Graduate Macro Theory II: Notes on Neoclassical Growth Model

Transcription

1 Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens. These agens are idenical, and so we can effecively rea hem as one. These agens consume, save in physical capial, and supply one uni of labor each period inelasically. Time runs from = 0,...,. The populaion a each poin in ime is N = 1+n) N 0, wih N 0 given. Households ge flow uiliy from consumpion, C, by an increasing and concave funcion, uc ). They discoun he fuure by 0 < β < 1. Firms produce oupu using capial and labor, where hese inpus are urned ino oupus hrough a consan reurns o scale producion funcion. The firms are owned by he households. There are wo oher exogenous inpus o producion: Z, which is called labor augmening echnology; and A, which is neural echnological progress. Assume ha Z = 1 + z) Z 0, Z 0 given. Suppose ha A follows some saionary sochasic process. The producion funcion is: Y = A F K, Z N ) 1) Oupu mus be eiher consumed or used as invesmen in new capial goods. aggregae accouning ideniy is: Hence he Y = C + I 2) Capial accumulaes according o he following equaion, wih 0 < δ < 1 he depreciaion rae on capial. There is an assumed one period delay beween when new capial is accumulaed and when i is producive: Equaions 1)-3) can be combined: K +1 = I + 1 δ)k 3) K +1 = A F K, Z N ) C + 1 δ)k 4) 1

2 The equilibrium of he economy can be expressed as he soluion o a social planner s problem we will discuss decenralized compeiive equilibria versus social planner s soluions laer): max C,K +1 E 0 β uc ) =0 s.. K +1 = A F K, Z N ) C + 1 δ)k There are wo sources of non-saionariy in he model populaion growh, n; and echnological progress, z. Sandard soluion mehodologies require ha he variables of he model be saionary. There are wo ways o proceed eiher ransform he economy and hen characerize he equilibrium condiions, or ge he equilibrium condiions and hen ransform hem o be consisen wih saionariy. I m going o impose a sandard funcional form specificaion on preferences, which is ha flow uiliy over consumpion is iso-elasic: u C ) = C1 σ 1 For wo funcions, fx) and gx), if lim x c fx) = lim x c gx) = 0 or +/, L Hopial s rule says: lim x c fx) gx) = lim f x) x c g x) L Hopial s rule is applicable in his case because, as σ 1, boh numeraor and denominaor of flow uiliy go o zero. The derivaive of C 1 σ wih respec o σ is ln C C 1 σ. As σ 1, his is jus ln C. The derivaive of wih respec o σ is -1. Hence he flow uiliy funcion simply converges o ln C as σ 1. Finally, we will almos always be working wih a Cobb-Douglas producion echnology, so ha: F K, Z N ) = K α Z N ) 1 α, 0 < α < 1 2 Firs Order Condiions We can find he firs order condiions necessary for an inerior soluion a couple of differen ways eiher he mehod of Lagrange mulipliers or by expressing he problem as a dynamic program. I ll begin wih he Lagrangian formulaion. 2

3 2.1 A Lagrangian Formulaion I wrie Lagrangians as curren value Lagrangians, which means ha he discoun facor muliplies he consrains. An alernaive formulaion is o use a presen value Lagrangian, in which case he discoun facor does no muliply he consrains. These yield idenical soluions and only differ in he inerpreaion of he mulipliers. The curren value Lagrangian for he planner s problem of he neoclassical growh model is is: L = =0 The firs order condiions are: C β 1 σ ) E 0 + λ A ) K α Z N ) 1 α C + 1 δ)k K +1 L = 0 C σ = λ 5) C L ) = 0 λ = βe λ +1 αa +1 K+1 α 1 K Z +1N +1 ) 1 α + 1 δ) 6) +1 L = 0 K +1 = A K α Z N ) 1 α C + 1 δ)k 7) λ This Lagrangian formulaion is called curren value because he Lagrange muliplier is he marginal uiliy of consumpion a ime. A presen value Lagrangian would have he muliplier equal o he marginal uiliy of consumpion a ime discouned back o he beginning of ime. In addiion o hese firs order condiions, which are necessary for an opimum, here is also he ransversaliy condiion, which combined wih he above FOC is necessary and sufficien for an opimum: lim T βt λ T K T +1 = 0 The ransversaliy condiion is mos easily undersand by supposing ha he problem is finie horizon. A he end of ime, you would never wan o leave any uiliy on he able, so o speak, since here is no more omorrow. K T +1, he amoun of capial lef over a he end of he ime period T is he las period, K T +1 is how much would be lef over afer producing in period T ), could always be consumed, yielding uiliy λ T K T +1. If he presen value of his is no zero, hen he household has over-saved and could no have been opimizing. 5) can be combined wih 6) o eliminae he muliplier, yielding: C σ ) = βe C+1 σ αa +1 K+1 α 1 Z +1N +1 ) 1 α + 1 δ) The variables of he model are growing, inheriing deerminisic rends from boh Z and N. To re-scale he firs order condiions so ha all he variables are he model are saionary, define lower case variables as hose divided by Z N : 8) 3

