time to build

8/20/2019 Time to Build

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STOR Time to Build and Aggregate Fluctuations

Finn E. Kydland, Edward C. Prescott

Econometrica, Volume 50, Issue !"o#., $%&'(, $)*5+$)0.

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3e researc3 was su44orted y t3e "ational -cience Foundation. @e are grate=ul to -ean Bec6etti, Fisc3er Blac6, oert -. C3irin6o, ar6 erso#it, C3risto43er A. -ims, and >o3n B.

l comments, to -umru Altug =or researc3 assistance, and to t3e 4artici4ants in t3e seminars at t3e se#eral uni#ersities at w3ic3 earlier dra=ts were 4resented.

E C ; " ; E T I C AV;1/E

50";VEBE ,

$%&'"/BE

TIE T; B/I1D A"D AEATE F1/CT/ATI;"- B< FI"" E. K <D1A"D A"D ED@AD C. PE-C;TT$

T3e euilirium growt3 model is modi=ied and used to e4lain t3e cyclical #ariances o= a set o=

economic time series, t3e co#ariances etween real out4ut and t3e ot3er series, and t3e autoco#ariance

o= out4ut. T3e model is =itted to uarterly data =or t3e 4ost+war /.-. economy. Crucial =eatures o= t3e

model are t3e assum4tion t3at more t3an one time 4eriod is reuired =or t3e construction o= new

4roducti#e ca4ital, and t3e non+time+se4arale utility =unction t3at admits greater intertem4oral

sustitution o= leisure. T3e =it is sur4risingly good in lig3t o= t3e model?s sim4licity and t3e small

numer o= =ree 4arameters.

1. I"T;D/CTI;"

TGAT @I"E I- ";T ADE in a day 3as long een recognied y economists !e.g., Bo3m+Bawer6 H(.

But, neit3er are s3i4s nor =actories uilt in a day. A t3esis o= t3is essay is t3at t3e assum4tion o=

multi4le+4eriod construction is crucial =or e4laining aggregate =luctuations. A general

euilirium model is de#elo4ed and =itted to /.-. uarterly data =or t3e 4ost+war 4eriod. T3e co+

mo#ements o= t3e =luctuations =or t3e =itted model are uantitati#ely consistent wit3 t3e corre+

s4onding co+mo#ements =or /.-. data. In addition, t3e serial correlations o= cyclical out4ut =or t3e

model matc3 well wit3 t3ose oser#ed.

;ur a44roac3 integrates growt3 and usiness cycle t3eory. 1i6e standard growt3 t3eory, a

re4resentati#e in=initely+li#ed 3ouse3old is assumed. As =luctuations in em4loyment are central to

t3e usiness cycle, t3e stand+in consumer #alues not only consum4tion ut also leisure. ;ne #ery

im4ortant modi=ication to t3e standard growt3 model is t3at multi4le 4eriods are reuired to

uild new ca4ital goods and only =inis3ed ca4ital goods are 4art o= t3e 4roducti#e ca4ital stoc6.

Eac3 stage o= 4roduction reuires a 4eriod and utilies resources. Gal==inis3ed s3i4s and =actories

are not 4art o= t3e 4roducti#e ca4ital stoc6. -ection ' contains a s3ort critiue o= t3e commonly

used in#estment tec3nologies, and 4resents e#idence t3at single+4eriod 4roduction, e#en wit3ad7ustment costs, is inadeuate. T3e 4re=erence+tec3nology+in=ormation structure o= t3e model is

4resented in -ection ). A crucial =eature o= 4re=erences is t3e non+time+se4arale utility =unction

t3at admits greater intertem4oral sustitution o= leisure. T3e eogenous stoc3astic com4onents in

t3e model are s3oc6s to tec3nology and im4er=ect indicators o= 4roducti#ity. T3e two tec3nology

s3oc6s di==er in t3eir 4ersistence.

$)*5



AEATE F1/CT/ATI;"- $)*

ation and Yj =or some ot3er industrial organiation, t3e same aggregate su44ly e3a#ior results

i= ' Yj 9 ' YJ N

T3e ad7ustment cost model, rat3er t3an assuming a linear 4roduct trans=ormation cur#e

etween t3e in#estment and consum4tion goods, im4oses cur#ature. T3is can e re4resented y

t3e =ollowing tec3nology2

G(C,I) < F(K,L), K= I-8K ,

w3ere G li6e F is increasing, conca#e, and 3omogeneous o= degree one. 1etting t3e 4rice o= t3e

consum4tion good e one, t3e 4rice o= t3e in#estment good q n t3e rental 4rice o= ca4ital r n and

t3e wage rate w t , t3e =irm?s 4rolem is to maimie real 4ro=its, C t q t I t ! w t L t ! r t K t ,

su7ect to t3e 4roduction constraint. As constant returns to scale are assumed, t3e distriution o=

ca4ital does not matter, and one can 4roceed as i= t3ere were a single 4rice+ta6ing =irm. Assuming

an interior solution, gi#en t3at t3is tec3nology dis4lays constant returns to scale and t3at t3e

tec3nology is se4arale etween in4uts and out4uts, it =ollows t3at I t 9 F"K n L t )# "q t ) 9 $ t #( q t ) ,

w3ere $ t is de=ined to e aggregate out4ut. T3e =unction # is increasing, so 3ig3 in#estment+

out4ut ratios are associated wit3 a 3ig3 4rice o= t3e in#estment good relati#e to t3e consum4tion

good. Figure $ de4icts t3e in#estment+consum4tion 4roduct trans=ormation cur#e and Figure '

FI/E $. FI/E '.



3e data on commercial construction and 4rice 4er suare =oot were =or $%& and were otained =rom F. @. Dodge Di#ision o= craw+Gill.

is oser#ation is due to Fumio Gayas3i.

$)*& F. E. K<D1A"D A"D E. C. PE-C;TT

To test w3et3er t3e t3eory is a reasonale a44roimation, we eamined cross+section state data.

