page, Harley, Hoover, Salyer, RBC Models: A User s Guide A User s Guide o Solving Real Business Cycle Models The ypical real business cycle model is based upon an economy populaed by idenical infiniely-lived households and firms, so ha economic choices are refleced in he decisions made by a single represenaive agen I is assumed ha boh oupu and facor markes are characerized by perfec compeiion Households sell capial, k, o firms a he renal rae of capial and sell labor, h, a he real wage rae Each period firms choose capial and labor subjec o a producion funcion o maximize profis Oupu is produced according o a consan-reurnso-scale producion funcion ha is subjec o random echnology shocks Specifically ( ) y = z f k, h, where y is oupu and z is he echnology shock (The price of oupu is normalized o one) Households decisions are more complicaed; given heir iniial capial sock, agens deermine how much labor o supply and how much consumpion and invesmen o purchase These choices are made in order o maximize he expeced value of lifeime uiliy Households mus forecas he fuure pah of wages and he renal rae of capial I is assumed ha hese forecass are made raionallya raional expecaions equilibrium consiss of sequences for consumpion, capial, labor, oupu, wages, and he renal rae of capial such ha facor and oupu markes clear While i is fairly sraighforward o show ha a compeiive equilibrium exiss, i is difficul o solve for he equilibrium sequences direcly Insead an indirec approach is aken in which he Pareo opimum for his economy is deermined (his will be unique given he assumpion of represenaive agens) As shown by Debreu (954), he Pareo opimum as characerized by he opimal sequences for consumpion, labor, and capial in his environmen will be idenical o ha in a compeiive equilibrium Furhermore, facor prices are deermined by he
page 2, Harley, Hoover, Salyer, RBC Models: A User s Guide marginal producs of capial and labor evaluaed a he equilibrium quaniies (For a deailed exposiion of he connecion beween he compeiive equilibrium and Pareo opimum in a real business cycle model, see Presco, 986 [4]We now provide an example of solving such a model I DERIVING THE EQUILIBRIUM CONDITIONS The firs sep in solving for he compeiive equilibrium is o deermine he Pareo opimum To do his, he real business cycle model is recas as he following social planner s problem: () max E β U( c, h ) = subjec o: ( ) ( δ ) c i = z f k, h y k = k i z ρ k is given = z ε where [ ] E denoes expecaions condiional on informaion a =, 0 < β < is agens discoun facor, c denoes consumpion, ( h ) is leisure (agens endowmen of ime is normalized o one), i is invesmen, and 0 < δ < is he depreciaion rae of capial The exogenous echnology shock is assumed o follow he auoregressive process given in he las equaion; he auocorrelaion parameer is 0 ρ and he innovaion o echnology is assumed o have a mean of one and sandard deviaion σ ε The firs wo consrains in () is he economy-wide resource consrain and he second is he law of moion for he capial sock Dynamic Programming Problem 2
page 3, Harley, Hoover, Salyer, RBC Models: A User s Guide This infinie horizon problem can be solved by exploiing is recursive srucure Tha is, he naure of he social planner s problem is he same every period: given he beginning-of-period capial sock and he curren echnology shock, choose consumpion, labor and invesmen Noe ha uiliy is assumed o be ime-separable; ha is he choices of consumpion and labor a ime do no affec he marginal uiliies of consumpion and leisure in any oher ime period Because of his recursive srucure, i is useful o cas he maximizaion problem as he following dynamic programming problem (for a discussion of dynamic programming, see Sargen (987)): sae variables a ime : ( k z ), conrol variables a ime : ( c h k ),, (2) ( ) ( c, k, h ) { ( ) β [ ( ) ]} (, ) ( δ) v k, z = max U c, h E v k, z subjec o c k = z f k z k and z = z ε ρ (Noe ha invesmen has been eliminaed by using he law of moion for he capial sock) A soluion o his problem mus saisfy he following necessary condiions and resource consrain: ( N) U = U z f 2,, 2, { [ ]} ( N 2) U, = β E U, z f, ( δ ) ( RC) k = z f ( k, h ) k ( δ ) c Where he noaion U i, ; i =, 2 denoes he derivaive of he uiliy funcion wih respec o he ih argumen evaluaed a he quaniies ( c h ), ; f i, ; i =, 2 has an analogous inerpreaion N represens he inra-emporal efficiency condiion (he labor-leisure radeoff)ii implies ha he marginal rae of subsiuion beween labor and consumpion mus equal he marginal produc of labor The second condiion, N2, represens he ineremporal efficiency condiion The lef-hand 3
