Chapter 11 New Classical Models of Aggregate Fluctuations

Transcription

1 George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 New Classical Models of Aggregae Flucuaions In previous chapers we sudied he long run evoluion of oupu and consumpion, real ineres raes and real wages, and he long run evoluion of he price level and inflaion. In order o focus on longerm rends we made he assumpion ha all markes are compeiive and in coninuous equilibrium, hrough he full adjusmen of prices, wages and ineres raes. However, as we noed in Chaper 1, economies are characerized by flucuaions in relaion o heir long-erm rends. In some periods oupu, consumpion and employmen grow a high raes, while a oher imes hey grow a low or even negaive raes. In some periods unemploymen is low and in ohers quie high. Inflaion displays significan flucuaions as well. Undersanding he deerminans of aggregae flucuaions is he second main objecive of macroeconomics. In his, and he chapers ha follow, we presen he main heories regarding he naure of aggregae flucuaions. In his chaper we sar by inroducing classical models of aggregae flucuaions. New classical models are essenially dynamic sochasic general equilibrium models (DSGE), based on opimizing households and firms, flexible wages and prices, and fully compeiive markes. Flucuaions in hese models are caused by real shocks o produciviy, household preferences and governmen expendiure, and he effecs of hese shocks are propagaed hrough endogenous dynamic processes, such as consumpion and invesmen. We sar wih he so called sochasic growh model, which is an exended sochasic version of he Ramsey model. The uiliy funcion of a represenaive household depends on boh consumpion of goods and services, and leisure, while random disurbances o real facors, such as produciviy, preferences and governmen expendiure cause aggregae flucuaions. 1 To be able o analyze he model, we make simplifying assumpions regarding he producion and uiliy funcions. Wihou hem he model becomes exremely complicaed. The dynamic analysis is conduced in discree raher han coninuous ime. Afer analyzing and characerizing his generalized new classical model, we also analyze a shor erm version of he model, wihou capial accumulaion. In his model, labor is he only variable facor of producion. This model, is comparable o he new keynesian models we shall analyze in subsequen chapers, and will help us pinpoin he main differences beween he new classical and he new keynesian approaches. This approach owes a lo o he papers by Kydland and Presco (1982), Long and Plosser (1983) and Presco (1986). 1 I is surveyed in King and Rebelo (1999). As we shall see, his approach has also influenced he new keynesian approach o aggregae flucuaions, which now also relies on dynamic sochasic general equilibrium models (DSGE).

2 As we shall see, in almos all classical models, he main impulses ha generae aggregae flucuaions are real, i.e shocks o produciviy, preferences and governmen expendiure. This is why his class of models is ofen referred o as he real business cycle model (RBC). Moneary shocks have no real effecs on oupu, employmen and capial accumulaion in his class of models, and only affec real money balances, and nominal variables such as he price level, inflaion and nominal wages and ineres raes The Naure and Key Characerisics of Aggregae Flucuaions As we have seen in Chaper 1, aggregae flucuaions are no characerized by some simple repeiive regulariy and seem o be characerized by randomness. The prevailing view oday, which daes back o Frisch (1930) and Slusky (1937), is ha economies are subjec o various kinds of random disurbances, which, hrough he operaion of economic ransmission mechanisms, affec oupu, employmen, real wages, real ineres raes, he price level and inflaion, and se in moion dynamic sochasic adjusmen processes. The way in which modern macroeconomiss approach and analyze economic flucuaions owns a lo o he following imporan observaion of Lucas (1977): Technically, movemens abou rend in gross naional produc in any counry can be well described by a sochasically disurbed difference equaion of very low order. These movemens do no exhibi uniformiy of eiher period or ampliude, which is o say, hey do no resemble he deerminisic wave moions which someimes arise in he naural sciences. Those regulariies which are observed are in he co-movemens among differen aggregaive ime series One is led by he facs o conclude ha, wih respec o he qualiaive behavior of co-movemens among series, business cycles are all alike. To heoreically inclined economiss, his conclusion should be aracive and challenging, for i suggess he possibiliy of a unified explanaion of business cycles, grounded in he general laws governing marke economies, raher han in poliical or insiuional characerisics specific o paricular counries or periods. (p. 9-10). This observaion by Lucas caused significan changes in he way in which all macroeconomic schools of hough are now approaching and ry o explain aggregae flucuaions. The radiional macroeconomic and macro-economeric models, from he 1950s o he 1970s, were deemed o have a number of weaknesses in he deailed sudy of economic flucuaions, and he impac of moneary and fiscal policy in relaion o he crieria of Lucas. The mos imporan of he weaknesses of radiional macroeconomic and macro-economeric models was ha he macroeconomic relaionships assumed were no explicily drawn from well defined microeconomic foundaions, based on iner-emporal opimizaion on he par of households and firms. Therefore one could no easily inerpre heir parameers, and be sure of heir sabiliy. This was he basis of he famous, and very poen, Lucas criique of economeric policy evaluaion. 2 See Lucas (1976) who concluded ha given ha he srucure of all economeric model consiss of opimal decision 2 rules of economic agens, and ha opimal decision rules vary sysemaically wih changes in he srucure of series relevan o he decision maker, i follows ha any change in policy will sysemaically aler he srucure of economeric models. (p. 41). He came o his conclusion afer having demonsraed he fragiliy of radiional economeric models of aggregae consumpion, invesmen and he Phillips curve.!2