T3e correlations etween t3e ratios o= commercial construction to eit3er state 4ersonal income or

state em4loyment and 4rice 4er suare =oot5 are ot3 M0.)5. @it3 4er=ectly elastic su44ly and

uncorrelated su44ly and demand errors, t3is correlation cannot e 4ositi#e. To e4lain t3is large

negati#e correlation, one needs a comination o= 3ig3 #ariaility in t3e cross+sectional su44ly

relati#e to cross+sectional demand 4lus a 4ositi#e slo4e =or t3e su44ly cur#e. ;ur #iew is t3at,

gi#en moility o= resources, it seems more 4lausile t3at t3e demand is t3e more #ariale.

Admitting 4otential data 4rolems, t3is cross+sectional result casts some dout u4on t3e adeuacy

o= t3e single ca4ital good ad7ustment cost model.

At t3e aggregate le#el, an im4lication o= t3e single ca4ital good ad7ustment cost model is t3at

w3en t3e in#estment+out4ut ratio is regressed on current and lagged q , only current q s3ould

matter. T3e =indings in H' are counter to t3is 4rediction.

In summary, our #iew is t3at neit3er t3e neoclassical nor t3e ad7ustment cost tec3nologies are

adeuate. T3e neoclassical structure is inconsistent wit3 t3e 4ositi#e association etween t3e

s3adow 4rice o= ca4ital and in#estment acti#ity. T3e ad7ustment cost tec3nology is consistent wit3t3is oser#ation, ut inconsistent wit3 cross+sectional data and t3e association o= in#estment wit3

t3e lagged as well as t3e current ca4ital s3adow 4rices. In addition, t3e im4lication t3at long+ and

s3ort+run su44ly elasticities are eual is one w3ic3 we t3in6 a tec3nology s3ould not 3a#e.

ost destructi#e o= all to t3e ad7ustment+cost tec3nology, 3owe#er, is t3e =inding t3at t3e time

reuired to com4lete in#estment 4ro7ects is not s3ort relati#e to t3e usiness cycle. ayer H', on

t3e asis o= a sur#ey, =ound t3at t3e a#erage time !weig3ted y t3e sie o= t3e 4ro7ect( etween t3e

decision to underta6e an in#estment 4ro7ect and t3e com4letion o= it was twenty+one mont3s.



l stoc6s are eginning+o=+t3e+4eriod stoc6s.

AEATE F1/CT/ATI;"- $)*%

;ur tec3nology assumes t3at a single 4eriod is reuired =or eac3 stage o= construction or t3at

t3e time reuired to uild new ca4ital is a constant. T3is is not to argue t3at t3ere are not

alternati#e tec3nologies wit3 di==erent construction 4eriods, 4atterns o= resource use, and total

costs. @e 3a#e =ound no e#idence t3at t3e ca4ital goods are uilt signi=icantly more ra4idly w3en

total in#estment acti#ity is 3ig3er or lower. 1engt3ening deli#ery lags !see H%( in 4eriods o= 3ig3

acti#ity may e a matter o= longer ueues and actual construction times may e s3orter. Premiums

4aid =or earlier deli#ery could #ery well e =or a more ad#anced 4osition in t3e ueue t3an =or a

more ra4idly constructed =actory. T3ese are, o= course, em4irical uestions, and im4ortant cyclical

#ariation in t3e construction 4eriod would necessitate an alternati#e tec3nology.

;ur time+to+uild tec3nology is consistent wit3 s3ort+run =luctuations in t3e s3adow 4rice o=

ca4ital ecause in t3e s3ort run ca4ital is su44lied inelastically. It also im4lies t3at t3e long+run

su44ly is in=initely elastic, so on a#erage t3e relati#e 4rice o= t3e in#estment good is inde4endent o=

t3e in#estment+out4ut ratio.

3. TGE ;DE1

%ec#no&o'

T3e tec3nology assumes time is reuired to uild new 4roducti#e ca4ital. 1et j t e t3e

numer o= 4ro7ects j stages or j 4eriods =rom com4letion =or j 9 $, . . . , M $, w3ere J 4eriods

are reuired to uild new 4roducti#e ca4acity. "ew in#estment 4ro7ects initiated in 4eriod t are

*>t. T3e recursi#e re4resentation o= t3e laws o= motion o= t3ese ca4ital stoc6s is

(3.1) + t & = ( & -8)+ t * ,



$)50 F. E. K<D1A"D A"D E. C. PE-C;TT

in#estment t & ! n and conseuently J

(3.3) i , 9 ' < j * j /, + 1 M , 0

+$

Total out4ut, t3at is, t3e sum o= consum4tion c t and in#estment, is constrained as =ollows2

(3.4) c, i ,<1(2, ,+ t , n, , t ) ,

w3ere n t is laor in4ut, 2 t a s3oc6 to tec3nology, and is a constant+returns+to+ scale 4roduction

=unction to e 4arameteried suseuently.

Treating in#entories as a =actor o= 4roduction warrants some discussion. @it3 larger in#entories,

stores can economie on laor resources allocated to restoc6ing. Firms, y ma6ing larger

4roduction runs, reduce eui4ment down time associated wit3 s3i=ting =rom 4roducing one ty4e

o= good to anot3er. Besides considerations suc3 as t3ese, analytic considerations necessitated t3is

a44roac3. I= in#entories were not a =actor o= 4roduction, it would e im4ossile to locally

a44roimate t3e economy using a uadratic o7ecti#e and linear constraints. @it3out suc3 an

a44roimation no 4ractical com4utational met3od currently eists =or com4uting t3e euilirium

4rocess o= t3e model.

T3e 4roduction =unction is assumed to 3a#e t3e =orm



e t3an6 asser -aidi =or suggesting t3is argument.

AEATE F1/CT/ATI;"- $)5$

T3e #ariale a t summaries t3e e==ects o= all 4ast leisure c3oices on current and =uture

4re=erences. I= n * = n t =or all * <t, t3en a t 9 nj t7, and t3e distriuted lag is sim4ly $ ! n t .

T3e 4arameters a 5 and t7 determine t3e degree to w3ic3 leisure is intertem4or+ ally

sustitutale. @e reuire 0 L t7 L $ and 0 L a 5 L $. T3e nearer a 5 is to one, t3e less is t3e

intertem4oral sustitution o= leisure. For a5 eual to one, time+ se4arale utility results. @it3 rj

eual to one, a t euals n t 6 7 . T3is is t3e structure em4loyed in H)). As t7 a44roac3es ero, 4ast

leisure c3oices 3a#e greater e==ect u4on current utility =lows.

