Macroeconomic Theory III: Competitive Equilibrium (Real) Business Cycles

Transcription

1 Macroeconomic Theory III: Compeiive Equilibrium (Real) Business Cycles Gavin Cameron Lady Margare Hall Michaelmas Term 2004

2 inroducion Real business cycle models are Walrasian hey feaure compeiive markes, and have no exernaliies or oher marke failures. RBC models derive from exensions o he Ramsey model in wo dimensions. Firs, hey need a source of real shocks, usually eiher hrough echnology or governmen spending. Second, hey need a propagaion mechanism, o explain why an iniial shock persiss for some period of ime. In paricular, earlier models such as Ramsey or Solow assume a consan supply of labour. Business cycles clearly feaure procyclical labour inpus, however.

3 he economics of Robinson Crusoe Robinson Crusoe lives on his own on an island and has he choice beween fishing (producion), making nes (invesmen) and swimming (leisure). He faces he following shocks: A big school of fish nearby (boom): producion rises because produciviy is high and Crusoe chooses less leisure. A rain-sorm (recession): producion falls because produciviy is low and Crusoe sis in his hu, possibly making nes (rise in invesmen), possibly eaching his parro o speak. Aack of he Cannibals (war-ime boom): producion (including defence) rises since he spends his whole ime defending himself. Invesmen is crowded ou since he has less ime o spare o make nes.

4 why real business cycles? Unil Kydland and Presco s groundbreaking 1982 paper, economiss ried o capure business cycles using disequilibrium models. Full-employmen is considered an equilibrium: ha is, as a siuaion where each worker s and each producer s preferences are saisfied and anyhing less han fullemploymen is a disequilibrium. The disequilibria is caused by differen hings in differen models, such as money-illusion, imperfec compeiion or some form of nominal rigidiy. In he RBC models, each sage of he business cycle is viewed as an equilibrium he rough as well as he peak. This is no o say ha workers prefer slumps o booms, jus ha slumps represen undesired, undesirable and unavoidable shifs in he consrains ha people face, bu ha given hose consrains, markes reac efficienly and people achieve he bes oucomes ha circumsances permi. Of course, even some New Keynesian models such as hose of coordinaion failures rea each possible oucome as an equilibrium. Bu hose equilibria can be Pareo-ranked, whereas RBC models ypically have a single, Pareoopimal equilibrium.

5 dynamic properies of GDP Wha would he sochasic process of GDP have o look like o generae a business cycle? Random walk in oupu: (1) Y = a+ Y + e 1 auoregression abou a rend: (2) Y = by + c+ e 1 wih b around Noe he parameers a and c capure he growh rend. Second-order auoregression: (3) Y = by dy + c+ e 1 2 when he firs lag has a posiive coefficien and he second is negaive, oupu can be hump-shaped. Therefore, a model ha produces an AR(2) oupu funcion can simulae a business-cycle.

6 producion in a baseline RBC model The producion funcion is Cobb-Douglas: (4) α 1 α Y = K ( AL) Oupu is divided among consumpion, invesmen and governmen purchases: (5) Y C + I + G Fracion δ of capial depreciaes each period. period +1 is: (6) K = K + I δ K = K + Y C G δ K + 1 Thus he capial sock in Labour and capial are paid heir marginal producs. Thus he real wage and he real ineres rae in period are α K α α w = (1 α) K ( AL) A = (1 α) A AL r AL K 1 α = α δ

7 households in a baseline RBC model The represenaive household maximizes he expeced value of (7) ρ N U = e u( c,1 ) = 0 H u(.) is he insananeous uiliy funcion of he represenaive member of he infiniely-lived household and ρ is he discoun rae. N is he populaion and H is he number of households; hus N /H is he household size. Populaion grows exogenously a rae n: N n (8) ln N = N + n N = e + where n<ρ Thus he level of N is given by. The insananeous uiliy funcion, u(.) has wo argumens. The firs is consumpion per household member, c. The second is leisure per member, which is he difference beween he ime endowmen (normalized o 1) and he amoun each member works. Since all households are he same, c=c/n and = L / N. For simpliciy, assume u(.) is log-linear in he wo argumens: (9) wih b>0 (marginal disuiliy of labour). u = ln c + bln(1 )