3 The realizaion of hese weaknesses, which applied equally o keynesian and monearis models, gradually gave rise o alernaive models based on more saisfacory dynamic microeconomic foundaions The Sochasic Growh Model The firs model we shall focus on is he so-called sochasic growh model. This is a compeiive dynamic sochasic general equilibrium model, wihou exernaliies, asymmeric informaion, fricions and oher imperfecions of markes. This model is essenially a generalizaion of he Ramsey model. I no only excludes any marke imperfecions, bu also all issues relaed o he heerogeneiy of economic agens. The exended Ramsey model is herefore he naural saring poin for he sudy of aggregae flucuaions, like he original Ramsey model is he naural saring poin for he sudy of he long run growh. There are wo direcions in which he Ramsey mus be exended in order o sudy aggregae flucuaions. Firs, one should allow for random disurbances, which can cause flucuaions. Wihou random disurbances, he Ramsey model converges o a unique seady sae. The disurbances usually inroduced in he Ramsey model are disurbances in oal facor produciviy (echnology shocks), as well as real demand shocks, such as shocks o he preferences of consumers or real governmen expendiure. Since boh kinds of shocks are real - unlike moneary or nominal shocks - his model urns ou o be a real business cycle model (RBC). Second, in order o allow he Ramsey model o explain flucuaions no only in oal oupu, bu also employmen, employmen mus become endogenous. This is achieved hrough he inroducion of employmen in he uiliy funcion of a represenaive household, in order o esablish an endogenous labor supply funcion. The exended Ramsey model which we end up wih is a dynamic sochasic general equilibrium model (DSGE), in which flucuaions are caused by real shocks. There are a number of idenical households and firms, so his is a compeiive represenaive household model. Firms use labor and capial in order produce a homogeneous produc. They choose invesmen and employmen in order o maximize heir profis, while households choose consumpion and labor supply in order o maximize heir iner-emporal uiliy. The key variables and parameers of he model are as follows: Y oal oupu K physical capial L employmen A labor efficiency (produciviy) C oal privae consumpion C G oal real governmen expendiure N oal populaion δ depreciaion rae of capial!3

4 $ George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 ρ pure rae of ime preference of households r real ineres rae W real wage per employee The Behavior of Firms The economy consiss of a large number of idenical households and firms, ineracing hrough compeiive markes. Oupu and facor prices are hus given for every household and every firm. The represenaive firm has a producion funcion wih consan reurns o scale, which akes he Cobb-Douglas form. Thus, he aggregae producion funcion is also Cobb Douglas.! Y = K α (A L ) 1 α 0<α<1 (11.1) The demand for oupu consiss of privae consumpion, invesmen and governmen consumpion. Governmen consumpion is financed hrough non disorionary axaion, and, in each period, axes are equal o governmen consumpion. Thus, he equilibrium condiion in he oupu marke is given by,! Y = C + C G + K +1 K + δ K (11.2) Solving (11.2) for K$ we ge a capial accumulaion equaion of he form, +1! K +1 = K + Y C C G δ K (11.3) To he exen ha oal savings Y-C-C G exceed depreciaion invesmen δκ, here is accumulaion of capial. Labor and capial are paid heir marginal produc, as firms maximize profis aking he real ineres rae r and he real wage W as given. α K! W = (1 α ) (11.4) A L A r = α A L K 1 α δ (11.5) (11.1)-(11.5) describe he behavior of firms. Firms employ workers up o he poin where he marginal produc of labor equaes he real wage, and capial up o he poin where he marginal produc of capial equals he user cos of capial, equal o he real ineres rae plus he depreciaion rae The Represenaive Household!4

5 The economy is inhabied by a large number of idenical households, each of which has an infinie ime horizon. The represenaive household maximizes is expeced iner-emporal uiliy funcion, which depends on he pah of real consumpion of goods and services and leisure. The uiliy funcion is defined by, +s 1! U = E (11.6) 1+ ρ u(c +s,1 l +s ) N +s s=0 H where E is he mahemaical expecaions operaor and u is he per capia insananeous uiliy of he represenaive household. Per capia consumpion is given by c=c/n and per capia employmen by l=l/n. We shall assume ha he insananeous uiliy funcion is log-linear.! u = lnc + bln(1 l ), b > 0 (11.7) This assumpion is made in order o arrive a simpler funcional relaionships. However, like all simplificaions, his assumpion implies specific properies for he model Populaion, Efficiency of Labor and Governmen Expendiure Populaion increases exogenously a a rae n per period. Consequenly,! ln N = n _ + n, n < ρ (11.8) The final assumpions of he model concern he behavior of he wo main exogenous variables. Boh produciviy (labor efficiency), and governmen expendiure are supposed o be subjec o random disurbances. The sochasic process describing he evoluion of he efficiency of labor is given by, 3! ln A = a _ + g + a (11.9) where, A! a = η A a 1 + ε -1<η$ <1 (11.10) ε A follows a whie noise sochasic process. A (11.9) and (11.10) imply ha labor efficiency grows a an exogenous rae g, bu ha i is subjec o random disurbances a ha follow a saionary firs order auoregressive process. The assumpions abou (11.10) imply ha he impac of a echnological disurbance is gradually reduced over ime. Similar assumpions are made regarding he sochasic process ha describes he evoluion of real governmen expendiure. We assume ha real governmen expendiure is growing a an average rae 3 Mahemaical Annex 4 conains an inroducion o sochasic processes.!5

6 !! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 n+g, i.e. ha on average i remains consan relaive o oal oupu. However, we also assume ha real governmen expendiure is subjec o disurbances follow a saionary firs order auoregressive process. More paricularly, _! lng = c g G + (n + g) + c (11.11) where,! c G = η G c G G 1 + ε -1<η$ G <1 (11.12) ε G follows a whie noise sochasic process. 4 These elemens complee he srucure of he model. The wo mos imporan differences from he original Ramsey model are, firs, he inroducion of leisure ime in he uiliy funcion of he represenaive household, which poenially allows for flucuaions in employmen, and, second, he inroducion of random disurbances o labor efficiency (produciviy) and governmen expendiure, which lead o flucuaions around a long-erm rend. Before we look a he general properies of he model, i is worh considering he implicaions for he behavior of he represenaive household of he inroducion of leisure in he uiliy funcion, as well as he implicaions of uncerainy, in he form of random disurbances Labor Supply of he Represenaive Household The firs difference of his model from he Ramsey model arises from he inroducion of leisure ime in he uiliy funcion of he household, which makes labor supply endogenous. To analyze he imporance of his addiion, le us firs consider he problem of a household ha lives for a single ime period and has no asses. The problem of ha household is defined as he maximizaion of, lnc + bln(1 l) under he consrain c = Wl. The Lagrangian is defined by,! Λ = lnc + bln(1 l) + λ(wl c) (11.13) The firs order condiions for c and l are, 1! (11.14) c λ = 0 b 1 l + λw = 0 (11.15) The assumpion ha he processes driving labor produciviy and real governmen expendiure are AR(1) are made for 4 simpliciy, and can of course be generalized.!6

