Business Cycle Theory I (REAL)

Business Cycle Theory I (REAL) I. Inroducion In his chaper we presen he business cycle heory of Kydland and Presco (1982), which has become known as Real Business Cycle heory. The real erm was coined because (1) he model does no include money and because (2) shocks o he TFP componen of he producion funcion are he source of he business cycle ha are emphasized in his research The mahemaics involved in solving ou his model are far oo advanced for his level. For his reason, we depar from our usual presenaion of he soluion mehods. Insead, we presen he basic elemens of he business cycle model, provide inuiion for he effec of produciviy shocks on he economy, and hen describe he calibraion exercise wih a summary of he findings of Kydland and Presco (1982). II. The Growh Model The model ha underlies real business cycle heory is jus he growh model sudied in Chaper X, wih hree noable differences. Firs, for he purpose of sudying he business cycle, he empirical counerpar of he model economy is a quarer of a year. This differen definiion of he period lengh does no aler he model s srucure in any way, bu does affec some of he calibraed parameers. The second difference is ha TFP in he producion funcion is a random variable. In all he previous models o dae, here were no sochasic elemens. The models were hus deerminisic. Deerminisic models are wha are called perfec-foresigh models because hey assume ha he households and firms know he values of all he economic variables over heir lives. 1

Models wih random variables are called sochasic models. The inclusion of random elemens means ha agens are no all knowing, i.e., have perfec foresigh of he fuure. The exisence of sochasic elemens means ha agens mus form an expecaion of he value of he fuure variables, and make a decision based on hose expecaions. Expeced Uiliy Wih he inroducion of random variables, here is he quesion of wha is he appropriae uiliy concep for households? The convenion in he lieraure is o use he von Neumann and Morgensern concep, which is jus he sum of he uiliies he household obains for each possible sae of naure weighed by he probabiliy he household pus on ha even happening. To illusrae, consider he following example. There are wo saes of naure, we label good and bad. The probabiliy assigned o he good sae by he household is π and he probabiliy assigned o he bad sae is hus, 1-π. Le c(g) be he consumpion ha household chooses in he case of he good sae and le c(b) be he household s consumpion in he case he bad sae is realized. Suppose he uliy funcion of he household is U(c). Then he household s expeced uiliy is (EU) U ( c( g)) (1 ) U( c( b)) In he conex of he growh model wih he infiniely lived consruc, he convenion is o no wrie ou he probabiliy of each even. Insead, he noaional convenion is o wrie expeced uiliy as 0 u( c, l ) 0 E. The subscrip on he expecaion operaor refers o he period in which he expecaion is formed. Here, i is he beginning period, ime =0. 2

III. Raional Expecaions Anoher issue we mus deal wih in economies wih random variables is how expecaions are formed. Specifically, wha is he basis for he subjecive probabiliies he household places on each sae of naure being realized? The convenion in sochasic macro models is o impose he assumpion ha expecaions are formed raionally. Raional Expecaions, as hey are known in he lieraure, are aribued o John Muh who in 1960 inroduced hem he sudy of finance. Rober E. Lucas and Edward C. Presco in 1970 inroduced Raional Expecaions o macroeconomics in heir paper, Invesmen under Uncerainy. The idea of raional expecaions is raher simple. People use all available informaion hey have o dae, and hen ac in a manner ha is consisen wih hose expecaions. Formally, Raional Expecaions is he condiional expecaions operaor. To illusrae, suppose we had a random variable, z, in each period. Then in period, here would be some informaion available o he household, call i I. Then he raional expecaion of z +1 would be he expeced value of z in period +1 given he available informaion I. To furher illusrae he concep, suppose ha a random variable, z, evolves according o he following law of moion (AR1) z 1. 95z 1. Such a process is called an auo regressive process of order 1. The order refers o he number of lagged values in he equaion. If here were an addiional erm z -1 on he righ hand side of he equaion, hen he AR would be of order 2. 3

