16. Real business cycle models, endogenous growth models and cyclical growth: a critical survey

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1 16. Real business cycle models, endogenous growh models and cyclical growh: a criical survey Davide Fiaschi and Serena Sordi INTRODUCTION Since he early conribuions o he opic, business cycles have been considered as essenially conneced wih he developmen of capialis economies. In he early 198s he issue of he relaionship beween growh and cycles was addressed wihin a marke clearing environmen by he real business cycle (RBC) lieraure, and, more recenly, in endogenous growh (EG) models. 1 In he aricles belonging o he RBC lieraure, which are based on he neoclassical model of (opimal) capial accumulaion augmened by echnology shocks, i is common o find asserions like our approach inegraes growh and business cycle heory (Kydland and Presco, 1982, p. 1345) or real business cycles heory (...) holds considerable promise for enhancing our undersanding of economic flucuaions and growh as well as heir ineracion (King, Plosser and Rebelo, 1988, p. 196). Thus, i appears ha such an inegraion of growh and business cycle heory is undersood as one of he mos, if no he mos, imporan achievemen of he analysis. The relaionship beween growh and cycles has also been ackled from a differen poin of view; some conribuions in he EG lieraure focus on he possibiliy of generaing non-linear (periodic) dynamics as he effec of inroducing endogenous growh ino an oherwise neoclassical growh model. In he whole of his variegaed lieraure on RBC and EG, one can disinguish a leas hree main differen approaches o he sudy of he ineracion beween growh and cycles: 1. Saring wih he conribuions by Kydland and Presco (1982) and Long and Plosser (1983), RBC heoriss have sudied he ineracion beween growh and cycles wihin a sochasic business cycles framework, where cycles are generaed by coninuous exogenous shocks o echnology; 36

2 Real business cycle models, endogenous growh models and cyclical growh Some EG heoriss analyse he implicaions of a sounder microfoundaion of echnological progress on he relaionship beween growh and cycles. Examples are Aghion and Sain-Paul (1998a), where cycles are generaed in a model wih a Schumpeerian flavour, and Sadler (199), where growh is generaed by a learning-by-doing process; 3. Finally, he possibiliy of muliple seady saes in EG models has given imporance o he analysis of ou-of-seady-sae dynamics (cycles) in a deerminisic framework (see Greiner and Semmler, 1996a, 1996b; and Benhabib and Perli, 1994). The aim of his chaper is o survey and compare hese conribuions, saring from he sochasic approaches 1 and 2 (Secions 16.2) and coninuing wih he deerminisic approach 3 (Secion 16.3). Secion 16.4 concludes and gives some suggesions for furher research GROWTH MODELS WITH STOCHASTIC BUSINESS CYCLES The firs aemps o explain business cycles on he basis of sochasic shocks are due o Frisch (1933) and Slusky (1937). While he laer showed how he sum of random componens generaes cycles similar o empirical flucuaions, he former presened echnical innovaions as exogenous perurbaions o he available level of echnological progress. In he same period Schumpeer (1939) idenified in he coninuous inroducion of new innovaions, and he resuling shocks o produciviy, he source of growh of a counry. In his view growh and business cycle are generaed by he same source and herefore hey mus be joinly analysed. Unlike Frisch, however, Schumpeer considered innovaions as driven by economic facors, i.e., as endogenous. 2 These conribuions are he main inspiraion of he modern heory of RBC, where he business cycle is seen as a phenomenon essenially due o shocks o he real par of he economy and long-run growh as he cumulaive sum of such shocks. The seminal conribuion by Kydland and Presco (1982) sars from he idea of analysing wihin he neoclassical framework of opimising agens he behaviour of an economy which is converging oward is long-run equilibrium bu which is coninually shocked by random disurbances. The sandard RBC model is essenially a neoclassical growh model in which exogenous echnological progress is modelled as a sochasic process (see Cooley and Presco, 1995). The aim is o simulae series whose properies are similar o hose of observed series. Their conclusion is ha he business cycle is essenially a real phenomenon and ha he neoclassical framework

3 38 The Theory of Economic Growh: a Classical Perspecive can accoun for mos of he cyclical componen. This conclusion, however, can be criicised from many poins of view. We focus on he assumpion of exogenous echnological change. Sadler (199) and Aghion and Sain-Paul (1998a) provide wo ineresing conribuions, whose findings subsanially differ from RBC resuls. The nex secion is devoed o he exposiion of a sandard RBC model, which hen will be used as a erm of comparison for he wo models of endogenous echnological progress presened in Secion A Basic Model of RBC In his secion we presen a basic RBC model, referring, in paricular, o he classical conribuion by Chrisiano and Eichenbaum (1992), in which he auhors analyse an RBC model wih governmen expendiure and endogenous labour supply. For he sake of simpliciy, in our presenaion, we ignore he governmen expendiure. The model economy considered is composed by an infiniely living represenaive agen, which maximises he discoun sum of insananeous uiliy. To keep hings simple we consider a log insananeous uiliy funcion, U = ln c + φv( l), where c is he per capia consumpion and l is leisure a period. The producion side of he economy is characerised by a compeiive marke where each firm produces homogeneous oupu according o a Cobb Douglas consan reurns o scale echnology: ( ) 1 θ θ y = Az h k (1) where y is per capia oupu, A, an index of long-run deerminisic echnological progress, z, an index of shor-run cyclical produciviy, h = L l are he worked hours, and k is per capia capial. Then i is assumed ha produciviy evolves according o he following sochasic process 3 2 where γ > is a consan drif and ε N (, σε ) z exp( ) = z 1 γ + ε (2). From he assumpion of compeiive markes i follows ha facors are paid according o heir marginal produciviy, so ha ( ) ( 1 θ ) 1 θ θ 1 r = θ Azh k δ 1 θ 1 θ θ θ w = A z h k where δ is he depreciaion rae of capial.