7 !! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 From he budge consrain c = Wl and (11.14) i follows ha λ = 1/(Wl). Subsiuing in (11.15), we ge ha, b 1 l + 1 l = 0 (11.16) From (11.16) i is apparen ha labor supply is independen of he real wage. This is because of he assumpion of logarihmic preferences, implying ha he elasiciy of subsiuion beween consumpion and leisure is equal o uniy. Thus, he subsiuion effec from a change in he real wage is couneraced by he income effec. However, his does no mean ha emporary changes in real wages do no affec labor supply. This can be seen if we look a he behavior of a household living for wo periods Iner-emporal Subsiuion in Labor Supply We shall nex analyze he behavior of a household living for wo periods, has no iniial wealh, and no uncerainy abou he real ineres rae or he real wage of he second period. The iner-emporal budge consrain of he household is given by,! c (11.17) 1+ r c = W l r W l 2 2 The Lagrangian is defined by, Λ = lnc 1 + bln(1 l) ρ lnc 2 + bln(1 l) 2 The household chooses consumpion and labor supply for each of he wo periods. From he firs order condiions for labor supply, b! = λw 1 (11.18) 1 l 1 b! = 1+ ρ (11.19) 1 l 2 1+ r λw 2 Dividing (11.19) by (11.18), ( ) + λ W 1 l r W l c r c 2 1 l! 1 = 1+ ρ W 2 (11.20) 1 l 2 1+ r W 1 (11.20) implies ha he relaive labor supply in he wo periods depends posiively on he relaive real wage in he wo periods. The higher he real wage of he firs period in relaion o he real wage of he second period, he higher he labor supply of he firs period, in relaion o ha of he second. The household subsiues labor beween periods, depending on relaive real wages beween!7

8 periods. Because of logarihmic preferences, he iner-emporal subsiuion elasiciy is equal o one. Moreover, he higher he real ineres rae r he greaer he labor supply of he firs period compared o he second period. The increase in he ineres rae increases he araciveness o work in he firs period and save for he second period, compared o working in he second period. I has he opposie effec of he pure rae of ime preference rae ρ. These effecs of relaive wages over ime and he real ineres rae on labor supply are known as iner-emporal subsiuion in labor supply. Such effecs obviously generalize o a muli period seing. Consequenly, flucuaions in real wages and he real ineres rae can cause flucuaions in employmen, alhough permanen changes in real wages do no affec labor supply in a model wih logarihmic preferences Uncerainy and he Behavior of he Represenaive Household The second elemen ha differeniaes he sochasic growh model from he Ramsey model is uncerainy arising from he sochasic disurbances. Therefore, he expecaions of he represenaive household for fuure developmens play a significan role. I can be shown ha, for he general case when he household maximizes he expeced ineremporal uiliy funcion (11.6), he Euler equaion for consumpion akes he form, 1! = 1 (11.21) c 1+ ρ E 1 ( 1+ r +1 ) c +1 The mahemaical expecaion of he produc of wo random variables is no equal o he produc of mahemaical expecaions. I is equal o he produc of mahemaical expecaions plus he covariance of wo random variables. Thus, (11.21) implies, 1! = 1 (11.22) c 1+ ρ E 1 1 E ( 1+ r +1 ) + Cov,( 1+ r +1 ) c +1 c +1 On he oher hand, from he firs-order condiions for consumpion and labor supply, he raio of consumpion o leisure is a posiive funcion of he real wage of he form, c! = W (11.23) 1 l b (11.23) links labor supply (leisure) and consumpion wih he real wage. I includes only curren variables, as here is no uncerainy in he curren period. 5 The concep of iner-emporal subsiuion in labor supply was firs analyzed in an imporan paper by Lucas and Rapping (1969). For an empirical invesigaion of is significance for flucuaions in employmen in he USA and he UK see Alogoskoufis (1987a,b).!8

9 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 Equaions (11.21) and (11.23) are he basic equaions ha describe he behavior of households in his model. We can now examine he properies of he model. This model is no easy o solve analyically, as i conains facors ha are linear, and facors ha are log-linear in is variables. The properies of he model can be described if we simplify i furher, or if we use a log-linear approximaion around he balanced growh pah, and solve i numerically for specific values of he parameers. In he Annex o his chaper, we presen he Campbell (1994) log-linear approximaion of he full model, around is balanced growh pah. This allows us o describe he full properies of he model hrough a numerical simulaion. In he remainder of his secion we shall concenrae on he properies of a simplified version of he model A Simplified Version of he Sochasic Growh Model To furher analyze he sochasic growh model, we will consider a special case wihou governmen expendiure and a depreciaion rae of 100%. The equaions ha describe he accumulaion of capial and he deerminaion of he real ineres rae are hen simplified o,! K +1 = Y C (11.24) 1+ r = α A L K 1 α (11.25) Because of he assumpion of compeiive markes and he absence of exernaliies, he equilibrium of he model is Pareo opimal. We shall define he properies of he model by solving for he compeiive equilibrium. We will focus on wo variables. Labor supply per person l, and he savings rae s. Defining he savings rae we also deermine aggregae consumpion as, C=(1-s) Y. We will focus on boh behavioral equaions of he represenaive household (11.21) and (11.23). Once we deermine labor supply and he savings rae, all he res follows auomaically eiher from equilibrium condiions or definiions. From (11.21), afer we use he relaions,! c = (1 s )Y / N,! 1+ r +1 = αy +1 / K +1,! K +1 = s Y we end up ha he savings rae is consan and given by,! s _ α(1+ n) = (11.26) 1+ ρ!9

