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Chapter 21

T H E F I N A N C I A L A C C E L E R A T O R I N A Q U A N T I T A T I V E

B U S I N E S S C Y C L E F R A M E W O R K

BEN S. BERNANKE, MARK GERTLER and SIMON GILCHRIST

Princeton University, N ew York University, and Boston Unicersity

on t e n t s

A b s t r a c t 1 3 4 2

K e y w o r d s 1 3 4 2

1. I n t r o d u c t i o n 1 3 43

2 . T h e m o d e l : o v e r v i e w a n d b a s i c a s s u m p t i o n s 1 3 46

3. T h e d e m a n d f o r c a p i t al a n d t h e r o l e o f n e t w o r t h 1 3 49

3.1. Contract terms when there is no aggregate risk 13 50

3.2. Contract terms when there is aggregate risk 13 52

3.3. Net worth and the optimal choice of capital 13 52

4 . G e n e r a l e q u i l i b r i u m 1 3 55

4.1. The entrepreneurial sector 13 56

4.2. The complete log-linearized model 13 60

4.2.1. Two extensions of the baseline model 13 65

4.2.1.1. Investment delays 13 65

4.2.1,2. Heterogeneous firms 13 66

5. M o d e l s i m u l a t i o n s 1 3 67

5.1. Mod el parametrization 13 67

5.2. Results 136 8

5.2.1. Response to a monetary polic y shock 13 68

5.2.2. Sho ck to technology, demand, and wealth 13 72

5.2.3. Inve stm ent delays and heterogeneous firms 13 73

6 . A h i g h l y s e l e c t e d r e v i e w o f t h e l i t e r a t u r e 1 3 75

7 . D i r e c t i o n s f o r f u r o r e w o r k 1 3 7 9

A p p e n d i x A . T h e o p t i m a l f i n a n ci a l c o n t r a c t a n d t h e d e m a n d f o r c a p i ta l 1 3 80

A. 1. The partial equilibrium contracting problem 13 80

A.2. The log-norm al distribution 13 85

A.3. Aggregate risk 138 5

Thanks to Michael Woodford, Don Morgan and John Taylor for helpful conanents, and to the NSF

and C.M Starr Center for financial support.

Each author is also affiliated with the National Bmeau of Economic Research.

Handbook o f Macroeconomics, Volume 1, Edi ted by J..B. lay lor and M. WoodJb~d

© 1999 Else vier Science B.V . Al l rights reserved

1341


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1 3 4 2

A p p e n d i x B . H o u s e h o l d , re t ai l a n d g o v e r n m e n t s ec t or s

B 1 H o u s e h o l d s

B 2 T h e r e t ai l s e c t o r a n d p r i c e s e t t in g

B 3 G o v e r n m e n t s e c t or

R e f e r e n c e s

B S B e r n a n k e e t a l

1387

1387

1388

1389

1390

bstract

Thi s chap t e r deve l ops a dynam i c gene r a l equ i l i b r i um m ode l t ha t i s i n t ended t o

he l p c l a r i f y t he r o l e o f c r ed i t m a r ke t f r i c t i ons i n bus i ne s s f l uc t ua t i ons , f r om bo t h

a qua l i t a tive and a quant i t a t ive s tandpo in t . Th e mod el i s a synthes i s o f the l eadin g

app roach es in the l it e ra ture . In par t i cu la r , the f ram ew ork exhib i t s a f inanc ia l

acce l e r a t o r , i n t ha t endogenous deve l opm en t s i n c r ed i t m a r ke t s w or k t o am pl i f y and

pr opaga t e shocks t o t he m ac r oeconom y. I n add i t i on , w e add s eve r a l f e a t u r e s t o t he

m ode l t ha t a r e de s i gned t o enhance t he em pi r i ca l r e l evance . F i r s t , w e i nco r po r a t e

m oney and p r i ce s t i ck i ne s s , w h i ch a l l ow s us t o s t udy how c r ed i t m a r ke t f r i c t i ons

m ay i n f l uence the t r ansm i s s i on o f m on e t a r y po l i cy . I n add i t ion , w e a l l ow f o r l ags i n

i n v e st m e n t w h i c h e n a b le s t h e m o d e l t o g e n e r a t e b o th h u m p - s h a p e d o u t p u t d y n a m i c s

and a l e ad - l ag r e l a t i on be t w een a s se t p r i ce s and i nves t m en t , a s i s cons i s t en t w i t h t he

da t a. F i na l ly , w e a l l ow f o r he t e r og ene i t y am o ng f ir m s t o cap t u r e t he f ac t t ha t bo r r ow er s

have d i f f e r en t i a l a cce s s t o cap i t a l m a r ke t s . U nde r r ea sonab l e pa r am e t r i za t i ons o f

t he m ode l , t he f i nanc i a l a cce l e r a t o r ha s a s i gn if i can t in f l uence on bus i ne s s cyc l e

dynam i cs .

Keywords

f i nanc ia l a cce l e r at o r , bus ine s s f l uc tua t i ons , m o ne t a r y po l i cy

J E L c l a s s i f i c a t i o n E 3 0 , E 4 4 , E 5 0


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1344 B S B e r n a n k e e t a l

r e l a t i ve l y s m a l l s hocks ( m odes t changes i n r ea l i n t e r e s t r a t e s i nduced by m one t a r y

po l i cy , f o r exam pl e , o r t he s m a l l ave r age changes i n f i r m cos t s i nduced by even

a r e l a t i ve l y l a r ge m ovem en t i n o i l p r i c e s ) c an neve r t he l e s s have l a r ge r ea l e f f ec t s .

I n t r oduc i ng c r ed i t - m ar ke t f r i c ti ons ha s t he added advan t age o f pe r m i t ti ng t he s t anda r d

m o d e l s t o e x p l a i n a b r o a d e r c l a ss o f i m p o r t a n t c y c l i c a l p h e n o m e n a , s u c h a s c h a n g e s

i n c r ed i t ex t ens i on and t he s p r eads be t ween s a f e and r i s ky i n t e r e s t r a t e s .

T h e s e c o n d r e a s o n f o r i n c o r p o r a t i n g c r e d i t -m a r k e t e f f e c ts in t o m a i n s t r e a m m o d e l s

is t h a t m o d e r n e m p i r ic a l r e s e a r c h o n t h e d e t e r m i n a n t s o f a g g re g a t e d e m a n d a n d ( to

a l e s s e r ex t en t ) o f agg r ega t e s upp l y ha s o f t en a s c r ibed an i m por t an t r o l e t o va r ious

c r ed i t - m ar ke t f r i c t i ons . Recen t em pi r i ca l wor k on cons um pt i on , f o r exam pl e , ha s

e m p h a s i z e d t h e im p o r t a n c e o f li m it s o n b o r r o w i n g a n d t h e c l o s e ly - r e la t e d b u f f e r

s t ock behav i o r [ Mar i ge r (1987) , Ze l de s ( 1989) , J appe l li ( 1990) , De a t on ( 1991) , Eb e r l y

( 1994) , Gour i nchas and Pa r ke r ( 1995) , Enge l ha r d t ( 1996) , Ca r r o l l ( 1997) , Ludv i gs on

( 1997) , Bac che t t a and G e r l ach ( 1997) ] . I n t he i nves t m en t l it e r at u r e , de s p i t e s om e r ecen t

r ehab i l i ta t i on o f a r o l e f o r neoc l a s s i ca l cos t - o f - cap i t a l e f f ec t s [ Cum m i ns , H as s e t t and

H u b b a r d ( 1 9 9 4 ) , H a s s e t t a n d H u b b a r d ( 1 9 9 6 ) ] , t h e r e r e m a i n s c o n s i d e r a b l e e v i d e n c e

f o r t he v i ew t ha t c a s h f l ow, l eve r age , and o t he r ba l ance - s hee t f a c t o r s a l s o have

a m a j o r i n f l uence on i nves t m en t s pend i ng [ Fazza r i , Hubba r d and Pe t e r s en ( 1988) ,

Hos h i , Kas hyap and Scha r f s t e i n ( 1991) , Whi t ed ( 1992) , Gr os s ( 1994) , Gi l ch r i s t and

Hi m m e l be r g ( 1995) , Hubba r d , Ka s hyap and W hi t ed ( 1995) ] 1. S i m i l a r conc l us i ons

a r e r e a c h e d b y r e c e n t s tu d ie s o f t h e d e t e r m i n a n t s o f i n v e n to r ie s a n d o f e m p l o y m e n t

[ Can t o r ( 1990) , B l i nde r and Macc i n i ( 1991) , Kas hyap , Lam on t and S t e i n ( 1994) ,

Sha r pe ( 1994), C a r pen t e r, Fazza r i and Pe t e r s en ( 1994) ] . Aggr eg a t e m ode l i ng , i f it is

t o de s c r i be t he dyn am i cs o f s pend i ng and p r o duc t i on r ea l is t ic a ll y , needs t o t ake t he s e

em pi r ica l f indings in to a cco un t 2 .

How does one go abou t i nco r po r a t i ng f i nanc i a l d i s t r e s s and s i m i l a r concep t s i n t o

m ac r oeconom i cs ? Whi l e i t s eem s t ha t t he r e ha s a l ways been an em pi r i ca l c a s e

f o r i nc lud i ng c r ed i t - m ar k e t f a c t o r s i n t he m a i ns t r eam m o de l , e a r l y wr i t e r s f o und i t

d i f f icu l t t o b r i ng s uch ap pa r en t l y d i ve r s e and chao t i c phenom ena i n t o t he i r f o r m a l

ana l ys es . As a r e su l t , advoc acy o f a r o l e f o r the s e f ac t o r s in agg r ega t e dyna m i cs

f e l l f o r t he m os t pa r t to e con om i s t s ou t s i de the US academ i c m a i ns t r eam , s uch a s

H ym an M i ns ky , and t o s om e f o r eca s t e r s and f i nanc i a l - m ar ke t p r ac t i t ione r s , s uch a s

Ot t o Eck s t e i n and A l l en S i na i ( l 986 ) , Al be r t W oj n i lower ( 1980) , and H en r y Kauf m a~

( 1986) . How eve r , ove r t he pa s t t wen t y - f ive yea r s , b r eak t h r oug hs i n t he e con om i cs

o f i n c o m p l e t e a n d a s y m m e t r i c i n f o r m a t i o n [ b e g in n i n g w i th A k e r l o f (1 9 7 0 ) ] a n d

t he ex t ens ive adop t i on o f t he s e i dea s i n co r po r a t e f i nance and o t he r app l i ed f i e ld s

[ e . g . , J ens en and Meck l i ng ( 1976) ] , have m ade pos s i b l e m or e f o r m a l t heo r e t i c a l

1 A critique of the cash-flow literature is given by Kap lan and Zingales (1997). See C hirinko (1993)

for a broa d survey of the em pirical literature in inveslment.

2 Contemporarymacroeconometric orecasting mo dels, such as the MPS m odel used by the Federal

Reserve, typically do incorporate factors such as borrowing constraints and cash-flow effects. See for

examp le Bray ton et al. (1997).


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h u m p - s h a p e d o u t p u t d y n a m i c s a n d a l e a d - l a g r e l a t io n s h i p b e t w e e n a s s e t p r i c e s a n d

inves tmen t , a s i s cons i s t en t w i th t he da t a . F ina ll y , we a l l ow fo r he t e ro gene i t y am on g

f i rms t o cap tu re t he r ea l -wor ld f ac t t ha t bo r rower s have d i f f e r en t i a l a cce s s t o cap i t a l

m a r k e t s . A l l t h e se i m p r o v e m e n t s s i g n if i ca n t ly e n h a n c e t h e v a l u e o f th e m o d e l f o r

quan t i t a t i ve ana ly s i s , i n ou r v i ew .