4 Real business cycle models, endogenous growh models and cyclical growh 39 In his economy he firs welfare heorem holds, so ha i is convenien o solve he compeiive allocaion as a social planner problem, ha is 4 { c, l } = = { φ ( ) } max W = E β ln c + V l (3) subjec o 1 θ θ ( ) ( 1 δ ) k = Azh k + c + k k 1 = k() Since z follows a sochasic pah, a closed form soluion o problem (3) does no exis in general, bu only for a paricular configuraion of he parameers (θ = δ = 1 and V(l ) = ln l ). The procedure generally used consiss in calculaing he seady-sae equilibrium (which corresponds o he locus which he economy converges o if i were no subjec o produciviy shocks) and in approximaing problem (3) around his seady sae. 5 In order o calculae he seady-sae equilibrium, i is useful o normalise each growing variable wih respec o is long-run growh rae. By so doing, we obain he following normalised variables: ˆ k c y z k =, cˆ =, yˆ =, zˆ = A z 1 1 Az Az z 1 The soluion o problem (3) is given by he following difference equaions h = q k ˆ zˆ k ˆ = q k ˆ zˆ (4) n m h h nk mk + 1 h ; + 1 k where he coefficiens q i, n i and m i for i = h, m are non-linear funcions of he original parameers of he model. The sysem of difference equaions (3) ogeher wih he sochasic process (2) fully describes he dynamics of model. As already sressed, a closed form soluion does no exis and q i, n i and m i mus be deermined by numerical simulaions. Chrisiano and Eichenbaum (1992, p. 441) provides a numerical simulaion for which per capia capial shows a sochasic rend (he average growh rae of which equals o he value of he drif of sochasic process). On he conrary, hours worked do no show any rend. Furher he sysem proves o be sable. The properies of he arificial model economy are analysed by means of numerical simulaions. In so doing, he purpose is o mach some empirical regulariies or sylised facs (see Canova, 1998). These regulariies generally refer o differences in he variance of some relevan variables and/or crosscorrelaions among he laer. Le us urn our aenion o he problem of he choice of proper indicaors for business cycles. We sar from he sandard deviaions of he variables. In

5 31 The Theory of Economic Growh: a Classical Perspecive Table 16.1 we repor he sandard deviaions of our simulaed series afer hey are logged and derended by he Hodridk Presco filer o exrac he ransiory componen a frequency 4 6 years and, for comparison, we also repor he corresponding saisics relaive o US daa (see Canova, 1998). 6 As we see, he simulaed series show sandard deviaions normalised wih respec o he sandard deviaion of oupu ha in some cases overesimae he real values (consumpion and produciviy) while in ohers underesimae hem (invesmen, hours worked and real wages). Table 16.1 Sandard deviaions, sources US esimaes: Canova (1998) Saisic of σ Simulaed daa US daa 1955:3 1986:3 σ c /σ y σ i /σ y σ h /σ y σ w /σ y σ y/n /σ y Anoher poin is he correc sign and magniude in he cross-correlaion among variables. In Table 16.2 we repor such cross-correlaion for simulaed series (firsly derended and hen filered by he Hodrick Presco procedure) and for US daa. Table 16.2 shows he main drawback of he RBC model, namely, he discordance beween simulaed and real series wih respec o he cross-correlaion of hours worked and produciviy wih oupu. The negaive empirical correlaion beween hours worked and produciviy suggess ha he labour marke does no work as a perfecly compeiive marke. Moreover, simulaions predic a srong correlaion beween produciviy and oupu whereas he empirical evidence shows only a slighly posiive correlaion. According o Canova (1998, p. 53) his correlaion changes over ime so ha here is he need for heoreical work o provide reasons for why his phenomenon occurs. Table 16.2 Correlaions, sources US esimaes: Canova (1998) Saisic of σ Simulaed daa US daa 1955:3 1986:3 corr(c, y) corr(i, y) corr(h, y) corr(w, y) corr(y/h, y) corr(h, w) corr(h, y/h)

6 Real business cycle models, endogenous growh models and cyclical growh 311 The laer poin suggess ha he curren RBC heory needs a sounder microfoundaion of echnological progress. This will be he argumen of he nex secion Microfoundaion of Technological Progress The exogeneiy of echnological progress was he mos serious criicism faced by exogenous growh models à la Solow. In he same way some auhors argue ha a business cycle heory needs a microfoundaion of innovaion. Moreover, as we showed in he previous secion, he RBC model predics a srong posiive correlaion beween produciviy and oupu, which conrass wih he empirical evidence. 7 Thus, endogenising echnological progress may help o undersand he causes of his phenomenon. Firsly, we analyse he model proposed by Sadler (199), in which echnological progress is viewed as a learning-by-doing process, i.e., as a byproduc of producion. In so doing, our purpose is o sress ha, even if produciviy is always srongly procyclical, his has crucial implicaions for business cycles analysis, in paricular for he effec of moneary policy on long-run produciviy. Secondly, we consider he conribuion by Aghion and Sain-Paul (1998b), in which echnological progress is endogenised wihin a Schumpeerian framework. The focus in his case is on he relaionship beween produciviy and he growh rae of oupu; i will be shown ha produciviy can be boh procyclical and counercyclical. Sadler (199) proposes a model in which growh is generaed only hrough learning by doing. Business cycles are generaed by shocks o produciviy similar o hose in he model presened above bu wihou a posiive drif. In he model economy here is no fixed capial and he only accumulaed facor is knowledge, which is accumulaed hrough learning by doing. The learning-by-doing process is exernal o he firm and is a byproduc of producion. In such a framework, he consumer plays no role in he allocaion of resources. As a consequence, he condiion for maximisaion of profis of he represenaive firm deermines he compeiive allocaion. The represenaive firm solves he following problem: w V = max E β y h (5) { h } = = p where < β < 1 is a discoun facor, 8 w, he moneary wage, p, he price level and h, he labour employed by he firm. I is assumed ha he producion funcion is:

