Growth Cycles and Market Crashes

Transcription

1 Federal Reerve Bank of Minneapoli Reearch Department Staff Report 279 September 2000 Growth Cycle and Market Crahe Michele Boldrin* Federal Reerve Bank of Minneapoli and Univerity of Minneota David K. Levine* Univerity of California, Lo Angele ABSTRACT Market boom are often followed by dramatic fall. To explain thi require an aymmetry in the underlying hock. A traightforward model of technological progre generate aymmetrie that are alo the ource of growth cycle. Auming a repreentative conumer, we how that the tock market generally rie, punctuated by occaional dramatic fall. With high rik averion, bad new caue dramatic increae in price. Bad new doe not correpond to a contraction of exiting production poibilitie, but to a lowdown in their rate of expanion. Thi economy provide a model of endogenou growth cycle in which recoverie and receion are dictated by the adoption of innovation. *We are grateful to Fundacion Marc Rich, DGICYT PB97-009, NSF grant SBR and the UCLA Academic Senate for upport. The paper benefited greatly from comment by two anonymou referee and from converation with Boyan Jovanovic, Joe Wynne and Bill Zame. Joe Wynne carried out the imulation. The view expreed herein are thoe of the author and not necearily thoe of the Federal Reerve Bank of Minneapoli or the Federal Reerve Sytem.

2 . Introduction Whenever there i a dramatic fall in the tock market, public dicuion center on the idea that invetor pychology or irrationality i the root caue. Cloely aociated with thi view i the idea that tock market crahe (and the boom that precede them) are a bad thing. Even within profeional circle, the fact that the tock market rie relatively moothly, but fall abruptly i ued a evidence for thi point of view. Indeed, to explain the aymmetry in rie and fall, a fundamentalit view of the tock market require an aymmetry either in the underlying technology hock that drive fundamental or in the information-proceing mechanim that characterize the market. The firt ource of aymmetry ha been recently tudied by Zeira [8]; the econd ource of aymmetry ha been tudied, for example, in a recent paper by Lee [3]. In thi paper we conider the firt cae, that i, an aymmetry in the underlying proce of technological improvement. On the face of it, uch an aymmetry eem implauible. Our goal i to argue, on the contrary, that a very natural model of technological progre under uncertainty lead to exactly the type of aymmetry required to explain gradual rie in the value of the capital tock, punctuated by harp decline, provided that trader are not terribly rik avere. Indeed our contention i that thi ame model can explain both aymmetric growth cycle with long recoverie and hort and harp receion, and aymmetric movement of tock market price. Thi i accomplihed without the need for the unpalatable aumption of technological regre. The evidence preented in Craft [5], David [6], Horntein and Kruell [], among other, ugget that large receion may be triggered by technological innovation and the adoption of new machinery. Greenwood and Jovanovic [8] and [9] make a related point. They argue that when a technology play out, the tock market decline. Thi may occur becaue incumbent firm reit the introduction of the new technology, or it may be becaue of the need to train labor a in Greenwood and Yorukoglu [0]. After a period of time, the new technology i uccefully introduced and the tock market rebound. They focu on the IT revolution a a cae tudy of the replacement of one technology (main frame computer) by another (PC). Unlike in our argument, however, the market doe not recover gradually, but rather uddenly, a the new technology come on line all at once.

3 2 The idea that a witch-over between the old and new technology lead to an economic lowdown can be found in many other place; Atkeon and Kehoe [2] i one example. Another i Zeira [8], which i cloely connected in pirit and baic intuition to thi paper. Like u, Zeira tree the role played by the arrival of information on tock market movement and the intrinic aymmetry between the good new, correponding to a continuation of the growth proce, and the bad new, correponding to a udden halt of the growth proce. Zeira, however, concentrate on the learning proce induced by the arrival of new information, while we focu on the endogenou growth and technological innovation ide of the problem. We hould alo note that we tudy a one input (capital) model of production, a for example, in Luca [4] or Mehra and Precott [6]. A a reult, our model doe not addre the controverial iue of correlation between labor input and technology hock, dicued, for example, in Gali [7]. Our baic model i that of an economy growing due to improving technology introduced in Boldrin and Levine [4]. In thi model, activity analyi i ued to model the exiting tate of knowledge a a et of currently available activitie. Thi et of available activitie grow over time due to technological progre. In general, to make ue of new activitie, it i neceary to introduce new kind of capital. In thi etting, it i natural to ditinguih between reearch and development the proce of introducing new technologie and the proce of improving old one. We model thi by auming that there are many type of capital, each one correponding to a different underlying technology. In addition, each technology (type of capital) i available in many generation. The mot rapid form of technological improvement i to upgrade exiting capital to a newer generation. Treating reearch a extenive and development a intenive, Jovanovic and Rob [2] have derived exactly thi reult in a fairly general micro-model of endogenou technological proce. Thi ditinction i alo conitent with evidence from the applied indutrial organization literature. Thi literature ugget that new technologie require a long time to be adopted on a large cale and go through prolonged period of productivity enhancement before improvement opportunitie die out. See, for example, Manfield [5].

