Animal Spirits Through Creative Destructionsocsci.uci.edu/~wbranch/Francois_Lloyd-ellis_2003.pdf · 2005. 10. 25. · Animal Spirits Through Creative Destruction ByPATRICKFRANCOISANDHUWLLOYD-ELLIS*

Animal Spirits Through Creative Destruction

By PATRICK FRANCOIS AND HUW LLOYD-ELLIS

We show how a Schumpeterian process of creative destruction can induce rationalherd behavior by entrepreneurs across diverse sectors as if fueled by ldquoanimalspiritsrdquo Consequently a multisector economy in which productivity improvementsare made by independent pro t-seeking entrepreneurs exhibits regular boomsslowdowns and downturns as part of the long-run growth process Our cyclicalequilibrium has higher average growth but lower welfare than the correspondingacyclical one We show how a negative relationship can emerge between volatilityand growth across cycling economies and assess the extent to which our modelmatches several features of actual business cycles (JEL E0 E3 O3 O4)

Are business cycles simply random shocksaround a deterministic trend or are there morefundamental linkages between short-run uctu-ations and long-run growth Macroeconomistshave tended to study the sources of uctuationsand the determinants of growth separately butthere are several reasons to question this stan-dard dichotomy First postwar cross-countryevidence suggests a signi cant negative partialcorrelation between volatility and growth aftercontrolling for standard growth correlates (egGary Ramey and Valerie A Ramey 1995) This

correlation is economically signi cant evenamong OECD countries Second while it isclear that some portion of aggregate volatility isthe result of exogenous disturbances the recur-ring asymmetry between the responses of theeconomy during upturns and downturns is sug-gestive of an endogenously determined compo-nent Third there is increasing evidence that thestrength of cyclical upturns is related to thedepth of preceding downturns Finally as ar-gued by Victor Zarnowitz (1998) even for uc-tuations that are typically associated withobvious aggregate shocks the causal links arenot always clear

The view that growth and cycles are inti-mately linked is often associated with JosephSchumpeter (1927) He argued that growth oc-curs through a process of ldquocreative destructionrdquomdashcompetition among entrepreneurs in the searchfor new ideas that will render their rivalsrsquo ideasobsolete This idea is central to modern theoriesof endogenous long-run growth starting withPaul S Segerstrom et al (1990) Gene Gross-man and Elhanan Helpman (1991) and PhilippeAghion and Peter Howitt (1992) HoweverSchumpeter also argued that this process ofentrepreneurial innovation is responsible for theregular short-run uctuations in economic ac-tivity which he termed the ldquonormalrdquo businesscycle The key to explaining such business cy-cles he argued was to understand why entre-preneurial activity would be clustered over time

A source of clustering suggested by Schum-peter himself has been recently formalized by a

Francois CentER Tilburg University The Nether-lands and Department of Economics University of BritishColumbia Vancouver Canada (e-mail pfrancoiskubnl)Lloyd-Ellis Department of Economics Queenrsquos UniversityKingston Ontario Canada K7L 3N6 (e-mail lloydellqedeconqueensuca) We are grateful to Daron AcemogluGadi Barlevy Paul Beaudry Francisco Gonzalez ElhananHelpman Michael Krause Joanne Roberts Shouyong ShiGregor Smith Gianluca Violante Alwyn Young and twoanonymous referees for helpful suggestions We also thankseminar participants at the University of Colorado-Boulderthe University of Illinois-Urbana Champain NorthwesternUniversity Queenrsquos University the University of Torontothe University of British Columbia the University of Lau-sanne the October 2000 Canadian Institute for AdvancedResearch (CIAR) meetings in Montreal the 2001 meetingson New Approaches to the Study of Economic Fluctuations(NASEF) in Hydra and the 2001 Canadian Macroeconom-ics Study Group (CMSG) meetings in Vancouver Theresearch of Francois is funded by a fellowship of the RoyalNetherlands Academy of Arts and Sciences Lloyd-Ellisgratefully acknowledges funding from Social Sciences andHumanities Research Council of Canada (SSHRCC) and theCIAR

530

number of authors (for a survey see Aghion andHowitt 1998) However such ldquoSchumpeteriancyclesrdquo rely on the arrival of major technolog-ical breakthroughs that in uence all sectorsmdashageneral purpose technology (GPT) While theGPT story may be consistent with ldquolongwavesrdquo there is little evidence to support thenotion that such economy-wide advances canexplain high-frequency business cycles An al-ternative theory of why activity in diverse sec-tors of the economy may be clustered isdeveloped by Andrei Shleifer (1986) He showsthat when imitation limits the longevity of mo-nopoly pro ts a strategic complementarityarises that could lead entrepreneurs to imple-ment innovations at the same time even if theinnovations themselves arrive uniformly throughtime The clustering of implementation resultsin a boom in labor demand which in turn gen-erates the high demand for output necessary tosupport the boom The temporary nature of theassociated monopoly pro ts induces entrepre-neurs to delay implementation until demand ismaximized so that a self-reinforcing cyclearises Shleifer interprets his theory as a formal-ization of John Maynard Keynesrsquo (1936) notionof ldquoanimal spiritsrdquo1

There are however several important limita-tions to Shleiferrsquos theory of implementationcycles Firstly since innovations arrive exog-enously long-run growth is exogenous so thetheory has no implications for the impact ofcycles on growth Secondly because of the mul-tiplicity of equilibria that arise in his model it isnot possible to obtain precise predictions evenfor the effect of growth on cycles Thirdly thetemporary nature of pro ts relies on the as-sumption of drastic but costless imitation It isnot clear how robust the results would be to aless abrupt erosion of pro ts Finally Shleiferrsquostheory depends critically on the impossibility ofstorage If they could innovators would chooseto produce when costs are low (ie before theboom) store the output and then sell it when

demand is high (ie in the boom) Such apattern of production would undermine the ex-istence of cycles

In this article we draw on the insights ofSchumpeter (creative destruction) and Shleifer(animal spirits) to develop a simple theory ofendogenous cyclical growth We show how amultisector economy in which sector-speci cproductivity improvements are made by inde-pendent pro t-seeking entrepreneurs can ex-hibit regular booms slowdowns and downturnsin economic activity as an inherent part of thelong-run growth process We establish the ex-istence of a unique cyclical growth path alongwhich the growth rate and the length and am-plitude of cycles are endogenously determinedOur theory does not rely on the arrival of GPTsnor on drastic imitation and allows for thepossibility of storage Speci cally we showthat the process of creative destruction itself caninduce endogenous clustering of implementa-tion and innovation

Creative destruction implies that the dissem-ination of knowledge caused by implementationeventually leads to improvements that limit asuccessful entrepreneurrsquos time of incumbencyAnticipating this entrants will optimally timeimplementation to ensure that their pro ts ar-rive at a time of nondepressed aggregate activityand that they maximize the length of their in-cumbency It is these effects which lead toclustering in entrepreneurial implementationand hence to an aggregate level boom If anentrepreneur implements before the boom hereveals the information underlying his produc-tivity improvement to potential rivals who mayuse this information in designing their own pro-ductivity improvements By delaying imple-mentation until the boom he delays reaping therewards but maximizes his expected reign ofincumbency During the delay entrepreneursrely on maintaining secrecy regarding the natureof the innovations that they hold2

Our cycle not only features clustering of im-plementation but also endogenous clusteringof innovation It is this feature which generates

1 The expression ldquoanimal spiritsrdquo is often associated withstochastic changes in the expectations of investors that turnout to be self-ful lling In the cyclical equilibrium that westudy however the behavior of entrepreneurs may have theappearance of being fueled by animal spirits but in factexpectations are deterministic

2 As Wesley M Cohen et al (2000) document rms doindeed view secrecy as the best form of protectionmdashpatenting is a less desired means of protecting knowledge

531VOL 93 NO 3 FRANCOIS AND LLOYD-ELLIS ANIMAL SPIRITS

the endogenous interactions between long-rungrowth and short-run uctuations After theboom wage costs are so high that it is initiallynot pro table to undertake new entrepreneurialactivities As the next boom approaches how-ever the present value of new innovationsgrows until at some point it becomes pro tableto allocate entrepreneurial effort to innovationAs labor effort is withdrawn from productionper capita output (and measured productivity)gradually decline Eventually it becomes prof-itable to implement the stock of innovations thathave accumulated during the downturn and thecycle begins again

We adopt a broad interpretation of innovationto include any improvement that is the outcomeof purposive design in search of pro t Entre-preneurs are the source of re nements to pro-cess organization and product improvementsthat increase productivity within narrowly de- ned sectors The knowledge created by suchentrepreneurial activity is both tacit and sectorspeci c Unlike RampD or scienti c knowledgethe improvements created may not be formallyexpressible (as in a blueprint or design) andneed not lend themselves to protection bypatent It is our view that such mundane entre-preneurial decisions are the major source ofhigh-frequency productivity improvements notthe patentable RampD improvements of a labora-tory which are often the focus in the growthliterature3 In modern production activitiesmuch of the entrepreneurial function has beenallocated to managers and other skilled workersIt may therefore be more useful to think ofinnovation as requiring a reallocation of laboreffort within rms This interpretation is thusnot unlike that of Robert E Hall (2000) whoemphasizes the role of ldquoreorganizationrdquo in arecession

