r e l a t e d t o
financial structure.

Tax r a t e s f a v o r debt issue, since corporate i n t e r e s t

payments are tax- deductib l e f o r f i r m s .

D i f f e r e n c e s between the personal

income t a x r a t e , the c a p i t a l gains tax r a t e , and the corporate tax r a t e have
a1 1 been c i t e d as determinants o f an optimal f i n a n c i a l s t r u c t u r e f o r firms and
t a x c l i e n t e l es among i n v e s t o r s (see M i 1l e r C19771).

Reasonable values f o r a1 1

t h r e e t a x r a t e s imply t h a t tax r a t e s f a v o r debt over e q u i t y .
The model u t i 1i z e s frameworks developed by Abel and Blanchard ( 1 983) and
Brock and Turnovsky (1981).

I n t h i s dynamic, general e q u i l i b r i u m model of

savings and investment, the r e l a t i o n between i n v e s t o r s ' p o r t f o l i o d e c i s i o n s
and the decisions o f f i r m s i s c l e a r l y exposed.

The l i n k between households

and f i r m s i s the c o s t o f c a p i t a l , d r i v e n by the r a t e s o f r e t u r n r e q u i r e d by
households.

Firms, f a c i n g t a x r a t e s f a v o r i n g debt, and a cost of debt, choose

the debt- to- equity r a t i o i n o r d e r t o t o minimize the cost o f c a p i t a l .
V a r i a t i o n s i n t a x r a t e s o r i n the i n t e r e s t r a t e a f f e c t the trade- off between
debt and e q u i t y and, thus, the optimal debt- to- equity r a t i o .

We f i n d t h a t even i f f i n a n c i a l s t r u c t u r e a f f e c t s f i r m value, q i s a
" s u f f i c i e n t s t a t i s t i c " f o r investment.

Thus, u n l i k e C h i r i n k o (1987), we

conclude t h a t f i n a n c i a l s t r u c t u r e does n o t e x p l a i n the poor performance of
T o b i n ' s q.

F i n a n c i a l s t r u c t u r e a f f e c t s q, however, because q i s the present

discounted value o f a f t e r - t a x marginal products o f c a p i t a l and because t h e
d i s c o u n t r a t e ( c o s t o f c a p i t a l 1 v a r i e s w i t h the debt- to- equi t y r a t i o .

Indeed,

the endogenous adjustment o f f i n a n c i a l s t r u c t u r e has r e a l e f f e c t s .

For

example, when i n t e r e s t r a t e s r i s e , f i r m s o f f s e t p a r t o f the e f f e c t on t h e c o s t
of c a p i t a l by i n c r e a s i n g the debt- to- equity r a t i o t o take advantage o f t h e
i n t e r e s t - d e d u c t i b i l i t y o f debt.

This i m p l i e s t h a t models i g n o r i n g r e a l

effects o f financial structure w i l l systematically e r r i n predicting
investment.
I n a d d i t i o n , one f a c t o r t h a t may p o t e n t i a l l y break the l i n k between
marginal q and investment i s uncovered.

U n a n t i c i p a t e d changes i n t h e

corporate t a x r a t e may cause marginal q and investment t o move i n o p p o s i t e
directions.

The comparative s t a t i c s and dynamics o f an u n a n t i c i p a t e d change

i n t h e corporate tax r a t e a r e analyzed t o demonstrate the s i g n i f i c a n c e o f t h e
endogenous adjustment o f f i n a n c i a l s t r u c t u r e .

11.

D e s c r i p t i o n o f the Model
The model i s a v a r i a n t o f the p e r f e c t f o r e s i g h t models o f general

e q u i l i b r i u m o f Abel-Blanchard (A-B> and Brock-Turnovsky (B-TI.

F o l l o w i n g B-T,

I i n d i c a t e the way i n which consumers' demand f o r savings i n f l u e n c e s t h e c o s t
o f capital.

The a p p r o p r i a t e form f o r the c o s t o f c a p i t a l i s determined

e x p l i c i t l y from the f i r m s assumed o b j e c t i v e .
costs i n t h e model i s i n t h e s p i r i t o f A-B.

The i n c o r p o r a t i o n o f adjustment
Costly financial structure

provides t h e unique l i n k between f i n a n c i a l markets and r e a l decisions.
The model c o n s i s t s o f t h r e e sectors:

consumers, f i r m s , and government.

Because a l l consumers are assumed t o be i d e n t i c a l , t h e a n a l y s i s i s conducted
i n terms o f a r e p r e s e n t a t i v e consumer.

S i m i l a r l y , a l l f i r m s are assumed t o be

i d e n t i c a l , and the corporate s e c t o r i s aggregated t o a s i n g l e f i r m .

Both t h e

consumer and the f i r m a r e i n f i n i t e l y l i v e d and solve e x p l i c i t maximization
problems by making f o r e c a s t s o f v a r i a b l e s r e l e v a n t t o these decisions.

I

assume t h a t a l l expectations o r f o r e c a s t s are f u l f i l l e d , t h a t a l l markets
c l e a r , and t h a t a l l supply and demand f u n c t i o n s o f f i r m s and households a r e
d e r i v e d from maximizing behavior under p e r f e c t competition.
The consumer saves i n the form o f e i t h e r bonds o r e q u i t i e s .

Being

concerned o n l y w i t h r a t e s of r e t u r n on a l t e r n a t i v e assets, t h e consumer's
problem i m p l i e s t h a t i f debt and e q u i t y are t o c o e x i s t , then a f t e r - t a x r a t e s
of r e t u r n must be equal.

Since t h e r e t u r n s are equal, the consumer would seem

t o be i n d i f f e r e n t about the debt- to- equity r a t i o .

The f i r m , however, i s n o t

i n d i f f e r e n t about the debt- to- equity r a t i o .
The f i r m maximizes i t s market value by choosing sequences o f n o t i o n a l
demands and supplies.

Given a s e t o f f i n a n c i a l / a c c o u n t i n g c o n s t r a i n t s , market

value maximization i s e q u i v a l e n t t o maximization o f a p a r t i c u l a r present
d i scounted value.

