Lectures 11 & 12: Real business cycles

Transcription

1 Lecures 11 & 12: Real business cycles 1. Solow and macroeconomic accouning 2. The real business cycle view: Kydland and Presco, Long and Plosser 3. Presco (86) 4. KPR (88) 5. Do produciviy shocks need o be large? 1

2 1. The aggregae producion funcion I is sandard for us o hink abou economic aciviy a he level of he firm or indusry using a producion funcion Bu i once was no sandard: i ook work by economiss like Paul Douglas o urn i from a heoreical device o one ha was an a pracical ool. Douglas noiced ha shares paid o facors (like wn/y) were relaively consan as levels of inpus and oupus changed. Working wih a UC mahemaician, he deermined a producion funcion ha had hese properies and explored is fi 2

3 3

4 Producion Funcion CD producion funcion log( Y ) = log( A) + α log( K ) + (1 α)log( N ) + e Oupu = [Producion Funcion] + error Producion Funcion explains mos of oupu 4

5 5

6 Solow Similar sraegy, bu very differen findings [Some elemens such as using unemploymen rae o correc for uilizaion are very imporan for year o year umbers, bu no for longer-erm evoluion] Figure is based on equaion below log( A ) = log( Y ) α log( K ) (1 α)log( N ) Residual (Produiviy) = Oupu - [Toal Facor Inpu] 6

7 2. The Real Business Cycle hypohesis F. Kydland and E.C. Presco, Time o Build and Aggregae Flucuaions, Economerica, J. Long and C.I. Plosser, Real Business Cycles, Journal of Poliical Economy,

8 Background Alhough real facors were suggesed o be an imporan of sources by various auhors (noably Hayek and Schumpeer in he 1920s and 1930s), he Keynesian revoluion in macroeconomics led o a siuaion in which (a) changes in aggregae demand were viewed as he major source of business cycle impulses; and (b) various privae marke failures involunary unemploymen and various marke fricions including sicky prices and wages were viewed as essenial o undersanding he posiive naure and he normaive consequences of economic flucuaions 8

9 Background Circa 1975, i was sandard for macroeconomiss o view equilibrium aciviy as a largely consan or slowly evolving value (a rend, perhaps) and o view cyclical variaions as disequilibrium phenomena Circa 1975, he main debaes among macroeconomiss were mainly abou wheher moneary or fiscal impulses (or rules) had larger effecs on economic aciviy and which should be used o sabilize flucuaions and o eliminae hem if possible. 9

10 Background: The New Simulae circa 1975 Educaion of a generaion of economiss on he dynamic models of Solow and Cass ha we jus sudied, focusing aenion on economic dynamics in heory and he role of producivy in growh. Rise of quaniaive (if saic) models of general equilibrium in public finance and oher areas. Lucas criique which suggesed ha exising policy models were inadequae and a new generaion of models were necessary. 10

11 Basic RBC papers The iniial se of dynamic general equilibrium models of he 1980s focused on produciviy shocks as a source of economic flucuaions The Kydland-Presco (henceforh KP) and Long-Plosser (henceforh LP) each described how a well-funcioning marke economy would respond o produciviy shocks, concluding ha i would display imporan characerisics of business cycles in he real world 11

12 Basic RBC papers Kydland and Presco sudied a varian of he onesecor neoclassical growh model, augmened o include variable labor supply, invesmen according o a ime o build invesmen process and various oher elemens aimed a producing a modern business cycle model. Their model was driven by an aggregae produciviy shock. Long and Plosser consruced a mulisecor neoclassical growh model, in which changes in secoral produciviy produced responses in oher secors and ulimaely a he aggregae level in he economy. 12

13 Impac These papers launched a revoluion in macroeconomics, which has resuled in mehodology ha is now used widely o sudy a wide range of problems (including in New Keynesian models ha we will discuss laer in he course). These papers simulaed much research ino he influence of produciviy on macroeconomic aciviy including work on is measuremen, is origin and is consequences. 13

14 2A. Kydland-Presco KP specified a version of he neoclassical growh model wih several complicaions of preferences and he producion funcion: An invesmen echnology designed o capure a range of phenomena described hrough he early secion of he aricle, from which he paper akes is name. Invenories (since invenories vary subsanially over business cycles) Variable labor (since labor varies over business cycles) A uiliy funcion allowing for nonseparabiliies in labor, designed o allow for differen shor-run and long-run labor supply elasiciies. 14

15 Time o Build Le s j be he number of invesmen projecs wih j sages o go unil hey are compleed and incremen he capial sock. Suppose ha here are J sages, so ha k k = s δ k s j 1, + 1 = s j, for j= 2,3, 4, J In his expression, s J is he number of sars, i.e., he number of new projecs iniiaed in period. Oher projecs simply move oward compleion (hey canno be cancelled) 15