4 C c 9) Z N Y y Z N 10) k K Z N 11) Muliply and divide boh sides of 8) by Z N as needed o make i saionary: ) σ ) σ ) α 1 Z N ) σ C = β Z +1N +1) σ C+1 K+1 E αa δ)) Z N Z +1 N +1 Z +1 N +1 Using he ransformed variable noaion and simplifying somewha, we have: c σ Z+1 N +1 = β Z N Noing ha Z +1 Z = 1 + z and N +1 N ) σ E c σ +1 αa+1 k+1 α δ)) = 1 + n, we can define γ = 1 + z)1 + n) and wrie his as: c σ = βγ σ E c σ +1 αa+1 k+1 α δ)) 12) Effecively he presence of rend growh changes he discoun facor ha shows up in he Euler equaion: insead of β, i is βγ σ. For plausible parameerizaions, here isn much difference. Now ransform he capial accumulaion equaion by muliplying and dividing as necessary: Simplifying: ) K α +1 K C Z +1 N +1 = A Z N Z N + 1 δ) K Z N Z +1 N +1 Z N Z N Z N γk +1 = A k α c + 1 δ)k 13) Again, his only differs from he regular specificaion by he facor γ on he lef hand side, which is close o one anyway. 12)-13), along wih he ransversaliy condiion which is he same expressed in efficiency unis, divided by Z N, or regular unis), characerize he soluion o he planner s problem. Because of he fac ha he equilibrium condiions of he ransformed economy do no differ much from he efficiency unis represenaion, i is common o jus ignore rend growh alogeher. I really has very lile effec on he numerical soluion of he model. A he end of he day, he soluion of he model is a sysem of wo difference equaions 12) and 13). These joinly describe he dynamics of he endogenous sae variable, k, and he conrol, or jump, variable c. Since capial is predeermined, you have o have iniial iniial condiion on capial exogenously given. The soluion of he model is a value of c, given k, ha is consisen wih he wo difference equaions. In general, here are many differen values of c consisen wih 4

5 he wo difference equaions a an inuiive level, hink abou i here are wo equaions bu only one real unknown since k is given). To ge a unique soluion, we use he ransversaliy condiion combined wih a feasibiliy consrain neiher c nor k can go o zero in he limi, or, colloquially, he economy canno explode. If he economy canno explode, we mus in he limi approach he seady sae if we weren a he seady sae, consumpion would eiher be forever decreasing hiing zero, which would drive he marginal uiliy of consumpion o infiniy, violaing he ransversaliy condiion) or capial would go o zero which would mean consumpion would have o go o zero for he allocaion o be feasible, which would also violae he ransversaliy condiion). So we can hink abou he soluion as picking c, given an iniial condiion on k, such ha i i) obeys he dynamics of he wo difference equaions, and ii) he sysem approaches he seady sae over ime. 2.2 A Value Funcion Represenaion An alernaive way o find he firs order condiions is by seing he problem up as a dynamic programming problem. Before beginning, here is an alernaive way o deal wih rend growh. Raher han characerizing he soluion in hen re-wriing i in efficiency unis, you can firs ransform he problem and hen ge he equilibrium condiions. For pracice, le s do i his way now. The problem is: max C,K +1 E 0 =0 s.. β C1 σ 1 K +1 = A K α Z N ) 1 α C + 1 δ)k We can re-wrie boh objecive funcion and consrain in erms of efficiency unis by fiddling around. Begin wih he objecion funcion, muliplying and dividing by Z N ) 1 σ. A dae flow uiliy is ignore he exra -1 in he objecive funcion, which is jus a consan and is here only so ha we can echnically apply L Hopial s rule in he case ha σ = 1): ) 1 σ C β C1 σ = β Z Z N N ) 1 σ Use he fac ha Z = 1 + z) Z 0 and N = 1 + n) N 0, and normalize he iniial condiions o Z 0 = N 0 = 1 which is fine because ha is effecively going o jus normalize flow uiliy by a consan, which does no maer) and we can wrie Z N = γ, where again γ = 1 + n)1 + z). Then we can re-wrie period flow uiliy as: Now go o he consrain: βγ 1 σ ) c 1 σ 5