The d i scount r a t e i ncorporates the r a t e o f r e t u r n r e q u i r e d

by t h e consumer, the t a x r a t e s f a c i n g the consumer and the f i r m , and the
agency c o s t o f debt.

Because o f t h e presence o f an agency c o s t and a t a x

s t r u c t u r e t h a t f a v o r s debt, t h e r e i s an optimal debt- to- equity r a t i o which
minimizes the cost o f c a p i t a l and l i e s between 0 and i n f i n i t y .

Without t h e

agency c o s t , the optimal debt- to- equity r a t i o would be 0 o r i n f i n i t y ,
depending on the tax r a t e s .
The government sets t h e t a x r a t e s

T,

T,,

and

consumption goods, b u t issues no money o r debt.
statement i d e n t i t y r e q u i r e s t h a t t h e lump-sum t a x

111.
A.

T,,

and consumes

The government income
T

v a r y endogenously.

The S t r u c t u r e
The Consumer

The u t i l i t y o f the consumer i s a f u n c t i o n o n l y o f consumption: t h e r e i s no
d i s u t i 1it y associated w i t h l a b o r .

The f i r m determines employment.

Individual

consumers view themselves as unable t o i n f l u e n c e the s i z e o f the lump sum
tax.

The household's o b j e c t i v e i s t o choose the sequence

t = CO,

{ c t , bz, E;),

m)

t o solve
m

(1)

MAX

S

e-"U(ct)dt

0
subject t o

(3)

lim
t-

get b t -> 0, l i m get ztEt 2 0,
t-

and
(4)

bt=o = bo,

E L 0

=

Eo.

c t = r e a l p r i v a t e consumption
t t = r e a l employment

b t = demand f o r r e a l corporate bonds
Et = number o f shares o f e q u i t y demand
z, = r e l a t i v e p r i c e o f e q u i t y i n terms o f o u t p u t

w t = r e a l wage r a t e
s t = r e a l i n t e r e s t r a t e on p r i v a t e bonds
r, = t a x r a t e on c a p i t a l gains
r, = t a x r a t e on wage and i n t e r e s t income

lump sum tax determined as r e s i d u a l from government income i d e n t i t y

r

=

d

= d i v i d e n d pay.out r a t e : = D/zE

D '= r e a l dividends
I3 = r a t e o f time discount

8 = r a t e o f r e t u r n on consumption, d e f i n e d below

The u t i l i t y f u n c t i o n U ( c t ) has the usual concavity p r o p e r t i e s :

< 0, U1(0)

0, U"(

>

Expression (2) i s the intertemporal budget

=

c o n s t r a i n t f a c i n g the consumer.

Income i s received as l a b o r income, i n t e r e s t

income, d i v i d e n d income, and c a p i t a l gains.
income are taxed a t the r a t e
2,.

U1(

T,.

Labor, i n t e r e s t , and dividend

C a p i t a l gains are taxed a t the r a t e

A l l c a p i t a l gains and losses on e q u i t y are immediately r e a l i z e d .

With t h i s income, the household consumes o r purchases bonds o r e q u i t i e s .
Expression ( 3 ) s t a t e s t h a t t h e values o f debt and e q u i t y are bounded.
bonds mature instantaneously.
stock o f debt and equities,

All

Expression ( 4 ) s t a t e s i n i t i a l conditions on t h e
The outcome o f t h i s o p t i m i z a t i o n problem w i l l be

sequences o f n o t i o n a l demands, where unambiguous t s u b s c r i p t s are omitted.
This problem leads t o t h e f o r m u l a t i o n o f the f o l l o w i n g Hamiltonian:

(5) H

= e-"

U { [ ( ~ O + ~ b ~ + d z E ~ ) ( l - r , )T-

- bd -

, ~ E ~

zid

- TI
+ a(bd) +

((id))

where a and ( are t h e c o s t a t e v a r i a b l e s associated w i t h b and E,
respectively.

I f both debt and e q u i t y are t o be h e l d by the household, then a f t e r - t a x
r a t e s o f r e t u r n must be equal.

The existence o f both debt and e q u i t y w i 11 be

optimal f o r the f i r m and, hence, both assets w i 11 be h e l d by the household.
This i m p l i e s t h a t c o n d i t i o n (6) w i l l hold:
(6)

~(1-T,)

=

8 = (~-T,)z/z

+ d(1-2,).

8 i s the r a t e of r e t u r n on consumption:

B.

8 = 13

-

U,,c/U,.

TheFirm
The f i r m maximizes t h e market value o f debt p l u s e q u i t y a t time t = 0,

V,=,

= bt,o

+

zt,oEt,o.

Given a s e t o f f i n a n c i a l and production

c o n s t r a i n t s , and numerical values f o r KO, b o , and Eo, maximizing V t Z o

i s e q u i v a l e n t t o maximizing a p a r t i c u l a r present discounted value.
equivalence i s demonstrated i n appendix A.

This

The appropriate discount r a t e o r

c o s t o f c a p i t a l i s determined by t h i s equivalence.

These c o n s t r a i n t s are:

y s = F(Kd, Id)

(7)

where
Id = r e a l l a b o r demand by the f i r m

K d = r e a l c a p i t a l demand by the f i r m
n = r e a l gross p r o f i t s
6 = exogenous r a t e o f d e p r e c i a t i o n

b s = r e a l supply o f corporate bonds

X

= bIzE, t h e debt- to- equity r a t i o

y s = r e a l o u t p u t , the supply o f goods
q( 1 = adjustment c o s t of i n v e s t i n g

a(X)b = c o s t o f m a i n t a i n i n g the bond p o r t f o l i o
RE = r e t a i n e d earnings

Es = r e a l supply o f e q u i t i e s
2, =

corporate t a x r a t e
Expression ( 7 ) i s a production f u n c t i o n w i t h p o s i t i v e b u t d i m i n i s h i n g

marginal p r o d u c t i v i t i e s and constant r e t u r n s t o scale.