4 3 In our model each type of capital (technology) i limited in how many generation of improvement can be utained: no matter how much effort we put into it, an improved bicycle never turn into an automobile. Neverthele the tage of improvement at which a capital tock i played out and can be improved no further i not known in advance. When a technology i played out, however, it i poible to introduce a new type of capital, but there i necearily a time delay before the new capital can compete on equal term with the old type of capital. We tudy an activity analyi economy with contant return to cale and zero profit. At each moment of time, capital i a fixed factor and i priced according to the expected dicounted value of future rent, which are in the form of conumption produced directly or indirectly from that capital. In equilibrium, thi price, computed from marginal rate of tranformation, i equal to the price computed from marginal rate of ubtitution, uch a thoe ued in Luca [4]. In thi economy the baic technology hock i the dicovery that an exiting type of capital i played out. What impact doe thi have on the value of the exiting capital tock? The playing out of an exiting type of capital doe not reduce the currently available activitie or capital in any way, but it doe make future production poibilitie le attractive than they would have been if the technology had not played out. The impact thi ha on the market value of the capital tock depend (in our cae of CES preference) on the coefficient of relative rik averion. If the coefficient of relative rik averion i poitive, then the value of capital increae; if it i negative, it decreae. Thi i due to the combined impact of the bad new on future interet rate and on future conumption flow. In the highly rik avere cae, interet rate drop ufficiently that even though the value of future conumption to which current capital i a claim goe down, the preent value actually goe up. Thi, maybe urpriing, feature of aet price i not pecial to our model. It applie to all conumption-baed aet pricing model which adopt a time-eparable, CES utility function to value uncertain conumption tream. For example, in imulation with the Mehra-Precott [6] model, we find that with the coefficient of relative rik averion greater than one, feeding oberved conumption hock into the model reult in tock price movement of an empirically relevant magnitude, but the wrong ign. Thi underlying mechanim alo mean that when

5 4 conumption growth hock are poitively correlated, the riky aet i a good hedge againt rik, o that with high rik averion the rik premium i actually negative. Thi wa firt pointed out in Boldrin, Chritiano and Fiher [3]. Given our pecification for preference, the cae of low rik averion i therefore the empirically intereting one, becaue it correpond to the tock market dropping on bad new. We explore in ome detail the time path of capital, conumption and price in the cae in which technology hock are relatively infrequent. Ordinarily there i no bad new, and the value of capital gradually increae over time due to the overall growth in conumption. Thi i the period of tock market boom. However, following a bad hock, the value of capital fall abruptly. At thi time, a new type of capital i introduced and ued to produce additional generation of the new type of capital, until eventually they are ufficiently productive to be ued in the production of conumption. At thi time, the old type of capital i retired, and the economy reume it previou growth. Of particular interet i the tranition. Although thi period correpond to one of technological innovation in the ene that a new kind of capital i being produced for the firt time, it i alo a receion. In particular during the tranition following the negative hock, conumption and GNP may rie or decline, but they rie le rapidly than in good time. During the receion, net invetment tall and a replacement proce from old to new machine take place. The tock of capital grow at the ame low rate that conumption doe. During thi receion, the tock market gradually recover from it initial fall. We alo examine the quantitative implication of the theory, howing that in a realitic range of parameter value, it i poible to generate fall in the tock market on the order of 0-20%, together with plauible buine cycle. Key to thi point of view i the idea that neither the tock market boom, the tock market crah, nor the receion that follow the crah are bad. Here they are part of a firt bet olution to the problem of maximizing the preent value of utility from conumption. Of coure, the negative hock i a bad thing in the ene that it would be better if it were poible for technology to continue improving. However, there i nothing in the working of the tock market or in the behavior of conumption after the crah that could be improved upon.

6 5 At the heart of our reult i the underlying idea that technology hock are properly thought of a aymmetric. Take, for example, Moore Law, the law that ay that the peed of microproceor double every 8 month. From baic phyical principle, we are confident that thi improvement will not continue forever. However, it i difficult to predict with certainty when it will come to a top. The dicovery that Moore law no longer hold, when it happen, may occur quite abruptly. Conider, on the other hand, the dicovery of a new technology, however, abrupt: there i necearily a delay while the new capital i built, the new technology deployed, and the practical experience (development) needed to ue it effectively take place. It i thi baic aymmetry that we capture here. Thi aymmetry i epecially important when we conider technological change in a broader perpective. For, at the aggregate level, technology i not merely the knowledge of how to build thing. It i alo the intitution and arrangement for trade, and capital include the knowledge of how bet to make ue of thoe intitution. So, for example, in a developing country, change in legal arrangement uch a trade policy or improvement in financial market can be viewed a the introduction of a new technology. However, it take ome time for producer and trader to learn how bet to make ue of thee new intitution. In our model thi correpond to upgrading the current type of capital. However, the benefit, for example, of trade liberalization, are not unlimited; in our model, it i the dicovery that benefit of liberalization have been exhauted that trigger a receion and a fall in the tock market. The ret of the paper i organized a follow. The next ection look at the behavior of the S&P 500 index over the twentieth century and conclude that the empirical evidence reject the hypothei that tock market movement are driven by a ymmetric Markov proce. Section 3 introduce our formal model and how it equilibria are efficient. Section 4 characterize the dynamic propertie of uch equilibria, howing, in particular, the differential impact on the value of the capital tock of bad and good new. We preent both qualitative and quantitative evidence of thi fact by imulating the growth cycle generated by the model. Section 5 conclude.