16 Invesmen Invesmen expendiures are disribued hrough ime on any given projec, wih weighs suggesed by daa on acual invesmen expendiure J = ϕ j j j= 1 i s ( m m ) where m is he predeermined sock of invenories a so m +1 -m is invenory invesmen and he oher par is fixed invesmen (invenory noaion differs from KP) 16

17 Producion funcion Invenories are facor of producion for KP, so ha y = f ( a, n, k, m ) where we use our sandard noaion wih a being produciviy, n being labor inpu Furher, he consrain on goods is hen c + i y 17

18 Uiliy Funcion KP posi a uiliy funcion of he form u( c, l ) j = 1 γ (1 γ ) η η j j= 1 l n n = 1 γ n (1 γ ) ηz z = (1 η) z + n

19 Uiliy Funcion (con d) This uiliy funcion imbeds he idea ha supply of work is more cosly if one has recenly worked a lo. I herefore allows for a higher response of work o emporary variaions in opporuniies (emporary changes in wages) han permanen ones I is se up o allow for dynamic programming: z is a sae variable a dae 19

20 Oher feaures no emphasized in lecure Two sage decision srucure, wih some variables chosen before ohers Incomplee informaion abou permanen or emporary naure of aggregae produciviy siuaion (originally included o allow a poenial source of moneary nonneuraliy). 20

21 Compuing Equilibrium KP se up a dynamic programming problem, essenially v( k,[ s ], m, z, ς ) = max{ u( c, l ) J 1 j j= 1 + β E v( k,[ s ], m, z, ς } J j, + 1 j= s c + i = f ( a( ς ), k, n, m ) i = ϕ s + ( m m ) j j + 1 j= 1 k k = s δ k J s = s for j= 2,3, 4, J j 1, + 1 j, l = 1 γ n (1 γ ) ηz z = (1 η) z + n

22 Compuing equilibria (con d) KP designed heir economy so ha i was linear in he consrains, once hey subsiued ou for consumpion from he firs consrain. They knew ha opimizaion models wih quadraic objecives and linear consrains (like hose which we sudied las erm) could be rapidly and easily compued, resuling in linear decision rules, so ha hey sough a quadraic approximaion o a reduced form objecive (or indirec uiliy funcion). 22

23 Compuing Equilibrium(con d) To deermine he equilibrium process for his model, we exploi he well-known resul ha, in he absence of exernaliies, compeiive equilibria are Pareo opima. Wih homogenous households, he relevan Pareo opimum is he one which maximizes he uiliy of he represenaive consumer subjec o he echnology consrains and he informaion srucure KP, pg

24 Compuing equilibria (con d) Approximaions are necessary before equilibrium decision rules are compued. Our procedure is o deermine he seady sae for he model wih no shocks o echnology. Nex, quadraic approximaions (of an objecive) are made in he neighborhood of he seady sae. Equilibrium decision rules of he resuling approximae economy are hen compued. These decision rules are linear, so he approximaed economy is generaed by a sysem of sochasic difference equaions for which covariances are easily deermined. KP

25 Tesing he heory KP propose a specific es of heir heory, explained on pages They begin by choosing mos of he parameers of heir model o mach micro observaions and seady sae facs, a mehod of parameer selecion which hey call calibraion. They hen evaluae how well heir model does a reproducing cerain facs abou business cycles (which hey refer o as he deviaions ). 25

26 Example The model is consisen wih he high (percenage) variabiliy in invesmen and he low variabiliy in consumpion and heir high correlaion wih oupu. KP, page

27 KP s conclusion The preference-echnology environmen was he simples one ha explained he quaniaive comovemens and serial correlaion properies of oupu. The fi of he model was surprisingly good in ligh of he model s simpliciy. A crucial componen of he model ha conribued o persisence of oupu movemens was he ime-o-build requiremen. 27

28 KP s conclusions (con d) KP sugges ha models of his class could be used o sudy design of policy rules, in a manner consisen wih he Lucas criique, bu ha hese analyses will require furher developmens of mehods, as models wih general policy rules are ypically no models in which compeiive equilibria are pareo opima. 28

29 2B. Long-Plosser Considered how wo key economic ideas could lead o business cycle phenomena. 1. The idea ha consumers generally desire variey of goods and so will spread incremens o wealh across many differen commodiies a many differen daes. 2. Tha producers generally require many differen produced inpus o efficienly produce final goods. 29