Expression (8) defines

r e a l gross p r o f i t s as revenue minus the wage b i l l and adjustment costs
associated w i t h gross investment.

Expression (9) i n d i c a t e s t h a t gross p r o f i t s

go t o bondholders as i n t e r e s t , t o the government as taxes, t o stockholders as

dividends, i n t o r e t a i n e d earnings, o r are absorbed by the agency cost.
Dividends a r e p a i d o u t a t a f i x e d r a t e d.
taxable income.

I n t e r e s t payments are deducted from

Expression (10) s t a t e s t h a t a l l investment must be f i n a n c e d

through r e t a i n e d earnings o r debt issue.

There i s no e q u i t y issue, although

there i s an i n i t i a l stock o f e q u i t y , Eo.

Expression (11) s t a t e s t h a t a l l

investment i s n e t investment o r replacement investment where d e p r e c i a t i o n
occurs a t a constant r a t e 6.

There i s no deduction f o r depreciation.

Expression (12) s t a t e s the i n i t i a l c o n d i t i o n s f o r c a p i t a l , e q u i t y , and debt.
The adjustment c o s t q(I,K)

i s deducted f r o m t a x a b l e income, since

increases i n the r a t e o f investment draw resources o f the f i r m away from
productive a c t i v i t i e s .

The adjustment c o s t f u n c t i o n q(I,K) i s assumed t o

be 1i n e a r l y homogenous so t h a t i t can be w r i t t e n as h ( I / K ) I w i t h the
p r o p e r t i e s h(0) = 0, hl(O> > 0, 2 h 1 (

+

( I / K ) h t ( 1 > 0.

This i s s i m i l a r t o

f o r m u l a t i o n s o f adjustment costs proposed by Hayashi (1 982) and o t h e r s .
The c o s t a(X)b i s assumed n o t t o reduce o u t p u t by drawing on p r o d u c t i v e
resources.

While o n l y the i n t e r e s t r a t e on debt i s tax- deductible, the gross

cost o f debt t o t h e f i r m i s Cs + g t X ) l b .

The s i z e o f the f i r m does n o t

a f f e c t t h i s cost d i r e c t l y , b u t t h e f i r m ' s value does.

I f the f i r m ' s value

increases due t o increases i n e q u i t y share p r i c e s , the debt- to- equity r a t i o
fa1 1s.

For a g i v e n f i r m value, increases i n debt lead t o an increase i n t h e

agency cost.
This c o s t i s b e s t viewed as an agency c o s t .

Smith and Warner (1979) p o i n t

o u t t h a t t h e existence o f bond covenants can be viewed as a method o f
c o n t r o l l i n g the c o n f l i c t between bondholders and stockholders.

I t i s assumed

here t h a t bond covenants succeed i n e l i m i n a t i n g t h e c o n f l i c t , and thus
firm- value maximization, r a t h e r than equity- value maximization, i s the proper
f i r m objective.

Without such r e s t r i c t i o n s ,

Bond covenants t y p i c a l l y r e s t r i c t debt issue.

For

stockholders would issue more debt and i n c u r g r e a t e r agency costs.
example, i t may be t h a t the i n c e n t i v e f o r stockholders t o s h i f t t o

higher- variance investment p r o j e c t s increases w i t h the face value of debt (see
Barnea, Haugen, and Senbet C19851).
of the f i r m ,

I f such a s h i f t decreases the t o t a l value

then bondholders, a n t i c i p a t i n g such actions, would demand h i g h e r

interest rates.

The increase i n the i n t e r e s t r a t e demanded by bondholders i s

a type o f agency cost.

Here I assume t h a t bond covenants are n e g o t i a t e d t o

r e s t r i c t the l e v e l o f debt, f o r a given value o f e q u i t y .

The higher t h e

debt- to- equity r a t i o , the more 1 i k e l y t h a t the covenant w i 11 be v i o l a t e d ,
r e s u l t i n g i n r e s t r i c t i o n s on investment a c t i v i t i e s and a decrease i n f i r m
value.

Thus, the c o s t o f i s s u i n g b u n i t s o f debt increases w i t h t h e

debt- to- equity r a t i o . I assume t h a t a(0) > 0, a ' ( 1 > 0, and a " (
Appendix A demonstrates how expressions (7)-(12)

can be used t o d e r i v e

expression (13):

Y, t h e cash f l o w , i s defined as

r,

t h e c o s t o f c a p i t a l , i s defined as

Now t h e f i r m ' s problem i s t o choose the sequence

a

(16) MAX
subject t o

J'e
0

- SiT(z>dz

=

Y( t 1 d t

KO, b t Z o = b o .

1 > 0.

Kt,

c:, bs)

Note t h a t Y i s s o l e l y a f u n c t i o n o f " r e a l " v a r i a b l e s Kt and It,
whereas

r

i s a f u n c t i o n o f o n l y " f i n a n c l a l " v a r i a b l e s summarized by 1,.

These considerations imply t h a t the f i r m can optimize i n t h e f o l l o w i n g
sequence:

first,

choose K: and

It

t o maximize Y(t),

then choose

At t o minimize T ( t ) .

I f a(X) = 0, the f i r m would choose X as f o l l o w s :
(17) s e t X =

rn

if (1 =

T,)

> (1 - ~ , )-(Tl ~

-

T,)

< (1

(18) s e t X = 0 i f (1
where

T,

= dr,+

(;/z)T,
d + ilz

-

~,)(1-

)

T,)

i s the marginal t a x r a t e on e q u i t y income.

Expressions (17) and (18) imply t h a t i f the a f t e r - t a x income from bonds
exceeds the a f t e r - t a x i ncome from e q u i t y , i n v e s t o r s p r e f e r debt f inance.
Given reasonable values o f d,

T,

T,,

and

T,,

(17) i s l i k e l y t o be

the case and w i l l be a ~ s u m e d . ~C o n d i t i o n (17) i n c o n j u n c t i o n w i t h a c o s t t o
debt a(X)b r e s u l t s i n an optimal X:

0 < X <

The f i r m ' s problem leads t o the f o l l o w i n g Hamiltonian f o r m u l a t i o n :

where q'and Q are the c o s t a t e v a r i a b l e s associated w i t h K and b,
respectively.