Although our model is rather stylized it hasclear predictions for the interactions of long-rungrowth and short-run uctuations Firstly the

cycle in our model shows a positive feedbackfrom both the duration and depth of downturnsto the magnitude of succeeding upturns Thisfeature is consistent with the evidence of PaulBeaudry and Gary Koop (1993) M HashemPesaran and Simon M Potter (1997) and Fil-ippo Altissimo and Giovanni L Violante(2001) Secondly the cycles generated by ourmodel exhibit asymmetries in upturns anddownturns that have some features in commonwith the evidence of Kenneth M Emery andEvan F Koenig (1992) Daniel E Sichel (1993)and Nathan S Balke and Mark A Wynne(1995) In particular business cycles typicallyexhibit rapid growth in output at the beginningof the boom a gradual slowdown and then adecline which occurs at a fairly constant rateThirdly consistent with the evidence of Rameyand Ramey (1995) and Kory Kroft and Lloyd-Ellis (2002) variation in the productivity ofentrepreneurship induces a negative relationshipbetween long-run growth and output volatility

Recently several authors have developed re-lated non-GPT models of endogenous growthand cycles Francois and Shouyong Shi (1999)modify the Grossman and Helpman (1991)growth model by allowing exogenous drasticimitation (as in Shleifer 1986) by introducing atechnological innovation process requiring ac-cumulated inputs through time and by treatingthe interest rate as exogenous That model alsoinherits Shleiferrsquos (1986) nonrobustness tostorage In Kiminori Matsuyama (1999) theclustering of innovations also results from theshort-term nature of monopoly rents thoughthrough a different channel In his frameworkgrowth arises due to increasing product varietyThus the upsurge in growth there arises throughdrastic innovations that represent wholly new(though partially substitutable) products and isdriven by a few leading sectors This againlends itself more easily to a long-cycle interpre-tation Scott Freeman et al (1999) develop amodel of cycles featuring a ldquotime to buildrdquocomponent in innovation As they emphasizethis technology describes ldquobigrdquo research or in-frastructure projects once again suggesting along wave application of the cycle

The present paper proceeds as follows Sec-tion I presents the economyrsquos fundamentalsde nes a general equilibrium and shows that

3 This view was shared by Schumpeter (1942 p 132)ldquo The function of entrepreneurs is to reform or revolution-ize the pattern of production by exploiting an invention ormore generally an untried technological possibility Thisfunction does not essentially consist in either inventinganything or otherwise creating the conditions which theenterprise exploits It consists in getting things donerdquo

532 THE AMERICAN ECONOMIC REVIEW JUNE 2003

one equilibrium of the model is an acyclicalgrowth path that is qualitatively identical to thatstudied by Grossman and Helpman (1991) Sec-tion II presents the main results of the paperWe posit a cycle and derive the equilibriumbehavior of households rms and entrepre-neurs that would be consistent with such a cy-cle We then derive the suf cient conditionsrequired for a unique cyclical equilibrium toexist and show that the cyclical equilibrium isstable Section III examines the implications ofour equilibrium growth process for the endog-enous relationship between long-run growthand short-run volatility We also compare thelong-run growth and welfare in the acyclicaland cyclical equilibria Section IV considers theimplications of our model for some qualitativefeatures of the business cycle and comparesthese to available evidence The nal sectionsummarizes and considers some potential ex-tensions of the model Technical details ofproofs and derivations are relegated to a not-for-publication Appendix which is available onthe AER web site at httpwwwaeaweborgaercontents or from the authors upon request

I The Model

A Assumptions

Time is continuous and indexed by t Weconsider a closed economy with no governmentsector Households have isoelastic preferences

(1) U~t 5 t

`

e2r ~s 2 tc~s1 2 s

1 2 sds

where r denotes the rate of time preference andwe assume that s [ (0 1) Each householdmaximizes (1) subject to the intertemporal bud-get constraint

(2) t

`

e2R~t 2 R~tc~t dt

B~t 1 t

`

e2R~t 2 R~tw~t dt

where w(t) denotes wage income B(t) denotesthe householdrsquos stock of assets at time t andR(t) denotes the discount factor from time zeroto t

Final output is produced by competitive rmsaccording to a Cobb-Douglas production func-tion utilizing intermediates k indexed by iover the unit interval

(3) y~t 5 expX 0

1

ln k i ~t diD

Final output is storable (at an arbitrarily smallcost) but cannot be converted back into aninput for use in production We let pi denote theprice of intermediate i

Output of intermediate i depends upon thestate of technology in sector i Ai(t) and thelabor resources devoted to production li in alinear manner

(4) k is~t 5 A i ~tl i ~t

Labor receives the equilibrium wage w(t)There is no imitation so the dominant entrepre-neur in each sector undertakes all productionand earns monopoly pro ts by limit pricinguntil displaced by a higher productivity rivalWe assume that intermediates are completelyused up in production but can be produced andstored for use at a later date Incumbent in-termediate producers must therefore decidewhether to sell now or store and sell later

Competitive entrepreneurs in each sector at-tempt to nd ongoing marginal improvementsin productivity by diverting labor effort awayfrom production and towards innovation4 They nance their activities by selling equity sharesto households The probability of an entrepre-neurial success in instant t is dxi(t) where d isa parameter and xi is the labor effort allocatedto entrepreneurship in sector i At any point intime entrepreneurs decide whether or not toallocate labor effort to innovation and if theydo so how much The aggregate labor effort

4 This process can equivalently be thought of as a searchfor product improvements process improvements organi-zational advances or anything else which creates a produc-tive advance over the existing state of the art


allocated to entrepreneurship is given by X(t) 5

01 xi(t) dtNew innovations dominate old ones by a

factor eg Entrepreneurs with innovations mustchoose whether or not to implement their inno-vation immediately or delay implementationuntil a later date Once they implement theknowledge associated with the innovation be-comes publicly available and can be built uponby rival entrepreneurs However prior to imple-mentation the knowledge is privately held bythe entrepreneur We let the indicator functionZ i(t) take on the value 1 if there exists a suc-cessful innovation in sector i which has not yetbeen implemented and 0 otherwise The set ofperiods in which innovations are implementedin sector i is denoted by i We let Vi

I(t) denotethe expected present value of pro ts from im-plementing an innovation at time t and Vi

D(t)denote that of delaying implementation fromtime t until the most pro table time in thefuture

Finally we assume the existence of arbitra-geurs who instantaneously trade assets to erodeany pro t opportunities There are three poten-tial assets in our economy claims to the pro tsof intermediate rms stored intermediate out-put and stored nal output As we shall see inall of the equilibria discussed below onlyclaims to the pro ts of intermediate rms willbe tradedmdashintermediate and nal output arenever stored However the potential for storedoutput to be traded imposes restrictions on thepossible equilibria that can emerge

In summary our model is formally identicalto that developed by Grossman and Helpman(1991) but with an elasticity of intertemporalsubstitution 1s that exceeds unity Howeverwe have expanded the set of possible strategiesby divorcing the realization of innovations fromthe decision to implement them (as in Shleifer1986) and by allowing intermediate output to bepotentially storable

B De nition of Equilibrium

Given an initial stock of implemented inno-vations represented by a cross-sectoral distribu-tion of productivities Ai(0)i50

1 and an initialdistribution of unimplemented innovations

Z i(0)i5 01 an equilibrium for this economy

satis es the following conditions

Households allocate consumption over timeto maximize (1) subject to (2) The rst-orderconditions of the householdrsquos optimizationrequire that

(5) c~ts 5 c~sseR ~t 2 R~s 2 r ~t 2 s t s

and that the transversality condition holds

(6) lims `

e2R~sB~s 5 0

Final goods producers choose intermediatesto maximize pro ts The derived demand forintermediate i is then

(7) k id~t 5

y~t

p i ~t

Intermediate producers set prices It followsthat the price of intermediate i is given by

(8) p i ~t 5w~t

e2gA i ~t

and the instantaneous pro t earned is

(9) pi ~t 5 ~1 2 e2g y~t

Note crucially that rm pro ts are propor-tional to aggregate demandLabor market clearing

(10) 0

1

l i ~t di 1 X~t 5 1

Labor market equilibrium also implies

(11) w~t~1 2 X~t 5 e2gy~t

Free entry into arbitrage For all assets thatare held in strictly positive amounts by house-


holds the rate of return between time t andtime s must equal [R(s) 2 R(t)](s 2 t)There is free entry into innovation Entrepre-neurs select the sector in which they innovateso as to maximize the expected present valueof the innovation Also

(12) d maxV iD~t V i

I~t w~t xi ~t $ 0

with at least one equality

In periods where there is implementation en-trepreneurs with innovations must prefer toimplement rather than delay until a later date

(13) V iI~t $ V i

D~t t [ i

In periods where there is no implementationeither there must be no innovations availableto implement or entrepreneurs with innova-tions must prefer to delay rather than imple-ment

(14) Either Z i ~t 5 0

or if Z i ~t 5 1 V iI~t V i

D~t

t i

The familiar acyclical growth path analyzedby Grossman and Helpman (1991) satis esthese conditions The key feature of this equi-librium is that innovation occurs every periodand implementation occurs immediately so thatZ i(t) 5 0 i t Growth is steady and con-sumption behavior is described by the familiardifferential equation

(15)csbquo ~t

c~t5

r~t 2 r

s

where r(t) 5 Rsbquo (t) denotes the instantaneousinterest rate The properties of this equilibriumare already well known so its derivation isrelegated to the unpublished Appendix but thefollowing will be useful for comparison with thecyclical equilibrium

PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1

then there exists an acyclical equilibrium witha constant growth rate given by

(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g

The rst inequality in (16) implies that r(t) ga(t) at every moment Along a balancedgrowth path this condition must hold for utilityto be bounded However this condition alsoensures both that no output is stored and thatthe implementation of any innovation is neverdelayed The return on storage is the growth inthe price of the intermediate good in noninno-vating sectors which in turn equals ga(t) Thussince r(t) ga(t) it never pays to store theintermediate

II The Cyclical Equilibrium

There is a second growth path that satis esthe equilibrium conditions above Along thispath innovations are implemented in clustersrather than in a smooth fashion In this sectionwe derive the optimal behavior of agents in sucha cyclical equilibrium and the evolution of thekey variables under market clearing We derivesuf cient conditions for existence and show thatmarket clearing implies a unique positive cyclelength and long-run growth rate

Suppose that the implementation of entrepre-neurial innovations occurs at discrete intervalsAn implementation period is denoted by Tvwhere v [ 1 2 ` and we adopt theconvention that the vth cycle starts in periodTv2 1 and ends in period Tv The evolution of nal output during a typical cycle between im-plementation periods is depicted in Figure 1 Aboom occurs when accumulated innovations areimplemented at Tv 2 1 After that there is aninterval during which no entrepreneurial effortis devoted to improvement of existing technol-ogies and consequently where all resources areused in production During this interval no newinnovations are implemented so that growth


slows to zero At some time TvE innovation

commences again but successful entrepreneurswithhold implementation until time Tv Entre-preneurial activity occurs throughout the inter-val [Tv

E Tv] and causes a decline in theeconomyrsquos production as resources are divertedaway from production towards the search forimprovements At Tv all successful entrepre-neurs implement and the (v 1 1)th cycle startswith a boom

Over intervals during which the discount fac-tor does not jump consumption is allocated asdescribed by (15) However as we will demon-strate here along the cyclical growth path thediscount rate jumps at the boom so that con-sumption exhibits a discontinuity during imple-mentation periods We therefore characterizethe optimal evolution of consumption from thebeginning of one cycle to the beginning of thenext by the difference equation

(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1

where the 0 subscript is used to denote values ofvariables the instant after the implementation

boom Note that a suf cient condition for theboundedness of the consumerrsquos optimizationproblem is that ln(c0(Tv)c0(Tv2 1)) R(Tv) 2 R(Tv 2 1) for all v or that

(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s

In our analysis below it is convenient tode ne the discount factor that will be used todiscount from some time t during the cycle tothe beginning of the next cycle This discountfactor is given by

(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship

Let Pi(s) denote the probability that sincetime Tv no entrepreneurial success has beenmade in sector i by time s It follows that theprobability of there being no innovation by timeTv1 1 conditional on there having been none bytime t is given by Pi(Tv1 1)Pi(t) Hence thevalue of an incumbent rm in a sector where noinnovationhas occurred by time t during the v thcycle can be expressed as

(21) V iI~t 5

t

Tv 1 1

e2 tt r~s dspi ~t dt

1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1

The rst term here represents the discountedpro t stream that accrues to the entrepreneurwith certainty during the current cycle and thesecond term is the expected discounted value ofbeing an incumbent thereafter

In the acyclical equilibrium the role of se-crecy is not relevant because innovators wouldalways prefer to implement even if it were pos-sible that by delaying they could protect their

FIGURE 1 THE CYCLICAL GROWTH PATH


knowledge Since simultaneous innovation canonly occur with a second-order probability inthat equilibrium it is assumed away In thecyclical equilibrium considered here secrecy(ie protecting the knowledge embodied in anew innovation by delaying implementation)can be a valuable option Innovations are with-held until a common implementation time sothat simultaneous implementation is a possibil-ity However as the following lemma demon-strates such duplications do not arise in thecyclical equilibrium

LEMMA 1 In a cyclical equilibrium success-ful entrepreneurs can credibly signal a suc-cess immediately and all innovation in theirsector will stop until the next round ofimplementation

Unsuccessful entrepreneurs have no incen-tive to falsely announce success As a result anentrepreneurrsquos signal is credible and other en-trepreneurs will exert their efforts in sectorswhere they have a better chance of becomingthe dominant entrepreneur5

In the cyclical equilibrium entrepreneursrsquoconjectures ensure no more entrepreneurship ina sector once a signal of success has been re-ceived until after the next implementation Theexpected value of an entrepreneurial successoccurring at some time t [ (Tv

E Tv) but whoseimplementation is delayed until time Tv is thus

(22) V iD~t 5 e2b~tV0i

I ~Tv

Since no implementation occurs during the cy-cle the entrepreneur is assured of incumbencyuntil at least Tv 1 1 Incumbency beyond thattime depends on the probability that there hasnot been another successful innovation in thatsector up until then The symmetry of sectorsimplies that innovative effort is allocated evenly

over all sectors that have not yet experienced aninnovation within the cycle Thus the probabil-ity of not being displaced at the next implemen-tation is

(23) P i ~Tv 5 expX 2Tv

E

Tv

dx i ~t dtDwhere xi(t) denotes the quantity of labor thatwould be allocated to entrepreneurship if noinnovation had been discovered prior to time tin sector i The amount of entrepreneurshipvaries over the cycle but at the beginning ofeach cycle all industries are symmetric withrespect to this probability Pi(Tv) 5 P(Tv) i

B Within-Cycle Dynamics

Within a cycle t [ [Tv 2 1 Tv] the state oftechnology in use is unchanging A criticalvariable is the amount of labor devoted toentrepreneurship the opportunity cost of whichis production In order to determine this we rst characterize wages paid to labor inproduction

LEMMA 2 The wage for t [ [Tv 2 1 Tv] ispinned down by the level of technology

(24)

w~t 5 e2gexpX 0

1

ln Ai ~Tv 2 1 diD 5 wv

Competition between rms attempting to hiredoes not drive the wage up to laborrsquos marginalproduct because rms earn monopolistic rentsHowever it does ensure that labor bene ts pro-portionately from productivity advancementsWe denote the improvement in aggregate pro-ductivity during implementationperiod Tv (andhence the growth in the wage) by eG v where

(25) G v 5 0

1

ln Ai ~Tv 2 ln A i ~Tv 2 1 di

Since wages are determined by the level of

5 With an arbitrarily small signaling cost the equilibriuminvolves strictly dominant strategies This equilibrium relieson the memoryless nature of the Poisson process governinginnovation However with memory a similar shutting downof innovation after a success would arise if innovative effortis directly observable However the direction of this effortcannot be observed or incumbents might allocate effort intheir own sectors to deter entry


technology in use and since this does notchange within the cycle wages are constantwithin the cycle

Following an implementation boom theeconomy passes through two distinct phases

The SlowdownmdashAs a result of the boomwages rise rapidly and the present value ofengaging in innovation falls below the wagedVD(t) w(t) During this phase no labor isallocated to entrepreneurship and no new inno-vations come on line so nal output must beconstant

(26) g~t 5wsbquo ~t

w~t5 0

With zero growth the demand side of the econ-omy dictates that the interest rate just equal thediscount rate

(27) r~t 5 sg~t 1 r 5 r

Since the economy is closed and there is noincentive to store either intermediate or naloutput when r(t) $ 0 it must be the case that

(28) c~t 5 y~t

During the slowdown the expected value ofentrepreneurship dVD(t) is necessarily grow-ing at the rate of interest r(t) 5 r but contin-ues to be dominated by the wage in productionSince the wage is constant during the cycledVD(t) must eventually equal w(t) At thispoint entrepreneurship commences The fol-lowing lemma demonstrates that it does sosmoothly

LEMMA 3 At time TvE when entrepreneur-

ship rst commences in a cycle wv 5 dVD(t)and X(Tv

E) 5 0

The DownturnmdashFor positive entrepreneur-ship to occur under free entry it must be thatwv 5 dVD(t) Since the wage is constantthroughout the cycle dVD(t) must also be con-stant during this phase Since the time until

implementation for a successful entrepreneur isfalling and there is no stream of pro ts becauseimplementation is delayed the instantaneousinterest rate must be zero

(29) r~t 5Vsbquo D~t

VD~t5

wsbquo ~t

w~t5 0

With a positive discount rate r 0 a zerointerest rate implies that consumption must bedeclining Since the economy is closed it fol-lows once again that because there is no incen-tive to store output (28) holds Hence percapita output must also decline

(30) g~t 5r~t 2 r

s5 2

r

s

This occurs during the downturn because labor ows out of production and into entrepreneur-ship (knowledge capital is being built) Using(11) (30) and the fact that X(Tv

E) 5 0 yieldsthe following expression for aggregate entrepre-neurship at time t

(31) X~t 5 1 2 e2~r s t 2 TvE

The proportion of sectors that have not yetexperienced an entrepreneurial success by timet [ (Tv

E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD

Recalling that labor is only devoted to entrepre-neurship in sectors which have not innovatedsince the start of the cycle the labor allocated toentrepreneurship in each sector is then

(33) x~t 5X~t

P~t

Differentiating (32) and substituting in (33) wethus obtain the aggregate rate of entrepreneurialsuccess