30 LP s focus on paricular aspecs of business cycles Business cycles involve srong posiive serial correlaion of aggregaes: persisence Over he course of business cycles expansions and conracions, many differen secors rise or fall ogeher: comovemen 30

31 LP s approach Describe general model in which ideas can be explored. Analyze in deail a paricular analyical soluion, obained by making specific resricions on preferences and echnology (essenially loglinear preferences and Cobb-Douglas echnology). We will discuss his model using noaion which is compaible wih he preceding presenaion of KP and he laer presenaion of oher models. 31

32 Preferences LP model elemens Depend on a J elemen vecor of consumer goods, wih c being he dae vecor and c j he quaniy of he jh good (jh elemen of vecor). Resriced so demand for each good increases wih wealh (posiive wealh elasiciies) Depend on he quaniy of leisure (do no inroduce dynamic elemens as in KP) U = = o β u( c, l ) 32

33 LP model elemens Secoral producion y i,+1 depends on labor allocaed o he secor n i he amoun of inermediae inpus from oher secors m ij, wih i being he secor of use and and j being he secor of origin a secor-specific shock a i,+1 y = f ( a, n,[ m ] ) J j, + 1 j j, + 1 j ij j= 1 33

34 LP elemens (indicaed by noaion on prior page) f j indicaes ha producion funcion in secors need no be same Daing of oupus (+1) and inpus () indicaes ha lengh of period is he amoun of ime ha producion requires, which is assumed consan across secors 34

35 LP elemens Wrie producion funcion compacly as y = F( a, n, m ) where y is he oupu vecor (J by 1) a is he produciviy vecor (J by 1) n is he labor inpu vecor (J by 1) m is he inermediae inpu marix (J by J) 35

36 LP model elemens There are consrains J c + m = y j ij j i= 1 1= l + J i= 1 n i 36

37 Equilibrium compuaion: Solving Robinson Crusoe s problem v( y, ς ) = max{ u( c, l ) + β E v( y, ς )} s.. c + ( φm )' = y y = F( a, n, m ) l φn = 1 wihφ being a J elemen row vecor of 1s so ha his sums elemens of labor vecor and ( φm )' is he column vecor of oal inpu use. 37

38 Example Preferences Producion J u( c, l ) θ log( l ) θ log( c ) = + 0 j j j=1 J n x i, + 1 = i, α + i i αij ij j= 1 log( y ) log( a ) log( n ) log( m ) 38

39 LP soluion Remarkable (posiively) in ha hey are able o acually generae an exac loglinear difference sysem as he soluion for he policy funcions. Generalizes soluion we explored in earlier compuaional dp lecure o many goods Remarkable (negaively) ha labor in he economy as a whole and o each secor is consan in his soluion, reflecing a complicaed general equilibrium paern of offseing wealh and subsiuion effecs. 39

40 LP soluion Takes he form log( y ) = κ + α log( y ) + log( a ) x x x where α = [ α ij ] Depends cenrally on inpu oupu informaion from marix of cos-shares α x Is invarian o naure of driving process (analogous o Sandmo s offseing effecs of reurn level and uncerainy) 40

41 Calibraion Choice of secors (degree of disaggregaion) Inpu-oupu informaion 41

42 Analysis Response of sysem o ransiory shocks: how does oupu shock in secor i respond o evens in secor j hrough ime? (depends on inpu-oupu srucure) How much persisence measured by serial correlaion do mechanisms produce? (some) How much comovemen measured by correlaion do mechanisms produce? (some) How does sysem respond o permanen produciviy shocks, e.g., in a simulaion? (looks more like macro daa) 42

43 LP conclusions Real shocks and real mechanisms may be imporan for business cycle (enduring) Secoral srucure and secoral shocks are imporan (largely unexplored, bu analizing research problem) Technical progress is possible in obaining explici soluions o models wih alernaive specificaions of preferences and echnology : 20 years laer, we don have many addiional ones Empirical research : acive wih highly aggregaed models as in KP bu relaively lile wih models of secoral ineracions and/or shocks 43

44 Quesions raised by iniial RBC sudies How should we bes measure produciviy change and o wha exen is i exogenous o business cycle? How imporan are various mechanisms sressed in individual sudies? (e.g., ime-o-build) How should we selec parameers for and evaluae he performance (fi) of simple dynamic models? How should we solve models wih subopimal equilibria? 44

45 3. Presco and Theory Ahead of Business Cycle Measuremen Derending of daa via HP filer Compuaion of momens sylized facs Exploraion of simple model via momens and sochasic simulaions Role of highly elasic labor supply 45