The above equaion is sochasic in ha ε is a random variable. As such, here is a probabiliy disribuion ha governs is value, wih a mean and variance. The convenion is o assume ha he random shock is drawn from he same disribuion each period and ha is value in period +1 is independen of any pas realized value. A ime, he value of A is given and hence par of he household s informaion se. The agen mus form a raional expecaion of z +1. The condiional expecaion is found by applying he expecaions operaor o boh sides of (AR1). Since he expecaions operaor is linear, we have (CE) E z }.95z E{ }. { 1 1 Assuming ha he mean of ε is zero, he condiional expecaion of z +1 is jus.95 z. This is he raional expecaion of z. The raional expecaion (i.e., he condiional expecaion) is very differen from he uncondiional expecaion. To calculae he uncondiional expecaion we ierae backwards on he erm A on he lef hand side of (CE). Given ha (AR1) mus apply in all periods, we can rewrie he equaion as z 2 1. 95[.95z 1 ] 1.95 z 1. 95 1. If we coninue his backward ieraion forever, we arrive a z 2 3. 95.95 1.95 2... 1 1 Now if we apply he expecaions operaor o boh sides, and mainain our assumpion ha he mean of he error erm is zero each period, hen we find he uncondiional expecaion of z +1 =0. 4

Tha is o say ha if you did no have any informaion a ime, (i.e., he value of z ), hen your bes guess for z +1 is zero. IV. Undersanding he Effecs of Produciviy Shocks We have used hroughou his book he uni elasiciy uiliy funcion, u( c, l ) ln c ln l. The jusificaion for his funcional form is he observaion ha since 1950 he real wage has shown consan growh whereas average hours worked has shown no secular rend. Wih he uni elasiciy subsiuion, he subsiuion effec offses he income effec so ha here is no price effec. This is acually a long-run resul, and implicily involves a permanen increase in he price of leisure. If here is a emporary change in he price of leisure, hen leisure and work hours will change in he shor-run. To see his, consider he wo key household opimizing condiions (H1) (H2) 1 h 1 c w c E c 1 1 1 i Using (H1) for boh period and period +1 and subsiuing ino (H2) we arrive a w l E (1 ) i 1. w 1 l 1 Now if here is a posiive produciviy shock oday, hen w will increase. Being emporary w +1 5

eiher does no change or increases less han w. If he ineres rae is no much changed (or increases), hen i follows ha he raio of l /l +1 mus fall, which is he same as saying ha work hours increase oday. Noice ha he increase in he real wage were permanen so ha w =w +1, here would no be any change in leisure. V. Calibraion- Esimaing Produciviy shocks Alhough he business cycle is he cener of he analysis, he calibraion of he model is essenially he same as before. Namely, he preferences and echnology parameers are calibraed so ha he deerminisic model s seady sae- mach he long-run observaions of he Unied Saes economy over he 20 h cenury. In doing his, he quaniaive exercise of he real business cycle school is o examine he shor-run implicaions of a model resriced o mach he long-run behavior of he US economy. There are a couple of noed changes in he calibraed echnology and preference parameers on accoun ha we are analyzing he economy a he much higher frequency of 3 monh inervals. Since oupu is a flow variable, is value over a quarer of a year is one fourh is value over he enire year. This means ha he observaion for k/y mus be muliplied by four for he purpose of calibraing he value of δ, around.025. Wih his much value for k/y and much lower calibraed value of δ, he impued real rae of ineres and renal rae of capial are lowered. This implies a much higher value for he subjecive ime discoun facor, β around.99. These are he only changes in Sep 4 of he calibraion. To complee he calibraion, he process for he echnology shocks needs o be esimaed. Kydland and Presco s sraegy was o underake a Solow Growh Accouning exercise so as o 6

impue a ime series for Solow Residual s {A }. Then having his daa, hey esimaed an (AR1) process of he form. To illusrae, le us rewrie he producion funcion as z y e k [( 1 ) h ] 1 Where z is given by (AR1) Then he Solow residual is z (1 ) y (SR) e (1 ). 1/ 3 2 / 3 k h Using quarerly daa from he US economy on GDP, he capial sock and oal work hours one can calculae he ime series for TFP, i.e., Solow Residuals. Nex, if we divide boh sides of (SR) by (1+γ) (1-θ) and ake he log of boh sides, we can ge he ime series for { z }. We hen use he ime series for {z } and esimae he (AR1) equaion. This gives us an esimae for he parameer, ρ, of.95. The calibraion is no complee because wha is criically imporan for he heory is he magniude of he shocks. This is deermined by he variance of he shock, ε. To assign he value o he variance, we compue he error erm in each period, namely, he difference beween he acual and prediced values for z. This gives us a ime series for he error erms, {ε }. Using hese numbers, we hen calculae he mean of he errors and he variance. Wih hese parameer values in hand, Sep 4 of he calibraion procedure is complee, and he heory can be esed. 7