"on+time+se4arale utility =unctions are im4licit in t3e em4irical study o= aggregate laor

su44ly in H'5. rossman H$' and 1ucas H'* discuss w3y a non+time+se4arale utility =unction is

needed to e4lain t3e usiness cycle =luctuations in em4loyment and consum4tion. A micro

7usti=ication =or our 3y4ot3esied structure ased on a Bec6erian 3ouse3old 4roduction =unction is

as =ollows.& Time allocated to non+mar6et acti#ities, t3at is ,, is used in 3ouse3old 4roduction. I=

t3ere is a stoc6 o= 3ouse3old 4ro7ects wit3 #arying out4ut 4er unit o= time, t3e rational 3ouse3old

would allocate & t to t3ose 4ro7ects wit3 t3e greatest returns 4er time unit. I= t3e 3ouse3old 3as

allocated a larger amount o= time to non+mar6et acti#ities in t3e recent 4ast, t3en only 4ro7ectswit3 smaller yields s3ould remain. T3us, i= a t is lower, t3e marginal utility #alue o= & t s3ould e

smaller.

Cross+sectional e#idence o= 3ouse3olds? willingness to redistriute laor su44ly o#er time is t3e

lum4iness o= t3at su44ly. T3ere are #acations and mo#ements o= 3ouse3old memers into and out

o= t3e laor =orce =or etended 4eriods w3ic3 are not in res4onse to large mo#ements in t3e real

wage. Anot3er oser#ation suggesting 3ig3 intertem4oral sustitutaility o= leisure is t3e large

seasonal #ariation in 3ours o= mar6et em4loyment. Finally, t3e =ailure o= Aowd and As3en=elter

H' to =ind a signi=icant wage 4remium =or 7os wit3 more #ariale em4loyment and earnings

4atterns is =urt3er e#idence. In summary, 3ouse3old 4roduction t3eory and cross+sectional



3e im4ortance o= 4ermanent and transitory s3oc6s in studying macro =luctuations is em43asied in H&.3e #alue used =or 4 in t3is study was 0.%5. T3e reason we restricted 4 to e strictly less t3an one was tec3nical. T3e t3eorem we em4loy to guarantee t3e eistence o= com4etiti#e euiliri

arity o= t3e s3oc6.

$)5' F. E. K<D1A"D A"D E. C. PE-C;TT

moti#ated y t3e =act t3at 3ouse3olds? allocation o= time to nonmar6et acti#ities is aout twice as

large as t3e allocation to mar6et acti#ities.

In 1orm ati on t r ct re

@e assume t3at t3e tec3nology 4arameter is su7ect to a stoc3astic 4rocess wit3 com4onents o=

di==ering 4ersistence. T3e 4roducti#ity 4arameter is not oser#ed ut t3e stand+in consumer does

oser#e an indicator or noisy measure o= t3is 4arameter at t3e eginning o= t3e 4eriod. T3is

mig3t e due to errors in re4orting data or 7ust t3e =act t3at t3ere are errors in t3e est or

consensus =orecast o= w3at 4roducti#ity will e =or t3e 4eriod. ;n t3e asis o= t3e indicator and

6nowledge o= t3e economy+wide state #ariales, decisions o= 3ow many new in#estment 4ro7ects

to initiate and o= 3ow muc3 o= t3e time endowment to allocate to t3e 4roduction o= mar6eted

goods are made. -useuent to oser#ing aggregate out4ut, t3e consum4tion le#el is c3osen wit3

in#entory in#estment eing aggregate out4ut less =ied in#estment and consum4tion.

-4eci=ically, t3e tec3nology s3oc6, 2 n is t3e sum o= a 4ermanent com4onent, 2 , and a transitory

com4onent,% 2 't2

(3.7) 2 t 9 2 2 t .

AEATE F1/CT/ATI;"- $)5)

oser#ed or deducile at t3e time o= t3ese decisions. T3e consum4tion+in#entory in#estment

decision, 3owe#er, is contingent u4on 2 t =or aggregate out4ut is oser#ed 4rior to t3is decision

and 2 t can e deduced =rom aggregate out4ut and 6nowledge o= in4uts.

T3e state s4ace is an a44ro4riate =ormalism =or re4resenting t3is recursi#e in=ormation

structure. Because o= t3e two+stage decision 4rocess, it is not a direct a44lication o= Kalman

=iltering. 1i6e t3at a44roac3 t3e se4aration o= estimation and control is e4loited. T3e general

structure assumes an unoser#ale state #ector, say 7 t , t3at =ollows a #ector autoregressi#e

4rocess wit3 inde4endent multi#ariate normal inno#ations2

(3.11) 7 t & = 97 t *+ Q0,, w3ere Q0R iV!0, : 5 ).



: 7 9 H#ar!= )(, and : 9 H0. @it3 t3ese de=initions, t3e in=ormation structure

(3.7) M!).$0( #iewed as de#iations =rom t3e mean and t3e re4resentation !).$$(+

(3.17) S2 = 2, - (;Yi;;i F2)-'522,.

$)5* F. E. K<D1A"D A"D E. C. PE-C;TT

Finally, =rom !).$$(,

(3.18) m0 t & =9 m t , and

(3.19) '0 9 9 9 *+ : 0.

T3e co#ariances '0, '$8 and '' are de=ined recursi#ely y !).$5(, !).$(, and

(3.19) . T3e matri : 0 eing o= =ull ran6 along wit3 t3e staility o= 9 are su==icient to insure

t3at t3e met3od o= successi#e a44roimations con#erges e4onentially =ast to a uniue solution.

T3e co#ariance elements '0, '$8 and '' do not c3ange o#er time and are t3ere=ore not 4art o= t3e

in=ormation set. T3e m 5 t , mu, and m t do c3ange ut are su==icient relati#e to t3e rele#ant

3istories =or =orecasting =uture #alues o= ot3 t3e unoser#ed state and t3e oser#ale 8 T, r > t ,

and =or estimating t3e current unoser#ed state.