(34) Psbquo ~t 5 2dx~tP~t 5 2dX~t

We characterize an equilibrium in which thecycle is never long enough that all sectors in-novate P(Tv) 0 The parameter restrictionsthat ensure this are discussed in subsection E ofthis section

The dynamic movement of variables impliedby our hypothesized cycle is sketched in Figure1 The resulting allocation of labor to entrepre-neurship (31) determines the size of the outputboom at the end of the cycle Denote the inter-val over which there is positive entrepreneur-ship by Dv

E 5 Tv 2 TvE Then we have

PROPOSITION 2 In an equilibrium wherethere is positive entrepreneurship only over theinterval (Tv

E Tv] the growth in productivityduring the succeeding boom is given by

(35) G v 5 dgDvE 2 dgX 1 2 e2~rsDv

E

rs D

Equation (35) tells us how the size of theproductivity boom depends positively on theamount of time the economy is in the entrepre-neurship phase Dv

E The size of the boom isconvex in Dv

E re ecting the fact that as theboom approaches the labor allocated towardsinnovation is increasing which also implies thatthe boom size is increasing in the depth of thedownturn

C Market Clearing During the Boom

For an entrepreneur who is holding an inno-vation VI(t) is the value of implementing im-mediately Just prior to the boom when theprobability of displacement is negligible thevalue of implementing immediately must equalthat of delaying until the boom

(36) dV I~Tv 5 dVD~Tv 5 w v

During the boom since entrepreneurs prefer toimplement immediately it must be the case thatV0

I (Tv) V0D(Tv) Thus the return to innova-

tion at the boom is the value of immediate(rather than delayed) incumbency It follows

that free entry into entrepreneurship at the boomrequires that

(37) dV0I ~Tv wv 1 1

The opportunity cost to nancing entrepre-neurship is the rate of return on shares in in-cumbent rms in sectors where no innovationhas occurred Just prior to the boom this isgiven by the capital gains in those sectors

(38) b~Tv 5 logX V0I ~Tv

V I~Tv D

Note that since the short-term interest rate iszero over this phase b(t) 5 b(Tv) t [ (Tv

ETv) Combined with (36) and (37) it followsthat asset market clearing at the boom requires

(39) b~Tv logX wv 1 1

wvD 5 Gv

Provided that b(t) 0 households willnever choose to store nal output from within acycle to the beginning of the next because it isdominated by the long-run rate of return onclaims to future pro ts However unlike naloutput the return on stored intermediate outputin sectors with no innovations is strictly posi-tive because of the increase in its price thatoccurs as a result of the boom Even thoughthere is a risk that the intermediate becomesobsolete at the boom if the anticipated priceincrease is suf ciently large households maychoose to purchase claims to intermediate out-put rather than claims to rm pro ts6

If innovative activities are to be nanced attime t households cannot be strictly better offbuying claims to stored intermediates Twotypes of storage could arise with equal returns

6 One may suppose that incumbents have an incentive tostore intermediate production and threaten to use it to un-dercut any future innovator in their sector However such athreat is not credible If faced with an innovator holding aproductive advantage that will be implemented at time T anincumbent would sell stockpiled intermediates before timeT and obtain a higher price than by delaying and selling itin competition with the new innovator


In sectors with unimplemented innovations in-novators could implement immediately but de-lay sales until the boom The best way to do thisis to produce an instant prior to the boom and tosell an instant afterwards Since the revenue isthe same the difference between producing be-fore and after the boom is that the former in-volves the current wage and the latter involvesthe higher future wage Thus the return onclaims to stored intermediates is ln(wv1 1wv) 5 Gv In sectors with no innovation sim-ilar trade in such claims leads to an identicalreturn on stored intermediates of ln(piv1 1piv) 5 Gv

It follows that the long-run rate of return onclaims to rm pro ts an instant prior to theboom must satisfy

(40) b~Tv $ Gv

Because there is a risk of obsolescence thiscondition implies that at any time prior to theboom the expected rate of return on claims tostored intermediates is strictly less than b(t)

Combining (39) and (40) and observing thatb(Tv) 5 sGv 1 rDv

E yields the followingimplication of market clearing during the boomfor the long-run growth path

PROPOSITION 3 Long-run asset marketclearing implies that

(41) Gv 5rDv

E

1 2 s

It follows that asset market clearing yields aunique relationship between the length of thedownturn and the size of the subsequent pro-ductivity boom

Figure 2 depicts the two conditions (35) and(41) graphically As shown by the solid linescombining the two conditions yields a unique(positive) equilibrium pair (G DE) that is con-sistent with the within-cycle dynamics and theasset market-clearing condition Combiningthem implies that DE must satisfy

(42) X 1 2r

dg~1 2 sD DE 51 2 e2~rsDE

rs

Note that although we did not impose any sta-tionarity on the cycles the equilibrium condi-tions imply stationarity of the size of the boomand the length of the downturn For a uniquepositive value of DE that satis es this conditionto exist it is suf cient that r dg(1 2 s)

D Optimal Entrepreneurial Behavior

It has thus far been assumed that entrepre-neurs are willing to follow the innovation andimplementation sequence hypothesized in thecycle The equilibrium conditions that we haveconsidered so far effectively assume that entre-preneurs who plan to innovate will implement atTv and that they start innovation at Tv

E How-ever the willingness of entrepreneurs to delayimplementation until the boom and to just startengaging in innovative activities at exactly Tv

E

depends crucially on the expected value of mo-nopoly rents resulting from innovation relativeto the current labor costs This is a forward-looking condition given G and DE the presentvalue of these rents depend crucially on thelength of the subsequent cycle Tv 1 1 2 Tv

Since Lemma 3 implies that entrepreneurshipstarts smoothly at Tv

E free entry into entrepre-neurship requires that

(43) dVD~TvE 5 de2b~Tv

EV0I ~Tv 5 w v

FIGURE 2 EQUILIBRIUM RECESSION LENGTH

AND BOOM SIZE


Since the increase in the wage across cyclesre ects only the improvement in productivitywv 1 1 5 eGwv and since from the asset market-clearing conditions we know that b(Tv

E) 5 Git immediately follows that the increase in thepresent value of monopoly pro ts from the be-ginning of one cycle to the next must in equi-librium re ect only the improvements inaggregate productivity

(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv

Equation (44) implies that given some initialimplementation period and stationary values ofG and DE the next implementation period isdetermined Notice once again that this station-arity is not imposed but is an implication of theequilibrium conditions Letting Dv 5 Tv 2Tv2 1 we therefore have the following result

PROPOSITION 4 Given the boom size G andthe length of the entrepreneurial innovationphase DE there exists a unique cycle length Dsuch that entrepreneurs are just willing to com-mence innovation DE periods prior to theboom

In the unpublished Appendix we show that theimplied cycle length is given by

(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence

The equilibrium conditions (12) (13) and(14) on entrepreneurial behavior also imposethe following requirements on our hypothesizedcycle

Successful entrepreneurs at time t 5 Tv2 1must prefer to implement immediately rather

than delay implementation until later in thecycle or the beginning of the next cycle

(E1) V0I ~Tv 2 1 V0

D~Tv 2 1

Entrepreneurs who successfully innovate dur-ing the downturn must prefer to wait until thebeginning of the next cycle rather than im-plement earlier

(E2) V I~t VD~t t [ ~TvE Tv

No entrepreneur wants to innovate during theslowdown of the cycle Since in this phase ofthe cycle dVD(t) w(t) this condition re-quires that

(E3) dV I~t w~t t [ ~Tv 2 1 TvE

The downturn is not long enough that allsectors innovate

(E4) P~T 0

The following proposition demonstrates thatthere is a nonempty parameter space such thatthe triple (DE D G) 0 solving (35) (41)and (45) also implies that conditions (E1) (E2)(E3) and (E4) are satis ed

PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg

then there exists a unique cyclical equilibriumgrowth path (DE D G)

To understand this proposition it is useful torefer to Figure 3 which illustrates the impliedevolution of VI(t) VD(t) and w(t)d The in-equality (1 2 e2 g)g(1 2 s) rd on the leftof (47) is suf cient for (E1) to hold It ensuresthat during the cycle the instantaneous interestrate always exceeds the instantaneous growthrate As a result at the beginning of a cycleimplementation is never delayed because any


gain in pro ts from delay is less then the rate atwhich it is discounted This is equivalent to thetransversality condition that the long-run inter-est rate exceeds the long-run growth rate

During the downturn the left-hand inequalityin (47) ensures that VI(t) approaches VD(t)from below so that implementation is delayeduntil the next boom (E2) To understand thisnote that the boom is the only time during thecycle at which the increment in output exceedsthe increment in the discount factor Althoughthe increment in productivity G exactly equalsthe increment in the discount factor b thereallocation of resources back into productionimplies that output increases by more than theincrement in productivity Thus the increase inpro ts at the boom exceeds the rate at whichthey are discounted and this will induce delayprior to the boom provided that the probabilityof being displaced is suf ciently low Hence theupper bound on gd required for this condition tohold

The right-hand inequality in (47) is necessaryand suf cient for condition (E4) to be satis edso that not all sectors innovate during the cycleThis is a possibility because through the down-turn an increasing amount of innovative effortX(t) is allocated across fewer and fewer sec-tors This condition effectively ensures that theentrepreneurial phase does not proceed for toolong which can be seen from equation (41) torequire a low value for rdg as implied by thecondition