To do his, Kydland and Presco fed in a sequence of TFPs ha are based on he esimaed AR1 process above wih ρ=.95 and wih compuer generaed random numbers ha are based on he variance and means of he error erms. Wih hese realized produciviy shocks hey hen solve ou he acual pah for he model economy. In his way hey simulaed he US economy. Now, i is quie possible ha he compuer random number generae produces a series of good shocks in each period, or a series of bad shocks every period. To eliminae he effec of a paricular draw on he conclusions, Kydland and Presco simulaed he economy over 20 imes. In his way, hey generae a sampling disribuion of saisics. This requires ha for each simulaion, he HP filer is applied in order o find he rend and deviaion componens. Nex, he volailiy for each simulaion is deermined. 1 The average of hese volailiies is hen compued. This is he relevan saisic o be compared wih he volailiy compued for he acual US economy. In comparing comovemens as prediced by he model and as displayed by he acual US economy, again we compue for each simulaion he correlaion coefficiens, hen we ake he average across he simulaions. This average comovemen is compared o he correlaion coefficien compued using he acual quarerly daa for he US economy. VI. Assessing he Real Business Cycle Theory. How much of he volailiy in US GDP can be accouned by produciviy shocks? The following able reprined from a 1992 Federal Reserve Bank of Minneapolis Quarerly Review aricle wrien by Gary Hansen and Randy Wrigh compares he model s predicions wih he daa boh in erms of volailiy and comovemens. The period of comparison is 1947:Q1-1992:Q3. 1 Recall, o calculae he measure of volailiy, we firs express he deviaion as a percenage of he rend, namely, define x = d/τ. Nex, we ake he sandard deviaion of he (percenage) deviaion from rend. This is he measure of volailiy. 8

Relaive o he ables in Chaper X on he Business Cycle Facs, he Hansen and Wrigh able presens he business cycle properies of he model and daa in a slighly differen way and considers fewer dimensions of he daa. Imporanly, excep for he volailiy of oupu, he volailiy of all oher variables are expressed relaive o he volailiy of oupu. Addiionally, he able includes a measure of produciviy defined as oupu divided by hours. This variable is indexed by he leer ω. US Time series % s.d. of Consumpion Invesmen hours Produciviy oupu σ y σ c /σ y σ c /σ y σ h /σ y σ ω /σ y σ h /σ ω Corr(h,ω) 1.92.45 2.78.87.51 1.76 -.07 Model 1.30.31 3.15.49.53.94.93 Noe: he las four columns is an average of he saisic implied by he Household Survey daa and he Esablishmne Survey daa. There are a number of key poins o be aken from he able. Firs, reurning o he quesion of he calibraion, produciviy shocks can accoun for roughly 2/3rds of he volailiy of oupu. This is acually quie amazing, given ha here are clearly oher shocks in he real world ha cause oupu o deviae from is long-run rend. In erms of consumpion and invesmen, he heory is judged o be a relaive success. The model predics a lile oo much by way of consumpion smoohing, and hence for ha reason, a lile oo much by way of invesmen volailiy. Where he model is deemed o be less successful relaes o he predicions wih respec o hours and produciviy. Firs, hours in he model are no volaile enough relaive o he daa. This is he main reason why he model does no accoun for 100 percen of he oupu volailiy. 9

