A Simple Introduction to Dynamic Programming in Macroeconomic Models

Transcription

1 Economics Deparmen Economics orking Papers The Universiy of Auckland Year A Simple Inroducion o Dynamic Programming in Macroeconomic Models Ian King Universiy of Auckland, ip.king@auckland.ac.nz This paper is posed a ResearchSpace@Auckland. hp://researchspace.auckland.ac.nz/ecwp/3

2 A Simple Inroducion o Dynamic Programming in Macroeconomic Models Ian King * Deparmen of Economics Universiy of Auckland Auckland New Zealand April (Ocober 987 Absrac This is inended as a very basic inroducion o he mahemaical mehods used in Thomas Sargen's book Dynamic Macroeconomic Theory. I assumes ha readers have no furher mahemaical background han an undergraduae "Mahemaics for Economiss" course. I conains secions on deerminisic finie horizon models, deerminisic infinie horizon models, and sochasic infinie horizon models. Fully worked ou eamples are also provided. * Financial suppor of he Social Sciences and Humaniies Research Council of Canada is graefully acknowledged. I am indebed o David Backus, Masahiro Igarashi and for commens. I would like o hank Tom McCurdy for his commens on earlier drafs, and for suggesing he las eample presened here.

3 FOREARD ( I wroe his guide originally in 987, while I was a graduae suden a Queen s Universiy a Kingson, in Canada, o help oher sudens learn dynamic programming as painlessly as possible. The guide was never published, bu was passed on hrough differen generaions of graduae sudens as ime progressed. Over he years, I have been informed ha several insrucors a several differen universiies all over he world have used he guide as an informal supplemen o he maerial in graduae macro courses. The qualiy of he phoocopies has been deerioraing, and I have received many requess for new originals. Unforunaely, I only had hard copy of he original, and his has also been deerioraing. Because he maerial in his guide is no original a all i simply summarizes maerial available elsewhere, he usual oules for publicaion seem inappropriae. I decided, herefore, o simply reproduce he handou as a pdf file ha anyone can have access o. This required reyping he enire documen. I am eremely graeful o Malliga Rassu, a he Universiy of Auckland for paienly doing mos of his work. Raher han oally reorganize he noes in ligh of wha I ve learned since hey were originally wrien, I decided o leave hem prey much as hey were wih some very minor changes (mainly references. I hope people coninue o find hem useful. INTRODUCTION (987 This noe grew ou of a handou ha I prepared while uoring a graduae macroeconomics course a Queen's Universiy. The main e for he course was Thomas Sargen's Dynamic Macroeconomic Theory. I had been my eperience ha some firs year graduae sudens wihou srong mahemaical backgrounds found he e heavy going, even hough he e iself conains an inroducion o dynamic programming. This could be seen as an inroducion o Sargen's inroducion o hese mehods. I is no inended as a subsiue for his chaper, bu raher, o make his book more accessible o sudens whose mahemaical background does no eend beyond, say, A.C. Chaing's Fundamenal Mehods of Mahemaical Economics. The paper is divided ino 3 secions: (i Deerminisic Finie Horizon Models, (ii Deerminisic Infinie Horizon Models, and (iii Sochasic Infinie Horizon Models. I also provides five fully worked ou eamples.

4 . DETERMINISTIC, FINITE HORIZON MODELS Le us firs define he variables and se up he mos general problem, (which is usually unsolvable, hen inroduce some assumpions which make he problem racable.. The General Problem: { v( U (,,, ; v,v,,v T T subjec o i G (,,, ; v,v,, T v T ii Ω for all T iii v iv T given here: v is a vecor of sae variables ha describe he sae of he sysem a any poin in ime. For eample, i could be he amoun of capial good i a ime. is a vecor of conrol variables which can be chosen in every period by he decision-maker. For eample j v could be he consumpion of good j a ime. U ( is he objecive funcion which is, in general, a funcion of all he sae and conrol variables for each ime period. G ( is a sysem of ineremporal consrains connecing he sae and conrol variables. Ω is he feasible se for he conrol variables assumed o be closed and bounded. In principle, we could simply rea his as a sandard consrained opimisaion problem. Tha is, we could se up a Lagrangian funcion, and (under he usual smoohness and concaviy assumpions grind ou he Kuhn-Tucker condiions.