Expressions (20) through (23) a r e the o p t i m a l it y c o n d i t i o n s f o r

the f i r m .

(24)

(TVC) l i m p t q t K t = 0 = l i m ptQtbt
tt-

-Sh T ( ~ > d z
where p , = e

Expression (21) s t a t e s t h a t marginal q d i f f e r s from one by the a f t e r - t a x
d e c l i n e i n r e a l cash f l o w due t o i n s t a l l a t i o n costs and i m p l i e s t h a t the
Expression (22) i s

investment r a t e i s an i n c r e a s i n g f u n c t i o n o f marginal q.

t h e a r b i t r a g e c o n d i t i o n t h a t t h e shadow r e t u r n from h o l d i n g c a p i t a l must equal
t h e r e q u i r e d r e t u r n on c a p i t a l ,

(r +

6)q.

Expression (22) can be

i n t e g r a t e d s u b j e c t t o the t r a n s v e r s a l i t y c o n d i t i o n (24) t o o b t a i n

Thus, q equals the present discounted sum o f a f t e r - t a x marginal products
of a u n i t o f c a p i t a l i n s t a l l e d a t time t.

T,

r

depends on tax r a t e s and

Expression (23) s t a t e s the r e l a t i o n between X,

t h e debt cost, so does q.

8, and

Since

when the f i r m chooses X so as t o minimize

r.

Since t h e r e

i s no e q u i t y issue, X i s adjusted by v a r y i n g the b versus RE f i n a n c i n g m i x .
Having chosen the p a t h o f the c a p i t a l stock, employment, and t h e
debt- to- equi t y r a t i o , the f i r m has maximized V t = o , and the r e s u l t i n g i n i t i a l
share p r i c e i s determined by the c o n d i t i o n b t Z o + zt,oEt=o = V t Z o .

C.

Government
The government i s c h a r a c t e r i z e d by an income statement i d e n t i t y :

(26)

r, [ w l

+

sb + dzEl

The government sets

T,

+
T,

T
E
;,
T,,

+

T,[Y

and g.

v a r i e s so as t o s a t i s f y the i d e n t i t y .
impact o f a change i n

T

- w l - q(I,K)

-

s b l + r = g.

The lump sum tax
Although

T

T

a d j u s t s endogenously, t h e

i s ignored i n the dynamic a n a l y s i s t h a t follows

because i n d i v i d u a l consumers take

T

as given p a r a m e t r i c a l l y .

IV.

Perfect Foresight Equilibrium
A p e r f e c t f o r e s i g h t e q u i l i b r i u m i s a sequence o f p r i c e s and q u a n t i t i e s f o r

which n o t i o n a l demands and suppl i e s f o r bonds and e q u i t i e s a r e equal i n t h e
p r e s e n t and f u t u r e .

Given t h e o p t i m a l i t y c o n d i t i o n s o f consumers and f i r m s

and t h e government income statement i d e n t i t y , expressions (27)-(32)
t h e sequence o f v a r i a b l e s , {Kt, c t ) , t=[O,

determine

a).

Here I have simp1 i f i e d t h e model b y s e t t i n g
assume t h a t employment i s g i v e n exogenously.

T, =

TC

= 0.

I a1 so

Together w i t h expression (201,

t h i s i s analogous t o assuming t h a t t h e r e a l wage a d j u s t s t o e q u i l i b r a t e t h e
l a b o r market.

So, c o n d i t i o n (20) i s n o t needed below.

Expression (28) i s a

r e s t a t e m e n t o f t h e m a t e r i a l balance c o n s t r a i n t which uses t h e f a c t s :
CXl(1

+

b =

X I I V , V = qK.

Expression (31) g i v e s t h e o p t i m a l d e b t - t o - e q u i t y r a t i o and i n d i c a t e s t h a t
a h i g h e r i n t e r e s t r a t e or c o r p o r a t e t a x r a t e means a h i g h e r d e b t - t o - e q u i t y
r a t i o as t h e v a l u e o f t h e i n t e r e s t d e d u c t i o n r i s e s .

To see t h a t (31) i m p l i e s

an o p t i m a l X, d e r i v e e x p r e s s i o n (33) f r o m e x p r e s s i o n (32) f o r g i v e n
and 8.

T,

Figure 1 shows how expression (33) i m p l i e s t h a t an increase i n

from

T,

T,O

t o .rPl leads t o an increase i n the optimal X from Xz t o 1:.
An increase i n

T,

makes a higher X optimal.

Expressions (31) and

(32) together show t h a t t h e minimized cost o f c a p i t a l i s equal t o
(34)

r = 8 -

~'(x)x'.

The decrease i n the c o s t o f c a p i t a l due t o an increase i n
by the

T,I

l i n e i n f i g u r e 1.

T,

i s indicated

However, expression (28) i n d i c a t e s t h a t an

increase i n the debt- to- equity r a t i o , which reduces t h e c o s t o f c a p i t a l , may
increase the debt cost.

The change i n a(X)b depends on t h e change i n X,

t h e response o f q, and t h e response o f K.

These r e l a t i o n s h i p s a r e examined

below.

V.

Steady State and Comparative Results
I n the steady s t a t e , K = 0 and c = 0. The f i r s t c o n d i t i o n i m p l i e s t h a t

I = 6K and thus I I K

=

6.

The second impl i e s t h a t 8 = B:

the steady

s t a t e i n t e r e s t r a t e i s the r a t e o f time preference o f consumers.

The

c o n d i t i o n t h a t 8 = B, t o g e t h e r w i t h (301, impl i e s t h a t q = 0. The
debt- to- equi t y r a t i o must a1 so be constant i n the steady s t a t e .
t h a t d=B.