10 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 Because of logarihmic preferences, he savings rae is independen of he real ineres rae and consan. From (11.23), afer we noe ha,! c = (1 s _ )Y / N,! W = (1 α )Y / (l N ) we end up wih he conclusion ha labor supply per household member is consan and given by,! l _ 1 α = (11.27) (1 α ) + b(1 s _ ) Labor supply is consan because he impac of he shocks in echnology (labor efficiency) on he real wage and he real ineres rae cancel each oher ou, so here is no iner-emporal subsiuion. This is due o he specific assumpion ha we made in order o simplify he model, and as one can see from he analysis of he full model in he Annex is no a general feaure of he model. We can now deermine flucuaions in oal oupu. Log-linearizing he producion funcion (11.1) we ge,! lny = α ln K + (1 α )(ln A + ln L ) (11.28) We know ha,! K = s _ Y 1 and ha,! L = l _ N. Therefore,! lny = α ln s^ + α lny 1 + (1 a)(ln A + lnl^+ ln N ) (11.29) We can subsiue he logarihm of A and N from equaions (11.8) and (11.9). This implies,! lny = α ln s^ + α lny 1 + (1 α ) (a _ + g) + v A + (lnl^+ n _ + n) (11.30) We can express (11.30), as,! y = α y 1 + (1 α )a (11.31) where,! y = lny lny _, are percenage deviaions of oupu from rend oupu. The logarihm of rend oupu, is defined as, lny _ = α 1 α ln s_ + lnl _ + a _ + n _ + (n + g) From (11.10) and (11.31), we end up wih,!10

11 A! y = (α + η A )y 1 αη A y 2 + (1 α )ε (11.32) From (11.32), he percenage deviaions of oal real oupu from rend follow a second order auoregressive process (AR(2)). Because α is low (abou 1/3), he dynamic behavior of oal real oupu depends primarily on he degree of persisence of produciviy shocks. If he persisence of produciviy shocks ηa is high, hen we have considerable persisence in he flucuaions of oupu. Oherwise, he persisence of oupu flucuaions around rend is low. For example, running a regression of he log of US real GDP on a linear rend, and is wo lags, using annual daa for he period , one ges, log(gdp) = log(gdp) log(gdp) (0.083) (0.086) (0.087) (0.0012) R 2 =0.999, DW=2.020 T=125 6 From his regression, which seems o fi he daa for he US real GDP quie well, he esimae of ηa is equal o (s.e ), and he esimae of α is equal o (s.e ). The esimae of g +n, he long run growh rae, is equal o (s.e ). From hese esimaes, his simplified sochasic growh model can accoun for flucuaions in US GDP if he persisence of produciviy shocks is of he order of 0.8. If he degree of persisence of produciviy shocks is zero, hen (11.32) would simplify o, A! y = α y 1 + (1 α )ε (11.33) which obviously canno accoun for he flucuaions of US real GDP. The simplified form of he model, summarized in equaion (11.32), conains many of is essenial elemens, and provides he basic new classical accoun of flucuaions in oal oupu (GDP) around rend, mainly on he basis of persisen produciviy shocks and capial accumulaion. However, many oher feaures of aggregae flucuaions are no adequaely described by his simplified version of he sochasic growh model. 1. The consan savings raio. This means ha consumpion will display he same degree of variabiliy as oupu and invesmen, which does no end o happen in realiy. 2. The consan employmen rae. In realiy, he employmen rae is no consan over he business cycle. Employmen is pro-cyclical, moving in he same direcion as oupu. 3. Real Wages over he Business Cycle. In he simplified sochasic growh model real wages are pro-cyclical and equally volaile as GDP per capia, which is no always he case. When one examines he more general form of he model, assuming a low depreciaion rae, as we do in he Annex o his chaper, many of hese weaknesses are correced, as savings, invesmen and 6 Sandard errors are in parenheses below esimaed coefficiens. R 2 is he coefficien of deerminaion, DW is he Durbin Wason saisic and T he number of observaions.!11

12 employmen also display flucuaions in response o produciviy shocks. For example, in he full sochasic growh model, analyzed in he Annex, he savings rae is no consan, and consumpion ends o be less variable han invesmen and oupu. In addiion, in he full sochasic growh model, he employmen rae is pro-cyclical, and moves in he same way as oupu. Moreover, he inroducion of public expendiure shocks or preference shocks could relax he sric dependence of flucuaions in real wages on flucuaions in aggregae produciviy A New Classical Model wihou Capial In wha follows we shall focus on an analyically simpler version of he new classical model of aggregae flucuaions, in which he only variable facor of producion is labor. We shall hus absrac from capial accumulaion. In his analyically simpler model we allow for a more general approach o he preferences of he represenaive household, and also disinguish beween nominal and real variables, in order o consider he deerminaion of he level of prices and wages, inflaion and nominal ineres raes, and he role of moneary facors in classical models The Represenaive Household The represenaive household is assumed o maximize, s 1! E (11.34) 1+ ρ u(c +s, L +s ) s=0 where C is consumpion and L is labor supply. We assume ha,! u C = u > 0,! u C C,!,!. (11.35) C = 2 u C 0 u 2 L = u 0 u L L L = 2 u C 0 2 The consrains under which he maximizaion akes place are given by, P C i B B 1 + W L T lim E B 0 T T (11.36) (11.37) where P is he price level, W he nominal wage, i he nominal ineres rae, B a nominal one period bond, and T an exogenous ransfer of nominal income o he household (dividends, governmen ransfers of axes). From he firs order condiions i follows ha, This is essenially a shor run classical model, he shor run being defined as he ime span for which he capial 7 sock is fixed. In he shor run firms care abou he uilizaion rae of heir capial sock, varying employmen, which is he only variable facor of producion. For a more exensive reamen of his shor run new classical model, see Gali (2008). This simplified model will in fuure chapers allow us o direcly compare he new classical wih he new keynesian approach.!12