Eq i &i ri m



AEATE F1/CT/ATI;"- $)55

-TEAD< -TATE, APP;IATI;", A"D C;P/TATI;" ;F EJ/I1IBI/

Variales wit3out suscri4t denote steady state #alues. T3e steady state interest rate is r 9 !$ M

t ) ? 1 t , and t3e steady state 4rice o= !non+in#entory( ca4ital q 9 'y9i0 r(S$!Py+ T3e latter is

otained y oser#ing t3at L47 units o= consum4tion must e =oregone in t3e current 4eriod, L4 '

!*.$(

= r *+ 8

I ! a+ 9 +.

e 4roduction =unction wit3 res4ect to ca4ital, sustituting =or =rom !*.$(, and euating to t3e

!*.'( + 9

, _ ( 1 - 9 -

v ) / v

1/0

2 & ? 5n 0 @ A B ? ? 5 n.

-teady+state out4ut as a =unction o= n is9 !$ ) ? D A A 7 ? n 9 2 & ? 5 n. In t3e steady state, net

in#estment is ero, so

(4.3) c 9 2 B? e n ! 8+ = ! - 8 @ ) A & ? 5 n.

+ 7




In t3e steady state, c t 9 c, & t = , and w t = w =or all t. a6ing t3ese sustitutions

and using t3e =act t3at t3e a t sum to one, t3ese e4ressions sim4li=y to

t 00

- !c$)')(y9 & ie, and !c$)')(< ' i ( 2 i = : -> > 9 l

n 9

1 + 2

9DDro 7im at io n 9 ot t# e te aH t at e

I= t3e utility =unction were uadratic and t3e 4roduction =unction linear, t3ere would e no

need =or a44roimations. In euilirium, consum4tion must e eual to out4ut minus in#estment.

@e e4loit t3is =act to eliminate t3e nonlinearity in t3e constraint set y sustituting!A, + , n , )

! i =or c in t3e utility =unction to otain (1(2,+ ,n , ) ! i ,n ,a ) . T3e net ste4 is to

a44roimate t3is =unction y a uadratic in t3e neig3or3ood o= t3e model?s steady state. As

in#estment i is linear in t3e decision and state #ariales, it can e eliminated suseuent to t3e

a44roimation and still 4reser#e a uadratic o7ecti#e.



@e e4erimented a little and =ound t3at t3e results were essentially t3e same w3en t3e second order Taylor series a44roimation was used rat3er t3an t3is =unction. 1arry C3ristiano H$0

uadratic a44roimation met3od t3at we em4loyed yields a44roimate solutions t3at are #ery accurate, e#en wit3 large #ariaility, =or a structure t3at, li6e ours, is o= t3e constant elast


tion error is ero at t3e 7 *+ & and 7 ! A w3ere t3e A selected corres4ond to t3e a44roimate

a#erage de#iations o= t3e 7 t =rom t3eir steady state #alues 7 r T3e #alues o= & ?7 t used =or A, + , ,

n , i , and a were ), $, ', ), 8, and 0.5 4er cent, res4ecti#ely.11

T3e a44roimation errors eing ero at t3e 7 *+ and 7 ! reuires t3at

t = (7 & ) ! (7 ! (2+, and

q 9 Hw! U$( M (7) *+ (7 ! & ) ! ii!(>'8.

T3e elements q i j i j , are selected to minimie t3e sum o= t3e suared a44roimation errors at

7 & J A 7 & ! J , 7 ! j , and 7 ! & ! K T3e a44roimation error at t3e =irst

4oint is

u(x+z‘ + zJ) - u(x) - -bjZj - q„zf - q zf - Iq^Zj.

-umming o#er t3e suare o= t3is error and t3e t3ree ot3ers, di==erentiating wit3 res4ect to q t j ,



3e limit o= t3e seuence o= #alue =unctions eisted in e#ery case and, as a =unction o= , was ounded =rom ao#e, gi#en 7c. T3is, along wit3 t3e staility o= t3e matri 9, is su==icient to ens

t3e o4timal #alue =unction and t3at t3e associated 4olicy =unction is t3e o4timal one !see H)0(.

$)5& F. E. K<D1A"D A"D E. C. PE-C;TT

su7ect to

(4.6) + t A = " A - 8)+, * ,

!*+( * J J i = * j ( j 9 $, $(,

(4.8) 7 t & = 97 t ,

(4.9) a t & 9 !$ + i7(a, n t ,

J

(4.10) i t 9 ' <D j * ? t t i-, ,

j +$

(4.11) 2 t 9 *+ '.



S Jt >

nt

AEATE F1/CT/ATI;"- $)5%

T3e net ste4 is to determine t3e o4timal in#entory carry+o#er decision rule. It is t3e & inear

=unction t & = (7 t t n t , * J t ) w3ic3 sol#es

(4.12) ma (7 ,, t ,n t , * J t , t ,( ?@4 (7 t ,, t ,(

t $

su7ect to !*.(+!*.%( and ot3 n t and * J t gi#en. Finally, t3e solution to t3e 4rogram

mat'! , t ,n t ,* J t ) ,

w3ere 4 is t3e #alue o= maimiation o= !*.$'(, is determined. T3e linear =unctions * J t = *(7 t , t )

and n t 9 n(7 t , t ( w3ic3 sol#e t3e ao#e 4rogram are t3e o4timal decision rules =or new 4ro7ects

and laor su44ly.

Because o= t3e se4aration o= estimation and control in our model, t3ese decision rules can e used

to determine t3e motion o= t3e stoc3astic economy. In eac3 4eriod t , a conditional e4ectation,

m5 n is =ormed on t3e asis o= oser#ations in 4re#ious 4eriods. An indicator o= t3e tec3nology

s3oc6 is oser#ed, w3ic3 is t3e sum o= a 4ermanent and a transitory com4onent as well as an

indicator s3oc6. T3e conditional e4ectation, m 2n o= t3e unoser#ed 7 t is com4uted according to

euation !).$*(, and * J t and n t are determined =rom

(4.13) * J t 9 *( m , t ) ,

(4.14) n, = n(m , t ) ,

w3ere 7 t 3as een re4laced y m . T3en t3e tec3nology s3oc6, 2 n is oser#ed, w3ic3 c3anges t3e

conditional e4ectation o= 7 t . From !).$(, t3is e4ectation is m n and t3e in#entory carry+o#er is

determined =rom

(4.15) & B = (m t , t , * J t ,n t ) .