Expressions (35)-(38)

describe the steady s t a t e :

This impl i e s

-

14

-

Figure 1

The E f f e c t o f an Increase i n

T,

on t h e Cost o f C a p i t a l ( T I

The impact o f i n c r e a s i n g the corporate t a x r a t e i s given by:

where

dq

=

-

Ch(S)+6h1(6>1 < 0,

dz P

a

a

@=
@ < 0,
< 0, and
dzP dX dzP
dl

The s i g n of dK/dzp i s unclear because o f two competing e f f e c t s .
F i r s t , the increase i n
X optimal.

T,

reduces the c o s t o f c a p i t a l by making a g r e a t e r

This i s c l e a r from expression (34).

Second, the r e d u c t i o n i n
The s i g n

the a f t e r - t a x r e t u r n from a u n i t o f c a p i t a l reduces steady s t a t e q.
of dc/dzp i s a l s o unclear.
would be -CF,-6-h(6)I

I n the absence o f the agency cost, d c / d ~ p

dK/dzP.

Then the requirement f o r steady s t a t e

c t o change i n the same d i r e c t i o n as K would be:

< 0, where A(X>H(

t h e analogous v e r s i o n i s FK-6-h(6>6-AtX>H(
the p e r - u n i t o f c a p i t a l agency cost.

FK-6-h(6)6 < 0.
)

Here,
is

This c o n d i t i o n , however, i s n o t

s u f f i c i e n t f o r steady s t a t e K and c t o move i n the same d i r e c t i o n , since the
increase i n z P a l s o a f f e c t s X, q, and K.

Note t h a t d c / d ~ pcan be

w r i t t e n as:

where the l a s t t h r e e terms equal dEa(X>bl/dz,.

Although X w i l l

increase, t o t a l agency costs may f a l l , since the value o f the f i r m may f a l l .

VI.

Dynamic Behavior
As i s w e l l known, i n dynamic r a t i o n a l - e x p e c t a t i o n s models, endogenous

v a r i a b l e s may overshoot the new steady s t a t e .

Consequently, i t i s useful t o

d i s t ingui sh between s hort- run and 1ong-run responses t o pol i cy changes.

In

t h i s s e c t i o n I describe the movement o f the system about the steady s t a t e i n
response t o an u n a n t i c i p a t e d increase i n

T,.

The movement can be

described by changes i n the r a t e o f investment, x = I I K , and the c a p i t a l
stock.

This s i m p l i f i c a t i o n recognizes t h a t the c o s t o f c a p i t a l v a r i e s w i t h

the i n t e r e s t r a t e and the debt- to- equity r a t i o , both o f which vary w i t h t h e
r a t e of change o f consumption and marginal u t i l i t y .

The f o l l o w i n g system i s

derived i n appendix B by expanding t h e system described by expressions
(27)-(32)

about the steady s t a t e :
X ~ X *

(42)

(47)

-

A1

n2

KIK
K *
o

C1 = 1

+

H(G,T,)KA'~K
zE

X- X*

K- K*

Ucc
Uc

Here I have made use o f t h e f o l l o w i n g d e f i n i t i o n s : 1) H O = l + ( l - ~ , ) q , = q;

2) H I 0 = dHO1dx; 3) r ' = d r l d e ; 4) A ' ( >

= dA(X)/dX;

and 5) h ' 0

=

dh(x)/dx.

Since investment (I)
can change instantaneously, x i s the "jump" v a r i a b l e
f o r which d i s c o n t i n u i t i e s can occur.

Since K i s predetermined w h i l e x depends

on the f u t u r e p a t h o f the system, i n o r d e r f o r the system t o be s t a b l e , t h e r e
must be one p o s i t i v e and one negative eigenvalue f o r the m a t r i x i n (42).

This

c o n d i t i o n assures me o f a jump o n t o the s t a b l e arm o f the system i n i t s
movement toward t h e new steady s t a t e .

< 0, -Kn2 < 0,

(48) d e t

From (43)-(471,

To assure t h i s s t a b i l i t y I assume t h a t :

n2 > 0.

i t i s c l e a r t h a t expression (49) i s s u f f i c i e n t f o r (48) t o

hol d.
(49)

C1 > 0
The second term i n C1 r e f l e c t s t h a t an increase i n consumption

accompanies a r i s i n g i n t e r e s t r a t e and an increase i n e q u i t y values.

An

increase i n e q u i t y values reduces the r a t e o f change o f the debt- to- equi t y
r a t i o , thus reducing the r a t e o f change o f the debt cost.

Condition (49)

s t a t e s t h a t the d i r e c t e f f e c t o f r i s i n g consumption on aggregate demand
outweighs t h e i n d i r e c t e f f e c t o p e r a t i n g through the debt cost.

So, t o assure

C1 > 0, I assume t h a t the r a t e o f change o f demand f o r goods v a r i e s
p o s i t i v e l y w i t h the r a t e o f change o f consumption.
Note t h a t f o r the s t a b l e eigenvalue Q < 0 and the corresponding
eigenvector, (XI,

X2I1,

Expression (50) imp1 i e s t h a t t h e movement o f x and K f r o m t h e i r steady
s t a t e values can be d e s c r i b e d b y expression (51):
(51) x t

-

x* = ylw2eat, K t - K* = Y , ( n

-

v1)eQt.

Y1 i s a c o n s t a n t determined f r o m t h e f a c t t h a t K remains f i x e d a t t h e
t i m e of t h e announcement and implementation o f an u n a n t i c i p a t e d change i n
Having determined Yl as d e s c r i b e d i n appendix B, t h e movements o f

2,.

x and K about t h e steady s t a t e a r e described b y expressions (52) and (53).
(52)

xt

-

x* =

(-

d K / d ~ , ) d ~ , w ~ e ' ~ / ( Q- w,)

(53)

Kt

-

K* =

(-

dK/d~,)d~,e~'

These i m p l y x t

-

x ZO as dK/d-~, ZO.

I n o t h e r words, i f t h e steady s t a t e

c a p i t a l s t o c k increases, investment i s above i t s steady s t a t e v a l u e d u r i n g t h e
adjustment p e r i o d .