6 Simplifying: ) K α +1 K C Z +1 N +1 = A Z N Z N + 1 δ) K Z N Z +1 N +1 Z N Z N Z N γk +1 = A k α c + 1 δ)k Hence, wrien in efficiency unis, he planner s problem is: max c,k +1 E 0 =0 βγ 1 σ ) c 1 σ s.. γk +1 = A k α c + 1 δ)k For he purposes of he Bellman equaion, he sae variables are A and k. The choice variable is c. The Bellman equaion wries he infinie horizon dynamic programming problem ino wo periods: V A, k ) = max c { c 1 σ } + βγ1 σ E V A +1, k +1 ) s.. γk +1 = A k α c + 1 δ)k I is easies o eliminae he choice variable and insead wrie he problem as choosing he fuure sae. We can do his by imposing ha he consrain holds and eliminaing c : V A, k ) = max k +1 { A k α + 1 δ)k γk +1 ) 1 σ + βγ 1 σ E V A +1, k +1 ) } The funcion V ) is unknown, bu assume i is differeniable. The max operaor ouside means ha k +1 needs o be chosen opimally. This implies aking a derivaive wih respec o k +1. The firs order condiion is: Re-arranging slighly yields: γc σ = βγ 1 σ E V A +1, k +1 ) k +1 c σ = βγ σ E V A +1, k +1 ) k +1 14) 6

7 Well, i isn so immediaely obvious why his helpful, because we have a derivaive of an unknown funcion on he righ hand side. Here we make use of he Benvenise-Scheinkman envelop condiion. Suppose ha he sequence of fuure capial socks has been chosen opimally i.e. we have aken care of he max operaor): V A, k ) = A k α + 1 δ)k γk +1 ) 1 σ Differeniae V ) wih respec o k : + βγ 1 σ E V A +1, k +1 ) V A, k ) k = c σ αa k α δ) γ dk ) +1 + βγ 1 σ V A +1, k +1 ) dk +1 E dk k +1 dk This can be re-arranged: V A, k ) k = c σ αa k α δ) ) ) γc σ βγ 1 σ V A +1, k +1 ) dk+1 E k +1 dk This can be furher re-wrien as: V A, k ) k = c σ αa k α δ) ) ) γ c σ βγ σ V A +1, k +1 ) dk+1 E k +1 dk Bu he FOC as wrien above ells us ha c σ = βγ σ V A E +1,k +1 ) k +1. This means ha he second erm drops ou, leaving: V A, k ) k = c σ αa k α δ) ) This is he derivaive of he value funcion wih respec o is argumen, he capial sock. The envelope condiion essenially ells us ha we can rea k +1 as fixed when evaluaing he derivaive of he value funcion wih respec o k. To use his in he firs order condiion 14) we have evaluae he argumen a k +1 and A +1. Plugging in o 14), we ge: c σ = βγ σ E c σ +1 αa+1 k+1 α δ)) 15) Noe his is exacly he same as 12), he firs order condiion we go using he Lagrangian. The value funcion/bellman equaion approach is a lile more widely applicable because i can beer deal wih non-lineariies and corner soluions, bu mos of he ime hey yield he same soluions. 7

8 3 The Seady Sae The soluion of he model is a policy funcion given he saes, wha is he opimal choice of he conrol. Excep for very special cases in paricular δ = 1), here is no analyical soluion for his model. We can analyically characerize he soluion for a special case in which he variables of he model are consan, however. This is called he seady sae. From our specificaion of he producion funcion i follows ha he per effecive worker producion funcion is hen: y = A k α 16) The non-sochasic seady sae is defined as a siuaion in which all variables are consan and where he only source of uncerainy which in his case is A ) is held consan a is uncondiional mean. In paricular, his requires ha k +1 = k and c +1 = c. Denoe hese values k and y, respecively. Le A denoe he seady sae value of A, which is equal o is uncondiional mean. We can analyically solve for he seady sae capial sock from equaion 8): c σ = βγ σ c σ αa k α δ) ) γ σ β 1 δ) = αa k α 1 k = γ σ β αa 1 δ) From he accumulaion equaion, we know ha: ) 1 1 α 17) A k α = c + γ 1 δ))k Hence we can solve for seady sae consumpion as: c = A αa γ σ /β 1 δ) ) α 1 α γ 1 δ)) αa γ σ /β 1 δ) ) 1 1 α We can consider a couple of simple comparaive saics. If A increases, hen k and c also increase. If β goes up households become more paien), hen k and c boh go up as well. 18) 4 The Phase Diagram As I discussed above, he soluion of he model is a value of c, given an iniial condiion on k, such ha i) he sysem evolves according o he dynamics of he wo difference equaions and ii) he 8