T h e r e s t o f th e c h a p t e r i s o r g a n i z e d a s f o ll o w s . S e c t i o n 2 i n t r o d u c es t h e m o d e l

ana ly zed i n t he p r e sen t chap t e r . S ec t i on 3 cons ide r s t he sou rce o f t he f i nanc i a l

a c c e l er a t o r: a c r e d i t - m a r k e t f r i c ti o n w h i c h e v o l v e s f ro m a p a r ti c u l a r f o r m o f a s y m -

m e t r i c i n f o r m a t i o n b e t w e e n l e n d e r s a n d p o t e n t i a l b o rr o w e r s . I t t h e n p e r f o r m s a p a r ti a l

e q u i li b r iu m a n a l y s is o f t h e r es u l t in g t e r m s o f b o r r o w i n g a n d o f f i r m s d e m a n d f o r

c a p i t a l , a n d d e r i v e s t h e l i n k b e t w e e n n e t w o r t h a n d t h e d e m a n d f o r c a p i t a l t h a t i s

t h e e s s e n c e o f t h e f i n an c i al a c c e l e ra t o r. S e c t i o n 4 e m b e d s t h e c r e d i t - m a r k e t m o d e l

i n a D y n a m i c N e w K e y n e s i a n ( D N K ) m o d e l o f t h e b u s i n e s s c y c l e , u s i n g t h e d e v i c e

p r o p o s e d b y C a l v o ( 1 9 8 3 ) t o i n c o r p o r a t e p r i c e s t i c k i n e s s a n d a r o l e f o r m o n e t a r y

po l i cy ; i t a l so cons ide r s s eve ra l ex t ens ions , such a s a l l owing fo r lags i n i nves tm en t and

fo r d i f f e r en ti a l c r ed i t a cce s s a c ro s s f i rms . Se c t i on 5 p r e sen t s s im u la t i on r e su l t s , d r awing

c o m p a r i s o n s b e t w e e n t h e c a s e s in c l u d i n g a n d e x c l u d i n g t h e c r e d i t - m a r k e t f ri c ti o n . H e r e

w e s h o w t h a t t h e f i n a n c i a l a c c e l e r a t o r w o r k s t o a m p l i f y a n d p r o p a g a t e s h o c k s t o t h e

ec on om y in a quan t i t a ti ve ly s i gn i f i c an t way . Sec t i on 6 t hen g ive s a b r i e f and s e l ec t i ve

su rv ey tha t de sc r ibe s how the f r am ew ork p re se n t f it s i n the l i t e ra tu r e . Sec t i on 7 then

desc r ibe s s eve ra l d i r ec t i ons fo r f u tu r e r e sea r ch . Two append ice s con t a in add i t i ona l

d i s c u s s io n a n d a n a l y s is o f th e p a r t i a l -e q u i l i b r i u m c o n t r a c ti n g p r o b l e m a n d t h e d y n a m i c

g e n e r a l e q u i l i b r i u m m o d e l i n w h i c h t h e c o n t r a c t i n g p r o b l e m i s e m b e d d e d .

2 T h e m o d e l : o v e r v i e w a n d b a s i c a s s u m p t i o n s

O u r m o d e l is a v a ri a nt o f t h e D y n a m i c N e w K e y n e s i a n ( D N K ) f r a m e w o r k , m o d i f i e d

t o a l l o w f o r f i n a n c i a l a c c e l e r a t o r e f f e c t s o n i n v e s t m e n t . T h e b a s e l i n e D N K m o d e l

i s e s s e n t i a l l y a s t o c h a s t i c g r o w t h m o d e l t h a t i n c o r p o r a t e s m o n e y , m o n o p o l i s t i c

c o m p e t i t i o n , a n d n o m i n a l p r i c e r i g id i ti e s. W e t a k e t h i s f r a m e w o r k a s th e s t a r ti n g p o i n t

fo r s eve ra l r e a sons . F i r s t, t h i s app ro ach ha s b eco m e wide ly accep t ed i n the l i t e r a tu r e 3

I t h a s t h e q u a l it a ti v e e m p i r i c a l a p p e al o f t h e I S - L M m o d e l , b u t i s m o t i v a t e d f r o m f ir s t

p r i n ci p l es . S e c o n d , i t i s p o s s i b l e to s t u d y m o n e t a r y p o l i c y w i t h t h i s fr a m e w o r k . F o r o u r

pu rpose s , t h i s means t ha t i t i s pos s ib l e t o i l l u s t r a t e how c red i t ma rke t impe r f ec t i ons

i n f lu e n c e th e t r a n s m i s s i o n o f m o n e t a r y p o l ic y , a t h e m e e m p h a s i z e d i n m u c h o f t h e

r ecen t l i te r a tu r e 4 . F ina l ly , i n t he l im i t i ng ca se o f pe r f ec t p r i c e f l ex ib il it y , the cyc l i c a l

p r o p e r t i e s o f th e m o d e l c l o s e l y r e s e m b l e t h o s e o f a re a l b u s in e s s c y c l e f r a m e w o r k . I n

3 See G oodfriend and K ing (1997) for an exp osition of the DN K approach.

4 For a review of the recent literature on the ro le of c redit ma rket fiqctions in the transmission of

mo netary policy, see Bernanke and Gertler (1995).


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C h 2 1 : T h e F i n a n c i a l A c c e l e r a t o r i n a Q u a n ti t a ti u e B u s i n e s s C y c l e F r a m e w o r k

1347

t h i s a p p r o x i m a t e s e n s e , t h e D N K m o d e l n e s t s t h e r e a l b u s i n e s s c y c l e p a r a d i g m a s a

spec i a l c a se . I t thus ha s t he v i r t ue o f ve r sa t il it y .

E x t e n d i n g a n y ty p e o f c o n t e m p o r a r y b u s i n e ss c y c l e m o d e l t o in c o r p o r a t e f in a n c ia l

acce l e r a to r e f f ec t s i s , howeve r , no t s t r a igh t fo rward . The re a r e two gene ra l p rob l ems :

F i r s t , b e c a u s e w e w a n t l e n d i n g a n d b o r r o w i n g t o o c c u r a m o n g p r i v a t e a g e n t s i n

e q u i l ib r i u m , w e c a n n o t u s e t h e r e p r e s e n t a ti v e a g e n t p a r a d i g m b u t m u s t i n s te a d g r a p p l e

w i t h th e c o m p l i c a t i o n s i n t ro d u c e d b y h e t e r o g e n e i t y a m o n g a g e n t s . S e c o n d , w e w o u l d

l i ke t he f i nanc i a l con t r ac t s t ha t agen t s u se i n t he mode l t o be mo t iva t ed a s f a r a s

poss ib l e f rom f i rs t p r i nc ip l e s . S ince f i nanc i a l con t r ac t s and i n s t i t u t ions a r e endogeno us ,

r e su l t s t ha t h inge on a rb i t r a ry r e s t r i c t i ons on f i nanc i a l r e l a t i onsh ips may be suspec t .

M o s t o f th e n o n s t a n d a r d a s s u m p t i o n s th a t w e m a k e i n s e t t i n g u p o u r m o d e l a r e

d e s i g n e d t o f a c i l i t a t e a g g r e g a t i o n ( d e s p i t e i n d i v i d u a l h e t e r o g e n e i t y ) a n d p e r m i t a n

endogenous f i nanc i a l s t r uc tu r e , t hus add re s s ing t he se two key i s sues .

T h e b a s i c s t ru c t u r e o f o u r m o d e l i s a s f o l lo w s : T h e r e a r e t h re e t y p e s o f a g en t s , c a l le d

househo lds , en t r ep reneu r s , and r e t a i l e r s . Househo lds and en t r ep reneu r s a r e d i s t i nc t

f r o m o n e a n o t h e r i n o r d e r t o e x p l i c i t l y m o t i v a t e l e n d i n g a n d b o r r o w i n g . A d d i n g

re t a i l e r s pe rmi t s u s t o i nco rpo ra t e i ne r t i a i n p r i c e s e t t i ng i n a t r a c t ab l e way , a s we

d i sc u s s. I n a d d i t io n , o u r m o d e l i n c l u d e s a g o v e r n m e n t , w h i c h c o n d u c t s b o t h f i s c al a n d

m o n e t a r y p o l i c y .

H o u s e h o l d s l i v e f o r e v e r ; t h e y w o r k , c o n s u m e , a n d s a v e. T h e y h o l d b o t h r e a l m o n e y

b a l a n c e s a n d i n t e r e s t - b e a r i n g a s s e t s . W e p r o v i d e m o r e d e t a i l s o n h o u s e h o l d b e h a v i o r

be low.

For i nduc ing t he e f f ec t we r e f e r to a s t he f i nanc i a l a cce l e r a to r , en t r ep reneu r s p l a y the

k e y r o l e i n o u r m o d e l . T h e s e i n d iv i d u a ls a r e a s s u m e d t o b e r i s k -n e u t r a l a n d h a v e f i n it e

ho r i zons : Spec i f i c a l l y , we a s sume tha t e ach en t r ep reneu r ha s a cons t an t p robab i l i t y y

o f su r v i v i n g t o t h e n e x t p e r io d ( i m p l y i n g a n e x p e c t e d l i f e t im e o f 1 @ ) T h e a s s u m p t i o n

o f fi n it e h o r i z o n s f o r e n t r e p r e n e u rs i s i n t e n d e d t o c a p t u r e t h e p h e n o m e n o n o f o n g o i n g

b i r t h s and d ea th s o f f i rms , a s we l l a s t o p r ec lude t he po s s ib i l i t y t ha t t he en t r ep reneu r i a l

s e c t o r w i l l u l t i m a t e l y a c c u m u l a t e e n o u g h w e a l t h t o b e f u l l y s e l f - f i n a n c i n g . H a v i n g

the su rv iva l p ro bab i l i t y be cons t an t ( i ndepende n t o f age ) f ac i l it a t e s agg rega t i on . We

a s s u m e t h e b i r t h r a te o f e n t r e p re n e u r s t o b e s u c h t h a t t he f r a c t i o n o f a g e n t s w h o a r e

en t r ep reneu r s i s cons t an t .

I n each pe r iod t en t r ep reneu r s a cqu i r e phys i ca l c ap i t a l . (En t r ep reneu r s w ho d i e

i n p e r i o d t a r e n o t a l l o w e d t o p u r c h a s e c a p i t a l , b u t i n s t e a d s i m p l y c o n s u m e t h e i r

a c c u m u l a t e d r e s o u r c e s a n d d e p a r t f r o m t h e s c e n e .) P h y s i c a l c a p i t a l a c q u i r e d i n p e ri o d

t is u sed i n co m bina t i on wi th h i r ed l ab o r t o p rodu ce o u tpu t i n pe r iod t + 1 , by

m e a n s o f a c o n s t a n t -r e t u r n s t o s c a l e te c h n o lo g y . A c q u i s i t io n s o f c a p i t al a r e f i n a n c e d

b y e n t r e p r e n e u r i a l w e a l t h , o r n e t w o r t h , a n d b o r r o w i n g .

T h e n e t w o r t h o f e n t r e p r e n e u r s c o m e s f r o m t w o s o u r ce s : p r o f it s ( in c l u d in g c a p i ta l

g a i ns ) a c c u m u l a t e d f r o m p r e v i o u s c a p i t a l i n v e s t m e n t a n d i n c o m e f r o m s u p p l y i n g l a b o r

(we a s sum e tha t en t r ep reneu r s sup p ly one un i t o f labo r i ne l a s t i c a l l y t o the gen e ra l

l abo r marke t ) . A s s t r e s sed i n t he li t e r a tu r e , en t r e p ren eu r s ' ne t w or th p l ays a c ri t ic a l r o l e

i n t h e d y n a m i c s o f t h e m o d e l . N e t w o r t h m a t t e r s b e c a u s e a b o r r o w e r ' s fi n a n ci a l p o s i t i o n


8/53


i s a k e y d e t e r m i n a n t o f h is c o s t o f e x t er n a l f i n an c e . H i g h e r l e v e ls o f n e t w o r t h a l l o w

fo r i nc r ea sed s e l f - f i nanc ing ( equ iva l en t l y , co l l a t e r a l i z ed ex t e rna l f i nance ) , m i t i ga t i ng

t h e a g e n c y p r o b l e m s a s s o c i a t e d w i t h e x t e r n a l f i n an c e a n d r e d u c i n g t h e e x t e r n a l f i n an c e

p r e m i u m f a c e d b y t h e e n t r e p r e n e u r i n e q u i l i b r i u m .