The right-hand inequality is also suf cient toensure that the value of immediate implemen-tation VI(t) declines monotonically during theslowdown The weaker necessary and suf cientcondition that (47) implies is rd eg 2 1This condition ensures that over this phaseforgone pro t by delaying implementation (pro-portional to eg 2 1) exceeds the bene t fromsuch delay a higher probability of retainingincumbency (proportional to rd) Since at thebeginning of the cycle dVI(Tv2 1) 5 wv con-dition (E3) follows

The parameter restrictions imposed by (47)are stronger than those needed for the existenceof the acyclical equilibrium in (16) Note how-ever that while they are suf cient for existencethey are not necessary In particular the cyclicalequilibrium can exist even if the condition thatg(1 2 s) 2 s rd is violated Table 1 givessome parameter examples that satisfy (47) andyield long-run growth rates in the 2ndash3 percentrange The increase in consumption at the boommust equal the increase in output which in turnmust exceed the increase in the discount factorThis is only possible here if s is small enoughand certainly less that unity Introducing phys-ical capital weakens this restriction since someof the output boom is diverted to investmentWe discuss such an extension in Francois andLloyd-Ellis (2003)

F Stability

There are two notions of stability that wemust consider The rst relates to the stability ofthe instantaneous equilibrium In every instantlabor chooses between entrepreneurship andproduction entrepreneurs choose between im-plementing today or delaying until tomorrowand incumbents must decide whether to sellnow or store It is straightforward to demon-strate stability by considering errors in the de-cisions of agents and showing that suchperturbations result in movement back towardsequilibrium in the next instant (in the unpub-lished Appendix)

A second notion of stability relates to thedynamic convergence of the economy to itslong-run growth path Like the acyclical growthpath the cyclical equilibrium is ldquojump stablerdquoAs our analysis demonstrates there is a unique

FIGURE 3 EVOLUTION OF VALUE FUNCTIONS AND WAGE


triple (G DE D) that is consistent with equi-librium Thus the economy necessarily jumps tothis long-run path since no other (G DE D) triplecan hold even in the short run without violat-ing the equilibrium conditions Note nally thatthere is one element of indeterminacy in thecycling equilibrium the length of the rst cycleD0 is indeterminate on the interval [DE D]since there is no previous entrepreneurshipphase to pin it down

III Implications for Growth Welfareand Volatility

In this section we compare the long-rungrowth rates in the cyclical and acyclical econ-omies and examine the impact of changes in theproductivity of innovative effort

A Growth and Welfare in Cyclical andAcyclical Economies

Let the average growth rate in the cyclingequilibrium be denoted

(48) g c 5G

D

and recall the acyclical equilibrium growth gagiven in (17) Then we have

PROPOSITION 6 The long-run growth ratein the cyclical equilibrium gc exceeds that in theacyclical equilibrium ga

The cyclical equilibrium yields higher aver-age growth because all entrepreneurship occursin the downturn when growth is negative andthe interest rate is low relative to the economyrsquoslong-run average Thus compared with the acy-clical economy where the interest rate is con-stant the same expected ow of pro ts for thesame expected length of incumbency has highervalue in the cycling economy thereby inducingmore entrepreneurship and higher growth

Although the long-run growth rate is higherin the cyclical equilibrium the same is not trueof welfare Consider two economies that startwith an identical stock of implemented technol-ogies and zero unimplemented innovations

Suppose one of the economies is in a cyclicalequilibrium at the beginning of a cycle and theother is in an acyclical equilibrium Then

PROPOSITION 7 Welfare in the acyclicaleconomy exceeds that in the cyclical one

There are three key differences that deter-mine relative welfare in the two economies (1)the long-run growth rate in the cyclical econ-omy is higher (2) the initial consumption in thecyclical economy is higher because some laboris allocated to innovation in the acyclical econ-omy whereas none is during this phase of thecyclical equilibrium and (3) until the nextboom the short-run growth rate in the cyclicaleconomy is zero or negative whereas it is pos-itive in the acyclical one Proposition 7 demon-strates this last factor dominates so that welfareis lower in the cyclical economy

B Impact of Entrepreneurial Productivity

Consider the impact of an increase in entre-preneurial productivity d on the cyclical growthpath

LEMMA 4 An increase in d results in shortercycles smaller booms and shorter recessions

To understand these results rst consider Fig-ure 2 For a given cycle length and downturnlength (D DE) an increase in d causes the sizeof the boom to be larger because entrepreneur-ship is now more productive This is illustratedby the outward shift in OA to OA9 Howevernow the economy would be to the right of OBso that the asset market is out of equilibriumwith b G just prior to the boom so that thereis an incentive to store Arbitrageurs would bewilling to offer incumbents and entrepreneursincentives to produce more intermediate out-put than needed to supply current demand Inparticular entrepreneurs with unimplementedinnovations would respond by bringing produc-tion forward slightly from the boom But if allentrepreneurs do this the boom would actuallyoccur earlier and the incentive to store woulddisappear Applying this argument recursivelyone can see that the length of the downturn (andhence the entire) cycle would fall until it is just


short enough to ensure that for the (smaller) sizeof the boom that results the incentive to pro-duce early and store has been removed (ieb 5 G just prior to the boom) Thus as noted inthe lemma the cycle length recession lengthand boom size would all fall

The economyrsquos volatility is also affected by achange in d Since the standard deviation is notwell-de ned in our context we measure vola-tility as the average absolute size of deviationsin long output from trend

(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r

s~t 2 ~D 2 DE 2 gct dt

The variable d affects this through numerouschannels D DE and gc Though the affects onphase lengths are unambiguous (see Lemma 4)the relationship between gc and is not analyt-ically clear The rst three rows of Table 1shows how growth volatility and cycle lengthvary with changes in d within the parameterspace given by (47) These numerical examplesillustrate what extensive simulations show is amore general result

The long-run relationship between growth gcand volatility across economies with differ-ent levels of entrepreneurial productivity isnegative

Thus the relationship between growth andvolatility across cyclical equilibria is at leastsuper cially consistent with the empirical re-sults of Ramey and Ramey (1995) and Kroftand Lloyd-Ellis (2002) Note however that thisrelationship does not represent the impact ofvolatility on growth nor the impact of growthon volatility Rather it is an induced relation-ship due to variation in the productivity ofentrepreneurship

Our emphasis on the consequences of cross-country variation in d rather than g stems fromour view that the former parameter capturescountry-speci c factors whereas the latter cap-tures (on average) characteristics of a techno-logical possibilities frontier which is commonto all countries Thus differences in entrepre-neurial productivitymay re ect variations in thequality of education institutional arrangementsand culture for example It is possible how-ever that d also re ects technological possibil-ities as well For example one refereespeculates that the combination of a low d anda high g may be characteristic of a maturetechnological stage If so then such a stagewould be associated with cycles whereas earlystages (high d low g) would involve steadiergrowth

IV Implications for the Business Cycle

In this section we consider the extent towhich some essential features of the impliedcyclical process are qualitatively consistentwith the facts In our conclusion we also discussseveral extensions to the model that we believewill allow it to match the data more closely

A The Downturn is Not a Consequence ofMismeasurement

The downturn in our cycle results from theallocation of labor to entrepreneurship in antic-ipation of the upcoming boom Since this real-location represents an investment in intangibleassets one may wonder whether the implieddownturn is really just a result of mismeasure-ment which would disappear if we includedintangible investment I(t) in computing aggre-gate GDP If we did so aggregate GDP duringthe downturn would be

TABLE 1mdashGROWTH AND VOLATILITY

Benchmark parameters Long-rungrowth

(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t

5 p~t 1 wv1 2 X~t 1 wvX~t

5 ~1 2 e 2 g y~t 1 wv

Thus the downturn does not arise from mis-measurementmdasheven though the wage is con-stant through the cyclemdashGDP declines becausepro ts decline This is because imperfect com-petition in the intermediate sector implies thatthe total marginal cost of labor leaving produc-tion (ie its marginal product) exceeds the pri-vate marginal cost (ie the wage) Althoughworkers are equally well off in the two activi-ties the reallocation has a negative externalityon current pro ts A similar implicationemerges from the GPT-driven downturn inHelpman and Manuel Trajtenberg (1998)

B Stock Market Implications

Our model also has predictions for the cycli-cal behavior of the stock market Here the stockmarket consists of three types of rms incum-bents in sectors where no new innovations haveoccurred ldquoterminalrdquo incumbents in sectorswhere innovations have occurred and new en-trants in those sectors that have not yet imple-mented but have value In the slowdown onlythe rst type of rm exists but during the down-turn all three are present At any point in timethe total value of rms on the stock market isgiven by

(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t

where VT(t) denotes the value of ldquoterminalrdquo rms who are certain to be made obsolete dur-ing the next wave of implementation The valueof these rms can be written as

(52) VT~t 5 V I~t 2P~Tv

P~tVD~t

Substituting into (51) yields

(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t

During the slowdown P(t) 5 1 so that(t) 5 VI(t) Immediately prior to the boom

P(t) 5 P(Tv) so that again (Tv) 5 VI(Tv)Thus the evolution of the aggregate value of thestock market during the cycle resembles that ofincumbent rms in sectors that have not yetinnovated VI(t) (see Figure 3) except that dur-ing the downturn it is always higher re ectingthe fact that incumbents with uncertain longev-ity are being replaced by new entrants who willhave incumbency for at least one full cyclelength