7 312 The Theory of Economic Growh: a Classical Perspecive y = z h a, wih < θ < 1, χ >, η > (6) χ 1 θ η where z is he produciviy shock and a he per capia accumulaed knowledge available o he firm. The laer is accumulaed boh by an exogenous componen and as a by-produc of producion: 9 a = a y (7) 1 δ a λ 1 1 where λ, δ a >. The produciviy shock follows a sochasic process similar o (2) bu wihou drif: 1 2 ( ) z = z exp( ε ) ε N, σε (8) Given he assumpion of exogenous knowledge accumulaion, he ineremporal maximizaion problem (5) of he represenaive firm becomes a saic opimisaion problem. Therefore he demand curve of he firm is given by he firs order condiion (FOC) o he problem of he maximisaion of profis a ime, ha is: h d ( 1 θ ) χ η za = w / p 1/ θ (9) Labour supply is assumed o be 1 φ2 w 1 φ1 φ2 p ( φ ) s h = exp wih, > (1) Sadler (199) hen assumes ha neiher consumer nor firm knows he level of price a period when hey bargain in he labour marke o se he wage a he beginning of he period. Thus, hey formulae heir choice according o heir expecaions of he level of prices a ime, given heir informaion se (which included he level of produciviy z 1 ). In our case, we obain he following expression for he log of aggregae supply y s e ( ) y = f + f z + f z + f a + f p p (11) s where ~ over a variable denoes he log of he variable, f =(1 θ)[φ 2 log(1 θ) +φ 1 ]/(1+φ 2 θ), f 1 =χ/θ, f 2 = (1 θ)χ/θ(1+φ 2 θ), f 3 =(1 θ)φ 2 η/(1+φ 2 θ) and f 4 =(1 θ)/θ. Hence aggregae supply depends on produciviy shocks, z, accumulaed knowledge, a, and on he e difference beween acual price and expeced price p p ; i is o be noed

8 Real business cycle models, endogenous growh models and cyclical growh 313 ha an expeced increase in demand can lead o an increase in supply of oupu. We nex deal wih his poin. On he basis of he quaniaive heory, following Sadler (199) we assume ha aggregae demand is given by y d m p = (12) where m is per capia money sock. m is assumed no o be observable a he beginning of period bu follows he sochasic process 2 ( γ ζ ) ζ ( σ ) m = m exp + N, ζ (13) 1 m where γ m > is he drif in money supply. In equilibrium we have: ζ f ε p p = e 1 1+ f 4 (14) Subsiuing (14) in (11) yields equilibrium (log) oupu: y = q + q z + q a + qζ + q ε (15) where q = [φ 2 log(1 θ) + φ 1 ]/(1+θφ 2 }), q 1 =χ(1+φ 2 )/(1+φ 2 θ), q 2 =η(1+φ 2 )/(1+φ 2 θ), q 3 = 1 θ and q 4 =χ. The dynamics of oupu is driven by moneary and real shocks (ζ and ε, respecively), by produciviy shocks, z, and by he sock of accumulaed knowledge, a. Equaion (15), provided ha a is ime-consan, makes i clear why moneary shocks canno have a long-run effec in an RBC model: in fac, only he level of produciviy z is relevan in deermining he longrun behaviour of y, while he moneary shock ζ has only a shor-run effec. However, if a incorporaes also he shor-run dynamics of y, as in our model, hen moneary shocks can have a long-run effec. Subsiuing (7) and (8) in (15) leads o: 1 j y= q + q1 εi + q2 ( 1 δa + λq2) a + λ ( 1 δa + λq2) i= 1 j= j 1 q + q1 εi + q3ζ j + q4ε j + q3ζ + q4ε i= 1 (16) From equaion (16) we can calculae he expeced level of long-run (log) oupu:

9 314 The Theory of Economic Growh: a Classical Perspecive δ λq E y q q a q a [ ] = + + ( 1 δa + λ ) 2 2 δa λq2 λq2 δa The exisence of a posiive long-run growh of oupu depends on he value of 1 δ a + λq 2. If 1 δ a + λq 2 > 1, hen he growh rae of oupu is ever increasing; oherwise, he long-run growh rae is zero and he level of oupu (in log) converges o q δ a /(δ a λq 2 ). From equaion (17) i follows ha if he economy achieves posiive longrun growh, i.e. 1 δ a + λq 2 > 1, hen boh moneary and real shocks have long-run effecs. However hey work hrough wo differen channels. Moneary shocks affec he long-run oupu hrough he sock of accumulaed knowledge, while real shocks work boh direcly on oupu and indirecly on he sock of accumulaed knowledge. In his regard assume η =, i.e. he sock of accumulaed knowledge is no relevan in producion; his implies ha q 2 = (he long-run growh rae is equal o zero). From (16) we have: 1 = + 1 εi + 3ζ + 4ε i= 1 y q q q q Therefore real shocks ε have long-run effecs, while moneary shocks ζ have only one-period effecs. This case corresponds o he sandard RBC model of Secion , provided ha he sochasic process of produciviy has a posiive drif. Finally, i is o be noed ha moneary shocks alone can accoun for he uni roo deeced in empirical daa. Consider he case where real shocks have no effec on oupu, ha is χ =, from which q 1 = q 4 =. From (16) we have j y = q 1+ λ ( 1 δa + λq2) + q2( 1 δa + λq2) a j= ( 1 ) j + λq δ + λq ζ + qζ 3 a 2 j 3 j= and, since 1 δ a + λq 2 > 1, moneary shocks ζ have long-run effecs and y presens a uni roo (or more han a uni roo in our case where growh rae is ever increasing) Endogenous growh heory and he Schumpeerian approach o innovaion The relaionship beween produciviy and growh is holy debaed in he lieraure. According o Schumpeer, recessions are required o eliminae inefficien firms from he marke, so ha he final effec of a decrease in