46 Derending of daa via Hodrick-Presco filer Filered oucomes are soluions o a minimizaion problem bes esimae of unobserved componen if here is a paricular probabiliy model More generally, a procedure ha eliminaes low frequency elemens from he daa J = = = d m y wih m m and m 0 j j j j j j= J j= J J

47

48 Sylized Facs (Momens)

49 Simple model Basic neoclassical model (Ramsey, Cass, Koopmans) Wih produciviy shock (as in Solow s empirics) Wih u(c,l): endogenous labor and leisure

50 Implicaions

51 Labor supply on he exensive margin

52 Labor supply on he inensive margin Sandard model: max u( c, l) s.. c+... w(1- l) +... Discree model max (1- π ) u( c,1) + πu( c,1 n) 1 2 s.. (1- π ) c + π c... wπ + n

53 Key poins Loeries Efficien consumpion equaed across individuals Labor supply elasiciy is effecively infinie (holding fixed marginal uiliy of consumpion) Works, bu wha does his correspond o in real world?

54 Implicaions for labor-oupu comovemen in simulaion

55 Conclusions of Presco Basic heory works in sense of explaining a grea deal of cyclical flucuaions Theory should be used o guide fuure measuremen (e.g., aggregaion of workers wih diverse skills) Sabilizaion policy is unnecessary or undesirable, as oucomes are approximaely opimal

56 4. KPR and he mechanics of Real Business Cycles: R.G. King, C.I. Plosser and S.T. Rebelo, Producion, Growh and Business Cycles, 2 papers in Journal of Moneary Economics, Anoher relaed paper: R.G. King and S.T. Rebelo, Resusciaing Real Business Cycles, in Handbook of Macroeconomics

57 Quesions posed by KPR Wha are he business cycle properies of he basic neoclassical model of capial accumulaion, when i is augmened by produciviy shocks? Wha are desirable sraegies for compuing linear approximaion (or loglinear approximaion) soluions o dynamic equilibrium models? Do hese mehods coninue o work if here is a sochasic rend in produciviy, i.e., here are permanen variaions in he level of produciviy If governmen policy or privae marke failure leads compeiive equilibrium o be subopimal, are enirely new mehods necessary? 57

58 Recipe: Par A 1. Sar wih model which includes echnical progress and describes a growing economy (depends on via deerminisic rend growh) 2. Resric preferences and echnology so ha seady sae growh is feasible 3. As in growh models, find a ransformed economy which is saionary (i.e, in which he equaions do no depend on excep as subscrip) 4. Wrie down a discree ime Lagrangian 5. Find he FOCs and TC for his economy 6. Find he saionary poin of hese FOCS (which surely saisfies he TC) 58

59 Recipe: Par B 1. Linear/loglinear approximaion 2. Solve he resuling linear RE model in general: {Y} depends on {X} 3. Calculae he soluion under a paricular driving process, obaining a soluion in sae space form 4. Use his soluion o calculae a. Comparaive dynamics (impulse responses) b. Populaion momens c. Sochasic simulaions 5. Use economic analysis o inerpre soluion: e.g., permanen income heory including Fisher s rule for consumpion growh and real ineres rae 59

60 Exensions of Recipe KPR88b: To subopimal equilibria KPR88b: To permanen variaions in produciviy (sochasic rend produciviy) moivaed by uni roo findings Laer: o all sors of real and moneary models, including New Keynesian models Now rouine: Wrie down FOCs and oher condiions of equilibrium Linearize around appropriaely defined saionary poin Ge dynamic oucomes using linear RE echniques 60

61 Recipe: Par A The original economy: producion wih deerminisic echnical progress expressible in labor augmening form Y = A F( K, X N ) and X = γ X + 1 Y y = A F( k, N ) wih y = and k = X [form necessary for ss growh] K X 61

62 Consumpion, Invesmen and Capial Accumulaion C + I = Y and K = (1 δ ) K + I + 1 c + i = y and γ k = (1 δ ) k + i wih c + 1 C and i = = X K I K K X (1 δ ) X X X X X = = = + 1 I X γ k

63 Labor and leisure Per-capia hours of work and leisure mus be consan in seady sae This comes despie fac ha wage rae will grow due o echnical progress W = wx = wx γ 0 63

64 Preference resricions necessary for ss growh o be opimal Need: consan consumpion growh in face of consan ss real ineres rae (implied by above) Need: consan work level in face of real wage growh Uiliy (general): u(c,l) 64

65 Preference resricions (someimes called KPR uiliy) Ineremporal MRS: mus be consan elasiciy (as in permanen income model) MRS beween work and leisure, mus display invariance o growing consumpion and wage rae (offseing income and subsiuion effecs) u c L 1 1 σ c v L or u c L c v L 1 σ (, ) = ( ) (, ) = log( ) + ( ) 65