13 ! u L = W (11.38) u C P 1! = 1 (11.39) 1+ i 1+ ρ E u C+1 P u C P +1 We assume ha he per period uiliy funcion is given by, 1 θ 1+λ! U(C, L ) = C, where θ>0 and λ>0 (11.40) 1 θ L 1+ λ The firs order condiions for he problem of he represenaive household in his case ake he form, W P! = C θ λ L (11.41) 1! = 1 θ (11.42) 1+ i 1+ ρ E C +1 P C P +1 (11.41) and (11.42) can be wrien in log-linear form, as,! w p = θc + λl (11.43) ( )! c = E (c +1 ) 1 (11.44) θ i E (π ) ρ +1 where w=lnw, p=lnp, c=lnc, l=lnl and! π = p p 1 is he inflaion rae The Represenaive Firm Producion of he represenaive firm is a posiive funcion of employmen, and is described by an aggregae producion funcion of he form, 1 α! Y = A L (11.45) where Α>0 and 0<α<1 are exogenous echnological parameers. α is a consan, while A follows an exogenous sochasic process. The represenaive firm chooses employmen in order o maximize profis, for given nominal wages and prices. Profis are deermined by,! P Y W L (11.46)!13

14 Profi maximizaion implies ha employmen will be deermined so as o equae he marginal produc of labor o he real wage. Thus, W P α! = (1 α )A L (11.47) One can solve he marginal produciviy condiion for he price level. The inerpreaion is ha he produc price is equal o marginal cos. W! P = (11.49) α (1 α )A L Log-linearizing he firs order condiion (11.47) we ge,! w p = a αl + ln(1 α ) (11.50) where a=lna. Log-linearizing he producion funcion (11.45) we ge,! y = a + (1 α )l (11.51) Having deermined he behavior of households and firms, we can now analyze he equilibrium in his model General Equilibrium In he basic form of his model we shall assume ha here is no invesmen or public consumpion. Accordingly, in produc marke equilibrium, consumpion will be equal o oal oupu.! y = c (11.52) The equilibrium condiion (11.52) will deermine he real ineres rae. The condiion for equilibrium in he labor marke will require ha labor demand, as implied by (11.50), should be equal o labor supply, as implied by (11.43). This will deermine he real wage. The model consiss of equaions (11.43), (11.44), (11.50) and (11.51) and he equilibrium condiion (11.52), and deermines employmen, oupu, consumpion, real wages and he real ineres rae, as funcions he exogenous labor produciviy a. The real ineres rae is defined by he Fisher equaion, as, 8 To quoe from Fisher (1896), When prices are rising or falling, money is depreciaing or appreciaing relaive o 8 commodiies. Our heory would herefore require high or low ineres according as prices are rising or falling, provided we assume ha he rae of ineres in he commodiy sandard should no vary. (p. 58). The rae of ineres in he commodiy sandard is he real ineres rae, and rising or falling prices are expeced inflaion.the Fisher equaion was furher elaboraed in Fisher (1930), where i was made even clearer ha Fisher referred o expeced inflaion.!14

15 ! r = i E (π +1 ) (11.53) Solving he model for he five endogenous variables, we ge,! l = η LA a + l _ (11.54) 1 θ where,! η LA = and,! l _ ln(1 α ) =. θ(1 α ) + α + λ θ(1 α ) + α + λ! y = c = η YA a + y _ (11.55) 1+ λ where,! η YA = 1+ (1 α )η LA = and,! y _ = (1 α )l _. θ(1 α ) + α + λ! w p = η WA a + ω _ (11.56) θ + λ where,! η WA = 1 αη LA = and,! ω _ = ( θ(1 α ) + λ)l _. θ(1 α ) + α + λ! r = ρ +θη YA E (Δa +1 ) (11.57) (11.54), (11.55), (11.56) and (11.57), along wih he produc marke equilibrium condiion (11.52), deermine he five endogenous variables, as a funcion of he exogenous shock o labor produciviy a. I worh noing ha flucuaions in employmen, oupu, consumpion and real wages are a funcion only of flucuaions in labor produciviy and flucuaions in he real ineres rae depend on flucuaions in he expeced rae of change of produciviy. Oupu, consumpion and real wages are posiive funcions of produciviy, while employmen is a posiive funcion of produciviy only if θ<1, i.e. if he iner-emporal elasiciy of subsiuion of consumpion is greaer han one. If θ>1, employmen is a negaive funcion of produciviy, while if θ=1, employmen is independen of produciviy. This is because if θ<1 he subsiuion effec dominaes over he income effec, afer a change in produciviy and real wages, and employmen rises. If θ>1 he income effec dominaes over he subsiuion effec, while in he case θ=1 he wo effecs cancel each oher ou, and employmen is no affeced. Only real facors, such as real produciviy, affec flucuaions in real variables. As in he sochasic growh model, moneary facors such as money supply and nominal ineres raes have no impac on he evoluion of real variables Moneary Facors in he New Classical Model!15