T o e n d o g e n o u s l y m o t i v a t e t h e e x i s t e n c e o f a n e x t e r n a l fi n a n ce p r e m i u m , w e

p o s t u l a t e a s i m p l e a g e n c y p r o b l e m t h a t i n tr o d u c e s a c o n f li c t o f in t e r e st b e t w e e n

a bo r rower and h i s r e spec t i ve l ende r s . The f i nanc i a l con t r ac t i s t hen de s igned t o

m i n i m i z e t h e e x p e c t e d a g e n c y c o s t s . F o r t r a c t a b i l i t y w e a s s u m e t h a t t h e r e i s e n o u g h

a n o n y m i t y i n f i n a n c i a l m a r k e t s t h a t o n l y o n e - p e r i o d c o n t r a c t s b e t w e e n b o r r o w e r s a n d

l e n d e r s a r e f e a s i b l e [ a s i m i l a r a s s u m p t i o n i s m a d e b y C a r l s t r o m a n d F u e r s t ( 1 9 9 7 ) ] .

A l l o w i n g f o r l o n g e r - t e r m c o n t r a c t s w o u l d n o t a f f e c t o u r b a s ic r e s u l ts 5. T h e t b r m o f

t h e a g e n c y p r o b l e m w e i n t r o d u c e , t o g e t h e r w i t h t h e a s s u m p t i o n o f c o n s t a n t re t u r n s

to s ca l e i n p roduc t i on , i s su f f i c i en t ( a s we sha l l s ee ) t o gene ra t e a l i nea r r e l a t i onsh ip

b e t w e e n t h e d e m a n d f o r c a p it a l g o o d s a n d e n t r e p re n e u r i a l n e t w o r th , w h i c h f a c il i ta t e s

agg rega t i on .

One compl i ca t i on i s t ha t t o i n t roduce t he nomina l s t i ck ine s s i n t r i n s i c t o t he

D N K f r a m e w o r k , a t l e a s t s o m e s u p p l i e r s m u s t b e p r i c e s e t t e r s , i . e . , t h e y m u s t

f a c e d o w n w a r d - s l o p i n g d e m a n d c u r v e s . H o w e v e r , a s s u m i n g t h a t e n t r e p r e n e u r s a r e

i m p e r f e c t c o m p e t i t o r s c o m p l i c a t e s a g g r e g a t i o n , s i n ce i n th a t c a s e th e d e m a n d f o r

c a p i t a l b y i n d i v id u a l f i r m s i s n o l o n g e r l i n e a r in n e t w o r t h . W e a v o i d th i s p r o b l e m b y

d i s t i ngu i sh ing be tw een en t r ep reneu r s an d o the r agen t s , c a l l ed ' r e t a il e r s . En t r e p rene u r s

p r o d u c e w h o l e s a l e g o o d s i n c o m p e t i t iv e m a r k e t s , a n d t h e n s e ll th e i r o u t p u t t o r e ta i l er s

w h o a r e m o n o p o l i s t i c c o m p e t i t o r s . R e t a i l e r s d o n o t h i n g o t h e r t h a n b u y g o o d s f r o m

en t r ep reneu r s , d i f f e r en t i a t e t hem ( cos t l e s s ly ) , t hen r e - s e l l t hem to househo lds . The

m o n o p o l y p o w e r o f r e ta i le r s p r o v i d e s t h e s o u r c e o f n o m i n a l s t i c k in e s s i n t h e e c o n o m y ;

o the rwi se , r e t a i le r s p l ay no ro l e . W e a s sum e tha t p ro f i ts f r om r e t a i l a c t i v i t y a r e

r e b a t e d l u m p - s u m t o h o u s e h o l d s . H a v i n g d e s c r i b e d t h e g e n e r a l s et u p o f th e m o d e l ,

w e p r o c e e d i n t w o s te p s . F ir s t, w e d e r iv e t h e k e y m i c r o e c o n o m i c r e la t i o n s h i p o f t h e

m o d e l : t h e d e p e n d e n c e o f a f i r m ' s d e m a n d f o r c a p it a l o n th e p o t e n t i a l b o r r o w e r ' s n e t

w o r t h . T o d o s o , w e c o n s i d e r t h e f i r m ' s ( e n tr e p r e n e u r 's ) p a r t ia l e q u i l i b ri u m p r o b l e m o f

j o i n t l y d e t e r m i n i n g i ts d e m a n d f o r c a p i ta l a n d t e r m s o f e x t e r n a l fi n a n c e in n e g o t i a t i o n

w i t h a c o m p e t i t i v e l e n d e r ( e . g . , a f i n a n c i a l i n t e r m e d i a r y ) . S e c o n d , w e e m b e d t h e s e

r e l a ti o n s h i p s n a n o th e 1 ~ i s e c o n v e n t i o n a l D N K m o d e l . O u r o b j e c t iv e i s t o s h o w h o w

f l u ct u a ti o n s in b o r r o w e r s ' n e t w o r t h c a n a c t t o a m p l i f y a n d p r o p a g a t e e x o g e n o u s s h o c k s

t o t h e s y s t e m . F o r m o s t o f t h e a n a l y s i s w e a s s u m e t h a t th e r e i s a s i n g le t y p e o f

f i rm ; h o w e v e r , w e e v e n t u a l ly e x t e n d t h e m o d e l t o a l l o w f o r h e t e ro g e n e o u s f i rm s w i t h

d i f f e r en t i a l a cce s s t o c r ed i t .

So lon g as b orrowers have finite horizons, net wo rth influences the term s of borrowing, eve n ai~er

allowing for nmlti-period contracts. See, for exam ple, Gertter (1992).


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C h 2 1 : T h e F i n a n c i a l A c c e l e r a t o r i n a Q u a n t i ta t i v e B u s i n e s s C y c l e F r a m e w o r k

1 3 4 9

3 T h e d e m a n d f o r c a p i t a l a n d t h e r o l e o f n e t w o r t h

We now s t udy t he cap i t a l i nves t m en t dec i s i on a t t he f i r m l eve l , t ak i ng a s g i ven t he

p r i ce o f c ap i t a l goods and t he e xpec t ed re t u r n t o cap it a l. I n t he s ubs equen t s ec t i on we

endo gen i ze cap i t a l p r i c e s and r e t u r ns a s pa r t o f a gene r a l equ i l i b r i um s o l u t i on .

At t i m e t , t he en t r ep r eneu r wh o m anages f i r m j pu r chas e s cap i ta l f o r u s e a t t + I .

The quan t i t y o f c ap it a l pu r cha s ed i s de no t ed K/+ I, w i t h t he s ubs c r i p t deno t i ng t he

pe r i od i n w h i ch t he cap i t a l i s a c t ua l ly u s ed , and t he s upe r s c r i p t j deno t i ng t he f i r m .

The p r i ce pa i d pe r un i t o f c ap i ta l i n pe r i od t i s Q t . Cap i t a l i s hom ogeneous , and s o

i t does no t m a t t e r whe t he r t he cap i t a l t he en t r ep r eneu r pu r chas e s i s newl y p r oduced

wi t h i n the pe r i od o r is o l d , dep r ec i a t ed cap i ta l . Hav i ng t he en t r ep r eneu r pu r chas e

( o r r epu r ch as e ) h i s en t i r e c ap i ta l s t ock each pe r i od i s a m ode l i ng d ev i ce t o ens u r e ,

rea l i s t i ca l ly , tha t l everage res t r i c t ions or o ther f inanc ia l cons t ra in t s apply to the f i rm

as a who l e , no t j u s t t o t he m ar g i na l i nves t m en t .

Th e r e t u r n t o cap i t a l is s ens i ti ve t o bo t h agg r ega t e and i d i o s ync r a t i c ri s k. T he ex pos t

g r os s r e t u r n on cap i t a l f o r f i r m j i s

t ~JPk

w here coy i s an id iosyn cra t i c d i s turbance to

* t + l ,

f i r m j ' s r e t u r n

nd

Rk+l i s the ex pos t aggrega te re turn to capi ta l ( i . e . , the gross re turn

ave r aged ac r os s f i r m s ) . The r andom va r i ab l e (.0 i s i . i . d , ac ross t ime and across f i rms ,

wi t h a con t i nuo us a nd once - d i f f e r en t i ab l e c . d .f . , F ( ~o ) , ove r a non - nega t i ve s uppor t ,

and E { { o J } = 1 . W e i m pos e t he f o l l owi ng r e s t r i c t ion on t he co r r e s po nd i ng haza r d

rate h((o) :

O coh o)))

0 a )

> 0 , 3 . 1 )

dF(~o)

wh ere h(co) _= ~ F(o~ This reg ular i ty con di t ion i s a re la t ive ly w eak res t r i c t ion tha t is

s a ti s fi ed by m o s t con ven t i ona l d i s tr i bu ti ons , i nc l ud i ng f o r exam pl e t he l og - no r m a l .

At t he end o f pe r i od t ( go i ng i n t o pe r i od t + 1 ) en t r ep r eneu r j ha s ava i l ab le ne t

wor t h ,

N / + 1

To f inance t he d i f f e r ence be t w een h i s exp end i t u r e s on cap i t a l goods and

h is n e t w o r t h h e m u s t b o r r o w a n a m o u n t B J < , g i v e n b y

B t J l K j J

Q, ,+, -N ,+ 1. (3.2)

The en t r ep r eneu r bo r r ows f r om a f i nanc i a l i n t e r m ed i a r y t ha t ob t a i n s i t s f unds f r om

hous eho l ds . The f i nanc i a l i n t e r m ed i a r y f ace s an oppor t un i t y cos t o f f unds be t wee n t

and t + 1 equa l t o t he econo m y ' s r i s k l e ss g r os s r a t e o f r e t u r n , R t + l . The r i sk less ra te i s

t he r e levan t oppo r t un i t y cos t becaus e i n the equ i l ib r i um o f ou r m ode l , t he i n t e r m ed i a r y

ho l ds a pe r f ec t l y s a fe po r t f o l i o ( it pe r f ec t l y d i ve rs i fi e s the i d i o s ync r a t i c r i s k i nvo l ved

i n lend i ng ) . B ecaus e en t r ep r eneu r s a r e r i sk - neu t r a l and hou s eho l ds a r e r i sk - ave r s e , the

l oan con t r ac t t he i n t e r m ed i a r y s i gns ha s en t r ep r eneu r s abs o r b any agg r ega t e r i s k , a s

we d i scuss be low.

To m ot i va t e a non t r i v i a l ro l e f o r f inanc i a l s tr uc t u re , w e f o l l ow a num ber o f p r ev i ous

pape r s i n a s s um i ng a cos t l y s t a te ve r i f i c a t i on ( CS V) p r ob l em o f t he t ype fi rs t


10/53

1 3 5 0

B S B e r n a n k e e t a l

a n a l y z e d b y T o w n s e n d ( 1 9 7 9) , i n w h i c h l e n d e r s m u s t p a y a f ix e d a u d i t i n g c o s t

i n o r d e r t o o b s e r v e a n i n d i v i d u a l b o r r o w e r ' s r e a l i z e d r e t u r n ( t h e b o r r o w e r o b s e r v e s

t h e r e t u r n f o r f r e e ) . A s T o w n s e n d s h o w e d , t h i s a s s u m p t i o n a l l o w s u s t o m o t i v a t e

w h y u n c o l l a t e r a l i z e d e x t e r n a l f i n a n c e m a y b e m o r e e x p e n s i v e t h a n i n t e r n a l f i n a n c e

w i t h o u t i m p o s i n g a r b i t ra r y r e s tr i c ti o n s o n t h e c o n t r a c t st r u ct u re . T h e r e a r e m a n y o t h e r

s p e c i fi c a ti o n s o f th e i n c e n ti v e p r o b l e m b e t w e e n t h e e n t r e p r e n e u r a n d o u t s id e l e n d e rs

t h a t c a n g e n e r a t e q u a l it a ti v e l y s i m i l a r r e s ul ts . T h e v i r tu e s o f th e T o w n s e n d f o r m u l a t i o n

a re i t s s imp l i c i t y and de sc r i p t i ve r ea l i sm .