9 sysem approaches he seady sae. A phase diagram allows us o hink abou he model soluion graphically. We can qualiaively characerize he full soluion o he model hrough a phase diagram. In a wo dimensional world, he phase diagram ypically pus he endogenous sae variable on he horizonal axis in his case k ) and he jump variable on he verical axis in his case c ). Phase diagrams are more naural in coninuous ime; we will proceed in discree ime wih one sligh abuse of noaion. We wan o find he ses of poins where he sae capial) and jump variables are no changing in k, c ) space. Call hese wo ses of poins he k +1 k = 1 isocline and he c +1 c = 1 isocline. There is only one value of k +1 consisen wih c +1 c = 1, which is he seady sae capial sock if A is a is mean. We can see his from he firs order condiion. c +1 = 1 : k +1 = c αa +1 γ σ /β 1 δ) ) 1 1 α 19) The k +1 k = 1 isocline is found by looking a he capial accumulaion equaion: Hence he isocline is defined by: γk +1 = A k α c + 1 δ)k k +1 = k c = A k α + 1 δ γ)k k +1 k = 1 : c = A k α + 1 δ γ)k 20) A complicaion arises because k shows up in he k +1 k = 1 isocline, whereas he c +1 c = 1 isocline depends on k +1. This problem would no be presen in coninuous ime, which is why coninuous ime is more naural for phase diagram. I m going o simply circumven his issue by assuming ha k +1 k, and will rea he k +1 in he c +1 c = 1 isocline as k. Given his simplifying assumpion, we can graph each of hese lines in a plane wih c on he verical axis and k on he horizonal axis. The c +1 c assuming ha A is a is seady sae). The k +1 k see ha is slope is dc dk = αa k α 1 = 1 isocline is a verical line a k = k = 1 isocline is a bi more complicaed. We can + 1 δ γ). When k is small i.e. near he origin), hen his slope is posiive because αa k α 1 slope will be negaive and will approach 1 δ γ) < 0 because αa k α 1 will be large. When k is large far away from he origin), he will go o zero, and ) 1 1 α, which is greaer han k when evaluaed a he γ 1). The peak occurs where k = αa δ+γ 1 seady sae value of A. The acual seady sae is where he wo isoclines cross. 9

10 The above picure shows he isoclines and he seady sae. I also shows he i) saddle pah and ii) some unsable dynamic lines. These dynamics can be derived as follows. Below he k +1 k = 1 isocline, c is oo small, and hence k +1 > k, and we draw arrows poining he righ, denoing he direcion in which k will be expeced o ravel. Above he k +1 k = 1 isocline, c is oo big, and k +1 < k, and hence we draw arrows poining lef. To ge he dynamic arrows relaive o he c +1 c = 1 isocline, we have o engage in a sligh abuse of erminology. Technically wha governs he evoluion of c is where k +1 is relaive o k, bu wha shows up in he diagram is k. Le s ignore his disincion and rea k k +1. To he righ of he c +1 c means ha αa +1 k α 1 +1 he righ of he c +1 c = 1 isocline, k is oo big ; his will be small, and consumpion will be expeced o decline. Hence, o = 1 isocline we draw arrows poining down, showing he expeced direcion of consumpion in ha region. To he lef of he c +1 c consumpion will be expeced o grow, so we draw arrows poining up. = 1 he opposie is rue; k is oo small, and Visually inspecing he picure, we see ha he arrows poin oward he seady sae when he sysem sis o norheas and souhwes of he seady sae. In he regions of he picure ha are o he norhwes or souheas of he seady sae, he arrows poin away from he seady sae. The idea of he policy funcion is o pick c given k such ha i) he firs order condiions hold; ii) he ransversaliy condiion holds; and iii) he soluion is feasible wih he consrains. Drawing in he dynamics as we have done presumes i). The ransversaliy condiion rules ou picking any value of c in he souheas region hose regions would evenually lead o 0 consumpion so λ ) and infinie capial so k +1 ), which leads o a violaion of he ransversaliy condiion. Picking any value of c in he norhwes region would violae iii) we would move owards infinie consumpion wih zero capial, which is infeasible. Hence, for any given k he sae), consumpion mus sar eiher in he souhwes or norheas regions. Any old value of c will no do here will be a unique value of c for each k such ha we ravel owards he seady sae. Any oher value of c shown by he explosive dynamic arrows), would evenually lead o a violaion of ii) or iii). The unique se of values of c consisen wih i) - 10