8 echnology in a baseline RBC model To capure rend growh, he model assumes ha echnology grows a a consan rae subjec o random disurbances. (10) ln A = A + g+ A where reflecs he effecs of he shocks. firs-order auoregression: A is assumed o follow a (11) A = ρ A + ε where 1<ρ A <1 A 1 A, where he ε A erms are whie-noise disurbances a series of mean zero shocks wih no serial correlaion. (11) shows ha he random componen of lna, A,equals fracion ρ A of is previous period s value plus a random erm. If ρ A is posiive, his means ha he effecs of a shock gradually disappear over ime. A

9 governmen in a baseline RBC model An alernaive driving variable for he model is he amoun of governmen purchases. The rend rae of per capia governmen purchases equals rend growh rae of echnology, oherwise hey would become arbirarily large or small. Thus, (12) ln G = G+ ( n+ g) + G (13) G = ρ G + ε where 1<ρ G <1 G 1 G, where he ε G erms are also whie-noise disurbances and are uncorrelaed wih he ε A disurbances.

10 ineremporal subsiuion of labour The wo mos imporan differences beween his model and he Ramsey model are he inclusion of leisure in he uiliy funcion and he inroducion of auoregressive echnology or governmen purchases. Because of he logarihmic form of he uiliy funcion (9), he ineremporal elasiciy of subsiuion of leisure is 1. A rise in wages oday causes an increase in labour supply oday relaive o supply omorrow. A rise in he ineres rae raises relaive labour supply oday as well. Inuiively, a rise in he ineres rae increases he araciveness of work oday and saving oday relaive o omorrow.

11 household opimizaion under uncerainy The household s opimizaion problem also differs from he Ramsey model because i faces uncerainy abou he pah of fuure wages and ineres raes (due o auoregressive echnology and governmen purchase shocks). Because of his uncerainy, he choices of consumpion and leisure a any poin depend upon all shocks up o ha dae, so he household does no choose deerminisic pahs for consumpion and labour supply. Wih uncerainy we can derive a Euler equaion relaing curren consumpion o expecaions concerning consumpion and ineres raes in he nex period. Consider a household ha reduces consumpion by a small amoun oday and uses is consequen greaer wealh o fund higher consumpion omorrow. If he household is behaving opimally, a small change of his ype mus leave uiliy unchanged. Hence, (14) 1 1 ρ = e E ( 1 r 1 ) c + + c + 1

12 radeoffs beween labour and consumpion The household chooses consumpion and leisure a each dae. A second firs-order condiion (o go wih he Euler equaion in consumpion) relaes is curren consumpion and labour supply. Specifically, imagine he household raising is labour supply oday by a small amoun and using he income o increase is consumpion oday. If he household is behaving opimally, is expeced uiliy mus be unchanged. A his poin, he raio of consumpion o leisure is an increasing funcion of he wage and a decreasing funcion of he marginal disuiliy of labour, b: c w (15) = 1 b

13 solving he model Basic problem is o maximize uiliy (7) subjec o he producion funcion (4), he oupu ideniy (5), he capial sock (6) and he ime endowmen. This yields a se of firs-order condiions which characerize marke equilibrium. As we have seen, he wo mos imporan of which are: he equaion which equaes he marginal uiliy of consumpion o is shadow price and one which equaes he marginal disuiliy of labour o is marginal produc. The soluion focuses on wo variables, labour supply per person and he fracion of oupu ha is saved. The basic sraegy is o rewrie he equaions of he model in log-linear form, subsiuing (1-s)Y for C whenever i appears.

14 a simple model Given explici forms for he uiliy and producion funcions, i is possible o solve for he ime pahs of consumpion, capial and labour. In order o obain a specific soluion we assume (McCallum, 1989) ha capial fully depreciaes, uiliy is log-linear and he producion funcion is Cobb-Douglas. In his case we find ha here is a consan opimal saving rae and ha labour supply is also consan: 1 α n sˆ = αe ˆ ρ = (1 α) + b(1 sˆ ) Despie household s desire o subsiue heir labour supply ineremporally, movemens in eiher echnology or capial have offseing impacs on he relaive-wage and ineres rae effecs on labour supply. An improvemen in echnology raises curren wages relaive o expeced fuure wages and hence raises labour supply. Bu i also raises he amoun saved and hence lowers he expeced ineres rae, which reduces labour supply. In his specific case, hese effecs exacly balance.

15 dynamics of oupu I When labour and saving are consan, we can examine he dynamics of oupu in he following way: The producion funcion implies (16) lny = α ln K + (1 α)(ln A + ln L) we know ha K = sy ˆ 1 and L = ˆ N ; hus (17) lny = αln sˆ + αln Y + (1 α)(ln A + ln ˆ + ln N ) 1 since only Y -1 and A are no deerminisic in he model, we can rewrie his as: (18) lny = α ln Y + (1 α ) A 1 where is he difference beween lny and he value i Y would ake if lna equalled A + g a each period.