10 Real business cycle models, endogenous growh models and cyclical growh 315 economic aciviy is an increase in overall produciviy (see Caballero and Hammour, 1991). There are however facors suggesing ha produciviy is procyclical, like learning by doing, demands spillovers and capial marke imperfecions ha consrain invesmen in he R&D secor (see Sigliz, 1993). Moreover, Bean (199) sresses ha, if he reorganizaion of he firm is cosly in erms of oupu, during a recession he opporuniy cos of such an aciviy is lower and herefore he procyclical paern of produciviy is magnified. The empirical evidence for he relaionship beween produciviy and oupu is mixed. Canova (1998), for example, shows ha produciviy was counercyclical up o he mid 196s and procyclical aferwards. In wha follows we discuss a simplified version of he Aghion and Sain- Paul (1998b) model, which provides an example of how produciviy can be counercyclical if innovaions are endogenised in a Schumpeerian fashion. The basic idea is ha he cos of innovaion is lower in recession han in a boom if he ac of innovaing drains resources from producion. This framework corresponds o he case analysed by Hall (1999), who assumes ha he increases in produciviy are he resul of inernal reorganizaion of a firm, which negaively affecs he curren level of oupu. 11 In he economy a period here are N firms and N differen goods; each firm is a monopolis in is own marke. Each is characerised by a level of produciviy z i, and i is assumed ha firms canno change heir employed workforce and producion capaciy. Thus, he gross produc of each firm is deermined only by is produciviy, which is given by exp(z i, ). Le v i, = dz i, /d be he change in produciviy. Each firm can modify is produciviy by sacrificing par of is producion; a change in produciviy equal o v i, involves a proporional drop of oupu equal o µ = µ(v i, ), where he auhors assume ha µ() =, µ >, µ >. For firm i, demand a period is given by: d D p i, i, = P P ψ (17) where ψ > 1, D is an aggregae demand index and P, an aggregae price index, defined by N 1/ ( 1 ψ ) 1 ψ i, (18) P = p di The ne oupu of firm i, y i,, is given by: ( ) ( ) y = 1 µ v exp z i, i, i, (19)

11 316 The Theory of Economic Growh: a Classical Perspecive Given ha he equilibrium in he monopolis marke i is given by y i, = d i,, we hen have: z p ( v ) Y P ψ 1/ ψ i, 1/ ψ ( ψ 1)/ ψ i, = 1 µ i, exp (2) where p i, is he price of good i produced by firm i. Given he acual level of produciviy, firm i mus choose he opimal increase in produciviy; he laer is he resul of he following problem: 12 { π } ( i, ) =max i, + ( 1 ) ( i, + i ) V z d rd E V z vd (21) vi where V is he value of he firm, r is he consan ineres rae a which agens can lend or ren heir resources. The fuure value of he firm depends on he level of prices, which, in urn, depend on he aggregae demand index; he expecaion operaor reflecs he possible uncerainy of he laer variable. To deermine he equilibrium, marke enry and exi condiions mus be specified. Following Aghion and Sain-Paul (1998b) we assume ha he firm mus bear a fixed cos equal o C o ener he marke, while he firm has a liquidaion value equal o τc, where τ < Finally, i is assumed ha new firms show he same level of produciviy. In he equilibrium he number of firms will be consan if he expeced value of a firm V is greaer han τc, bu lower han C, ha is [ τ ] N = V C, C (22) In equilibrium we expec ha V [τc, C] since V greaer han C drives new firms o ener he marke, causing V o decrease up o C, while a firm s value lower han τc leads firms o exi, causing V o increase up o τc. Consider he symmeric equilibrium where p i, = p and z i, = z and herefore v i, = v for all i. 14 From (22) we obain he level of profis for each firm: D π = = d (23) N where d = D /N is he demand for he single firm. Aghion and Sain-Paul (1998b) assume ha cycles are generaed by flucuaions in he level of aggregae demand D. In paricular, i is assumed ha here exis only wo saes, E, expansion and R, recession, and ha he probabiliy of jumping from sae R o E follows a Poisson process and is given by ε, while from sae E o R i is given by ζ.

12 Real business cycle models, endogenous growh models and cyclical growh 317 In he sochasic seady sae for each possible sae E and R, all variables, excep for z and p, are consan and he economy says for a fracion of ime equal o ε/(ζ + ε) in expansion and for a fracion equal o ζ/(ζ + ε) in recession. In such a framework he problem of he firm (21) becomes: 15 { π ( ) ( ε ) + ε + } π ( ) ( ζ ) ζ R R R E V maxv d 1 rd 1 d V d dv = + + d E E E R V = maxv d + 1 rd 1 d V d + dv d { + + } (24) j j j where π = D / N = d for j = E, R. To close he model we mus calculae he level of per capia demand in he wo saes; in fac, since he number of firms is endogenous, a higher level of aggregae demand may no be mached by a higher level of per capia demand. We assume ha recession is deep enough o lead some firms o exi; in his case he free enry condiion implies ha R E V = τc; V = C (25) By insering (25) ino problem (24), we obain he levels of profis for each of he wo saes: E ( 1 ) ; C r ( 1 ) = C r = + R π τ ε τ π ζ τ from which he levels of per capia demand also easily follow (see equaion (23)): E ( 1 ) ; ( 1 ) = τ ε τ = + ζ τ (26) R d C r d C r From hese expressions, i follows ha d E > d R, i.e. per capia demand is higher in expansion han in recession. From equaions (26) and he FOCs of problem (24), we obain: 1 µ 1 µ R ( v ) = R ( v ) rτ ε( 1 τ) E ( v ) 1 = E ( v ) r + ζ ( 1 τ) µ τ µ (27) which provide implici soluions for v R and v E. From (27), given he assumpions on µ, i follows ha v R > v E : his means ha produciviy in his model is counercyclical. The inuiion is sraighforward: he opporuniy