16 ! George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 In order o examine he impac of moneary facors in he new classical model, we shall assume he exisence of a money demand funcion by households and firms, which, in logarihms, akes he form, m p = y ηi (11.58) where η is he semi-elasiciy of money demand wih respec o he nominal ineres rae. From he definiion of he real ineres rae hrough he Fisher equaion (11.53), he nominal ineres rae is equal o,! i = r + E (π +1 ) (11.59) where he real ineres rae r is deermined by (11.57) and is independen of moneary facors. We will show ha, like in he case of he models analyzed in Chaper 10, when he cenral bank follows a rule for he money supply, hen he model deermines he price level and he level of inflaion and nominal ineres raes. If he cenral bank pegs he nominal ineres rae, he price level and he level of he nominal money supply canno be deermined, unless he nominal ineres rae reacs sufficienly srongly o changes in he price level Exogenous Pah for he Money Supply If he cenral bank deermines an exogenous pah for he money supply, hen, from (11.58) and (11.59) i follows ha,! p = η (11.60) 1+ η E (p ) η m 1 ( 1+ η y ηr ) Under he assumpion ha η>0, he soluion of (11.59) implies ha, j ( )! p = 1 η (11.61) j=0 1+ η 1+ η E m + j y + j + ηr + j From (11.61), he price level and inflaion are deermined as funcions of he exogenous pah of he money supply, and he pahs of real oupu and he real ineres rae, which, as we have seen, are independen of moneary facors in he new classical model. The nominal ineres rae is deermined endogenously from (11.61) and (11.59) Exogenous Pah for he Nominal Ineres Rae If we assume ha he cenral bank follows an exogenous pah for he nominal ineres rae, hen, from he Fisher equaion (11.59), i follows ha,! E (π +1 ) = i r (11.62)!16

17 (11.62) does no deermine inflaion, bu expeced inflaion, given he exogenous pah of nominal ineres raes. (11.62) is consisen wih any price level pah ha saisfies,! p +1 = p + i r + ξ +1 (11.63) ( ) = 0 where ξ is any shock ha saisfies! E ξ +1. (11.63) suggess ha here are muliple equilibria for he price level and inflaion, depending on ξ. This price level indeerminacy when he cenral bank follows an exogenous pah for he nominal ineres rae is also ransferred o he money supply, hrough he money demand funcion (11.58). Consequenly neiher he money supply nor he price level can be deermined uniquely when he cenral bank follows an exogenous pah for he nominal ineres rae. 9 Such an equilibrium is ofen referred o as a bubble or a sunspo equilibrium, since he equilibrium depends on exernal facors ha have nohing o do wih economic fundamenals. However, no all ineres rae rules resul in price level indeerminacy. As suggesed more han a cenury ago by Wicksell (1898), and we demonsraed in Chaper 10, if he cenral bank condiions is nominal ineres rae on he price level, or inflaion, hen price level and inflaion indeerminacy does no necessarily follow An Inflaion Based Nominal Ineres Rae Rule Cenral banks predominanly use he nominal ineres rae as heir preferred moneary insrumen. However, hey follow policies according o which he pah of nominal ineres raes is no exogenous, bu depends on pas, curren and expeced fuure economic developmens, mainly inflaion. For example, if inflaion rises, cenral banks usually raise nominal ineres raes in order o reduce i, and vice versa. This was afer all he essence of he Wicksell rule. Le us herefore assume he following rule for deermining nominal ineres raes, 10! i = ρ + φπ (11.64) where φ>0 is he reacion of he cenral bank nominal ineres rae o inflaion. From (11.59) και (11.64) we herefore have ha, 9 The classic analysis of he appropriae choice of moneary insrumens in a simple sochasic macro model was Poole (1970). As we argued in Chaper 10, Sargen and Wallace (1975) demonsraed ha under raional expecaions, a nonconingen ineres rae arge leads o price level indeerminacy and insabiliy in such a model. However, i is now acceped ha his problem does no arise in he case of coningen ineres rae rules ha make he nominal ineres rae depend on he price level (McCallum 1981), or a sufficienly sensiive posiive funcion of inflaion. See Woodford (2003) Ch.1 for he relevan argumens. In addiion, cenral banks, have been consisenly using ineres raes and no moneary aggregaes as heir main moneary policy insrumen, especially in he las hiry years. As noed by Bernanke (2006), In pracice, he difficuly has been ha, deregulaion, financial innovaion, and oher facors have led o recurren insabiliy in he relaionships beween various moneary aggregaes and oher nominal variables.. Wicksell (1898) was probably he earlies advocae of a such a sabilizing ineres rae rule. We have shown how 10 Wicksell's rule resuls in price level deerminacy in he models of Chaper 10. Here, we use a version of Wicksell s rule for inflaion raher han he price level.!17

18 ( ) + 1 ( φ r ρ)! π = 1 (11.65) φ E π +1 where he real ineres r depends only on real facors, as is he implicaion of new classical models. 11 Solving (11.65) under raional expecaions, s+1 1! π =, if φ>1 (11.66) s=0 φ E ( r +s ρ) ( ) + ξ +1! π +1 = φπ r ρ, if φ 1 (11.67) Thus, if he reacion of he cenral bank nominal ineres raes o inflaion is sufficienly pronounced (φ>1), here is no indeerminacy problem for inflaion. The fundamenal soluion is given by (11.66). If he reacion of he nominal ineres raes o inflaion is no sufficienly pronounced (φ 1), hen he problem of inflaion indeerminacy and he possibiliy of price bubbles remains. In any case, as we have already shown, in he new classical model of aggregae flucuaions only real facors affec flucuaions in real variables. Moneary facors and moneary policy only affec real money balances and nominal variables such as he price level and inflaion, nominal ineres raes and he nominal money sock Conclusions New classical models of aggregae flucuaions imply ha aggregae flucuaions are caused by real facors. This is why such models are ofen called real business cycle models. New classical models are dynamic sochasic general equilibrium models (DSGE) based on opimizing behavior by boh households and firms, flexible prices, and fully compeiive markes. Households maximize heir iner-emporal uiliy, firms maximize he presen value of heir profis, and markes funcion efficienly. If he compeiive general equilibrium models of his kind could explain all he feaures of aggregae flucuaions, hen here would be no need for models ha sress disorions in produc and labor markes, and oher marke inefficiencies. However, new classical models have a number of weaknesses as models of aggregae flucuaions. Firs, hese models canno accoun for he real effecs of nominal and moneary shocks. For example, i is widely acceped ha he Grea Depression of he 1930s was caused by moneary and no real shocks. Similar views are prevalen regarding he recession of , which was one of he deepes pos World War II recessions. 11 See for example, equaion (11.57). We shall furher assume ha he process deermining he real ineres rae is saionary.!18