Produciviy shocks do no bring abou enough of an increase in hours in he model relaive o he daa, so ha he model does no generae a larger deviaion in oupu. For he purpose of undersanding his issue as well as he greaes failure of he model, he correlaion beween hours and produciviy, i is useful o show he implicaions of he RBC heory using he labor marke diagram. Labor demand is jus he Marginal produc of labor from he firm s profi maximizaion problem. A posiive TFP shock shifs he labor demand curve o he righ, implying boh a rise in he real wage and an increase in hours. Noice, imporanly, he he size of he hour increase depends on he seepness of he labor supply curve, wih a larger response being associaed wih a flaer curve. This is shown in Figure 1. Here we have drawn an increase in he Labor Demand curve associaed wih a posiive TFP shock and compare he effec on he equilibrium hours for wo labor supply curves ha differ in heir seepness. Figure 1: Change in Hours Depends on Slope of Labor Supply w h s h d hours 10

Recall ha he labor supply is upward sloping because he higher wage is viewed as a emporary phenomenon. The slope hen reflecs he degree which he household is willing o ineremporally subsiue beween oday and omorrow. Inuiively, his deermined by he curvaure, or concaviy of he uiliy funcion, which in his case has been specified as he log. Log uiliy alhough concave, simply implies oo small of an ineremporal subsiuion effec, and for his reason, hours respond by far less in he model compared o he daa. Alhough predicing less volailiy in hours han observed in acualiy, he model is sill successful along his dimension. Much harder o reconcile is he models predicion for he correlaion of hours and produciviy. In he daa, i is essenially zero; in he daa i is close o 1. Figure 2: Posiive Produciviy Shock wih Increase in Labor Supply w h s h d hours The reason why he model predics a correlaion close o one is eviden from he labor diagram above. Wih he increase in labor demand, hours and wages move ogeher. Since produciviy is jus he wage divided by he labor share, he hours and produciviy move more or less he same. Wha would have o happen o ge (1) hours o increase wih produciviy shocks and (2) no 11

change in he wage? One way o achieve (1) and (2) is o have he labor supply increase a he same ime as labor demand increases. Noe ha his shif would no only solve he correlaion issue bu generae a bigger response in hours. IV. Exensions A number of exensions o he Real Business Cycles were pu forward in an effor o show ha he heory could be reconciled wih he volailiy of work hours over he cycle and he correlaion beween hours and labor produciviy. Non-Separable Leisure This is a fix o he firs deficiency of he sandard RBC model, namely, is inabiliy o generae enough volailiy in work hours. I was explored in a paper by Hoz, Kydland and Sedlacek (1988). Recall from Figure 1 ha greaer volailiy is implied if he oday s supply of labor responds more o a higher wage, i.e., he labor supply curve is flaer, or he elasiciy of subsiuion is higher. Wih non-separable leisure, oday s uiliy does no depend jus on oday s leisure bu a weighed sum of he household s pas leisure choices. Specifically, curren period uiliy is (N-SUiliy) u( c, L ) ln c ln L. The key difference is ha L is he weighed sum of his period s and all pas period s leisure, namely, 12

L 0l 1l 1 2l2 3l3... One inerpreaion of his ype of preference is ha people are willing o work a lo of hours in his quarer if hey had enjoyed a nice vacaion 3 or 6 monhs earlier. I is he idea of your universiy giving you a Spring break- wih a week off, you will be beer able o focus and sudy longer hours as finals approach. Hoz, Kydland and Sedlacek (1988) impose he following resricion on he weighs so ha i1 i The implicaion of his weighing sysem is ha he weighs decline geomerically by he facor ρ each period. Wih respec o he calibraions, non-seperable leisure inroduces wo new parameers whose values mus be resriced in sep 4 of he calibraion procedure, namely, η 0 and ρ. The auhors use employmen hisories for whie male head of households aken from he Panel Sudy of Income Dynamics Using o resric he value of hese wo parameers. The esimaed parameers are found o be η 0 =.35 and ρ=.90. Home Producion Benahabib, Rogerson and Wrigh (1991) considered he role of home producion in he conex of he business cycle. Home producion is he goods and services ha are done by households ouside he marke. These goods and services do no receive any compensaion and are generally 13