7 6 2. I There An Aymmetry Between Rie and Fall? The firt iue i whether there i evidence of the conventional widom of aymmetry in rie and fall of the tock market. The purpoe of the imple data analyi we are going to preent next i not to tet our model, although detailed teting would be valuable in future work, but rather to how that aymmetry i potentially a real puzzle that need to be explained. We hould point out that our explanation of aymmetric hock or repone to hock i not the only one ee, for example, Zeira [8] and Lee [3] for intereting model of how informational dynamic can lead to tock market overhooting and correponding aymmetrie. In the Appendix, we reproduce from Shiller [7] the real Standard and Poor Index annually from We alo calculate the logarithmic difference from one year to the next a a deviation from the mean. The mean logarithmic growth rate during the period wa.0%. The common widom we wih to verify i that increae are maller and more peritent than decreae. In the table below we compute the length of run of deviation above (+) and below (-) the mean growth. Run Thi provide ome preliminary evidence of aymmetry: run of poitive deviation are longer than thoe with negative deviation, and there are more poitive than negative deviation. The fact that there are more poitive than negative deviation i alo reflected in the fact that poitive deviation are on average maller than negative

8 7 deviation: the average poitive deviation i 2.3%, while the average negative deviation i -6.0%. Thi difference of 3.7% i reaonably large in economic term. If hock to growth are ymmetric around the mean and i.i.d., how likely i it that thee difference are due to ampling error? A imple tet i to fit a two tate Markov model in which the probability of the next ign depend on the previou ign. The maximum likelihood etimate are derived by computing the frequency of + eparately, conditional on the previou ign. If the previou ign wa +, the probability of another + i 53%; if the previou ign wa -, the probability of a + i 6%. We can alo compute tandard error for each of thee coefficient, ubtract 50% and normalize: conditional on + the normalized value i 4%; conditional on - the normalized value i 4%. The probability that the firt value i 4% or more (uing the normal approximation to the binomial) i 34%; the probability that the econd value i 4% or more i 8%. Since the two value are independent of one another under the i.i.d. aumption, the probability that both event occur i jut the product of the two probabilitie: 2.7%. So if the data are truly ymmetric and i.i.d. there i only a 2.7% chance of the oberved dicrepancy. 3. The Model 3.. The State Space We ue a tandard infinite horizon event tree model of time and uncertainty. Date are denoted by t = 02,,,K. At each date t there are poible a countable number of tate h t ³ I. A tate hitory = ( h0, h, K, ht ) i a finite hitory of tate. The (countable) et of all tate hitorie i denoted by S. For a given tate hitory = ( h0, h, K, ht ), we let t ()= tdenote the terminal time and η t the terminal tate. We alo order tate hitorie o that ~ if t ( ~ ) t ( ) and h ~ = h for t ˆ t ( ). We write - for the immediate t t predeceor of and + for the et of immediate follower of. We let p denote the probability of tate hitory, o that Ê π =. t :( ) = t 3.2. Houehold There i a ingle repreentative houehold, who following the tate hitory t conume c ³ +. Thi houehold receive total lifetime utility of Ê d ( ) uc ( ) p ³ S, where 0 δ < i a ubjective dicount factor. Rather more trongly, we will aume that

9 8 the period utility function i mooth, concave, bounded below, and, at leat for level of conumption above a ubitence level c c, ha the CES (or contant relative rik -q averion) form uc () =-(/ q) c, where θ + 0 i the coefficient of relative rik averion. Note that for q 0 the CES function i not bounded below, which i why we require thi functional form only for level of conumption above ubitence. A we are intereted in the theory of growth, not the theory of ubitence, the behavior of u for mall quantitie of conumption i not terribly intereting. In particular, in the equilibria we conider, conumption will be uniformly bounded away from zero, and we will aume that thi bound exceed the ubitence level. Thi mean that we can aume that utility i bounded below, which i technically ueful, while neverthele having the convenience of the CES functional form Production Poibilitie There are a countable number of type of capital w = 0,,K. Each type of capital i potentially available in a countable number of generation i = K, -0,,, K. We denote by k ωi the amount of type w capital of generation i. We refer to a type/generation pair a a kind of capital. Production take place through linear activitie, uing capital a input. We begin by decribing all potential activitie, although not all of thee activitie are available at any moment of time, and indeed, ome activitie may never be available. Each activity take a input one unit of capital of a pecific kind and produce a ingle output in the following period. Type w capital of generation i may be ued a the input into 4 ditinct activitie, each of which i characterized by the output obtained: ) γ i unit of the conumption characteritic, g > 2) b > unit of the ame kind of capital: type w capital of generation i 3) ρβ : > ρ> unit of the next generation of capital: type w capital of generation i + 4) unit of type w + capital of generation i L, L > 0. We aume that gr > b, o that more conumption can be obtained by moving to the next generation of capital than by remaining with the current generation. Notice that we do not preume that the increaed conumption from higher generation of capital take place

10 9 through the phyical production of increaed unit of ome commodity. The increaed amount of characteritic may take place through the production of more advanced commoditie which contain more of the characteritic that i actually conumed. However, to keep notation to a minimum, we will not explicitly introduce the ditinction between the amount of commoditie produced and purchaed and the amount of characteritic conumed. At each moment of time, only a finite number of thi countable collection of activitie are actually available. The et of available activitie i determined by the tate, which we take to be a lit of thoe activitie that are available. We now decribe both the et of poible tate and the tate tranition proce that determine p, the probabilitie of tate hitorie. A tate h ³I conit of a latet type of capital w( h), and for each type of capital w ˆ w( h) a latet generation of capital i ( ωη,, ) plu a number 0 ˆ m( h ) (thi number i ued to track the time ince the latet type of capital wa introduced). The available activitie are thee: if ω ω( η), i i ( ω, η) then all four activitie making ue of k ωi are available, with the following exception: ) if ω < ω( η), i = i ( ω, η) then activity 3 producing the next generation of capital i not available 2) if ω = ω( η), i i ( ω, η) then activity 4 producing the next type of capital i not available. On the other hand, if w > w( h) or i> i( ω, η ) then no activitie making ue of k ωi poible. The interpretation of the tate i thi: except for the latet type of capital, improvement on other type of capital have talled, and it i no longer poible to produce new generation. New type of capital can be introduced, from the very latet type of capital and it latet generation, only when the opportunity for producing new generation of the latet type of capital ha vanihed. The latter aumption deerve ome comment. It amount to aying that experimentation with future technologie cannot take place while the current technology i being ued and improved. If thi were are