13 318 The Theory of Economic Growh: a Classical Perspecive cos of innovaing, measured in erms of lower oupu, is higher in expansion because he level of demand (and herefore of profis) is greaer han in recession. This is he main finding of he model: recession can have a posiive impac on he growh rae of produciviy. Finally, from (27) i follows ha v j / r <, j = E, R, v R / ε >, v R / ζ <, v R / τ < and v E / τ >. The business cycle affecs he growh rae of he economy because of he differen increases in produciviy which characerise he wo saes. The average growh rae of he economy is given by: ζ ε = + ζ + ε ζ + ε R E g v v (28) Wih regard o he average growh rae of he economy, Aghion and Sain- Paul (1998b, p. 333) underline he imporance of he following hree effecs: 1. Composiion effec: he ime ha he economy spends in expansion, ε/(ζ + ε), and in recession, ζ/(ζ + ε), given v R > v E, crucially affecs he average growh rae; 2. Reurn effec: in expansion per capia demand is higher han in recession and herefore he longer he economy spends in expansion, he higher are he increases in produciviy; 3. Cos of capial effec: since he firm does no recoup all coss of enry in he case of exi, boh he ime i spends in expansion and in recession and he liquidaion value affec he incenive o increase produciviy. Given all his, we can conclude ha he hree main parameers (ε, ζ, τ) of he model affec he average growh rae in he following way: g/ ε > if dv R /dε > (v R v E )/(ζ + ε); in he case of an increase in ε, he composiion effec is negaive (ζ(v E v R )/(ζ + ε) 2 < ), whereas he sum of he reurn and of he cos of capial effecs is posiive ([ζ/(ζ + ε)]/(dv R /dε)>), so ha g/ ε > if he laer is greaer han he former. g/ ζ < if dv E /dζ > (v R v E )/(ζ+ε); in he case of an increase in ζ he composiion effec is posiive ε(v E v R )/(ζ + ε) 2 > ), whereas he sum of he reurn and of he cos of capial effecs are negaive ([ε/(ζ + ε)](dv E /dζ)<), so ha g/ ζ< if he former is greaer han he laer. g/ τ > if dv E /dτ > (ζ/ε)(dv R /dτ); he cos of capial effec is posiive for v E ([ε/(ζ + ε)](dv E /dτ)>) and negaive for v R ([ζ/(ζ ε)]dv R /dτ)<), so ha g/ τ> if he former is greaer han he laer.

14 Real business cycle models, endogenous growh models and cyclical growh 319 To sum up, he idea ha expansion is beer ha recession for growh is challenged in his model; our inuiion is based on he fac ha during a recession he reorganizaion of a firm in order o increase is produciviy is less cosly; his has implicaions for boh he cross-correlaion beween oupu and produciviy (which is negaive raher han posiive) and he effec of recession on average growh rae (which is posiive raher han negaive) ENDOGENOUS GROWTH MODELS WITH DETERMINISTIC CYCLES Since he lae 198s, following seminal conribuions by Lucas (1988) and Romer (1986a, 199), a large number of aricles have focused on he dynamics of EG models, in a deerminisic conex. This line of research was srongly moivaed by he fac ha he original conribuions by Romer and Lucas focused on seady sae only and, in addiion, negleced he sabiliy properies of he seady sae. We now urn o he analysis of his lieraure. Our main purpose is o check wheher here exis resuls concerning he emergence of persisen cycles similar o hose obained by Benhabib and Nishimura (1979) for he convenional exogenous growh models. A posiive answer o his quesion would imply ha EG models offer an addiional (deerminisic) approach o he sudy of cyclical growh in a marke-clearing conex, alernaive o he sochasic approach presened in Secion In reviewing his lieraure on EG, i is useful o disinguish beween onesecor models and wo-secor models. In models belonging o he firs class (e.g., Romer, 1986a), he accumulaion of knowledge is only a by-produc of producion aciviies and EG is generaed by mechanisms of learning by doing, by exernaliies or by increasing reurns. The models belonging o he second class, on he oher hand, saring eiher from Lucas (1988) or from Romer (199), generae EG by assuming an inenional allocaion of resources for he accumulaion of human capial or an inenional R&D effor for increasing he level of echnological progress. In wha follows, we firs review he exising lieraure on his opic wih regard o one-secor EG models (Secion ). We will focus in paricular on recen conribuions by Greiner and Semmler (e.g., 1996a, 1996b), which aim o show ha a basic model of EG wih learning by doing (which is a modified version of he Romer 1986a model) may produce a rich array of oucomes, such as muliple seady saes, indeerminacy of equilibria or even persisen cycles of he sae variables. Then, in Secion we ackle he same problem wih regard o wo-secor EG models. In his case, we discuss he classical conribuion by Benhabib and Perli (1994) and we briefly

15 32 The Theory of Economic Growh: a Classical Perspecive presen he resuls of a recen conribuion by Maana and Venuri (1999), in which hey show ha periodic soluions may emerge in he Lucas model Persisen Cycles in One-Secor EG Models The model considered by Greiner and Semmler (1996a, 1996b) is a onesecor EG model of he Romer ype (wih learning by doing) in which, however, i is assumed ha one uni of invesmen has differen effecs concerning he building up of physical capial and knowledge. This implies ha he wo variables canno be merged ino a single variable. The producion possibiliies of he model economy (in per capia erms) are given by Y L ba k b ( ) α 1 α =, >, α,1 (29) where A sands for he sock of knowledge, K, he sock of physical capial, L, labour force, k = K/L. In wha follows, o simplify, we choose b = 1. Assuming ha L grows exponenially in ime a a consan rae equal o n >, he equaion for he evoluion of k is he following: ( δ ) k = i + n k where δ > is he rae of depreciaion, i = I/L and I, gross invesmen. Wih regard o he sock of knowledge, i is assumed ha i accumulaes according o a learning-by-doing process à la Arrow, in he formulaion given by Levhari (1966). In addiion, i is assumed ha he conribuion o he formaion of knowledge of gross invesmen furher back in ime is smaller han ha of recen gross invesmen. Hence (see Greiner and Semmler, 1996a, p. 82): or () ( ) () A = ρ exp ρ s isds, ρ > (3) α 1 α ( ) ρ( ) A = ρ i A = A k c A (31) where ρ > represens he weigh given o more recen levels of gross invesmen and c sands for per capia consumpion. 16 Following Greiner and Semmler (1996a, p.82), we limi ourselves o analysing he compeiive siuaion, in which he evoluion of knowledge is no explicily aken ino accoun by he represenaive agen when solving he