I n f a c t , x must immediately i n c r e a s e a t t i m e t = 0.

The

jump is computed as :

VII.

Dynamic R e l a t i o n s h i p s Between T o b i n ' s q , t h e Debt- to- Equity R a t i o , and

t h e Rate o f Investment
Because t h e movements o f q and x a r e c l o s e l y r e l a t e d by e x p r e s s i o n (211,
on t h e s t a b l e adjustment path, s i g n

[;I =

sign

Cql.

The new steady s t a t e

v a l u e o f q w i 11 be lower, s i n c e z P has increased.
I n i t i a l l y , however, q and x may jump i n d i f f e r e n t d i r e c t i o n s .

The

i n c r e a s e i n t h e r a t e a t which c u r r e n t r e t u r n s a r e taxed tends t o decrease q a t
t h e t i m e o f t h e announcement.

The movement o f q a l o n g t h e s t a b l e arm can be

d e s c r i b e d by expresssion (561, where q: i s t h e new steady s t a t e value:

The jump i n q a t t = 0 i s

If steady s t a t e K increases, q may e i t h e r r i s e o r f a l l i n i t i a l l y .

These r e s u l t s a r e

However, i f steady s t a t e K f a l l s , q must immediately f a l l .
depicted i n f i g u r e 2.

The i n i t i a l movement i n X can be i n f e r r e d d i r e c t l y from the jump i n q
u s i n g the f o l l o w i n g r e l a t i o n s :

a) X = b/zE, b) V = b + zE, c) V = qK.

v a r i a b l e s b, E, and K a r e predetermined.
opposite i n s i g n t o t h a t o f q.

The

These i m p l y t h a t t h e jump i n . X i s

An increase i n the market value o f e x i s t i n g

c a p i t a l w i l l be immediately r e f l e c t e d i n a higher p r i c e o f e q u i t y and a lower
debt- to- equity r a t i o .
Once t h i s i n i t i a l adjustment has been made, however, subsequent movements
i n X are govern,ed by r e l a t i o n (31):

82, = a(X) + al(X)X(l + X I .

In

f a c t , the i n i t i a l r e a c t i o n o f the i n t e r e s t r a t e i s a l s o governed by (31).
Thus, X and 8 may i n i t i a l l y jump i n opposite d i r e c t i o n s .

To demonstrate

the dynamics o f these r e l a t i o n s , two p o s s i b l e cases a r e examined below.

Case I: d K / d ~ , > 0.
As noted above, x w i l l immediately increase w h i l e t h e movement of q i s

unclear.

I f q immediately increases, then the debt- to- equity r a t i o drops.

Then, from (311, i n o r d e r t o be on t h e s t a b l e path, t h e i n t e r e s t r a t e must
i n i t i a l l y f a l l , given t h e h i g h e r
consumption i s f a l l i n g .

T,.

When the i n t e r e s t r a t e i s below 6,

This i s seen from expresssion (27):

As t h e i n t e r e s t r a t e r i s e s toward O , X r i s e s toward i t s higher steady s t a t e
level.

The s i g n o f t h e i n i t i a l change i n c i s indeterminate, as i s t h e s i g n

of t h e comparative s t a t i c r e s u l t , because o f the presence o f debt costs t h a t
may move i n the o p p o s i t e d i r e c t i o n o f the c a p i t a l stock.

The former case i s

depicted i n f i g u r e 3.

Case 11:

dK/dr,

< 0.

Here t h e i n i t i a l impact on q i s c l e a r .

Both t h e drop i n investment and

t h e higher t a x r a t e r e q u i r e q t o f a l l .

I n t h i s case, i t i s c l e a r t h a t t h e new

steady s t a t e f i r m value w i l l be lower.

As a r e s u l t o f t h e change i n

T,,

t h e r e l a t i o n s h i p between X and 8 has been a l t e r e d , and the d i r e c t i o n of
t h e jump i n 8 i s indeterminate.
4.

Two p o s s i b i l i t i e s a r e p i c t u r e d i n f i g u r e

The ambiguity r e g a r d i n g t h e path o f X and 8 i n f i g u r e 4 can be

r e s o l v e d by r e s t r i c t i n g t h e response o f the system t o t h e change i n

T,.

Expressions (87) and (88) i n appendix B imply expression (58):

Here

IT,

> 0 and C1 > 0 i m p l y t h a t i f d K / d ~ , > 0, as K increases

toward the new steady s t a t e , c > 0.
determined through i n s p e c t i o n o f B 2 .

The s i g n o f

IT,

=

B2/B1 can be

B2 i s t h e response o f q t o an
a

increase i n x.

I f an increase i n t h e l e v e l o f investment decreases x, q w i l l

a l s o decrease.

This i s c l e a r from expression (29) and w i l l be assumed,

i m p l y i n g n , < 0.

Figure 3

The E f f e c t o f an Increase i n T, on q, X, x , and 8
When d K / d ~ , > 0 and q,-q* 1 ,,, > 0

Figure 4

The E f f e c t o f an I n c r e a s e i n T, on q, A, x , and 8
When dK/dr, and q,-q* 1 t = o > 0

The f i r s t term i n t h e c o e f f i c i e n t f o r (x

-

x*) i n expression (58)
This

represents t h e response o f excess supply o f goods t o an increase i n K.

w i l l be assumed p o s i t i v e , assuring us t h a t (c

-

c*) v a r i e s p o s i t i v e l y w i t h

Thus, (58) i m p l i e s t h a t i f d K / d ~ , < 0, (c

(x-x*).

-

c * ) > 0. This, i n

turn, t e l l s us t h a t p a t h A i s the a p p r o p r i a t e path f r o m f i g u r e 4.

VIII.

The D i s t i n c t i o n Between the Cost o f C a p i t a l and t h e I n t e r e s t Rate

This model p r e d i c t s t h a t the i n t e r e s t r a t e and t h e debt- to- equity r a t i o
are p o s i t i v e l y c o r r e l a t e d .