5 In general hough, he firs order condiions will be non-linear funcions of all he sae and conrol variables. These would have o be solved simulaneously o ge any resuls, and his could be eremely difficul o do if T is large. e need o inroduce some srong assumpions o make he problem racable.. Time Separable (Recursive Problem: Here i is assumed ha boh he U ( and he ( G funcions are ime-separable. Tha is: (,, ;v,v U (,v + U (,v + + U (,v S( U + T, T T T T where S ( T is a "scrap" value funcion a he end of he program (where no furher decisions are made. Also, he G ( funcions follow he Markov srucure: T T G G (, v G (, v Transiion equaions v T ( T, T Noe: Recall ha each is a vecor of variables i where i indees differen kinds of sae variables. Similarly wih v. Time separabiliy sill allows ineracions of differen sae & conrol variables, bu only wihin periods. The problem becomes: v ;,,..., T v Ω T U (, v + S( i i subjec o i G (, v i n + T and,..., T ii i given i n i Once again, in principle, his problem could be solved using he sandard consrained opimisaion echniques. The Lagrangian is: 3

6 T i [ + ] T n i i (,v + S( + G (,v L U λ T i This problem is ofen solvable using hese mehods, due o he emporal recursive srucure of he model. However, doing his can be quie messy. (For an eample, see Sargen (987 secion.. Bellman's "Principle of Opimaliy" is ofen more convenien o use..3 Bellman's Mehod (Dynamic Programming: Consider he ime-separable problem of secion. above, a ime. Problem A: v ;,,..., T v Ω T U (, v + S( T and,..., T i i subjec o i G (, v i n + ii i given i n i Now consider he same problem, saring a some ime >. Problem B: v ;,..., T v Ω T U (, v + S( i i subjec o i G (, v i n T and,..., T + ii i i given i n Le he soluion o problem B be defined as a value funcion (,T "Principle of Opimaliy" assers: V. Now, Bellman's Any soluion o Problem A (i.e. on he range T which yields mus also solve Problem B (i.e.: on he range T. 4

7 (Noe: This resul depends on addiive ime separabiliy, since oherwise we couldn' "break" he soluion a. Addiive separabiliy is sufficien for Bellman's principle of opimaliy. Inerpreaion: If he rules for he conrol variables chose for he problem are opimal for any given, hen hey mus be opimal for he * of he larger problem. Bellman's P. of. allows us o use he rick of solving large problem A by solving he smaller problem B, sequenially. Also, since is arbirary, we can choose o solve he problem T firs, which is a simple -period problem, and hen work backwards: Sep : Se T, so ha Problem B is simply: { v T U (,v S( T T T + T subjec o: i G ( v T T T, T ii T T given One can easily subsiue he firs consrain ino he objecive funcion, and use sraighforward calculus o derive: v ( T Conrol rule v T ht for T This can hen be subsiued back ino he objecive fn o characerize he soluion as a value funcion: V (, U (,h ( S( G (,h ( T T T T T + T T T T 5

8 & % Sep : Se T so ha Problem B is:!" v v T T { U (, v + U ( v + S( T T T T T, subjec o: i G (, v T T T T T ii G (, v T T T T iii T T given Bellman's P.O. implies ha we can rewrie his as: '( ( T, vt + { U T ( T, vt S( T vt #$ U T vt - T subjec o (i, (ii and (iii. Recall ha sep has already given us he soluion o he inside maimizaion problem, so ha we can re-wrie sep as: { v T { U (, v V (, T T T + T subjec o: i G (, v T T T T ii T T given Once again, we can easily subsiue he firs consrain ino he objecive funcion, and use sraighforward calculus o derive: v ( T Conrol rule v T ht for T This can be subsiued back ino he objecive fn. o ge a value funcion: V ( { U (, v V (, T, T T T + T { v U T (,h ( V ( G (,h ( T T T T + T T T T, 6