page 4, Harley, Hoover, Salyer, RBC Models: A User s Guide side represens he marginal cos in erms of uiliy of invesing in more capial while he righ-hand side represens he expeced marginal uiliy gain; a an opimum hese coss and benefis mus be equal To simplify he analysis (again, see Presco (986 [4]) for a jusificaion), assume he following funcional forms: (, ) ln ( ); (, ) U c h = c A h f k z = z k h α α (The assumpion ha uiliy is linear in leisure is based on Hansen s (985 [8]) model Then he hree equilibrium condiions become (3) α α [( α) ] α α β [ α ( δ )] α α = ( δ ) c = z k h A { } c = E c z k h k z k h k c A seady-sae equilibrium for his economy is one in which he echnology shock is assumed o be consan so ha here is no uncerainy, ha is z = for all, and he values of capial, labor, and consumpion are consan, k = k, h = h, c = c for all Imposing hese seady-sae condiions in (3), he seady-sae values are found by solving he following seadysae equilibrium condiions: ( ) ( ) = ( α) SS c A k h α α ( SS2) α k h α β δ α α ( y k ) ( 3) = = α α SS δ k = k h c = y c In he above expressions, y denoes he seady-sae level of oupu Calibraion 4
page 5, Harley, Hoover, Salyer, RBC Models: A User s Guide The nex sep in solving he model is o choose parameer values for he model This is done hrough calibraion: he se of parameers ( δ, β,, α) A are chosen so ha he seady-sae behavior of he model mach he long-run characerisics of he daa The feaures of he daa which do no exhibi cyclical characerisics are: () ( α ) = labor s average share of oupu (2) β = average risk-free real ineres rae (3) Given ( α, β) choose δ so ha he oupu-capial raio (from (SS2)) is consisen wih observaion (4) The parameer A deermines he ime spen in work aciviy To see his, muliply boh sides of (SS) by h and rearrange he expression o yield: h [( ) A]( y c) = α Bu he seady-sae resource consrain, (SS3), implies ha y c = k δ y so ha he oupu-consumpion raio is implied by he parameer values chosen in he previous hree seps Hence, he choice of A direcly deermines h Typical parameer values based on poswar US daa (see Hansen and Wrigh (992 [4]) are: α = 0 36 implying labor s share is 64%, β = 0 99 implying an annual riskless ineres rae of 004%, δ = 0 025 implying he capial-oupu raio (where oupu is measured on a quarerly basis) of roughly 0, and A = 3 which implies ha roughly 30% of ime is spen in work aciviy (These values will be used laer in Secion IV below) II LINEARIZATION The soluion o he social planner s problem is characerized by a se of policy funcions 5
page 6, Harley, Hoover, Salyer, RBC Models: A User s Guide for capial, consumpion, and labor; moreover, he soluion exiss and is unique; (see Presco (986 [4]) There is, however, no analyical soluion To make he model operaional, herefore, an approximae numerical soluion is found One of he simples mehods is o ake a linear approximaion (ie a firs-order Taylor series expansion) of he hree equilibrium condiions and he law of moion of he echnology shock around he seady-sae values ( c, k, h, z) Provided he sochasic behavior of he model does no push he economy oo far from he seady-sae behavior, he linear approximaion will be a good one (The discussion below follows closely ha of Farmer (994)) This echnique is demonsraed below: Inraemporal efficiency condiion: The opimal labor-leisure choice is represened by condiion N: c [( α) ] = A z k h α α Linearizing around he seady-sae values ( c, k, h, z) : (4) α α α α α [ α ] ( ) α [( α) ] ( ) α α [ A] k h ( z z) ( k k ) ( h h α α α α ) α [( α) A] k h α [( α) A] k h ( ) ( ) ( α) c c = A k h k k A k h h h = [( α) ] α A k h α k ( z z) z h [ ] Noe ha in he las expression, all variables have been expressed as percenage deviaions from he seady-sae (he firs wo erms modify he respecive derivaives while he las erm uses he fac ha z = in seady-sae) Consumpion can be expressed as a percenage deviaion from seadysae by using he seady-sae condiion c = ( α) α A k h α ; dividing boh sides of he Recall ha he general form for he Taylor series expansion of a funcion around a poin x* is: ( ) ( ) ( )( ) ( ) ( x x * ) f ( x ) ( x x * * = ) * * * * f x f x f x x x f x where N! denoes facorial 2 3 2! 3! 6