11 0 not the cae, a benevolent planner could take advantage of the contant return to cale nature of our activitie and run the technology ω + at an infiniteimal cale to acquire valuable information about the generation at which it will play out. Thi information i valuable becaue it eliminate the uncertainty about future conumption growth rate. A a reult, it alo modifie the current market evaluation of the exiting tock of capital and make tock market price a monotone determinitic proce. Thi aumption i therefore needed to retain a minimum of uncertainty about future technological evolution. The tate follow a Markov proce. Fix an initial tate h. We define the next generation of capital a the tate η g in which ωη ( g ) = ωη ( ), i( ωη, ) = i( ωη, ) for w < w( h), i( ωη ( ), η) = i( ωη ( ), η) +, and m( η ) = m( η) +. We define the next type of g capital a the tate η b in which ωη ( b ) = ωη ( ) +, i( ωη, ) = i( ωη, ) for w < w( h), i( ωη ( ), η) = i( ωη ( ), η) +, i( ωη ( ), η) = i( ωη ( ), η) L, and m( η b ) = 0. Starting at b b b h, only η, η can be reached with poitive probability. If m( h) ˆ M then, with g b probability, η g i reached. On the other hand, if m( h ) > M with probability - p the tate move to η g and with probability p the tate move to η b. Thi mean that for M period following a tranition from tate h to tate η b there i no uncertainty about the ability to build new generation of the new type of capital. Since we view p a relatively mall, thi aumption doe have much economic import. However, by carefully chooing M we can greatly implify computation a we indicate below. We will frequently refer to that tranition from h to η b a a negative or bad hock to technology. Similarly, we will call η b, which i characterized by m( η ) = 0, the negative or bad tate. But notice that the technological poibilitie at η b are a trict uperet of thoe available at h. All activitie that were available at h are till available at η b. In particular, we do not aume that a negative hock imply caue the current capital tock to retrogre or become le productive. Indeed, we may wonder why it would ever be deirable to witch to a new kind of capital, which i, after all, eentially identical to the current kind of capital. The anwer i that depite the fact that the new kind of capital i actually inferior to the old kind of capital, development of new generation of the old kind of capital i impoible. Switching to the new kind of capital make it at leat poible to introduce higher generation of capital. g b g b

12 3.4. Production Plan At any moment of time there are in principle a countable number of different type of capital. Let k ³ X be the vector of different kind of capital when the tate hitory i. The component of k are k ω i. An activity may be thought of a a triple ( xyc,,, ) where xy, ³ Xare the capital input in period t and output in period t +, and c ³ + i the amount of characteritic output in period t +. Let A be the et of all potential activitie, and recall that h are thoe actually available when the tate i h. An activity vector l i a map l : A +. Feaible activitie are given by et A A, with A 0 and for ~ that A~ A. A production plan conit of a map from tate hitorie S to capital vector k, conumption of characteritic c (for t ()> 0) and activity level λ A. We ay that the production plan i feaible with repect to the initial condition k 0,h 0 if for every tate hitory with p > 0 Ê Ê a³ η a³ η - - λ λ - - ( aya ) ( ) k ( aca ) ( ) c k Ê λ - a³ η - - ( a) x( a). We call an activity viable for the tate hitory and initial condition k 0 if there exit a ocially feaible production plan uch that l ( a)> 0. We denote the et of viable activitie for tate hitory by A ( k, ) 0 h 0. We aume that the initial et of activitie A ( k, η ) i non-empty for k 0 0 and that for ~, A~( k, η ) A ( k, η ). Note that A ( k, ) 0 h 0 may be a proper ubet of the et of feaible activitie in h. Similarly, we call a kind of capital viable for the tate hitory and initial condition k 0,h 0 if there exit a ocially feaible production plan uch that there i a poitive amount of it available at. Finally, we ay that a production plan olve the ocial planner problem for initial t( ) condition k 0,h 0 if they olve max δ u( λ ( ) ( )) π S a η a c a ubject to ocial feaibility Price Let q be the preent value price of different kind of capital when the tate hitory i, and let p be the correponding preent value price of the characteritic. In a competitive equilibrium, thee price hould atify two condition: they hould yield