F o l l o w i n g th e C S V a p p r o a c h , w e a s s u m e t h a t th e l e n d e r m u s t p a y a c o s t i f h e o r

she w i shes t o obse rve t he bo r rower ' s r e a l i z ed r e tu rn on cap i t a l . Th i s aud i t i ng cos t i s

i n t e rp r e t a b l e a s th e c o s t o f b a n k r u p t c y ( i n c lu d i n g f o r e x a m p l e a u d i ti n g , a c c o u n t i n g ,

and l ega l cos t s , a s we l l a s l o s se s a s soc i a t ed w i th a s se t l i qu ida t i on and i n t e r rup t i on

o f b u s in e s s ) . T h e m o n i t o r i n g c o s t i s a s s u m e d t o e q u a l a p r o p o r t i o n / ~ o f th e r e a l i z ed

k . i

gros s pa yo f f t o t he f i rm ' s c ap i ta l , i .e . , the m on i to r i n g cos t equa l s /~ ~o R t + l Q t K i + I .

A l t h o u g h o n e m i g h t e x p e c t th a t th e r e w o u l d b e e c o n o m i e s o f s c a le i n m o n i t o r in g , t h e

p r o p o r t i o n a l i t y a s s u m p t i o n i s v e r y c o n v e n i e n t i n o u r c o n t e x t a n d d o e s n o t s e e m t o o

unrea sonab l e .

3 . 1 . C o n t r a c t t e r m s w h e n t h e r e i s n o a g g r e g a t e r i s k

To desc r i be t he op t ima l con t r ac tua l a r r angemen t , i t i s u se fu l t o f i r s t work t h rough t he

case whe re t he agg rega t e r e tu rn t o cap i t a l R t : + l i s known in advance . I n t h i s i n s t ance

the on ly unce r t a in ty abou t t he p ro j ec t ' s r e t u rn i s i d io sync ra t i c t o t he f i rm , a s i n t he

c o n v e n t i o n a l v e r s i o n o f t h e C S V p r o b l e m .

Absen t any agg rega t e unce r t a in ty , t he op t ima l con t r ac t unde r cos t l y s t a t e ve r i f i c a t i on

l o o k s v e r y m u c h l ik e s t a n d a rd r i s k y d e b t ( s e e A p p e n d i x A f o r a d e t a il e d a n a l y s i s o f

t h e c o n t r a c t in g p r o b l e m ) : I n p a r t i cu l a r, t h e e n t r e p r e n e u r c h o o s e s t h e v a l u e o f f i rm

capi ta l , J

t K i + t , and t he a s so c i a t ed l eve l o f bo r row ing , B /+ L, p r i o r t o t he r ea l i z a t i on

o f t he i d io sync ra t i c shock . G iv en / k

t K i + l , B [ + I , and R t+ l , t he op t im a l co n t r ac t m ay

b e c h a r a c t e r i z e d b y a g r o s s n o n - d e f a u l t l o a n r a t e , Z / ~ I , a n d a t h r e s h o l d v a l u e o f th e

id io sync ra t i c shoc k ~ ) i, c a ll i t ~ sJ , such t ha t f o r va lue s o f t he i d io sync ra t i c shock

g r e a t e r t h a n o r e q u a l t o ~ J , t h e e n t r e p r e n e u r i s a b le t o r e p a y t he l o a n a t th e c o n t r a c t u a i

ra te , Z j 1. Th a t i s , N. / i s def in ed b y

. . . . ~ - j . / . J

~ R~ l t K i ~ 1 = Z t ~ l B t ~1

3 . 3 )

W hen ~o/ ) c~ j, unde r t he op t ima l con t r ac t the en t r ep reneu r r ep ays t he l ende r t he

r j j j ~ j - j j

p r o m i s e d a m o u n t Z i + l B t + 1 and k eep s t he d i f f e r ence , equ a l t o c o R t + l Q t K i ~ . ~ - Z i + B : , 1 .

I f coy < N : , t he en t r ep reneu r canno t pa y the con t r ac tua l r e t u rn and t hus dec l a r e s

de fau l t , i n t h i s s i t ua t i on t he l end ing i n t e rmed ia ry pays t he aud i t i ng cos t and ge t s t o

keep wha t i t f i nds . Tha t i s , t he i n t e rmed ia ry ' s ne t r e ce ip t s a r e ( 1 - l ~ ) v ) R ~ + ~ Q ~ K / + 1 A

d e f a u l t in g e n t r e p r e n e u r r e c e i v e s n o t h i n g .


11/53

C h . 2 1 : T h e t ,3 n a n c i a l A c c e l e r a t o r i n a Q u a n t i t a t iv e B u s i n e s s C y c l e F r a m e w o r k 1351

T h e v a l u e s o f N J a n d Z /~ 1 u n d e r t h e o p t i m a l c o n t r a c t a r e d e t e r m i n e d b y t h e

r e q u i r e m e n t t h a t t h e f i n an c i a l i n t e r m e d i a r y r e c e i v e a n e x p e c t e d r e t u r n e q u a l t o t h e

o p p o r t u n i t y c o s t o f i ts f u n d s . B e c a u s e t h e l o a n r i s k i n t hi s c a s e i s p e r f e c t l y d iv e r s if i a bl e ,

t h e r e l e v a n t o p p o r t u n i t y c o s t t o t h e i n t e r m e d i a r y i s t h e r i s k l e s s r a te , R t + l . A c c o r d i n g l y ,

t h e l o a n c o n t r a c t m u s t s a t i s f y

~ o

P

• / ~ k j '

[ 1 - F ~ S ) I Z / + I B / + , + ( 1 - ~ t ) ~ o R t + l Q t K i + 1 d F ( ~ o ) = R , + , S / + t, ( 3 .4 )

w h e r e t h e l e f t - h a n d s i d e o f E q u a t i o n ( 3 .4 ) is t h e e x p e c t e d g r o s s r e t u r n o n t h e l o a n t o

t h e e n t r e p r e n e u r a n d t h e r i g h t s i d e is t h e i n t e r m e d i a r y ' s o p p o r t u n i t y c o s t o f l e n d i n g.

N o t e t h a t F ~ j ) g i v e s t h e p r o b a b i l i t y o f d e f a u l t.

C o m b i n i n g E q u a t i o n s ( 3 . 2 ) a n d (3 . 3 ) w i t h E q u a t i o n ( 3 . 4 ) y i e l d s t h e f o l l o w i n g

e x p r e s s io n f o r ~ 5 i :

p

[1 - F ~ J ) ] ~ j - + - 1 -: -~ ) . /0 (o dF (co ) R ~ + I Q t K /+ , - - R t + I Q t K J 1 - N,{ 1). (3 .5)

B y u s i n g E q u a t i o n ( 3 . 4 ) t o e l i m i n a t e Z ~ I , w e a r e a b l e t o e x p r e s s t h e l e n d e r ' s e x p e c t e d

r e t u r n s i m p l y a s a f u n c t i o n o f t h e c u t o f f v a l u e o f t h e f i r m ' s i d i o s y n c r a t i c p r o d u c t i v i t y

shock , k5s . T h e r e a r e t w o e f f e c ts o f c h a n g i n g ~ on t h e e x p e c t e d r e t u r n , a n d t h e y

w o r k i n o p p o s i t e d i r e c t io n s . A r i s e i n N J i n c r e a s e s t h e n o n - d e f a u l t p a y o f f ; o n t h e

o t h e r h a n d , i t a l s o r a i s e s t h e d e f a u l t p r o b a b i l i t y , w h i c h l o w e r s t h e e x p e c t e d p a y o f f .

T h e a s s u m e d r e s t r i c t i o n s o n t h e h a z a r d f u n c t i o n g i v e n b y E q u a t i o n ( 3 . 1 ) i m p l y t h a t

t h e e x p e c t e d r e t u r n r e a c h e s a m a x i m u m a t a n u n i q u e i n t er i o r v a lu e o f N i : A s N J

r i s e s a b o v e t h i s v a l u e t h e e x p e c t e d r e t u r n d e c l i n e s d u e t o t h e i n c r e a s e d l i k e l i h o o d

o f d e f a u l t 6 . F o r v a l u e s o f ~O b e l o w t h e m a x i n m m , t h e f u n c t i o n i s i n c r e a s i n g a n d

c o n c a v e 7 . I f th e l e n d e r ' s o p p o r t u n i t y c o s t i s s o l a rg e t h a t t h e r e d o e s n o t e x i s t a v a l u e

o f N J t h a t g e n e r a t e s t h e r e q u i r e d e x p e c t e d r e t u r n , t h e n th e b o r r o w e r i s r a t i o n e d f r o m

t h e m a r k e t . A p p e n d i x A p r o v i d e s d e t a il s . F o r si m p l i c i ty , i n w h a t f o l l o w s , w e c o n s i d e r

o n l y e q u i l i b r i a w i t h o u t r a t i o n in g , i .e ., e q u i l i b r ia i n w h i c h t h e e q u i l i b r iu m v a l u e o f b 5

a l w a y s l ie s b e l o w t h e m a x i m u m f e a s i b le v a l u e a . U n d e r t h e p a r a m e t r i z a t i o n s w e u s e

l a t e r , t h i s c o n d i t i o n i s i n f a c t s a t i s f i e d .

r' Flb see that the m aximm n m ust be in the interior of the support o f co, note th at as cO approaches its

upper bound, the default probability converges to u nity. Appe ndix A show s that the interior optimum is

unique.

7 The cha nge in the expected payoff fl'om a unit increase m cO is {[ 1 -F(cOJ)] -#cO/dF(cOJ)}R~+IQtK i i 1

The fi rst term in the expression in brackets reflects the r ise in the non-default payoff. The second

term ref lects the r ise in expected defau lt costs . Note that we can rewrite th is expression as

{1 - ~ S J h U S J ) } [ 1 - F(?O/)]RI+ QtK/+I, where h(a0 = dF(co)

- ~ is the hazard rate. Given Equation (3.i),

the derivative o f this ex pression is negative for values of CO be low the m axinm m one feasible, implying

that the expected payoff is concave in th is range.

8 Note also that since w e are restricting attention to non-rationing equilibria, the lende r's expected return

is alw ays increasin g in COJ.


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3 . 2. C o n t r a c t t e r m s w h e n t h e r e i s a g g r e g a t e r i s k

W i t h a g g r e g a t e u n c e r t a i n t y p r e s e n t , N J w i l l i n g e n e r a l d e p e n d o n th e e x p o s t r e a l i z a t i o n

o f R )+ ~. O u r a s s u m p t i o n t h a t t h e e n t r e p r e n e u r i s r i s k - n e u tr a l l e a d s to a s i m p l e c o n t r a c t

s t r u c tu r e , d e s p i t e th i s c o m p l i c a t i o n . B e c a u s e h e c a r e s o n l y a b o u t th e m e a n r e t u r n o n h i s

w e a l t h , t h e e n t r e p r e n e u r i s w i l l i n g t o b e a r a l l t h e a g g r e g a t e r i s k 9 . T h u s h e i s w i l l i n g t o

g u a r a n t e e t h e l e n d e r a r e t u r n t h a t i s f r e e o f a n y s y s t e m a t i c r is k , i .e ., c o n d i t i o n a l o n t h e

e x p o s t r e a l i z a t i o n o f R ~ + l , t h e b o r r o w e r o f f e r s a ( s t a t e - c o n ti n g e n t ) n o n - d e f a u l t p a y m e n t

t h a t g u a r a n t e e s t h e l e n d e r a r e t u r n e q u a l i n e x p e c t e d v a l u e t o t h e r is k l e s s r a te . ( N o t e

t h a t t h e o n l y r e si d u a l r i s k t h e l e n d e r b e a r s a r i s e s f r o m t h e i d i o s y n c r a t i c s h o c k o )/+ 1, a n d

i s t h u s d iv e r s i f i a b l e .) P u t d i f f e r e n tl y , E q u a t i o n ( 3 .5 ) n o w im p l i e s a s e t o f r e s t r i c t i o n s ,

ko n e f o r e a c h r e a l i z a t i o n o f R t+ 1. T h e r e s u l t i s a s c h e d u l e f o r 75 , c o n t i n g e n t o n t h e

r e a l i z e d a g g r e g a t e s t a t e . A s w e a r e r e s t r i c t i n g a t t e n t i o n t o n o n - r a t i o n i n g e q u i l i b r i a ,

w e c o n s i d e r o n l y p a r a m e t r i z a t i o n s w h e r e t h e r e i n fa c t e x is t s a v a l u e o f N / f o r e a c h

a g g r e g a t e s ta t e t h a t s a t is f ie s E q u a t i o n ( 3 .5 ) . D i v e r s i f i c a t io n b y i n t e r m e d i a r i e s i m p l i e s

t h a t h o u s e h o l d s e a r n t h e r i s k l e s s r a t e o n t h e i r s a v i n g .