16 Real business cycle models, endogenous growh models and cyclical growh 321 opimisaion problem. Normalising so as o have L() = 1, he laer is he following: 17 ( β n) u( c ) Max exp ( ) d, β > (32) c α 1 α subjec o: = ( δ + ) k A k c n k. From he curren-value Hamilonian for problem (32), we hen obain he following se of necessary FOCs for an opimum: 18 () λ u () c u c = = λ (33) A λ = λ( δ + β) λ( 1 α) (34) k We are now in a posiion o derive he differenial equaions sysem which describes he dynamics of he model economy. From (33) (34), one obains () () u c c c c σ ( δ β) ( 1 α) A = = + u c c c k α α which, ogeher wih (31) and (33), gives ( 1 ) α c α A δ + β = c σ k σ α k A c = k + k k ( δ n) (35) (36) α 1 A A c = ρ ρ ρ (37) A k A Following a sandard pracice, he order of sysem (35) (37) can be reduced by performing a change of variables wih ka = k/ A and ca = c/ A. Hence: c c k k A A A A c = k + n k + + c 1 α ( δ ) ρ ρ( 1 ) α A A A A ka ( 1 α) ( δ + β) ( 1 ) = k ρk + ρ + c σ σ α 1 α A A A (38) (39)

17 322 The Theory of Economic Growh: a Classical Perspecive A balanced growh pah for he original sysem is obained as a res poin of he reduced sysem a which k A / ka= c A / ca= so ha A / A= k / k = c / c. As shown by he auhors (see Proposiion 2 in Greiner and Semmler 1996a, p. 85 and Theorem 1 and Theorem 2 in Greiner and Semmler 1996b, p. 11), in he case of posiive per capia growh, (i) if δ + n ( δ + β)/ σ, here exiss a unique seady sae proves o be saddle sable and such ha: A ( ) k ρk δ + n k + ρk c = ( 1 ) ( ) ( ) 1 α 2 α A A A A 1 ρka c k which ( A, A) ( ) α δ + β δ + β 1+ α ρka 1 + ka ρ + ρka ( δ + n) + = σ σ σ (ii) if δ + n < ( δ + β)/ σ, here exis wo seady saes, ( c A1, k A1) and ( ca2, k A2) wih k A1< ka2, such ha he pah associaed wih he second can be anyhing excep a saddle pah. Given our aim, we are mainly ineresed in case (ii) in which he second of he wo seady saes is eiher compleely sable or unsable. To undersand why i is so, le us noe ha complee sabiliy requires ha he Jacobian of he linearised sysem a ( c, k ) is such ha A2 A2 α 1 α αka2 (1 α) ρka2 + ca2 / ka2 ρka2 1 J 2 = α 1 α α( 1 α) / σ ca2ka2 ( 1 α) ρca2ka2 ρca2 c r J = (1 ) + + < α 1 α A2 2 αk A2 α ρka2 ρc A2 ka2 α 1 α c A2 de J2 = αka2 (1 α) ρka2 + ρca2 ka2 α ( 1 α) α 1 α + ( ρka2 1) ca2ka2 + ( 1 α) ρca2ka2 > σ The basic fac is ha, as one of he parameers, e.g. ρ, varies, here may exis a value ρ H for which rj 2 ( ρ H ) = and de J 2 ( ρ H ) >. When his happens, he dynamics of he sysem may undergo a qualiaive change,

18 Real business cycle models, endogenous growh models and cyclical growh 323 known in he lieraure as Hopf bifurcaion: 19 he model economy does no reach he seady-sae growh rae, bu raher persisenly flucuaes around i. In erms of he parameers of he model, we have: 2 ( 1 ) αk + α ρ k rj (4) α 1 α A2 H A2 2 = ca2 = 1 ( ρh + ka2 ) ( 1 ) α 1 α A2 H A2 2 > ca2 = 1 de J αk ( ρh + ka2 ) α(1 α)(1 ρhka2) + ρhσka2 > ρσ H ka < 2 2 ( α σ) αρ k ρ σk ( α) α( α) H A2 H A2 + α ρ k The exisence of a Hopf bifurcaion also requires ha 2 ρ ρ= ρh (41) d rj ( ρ) (42) d Using parameer values ha saisfy condiions (41) and (42), numerical simulaions show ha indeed he model may generae limi cycles which, in addiion, prove o be sable (see Greiner and Semmler, 1996a, pp and 1996, pp ). The conclusion is ha he model may generae persisen cycles in he growh rae of per capia variables, e.g. per capia oupu, namely, persisen growh cycles Persisen Cycles in Two-Secor EG Models I is a fac ha, wih few excepions of he kind considered in he previous subsecion, he conribuions on ou-of-seady-sae dynamics in EG models have focussed on wo-secor models, for boh he case wih human capial (see, for example, Caballé and Sanos, 1993, Mulligan and Sala-i-Marin, 1993, Benhabib and Perli, 1994, Boldrin and Rusichini, 1994, Xie, 1994, Barro and Sala-i-Marin, 1995, ch. 5, and Arnold, 1997) and wih R&D (see, for example, Benhabib, Perli and Xie, 1994, Asada, Semmler and Novak, 1998, and Arnold, 2a, 2b). The resul is a large body of lieraure in which i is possible o find some clear-cu resuls. Firs, no ineresing ou-ofseady-sae dynamics is usually found for he social planner soluion of he models. This is shown, for example, by Caballé and Sanos (1993) who find saddle-pah sabiliy for he Lucas model in he absence of exernaliies (see