However, t h i s i s n o t the r e s u l t o f bankruptcy

r i s k , b u t r a t h e r the i n t e r e s t deduction f o r debt.

While t h e c o s t o f c a p i t a l

i s p o s i t i v e l y c o r r e l a t e d w i t h the debt- to- equity r a t i o , t h e r a t i o between t h e
c o s t o f c a p i t a l and the i n t e r e s t r a t e i s given by expression (34).
(34)

r

= 8 - a1(X)X2.

Since the debt- to- equity r a t i o r i s e s w i t h the i n t e r e s t r a t e , the gap
between the c o s t o f c a p i t a l and the i n t e r e s t r a t e a l s o r i s e s w i t h 8.

This

i m p l i e s t h a t models t h a t use the i n t e r e s t r a t e as t h e discount r a t e o v e r s t a t e
the cost o f c a p i t a l by an i n c r e a s i n g amount as the i n t e r e s t r a t e r i s e s .

We

would expect t h a t these models would understate investment.
Using expressions (231, (341, and (581, i t i s s t r a i g h t f o r w a r d t o show how
the gap between

r

and 8 v a r i e s .

Together w i t h (581, (59) i m p l i e s :

Expressions (23) and (34) imply:

Given the assumptions above, i f d K / d ~ , > 0, dCT

r-

- 81 <

0.

Since

8 = -a1(X)X2 < 0, i f t h e c a p i t a l stock i s r i s i n g , t h e absolute

value of the gap between t h e c o s t o f c a p i t a l .and the i n t e r e s t r a t e i s r i s i n g .

IX.

Summary
Previous q models o f investment have imp1i c i t l y assumed t h a t f i n a n c i a l

decisions have no impact on t h e c o s t o f the c a p i t a l .

Here we show how

f i n a n c i a l s t r u c t u r e a f f e c t s t h e c o s t o f c a p i t a l and t h e time paths o f
investment, q and X.

I n t h i s paper, the presence o f a debt c o s t together

w i t h a tax code t h a t f a v o r s debt r e s u l t s i n an optimal debt- to- equi t y r a t i o
t h a t covaries w i t h the i n t e r e s t r a t e .

We f i n d t h a t q i s s t i l l a " s u f f i c i e n t

s t a t i s t i c " f o r investment, although f i n a n c i a l s t r u c t u r e has r e a l e f f e c t s .

In

a d d i t i o n , the gap between t h e c o s t o f c a p i t a l and the i n t e r e s t r a t e covaries
w i t h i n t e r e s t r a t e s and f i n a n c i a l s t r u c t u r e .

As the debt- to- equity r a t i o

r i s e s , however, the increase i n the cost o f debt reduces t h e amount o f o u t p u t
avai 1able f o r consumption o r investment.
The a n a l y s i s i m p l i e s t h a t models t h a t ignore the endogenous adjustment o f
f in a n c i a i s t r u c t u r e w i 11 systematical l y e r r i n p r e d i c t i n g the response o f

investment t o changes i n t a x r a t e s o r i n t e r e s t r a t e s .

For example, a change

i n t h e corporate tax r a t e a f f e c t s both the c o s t o f c a p i t a l and the gap between
t h e c o s t o f c a p i t a l and t h e i n t e r e s t r a t e .

I n a d d i t i o n , t h e presence of r e a l

c o s t s t o f i n a n c i a l s t r u c t u r e may imply t h a t the c a p i t a l stock would d e c l i n e i n
s p i t e o f a lower c o s t o f c a p i t a l .

Appendix A

Here I d e r i v e t h e a p p r o p r i a t e form f o r the c o s t of c a p i t a l through t h e use
o f the c o n s t r a i n t s (7)-(12)

V(0).

maximization o f

F i r s t , use (8) and ( 9 ) t o w r i t e

-

1 , K l - T
,
Note t h a t V = b + zE i m p l i e s
A

and t h e assumed f i r m o b j e c t i v e :

y

w

-

)

=

-

(s(l

T,)

+ a(X))b + dzE + RE.

*

Expressions (10) and (B1) imply (83):

where ( y

-

wC

-

*(I,K))(l

f l o w o f the f i r m .

- r,)

-I

i s d e f i n e d as Y, t h e r e a l cash

Using (61, (A41 can be r e w r i t t e n as

Using the d e f i n i t i o n o f 1, (A51 can be w r i t t e n as

F i n a l l y , d e f i n i n g the c o s t o f c a p i t a l

r

as

(A61 can be w r i t t e n as

which i s a l i n e a r d i f f e r e n t i a l equation i n V t h a t can be i n t e g r a t e d t o show
that

r

i s t h e discount f a c t o r which maintains the e q u a l i t y between the

i n t e g r a l i n expression (13) and b t = o + Z,=~E,=,.

Appendix B

Here I d e r i v e expressions (42)-(47)
(271432) about t h e steady s t a t e .

where H(x,T,>

=

1

+ (1-2,)

o f t h e t e x t by expanding t h e system

F i r s t note t h a t (29) i m p l i e s

[h(x) + h l ( x ) l and H'(x,T,)

= dH( )/dx.

Together w i t h (301, t h i s i m p l i e s

Since €3 = I3

-

Uccc/Uc, i n o r d e r for (B2) t o y i e l d a r e l a t i o n f o r x, I

r e q u i r e an expression for

;.

An expression for

;can

be derived f r o m (28).

Take t h e time d e r i v a t i v e o f (28):

Here I have used t h e f a c t t h a t K = K(x-6) and I = xK.

The l a s t term on

t h e r i g h t i s an expression f o r qKdCA(X>l/dt t h a t makes use o f t h e f o l l o w i n g
facts:

These y i e l d the f o l l o w i n g expression f o r ;:

Now l i n e a r i z e (2) about t h e steady s t a t e t o o b t a i n an expression f o r ( x

where A l = (x,~,),

A2 =

(r +

&)H(X,T~)

. x*):

-

- (1 - T ~ ) ( F K + x Z h ' ( x ) )

and, since a l l c o e f f i c i e n t s are evaluated a t the steady state,
A, = q = 0. So we have:

.