19 Second, even hough new classical models can accoun for employmen flucuaions, hey only do so on he basis of iner-emporal subsiuion in labor supply. This explanaion, however, is no sufficien o explain he phenomenon of high and persisen unemploymen and he widely held view ha unemploymen is an involunary condiion for hose who experience i, and no he resul of a volunary raional choice. For hese reasons, and despie he fac ha new classical models are heoreically consisen, many economiss consider hem exreme as he sole explanaion of aggregae flucuaions. The alernaive class of models are new keynesian models, which assume ha nominal wages and/or prices canno adjus immediaely in order o equilibrae labor and produc markes. Thus, following nominal shocks, quaniies have o adjus oo, resuling in flucuaions in real variables, deviaions of oupu and oher real variables from heir seady sae values and involunary unemploymen. I is o hese models ha we now urn.!19

20 Annex o Chaper 11: A Log-Linear Approximaion o he Sochasic Growh Model This Annex ses ou he sochasic growh model and derives an approximae analyical soluion, based on a log-linear approximaion around he seady sae equilibrium. Following Campbell (1994), he model is hus ransformed ino a sysem of log-linear sochasic difference equaions, which can be solved by he mehod of undeermined coefficiens. We shall solve he model assuming ha governmen expendiure is equal o zero. The firs equaion of he model is he producion funcion,! Y = K α (A L ) 1 α, 0<α<1 (A.11.1) The second is he capial accumulaion process,! K +1 = (1 δ )K + Y C (A.11.2) Finally here is a represenaive household which maximizes, 1! U = E (A.11.3) 1+ ρ ( lnc + bln(n L )) =0 subjec o he accumulaion process (A.11.2). Firms maximize profis, subjec o he producion funcion, and se he marginal produc of capial and labor equal o he real ineres rae and he real wage respecively. I hus follows ha, α K! w = (1 α ) (A.11.4) A L A! r = α A L (A.11.5) δ K 1 α From he firs order condiions for he maximizaion of he uiliy of he represenaive household i also follows ha, 1! = 1+ n (A.11.6) C 1+ ρ E 1 ( 1+ r +1 ) C +1 C! = w (A.11.7) N L b (A.11.6) is a sochasic version of he Euler equaion for aggregae consumpion, and (A.11.7) relaes curren consumpion and leisure o he real wage.!20

21 A.11.1 Seady Sae In he seady sae, aggregae variables such as oupu, effecive labor, capial and consumpion grow a a rae g+n. Thus, from (A.11.6), he seady sae real ineres rae is deermined by he condiion,! 1+ r = (1+ ρ)(1+ g) (A.11.8) (A.11.8) implies ha,! r = (1+ ρ)(1+ g) 1! ρ + g (A.11.9) From he marginal produciviy condiion for capial (A.11.5), he seady sae raio of oupu o capial is deermined by, 1 α Y! (A.11.10) K = AL K = r + δ α = ρ + g + δ α From (A.11.10), he seady sae raio of effecive labor o capial is deermined by, 1 AL! (A.11.11) K = ρ + g + δ 1 α α Finally, from he capial accumulaion process (A.11.2) and (A.11.10), he seady sae consumpion o oupu raio is given by, C! (A.11.12) Y = 1 α n + g + δ ρ + g + δ Noe ha he las erm in (A.11.12) is he seady sae savings rae. A.11.2 Log-linearizing he Model around he Seady Sae We shall consider flucuaions of he endogenous variables around he seady sae. Ouside he seady sae he model is a sysem of non-linear equaions in he logs of produciviy, capial, labor, oupu and consumpion. Non-lineariies arise because of he depreciaion rae, he equaion for capial accumulaion, he variable savings rae and he variable employmen rae. Unlike he simplified model we examined in he ex, we shall seek an approximae analyical soluion, by aking a log-linear approximaion of all equaions around he seady sae. The Cobb Douglas producion funcion can be log-linearized direcly. From (A.11.1), i follows ha,! y = αk + (1 α )(a + l ) (A.11.13)!21

22 where lowercase leers denoe he difference of he log of he relevan variable from is seady sae value. The capial accumulaion equaion (A.11.2) is obviously no log-linear. Dividing by K, i can be wrien as, K! +1 (1 δ ) (A.11.14) K = Y 1 C Taking logs, (A.11.14) can be wrien as,! ln[e Δk +1 (1 δ )] = y k + ln[1 e(c y ) ] (A.11.15) Taking a firs order Taylor approximaion of (A.11.15) around he seady sae, and using he loglinear version of he producion funcion (A.11.13), we end up wih he following log-linear approximaion of he accumulaion equaion around he seady sae,! k +1! λ 1 k + λ 2 (a + l ) + (1 λ 1 λ 2 )c (A.11.16) where,! λ 1 = 1+ ρ + g (1 α )(ρ + g + δ ),! λ 2 =. 1+ g α(1+ g) We nex urn o he deerminaion of he real ineres rae and he Euler equaion for consumpion. From he marginal produciviy condiion for capial, (A.11.5), i follows ha,! 1+ r +1 = α A L (A.11.17) + (1 δ ) Taking a log-linear approximaion of (A.11.17) around he seady sae, we ge ha,! r +1! λ 3 a +1 + l +1 k +1 (A.11.18) where, K +1 K 1 α ( ) (1 α )(ρ + g + δ )! λ 3 =. 1+ ρ + g Y Subsiuing (A.11.18) in he Euler equaion for consumpion (A.11.6), and assuming he variables on he righ hand side are joinly log-normal and homoskedasic, he Euler equaion for consumpion can be wrien as,! c = E c +1 E r +1 = E c +1 λ 3 E (a +1 + l +1 k +1 ) (A.11.19)!22