To summarie, t3e euilirium 4rocess go#erning t3e e#olution o= our economy is gi#en y !).$(+



ms H)* 3as estimated unrestricted aggregate #ector autoregressi#e models.


4ro4ortionately w3ile t3ere is little c3ange in em4loyment !all #ariales are in 4er+3ouse3old

terms( w3en t3e tec3nology 4arameter 2 grows smoot3ly o#er time. T3ese are 7ust t3e steady

state 4ro4erties o= t3e growt3 model wit3 w3ic3 we egan.

Juantitati#ely e4laining t3e co+mo#ements o= t3e de#iations is t3e test o= t3e underlying

t3eory. For want o= etter terminology, t3e de#iations will e re=erred to as t3e cyclical

com4onents e#en t3oug3, wit3 our integrated a44roac3, t3ere is no se4aration etween =actors

determining a secular 4at3 and =actors determining de#iations =rom t3at 4at3. T3e statistics to e

e4lained are t3e co#ariations o= t3e cyclical com4onents. T3ey are o= interest ecause t3eir

e3a#ior is stale and is so di==erent =rom t3e corres4onding co#ariations o= t3e smoot3ed series.

T3is is 4roaly w3y many 3a#e soug3t se4arate e4lanations o= t3e secular and cyclical

mo#ements.

;ne cyclical oser#ation is t3at, in 4ercentage terms, in#estment #aries t3ree times as muc3 as

out4ut does and consum4tion only 3al= as muc3. In s3ar4 contrast to t3e secular oser#ations,

#ariations in cyclical out4ut are 4rinci4ally t3e result o= #ariations in 3ours o= em4loyment 4er

3ouse3old and not in ca4ital stoc6s or laor 4roducti#ity.

T3e latter oser#ation is a di==icult one to e4lain. @3y does t3e consum4tion o= mar6et

4roduced goods and t3e consum4tion o= leisure mo#e in o44osite directions in t3e asence o= any

a44arent large mo#ement in t3e real wage o#er t3e so+called cycle8 For our model, t3e real wage

is 4ro4ortional to laor?s 4roducti#ity, so t3e crucial test is w3et3er most o= t3e #ariation in

cyclical out4ut arises =rom #ariations in em4loyment rat3er t3an =rom #ariations in laor?s

4roducti#ity.



AEATE F1/CT/ATI;"- $)$

reduce dramatically t3e numer o= =ree 4arameters t3at will e #aried w3en searc3ing =or a set

t3at results in cyclical co#ariances near t3ose oser#ed. In e4laining t3e co#ariances o= t3e

cyclical com4onents, t3ere are only se#en =ree 4arameters, wit3 t3e range o= two o= t3em eing

se#erely constrained a 4riori.

Ca4ital =or our model re=lects all tangile ca4ital, including stoc6s o= 4lant and eui4ment,

consumer durales and 3ousing. Consum4tion does not include t3e 4urc3ase o= durales ut does

include t3e ser#ices =rom t3e stoc6 o= consumer durales. Di==erent ty4es o= ca4ital 3a#e di==erent

construction 4eriods and 4atterns o= resource reuirements. T3e =indings summaried in -ection

' suggest an a#erage construction 4eriod o= nearly two years =or 4lants. Consumer durales,

3owe#er, 3a#e muc3 s3orter a#erage construction 4eriods. Ga#ing ut one ty4e o= ca4ital, we

assume, as a com4romise, t3at =our uarters are reuired, wit3 one+=ourt3 o= t3e #alue 4ut in

4lace eac3 uarter. T3us J = and c4i 9 c4' 9 L4) = L4* 9 0.'5.

A44roimately ten 4er cent o= national income account "P is t3e ca4ital consum4tion

allowance and anot3er ten 4er cent ecise ta. To "P s3ould e added t3e de4reciation o=

consumer durales w3ic3 3as t3e e==ect o= increasing t3e s3are o= out4ut going to owners o=

ca4ital. In $%, com4ensation to em4loyees 4lus 4ro4rietary income was a44roimately * 4er

cent o= "P 4lus consumer durales de4reciation less indirect usiness ta, w3ile owners o=

ca4ital recei#ed aout ) 4er cent. As laor s3are is , we set = 0.*.

Di==erent ty4es o= ca4ital de4reciate more ra4idly t3an ot3ers, wit3 durales de4reciating more

ra4idly t3an 4lant and 3ousing, and land not de4reciating at all. As a com4romise, we set t3e

de4reciation rate eual to $0 4er cent 4er year. @e assume a su7ecti#e time discount rate o= =our

4er cent and astract =rom growt3. T3is im4lies a steady+state ca4ital to annual out4ut ratio o= '.*. ;= total out4ut * 4er cent is wages, '* 4er cent de4reciation, and $' 4er cent return on

ca4ital w3ic3 includes consumer durales.



it3 t3e Godric6+Prescott met3od, t3e smoot3 4at3 Ws,X =or eac3 series " t M minimied

ength of the sample pero! " oth for the mo!el an! for the #.S. e$onom% s 118 &arters.

$)' F. E. K<D1A"D A"D E. C. PE-C;TT

o= 4er ca4ita consum4tion 3as een aout two 4er cent and t3e real return on 43ysical ca4ital si

to eig3t 4er cent, t3e ris6 a#ersion 4arameter, y, is constrained to e etween minus one and

ero.$*

T3e 4arameters a0 and rj w3ic3 a==ect intertem4oral sustitutaility o= leisure will e treated

as =ree 4arameters =or we could =ind no estimate =or t3em in t3e laor economics literature. As

stated 4re#iously, t3e steady+state laor su44ly is inde4endent o= t3e 4roducti#ity 4arameter .