page 7, Harley, Hoover, Salyer, RBC Models: A User s Guide equaion by his expression and denoing percenage deviaions from seady-sae as x, eq (4) can be wrien as: (5) c = α k α h z Ineremporal Efficiency Condiion: This efficiency condiion is given by N2 α { [ ( )]} α c = β E c α z k h δ Again, linearizing around he seady-sae and expressing all variables as percenage deviaions from seady-sae yields: α α α α [ ( )] ( ) ( ) ( ) c c = c k h E c β α δ β c α α k h E k β c α α k h E h β c α k h E z α α α α ( ) ( ) ( ) Muliplying each side of he equaion by c and using he seady-sae condiion (SS2) ha yields α α [ k h ( δ) ] = β α (6) α α ( ) β ( α ) α ( ) α α β ( α) α k h E ( h ) = c E c k h E k α α β α k h E z ( ) Resource Consrain Following he same procedure as before, linearizing he resource consrain around he seady-sae yields 7
page 8, Harley, Hoover, Salyer, RBC Models: A User s Guide (7) [ α ( δ) ] ( α) α α α α α α k = k h k k h h k h z ( ) c k c Technology Shock Process The criical difference beween he seady-sae model and he real business cycle model is he assumpion ha echnology shocks are random - he shocks follow he auoregressive process described in eq () Linearizing he auo-regressive process for he echnology shock resuls in: (8) z = ρ z ε Taking expecaions of boh sides: (9) ( ) E z z = ρ III SOLUTION METHOD The equaions ha define a raional expecaions equilibrium (eqs 5, 6, 7, 9) can be wrien as a vecor expecaional difference equaion Le u c k = where bold prin denoes a vecor, hen h z he linear sysem of equaions can be wrien as: (0) Au BE ( u ) = The marices A and B are: 8
page 9, Harley, Hoover, Salyer, RBC Models: A User s Guide α α A = 0 0 0 c k α k h δ ( α) k h k h 0 0 0 ρ α α α α α α 0 0 0 0 ( ) ( ) B = β α α k h β α α k h β α k h 0 0 0 0 0 0 Premuliplying boh sides of eq (0) by A yields: α α α α α α () u = A B E ( u ) The marix A B can be decomposed as (see Hamilon (994) for deails): (2) A B = QΛ Q where Q is a marix whose columns are he eigenvecors of A whose diagonal elemens are he eigenvalues of A B and Λ is a diagonal marix B Using his decomposiion and premuliplying boh sides of he resuling expression in eq () by Q yields: (3) Q u d = ( d ) = ( Q Λ E Λ E u ) Noe ha he elemens of he defined (4 x ) column vecor d are consruced from a linear combinaion of he elemens in he rows of he (4 x 4) marix Q and he elemens of he (4 x ) 9
page 0, Harley, Hoover, Salyer, RBC Models: A User s Guide column vecor u Since Λ is a diagonal marix, eq (3) implies four independen equaions: d, = λ E d, ; i =, 2, 3, 4 (4) i i ( i ) Since he equaions in (4) mus hold every period, i is possible o recursively subsiue he expressions forward for T periods o yield: d, = λ E d, ; i =, 2, 3, 4 T (5) i i ( i T ) The λ i are four disinc eigenvalues associaed wih he four equilibrium condiions (eqs 5-8) Since one of hese condiions is he law of moion for he exogenous echnology shock (eq (8)), one of he eigenvalues will be ρ Also, he firs rows of he marices A and B are deermined by he inraemporal efficiency condiion; since his is no a dynamic relaionship, one of he eigenvalues will be zero The remaining wo eigenvalues will bracke he value of uniy as is ypical for a saddle pah equilibrium implied by he underlying sochasic growh framework As implied by eq (5), he sable, raional expecaions soluion o he expecaional difference equaion is associaed wih he eigenvalue wih a value less han one Tha is, if λ i > hen ieraing forward implies d i, which is no a permissible equilibrium Furhermore, for eq (5) o hold for all T (again aking he limi of he righ-hand side), in he sable case when λ <, i mus be he rue ha d i, = 0 ; his resricion provides he desired soluion Tha is, d i, = 0 imposes he linear resricion on ( c, k, h, z ) which is consisen wih a raional expecaions soluion (Recall ha d i, represens a linear combinaion beween he elemens of a paricular row of Q and he elemens of he vecor u ) 0