D e s c r i p t iv e l y , t h e e x i s t e n c e o f a g g r e g a t e u n c e r t a i n t y e f f e c t iv e l y t i es t h e r i s k y

l o a n r a t e Z /+ ~to m a c r o e c o n o m i c c o n d i t i o n s . I n p a rt i c u la r , t h e l o a n r a t e a d j u s t s

c o u n t e r c y c l i c a l ly . F o r e x a m p l e , a re a l i z a t i o n o f

R~k+l

t h a t i s lo w e r t h a n e x p e c t e d r a i s e s

Zi /~

; t h a t is , t o c o m p e n s a t e f o r th e i n c r e a s e d d e f a u l t p r o b a b i l i t y d u e t o t h e l o w a v e r a g e

r e t u r n t o c a p it a l, t h e n o n - d e f a u l t p a y m e n t m u s t r i se . T h i s i n t u r n i m p l i e s a n i n c r e a s e i n

t h e c u t o f f v a l u e o f t h e i d i o s y n c r a t i c p r o d u c t i v i t y s h o c k , ~5 . T h u s t h e m o d e l i m p l i e s ,

r e a s o n a b l y , t h a t d e f a u l t p r o b a b i l i t i e s a n d d e f a u l t p r e m i a r i se w h e n t h e a g g r e g a t e r e t u r n

to c a p i t a l i s l o w e r t h a n e x p e c t e d ~0

3 .3 . N e t w o r t h a n d t h e o p t i m a l c h o i c e o f c a p i ta l

T h u s f a r w e h a v e d e s c r i b e d h o w t h e s t a t e - c o n t i n g e n t v a l u e s o f N / a n d z iJ ~ a r e

d e t e r m i n e d , g i v e n t h e e x p o s t r e a l i z a t i o n

o f R~/ +l

a n d t h e e x a n t e c h o i c e s o f

Q : K i

a n d

B/~ I.

W e n o w t u r n t o t he e n t r e p r e n e u r s g e n e r a l p r o b l e m o f d e t e r m i n i n g h i s d e m a n d

for cap i ta l .

9 The entrepreneur s value function can be show n to be linear in wealth because (i) his utility is linear in

consum ption and (ii) the project he is investing in exhibits constant returns to scale. [See, e.g., Bernanke

and G ertler (1989, 1990).]

10 T hi s kin d of state-contingent financial arrangeme nt is a bit stylized, but may be thou ght o f as

corresponding to the follow ing scenario: Let the maturity o f the d ebt be sho rter than the m aturity of the

firm s project. The de bt is th en rolled o ver afte r the realization of the aggregate m~certainty. If there is

bad aggregate news, then the new loan rate is higher than w ould be otherw ise. To implement the sort o f

risk-sharing arrangement imp lied by the m odel, therefore, all tha t is necessary is that som e com ponent

o f the financing have a shorter maturity than that o f the project.


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C h . 2 1 . T h e F i n a n c i a l A c c e l e r a t o r in a Q u a n t i t a ti v e B u s i n e s s C y c l e F r a m e w o r k

1353

G i ven t he s t a t e - con t i ngen t deb t f o r m o f t he op t i m a l con t r ac t , the ex pec t ed r e t u r n t o

t h e e n t r e p r e n e u r m a y b e e x p r e s s e d as

{ f i

j

E o ) R t+ 1 Q t K i + 1 dF (co ) - (1 - F ( ~ J ) ) N J R ~ I Q t K /+ l , (3.6)

k

wher e expec t a t i ons a r e t aken wi t h r e s pec t t o t he r andom va r i ab l e ,

R t + l ,

and i t i s

unde r s t ood t ha t

~ /

m ay be m ade con t i ngen t on t he r ea l i z a t i on o f th i s va r iab l e .

Co m bi n i ng t h i s re l a t i on wi t h Eq ua t i on ( 3 . 5 ) a l lows u s t o s i m p l i f y the en t r ep r eneu r s

o b j e c ti v e to m a x i m i z a t i o n o f

{ o }

9

L 1

- U ; O , J

3 . 7 )

k k

w h e r e U [a ~I =_ R t + j / E { R t + 1} i s the r a t i o o f the r ea l i z ed r e t u r n t o cap i ta l t o t he expe c t ed

r e tu r n . G i ven t ha t the i n t e r m ed i a r y m us t r e ce i ve a com pe t i t i ve r e t u r n , the en t r ep r ene u r

i n t e r na li z e s t he ex pec t ed de f au l t cos ts , a s E qua t i on ( 3 . 7 ) s ugges t s.

T h e f o r m a l i n v e s t m e n t a n d c o n t r a ct in g p r o b l e m t h e n r e d u c e s t o c h o o s i n g K /+ I an d

a s chedu l e f o r N / ( a s a f unc t ion o f t he r ea l iz ed va l ue s o f R) +I ) t o m ax i m i ze E qua -

t ion ( 3 . 7 ) , s ub j ec t t o the s e t o f s t a t e - con t i ngen t cons t r a in t s i m p l i ed by E qua t i on ( 3 .5 ) .

The d i s tr i bu t ions o f t he agg r ega t e and i d i o s ync r a t i c r i sk s t o t he r e t u r n t o cap i ta l , t he

p r i ce o f c ap i t al , and t he qu an t i ty o f ne t wor t h t ha t t he en t r ep r eneu r b r i ngs t o t he t ab le

a r e t aken a s g i ven i n t he m ax i m i za t i on .

L e t s t ~ E { R ~ + I / R t + I } be t he exp ec t ed d i s coun t ed r e t u r n t o cap i ta l . Fo r en t r ep r eneu r s

to pu rcha se capi ta l in the com pet i t ive equi l ibr ium i t m us t be the case tha t s t ~> 1 . G iven

s :

~> 1 , the f i r s t -order necessary condi t ions y ie ld the fo l lowing re la t ion for opt imal

cap i ta l pu r chas e s ( s ee App end i x A f o r de t ai ls ) :

Q a K / ~ , = * p ( s t ) N / + j ,

w i t h W ( 1 ) == l, W :( ) > O. (3 .8 )

Equ a t i on ( 3 . 8 ) de s c r i be s t he c r i t ic a l l ink be t w een cap i t a l expe nd i t u r e s by t he f i r m and

f i nanci a l cond i t i ons , a s m eas u r e d by t he w edge be t w een t he expec t ed t he r e t u rn t o

capi ta l and the sa fe ra te , s t , and b y e nt repren eur ia l n e t w or th , Art/1 JL. Given the value

o f K / + l

t ha t s a ti s fi e s Equa t i on ( 3 . 8 ) , the s ched u l e f o r NJ i s p i nn ed dow n un i que l y by t he

s t a t e - con t ingen t cons t r a i n t on t he e xpec t ed r e t u r n t o deb t , d e f i ned by Equ a t i on ( 3 . 5 ).

Equa t i on ( 3 . 8 ) is a k ey r e l a t ions h i p i n t he m ode l : I t s how s t ha t c ap i t a l expend i t u r e s

by each f i r m a r e p r opor t i ona l t o t he ne t wor t h o f t he owne r / en t r ep r eneu r , wi t h a

p r opo r t i ona l i t y f ac t o r t ha t i s i nc r ea s ing i n t he exp ec t ed d i s co un t ed r e t u r n t o cap it a l,

s t . Eve r y t h i ng e l s e equa l , a r i s e i n t he expec t ed d i s coun t ed r e t u r n t o cap i t a l r educes

t he expec t ed de f au l t p r obab i li t y . As a con s equence , t he en t r ep r ene u r car l t ake on m o r e

l J In the co stly enforcememmod el of K iyotakiand M oore (1997 ), ~p(.) - 1, implyingQ , K , . t - N i+ .


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B . S . B e r n a n k e e t a l .

E

2

c~

Demand 0 ~ , i . , , / / / , , .,.'

c5

f [

I i ~ T

6 7 8 9 10 11 12

CapitalStock

Fig. 1. Effect of an increase in net worth.

d e b t a n d e x p a n d t h e s i z e o f h is f i rm . H e i s c o n s t r a i n e d f r o m r a i s i n g t h e s i ze o f t h e f i r m

i n d e f i n it e l y b y th e f a c t t h a t e x p e c t e d d e f a u l t c o s t s a l s o r i se a s t h e r a ti o o f b o r r o w i n g

t o n e t w o r t h i n c r e a s e s .

A n e q u i v al e n t w a y o f e x p re s s i n g E q u a t i o n ( 3 .8 ) i s

N j

\

E { R f . l } = s [ ~ ,+1. | R ,~ , , s ( . ) < 0 . (3 .9 )

\ o,K/ ,]

F o r a n e n t r e p r e n e u r w h o i s n o t f u l l y s e l f - f i n a n c e d , i n e q u i l i b r i u m t h e r e t u r n t o c a p i t a l

w i l l b e e q u a t e d t o th e m a r g i n a l c o s t o f e x t e rn a l f i n a n c e. T h u s E q u a t i o n ( 3 . 9 ) e x p r e s s e s

t h e e q u i l i b r i u m c o n d i t i o n t h a t t h e r a ti o s o f t h e c o s t o f e x t e r n a l f i n a n c e t o t h e s a f e

r a t e - w h i c h w e h a v e c a l l e d t h e d i s c o u n t e d r e t u r n t o c a p i ta l b u t m a y b e e q u a l l y w e l l

i n t e r p r e t e d a s t h e e x t e rn a l f i n a n c e p r e m i u m - d e p e n d s i n v e r s e l y o n t h e s h a r e o f t h e

f i n n s c a p i ta l i n v e s t m e n t t h a t i s f i n a n c e d b y t h e e n t r e p r e n e u r s o w n n e t w o r t h .

F i g u r e 1 i l lu s t ra t e s t h i s r e l a ti o n s h i p u s i n g t h e a c t u a l c o n t r a c t c a l i b r a t e d f o r m o d e l

a n a l y s i s i n t h e n e x t s e c t i o n . F i r m j s d e m a n d f o r c a p it a l i s o n t h e h o r i z o n t a l a x i s

a n d t h e c o s t o f f u n d s n o r m a l i z e d b y t h e s a f e r a te o f r e t u r n i s o n t h e v e r t ic a l a x i s .

F o r c a p i t al s t o c k s w h i c h c a n b e f i n a n c e d e n t i r e ly b y t h e e n t r e p r e n e u r s n e t w o r t h ,

i n t h i s c a s e K < 4 .6 , t h e f i r m f a c e s a c o s t o f f u n d s e q u a l t o t h e r i s k f r e e r a te . A s

c a p i t a l a c q u i s i t i o n s r i s e i n t o t h e r a n g e w h e r e e x t e r n a l f i n a n c e i s n e c e s s a r y , t h e c o s t -

o f - f u n d s c u r v e b e c o m e s u p w a r d s l o p i n g , r e f l e c t i n g t h e i n c r e a s e i n e x p e c t e d d e f a u l t

c o s t s a s s o c i a t e d w i t h t h e h i g h e r r a t io o f d e b t t o n e t w o r t h . W h i l e t h e s u p p l y o f

f u n d s c u r v e i s - u p w a r d s l o p i n g , o w i n g t o c o n s t a n t r e t u r n s t o s c a l e , t h e d e m a n d f o r

c a p i t a l i s h o r i z o n t a l a t a n e x p e c t e d r e t u r n 2 p e r c e n t a g e p o i n t s a b o v e t h e r i s k f r e e r a t e .


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B.X Bernanke et al .

4 . 1. T h e e n t r e p r e n e u r i a l s e c t o r

R e c a l l t h a t e n t r e p r e n e u r s p u r c h a s e c a p i t a l i n e a c h p e r i o d f o r u s e i n t h e s u b s e q u e n t

p e r i o d . C a p i t a l is u s e d i n c o m b i n a t i o n w i t h h i r e d l a b o r t o p r o d u c e w h o l e s a l e ) o u t p u t .