Thus the stock market falls during the slow-down in anticipation of the subsequent reces-sion and rises during the downturn inanticipation of the subsequent boom This cy-clical anticipation of future pro ts implicit inaggregate stock prices accords well with the ndings of Hall (2001) who compares thegrowth rate of cash ows implicit in securitiesvalues with the actual ve-year forwardgrowth rate of cash ow (see his Figure9) While of course much of the variation inpro ts re ect unexpected shocks to the econ-omy Hallrsquos results are consistent with theview that a signi cant component of cyclical uctuations is indeed anticipated by equitymarkets

C The Clustering of Implementation

As in the pure real business cycle (RBC)model productivity improvements in our cycleare clustered over time and are procyclicalSusanto Basu (1996) nds that once onetakes account of variations in capital utiliza-tion and labor hoarding over the cycle theimplied movements in total factor productiv-ity (TFP) may be small and not strongly pro-cyclical However Robert G King and SergioRebelo (1999) argue that once one endog-enizes variations in factor utilization large uc-tuations in output can result from small changesin TFP Moreover the implied TFP movementsare hardly ever negative and hence more


consistent with their interpretation as technol-ogy improvements

There is however more direct evidence ofprocyclical clustering of implementation dur-ing booms Paul A Geroski and Chris FWalters (1995) investigate high-frequencymovements in both the granting of patents inthe United States to UK rms and the im-plementation of major innovations in theUnited Kingdom for the period 1948ndash1983They nd that the implementation of innova-tions and patenting activity are procyclical7

and that they occur in small (several year)clusters More generally Zvi Grilichesrsquo(1990) survey on patents also concludes thatthe basic procyclicality of patenting rst sug-gested by Jacob Schmookler (1966) has notbeen overturned

D The Countercyclicality of Innovative Effort

One often-used measure of innovative effortis RampD expenditure Antonio Fatas (2000) doc-uments that growth in real RampD expenditures inthe United States is positively correlated withreal GDP growth However if we consider onlyprivately funded RampD and distinguish betweenbasic research which is not generally driven bycommercial considerations (and is a small pro-portion of the total) and applied research whichis then this stylized fact is not so clear There isin fact no signi cant correlation betweengrowth rates in real applied research (NSF data)and real GDP for the United States over theperiod 1953 to 1999 As may be seen in Figure4 although the big increases during the 1960rsquosappear to occur during booms those of 1973ndash1974 1981 and 1991 occur during recessions

On the whole then the evidence on RampDexpenditures is not strongly supportive of eithera pro- or countercyclical view However inno-vation is a much broader concept than thatmeasured by RampD investment Much of the

7 Though Geroski and Walters term their observationsldquoinnovationsrdquo their data involves the actual implementationof innovations

FIGURE 4 APPLIED RESEARCH AND GROWTH IN THE UNITED STATES

Source National Science Foundation


entrepreneurial function in modern productionis undertaken by skilled workers and managerswithin industries8 Since much of their innova-tive effort occurs without separately measuredexpenditures or occupational reallocation theusual aggregate data sets are not helpful In-stead what we require is detailed informationabout plant-level activities Although the evi-dence so far is disparate a number of studies(discussed below) have used either specializeddata sets based on surveys or proxies to obtainrelated estimates9

A frequently emphasized feature of businesscycles is the apparent employment of labor dur-ing recessions beyond that which is technolog-ically necessary to meet regular productionrequirements Recently the RBC literaturehas argued that this behavior re ects ldquolaborhoardingrdquomdashlike capital there are signi cantcosts to adjusting labor (eg hiring and ringcosts) which cause rms to hold on to skilledlabor during recessions However another in-terpretation is that this labor is actually doingsomething productivemdashcoming up with newideas and approaches that will be useful in thefuture In a survey of US manufacturing plantsJon A Fay and James L Medoff (1985) ndthat during a trough quarter the typical plantpaid for about 8 percent more labor hours thantechnologically necessary Only half of this washoarded labormdashthe remainder was used in otherproductive activities Of the respondents thatreassigned workers during recessions (morethan half of respondents) about one-third allo-cated them to ldquoreworking outputrdquo and anotherthird to ldquotrainingrdquo

One might suspect that innovative activitiesare more likely to require skilled nonproduc-tion workers so that during downturns the ratioof skilled to unskilled workers should rise Al-though this is typically the case in the data it is

possible that this is motivated by labor hoardingsince the costs of adjustment for skilled workersare relatively high However such a motivationwould not lead to an absolute increase in skilledemployment during downturns which has beendocumented by Victor Aguirregabiria and CesarAlonso-Borrego (2001) Using Spanish manu-facturing data (a balanced panel of 1080 non-energy manufacturers from 1986ndash1991) they nd the employment of white collar workers tobe signi cantly countercyclical

More direct evidence of what managers aredoing during downturns is provided by StephenNickell et al (2001) They investigate whethermanagerial innovations occur in downturns us-ing two unique data sets The rst based on theConfederation of British Industries Pay Data-bank (66 manufacturing rms during the period1981ndash1986) includes information on two mea-sures of innovationmdashthe removal of restrictivepractices and the introduction of new technol-ogy The second data set includes small- tomedium-sized manufacturing rms in engineer-ing plastics electronics and food drink andtobacco This categorizes levels of managerialinnovation from 1991ndash1994 and compares themto lagged performance variables from the period1988ndash199110 Both data sets support the viewthat when demand is slack and pro tability lowmanagers and workers devote more time toinnovation

E Downturns and Subsequent ProductivityGrowth

In the cyclical process implied by our modellarger downturns are associated with biggerbooms This is broadly consistent with the ag-gregate empirical characterizations of Pesaranand Potter (1997) and Altissimo and Violante(2001) More direct evidence is provide by Vin-cenzo Atella and Beniamino Quintieri (1998)Using Italian data for nine industries from 1967to 1990 they nd strong evidence that down-

8 This interpretation of innovation is not unlike the ldquore-organizationrdquo activity emphasized by Hall (2000) Note thatalthough incumbents in our model would not engage ininnovation within their own product line they may innovatein other product lines

9 If we interpret innovative effort even more broadlyto include any withdrawal of labor from production toproductivity-enhancing activities it is also suggestive thatpostsecondary educational investments are countercyclical(Plutarchos Sakellaris and Antonio Spilimbergo 2000)

10 The managerial innovation variables included signif-icant change in structure organization leaner as result ofchange signi cant changes resulting in more decentralizedorganization signi cant changes in human resources man-agement practices and industrial relations and the imple-mentation of just-in-time technologies


turns tend to be followed by subsequent in-creases in TFP The correlation of negativedemand movements with subsequent TFPgrowth was greater than either public capital orRampD expenditures In a similar study for 2-digitSIC industries in the United States (using theNBER annual productivity database from1958ndash1991) Jim Malley and V Anton Mus-catelli (1999) nd that demand reductions tomanufacturing as a whole are signi cantly pos-itively correlated with subsequent TFP growth

V Concluding Remarks

This paper has established the existence ofcycles along a balanced growth path of a com-pletely standard multisectoral Schumpeteriangrowth model that allows for the possibility ofdelayed implementation and storage Speci -cally we show that even with multiple sectorsin general equilibrium with reasonable assump-tions on preferences technology and marketcompetition no static increasing returns toscale no stochastic expectations no thresholdeffects and rational forward-looking behaviorthere exists a business cycle that is interlinkedwith the economyrsquos growth process Moreoverwe establish conditions under which a uniquecycling equilibrium arises

The endogenous cycles generated by ourmodel have several features that we believe arecrucial to understanding actual business cyclesFirst and foremost the cyclical uctuations arethe result of independent actions by decentral-ized decision makers They are not the result ofeconomy-wide shocks or economy-wide tech-nological breakthroughs but emerge as a resultof pecuniary demand externalities that inducecoordination This is true of both the boomwhich re ects Shleiferrsquos (1986) formalizationof ldquoanimal spiritsrdquo in the joint implementationof innovations and of the downturn which re- ects the common incentives of entrepreneursin anticipation of the upcoming boom Secondas in our cycle the quantitative analyses ofEmery and Koenig (1992) Sichel (1993) andBalke and Wynne (1995) suggest that the aver-age cycle starts with a growth spurt which isthen followed by a growth slowdown before theeconomy enters a period of relatively constantdecline during the downturn Thirdly as is con-

sistent with the ndings of Pesaran and Potter(1997) and Altissimo and Violante (2001) thereis a positive feedback from downturns to sub-sequent cyclical upturns Finally the equilib-rium relationship between growth and volatilityis negative which is consistent with the cross-country evidence of Ramey and Ramey (1995)

A valuable feature of the model developedhere is its parsimony Apart from a slight gen-eralization of preferences the model is identicalto Grossman and Helpman (1991 Ch 4) Theultimate value of theoretical endeavors aimed atunderstanding the interactions between growthand cycles will be in their ability to provide aconvincing account of the high-frequency dataWhile the model ts some features of the ldquonor-malrdquo business cycle we do not claim to havedone that yet However the modelrsquos simplicityallows it to be used as a platform for these moreempirically motivated extensions The centralmechanism described here is robust to exten-sions which shall be explored in future workand which we brie y describe below