13 2 zero profit and upport the preference. Specifically, we ay that price yield nonnegative profit for the initial condition k 0,h 0 if Ê pca σ ( ) + qya σ ( ) -qxa ( ) ˆ 0, " a³ A ( k0, η0 ),", ³ + σ and if thi hold with equality, we ay that the activity a yield zero-profit. We ay that the price upport the conumption plan $c if $c i a olution to t( ) arg max δ uc ( ) π ubject to Ê S pc Ê ˆ pc$. ³ S ³ S The firt order condition for the conumer are atified if $ m > 0 p µδ π u ( c ) with equality unle c = 0. t( ) Here µ i the marginal utility of conumption. Note that thi firt order condition i valid for all value of c > 0, but that for c < c the period utility function u doe not have the CES form, a we dicued above. The firt order condition i mot ueful in the cae of an interior olution where c, c, in which cae - > 0 δ π π u ( c) = u ( c ) p p Finally, we ay that price and capital k atify the tranverality condition if = 0. lim qk T t ( ) = T Efficiency We how how to characterize the olution to the planner problem by a relatively tandard price decentralization cheme, depite the fact that there are infinitely many kind of capital, and thoe actually available grow over time. Propoition 3.: Suppoe that l $, $ k, c $ are ocially feaible for the initial condition t( ),h and that δ ( $ ) π <. Then the following three condition are equivalent k 0 0 uc S () l $, $ k, c $ olve the ocial planner problem for initial condition k 0,h 0

14 3 (2) there exit price q$, p $ that atify the non-negative profit condition, yielding zero profit for all activitie for which $ λ ( a ) > 0 and upport the conumption plan $c with Ê pc $$ <Š ³ S (3) there exit price q$, p $ that atify the non-negative profit condition, yielding zero profit for all activitie for which $ λ ( a ) > 0, the firt order condition and the tranverality condition. Proof: Firt we oberve that if the profit condition hold, then the tranverality condition i equivalent to the conumption plan having finite value. Indeed, from the zero profit condition qk $ T T 0 0 qk $ = q k qk pc $ $ = $. t ( ) = T+ t= 0 t ( ) =+ t t= 0 t ( ) = t+ Next we conider the T-truncated utility economy with utility function δ uc ( ) π + qk. t( ) t ( ) T t ( ) = T+ We oberve that when we eliminate tate with zero probability and non-viable kind of capital, thi i an ordinary finite economy. By tandard method, a production plan l $, $ k, c $ olve the ocial planner problem if and only if it truncation olve thi truncated problem. The equivalence of (2) and (3) to zero now follow immediately from the correponding fact for the finite horizon cae. ã 4. Analyi of the Model Let u begin by defining the mot advanced kind of capital in tate h to be capital of type w( h) and generation i( ω( η), η ) if m( h ) > 0, and capital of type w( h)- and generation i( ω( η), η) if m( h ) = 0. Fix the initial condition h 0. We will now analyze the model under three aumption: firt, that the parameter rg,, d atify the hypothei of Propoition 3. o that an optimum exit; econd, that M i relatively large compared to L; and third, that the initial tock of capital conit olely of the mot advanced kind of capital in tate h 0.

15 4 4.. Baic Reult It will be helpful to begin with a very imple problem: uppoe that capital of a particular type i talled. More pecifically, uppoe that there i a ingle unit of generation 0 capital of thi particular type, o that we are contemplating producing ome output of the characteritic t > period from now on uing one of two method. Firt, we can imply build more of the exiting type and generation of capital until period t -, yielding b t - unit of characteritic. Second, we can build a unit of the next type of capital of generation -L, then ue that to build generation - L + and o forth until t-2 t-2-l period t -. Thi yield ρ γ unit of characteritic. If we define ( L + )logγ + logρ m* = + log ρ+ logγ -log β and make the generic aumption that thi i not an integer, then for t prefer to ue the old kind of capital, while for t of capital. < m* we would > m* we would like to ue the new kind Propoition 4.: If M > m* - and the initial tock of capital conit olely of the mot advanced kind of capital in tate h 0, then with probability in every ubequent tate h there will be a poitive amount of the mot advanced kind of capital available. Moreover, there will be a poitive amount of at mot one other kind of capital, and thi will be of the type immediately preceding the mot advanced kind. Moreover, the negative tate m( h ) = 0 can occur only when there i a ingle type of capital. Remark: A we hall ee below, the equilibrium witche between uing a ingle type of capital to produce conumption and new capital. Then, when that kind of capital i played out, the older kind of capital i ued to produce the conumption good and reproduce itelf, while the new kind of capital i ued olely to produce new generation of itelf. Finally, when ufficiently advanced new generation of the new kind of capital are produced, the old kind of capital i abandoned completely, and the cycle begin anew. Of particular interet i the tranition when two kind of capital are in ue. We explicitly olve for price, output and o forth below. However, we hould give the intuitive idea of what happen during the tranition.

16 5 During the tranition, the new kind of capital i relatively unproductive at producing current conumption. However, it i quite productive at producing (indirectly) conumption a number of period into the future. At equilibrium price, firm are indifferent between uing activitie involving either producing conumption from the old kind of capital or activitie producing new generation of the new kind of capital. The price at which thoe new generation can be old reflect the fact that ultimately they will be ued to produce conumption good. If we think of the new generation of capital a being retained within the firm rather than old in the market, the firm producing the new generation of capital will have negative cah flow. Since the new kind of capital i not being ued to produce conumption, the activity of producing new generation can be viewed a development learning to produce more advanced generation of capital. The output from thi D part of R&D i more advanced capital, which i either old or retained within the firm. The activitie of Internet tartup firm, which have high capitalized value, yet have negative cah flow, can be interpreted a engaging in thi development activity: IPO correpond to elling the relatively advanced generation of capital that ha been produced over a number of year within the firm. Proof: In the good tate if there i one kind of capital, then, ince rg > b, the next generation hould be the only kind produced. On the other hand if the bad tate occur at time t, then there i no uncertainty about the technology available through period t + M. Oberve that if we produce conumption for period t + t by producing the old kind of capital until period t ˆ t + T < t + t- then output of conumption at t + t i T t-2-t t-2-t-l proportional to β ρ γ, which, ince rg > b, i trictly decreaing in T. Thi implie that it i alway bet to produce the new capital right away. Now conider t the leat integer greater than m*. Since M > m* - there i no uncertainty about the availability of generation t- L of the new type of technology to produce conumption for period t and all ubequent period. It follow that no capital of the old kind hould be produced beginning in period t - (or of coure any later period). Alo ince M > t-, it i impoible to produce any capital of a higher kind than the newet kind in period t -. Conequently in period t + the only type of capital i the mot advanced kind in that tate, giving the deired reult. ã