23 Log-linearizing he marginal produciviy condiion for labor (A.11.4), and aking deviaions from seady sae, we ge ha deviaions of he log of he real wage from seady sae are given by,! w = α(k l ) + (1 α )a (A.11.20) Finally, log-linearizing he firs order condiion for consumpion and leisure, (A.11.7), using he marginal produciviy condiion for employmen (A.11.20) o subsiue for he logarihm of he real wage, we ge,! l = ν αk + (1 α )a c (A.11.21) where, ( ) 1 (L _ / N)! ν =. (L _ / N) α(1 (L _ / N))! L _ / N is he seady sae labor supply as a percenage of oal available ime. Following Presco (1986) we shall assume ha his is equal o one hird. In order o close he model, we need only specify he exogenous sochasic process driving produciviy a. We shall coninue o assume, as in he main ex, ha i follows an AR(1) process of he form, A! a = η A a 1 + ε,! 0 < η A < 1 (A.11.22) A.11.3 Solving he Model The model consiss of equaions (A.11.13), (A.11.16), (A.11.18), (A.11.19), (A.11.20), (A.11.21), and deermines flucuaions around he seady sae for oupu, he capial sock, consumpion, employmen, he real ineres and he real wage. The exogenous shock driving he flucuaions is a produciviy (echnological) shock, ha follows he AR(1) process in (A.11.22). We can firs solve he sub-sysem of (A.11.16), (A.11.19) and (A.11.21) for capial, employmen and consumpion, and hen subsiue in he oher hree equaions o deermine oupu, he real ineres rae and he real wage. The easies way o solve he model analyically is o use he mehod of undeermined coefficiens. We sar from he equaion for consumpion, and conjecure ha consumpion will be a linear funcion of he wo sae variables k and a, of he form,! c = η CK k + η CA a (A.11.23) where ηck and ηca are coefficiens o be deermined. Subsiuing (A.11.23) in he employmen equaion (A.11.21), we ge he soluion for employmen as,!23

24 ! l = η LK k + η LA a (A.11.24) where,! η LK = να η CK, and! η LA = 1 α η CA. Subsiuing (A.11.23) and (A.11.24) in he capial accumulaion equaion (A.11.16), and making use of he exogenous process (A.11.22), we ge he soluion for he accumulaion of capial as,! k +1 = η KK k + η KA a (A.11.25) where,! η KK = λ 1 + (1 λ 1 λ 2 )η CK + λ 2 η LK, and,! η KA = λ 2 + (1 λ 1 λ 2 )η CA + λ 2 η LA. Finally, we can subsiue (A.11.24) and (A.11.25) in he Euler equaion for consumpion (A.11.19), and ake he raional expecaions soluion, using he exogenous process (A.11.22) as well. We hen find ha, ( )! c = λ 3η KK (1 η LK ) k λ 3 η A (1+ η ) η (1 η ) LA KA LK a (A.11.26) 1 η KK 1 η A Comparing coefficiens beween (A.11.26) and (A.11.23), we can deermine he undeermined coefficiens ηck and ηca. A.11.4 Aggregae Flucuaions around he Seady Sae. We can now use he soluion we have obained o characerize he flucuaions of he various aggregaes around he seady sae. From (A.11.25) and (A.11.22), flucuaions in he capial sock are deermined by, A! k +1 = (η KK + η A )k η KK η A k 1 + η KA ε (A.11.27) Flucuaions of he capial sock around is seady sae value follow a saionary AR(2) process. Subsiuing (A.11.27) and (A.11.22) in he consumpion equaion (A.11.23), we can see ha flucuaions in consumpion around is seady sae follow a saionary ARMA(2,1) process of he form,! c = (η KK + η A )c 1 η KK η A c 2 + η CA ε A A + ( η CK η KA η CA η KK )ε 1 (A.11.28) Subsiuing (A.11.27) and (A.1.22) in he employmen equaion (A.11.24), we can see ha flucuaions in employmen around is seady sae follow a saionary ARMA(2,1) process of he form,! l = (η KK + η A )l 1 η KK η A l 2 + η LA ε A A + ( η LK η KA η LA η KK )ε 1 (A.11.29)!24

25 Finally, subsiuing (A.11.27) and (A.11.28) in he log-linear version of he aggregae producion funcion (A.11.13), flucuaions of oupu around is seady sae follow,! y = (η KK + η A )y 1 η KK η A y 2 + (1 α )(1+ η LA )ε A A (1 α )( η KK (1+ η LA ) η LK η KA )ε 1 (A.11.30) Thus, flucuaions of oupu around is seady sae follow an ARMA(2,1) process as well. In Figure 11.1 we presen he resuls of a dynamic simulaion of he model, for a 1% posiive shock in produciviy a. The parameer values we used in he simulaion were α=0.333, ρ=0.02, g=0.02, δ=0.03, ν=2, ηa=0.90. As can be seen from he simulaions, all real variables move pro-cyclically, as innovaions in produciviy affec oupu, he capial sock, consumpion and employmen in he same direcion. Real wages and he real ineres rae also move pro-cyclically. Gradually, all variables converge o he seady sae unless he sysem is disurbed by anoher shock.!25

26 George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chaper 11 Figure 11.1 Dynamic Simulaion of he Sochasic Growh Model following a 1% persisen shock o produciviy!26

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