Tangible capital assetsmdashAlthough we allowfor saving through intangible assets and forthe possibility of storing output we assumeaway physical capital as a vehicle for smooth-ing aggregate consumption over time Intro-ducing physical capital that is completelyliquid would destroy the cyclical equilibriumbecause households would try to consume theanticipated bene ts of the boom in advanceby dis-saving However suppose (realisti-cally) that capital exhibits ldquoputty-clayrdquo char-acteristics and the capitalndashlabor ratio cannotbe adjusted fully except through expansionthen the cyclical equilibrium would still existDuring the downturn capital would be leftidle as complementary labor resources shiftout of production Because of the high oppor-tunity cost (the return on intangible assets)investment in new capital would be delayeduntil after the initial boom that is associatedwith implementation and the increased utili-zation of existing capitalAbruptness of the boommdashThe growth spurtand the start of the slowdown are unrealisti-cally abrupt In reality expansions tend to bespread out over time so that positive growthis more common than zero or negative


growth The introduction of tangible physicalcapital will also help here The innovationboom will lead to a rise in capitalrsquos marginalproduct and trigger a sustained period of in-vestment in which output grows smoothlyand continuously as capital is accumulatedAggregate uncertaintymdashThe length andother characteristics of actual business cyclesvary from cycle to cycle and look rather dif-ferent from the deterministic equilibrium cy-cle described here However introducing adegree of aggregate uncertainty would bepossible without changing the basic analysisFor example the stochastic arrival of GPTsthat raises productivity in all sectors saywould cause the size and length of booms andrecessions between GPTs to vary over time

REFERENCES

Aghion Philippe and Howitt Peter ldquoA Modelof Growth through Creative DestructionrdquoEconometrica March 1992 60(2) pp 323ndash51

Endogenous growth theory Cam-bridge MA MIT Press 1998

Aguirregabiria Victor and Alonso-Borrego Ce-sar ldquoOccupational Structure TechnologicalInnovation and Reorganization of Produc-tionrdquo Labour Economics January 20018(1) pp 43ndash73

Altissimo Filippo and Violante Giovanni L ldquoTheNon-linear Dynamics of Output and Unem-ployment in the USrdquo Journal of AppliedEconometrics JulyndashAugust 2001 16(4) pp461ndash86

Atella Vincenzo and Quintieri Beniamino ldquoPro-ductivity Growth and the Effects of Reces-sionsrdquo Giornale degli Economisti e Annali diEconomia December 1998 57(3ndash4) pp359ndash86

Balke Nathan S and Wynne Mark A ldquoReces-sion and Recoveries in Real Business CycleModelsrdquo Economic Inquiry October 199533(4) pp 640ndash63

Basu Susanto ldquoProcyclical Productivity In-creasing Returns or Cyclical UtilizationrdquoQuarterly Journal of Economics August1996 111(3) pp 719ndash51

Beaudry Paul and Koop Gary ldquoDo RecessionsPermanently Change Outputrdquo Journal of

Monetary Economics April 1993 31(2) pp149ndash63

Cohen Wesley M Nelson Richard R andWalsh John P ldquoProtecting Their IntellectualAssets Appropriability Conditions and WhyUS Manufacturing Firms Patent (or Not)rdquoNational Bureau of Economic Research(Cambridge MA) Working Paper No 7552February 2000

Emery Kenneth M and Koenig Evan F ldquoFore-casting Turning Points Is a Two-StateCharacterization of the Business CycleAppropriaterdquo Economics Letters August1992 39(4) pp 431ndash35

Fatas Antonio ldquoDo Business Cycles Cast LongShadows Short-Run Persistence and Eco-nomic Growthrdquo Journal of EconomicGrowth June 2000 5(2) pp 147ndash62

Fay Jon A and Medoff James L ldquoLabor andOutput over the Business Cycle Some DirectEvidencerdquo American Economic Review Sep-tember 1985 75(4) pp 638ndash55

Francois Patrick and Lloyd-Ellis Huw ldquoCo-Movement Capital and Contracts Endoge-nous Business Cycles Through CreativeDestructionrdquo Mimeo Queenrsquos University2003

Francois Patrick and Shi Shouyong ldquoInnova-tion Growth and Welfare-Improving Cy-clesrdquo Journal of Economic Theory April1999 85(2) pp 226ndash57

Freeman Scott Hong Dong Pyo and Peled DanldquoEndogenous Cycles and Growth with Indi-visible Technological Developmentsrdquo Re-view of Economic Dynamics April 19992(2) pp 403ndash32

Geroski Paul A and Walters Chris F ldquoInnova-tive Activity over the Business Cyclerdquo Eco-nomic Journal July 1995 105(431) pp916ndash28

Griliches Zvi ldquoPatent Statistics as Economic In-dicators A Surveyrdquo Journal of Economic Lit-erature December 1990 28(4) pp 1661ndash707

Grossman Gene and Helpman Elhanan Innova-tion and growth in the global economy Cam-bridge MA MIT Press 1991

Hall Robert E ldquoReorganizationrdquo Carnegie-Rochester Conference Series on Public Pol-icy June 2000 52 pp 1ndash22

ldquoStruggling to Understand the StockMarketrdquo American Economic Review May


2001 (Papers and Proceedings) 91(2) pp1ndash11

Helpman Elhanan and Trajtenberg Manuel ldquoATime to Sow and a Time To Reap GrowthBased on General Purpose Technologiesrdquo inElhanan Helpman ed General purposetechnologies and economic growth Cam-bridge MA MIT Press 1998 pp 55ndash83

Keynes John Maynard The general theory ofemployment interest and money LondonMacmillan 1936

King Robert G and Rebelo Sergio ldquoResuscitat-ing Real Business Cyclesrdquo Handbook of mac-roeconomics volume 1B Amsterdam North-Holland 1999

Kroft Kory and Lloyd-Ellis Huw ldquoFurtherCross-Country Evidence on the Link betweenGrowth Volatility and Business CyclesrdquoMimeo Queenrsquos University 2002

Malley Jim and Muscatelli V Anton ldquoBusinessCycles and Productivity Growth Are Tem-porary Downturns Productive or WastefulrdquoResearch in Economics December 199953(4) pp 337ndash64

Matsuyama Kiminori ldquoGrowing Through Cy-clesrdquo Econometrica March 1999 67(2) pp335ndash47

Nickell Stephen Nicolitsas Daphne and Patter-son Malcolm ldquoDoes Doing Badly EncourageManagement Innovationrdquo Oxford Bulletin ofEconomics and Statistics February 200163(1) pp 5ndash28

Pesaran M Hashem and Potter Simon M ldquoAFloor and Ceiling Model of US Outputrdquo

Journal of Economic Dynamics and ControlMay 1997 21(4 ndash5) pp 661ndash96

Ramey Gary and Ramey Valerie A ldquoCross-Country Evidence on the Link Between Vol-atility and Growthrdquo American EconomicReview December 1995 85(5) pp 1138ndash51

Sakellaris Plutarchos and Spilimbergo AntonioldquoBusiness Cycles and Investment in HumanCapital International Evidence on HigherEducationrdquo Carnegie-Rochester ConferenceSeries on Public Policy June 2000 52 pp221ndash56

Schmookler Jacob Invention and economicgrowth Cambridge MA Harvard UniversityPress 1966

Schumpeter Joseph ldquoThe Explanation of theBusiness Cyclerdquo Economica December1927 (21) pp 286ndash311

Capitalism socialism and democracyNew York Harper 1942

Segerstrom Paul S Anant TCA and Dinopou-los Elias ldquoA Schumpeterian Model of theProduct Life Cyclerdquo American Economic Re-view December 1990 80(5) pp 1077ndash91

Shleifer Andrei ldquoImplementation CyclesrdquoJournal of Political Economy December1986 94(6) pp 1163ndash90

Sichel Daniel E ldquoBusiness Cycle Asymmetry ADeeper Lookrdquo Economic Inquiry April1993 31(2) pp 224ndash36

Zarnowitz Victor ldquoHas the Business CycleBeen Abolishedrdquo Business Economics Oc-tober 1998 33(4) pp 39ndash45


number of authors (for a survey see Aghion andHowitt 1998) However such ldquoSchumpeteriancyclesrdquo rely on the arrival of major technolog-ical breakthroughs that in uence all sectorsmdashageneral purpose technology (GPT) While theGPT story may be consistent with ldquolongwavesrdquo there is little evidence to support thenotion that such economy-wide advances canexplain high-frequency business cycles An al-ternative theory of why activity in diverse sec-tors of the economy may be clustered isdeveloped by Andrei Shleifer (1986) He showsthat when imitation limits the longevity of mo-nopoly pro ts a strategic complementarityarises that could lead entrepreneurs to imple-ment innovations at the same time even if theinnovations themselves arrive uniformly throughtime The clustering of implementation resultsin a boom in labor demand which in turn gen-erates the high demand for output necessary tosupport the boom The temporary nature of theassociated monopoly pro ts induces entrepre-neurs to delay implementation until demand ismaximized so that a self-reinforcing cyclearises Shleifer interprets his theory as a formal-ization of John Maynard Keynesrsquo (1936) notionof ldquoanimal spiritsrdquo1