16 dynamics of oupu II Noe ha since (18) holds each period, i implies ha (19) lny = α ln Y + (1 α ) A or 1 (20) A = 1 ( Y αy 1 2 ) 1 α and since (11) saes ha A = ρ A +, ε we can subsiue hese wo A 1 A, equaions ino (18) we obain (21) = αy + (1 α)( ρ A + ε ) Y 1 A 1 A, = αy + ρ ( Y αy ) + (1 α) ε 1 A 1 2 A, = ( α + ρ ) Y αρ Y + (1 α) ε A 1 A 2 A, Thus, deparures of log oupu from is normal pah follow a second order auoregression ha is, oupu is a linear combinaion of is wo previous values plus a whie-noise disurbance.

17 a hump-shaped cycle This can lead o a hump-shaped response o shocks. Consider a shock of 1/(1-α) o ε A when α=1/3 and ρ A =0.9. This raises oupu by 1 in he firs period (1- α imes he shock), 1.23 in he nex (α+ρ A imes 1), 1.22 in he following (α+ρ A imes 1, minus α ρ A imes 1) hen 1.14, 1.03, 0.94 Noe ha he persisence of shocks is being driven by ρ A. Unforunaely, since saving and labour are consan, his version of he model is no very realisic.

18 a general mehod Papers in his general area canno be solved analyically. Insead, hey are usually solved numerically: parameer values are chosen and he model s quaniaive implicaions for he variances and correlaions of macroeconomic variables are discussed. An alernaive, recommended by Campbell (1994) is o ake firsorder Taylor approximaions of he equaions of he models in he logs of he relevan variables around he model s balanced growh pahs in he absence of shocks, and hen look a he properies of hese approximae models. Campbell also emphasizes ha you should look a he impulse response funcions raher han jus he variances and correlaions. In he case of he earlier model, when depreciaion is less han 100%, invesmen and employmen respond more o shocks. When depreciaion is no complee, a rise in echnology raises he marginal produc of capial and hence makes i opimal for households o save more. Since saving is emporarily high, we know ha he ineres rae mus be higher. Bu a higher ineres rae raises curren labour supply. So, invesmen and employmen respond more o shocks.

19 problems wih RBC models I The ineremporal subsiuion of labour Some economiss argue ha he ineremporal subsiuion of labour is no an imporan phenomenon since desired employmen is no very sensiive o he real wage and ineres rae. I is he unemploymen rae ha flucuaes over he business cycle. Why would so many people chose o work zero hours in recessions? Sudies of labour supply (such as Ball 1990) sugges ha expeced changes in real wages lead o only small hours responses. The Solow residual Alhough he Solow residual does flucuae significanly over ime, i is hard o believe ha he year on year measured changes represen echnology changes raher han changes in labour and capial uilisaion. As we saw, we for RBC models o generae plausible cycles we need a high degree of persisence. Empirical evidence for he UK suggess a coefficien of lagged adjused quarerly TFP of around 0.25, which means ha afer four quarers less han one per cen of he shock survives. Indeed, how are we o inerpre a negaive echnology shock? Neuraliy of money Evidence does no suppor he neuraliy of money (which is imporan in RBC models). Romer & Romer (1989) look a occasions where he FOMC has ighened money wihou any major change in condiions and find ha employmen and income fall.

20 problems wih RBC models II Flexibiliy of wages and prices Mos of he evidence is ha wages and prices are adjused infrequenly and ha here are paricular downward rigidiies. Do he models fi he facs? K&P sress ha RBC models are unrealisic in ha hey aim only o capure cerain feaures of he daa raher han a complee explanaion. RBC models ry o explain he relaionships among a number of series using jus a echnology shock. Millard, Sco and Sensier conclude from simulaions of six differen RBC models ha none can give a coheren accoun of UK labour marke developmens. They all undersae he volailiy and persisence of employmen and especially of unemploymen. There is also usually oo high a correlaion beween unemploymen and wages. Is a represenaive agen model appropriae? While represenaive agen models can be useful in modelling growh and invesmen, are hey really appropriae for modelling business cycles? One of he mos sriking feaures of business cycles is how oucomes differ across agens, for example, he unemployed are differen from he employed.