9 Sep 3: Using an argumen analogous o ha used in sep we know ha, in general, he problem in period T-k can be wrien as: V "Bellman' s" ( k { U (, v + V (, k Equaion T k, T k T k T k T k + { vt k subjec o: i G (, v ii T k + T k T k T k T k given T k This maimizaion problem, given he form of he value funcion from he previous round, will yield a conrol rule: v T k h T k ( T k Sep 4: Afer going hrough he successive rounds of single period maimizaion problems, evenually one reaches he problem in ime zero: V (, T { v { U (, v + V (, T subjec o: i G (, v ii given This will yield a conrol rule: v ( h Now, recall ha is given a value a he ouse of he overall dynamic problem. This means ha we have now solved for v as a number, independen of he 's (ecep he given. 7

10 Sep 5: Using he known and v and he ransiion equaion: (, G v i is simple o work ou, and hence v from he conrol rule of ha period. This process can be repeaed unil all he overall problem A will hen be solved. and v values are known. The.4 An Eample: This is a simple wo period minimizaion problem, which can be solved using his algorihm. Min { v [ + v ] + subjec o: i + ( + v ( ii (3 In his problem, T. To solve his, consider firs he problem in period T- (i.e.: in period : Sep : { v { + v Min + subjec o: i + v (5 ii (4 given (6 Subsiuing 5 and 6 ino 4 yields: { v { + v + [ v ] Min + FOC: v + [ + v ] * v Conrol rule in period (7 8

11 .-, Now subsiuing 7 back ino 4 yields (using 5: V (, + + ( * (, 3 V (8 Sep : In period T- (i.e.: period Bellman's equaion ells us ha he problem is: { v { + v V (, Min + (9 subjec o: i + v ( ii ( Subsiuing 8 ino 9, and hen and ino 9 yields: { v { + v + 3[ v ] Min + FOC: v + 6[ + v ] + 3 Conrol value in period ( v Sep 5: Subsiuing and ino gives: / (3 Now subsiue 3 ino 7 o ge: v Conrol value in period (4 Finally, subsiue 3 and 4 ino 5 o ge: + (5 Equaions -5 characerize he full soluion o he problem. 9

12 5. DETERMINISTIC, INFINITE HORIZON MODELS.. Inroducion: One feaure of he finie horizon models is ha, in general he funcional form of he conrol rules vary over ime: v h ( Tha is, he h funcion is differen for each. This is a consequence of wo feaures of he problem: i The fac ha T is finie ii The fac ha U (, v and (, v G have been permied o depend on ime in arbirary ways. In infinie horizon problems, assumpions are usually made o ensure ha he conrol rules o have he same form in every period. Consider he infinie horizon problem (wih ime-separabiliy: 67 ;,..., 8 v Ω v 43 U (, v subjec o: i G (, v + ii given For a unique soluion o any opimizaion problem, he objecive funcion should be bounded away from infiniy. One rick ha faciliaes his bounding is o inroduce a discoun facor β where β <. A convenien simplifying assumpion ha is commonly used in infinie horizon models is saionariy: Assume i β β ii U (, v β U (, v iii G (,v G(, v

13 < A furher assumpion, which is sufficien for boundedness of he objecive funcion, is boundedness of he payoff funcion in each period: Assume: U (,v < M < where M is any finie number. This assumpion, however, is no necessary, and here are many problems where his is no used. The infinie horizon problem becomes: > ;,...,? v Ω v ;9: β U (, v subjec o: i G(, v + ii given The Bellman equaion becomes: ( { v { β U (,v + V ( + + V subjec o (i and (ii This equaion is defined in presen value erms. Tha is, he values are all discouned back o ime. Ofen, i is more convenien o represen hese values in curren value erms: Define: ( v β ( Muliplying boh sides of he Bellman equaion by β yields: ( { U (, v + β + ( + { v (* subjec o (i and (ii. A suble change in noaion has been inroduced here. From his poin on, ( V represens he form of he value funcion in period.