W e a s s u m e t h a t p r o d u c t i o n i s c o n s t a n t r e t u r n s t o s c a l e , w h i c h a l l o w s u s t o w r i t e t h e

p r o d u c t i o n f u n c t io n a s a n a g g r e g a t e r e la t io n s h ip . W e s p e c i f y t h e a g g r e g a t e p r o d u c t i o n

f u n c t i o n r e l e v a n t t o a n y g i v e n p e r i o d t a s

Y t = A t K a L ] - a ,

4 .1 )

w h e r e Y t i s a g g r e g a t e o u t p u t o f w h o l e s a l e g o o d s , K t i s t h e a g g r e g a t e a m o u n t o f c a p i t a l

p u r c h a s e d b y e n t r e p r e n e u r s i n p e r i o d t - 1 , L~ i s la b o r i n p u t , a n d A t is a n e x o g e n o u s

t e c h n o l o g y p a r a m e te r .

L e t I t d e n o t e a g g r e g a t e i n v e s t m e n t e x p e n d i t u r e s . T h e a g g r e g a t e c a p i ta l s t o c k e v o l v e s

a c c o r d i n g t o

K ,+ I = k ,K , j K t + 1 - 6 ) K t ,

4 .2 )

w h e r e /5 i s t h e d e p r e c i a t i o n r a te . W e a s s u m e t h a t t h e re a r e i n c r e a s i n g m a r g i n a l

a d j u s t m e n t c o s t s i n th e p r o d u c t i o n o f c a pi ta l , w h i c h w e c a p t ur e b y a s s u m i n g t h a t

a g g r e g a t e i n v e s t m e n t e x p e n d i tu r e s o f L y i e l d a g r o s s o u t p u t o f n e w c a p i ta l g o o d s

• I ~ / K t ) K t ,

w h e r e q ~ .) is i n c r e a s i n g a n d c o n c a v e a n d q ~ 0 ) = 0 . W e i n c l u d e

a d j u s t m e n t c o s t s to p e r m i t a v a r i a b l e p r i c e o f c a p i ta l . A s i n K i y o t a k i a n d M o o r e 1 9 9 7 ) ,

t h e i d e a i s t o h a v e a s s e t p r i c e v a r i a b i l i ty c o n t r i b u t e t o v o l a t il i t y i n e n t r e p r e n e u r ia l n e t

w o r t h . I n e q u i l ib r i u m , g i v e n t h e a d j u s t m e n t c o s t f u n c t i o n , t h e p r ic e o f a u n i t o f c a p i ta l

i n t e r m s o f t h e n u m e r a i r e g o o d , Q t , i s g i v e n b y 12

= . 4 .3 )

W e n o r m a l i z e t h e a d j u s t m e n t c o s t f u n c t i o n s o t h a t t h e p r i c e o f ca p i t a l g o o d s i s u n i t y

in t h e s t e a d y s t a t e .

A s s u m e t h a t e n t r e p r e n e u r s s e ll t h e i r o u t p u t t o re t a il e rs . L e t 1 /X ~ b e t h e r e l a t i v e p r i c e

o f w h o l e s a l e g o o d s . E q u i v a l e n t l y , X t i s th e g r o s s m a r k u p o f re t a il g o o d s o v e r w h o l e s a l e

~2 1b implement investment expenditures in the decentralized equilibrium, think of there being

competitive capital producing firms that purchase raw output as a materials input, I~ and combine it

with rented capital, K to produce new capital goods via the production function I,3 g~ K . These capital

goods are t he n sold at the pric e Qt. S ince the cap ital-producing technology assumes constant returns to

scale, these capital-producing firms earn zero profits in equilibrium. Equation 4.3) is derived from the

first-order condition for investm ent for one of these firms.


17/53

Ch. 21: The Financial Accelerator in a Quan ti tative Business Cycle Fram ework

1357

g o o d s . T h e n t h e C o b b - D o u g l a s p r o d u c t i o n t e c h n o l o g y i m p l i e s t ha t t h e re n t p a i d to a

u n i t o f c a p i t a l i n t + 1 ( fo r p r o d u c t i o n o f w h o l e s a l e g o o d s ) i s 13

1 aY l l

Xt l

Kt l

T h e e x p e c t e d g r o s s r e t u r n t o h o l d i n g a u n i t o f c a p i t a l f r o m t t o t + 1 c a n b e w r i t t e n

1 aYt l }

k x ,2 , x,+~ + Q t + l ( 1 - 6 )

E { R z + 1} = E Ot '

( 4 .4 )

S u b s t i t u t i o n o f E q u a t i o n s ( 4 .1 ) a n d ( 4 .3 ) i n t o E q u a t i o n ( 4 . 4) y i e l d s a r e a s o n a b l y

c o n v e n t i o n a l d e m a n d c u r v e f o r n e w c a p i t a l. A s u s u a l , th e r e t u r n o n c a p i t a l d e p e n d s

i n v e r s e l y o n t h e l e v e l o f in v e s t m e n t , r e f l e c t in g d i m i n i s h i n g r e t u r n s .

T h e s u p p l y c u r v e f o r i n v e s t m e n t f i n a n c e is o b t a i n e d b y a g g r e g a t i n g E q u a t i o n ( 3 . 8 )

o v e r f ir m s , a n d i n v e r t i n g t o o b t a i n :

l, I : N t + l

E { R t + I } = s

( 4 . 5 )

A s i n E q u a t i o n ( 3 .9 ) , t h e f u n c t i o n s ( . ) is th e r a t i o o f t h e c o s t s o f e x t e r n a l a n d i n t e r n a l

f i n a n c e ; i t i s d e c r e a s i n g i n N t + l / Q t K t + l f o r N t ~ l < Q t K t + l . T h e u n u s u a l f e a t u r e o f

t h is s u p p l y c u r v e , o f c o u r s e , i s t h e d e p e n d e n c e o f t h e c o s t o f f u n d s o n t h e a g g r e g a t e

f i n a n c i a l c o n d i t i o n o f e n t re p r e n e u r s , a s m e a s u r e d b y t h e r a t io N t + l / Q t I t + l .

T h e d y n a m i c b e h a v i o r o f c a p i ta l d e m a n d a n d t h e r e t u r n t o c a p it a l d e p e n d o n

t h e e v o l u t i o n o f e n t r e p r e n e u r i a l n e t w o r t h , N :+ l . N : +I r e f l e c t s t h e e q u i t y s t a k e t h a t

e n t r e p r e n e u r s h a v e i n t h e i r f ir m s , a n d a c c o r d i n g l y d e p e n d s o n f i r m s ' e a r n i n g s n e t o f

i n t e r e s t p a y m e n t s t o l e n d e r s . A s a t e c h n i c a l m a t t e r , h o w e v e r , i t is n e c e s s a r y t o s t a rt

e n t r e p r e n e u r s o f f w i t h s o m e n e t w o r t h i n o r d e r t o a l l o w t h e m t o b e g i n o p e r a t i o n s .

F o l l o w i n g B e r n a n k e a n d G e r t l e r ( 1 9 8 9) a n d C a r l s t r o m a n d F u e r s t ( 1 99 7 ), w e a s s u m e

t~ To be con sistent with our assum ption that adjustme nt costs are external to the firm , we assum e that

entrepreneurs sell their capital at the end of p eriod t + 1 to the investm ent sector at price Q~+I. Thus

capital is then used to produce new investment goods and resold at the price Q ,j . The rental rate

(Q , 1 - Qt+l) reflects the influence of capital accumu lation on adjustm ent costs. Th is rate is determh led

by the zero-profit condition

~2 / 1 \ I t

Q, Q , =o.

In steady state q~ ( ~ ) = 6 and ~ ' ( : t )~ = 1, implying that Q = Q = 1. Around tile steady state,

the diffbrence between Q t~l and Qt is second ord er. We therefore om it the rental term and express

Equation (4.4) using Q:~ I rather than Qt+l-


18/53

1358 B.X Berv~anke et al.

t h a t, i n a d d i t io n t o o p e r a t i n g f i r m s , e n t re p r e n e u r s s u p p l e m e n t t h e ir i n c o m e b y w o r k i n g

i n t h e g e n e r a l l a b o r m a r k e t . T o t a l l a b o r i n p u t L t i s ta k e n t o b e t h e f o l lo w i n g c o m p o s i t e

o f h o u s e h o l d la b o r , H I , a n d e n t r e p r e n e u r i a l la b o r , H I :

L ,

= H t + + ( H / ) t - + .

4 , 6 )

W e a s s u m e f u r th e r t h a t e n t r e p r e n e u rs s u p p l y t h e i r l a b o r in e la s ti c al ly , a n d w e n o r m a l i z e

to t a l en t r ep reneu r i a l l abo r t o un i t y 14. I n t he ca l i b r a t i ons be low we k eep t he sha r e

o f i n c o m e g o i n g t o e n t r e p r e n e u r i a l l a b o r s m a l l ( o n th e o r d e r o f . 0 1 ), s o t h a t th i s

m o d i f i c a ti o n o f t h e s t a n d a r d p r o d u c t i o n f u n c t i o n d o e s n o t h a v e s i g n i fi c a n t d ir e c t e f f e c t s

on the resu l t s .

Le t V t be en t r ep reneu r i a l equ i t y ( i . e . , wea l t h accumula t ed by en t r ep reneu r s f r om

ope ra t i ng f i rms ) , le t W [ deno t e t he en t r ep re neu r i a l w age , an d l e t ? st deno t e t he s ta te+

con t i ng en t va lue o f ?5 s e t i n pe r i o d t . The n agg rega t e en t r ep ren eu r i a l ne t wo r th a t t he

end o f pe r i od t , N1+1 , i s g iven by

N + + I = y V t + W [

(4 .7)

w i t h

g t = R ) Q t 1 K t - R t

~ f O °

~ R ) Q t - I K t d F o ) ~

4- ~ ~ - t 5 ] (Qt-IK - ~Vt-1) '

4 . 8 )

wh ere g V~ i s t he equ i t y he ld b y en t r ep reneu r s a t t - 1 wh o a r e s t i ll i n bus ine s s

a t t. (En t r ep reneu r s w ho f a i l i n t con sum e the r e s idua l equ i t y (1 - 7 )V , Th a t is ,

C 7 = ( 1 - y ) V , ) E n t r e p r e n e u r i a l e q u i t y e q u a l s g r o s s e a r n i n g s o n h o l d i n g s o f e q u i t y

f r o m t - 1 t o t l es s r e p a y m e n t o f b o r r o w i n g s . T h e r a t i o o f d e f a u l t c o s t s t o q u a n t i ty

b o r r o w e d ,

f ~ , k( e R r Q t j K t

dF co)

Qt 1Kt

- -

N t - i

r e f l e c t s t he p r emium fo r ex t e rna l f i nance .

C lea r l y , unde r any r ea sonab l e pa r ame t r i z a t i on , en t r ep reneu r i a l equ i t y p rov ides t he

m a in sou rce o f va r i a t i on i n N t+ l. F u r the r , t h i s equ i t y ma y be h igh ly s en s i t ive t o

une xpe c t ed sh i f t s in a s se t p r i c e s , e sp ec i a l l y i f f irms a r e l eve r aged . To i l lu s t r a te , l e t

U [k =- R - E t j { R~} be t he unexpec t ed sh i f t i n t he g ros s r e tu rn t o cap i t a l , and l e t

14 N ote that entrepreneurs do not have to wo rk only on their own projects (such an assumiption would

violate aggregate returns to scale, given that individual projccts can b e of different sizes).


19/53

Ch. 21. The Financial Accelerator in a Quanti tative Business C ycle Framew ork

1359

U /p =-- f o ' o ) Q~_~Kt d F ( co ) - E t . i { ~ ' ~oQ t 1Kt dF( ~o) } be t he unexpec t ed sh i f t i n t he

cond i t i ona l ( on t he agg r ega t e s t a te ) de f au l t cost s . We can exp r e s s

Vt

as

V , = [ U ~ k ( 1 - t t U / P ) ] Q t 1 K t + E , l{V~}.

(4.9)

N ow cons i de r t he i m p ac t o f a unex pec t ed i nc r ea se i n t he ex pos t r e t u r n t o cap i ta l .