66 Original and modified uiliy For firs uiliy funcion above 1 σ 1 σ β (, ) ( 0) ( βγ ) (, ) = 0 = 0 U = u C L = X u c L For boh uiliy funcions: (i) X 0 affecs welfare bu no preferences (over c,l), so se X 0 =1 for convenience and absrac from i (ii) modificaion of discoun facor 66

67 Solving opimizaion problem Via Lagrangian L = * ( β ) u( c, L ) = 0 * ( β ) λ [ A f ( k, N ) (1 δ ) k γ k+ 1 c ] = * ( β ) ω[1 N L ] = 0 + Conceps β λ is shadow price of c y * : ( ), β ω is shadow price of N L * ( ), 67

68 FOCs+TC c L N * : ( β ) [ D1u ( c, L ) λ ] = 0 * : ( β ) [ D2u( c, L ) ω ] = 0 * : ( β ) [ ω + λ A D2 f ( k, N )] = 0 k : ( β ) [ λ + β λ ( A D f ( k, N ) + (1 δ ))] = 0 * * * ( β ) λ :[ A f ( k, N ) + (1 δ ) k γ k+ 1 c ] = 0 * ( β ) ω :[1 N L ] = 0 TVC * : lim ( β ) λ k + =

69 Saionary poin Consrained by c :[ D u( c, L) λ] = 0 1 L :[ D u ( c, L ) ω ] = 0 2 N : ω+ λ AD f ( k, N) = 0 k 2 * :[ λ+ β λ( AD1 f ( k, N) + (1 δ ))] = 0 λ :[ Af ( k, N) + (1 δ ) k γ k c] = 0 69

70 Recipe: Par B Linearizaion/loglinearizaion ξ cɵ ɵ ɵ + ξ L λ = 0 cc cl ξ cɵ ɵ + ξ L ω = 0 Lc LL ω ɵ ɵ + A + ξ k + ξ k = 0 nk N * N * ɵ + L L = 0 nk ɵ λ ɵ ɵ + λ + 1+ η A+ 1+ η k + 1+ η N + 1 = 0 A k N [ A ɵ ɵ ɵ + s N + s k + 1] + [ s c ɵ + sφk + 1 s ( φ 1) k ] = 0 N k c i i 70

71 Soluion of Linear Model Sage 1: Relae endogenous variables o sequences of exogenous variables (as in Blanchard-Kahn) ɵ ɵ j k + 1= µ 1k + ψ1 A + ψ 2 µ 2 E A+ j+ 1 j=0 Sage 2: Assuming driving process and evaluae discouned sums A = ρ A 1+ e ɵ ɵ ρ k ɵ + 1= µ 1k + [ ψ1+ ψ 2 ] A = µ 1k + π ka A 1 ( ρ / µ ) 2 71

72 Y =Π s s = Ms + Ge 1 Linear difference sysem (Bellman) Sae space model (Chow/Hamilon) RE soluion (from BK or KW) Used for calculaions of responses o one-ime shocks (impulse responses); sochasic simulaions; calculaions of momens 72

73 y π yk π ya c k i = a n... k a µ π k 0 ka 1 = 0 ρ a e 73

74 Subsanive conclusions from KPR Produciviy shocks (A) mus be persisen o generae business cycle phenomena: hese give rise o imporan wealh effecs on consumpion. Alhough labor is invarian o rend growh in produciviy (X), i varies sharply in response o produciviy shocks (A) Large produciviy variaions are necessary o produce subsanial volailiy in oupu [no sressed in paper, bu implici in graphs]. 74

75 Do we need large produciviy shocks? KR in Handbook of Macroeconomics review developmens from 1988 o 1999 when here was: A huge number of RBC sudies A rising concern ha produciviy shocks measured via Solow residual were oo big and o imporan o RBC modeler conclusions 75

76 Swich in model presenaion Recursive opimizaion = + β * V ( k A ) max{ u( c, L ) E V ( k, A )} A f ( k, N ) + (1 δ ) k γ k+ 1 c ] 0 [1 N L ] 0

77 FOCs + ET

78 FOCs + ET

79 KR volailiy enhancing sraegy Modified basic 1988 model o inroduce several key feaures ha had previously been sudied separaely Labor choice on exensive margin (o work or no) raher han inensive margin (how many hours o work) Varying uilizaion of capial 79

80 Implicaions Model economy: Solow residual as poor proxy for acual produciviy shock (uilizaion no observed) High response o acual produciviy shocks (so he implied shocks could be smaller) Alernaive mehod of exracing shocks 80