excluded from he NIPA. Examples are dinners you prepare a home, laundry, cleaning, yardwork, driving yourself and family members o heir jobs and aciviies. Inuiively, he addiion of home producion allows households o subsiue beween marke and non-marke aciviies. Specifically, when here is a posiive shock o TFP in he producion of marke goods and services, households will find i opimal o work more in he marke and less a home. Essenially, when here is a posiive shock o marke producion and a higher wage is offered, you sar eaing ou more as well as having your clohes laundered a he dry cleaner. Governmen Spending Chrisiano and Eichenbaum (1992) proposed ha governmen spending shocks could explain why labor is more volaile han in he sandard model and why he correlaion beween hours and labor produciviy is close o zero. The idea is o formulae a way by which such shocks would lead o a shif in he labor supply curve. In he Chrisiano and Eichenbaum model, governmen expendiures are paid for by lump-sum axes. Thus, a higher shock would imply larger lump-sum axes in he curren period. This would reduce he household s wealh, and as we saw in our model of labor supply decision, his makes he household choose less leisure, i.e., supply more work hours. The law of moion for governmen expendiures was specified as (G Shocks) ln g 1 (1 )ln( g) ln g 14

In (G Shocks), g denoes he seady-sae level of governmen spending. The variable μ represens he shock o governmen spending. I is assumed o be independenly and idenically disribued wih mean zero and wih variance σ μ. Imporanly, he shock o governmen expendiures is assumed o be independen of he echnology shock. In his model, he governmen is assumed o run a balanced period each period. Expendiures are paid for by lump-sum axes. The inclusion of governmen expendiures in he model has some imporan implicaions for sep 3 of he calibraion procedure. Recall, in he Solow model and Neoclassical Growh Model wihou governmen, we moved governmen consumpion in he NIPA ino he model consumpion caegory, and we moved governmen invesmen in o he invesmen caegory. Now, we no longer would make hose adjusmens. This differen reorganizaion would change some of he parameer values from he sandard RBC model, bu hese are no paricularly imporan in he conex of his discussion. Imporan o deermining he model s implicaions for business cycle is he assignmen of he parameers in he (G Shock) Equaion. This is done by using Ordinary Leas Squares o esimae he law of moion for (G Shocks). Imporanly, he value of g is calibraed o he long-run raio of governmen expendiures o GDP, which is roughly 20%. Chrisiano and Eichenbaum (1992) repor esimaes for λ=.96 and σ μ =.021. The governmen spending shocks and TFP shocks are hen fed ino he model, and he equilibrium pah is compued. This is done 200 imes, and hen he average volailiy and correlaions are compared o he daa. 15

US Series Time % s.d. of oupu σ y Consumpion σ c /σ y Invesmen σ c /σ y hours σ h /σ y Produciviy σ ω /σ y σ h /σ ω Corr(h,ω) 1.92.45 2.78.87.51 1.76 -.07 Model 1.30.31 3.15.49.53.94.93 Non- Separable Leisure Governmen Spending Home Producion 1.51.29 3.23.65.40 1.63.80 1.24.54 3.08.55.61.90.49 1.71.51 2.73.75.39 1.92.49 VI. Conclusion In his chaper we have sudied he Real Business Cycle Theory paradigm iniiaed by Finn Kydland and Ed Presco and exended by many oher economiss. The findng of his research agenda is ha produciviy shocks can accoun for a large porion of he business cycle, alhough no all. Despie his success, here are many economiss who do no appreciae he conribuions of his lieraure. As we shall discuss in he nex chaper, hese economiss do no like a number of aspecs peraining o he Real Business Cycle Theory such as he lack of any fricions. The RBC heory assumes ha all markes clear. Relaedly, opponens of his heory criicize i for he implicaion ha recessions are jus bough of laziness, ha anyone no working ou here is volunarily unemployed. Finally, here is no room for policy, paricularly moneary policy. In par his reflecs ha only real prices maer and ha he equilibrium is Pareo Opimal. In he 16

nex chaper, we will urn o he prooypical model ha supposedly avoids hese failures- he so called New Keynesian Dynamic Sochasic General Equilibrium model. Assignmen: 1. Flip of coin- AR(1) 2. (i) Calculae Solow Residuals for US economy (Quarerly Daa?), Capial, GNP, and Toal work hours. (2) HP filer and (3) simple linear AR1 regression 17