D i f f e r en t i a t ing Equ a t i on ( 4 .9 ) y i e l d s an exp r e s s i on f o r the e l a s t i c it y o f en t r ep r eneu r i a l

equ i t y w i t h r e spec t t o an unan t i c i pa t ed m ovem en t i n t he r e t u r n t o cap i t a l :

o v t / E ,

l{v }

_ E t I { R ~ } Q t - I K t

o r ; k i n , 1 R ? } E , , {

/> 1. (4. 10 )

A c c o r d i n g t o E q u a t i o n ( 4 . 1 0 ), a n u n e x p e c t e d o n e p e r c e n t c h a n g e i n t h e e x p o s t r e tu r n

t o cap i t a l l e ads t o a pe r cen t age change i n en t r ep r eneu r i a l equ i t y equa l t o t he r a t i o

o f g r os s ho l d i ngs o f c ap it a l t o equ i ty . To the ex t en t t ha t en t r ep r eneu r s a r e l eve r aged ,

t h is r a t i o excee ds un i t y , im p l y i ng a m agn i f ic a t i on e f f ec t o f unex pec t ed a s se t r e t u r ns

on en t r ep r eneu r i a l equ i t y . The key po i n t he r e i s t ha t unexpec t ed m ovem en t s i n a s se t

p r i ce s , w h i ch a r e l ike l y the l a r ge s t sou r ce o f unex pec t ed m ovem en t s i n g r os s r e tu r ns ,

can have a subs tant ia l e f fec t on f i rms f inanc ia l pos i t ions .

I n t he gene r a l equ i l ib r i um , f u r t he r, t he r e i s a k i nd o f m u l t i p l i e r e f fec t , as w e sha ll

s ee . A n unan t i c i pa t ed r i s e i n a s se t p r i ce s r a i s e s ne t w or t h m or e t han p r opor t i ona t e l y ,

w h i ch s t i m u l a t e s i nves t m en t and , i n t u r n , r a i s e s a s se t p r i ce s even f u r t he r . A nd so on .

T h i s p h e n o m e n o n w i l l b e c o m e e v i d e n t i n th e m o d e l s im u l a ti o n s.

W e n e x t o b t a i n d e m a n d c u r v e s f o r h o u s e h o l d a n d e n t r e p r e n e u r i a l l a b o r , f o u n d b y

equa t i ng m ar g i na l p r oduc t w i t h t he w age f o r e ach ca se :

1 - a ) ~ = x ,

N

1 - a ) 1 - ~ ) ~ = x , w f ,

(4 .11)

(4 .12)

w h e r e W~ i s t he r ea l w age f o r house ho l d l abo r and Wf i s t he r ea l w age f o r

ent repreneur ia l l abor .

Co m b i n i ng E qua t i ons ( 4 . 1 ) , (4 . 7 ) , (4 . 8 ) , and ( 4 .12 ) an d i m po s i ng t he cond i t i on t ha t

en t r ep r eneu r i a l l abo r i s f i xed at un i ty , y i e l d s a d i f f e r ence equ a t i on f o r Nt+l:

+ (1 - a)(1 O ) A t K ~ H ~ l- ) o.

(4 .13)

Equa t i on ( 4 . 13 ) and t he supp l y cu r ve t b r i nves t m en t f unds , Equa t i on ( 4 . 5 ) , a r e t he

t w o bas i c i ng r ed i en t s o f t he f i nanc ia l a cce le r a to r . The l a t te r equa t i on d e sc r i be s how


20/53

1360 B.S. Bernanke et al.

m ovem en t s i n ne t wor t h i n f l uence t he cos t o f c ap i t a l . The f o r m er cha r ac t e r i ze s t he

endogenous va r i a t i on i n ne t wor t h .

T h u s f a r w e h a v e d e t e r m i n e d w h o l e s a l e o u t p u t , i n v e s t m e n t a n d t h e e v o l u t i o n o f

cap i ta l , t he p r i ce o f c ap it a l, a nd t he evo l u t i on o f ne t w or t h , g i ven t he r i sk l e s s r ea l

in te res t r a te

R t + l ,

t he hous e ho l d r ea l wag e Wt, and t he r e l a ti ve p r i ce o f wh o l e s a l e

g o o d s I / X , T o d e te r m i n e th e s e p ri c e s a n d c o m p l e t e t h e m o d e l , w e n e e d t o a d d t h e

hous eho l d , r e t a i l , and gove r nm en t s ec t o r s .

4 .2 . T h e c o m p l e t e l o g - l i n e a r i z e d m o d e l

W e n o w p r e s e n t th e c o m p l e t e m a c r o e c o n o m i c f r a m e w o r k . M u c h o f th e d e r i v a ti o n i s

s t anda r d and n o t c en t r a l to t he d eve l o pm en t o f t he f i nanc i a l a cce le r a to r . We t he r e f o r e

s i m p l y w r i te t he com pl e t e l og - l i nea r i zed m o de l d i rec tl y , and de f e r m os t o f t he de t a i ls

t o A p p e n d i x B .

A s w e h a v e e m p h a s i z e d , th e m o d e l i s a D N K f r a m e w o r k m o d i f ie d to a l l o w f o r

a f i nanc i a l a cce l e r a t o r . I n t he backgr ound , a l ong wi t h t he en t r ep r eneu r s we have

des c r i bed a r e hous eho l ds an d r e t a il e r s. H ous eh o l ds a r e i n f i n i t e ly - l ived agen t s w ho

c o n s u m e , s av e, w o r k , a n d h o l d m o n e t a r y a n d n o n m o n e t a r y a ss et s. W e a s s u m e t h a t

hou s eho l d u t il i ty i s s epa rab l e ove r t i m e an d o ve r cons um pt i on , r e a l m on ey ba l ances ,

and l e is u r e . M om en t a r y u t il it y , fu r t he r , is l oga r i t hm i c i n each o f t he s e a r gum en t s is .

As i s s t anda r d i n t he l i t e r a t u r e , t o m o t i va t e s t i cky p r i ce s we m od i f y t he m ode l t o

a l l ow f o r m onopo l i s t i c com pe t i t i on and ( i m p l i c i t ) cos t s o f ad j us t i ng nom i na l p r i ce s .

I t i s inconve n i en t t o a s s um e t ha t the en t r ep r ene u r s who pu r ch as e cap i t a l and p r odu ce

ou t pu t i n t h i s m ode l a r e m onopo l i s t i c a l l y com pe t i t i ve , s i nce t ha t a s s um pt i on wou l d

com pl i ca t e the ana l ys e s o f t he f i nanc i a l con t r ac t wi t h l ende r s and o f t he evo l u t i on o f

ne t wor t h . To avo i d t h is p r ob l em , w e i n s t ead a s s um e t ha t t he m onop o l i s t ic com pe t i t i on

occu r s a t t he r e t a i l l eve l. Spec i fi c a ll y , we a s s um e t he r e ex i s ts a con t i nu um o f r e t a il e r s

( o f m e a s u r e o n e ) . R e t a il e rs b u y o u t p u t f r o m e n t r e p r e n e u r - p r o d u c e r s in a c o m p e t i t iv e

m ar ke t , t hen s l i gh t l y d i ff e r en t i a te t he o u t pu t t hey pu r chas e ( say, by pa i n t i ng i t a un i que

co l o r o r add i ng a b r and n am e ) a t no r e s o u r ce cos t . Becaus e t he p r oduc t i s d i f f e ren t i a t ed ,

each r e t a i le r ha s a b i t o f m a r ke t power . Hous eho l ds and f i rm s t hen pu r chas e CE S

aggr ega t e s o f t he s e r e t a il goo ds . I t is t he s e C ES aggr ega t e s t ha t a re co nve r t ed i n t o

cons um pt i on and i nves t m en t goods , and whos e p r i ce i ndex de f i ne s t he agg r ega t e p r i ce

l eve l. P r o f it s f r om r e t ai l a c t iv i t y a r e r eba t ed l um p- s um t o hous eho l ds ( i . e. , hous eho l ds

are the u l t imate ow ners o f re ta i l out l e t s . )

To i n t r oduce p r i ce i ne r ti a , we a s s um e t ha t a g i ven r e t a i le r i s f r ee t o cha nge h i s p r i ce

i n a g i v e n p e r io d o n l y w i th p r o b a b i l i ty 1 - 0 . T h e e x p e c t e d d u r at io n o f a n y p r ic e c h a n g e

i s 1@0- Th i s dev i ce , f o l l ow i ng Ca l vo ( 1983) , p r ov i des a s i m p l e way t o i nco r po r a t e

s t agge r ed l ong - t e r m nom i na l p r i ce s e t t i ng . Becaus e t he p r obab i l i t y o f chang i ng p r i ce

i s i ndependen t o f h i s to r y , the a gg r eg a t i on p r ob l em i s g r ea t l y si m p l if i ed . O ne ex t r a

~5 In particular, househ old atility s given by £ { ~ ;~ 0

[ ~

[ln(C~,k) + _~

n M t. JP , .-/~ + ~

In{ 1 l[,+i,)] .


21/53

Ch. 21. The Financ ial Acce lerator in a Quant i ta t ive Busine ss Cyc le Fra mew ork 1 3 6 1

t w i s t , f o l l o w i n g B e r n a n k e a n d W o o d f o r d ( 1 9 9 7 ) , i s t h a t f i r m s s e t t i n g p r i c e s a t t a r e

a s s u m e d t o d o s o p r i o r to t h e r e a li z a t io n o f a n y a g g r e g a t e u n c e r t a i n t y a t ti m e t.

L e t l o w e r c a s e v a r i a b l e s d e n o t e p e r c e n t d e v i a t i o n s f r o m t h e s t e a d y s t a t e , a n d l e t

r a ti o s o f c a p it a l l e tt e r s w i t h o u t t i m e s u b s c r ip t d e n o t e s t h e s t e a d y s t a te v a l u e o f t h e

r e s p e c t i v e ra t io . F u r t h e r , l e t ~ d e n o t e a c o l le c t i o n o f t e r m s o f s e c o n d - o r d e r i m p o r t a n c e

i n t h e e q u a t i o n f o r a n y v a r i a b l e z , a n d l e t e [ b e a n i . i . d , d i s t u r b a n c e t o t h e e q u a t i o n

f o r v a r i a b l e z . F i n a l l y , l e t G t d e n o t e g o v e r n m e n t c o n s u m p t i o n , : r t = - p ~ - p t - i t h e r a t e

o f i n f l a t io n f r o m t - 1 t o t , a n d

r /+1 ==- r ~+ ~ + E { p t + l - P t }

b e t h e n o m i n a l i n t e r e s t

r a te . I t i s t h e n c o n v e n i e n t t o e x p r e s s t h e c o m p l e t e l o g - l i n e a r i z e d m o d e l i n t e r m s o f

f o u r b l o c k s o f e q u a t io n s : ( 1 ) a g g r e g a t e d e m a n d ; ( 2 ) a g g r e g a t e s u p p l y ; (3 ) e v o l u t i o n

o f t h e s t a te v a r i a b le s ; a n d ( 4 ) m o n e t a r y p o l i c y r u l e a n d s h o c k p r o c e s s e s . A p p e n d i x B

p r o v i d e s d e t a i l s .

( 1 ) A g g r e g a t e d e m a n d

C 1 G C e

y ~ = ] 7 c , + ~ i t + ~ g t + - -c ~ 'y + + q ~, ( 4 . 1 4 )

ct = - rt+L + E t {ct+l } , (4 . 15 )

c~ = n t , l + +0~ :~ , ( 4 .16 )

E t { r ~ ÷ I }

- F z + I - - u [ ¥ 1 t + l

( q t + k t ÷ l ) ] ,

( 4 . 1 7 )

r ) + t - ( 1 - - e ) @ t k l -

k t - - i X t + l ) @

e q t ~ 1 q ,

( 4 . 1 8 )

qt = cp( i t - k t ) . (4 .19 )

( 2 ) A g g r e g a t e S u p p l y

y t = a, + a k t + ( 1 - a ) g 2 h ,

( 4 . 2 0 )

Y t - h t - x t - c t = ~ - I h t ,

4 . 2 1 )

YL) : E t l { l~ ( - X t ) + ~ 2T t+ l } .

( 4 . 2 2 )

3 ) E v o l u t i o n o f S t a t e V a r i ab l e s

k t + l = 6 i t + ( 1 - 6 ) k , ( 4 . 2 3 )

y R K t

4 . 2 4 )


22/53

1362

B .X B e r n a n k e e t a l .