page, Harley, Hoover, Salyer, RBC Models: A User s Guide IV A PARAMETRIC EXAMPLE In his secion, a parameerized version of he RBC model described above is solved The following parameer values are used: ( β = 0 99 α = 0 36 δ = 0 025 = 3),,, A These imply he following seady-sae values: ( c = 0 79 k = 0 90 h = 0 29 y = 06),,, Noe ha hese values imply ha agens spend roughly 30% of heir ime in work aciviies and he capial-oupu raio is approximaely 0 (oupu is measured on quarerly basis); boh of hese values are broadly consisen wih US experience (see McGraan, 994) The remaining parameer values deermine he behavior of he echnology shock These are esimaed by consrucing he Solow residual 2 and hen derending ha series linearly Specifically, he Solow residual is defined as = α ( α) Z ln y ln k lnh The Z series can hen be regressed on a linear ime rend (which is consisen wih he assumpion of consan echnological progress) and he residual is idenified as he echnology shock z Using his procedure on quarerly daa over he period 60-944 resuled in an esimae of he serial correlaion of z (he parameer ρ ) o be 0 95 The variance of he shock o echnology (ie he variance of ε in eq (8)) was esimaed o be 0007 Noe ha he variance of he echnology shock is no relevan in solving he linearized version of he model -- however, when he soluion of he model is used o generae arificial ime series in he simulaion of he economy, his parameer value mus be sipulaed These values generaed he following enries ino he A and B marices: 2 The use of he Solow residual as a measure of echnology shocks is discussed in Hoover and Salyer (996)
page 2, Harley, Hoover, Salyer, RBC Models: A User s Guide 2 0 36 0 36 0 0 0 0 072 00 0 062 0 098 0 0 0 0 95 0 0 0 0 0 022 0 022 0 035 0 0 0 0 0 0 = c k h z E c k h z Following he seps described in he previous secion (pre-muliplying by A ) yields he following: c k h z E c k h z = 0 022 0 022 0 035 0 23 0 94 0 005 0 27 2 55 087 0 057 2 75 0 0 0 05 Nex, decomposing A B ino Q Q Λ and hen pre-muliplying by Q yields ( ) Q u Q u = = = c k h z E E 28 0 048 0 048 24 26 0 0 0 23 0 2 50 36 0 056 0 2 62 0 94 0 94 2 62 062 0 0 0 0 05 0 0 0 0 0 93 0 0 0 0 0 28 0 048 0 048 24 26 0 0 0 23 0 2 50 36 0 056 0 2 62 0 94 Λ 0 94 2 62 c k h z The enries in he marix Λ (ie he eigenvalues of A B ) deermine he soluion Noe ha he second diagonal enry is (accouning for rounding error) ρ The fourh row of Λ is associaed wih he inraemporal efficiency condiion These values are proporional o hose given in he firs row of he A marix; consequenly dividing all enries by ( ) 2 62 reurns he original inraemporal efficiency condiion The remaining wo enries in he Λ marix are hose relaed o he saddle pah properies of he seady-sae soluion Since a sable raional expecaions soluion is associaed wih an eigenvalue less han uniy, he hird row of he Q marix provides he linear resricion we are seeking Tha is, he raional expecaions soluion is:
page 3, Harley, Hoover, Salyer, RBC Models: A User s Guide Or, 2 50 c 36 h 0 056k 0 z = 0 (6) c = 054 h 0 02k 0 44 z The law of moion for he capial sock (he parameer values are given in he hird row of he A marix) and he inraemporal efficiency condiion provides wo more equilibrium condiions: (7) (8) k c k h = 0 07 0 0 06 00 z h = 2 78c k 2 78 z A random number generaor can nex be used o produce a sequence of echnology shocks The above equilibrium equaions can hen be used o produce ime series for capial, consumpion, labor, and oupu V ANALYZING OUTPUT FROM THE ARTIFICIAL ECONOMY The soluion o he model is characerized by eqs (6)- (8) - given iniial values for capial, and nex generaing a pah for he exogenous echnology shock ( z ), hese equaions will produce ime-series for ( c, k, h ) Two oher series ha mos macroeconomiss are ineresed in, namely oupu and invesmen, can be generaed by linearizing he producion funcion and he resource consrain, respecively Specifically, for oupu, linearizing he assumed Cobb-Douglas producion funcion (ie y α = z k h α and using he calibraed value ha α = 0 36 ) yields he following equaion: (9) y = z 0 36 k 0 64 h Finally, a linear approximaion of he condiion ha, in equilibrium, oupu mus equal he sum of 3