There are however several important limita-tions to Shleiferrsquos theory of implementationcycles Firstly since innovations arrive exog-enously long-run growth is exogenous so thetheory has no implications for the impact ofcycles on growth Secondly because of the mul-tiplicity of equilibria that arise in his model it isnot possible to obtain precise predictions evenfor the effect of growth on cycles Thirdly thetemporary nature of pro ts relies on the as-sumption of drastic but costless imitation It isnot clear how robust the results would be to aless abrupt erosion of pro ts Finally Shleiferrsquostheory depends critically on the impossibility ofstorage If they could innovators would chooseto produce when costs are low (ie before theboom) store the output and then sell it when

demand is high (ie in the boom) Such apattern of production would undermine the ex-istence of cycles

In this article we draw on the insights ofSchumpeter (creative destruction) and Shleifer(animal spirits) to develop a simple theory ofendogenous cyclical growth We show how amultisector economy in which sector-speci cproductivity improvements are made by inde-pendent pro t-seeking entrepreneurs can ex-hibit regular booms slowdowns and downturnsin economic activity as an inherent part of thelong-run growth process We establish the ex-istence of a unique cyclical growth path alongwhich the growth rate and the length and am-plitude of cycles are endogenously determinedOur theory does not rely on the arrival of GPTsnor on drastic imitation and allows for thepossibility of storage Speci cally we showthat the process of creative destruction itself caninduce endogenous clustering of implementa-tion and innovation

Creative destruction implies that the dissem-ination of knowledge caused by implementationeventually leads to improvements that limit asuccessful entrepreneurrsquos time of incumbencyAnticipating this entrants will optimally timeimplementation to ensure that their pro ts ar-rive at a time of nondepressed aggregate activityand that they maximize the length of their in-cumbency It is these effects which lead toclustering in entrepreneurial implementationand hence to an aggregate level boom If anentrepreneur implements before the boom hereveals the information underlying his produc-tivity improvement to potential rivals who mayuse this information in designing their own pro-ductivity improvements By delaying imple-mentation until the boom he delays reaping therewards but maximizes his expected reign ofincumbency During the delay entrepreneursrely on maintaining secrecy regarding the natureof the innovations that they hold2

Our cycle not only features clustering of im-plementation but also endogenous clusteringof innovation It is this feature which generates

1 The expression ldquoanimal spiritsrdquo is often associated withstochastic changes in the expectations of investors that turnout to be self-ful lling In the cyclical equilibrium that westudy however the behavior of entrepreneurs may have theappearance of being fueled by animal spirits but in factexpectations are deterministic

2 As Wesley M Cohen et al (2000) document rms doindeed view secrecy as the best form of protectionmdashpatenting is a less desired means of protecting knowledge











I The Model

A Assumptions


(1) U~t 5 t

`

e2r ~s 2 tc~s1 2 s

1 2 sds


(2) t

`

e2R~t 2 R~tc~t dt

B~t 1 t

`

e2R~t 2 R~tw~t dt



(3) y~t 5 expX 0

1

ln k i ~t diD























(6) lims `

e2R~sB~s 5 0


(7) k id~t 5

y~t

p i ~t


(8) p i ~t 5w~t

e2gA i ~t


(9) pi ~t 5 ~1 2 e2g y~t


(10) 0

1

l i ~t di 1 X~t 5 1


(11) w~t~1 2 X~t 5 e2gy~t





I~t w~t xi ~t $ 0



(13) V iI~t $ V i

D~t t [ i




D~t

t i


(15)csbquo ~t

c~t5

r~t 2 r

s


PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1


(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g










(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES




















































I The Model

A Assumptions


(1) U~t 5 t

`

e2r ~s 2 tc~s1 2 s

1 2 sds


(2) t

`

e2R~t 2 R~tc~t dt

B~t 1 t

`

e2R~t 2 R~tw~t dt



(3) y~t 5 expX 0

1

ln k i ~t diD























(6) lims `

e2R~sB~s 5 0


(7) k id~t 5

y~t

p i ~t


(8) p i ~t 5w~t

e2gA i ~t


(9) pi ~t 5 ~1 2 e2g y~t


(10) 0

1

l i ~t di 1 X~t 5 1


(11) w~t~1 2 X~t 5 e2gy~t





I~t w~t xi ~t $ 0



(13) V iI~t $ V i

D~t t [ i




D~t

t i


(15)csbquo ~t

c~t5

r~t 2 r

s


PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1


(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g










(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES












































I The Model

A Assumptions


(1) U~t 5 t

`

e2r ~s 2 tc~s1 2 s

1 2 sds


(2) t

`

e2R~t 2 R~tc~t dt

B~t 1 t

`

e2R~t 2 R~tw~t dt



(3) y~t 5 expX 0

1

ln k i ~t diD























(6) lims `

e2R~sB~s 5 0


(7) k id~t 5

y~t

p i ~t


(8) p i ~t 5w~t

e2gA i ~t


(9) pi ~t 5 ~1 2 e2g y~t


(10) 0

1

l i ~t di 1 X~t 5 1


(11) w~t~1 2 X~t 5 e2gy~t





I~t w~t xi ~t $ 0



(13) V iI~t $ V i

D~t t [ i




D~t

t i


(15)csbquo ~t

c~t5

r~t 2 r

s


PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1


(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g










(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES


























































(6) lims `

e2R~sB~s 5 0


(7) k id~t 5

y~t

p i ~t


(8) p i ~t 5w~t

e2gA i ~t


(9) pi ~t 5 ~1 2 e2g y~t


(10) 0

1

l i ~t di 1 X~t 5 1


(11) w~t~1 2 X~t 5 e2gy~t





I~t w~t xi ~t $ 0



(13) V iI~t $ V i

D~t t [ i




D~t

t i


(15)csbquo ~t

c~t5

r~t 2 r

s


PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1


(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g










(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES













































I~t w~t xi ~t $ 0



(13) V iI~t $ V i

D~t t [ i




D~t

t i


(15)csbquo ~t

c~t5

r~t 2 r

s


PROPOSITION 1 If

(16) ~1 2 e2gg~1 2 s r

d eg 2 1


(17) ga 5d~1 2 e2g 2 re2gg

1 2 g~1 2 se2g










(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES















































(18) s lnc0 ~Tv

c0 ~Tv 2 1

5 R~Tv 2 R~Tv 2 1 2 r~Tv 2 Tv 2 1



(19)1

Tv 2 Tv 2 1ln

c0 ~Tv

c0 ~Tv 2 1

r

1 2 s


(20)

b~t 5 R~Tv 2 R~t

5 R~Tv 2 R~Tv 2 1 2 Tv 2 1

t

r~s ds

A Entrepreneurship


(21) V iI~t 5

t

Tv 1 1


1P i ~Tv 1 1

Pi ~te2b~tV0i

I ~Tv 1 1











I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES

















































I ~Tv




E

Tv





(24)

w~t 5 e2gexpX 0

1



(25) G v 5 0

1








(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES














































(26) g~t 5wsbquo ~t

w~t5 0


(27) r~t 5 sg~t 1 r 5 r


(28) c~t 5 y~t




E) 5 0




VD~t5

wsbquo ~t

w~t5 0


(30) g~t 5r~t 2 r

s5 2

r

s



(31) X~t 5 1 2 e2~r s t 2 TvE


E Tv) is given by

(32) P~t 5 expX 2Tv

E

t

dx~t dtD


(33) x~t 5X~t

P~t










E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES


















































E

rs D











(37) dV0I ~Tv wv 1 1



V I~Tv D




wvD 5 Gv







(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES













































(40) b~Tv $ Gv





(41) Gv 5rDv

E

1 2 s



(42) X 1 2r


rs





E





EV0I ~Tv 5 w v


AND BOOM SIZE




(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES













































(44) V0I ~Tv 1 1 5 eGV 0

I ~Tv




(45) D 5 DE 11

rln1 1 mDE

where

(46) m 5X r

dg~1 2 s2 ~1 2 e2gD

X 1 2 e2g

r2

e2g

d D 0

E Existence




(E1) V0I ~Tv 2 1 V0

D~Tv 2 1






(E4) P~T 0


PROPOSITION 5 If

(47) max~1 2 e2gg~1 2 s g~1 2 s 2 s

r

d g~1 2 s 2 s ~1 2 e2~1 2 ssg









F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES
















































F Stability










(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES
















































(48) g c 5G

D















(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES













































(49) 51D

0

D 2 DE

zG 2 g ctz dt

11D

D 2DE

D

G 2r












(percent)Volatility

Cyclelength

Dd g r s

20 0120 0025 025 267 0194 3824 326 0186 2618 238 0200 49

0115 244 0190 440125 291 0201 34

0022 270 0189 330028 264 0199 44

020 268 0202 27027 266 0191 44


(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES











































(50) GDP 5 c~t 1 I~t 5 y~t 1 w v X~t


5 ~1 2 e 2 g y~t 1 wv




(51) ~t 5 ~1 2 P~tVT~t 1 VD~t

1 P~tV I~t


(52) VT~t 5 V I~t 2P~Tv

P~tVD~t


(53) ~t 5 V I~t 1 ~1 2 P~t

3 1 2P~Tv

P~tVD~t





































REFERENCES









































































REFERENCES































































REFERENCES




















































REFERENCES












































REFERENCES































































Documents

Animal Spirits Through Creative Destructionsocsci.uci.edu/~wbranch/Francois_Lloyd-ellis_2003.pdf · 2005. 10. 25. · Animal Spirits Through Creative Destruction ByPATRICKFRANCOISANDHUWLLOYD-ELLIS*