19 324 The Theory of Economic Growh: a Classical Perspecive also Barro and Sala-i-Marin, 1995 and Arnold, 1997), whereas Asada, Semmler and Novak (1998) obain he same resul for he social planner problem of he Romer model. Second, saring wih he imporan conribuions by Benhabib and Perli (1994) and Xie (1994), ineresing dynamics (included muliple equilibria, indeerminacy of equilibria and even he emergence of periodic soluions) has been shown for he marke soluion of he Lucas model when exernaliies from human capial are considered. Given he aim of he presen chaper, we now urn o a brief descripion and analysis of his second se of resuls. In he model by Lucas (1988) he opimisaion problem ha has o be solved by he represenaive agen is he following exp( β) d 1 σ (43) 1 σ c 1 max c (),() u 1 subjec o: k = k α ( uh) α h γ c h = δ h ( 1 u ) ( ) ( ) a k = k >, h = h > where c is per capia consumpion, β, a posiive discoun facor, σ, he inverse of he ineremporal elasiciy of subsiuion, k, physical capial, h, human capial, u, he fracion of labour allocaed o he producion of physical capial (so ha uh is he fracion of effecive labour), δ, a posiive echnology parameer, 1 α, he share of capial, and γ, a posiive exernaliy parameer in he producion of human capial. For problem (43) we obain he following se of necessary FOCs for an inerior soluion: 21 α c σ = λ 1 (44) σ 1 α α+ γ 1 α c k h u δ ( ) = λ (45) 2 1 k α h α γ u α λ 1= βλ1 λ1 α + (46) 1 α α+ γ 1 α ( ) λ = βλ λαk h u λ δ u (47) whereas he ransversaliy condiion is 22 ( β)[ λ k λ h] lim exp + = 1 2 From (44) (47), wih simple algebraic manipulaion, we derive he dynamical sysem of he model. In order o do so, noe ha from (44) and (46) we obain:

20 Real business cycle models, endogenous growh models and cyclical growh 325 whereas from (45) and (47) 1 c α α α+ γ α β = ck h u c σ σ (48) ( 1 ) ( 1) ( 1 ) σc c+ α k k+ α+ γ h h α u u = β δ Insering in he laer expression (48) and he consrains of (43), we finally obain x = ηx + αψ x x x 1 α where x 2 = u, x 3 = c/k, η = (1 α γ)δ/(1 α), ψ = (α + γ)δ/α <. We hen define = ( α+ γ )/ α x1 h k from which, aking he derivaive wih respec o ime of boh sides and insering he wo consrains of (43) Finally where ( ) x = ψ (1 x ) x x x + x x α x = x + φx x x ξx 2 α α φ= 1 α / σ 1 and ξ = β / σ. Thus he resuls of Benhabib and Perli (1994) can be discussed by analysing he following reduced dynamical sysem he seady-sae values of which are 23 ( 1 ) x = x x + ψ x x x x (49) 1 α α ψα x 2 = ηx2 x2 x2x3 (5) φ x = x + φx x x ξx (51) 2 α α ( 1 α) ξ δ( α + γ) δ( 1 α γ) ( 1 α) φ 1/ α x 2 1 = x2 x

21 326 The Theory of Economic Growh: a Classical Perspecive x 2 1 ( δ) ( ) α β =, δ γ σ α + γ x 3 2 ( + γ) δ α = ηx + 1 α The sudy of he local sabiliy of he linearised sysem leads o some major resuls ha can be summarised as follows (see Proposiions 1 and 2 in Benhabib and Perli, 1994, p ) 1. For values of he parameers such ha < β < δ and σ > (1 + β/ψ), he Jacobian of he sysem has one eigenvalue wih a negaive real par and wo eigenvalues wih a posiive real par. In his case he compeiive equilibrium pah is locally unique: given he iniial condiions of k and h, here exiss only one value of u and c ha drives he economy owards is (deerminae) seady-sae pah; 2. For some oher values, namely for δ < β < ψ, σ < (1 + β/ψ) and γ > 1 α, one eigenvalue has a posiive real par and wo eigenvalues have a negaive real par. In his case here is always a coninuum of equilibria in he neighbourhood of he indeerminae seady-sae pah: given he iniial condiions of k and h, here exis infiniely many values of u and c ha drive he economy owards he BGP; 3. When β and σ are as in he previous case, bu < γ 1 α, here is eiher (i) one posiive eigenvalue and wo eigenvalues wih negaive real pars or (ii) hree eigenvalues wih posiive real pars. In he laer case here is complee insabiliy and no equilibrium pah leads o he BGP. As sressed by Benhabib and Perli (1994, p. 124), of paricularly ineres is Case 3, in which here is a basic change in he roos srucure of he Jacobian. This happens when, as one of he parameers (for example, he exernaliy parameer γ) changes, he real pars of he wo complex eigenvalues change sign. A he value γ = γ H for which he real pars of he wo complex eigenvalues are zero he sysem may undergo a Hopf bifurcaion such ha i persisenly flucuaes around he seady-sae pah. However, alhough his possibiliy is menioned by Benhabib and Perli (1994, p. 124), hey do no invesigae he maer any furher bu raher concenrae on he indeerminacy resuls. An ineresing analysis of he case is insead conained in a recen conribuion by Maana and Venuri (1999), where he auhors, afer a general presenaion of Benhabib and Perli s resuls, concenrae on he possibiliy of emergence of periodic soluions and achieve resuls worh menioning. They esablish analyically he exisence of closed orbis (see Theorem 1 in Maana and Venuri, 1999, p. 27) for he reduced sysem (46) (48). Moreover (see Theorem 2), hey esablish, wih he help of numerical simulaion, ha he closed orbis emerging from he seady sae