17 6 Suppoe now that only the mot advanced kind of capital i available, and denote z( h ) the mot advanced kind of capital in tate h by z( h ). Let V( h, k ) be the lifetime value of future conumption beginning in thi tate with k z h ( ) unit of the mot advanced capital. Notice that having k i unit of generation i capital will imply reult in γ i k i time a much conumption in every future time and tate a with a ingle unit of generation 0 capital. If we take V m to be the value of a ingle unit of generation 0 capital (of any type) when m( h ) = m, we can write for m > 0 and for m = 0 z( ) i( ( ), ) z( ) ( η, k η ) = γ ω η η k η V θ 4 9, m( η ) z i z ( η, k ( η ) ) γ ( ω( η ) =, η ) k ( η ) V θ Notice alo for m> M that V = V. m M Notice that capital i produced for period t before the tate at period t i known, but i ued in period t after the tate i determined. Conequently, we can compare the value of the capital tock (relative to the price of conumption) at time t immediately before and immediately after the tate i realized. Our goal i to how that good new ha a marginal impact on the value of capital, but bad new caue it to change abruptly. Propoition 4.2: Suppoe M > m* - and the initial tock of capital conit olely of the mot advanced kind of capital in tate h 0. When θ < 0, new of a negative hock m( h ) = 0 caue the value of the capital tock to fall immediately. If q > 0 the value of the capital tock rie immediately. Preciely, the ratio of the value of the capital tock after the announcement to that before i given by V0 p + ( -p). V M Remark: The ame method of proof how that good new caue the tock market to change by

18 7 ( - p) + p V. VM From an economic perpective, the intereting cae i when time interval are relatively hort, o that p i mall. Notice, however, that it i not the cae that horter time interval lead V 0 / V M : the need to witch to a new technology ha an appreciable utility cot regardle of how time i meaured. Thi mean that for hort time interval, to a good approximation, good new ha little effect on the value of the capital tock, while bad new caue it to change by V 0 / V M. Proof: Since the capital tock and current conumption are both fixed, the only quetion i what happen to the price of the capital tock. If in the current period m( h ) = 0 it mut be that in the previou period m( h ) > M. In addition, by Propoition 4. in the current period there i a ingle type of capital of the mot advanced type. Suppoe that the z( amount of thi capital i k η ). If there i not a negative hock in the current period then the value will be 0 γ ω η η η θ i ( ( g), g ) z( ) 4 k 9 VM ; if there i a negative hock the value will be γ ω η η η θ i b b k ( ( ), ) z( ) 4 V 9 0. z( η) The correponding price at which capital k i traded are determined by differentiating thee value with repect to k η ) z(. Notice that i( ωη ( g), ηg) = i( ωη ( b), ηb), ince the ability of the capital to produce next period conumption i not changed by the hock. Conequently the price of capital without the negative hock i proportional to -qv M, and with the negative hock to -qv 0. Clearly V 0 < V M. If q < 0 thi mean that - qvm > -qv0, while if q > 0 thi mean that - qv <- qv, which i the deired reult. M 0 ã Propoition 4.2 how that when time interval are hort, good new ha a marginal impact on tock price, while bad new caue them to change abruptly. We reinforce thi by howing that once the good tate i ufficiently well etablihed,

19 8 meaning m( h) M, the value of the capital tock from period to period rie exponentially and, in the continuou time limit, continuouly. Propoition 4.3: Suppoe M > m* - and the initial tock of capital conit olely of the mot advanced kind of capital in tate h 0. When m( h ) M z( h) z( h) q k / p z( h-) z( h-) q k / p q - q + V q + q ( rd ) g + 0 = ( - p) + p. V Proof: Oberve that ince m( h) M, the economy i following balanced growth, and in particular the relative price of capital to conumption i not changing. So the growth in the value of capital i due entirely to the increaed ize of the capital tock: M z( h) z( h) q k / p z( h ) z( h ) q k / p k = k z( h ) - - z( h-) Without lo of generality, take k z equation i ( h- - ) =,i( ωη ( ), η ) = 0. Then the Bellman q VM = d max f u( f) + rg ( -f) ( - p) VM + pv0 The firt order condition for the optimum i while from the Bellman equation or f q q rg q f q 0 + ( ) - ( - p) + p 0 V V V M V 6 6 M = -q -q V0 =- d f + d rg( -f) ( - p) + p q V M, M V M V M -q d f =- q -q V0 -drg ( -f) 6 ( - p) + p V M. Subtituting back into the firt order condition, we find