11 iii) being saisfied and holding A fixed is given by line wih all he arrows poining oward he seady sae his is he saddle pah. This can be inerpreed as he policy funcion when A is a is uncondiional mean for any given curren k, i ells you he value of c consisen wih opimizaion. In oher words, he saddle pah is he soluion of he model i ells us wha c should be for every k such ha he dynamics of he wo difference equaions are obeyed and such ha we end up approaching he seady sae. 5 Dynamic Effecs of Shocks In his secion I work hrough wo differen exercises: 1) an unexpeced permanen change in A ; and 2) an unexpeced bu emporary change in A. For 2), suppose ha A increases immediaely and is expeced o remain a ha level unil ime T, a which poin is goes back o is iniial saring value. For hese shifs, always assume ha he economy begins in is seady sae. The dynamics of hese sysems always work as follows. Whenever somehing exogenous changes, he jump variable in his case consumpion) mus jump in such a way ha i rides he new sysem dynamics for as long as he change in he exogenous variable is in effec, and in expecaion he sysem mus hi a seady sae evenually eiher reurning o he original seady sae or going o a new seady sae, depending on wheher he change in he exogenous variable is permanen or no). The sae variable canno jump immediaely, bu will follow in expecaion) he dynamics of he sysem hereafer. We begin wih he permanen increase in A. This clearly shifs boh isoclines. In paricular, he c +1 c = 1 isocline shifs o he righ, and he k +1 k = 1 isocline shifs up and o he righ. The new isoclines are shown as dashed lines; he original isoclines are solid. I is clear ha he seady sae values of boh consumpion and he capial sock are higher. There s also a new saddle pah/policy funcion. Depending on he slope of he saddle pah, his could cross he original value of k eiher above, below, or exacly a he original value of c. This means ha he iniial jump in consumpion is ambiguous. I have shown his sysem wih consumpion iniially jumping up, which is wha will happen under plausible parameerizaions of σ. Thereafer he sysem mus ride he new dynamics and approach he new seady sae. Since k +1 > k, invesmen increases on impac, so ha consumpion does no increase as much as oupu. Below he phase diagram I show he impulse responses, which race ou he dynamic responses of consumpion and he capial sock o he shock. Again, noe ha consumpion jumps on impac, whereas he capial sock does no, and from hereafer hey ride he dynamics o a seady sae. 11

12 Now consider an unexpeced bu emporary increase in A. Here he new isoclines shif in he same way hey do in he case of a permanen shif in A. I show he new isoclines as dashed lines, and also show he new saddle pah as a dashed line. Here, however, consumpion canno jump all he way o he new saddle pah as i did in he case of he permanen change in A. Consumpion mus jump and ride he unsable new sysem dynamics unil ime T, when A goes back o is original value, a which poin in ime i mus be back on he original saddle pah. From here on i mus follow he original saddle pah back ino he original seady sae. 12

13 The figure above shows he ime pah of c wih a scrachy line. I jumps up, rides he unsable dynamics of he new sysem, and his he original saddle pah a exacly ime T. Imporanly, we see from he picure ha i jumps less han i would if he change in A were permanen if i were permanen i would jump all he way o he new saddle pah). Since c jumps by less on impac bu he change in oupu is he same as if he shock were permanen, invesmen mus jump by more. We can back ou he ime pah of invesmen from he impulse response for he capial sock, shown in he figure below he phase diagram. These resuls are consisen wih he inuiion from he permanen income hypohesis. This exercise urns ou o provide imporan insigh ino more complicaed problems. In hese dynamic sysems you can almos never find he analyical soluion for a general case, hough you can do so for he i) seady sae and ii) wha he soluion would look like if he jump variable didn jump a all. For he case of emporary bu persisen exogenous shocks, he full soluion is somewhere in beween hose wo exreme cases beween i) and ii)). If he shock is no very persisen, a good approximaion o he soluion would have he jump variable no changing a all. 13