22 Real business cycle models, endogenous growh models and cyclical growh 327 can be eiher sub-criical (i.e., repelling) or super-criical (i.e., aracing). In paricular, when σ is small, he numerical simulaions show ha he supercriical seems o prevail CONCLUSIONS In his paper we explored he possibiliy of generaing persisen growh cycles in a marke-clearing framework, disinguishing beween sochasic and deerminisic models. As regards he firs class of models, we focussed on he novely originaing from he new growh heory. In paricular, we sressed boh he role played by moneary facors and he consequences of modelling R&D aciviy. Wih regard o he second class, we analysed he condiions for he emergence of periodic orbis in EG models which, in such a framework, represens growh cycles. The inverse empirical relaionship beween produciviy and oupu is a major challenge for business-cycle researchers. The represenaive agen framework, which is common o all conribuions considered in his chaper, does no appear well-suied o his goal. As suggesed by Lippi (1993), his may require he consideraion of he dynamics of he produciviy of firms and of heir ineracion. Finally, analysis of ou-of-seady-sae dynamics in EG models has shown ineresing resuls, bu he laer appear o be applied o a very narrow se of EG models. A direcion for fuure research is o explore he possibiliy of periodic orbis in oher ypes of deerminisic EG models, such as R&D Schumpeerian models, in an aemp o bridge he gap beween sochasic and deerminisic approaches. NOTES 1. In anoher conribuion o his volume, Sordi analyses he problem of he inerrelaion beween growh and cycle in a non-marke clearing framework. 2. Frisch (1933) argued ha his heory of he business cycle was suppored by he Schumpeerian idea of innovaions as a cause of economic flucuaions. However, he hough ha Schumpeer was considering as exogenous he process of innovaion, while Schumpeer was acually disinguishing beween scienific discovery, no driven by economic forces, and innovaion, i.e. he economic implemenaion of a scienific discovery, which depends on economic facors. 3. The form of his sochasic process is suggesed by he empirical evidence abou he presence of a uni roo in ime series. 4. We assume ha populaion is consan. In he case in which he populaion increases a a consan rae equal o n, he represenaive agen s problem becomes:

23 328 The Theory of Economic Growh: a Classical Perspecive { c, l} = = ( ) { φ ( )} max W = β 1+ n ln c + V l d subjec o he ineremporal budge consrain In his case, wriing β ˆ β( 1 n ) 1 θ θ ( ) ( δ ) k+ 1 Az h k c 1 n k = + = + and δ ˆ = δ + n, he problem becomes equivalen o (3). 5. The deails of such a procedure are in an appendix available upon reques. 6. The purpose in filering simulaed series is o make hem comparable wih he acual ime series, ha is o consider cycles of he same frequencies for boh. 7. As shown by Chrisiano and Eichenbaum (1992), his resul holds for a wide range of parameers. 8. β depends on he ineres rae and, in a model wih capial accumulaion, would be an endogenous variable; however, in he framework considered by Sadler, wihou fixed capial, he consancy of β does no appear a resricive assumpion. 9. To avoid scale effecs, which are ypical in endogenous growh models, we suppose ha every firm has a posiive exernaliy proporional o per capia capial knowledge sock available in he economy. 1. A possible microfoundaion of such labour supply can be performed by considering a represenaive agen ha maximises an insananeous uiliy funcion whose argumens are consumpion and leisure; moreover labour is assumed o be paid o he real wage rae w/p, which represens he only source of income for he agen. The choice of he labour supply (1) is due o is analyical racabiliy. We also noice ha he oal amoun of labour which is supplied has no upper bound; a more plausible formulaion would enail oal available labour being fixed and finie. For example a possible formulaion for he labour supply s ν 1 would be h [1 ( w / p ) ] L ν ν problem: ( - ), s. o ( / ) = +, which is he soluion o he following consumer U= c L h c = w p h. 11. The increase in produciviy, however, could also be he resul of employing in he producion process new resources brough ono he marke (see Sigliz, 1993). We analyse his more general case in an Appendix available on he auhors websies, in which he firm can choose beween hese wo mehods of increasing is produciviy. 12. Problem (21) is derived from he sandard Bellman equaion for a firm whose goal is o maximizes is value, given by he discouned sum of fuure expeced profis a rae r. 13. Aghion and Sain-Paul (1998b) highligh how he presence of a liquidaion value independen of produciviy can generae an exi effec, which leads firms o underinvesing; for he sake of simpliciy we ignore his aspec. 14. Noice ha by resricing our aenion o symmeric equilibrium we are excluding analysis of he dynamics of cross-secion produciviy along business cycles. A number of oher empirical conribuions focus on his poin. See, for example, Caballero and Hammour (1994) and Davis and Haliwanger (1992). 15. In he formualion of he firm problem we have implicily assumed ha here is no exi effec, in oher words, ha he value of he firm in recession is never below τ C. See Aghion and Sain-Paul (1998a, p. 326). 16. The relaion beween his formulaion and he original approach by Romer (1986a) is discussed in Greiner and Semmler (1996a, p. 82). 17. As usual, i is assumed ha he per capia uiliy funcion is such ha u ( ) >, u ( ) <, lim c u ( ) =, u ( c ) c / u (c ) σ, consan. 18. Clearly, condiions (33) (34) are also sufficien for an opimum if he following ransversaliy condiion is saisfied

24 Real business cycle models, endogenous growh models and cyclical growh 329 [ ( β ) ] λ()( ) lim exp n k() k () where k is he opimal value of per capia capial sock. 19. For an inroducion o he concep of Hopf bifurcaion, see Gandolfo (1997, ch. 25). 2. From condiion (41) i follows ha α + σ < 1 is a necessary condiion for a Hopf bifurcaion o occur. 21. To obain hese condiions we have aken accoun of he fac ha, in equilibrium, we mus have h = h a. 22. Benhabib and Perli (1984, p ) have shown ha he maximised Hamilonian is joinly concave in (k, h). This implies ha he above condiions are also sufficien for problem (43). 23. I is possible o show ha hese values saisfy he ransversaliy condiion. See Benhabib and Perli (1994, p. 122).