T3e remaining 4arameters are t3ose s4eci=ying t3e 4rocess on 2 t and t3e #ariance o= t3e

indicator. T3ese t3ree 4arameters are #ar!= 7(, #ar!= '(, and #ar!= )(. ;nly two o= t3ese are =ree

4arameters, 3owe#er. @e restricted t3e sum o= t3e t3ree #ariances to e suc3 t3at t3e estimate o=

t3e #ariance o= cyclical out4ut =or t3e model eualled t3at o= cyclical out4ut =or t3e /.-. economy

during t3e sam4le 4eriod.

In summary, t3e 4arameters t3at are estimated =rom t3e #ariance+co#ariance 4ro4erties o= t3e

model are t3ese #ariances 4lus t3e 4arameter 4 determining sustitutaility o= in#entories and

ca4ital, t3e 4arameters a 5 and rj determining intertem4oral sustitutaility o= leisure, and t3e

ris6 a#ersion 4arameter y. For eac3 set o= 4arameter #alues, means and standard de#iations were

com4uted =or se#eral statistics w3ic3 summarie t3e serial correlation and co#ariance 4ro4erties

o= t3e model. T3ese numers are com4ared wit3 t3ose o= t3e actual /.-. data =or t3e 4eriod $%502

$ to $%% 2' as re4orted in Godric6 and Prescott H$&. A set o= 4arameter #alues is soug3t w3ic3

=its t3e actual data well. Ga#ing only si degrees o= =reedom to e4lain t3e oser#ed co#ariances

im4oses considerale disci4line u4on t3e analysis.

T3e statistics re4orted in H$& are not t3e only way to uantitati#ely ca4ture t3e co+mo#ements

o= t3e de#iations.$5

T3is a44roac3 is sim4le, in#ol#es a minimum o= 7udgment, and is roust toslowly c3anging demogra43ic =actors w3ic3 a==ect growt3, ut are not t3e concern o= t3is t3eory.$

In addition, t3ese statistics are roust to most measurement errors, in contrast to, say, t3e

correlations etween t3e =irst di==erences o= two series. It is im4ortant to com4ute t3e same

statistics =or t3e /.-. economy as =or t3e model, t3at is, to use t3e same =unction o= t3e data. T3is

is w3at we do.

' (,- L(' $00 ' t;Li+ >( S ( * t3

t = 1 t = \

T3e de#iations =or series " t M are " t ! s,X. T3e numer o= oser#ations, % , was $$&. T3e solution to t3e ao#e

4rogram is a linear trans=ormation o= t3e data. T3us, t3e standard de#iations and correlations re4orted are well+

de=ined statistics.

16 -ee, =or eam4le, H$$.

AEATE F1/CT/ATI;"- $))

For 4arameters wit3 a time dimension, t3e unit o= time is a uarter o= a year.

ters 4 , a0, , and y. -imilarly, t3e conditional e4ectations o= t3e 4ermanent and transitory

s3oc6s w3ic3 enter t3e decision rules de4end only on t3e #ariances o= t3e t3ree s3oc6s and not

u4on t3e 4arameters o= 4re=erences and tec3nology.

For eac3 set o= 4arameter #alues t3e =ollowing statistics are com4uted2 t3e autocorrelation o=

cyclical out4ut =or u4 to si 4eriods, standard de#iations o= t3e cyclical #ariales o= interest, and

t3eir correlations wit3 cyclical out4ut. In H$& t3e #ariales !ece4t interest rates( are measured in

logs w3ile we use t3e le#els rat3er t3an t3e logs. T3is is o= conseuence only in t3e measurement o=

am4litudes, so in order to ma6e our results com4arale to t3eirs, our standard de#iations !ece4t

=or interest rates( are di#ided y t3e steady states o= t3e res4ecti#e #ariales. ;ne can t3en

inter4ret t3e cyclical com4onents essentially as 4ercentage de#iations as in H$&.

T3e 4arameter #alues t3at yielded w3at we considered to e t3e est =it are re4orted in Tale I.

T3ey were determined =rom a grid searc3 o#er t3e =ree 4arameters. In t3e case o= 4, we tried t3e

#alues ', ), *, and 5. T3e 4arameters a5 and /B were 7ust constrained to e etween ero and one.

;nly t3e #alues M $,

0.5, and M0.$ were considered =or t3e ris6 a#ersion 4arameter y. T3e last #alue is close to t3e

limiting case o= y 9 0 w3ic3 would corres4ond to t3e logarit3mic utility =unction.

$)* F. E. K<D1A"D A"D E. C. PE-C;TT

TAB1E II A/T;C;E1ATI;"- ;F

;/TP/TU



h of the sample p ero! "oth for the mo!el an! for the #.S. e$onom% s 118 &arters. " *easre! n per $ent.

o= t3is #ariaility, we re4ort t3e standard de#iations o= sam4le distriutions =or t3e model?s statistics w3ic3, li6e t3e

estimates =or t3e /.-. economy, use only $$& oser#ations. T3ese are t3e numers in t3e

4arent3eses in Tales II and III.

TAB1E III

;DE1?- -TA"DAD DEVIATI;"- A"D C;E1ATI;"- @ITG EA1 ;/TP/T5$


TAB1E IV

-AP1E -TA"DAD DEVIATI;"- A"D C;E1ATI;"- @ITG EA1 ;/TP/T /.-. EC;";<

$%502 $+$%%2 '



t3e #alue o= uncom4leted ca4ital goods to t3e model?s in#entory #ariale to otain w3at we call in#entories 4lus.

T3is corres4onds more closely to t3e /.-. in#entory stoc6 #ariale, wit3 its standard de#iation and

correlation wit3 real out4ut eing consistent wit3 t3e /.-. data.

In Tale III we include results =or t3e im4licit real interest rate gi#en y t3e e4ression r t 9

!&w&c,(!8is!&w0c, $(( M $. T3e e4ectation is conditional on t3e in=ormation 6nown w3en t3e

allocation etween consum4tion and in#entory carry+o#er is made.