- H( ) r t ( - u c c ~ / u c ) (-~ CI*).
From (B4) I derive:

This expression makes use o f the f a c t t h a t i n t h e steady s t a t e
x-6 = x = d-8 = 0. d-8 = 0 stems from the requirement t h a t i n the steady
s t a t e , X = 0, and thus z/z = 0. This i m p l i e s t h a t 8 = 8 = d.
r e s u l t i n g expression f o r ( x
(88)

( x 1 x*) = r l ( x

-

1 x*)

The

is:

x*) + n 2 ( K - K*), where rl and n2 are

defined as i n t h e t e x t .
I n t h i s s e c t i o n t h e d e t e r m i n a t i o n o f the constant Y, i n expression
(51) i n the t e x t i s described.

Expressions (89)-(811)

are i m p l i e d by the

f a c t t h a t c a p i t a l stock remains f i x e d a t t h e time o f t h e announced change, b u t
e v e n t u a l l y changes by ( d K / d ~ , ) d ~ , .
(B9)

Kt

-

Kl 1 t , o

= K*

(B10)

Kt

-

Kl It = o

=

(B11)

Y1 = (-dKld~,)d~,/(Q - nl)

-

K: = (-dK/dz,)d~,

YltQ -

IT,

= (-dK/d~,)d~,

Footnotes
1.

As an example o f more r e c e n t developments, Bernanke and G e r t l e r (1986)
have incorporated f i nanci a1 s t r u c t u r e i n t o a s t o c h a s t i c general
e q u i l i b r i u m model.

2.

C h i r i n k o (1 987) and Hayashi (1985) have incorporated f i n a n c i a l s t r u c t u r e
i n t o q frameworks. Chirinko, however, ignores tax e f f e c t s and
i n v e s t i g a t e s whether the debt versus e q u i t y choice can e x p l a i n the poor
performance o f q, defined as the market value o f e q u i t y d i v i d e d by t h e
replacement value o f c a p i t a l . The a n a l y s i s o f t h i s paper, as do o t h e r s ,
defines q as the market value o f debt and e q u i t y d i v i d e d .by the
replacement cost o f c a p i t a l . U t i l i z i n g a p a r t i a l e q u i l i b r i u m framework,
Hayashi (1985) shows how the f i r m ' s choice o f f i n a n c i a l p o l i c y depends on
the l e v e l o f p r o f i t s r e l a t i v e t o investment. He f i n d s t h a t o n l y when
incremental investment i s e n t i r e l y debt- financed i s t h e l i n k between q and
investment broken. C h i r i n k o and King (1985) take i n t o account the r o l e o f
debt i n the response o f investment by equity- maximizing f i r m s t o changes i n
inflation.

3.

A series o f a r t i c l e s (see Auerbach C19831 f o r a review) has concluded t h a t
bankruptcy r i s k and tax r a t e s t h a t vary across i n v e s t o r s and types of
income can l e a d t o a determinate debt- to- equi t y r a t i o and a c o s t of c a p i t a l
t h a t v a r i e s w i t h the debt- to- equity r a t i o . However, i f w i t h bankruptcy
r i s k a f i r m ' s decisions i n f l u e n c e the i m p l i c i t p r i c e s o f the Arrow-Debreu
s t a t e contingent commodities, the cost of c a p i t a l i s n o t well- defined.

4.

I f a(X>b were t r e a t e d as a deduction from taxable income o r as a c o s t
t h a t reduced taxable revenue, the form f o r the optimal c o s t of c a p i t a l
would be somewhat d i f f e r e n t . To see t h i s , replace (14) by the f o l l o w i n g :

and proceed as i n appendix B.

The r e s u l t i n g expression f o r

r is

A t the optimal debt- to- equity r a t i o , the c o n d i t i o n t h a t dT/dX = 0
implies t h a t

which y i e l d s r = 8 - ( l - - t , ) ~ ~ a ' ( ~ ) .I t can be shown t h a t an
increase i n 2, s t i 11 w i 11 decrease the steady s t a t e c o s t o f c a p i t a l .
5.

The f o r m u l a t i o n o f the adjustment
t h e form G(B,K,I) w i t h Ga > 0, G K
GnQ<O and G(I,K,I) homogeneous o f
assumption t h a t G a l = 0, G(8,K,I)
F(K,Q> - q(I,K).

cost i m p l i e s a p r o d u c t i o n f u n c t i o n o f
< 0, G I < O , G K K < 0,
degree one. Then w i t h the
can be w r i t t e n as G(Q,K,I) =

6.

C h i r i n k o (1982) estimates T C a t .033 when r a t e s a p p l i c a b l e t o v a r i o u s
i n v e s t o r classes a r e weighted by t r a n s a c t i o n s shares. The e s t i m a t e when
weights r e f l e c t SEC market value data i s .031. Feldstein, Poterba, and
Dicks- Mireaux (1981) estimate z c a t .083. While Gordon and M a l k i e l
(1981) argue t h a t t h e r a t e on d i v i d e n d and i n t e r e s t income should be
approximately equal f o r r e l e v a n t i n v e s t o r s , o t h e r researchers disagree.
F e l d s t e i n , Poterba, and Dicks-Mireaux estimate t h e dividend t a x r a t e a t
.349 i n 1979, and estimate t h e t a x r a t e on i n t e r e s t income a t .317.
C h i r i n k o estimates the d i v i d e n d t a x r a t e a t . I 6 8 w i t h t r a n s a c t i o n s
weights, and .278 when r a t e s are weighted by market shares. Gordon and
M a l k i e l estimate t h e dividend and i n t e r e s t income t a x r a t e a t .44. The
f e d e r a l corporate t a x r a t e i s .48. F e l d s t e i n , Poterba, and Dicks- Mireaux
estimate the corporate t a x r a t e i n c l u s i v e o f s t a t e and l o c a l government
corporate taxes a t .472.

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