page 4, Harley, Hoover, Salyer, RBC Models: A User s Guide consumpion and invesmen can be expressed in he form as a percenage deviaion from he seady sae as: (20) y i i y c = i c Using he seady-sae values employed in he numerical soluion, he invesmen equaion becomes: (2) 06 0 79 i y c y c = = 3 92 2 92 027 0 27 Hence, equilibrium in his economy is described by he following se of equaions (E) c = 0 54h 0 02k 0 44z k = 0 07c 0k 0 06h 0 0z h c k = 278 278z y = z 0 36 k 0 64h i y c = 3 92 2 92 z = 0 95 z ε To generae he ime series implied by he model, i is necessary o firs generae a series for he innovaions o he echnology shock, ie ε These are assumed o have a mean of zero and a variance ha is consisen wih he observed variance for he innovaions, which, as menioned above, is roughly 0007 Then, iniializing z = 0 and using a random number generaor in order o generae he innovaions, a pah for he echnology shocks is creaed Nex, assuming ha all remaining values are iniially a heir seady-sae (which implies ha all iniial values are se o zero), he sysem of equaions above can be solved o produce he ime pah for he endogenous variables 4
page 5, Harley, Hoover, Salyer, RBC Models: A User s Guide We generae arificial ime pahs for consumpion, oupu, and invesmen (3000 observaions were creaed and only he las 20 were examined) These are shown in Figure I is clear from Figure, as is also rue in he acual daa,ha, he volailiy of invesmen is greaer han ha of oupu, which is greaer han ha of consumpion To see his more precisely, he sandard deviaion of consumpion, labor, and invesmen relaive o oupu is repored in Table along wih he correlaions of hese series wih oupu Table : Descripive Saisics for US and RBC Model 3 relaive volailiy Corr(x, y) consumpion model US daa 052 049 082 076 invesmen model US daa 286 302 095 080 labor model US daa 065 096 089 088 3 Saisics for US daa are aken from Kydland and Presco (990 [2]), Tables I and II, p 0-5
page 6, Harley, Hoover, Salyer, RBC Models: A User s Guide % deviaion from seady-sae Figure : Oupu, Consumpion, and Invesmen in RBC Model 03 02 invesmen 0 oupu 00 consumpion -0-02 Time 6
page 7, Harley, Hoover, Salyer, RBC Models: A User s Guide REFERENCES Debreu, Gerard (954) "Valuaion Equilibrium and Pareo Opimum," Proceedings of he Naional Academy of Science, 40: 588--92 Farmer, Roger E A (993) The Macroeconomics of Self-fulfilling Prophecies, Cambridge, MA: MIT Press Hamilon, James D (994) Time Series Analysis, Princeon, New Jersey: Princeon Universiy Press Hansen, Gary D (985) "Indivisible Labor and he Business Cycle," Journal of Moneary Economics, 6(3), November: 309--28, reprined here in Chaper 8 Hansen, Gary D and Randall Wrigh (992) The Labor Marke in Real Business Cycle Theory, Federal Reserve Bank of Minneapolis Quarerly Review, 6(2), Spring: 2- -2, reprined here in Chaper 9 Hoover, Kevin D and Kevin D Salyer (996) Technology Shocks or Colored Noise? Why Real Business Cycle Models Canno Explain Acual Business Cycles, unpublished manuscrip Kydland, Finn E and Edward C Presco (990) "Business Cycles: Real Facs and a Moneary Myh," Federal Reserve Bank of Minneapolis Quarerly Review, 4(2), Spring: 3--8, reprined here in Chaper 2 McGraen, Ellen R (994) A Progress Repor on Business Cycle Models, Federal Reserve Bank of Minneapolis Quarerly Review, 8(4), Fall: 2--6 Presco, Edward C (986) "Theory Ahead of Business Cycle Measuremen," in Federal Reserve Bank of Minneapolis Quarerly Review, 0(4), Fall: 9--22, reprined here in Chaper 4 7
page 8, Harley, Hoover, Salyer, RBC Models: A User s Guide Sargen, Thomas J (987) Dynamic Macroeconomic Theory, Cambridge, Mass: Harvard Universiy Press 8