T3e model dis4lays more #ariaility in 3ours t3an in 4roducti#ity, ut not y as muc3 as t3edata s3ow. In lig3t o= t3e di==iculties in measuring out4ut and, in 4articular, em4loyment, we do

not t3in6 t3is discre4ancy is large. For eam4le, all memers o= t3e 3ouse3old may not e eually

roducti#e, say due to di==ering stoc6s o= 3uman ca4ital. I= t3ere is a greater re4resentation in t3e

or6 =orce o= t3e less 4roducti#e, =or eam4le less e4erienced yout3, w3en out4ut is 3ig3, 3ours



$) F. E. K<D1A"D A"D E. C. PE-C;TT

some tec3nological c3ange may e emodied in new ca4ital, and only a=ter t3e ca4ital ecomes 4roducti#e is t3ere

t3e increment to measured 4roducti#ity. -uc3 s3oc6s induce #ariation in in#estment and

em4loyment wit3out t3e #ariaility in 4roducti#ity. T3is is a uestion t3at warrants =urt3er

researc3.

@e also eamined lead and lag relations3i4s and serial correlation 4ro4erties o= aggregate

series ot3er t3an out4ut. @e =ound t3at, ot3 =or t3e 4ost+war /.-. economy and t3e model,

consum4tion and non+in#entory in#estment mo#e contem4oraneously wit3 out4ut and 3a#e serial

correlation 4ro4erties similar to out4ut. In#entory and ca4ital stoc6s =or t3e model lag out4ut,

w3ic3 also matc3es well wit3 t3e data. -ome o= t3e in#entory stoc6?s cross+serial correlations wit3

out4ut de#iate signi=icantly, 3owe#er, =rom t3ose =or t3e /.-. economy. T3e one #ariale w3ose

lead+lag relations3i4 does not matc3 wit3 t3e data is 4roducti#ity. For t3e /.-. economy it is a

leading indicator, w3ile t3ere is no lead or lag in t3e model. T3is was not une4ected in #iew o=

our discussion ao#e wit3 regard to 4roducti#ity. T3us, e#en t3oug3 t3e o#erall =it o= t3e model is

#ery good, it is not sur4rising, gi#en t3e le#el o= astraction, t3at t3ere are elements o= t3e =ine

structure o= dynamics t3at it does not ca4ture.

%#e moot#eH erie*

T3e smoot3ed out4ut series =or t3e /.-. 4ost+war data de#iated signi=icantly =rom t3e linear

time trend. During t3e $$&+uarter sam4le 4eriod t3is di==erence 3ad two 4ea6s and two troug3s.

T3e times etween suc3 local etremes were )0, )$, and )' uarters, and t3e corres4onding

di==erences in #alues at ad7acent etremes were 5.00, .'5, and 5.%0 4er cent, res4ecti#ely.

T3ese oser#ations matc3 well wit3 t3e 4redictions o= t3e model. T3e mean o= t3e model?s

sam4ling distriution =or t3e numer o= 4ea6s and troug3s in a $$&+uarter 4eriod is *.0Mw3ic3



AEATE F1/CT/ATI;"- $)

tec3nology s3oc6. T3is would 3a#e resulted in cyclical em4loyment #arying less t3an cyclical

4roducti#ity w3ic3 is inconsistent wit3 t3e data.

;= 4articular im4ortance =or t3e model is t3e de4endence o= current utility on 4ast leisure

c3oices w3ic3 admits greater intertem4oral sustitution o= leisure. T3e 4ur4ose o= t3is

s4eci=ication is not to contriute to t3e 4ersistence o= out4ut c3anges. I= anyt3ing, it does 7ust t3e

o44osite. T3is element o= t3e model is crucial in ma6ing it consistent wit3 t3e oser#ation t3at

cyclical em4loyment =luctuates sustantially more t3an 4roducti#ity does. For t3e 4arameter

#alues in Tale I, t3e standard de#iation o= 3ours wor6ed is $& 4er cent greater t3an t3e de#iation

o= 4roducti#ity. T3e s4ecial case o= a0 9 $ corres4onds to a standard time+se4arale utility

=unction. For t3is case, wit3 t3e 4arameters ot3erwise t3e same as in Tale I, t3e standard

de#iation o= 3ours is '* 4er cent less t3an t3e de#iation o= 4roducti#ity.

Im Dor tance o1 %i me to ;i &H

;= 4articular interest is t3e sensiti#ity o= our results to t3e s4eci=ication o= in#estment

tec3nology. T3e 4rominent alternati#e to our time+to+uild tec3nology is t3e ad7ustment+cost

structure. I= only one 4eriod is reuired =or t3e construction o= new 4roducti#e ca4ital, we can

write t3e law o= motion =or t3e single ca4ital good as + t 7 = !$ M 8)+ t * t , w3ere * t is t3e

amount o= in#estment in 4roducti#e ca4ital in 4eriod t . @e can t3en introduce cost o= ad7ustment

into t3e model y modi=ying t3e resource constraint !).*( as =ollows2

c, i , Y!7, + 8+,) < 1( A ,+ t ,n t , t ),



a4ital 4lays an im4ortant role in creating 4ersistence in t3e analysis o= 1ucas H') as well as in t3ose o= Blinder and Fisc3er H5 and 1ong and Plosser H''. In H') gradual di==usion o=

ys a crucial role. T3is is not t3e case in our model, 3owe#er, as agents learn t3e #alue o= t3e s3oc6 at t3e end o= t3e 4eriod. Townsend H) analyes a model in w3ic3 decision ma6ers

ts o= ot3ers, w3ic3 gi#es rise to con=ounding o= laws o= motion wit3 =orecasting 4rolems, and results in 4ersistence in ca4ital stoc6 and out4ut mo#ements.

An alternati#e way o= otaining 4ersistence is t3e use o= long+term staggered nominal wage contracts as in H)5.

$)& F. E. K<D1A"D A"D E. C. PE-C;TT

nearly douled =or consum4tion and reduced y a =actor o= two =or in#estment e4enditures,

ma6ing t3e am4litudes o= t3ese two out4ut com4onents muc3 too close as com4ared wit3 t3e data.

In addition, t3e standard de#iation o= ca4ital stoc6 was reduced y more t3an one 3al=. T3e results

were e#en worse =or larger #alues o= Y.