20 which i the deired reult. 9 6 ( ) ( ) q - q + q + q 0 r - f = rd g + V - p + p, V M ã 4.2. Dynamic Analyi We now examine in greater detail the time path of conumption and capital following a negative hock. Without lo of generality we may begin at time 0 in tate h 0, auming that a negative hock ha jut occurred. We may alo aume that there i only one kind capital, that thi i type 0 generation 0, and that there i unit of thi capital, a well a c 0 unit of characteritic produced from lat period. We alo aume that current capital i the mot advanced poible given the technology, o the initial tate ha w( h0) =, i( 0, η 0) = 0, i(, η 0 ) = L and m( h 0 ) = 0. In thi analyi it i convenient to take M to be the larget integer maller than m*. Thi mean that a oon a the economy witche from two kind of capital to one, it i poible to have negative hock once again. Thi implifie computation without detracting a great deal from the economic. Suppoe that 0 < t ˆ M. Notice that during thi period, only the old kind of capital i ued to produce conumption, and o from the zero profit condition the market - dicount factor from period t to period t + i pt+ / pt = b. In the interior the houehold mut be on the margin between conuming in period t and t +, o by the firt order condition d ct c t+ q+ = From thi we can derive the growth rate of conumption from period to M. b -. ct c + t = db6 q +. During thi period the new type of capital will grow at the rate r, a it i not ued to produce conumption. We ummarize thi information in the table below.

21 20 t c t k t 00 ( t-- L) k t 0 c 0 0 c k 00 2 bd6 q + c 00 k2 k ( - L) rk ( - L) M M M M M bd6 M - c q + k M 00 ρ M ( k M L ) M + bd6 M c q + 0 ρ M M L k ( ) We can alo work out the amount of old capital required for 0 < From thi we can find k k = c + b k. t t+ t+ M - - M --t = cê b bd t = 0 M -t q + ( ) 6. t ˆ M : We alo require that period output be produced from period 0 capital ( L) () k = - c + b - -M bd6 M q + - M - + bbd q bbd 6q 6 Finally, oberve that initial capital can be ued either to produce period conumption, yielding a marginal lifetime utility of δ u ( c ), or to produce period M + type capital of generation M - L. Thi yield an equal marginal lifetime utility of δ 3 8 M L M ( L) M + -θ - - d γ ρ k VM = δu ( c ( - L) ) dk.

22 2 or -q -q- -q M M M- L M - L (2) - d q g r k ( ) V = c Next we mut calculate lifetime utility beginning in the negative tate: θ M+ + δβδ θ θ M+ M L M ( L) θ (3) V0 = θ ( c) δ + δ γ ρ k V θ δβδ θ +! 6 6 " $#. 2 7 M. Notice alo that in the good tate, the fraction of the capital tock devoted to conumption f mut be contant. Conequently one unit of capital in the good tate will produce φγ i unit of the characteritic and ( - fr ) unit of the next generation of capital. Thi lead to the Bellman equation q q VM = d max f u( f) + ( -p) gr( - f) VM + p r( -f) V0. The firt order condition determining f i or 6 6 q q q f + qgr( -p) gr( - f) V + qrp r( - f) V = M 0 (4) f = -q - q q + - q - p gr VM + qpr V0 4 ( )6 9, while the Bellman equation for thi value of f give V =- dq f q + d ( -f) q ( - p) gr q V + d ( -f) q pr q V Solving thi for V 0 we find M (5) V 0 = 6 M q -q - -q 4-d( -f) ( - p) gr 9 V M + dq f. -q -q d( - f) pr ( - L) The 5 numbered equation mut be olved for the unknown c, k, f, VM, V0. By directly ubtituting () into (2) and (3), (3) into (4) and (5), and (4) into (5) thi may be reduced to two equation in the unknown c, V M.

23 Quantitative Apect of the Theory To get a handle on the implication of the theory, it i ueful to conider the extreme cae in which bad hock are rare, o that p 0, and in which new technologie are difficult to get on line, o that L Š. Thi will reult (for given value of the other parameter) in the mot extreme tock market fall. Uing (4) and (5) above, we may explicitly olve for f and V M when p = 0 : f = d gr q + q ( ) V M =-q - d dgr q + q ( ) q +. Let g denote the growth rate of conumption (in the good tate); then g = gr( -f), and q + we may ubtitute in the expreion for - f to find gr = g / d. Thi give the value of the capital tock in term of the growth rate of conumption and the ubjective dicount factor V =- - d M q - g -q + q -q d For the cae where L i large, once the technology play out, the time it take for a new kind of capital to come on line i exceively long. So we can regard the bad tate a imilar to the good tate, but with a potential growth rate of conumption of b rather than gr. If we let g b denote the eentially fixed growth rate of conumption in thi tate, we can find V 0 by replacing g with g b in V M. Thi yield V V 0 M = δ δ θ + θ θ g θ + θ θ gb θ + q +. θ + Notice that a the logarithmic cae i approached, o q 0, that V. 0 V M / a aerted by Propoition 4.. Notice, however, that it i alo the cae that a q - (o

24 23 the utility function approache linearity) V 0 / V M. So it i the intermediate range between the logarithm and linear utility function that i of interet. q Notice that the market rate of interet i d / g + in the good tate, and imilarly it θ i δ / g + b in the bad tate. To explore more carefully, let u uppoe that g b = o that there i no growth following a negative hock. In thi cae the market rate of interet in the bad tate i imply the ubjective dicount factor. Firt take thi to be 5%, o that d = If the rate of growth of conumption in the good tate i 5%, we may calculate the variation in the market value of the tock of capital caued by the bad hock for different level of rik averion. Thi i reported in the next table, along with the interet rate in the good and bad tate. q V0 / V M 6.8% 9.9% 9.6%.4% Interet at M 9.% 8.6% 7.7% 5.9% Interet at 0 5.0% 5.0% 5.0% 5.0% Notice that the ize of the market drop in the intermediate range of q i not terribly enitive to the value of q. In thi range the market drop roughly 0% in repone to bad new. If we conider a more dratic fall in the rate of conumption growth from 0% to 0%, we find the following. q V0 / V M 3.4% 9.8% 9.7% 2.9% Interet at M 2.8% 2.0% 0.3% 6.8% Interet at 0 5.0% 5.0% 5.0% 5.0% Here, the market drop on the order of 20%.