4 ) M o n e t a r y P o l ic y R u l e a n d S h o c k P r o c e s s e s

17 I t~

r , - p r t 1 + g ac ~ I + e l , (4 .25)

& = p g g t - i + ~ , (4 .26)

a L = p , , a t - i + g / , (4 .27)

wi t h

; ) l

log Iz e ) d F ( ~ o ) R ~ Q t _ I K / D K ,

P

? O

D # j ) 6o dF(~o)R/ ',

O~'e = l o g ( 1 - C e + l / N i + ~ )

i T c ~T; /N

( R i l R -

1 - a ) 1 - [ 2 ) Y /X )

q~t = N 1 )K (r~ + q ~ -i + k t ) + N Y t - x ~ ,

l p ( R k / R ) 1 - b

~ ' ( R k / R ) '

(1 - cs) +

a Y / ( X K ) '

( ~ ( I / K ) 1 )t ( 1 ~ _ )

q ) ~ D I / K )

1 ) , , ' K ~ ( 1 - - 0 / ~ ) .

Equ a t i on ( 4 . 14 ) i s t he l og - l i nea f i zed ve r s i on o f t he r e s ou r ce cons t r a i n t. T he

p r i m ar y de t e r m i nan t s o f t he va r i a t ion i n agg r ega t e expend i t u r e s

Y t

a r e h o u s e h o l d

cons um pt i on c t , i nves t m en t i t , a n d g o v e r n m e n t c o n s u m p t i o n gL. O f l e s se r i m p o r t a n c e

i s var ia t ion in e n t repren eur ia l con sum pt io n c~ 16. Fina l ly , var ia t ion in reso urce s dev oted

m o n i t o r i ng cos t, em bed ded i n t he t e r m ~ , a l s o m a t t e r s in p r i nc ip l e . U nde r r ea s onab l e

pa r am e t r i za t i ons , howeve r , t h i s f a c t o r ha s no pe r cep t i b l e i m pac t on dynam i cs .

H o u s e h o l d c o n s u m p t i o n i s g o v e r n e d b y t h e c o n s u m p t i o n E u l e r r e l a t i o n , g i v e n b y

Eq ua t i on ( 4 .15 ) . Th e un i t coe f f i c i en t on t he r ea l i n t e re s t r a te ( i. e ., t he i n t e r t em p or a l

e las t i c i ty of subs t i tu t ion) re f l ec t s the assumpt ion of logar i thmic u t i l i ty over con . -

s um pt i on . By en f o r c i ng t he s t anda r d cons um pt i on Eu l e r equa t i on , we a r e e f f ec t i ve l y

a s s um i ng t ha t fi nanc i a l m a r ke t f r i c ti ons do no t i m ped e hous e ho l d behav i o r. Num er ous

au t ho r s have a r gued , howeve r , t ha t c r ed i t cons t r a i n t s a t t he hous eho l d l eve l i n f l uence

a non- t r iv i a l po r t i on o f agg r ega t e co ns um pt i on s pend i ng . An i n t e re s t i ng ex t ens i on o f

16 N ote that each variable in the log-lmearized esou rce constraint is weighted by the variable's share

o f ou tput in the stea dy state. Und er any reasonable parametrization o f the m odel, c~ has a relatively ow

weight.


23/53

C h . 2 1. T h e F i n a n c i a l A c c e l e r a t o r i n a Q u a n t i t a t ie e B u s i n e s s C y c l e F r a m e w o r k

1363

t h is m o de l w ou l d be t o i nco r po r a t e hous eho l d bo r r ow i ng an d a s s oc i a t ed f ri c ti ons . W i t h

s om e s l igh t m o d i f i c a ti on , the f i nanc i a l a cce l e r a to r wou l d t he n a l s o app l y t o hous eh o l d

spending , s t rengthening the overa l l e f fec t .

S i nce en t r ep r eneu r i a l cons um pt i on i s a (s m a l l ) f ixed f r ac t i on o f agg r ega te ne t w or t h

( r eca l l t ha t en t r ep r eneu r s who r e t i r e s i m p l y cons um e t he i r a s s e t s ) , i t s i m p l y va r i e s

p r opo r t i ona t e l y wi t h agg r ega t e ne t wor t h , a s Eq ua t i on ( 4 . 16 ) i nd i ca t e s .

Equa t i ons ( 4 . 17 ) , ( 4 . 18 ) , and ( 4 . 19 ) cha r ac t e r i ze i nves t m en t dem and . They a r e

t he l og - l i nea r i zed ve r s i ons o f Equ a t i ons ( 4 .5 ) , ( 4 . 4 ) and ( 4 . 3 ), r e s pec ti ve ly . Equa -

t i on ( 4 . 17 ) , i n pa r t icu l a r , cha r ac t e r i ze s the i n f l uence o f ne t wo r t h on i nves tm en t . I n

/,

t he abs en ce o f c ap i t a l m a r ke t f r i c ti ons , th i s r e l a t i on co l l aps e s t o

E t { r f + ~

} - r t + l = 0 :

I nves t m en t i s pus hed t o t he po i n t wh e r e t he ex pec t ed r e t u r n on cap it a l, E t { r ~ + 1 }, equa l s

the opp or tu ni ty cos t o f funds r t+117. W i th capi ta l ma rke t f r i c t ions present , howev er , the

cos t o f ex t e r na l f unds depends o n en t r ep r eneu r s ' pe r cen t ag e equ i t y ho l d i ng , i . e. , ne t

w or th re la t ive to the g ross v a lue o f capit a l , n t~ l - (qr + k t f- l) . A r i se in th i s ra t io reduce s

t he cos t o f ex t e r na l f unds, i m p l y i ng t ha t i nves t m en t wi l l r is e . W hi l e Equ a t i on ( 4 . 17 )

em beds t he f i nanc i a l a cce l e r a t o r , Equa t i ons ( 4 . 18 ) and ( 4 . 19 ) a r e conven t i ona l ( l og -

l i nea r i zed ) r e l a t i ons f o r t he m ar g i na l p r oduc t o f c ap i t a l and t he l i nk be t ween a s s e t

p r i ce s and i nves t m en t .

Equa t i ons ( 4 . 20 ) , ( 4 . 21 ) and ( 4 . 22 ) cons t i t u t e t he agg r ega t e s upp l y b l ock . Equa -

t i on ( 4 . 20 ) i s t he l i nea r i zed ve r s i on o f t he p r odu c t i on f un c t i on ( 4 . 1 ) , a f te r inco r po r a t i ng

t he a s s um pt i on t ha t t he s upp l y o f en t r ep r eneu r i a l labo r i s f i xed . Equ a t i on ( 4 . 21 )

cha r ac t e r i ze s l abo r m ar ke t equ i l i b r i um . The l e f t s i de i s t he m ar g i na l p r oduc t o f

l abo r w e i gh t ed by t he m ar g i na l u t i li t y o f cons u m p t i on 18. I n equ i l ib r i um , i t va r ie s

p r opo r t i ona t e l y wi t h t he m ar kup o f r e t ai l goods ov e r who l e s a l e g oods ( i . e. , t he i nve r s e

o f the r e l a t ive p r i ce o f who l e s a l e goods . )

Equa t i on ( 4 . 22 ) cha r ac t e r i ze s p r i ce ad j us t m en t , a s i m p l i ed by t he s t agge r ed p r i ce

s e t ti ng f o r m ul a t i on o f Ca l vo ( 1983) t ha t we de s c r i bed ea r l i e r [ a long wi t h t he

m o d i f i c a t i o n s u g g e s t e d b y B e r n a n k e a n d W o o d f o r d ( 1 9 9 7 ) ] . T h i s e q u a t i o n h a s t h e

f l avo r o f a t r ad i t i ona l Ph i l l i p s cu r ve , once i t i s r e cogn i zed t ha t t he m ar kup x t va r i e s

i nve r s e l y wi t h t he s t a te o f dem and . W i th nom i na l p r i ce r i g i d i t ie s , t he r e t a il f i rm s t ha t

ho l d t he i r p r i ce s f ixed ove r t he pe r i od r e s pon d t o i nc r ea s ed d em and b y s e l li ng m or e . To

a c c o m m o d a t e t h e r is e i n s al es t h e y i n c r e a s e th e i r p u r c h a se s o f w h o l e sa l e g o o d s f r o m

en t r ep r eneu r s , wh i ch b i ds up t he r e l a t i ve who l e s a l e p r i ce and b i ds down t he m ar kup .

i t i s t b r t h i s r e a s on t ha t

- x t

pr ov i des a m eas u r e o f dem an d w hen p r i ce s a r e s ticky . I n

t u rn , t he s ens i ti v i ty o f in f l at ion to dem and depends on t he deg r ee o f p r i ce i ne r ti a : Th e

s l ope coe f f i c i en t t¢ can be s how n t o be dec r ea s i ng i n 0 , t he p r o bab i l i t y an i nd i v idua l

p r i ce s t ays fi xed f r om p e r i od t o pe r i od . O ne d i f f e r enc e be t w een E qua t i on ( 4 . 22 ) and

17 In the abse nce of capital ma rket frictions, the first-order con dition fro m the entrepreneur's partial

equilibrium capital ch oice decision yields E { R ) + I } = R t + L In this instance if E { R ~ '4 l } > R , j , the

entrepreneur wo uld buy an infinite am oun t of capital, and if E{R~+1} < R~+l, he would buy none. When

E { R t + I } - R ~ 1 , he is indifferentabou t the scale of o peration of his firm.

i~ G iven logarithmic preferen ces, he margina l utility of consum ption is sim ply - .


24/53


25/53

C h . 2 1 : T h e t q n a n c i a l A c c e l e r at o r i n a Q u a n t i t a t i o e B u s i n e s s C y c l e F r a m e w o r k 365

4 .2 .1 . T w o e x t e n s i o n s o f t h e b a s e l i n e m o d e l

Two mo d i f i c a t i ons t ha t we cons ide r a r e : 1 ) a l l ow ing fo r de l ays i n i nves tmen t ; and

2 ) a l l owing fo r f i rms w i th d i f f e r en t i a l a cce s s t o c r ed i t . The f i r s t mod i f i c a t i on pe rmi t s

t h e m o d e l t o g e n e r a t e t h e k i n d o f h u m p - s h a p e d o u t p u t d y n a m i c s t h at a re o b s e r v e d in

the da t a . The s econd i s mean t t o i nc r ea se de sc r i p t i ve r ea l i sm .

4 . 2 .1 . 1 . I n o e s t m e n t d e l a y s . D i s t u r b a n c e s t o t h e e c o n o m y t y p i c a l l y a p p e a r t o g e n e r a te a

d e l a y e d a n d h u m p - s h a p e d r e s p o n s e o f o u tp u t . A c l a s s i c e x a m p l e i s t h e o u t p u t re s p o n s e

t o a m o n e t a r y p o l i c y s h o c k [ s ee , e . g ., C h r is t ia n o , E i c h e n b a u m a n d E v a n s 1 9 9 6 ) a n d

B e r n a n k e a n d M i h o v 1 9 9 8 )] . I t t a k e s ro u g h l y tw o q u a r t e r s b e f o r e a n o r t h o g o n a l i z e d

innova t i on i n t he f ede r a l f unds r a t e , f o r example , gene ra t e s a s i gn i f i c an t movemen t

i n o u tp u t . T h e p e a k o f t h e o u t p u t r e s p o n s e o c c u r s w e l l a f t e r t h e p e a k i n th e f u n d s

r a te d e v ia t i o n . R o t e m b e r g a n d W o o d f o r d 1 9 9 7 ) a d d r e s s t h is i s s u e b y a s s u m i n g t h a t

c o n s u m p t i o n e x p e n d i t u r e s a re d e t e rm i n e d t w o p e r i o d s in a d v a n c e i n a m o d e l in w h i c h

n o n - d u r a b l e c o n s u m p t i o n i s th e o n l y t y p e o f p r i v a t e e x p e n d i t u r e ). W e ta k e a n a p p r o a c h

tha t is s im i l a r in sp i r it , bu t i n s t ead a s su m e tha t i t i s i nves tm en t expen d i t u r e

b gg handbook

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