T3e etreme case o= Z 9 0 corres4onds to t3e s4ecial case o= 9 $ in our model. T3us, neit3er

time to uild nor cost o= ad7ustment would e an element o= t3e model. T3e iggest c3anges in t3e

results =or t3is #ersion as com4ared wit3 Tale III are t3at t3e correlation etween ca4ital stoc6

and out4ut ecomes 4ositi#e and o= siale magnitude !0.*) i= t3e 4arameters are ot3erwise t3e

same as in Tale I(, and t3at t3e correlation etween in#entory stoc6 and out4ut ecomes negati#e

! M 0.50 =or our 4arameter #alues(. Bot3 o= t3ese correlations are inconsistent wit3 t3e

oser#ations. Also, t3e 4ersistence o= mo#ements in in#estment e4enditures as measured y t3e

autocorrelations was sustantially reduced.

For our model wit3 multi4le 4eriods reuired to uild new ca4ital, t3e results are not o#erly

sensiti#e to t3e numer o= 4eriods assumed. @it3 a t3ree or =i#e+uarter construction 4eriod

instead o= =our, t3e =it is also good.

6. C;"C1/DI" C;E"T-

A com4etiti#e euilirium model was de#elo4ed and used to e4lain t3e autoco#ariances o= real

out ut and t3e co#ariances o= c clical out ut wit3 ot3er a re ate economic time series =or t3e



AEATE F1/CT/ATI;"- $)%

ences de=ined on 3ours wor6ed 4er wee6, s3ould 3el4. Introducing more t3an a single ty4e o=

4roducti#e ca4ital, wit3 di==erent ty4es reuiring di==erent 4eriods =or construction and 3a#ing

di==erent 4atterns o= resource reuirement, is =easile. It would t3en e 4ossile to distinguis3

etween 4lant, eui4ment, 3ousing, and consumer durales in#estments. T3is would also 3a#e t3e

ad#antage o= 4ermitting t3e introduction o= =eatures o= our ta system w3ic3 a==ect trans=orma+

tion o44ortunities =acing t3e economic agents !see, e.g., H$*(. Anot3er 4ossile re=inement is in t3e

estimation 4rocedure. But, in s4ite o= t3e considerale ad#ances recently made y Gansen and

-argent H$5, =urt3er ad#ances are needed e=ore =ormal econometric met3ods can e =ruit=ully

a44lied to testing t3is t3eory o= aggregate =luctuations.

odels suc3 as t3e one considered in t3is 4a4er could e used to 4redict t3e conseuence o= a

4articular 4olicy rule u4on t3e o4erating c3aracteristics o= t3e economy.$% As we estimate t3e

4re=erence+tec3nology structure, our structural 4arameters will e in#ariant to t3e 4olicy rule

selected e#en t3oug3 t3e e3a#ioral euations are not. T3ere are com4utational 4rolems,

3owe#er, associated wit3 determining t3e euilirium e3a#ioral euations o= t3e economy w3en

=eedac6 4olicy rules, t3at is, rules t3at de4end on t3e aggregate state o= t3e economy, are used.

T3e com4etiti#e euilirium, t3en, will not maimie t3e wel=are o= t3e stand+in consumer, so a

4articular maimiation 4rolem cannot e sol#ed to =ind t3e euilirium e3a#ior o= t3e

economy. Instead, met3ods suc3 as t3ose de#elo4ed in H'0 to analye 4olicy rules in com4etiti#e

en#ironments will e needed.

Carne'ie-Ne&&on ni4er*i t anH

ni4er*i t o1 Ninne*ota

Nan*criDt recei4eH Janar, 8O re4i*ion recei4eH Janar, 8.

19 Eam4les o= suc3 4olicy issues are descried in H'$. -ee also Barro !e.g., H)(, w3o em43asies t3e di==erences

in e==ects o= tem4orary and 4ermanent c3anges in go#ernment e4enditures.

EFEE"CE-

[ \




[8] +R#R , K., A. #/0R*, A. G. *3T4R [-tag=lation, Persistent /nem4loyment and

t3e Permanence o= Economic -3oc6s,\ Jorna& o1 Nonetar Economic*, !$%&0(, *+*%'.

[9] R3TO, . @.2 [Contracts, Price igidity, and ar6et Euilirium,\ Jorna& o1 o&itica&

Econom, &!$%%(, $0)*+$0'.

[10] 6R0ST0O , 1. >.2 [;n t3e Accuracy o= 1inear Juadratic A44roimations2 An Eam4le,\

@or6ing Pa4er, /ni#ersity o= C3icago, -e4temer, $%&$.

[11] 0SO, E. F. 9ccontin' 1or niteH tate* Economic Growt# -Q. @as3ington, D.C.2

Broo6ings Institution, $%*.

[12] 7ROSS* , 6. I.2 [Aggregate Demand, >o -earc3, and Em4loyment,\ Jorna& o1 o&itica&

Econom, &$!$%)(, $)5)+$)%.

[13] 633, . E.2 [In#estment, Interest ates, and t3e E==ects o= -tailiation Policies,\ ;roo+in'*

aDer* on Economic 9cti4it, !$%(, $+$00.

[14] 633, R. E., . @. OR7SO [Ta Policy and In#estment Be3a#ior,\ 9merican

Economic Re4iew, 5!$%(, )%$+*$*.

[15] 6S, 1. P., T. >. SR7T [Formulating and Estimating Dynamic 1inear ational

E4ectations odels,\ Jorna& o1 Economic Snamic* anH Contro& , '!$%&0(, +*.

[16] 6S, 1. P., K. . S 073TO [eneralied Instrumental Variales Estimation o=

"onlinear ational E4ectations odels,\ Econometrica, 50!$%&'(, $'%+$'&.

[17] 69S60, F.2 [Toin?s arginal q and A#erage q A "eoclassical Inter4retation,\

Econometrica, 50!$%&'(, '$)+''*.

[18] 6OR0/ , R. ., E. C. :RSOTT [Post+@ar /.-. Business Cycles2 An Em4irical In#estiga

tion, \ @or6ing Pa4er, Carnegie+ellon /ni#ersity, re#ised "o#emer, $%&0.

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