25 24 However, in thee example, the market interet rate in the bad tate i 5%, which i high by hitorical tandard. If we reanalyze a change from 5% to 0% growth with d =.98, we find the following. q V0 / V M 7.8% 26.4% 27.8% 2.9% Interet at M 6.2% 5.8% 4.8% 4.4% Interet at 0 2.0% 2.0% 2.0% 3.0% Thi mean a nearly 30% drop in the market. The reaon that greater patience lead to a more ignificant market fall i that the bad hock ha no conequence for next period conumption, but the difference in future production poibilitie grow over time. While we cannot pretend that the preent, implified model provide a realitic account of oberved movement in tock market price, it i till worthwhile to ae the extent to which the model i roughly conitent with the hitorical evidence. To thi end, we firt conider the quality of our imple approximation and then preent the reult of a imulation of the model for parameter value that are not altogether impoible. We have not tried to calibrate our model to available U.S. data, a our framework doe not incorporate an explicit labor upply rule, nor doe it contain other ource of hock beyond the large and infrequent hock provoked by major technological change. During the majority of period in which the exiting technology i being improved, there are no other hock in our model and both quantitie and price follow a determinitic path. Real economie are affected by many other maller but more frequent hock that lead to large fluctuation in hour worked, quantitie produced and market price. Further, we are uing a very aggregate model in which only two type of capital are in ue at each moment of time. It i hard to imagine that we can approximate real market fluctuation terribly well with a model in which there i a ingle ector of production and a ingle technology which i occaionally dimied to be replaced with a

26 25 new one. A more realitic model would require everal capital ector, each with it own technology hock, and a more elaborate technology for the production of the final conumption good involving variou kind of capital a well a labor. Thi would, of coure, require an explicit analyi of the flow of reource between ector in repone to hock, a well a an explicit evaluation of the degree of ubtitutability among input in the variou ector. Thi i a very intereting tak which, neverthele, goe beyond the cope of the preent, theoretical, paper and i left to a future invetigation. To check the quality of the approximation conider the cae in which d = 095., θ =.4 and a rate of growth of conumption in the good tate of 5%. For comparion, we conider π = 0. 05, M = 2, L= 20, β = 052., ρ = 05., and γ = The growth rate + in the bad tate i ill defined, o we take g b = ( δβ) /( θ ), which i correct aymptotically. Both et of reult are hown in the table below. q approximation exact -V0 / V M 9.6% 9.0% g 5.0% 5.0% g b 0.0% 0.0% Notice the relatively mall reduction in the ize of the tock market drop, due to the fewer number of period required to get back to the level of efficiency of the previou technology. (Here L = 20, a oppoed to L = in the approximation.) Similar reult can be obtained with other et of more or le credible parameter value. In particular, we have looked at variou configuration of parameter value that are conitent with the choice of a quarter a the unit of time. In thi cae, for dicount factor in the range [. 98,. 99 ] and choice of β, γ, ρ and M yielding growth rate of conumption g [ 003.,. ] and g b [., ], we obtain drop in the tock market index that range between 3 and 9 percent over the quarter after the bad new i received. Finally we have imulated the time path of aggregate quantitie generated by our model over extended period of time to get an idea of their buine cycle propertie. The

27 26 figure below report the time path of the tock of capital and the flow of conumption generated by a imulation calibrated uing the parameter from the lat table. Over the time period conidered we oberve the introduction and conecutive replacement of three different type of capital tock. The overall pattern i clear and qualitatively acceptable. Accumulation occur lowly, while dimial i fat and, in the initial period, abrupt. Oberve that when the bad new arrive the proce of accumulation top for a while. During thi period only replacement of machine take place: the old type of capital i ued only to produce conumption, while the new kind of machine are accumulated and the um of the two tock remain contant from one period to the next. At the end of the replacement period the tock of new machine i equal to the tock of old machine when the receion began, after which net accumulation of capital reume again. The path of conumption follow along. A predicted, conumption receion tart at the time at which the old technology cannot be improved any further and continue after that for a relatively long number of period, while the new capital good i improved and accumulated.

28 27 Capital Stock dotted line i conumption Period One can ditinguih the conumption receion period, in which there i no growth, and the good period of high growth beginning when the replacement of the old technology i completed. The cyclical growth pattern of aggregate output can eaily be inferred from the eparate behavior of c t and k t by recalling that, in the practice of National Income Accounting, output i computed by normalizing relative price in a bae year. In our cae, we can, for example, ue the relative price of capital at the beginning of the receion period to compute net aggregate output a the um of the conumption flow plu the net invetment flow. The latter i zero during the tranition; therefore, output growth would be alo nil during thi period. When recovery begin, output of the invetment good become poitive and conumption growth reume; hence, aggregate output growth would increae harply and converge toward it balanced growth level until the next receion trike. If, contrary to National Income Accounting practice, one take into account the change in the relative price of capital good, output growth will be lightly poitive even during the receion period, but till much lower than during the period of high growth, and peritent ocillation in growth rate will till appear.