14 I can be shown ha he funcion defined in (* is an eample of a conracion mapping. In Sargen s (987 appendi, (see, also, Sokey e al, (989 he presens a powerful resul called he conracion mapping heorem, which saes ha, under cerain regulariy condiions, ieraions on (* saring from any bounded coninuous (say, will cause o converge as he number of ieraions becomes large. Moreover, he (. ha comes ou of his procedure is he unique opimal value funcion for he infinie horizon maimizaion problem. Also, associaed wih (. is a unique conrol rule h( solves he maimizaion problem. v which This means is ha here is a unique ime invarian value funcion (. which saisfies: ( { v { U ( v + β (, + subjec o: i G(, v + ii given. Associaed wih his value funcion is a unique ime invarian conrol rule: v h (. How To Use These Resuls: There are several differen mehods of solving infinie horizon dynamic programming problems, and hree of hem will be considered ou here. In wo, he key sep is finding he form of he value funcion, which is unknown a he ouse of any problem even if he funcional forms for U(. and G(. are known. This is no required in he hird. U v is bounded, coninuous, sricly increasing and sricly concave. Also, he producion echnology implici in G (, v mus be coninuous, monoonic and concave. Finally, he se of feasible end of period mus be compac Sokey e al., (989 chaper 3 spell hese condiions ou. Roughly, hese amoun o assuming ha (,

15 C Consider he general problem: DE v ;,..., F v Ω B@A β U (, v subjec o: i G(, v + (also regulariy condiions ii given Mehod : Brue Force Ieraions Firs, se up he Bellman equaion: ( { v { U( v + β (, + + s.. i G(, v + ii given. Se (, + + and solve he maimizaion problem on he righ side of he Bellman equaion. This will give you a conrol rule v h (. Subsiue his back ino U (,v, o ge: ( U (,h ( Ne, se up he Bellman equaion: ( { v { U (, v + βu (, h( s.. i G(, v ii given. Subsiue consrain (i ino he maimand o ge: ( { v { U (, v + βu( G(, v, h( G(, v 3

16 Solve he maimizaion problem on righ, o ge: v h ( Now subsiue his back ino he righ side o ge: (. Coninue his procedure unil he (. 's converge. The conrol rule associaed wih he (. a he poin of convergence solves he problem. Mehod : Guess and Verify. This mehod is a varian of he firs mehod, bu where we make an informed guess abou he funcional form of he value funcion. If available, his mehod can save a lo compuaion. For cerain classes of problems, when he period uiliy funcion U, v ( lies in he HARA class (which includes CRRA, CARA, and quadraic funcions, he value funcion akes he same general funcional form. 3 Thus, for eample, if he period uiliy funcion is logarihmic, we can epec he value funcion will also be logarihmic. As before, we firs se up he Bellman equaion: ( { v { U( v + β (, + subjec o: i G(, v + ii given. Second, guess he form of he value funcion, G ( Bellman equaion: +, and subsiue he guess ino he ( { v G { U ( v + β (, + s.. (i and (ii 3 See Blanchard and Fischer (989, chaper 6, for a discussion. 4

17 Third, perform he maimizaion problem on he righ side. Then subsiue he resuling conrol rules back ino he Bellman equaion o (hopefully verify he iniial guess. (i.e.: The G guess is verified if ( (. + If he iniial guess was correc, hen he problem is solved. If he iniial guess was incorrec, ry he form of he value funcion ha is suggesed by he firs guess as a second guess, and repea he process. In general, successive approimaions will bring you o he unique soluion, as in mehod. Mehod 3: Using he Benvenise-Schienkman Formula. This is Sargen's preferred mehod of solving problems in his Dynamic Macro Theory ebook. The advanage of using his mehod is ha one is no required o know he form of he value funcion o ge some resuls. From Sargen [987] p., he B-S formula is: ( U (, h( β G(, h( ( G(, h( + + B-S Formula Sargen shows ha if i is possible o re-define he sae and conrol variables in such a way as o eclude he sae variables from he ransiion equaions, he B-S formula simplifies o: ( U (,h( B-S Formula The seps required o use his mehod are: ( Redefine he sae and conrol variables o eclude he sae variables from he ransiion equaions. ( Se up he Bellman equaion wih he value funcion in general form. (3 Perform he maimizaion problem on he r.h.s. of he Bellman equaion. Tha is, derive he F.O.C.'s. (4 Use he B-S Formula o subsiue ou any erms in he value funcion. As a resul, you will ge an Euler equaion, which may be useful for cerain purposes, alhough i is essenially a second-order difference equaion in he sae variables. 5

18 H M I M.3. Eample : The Cass-Koopmans Opimal Growh Model. { c, k G + α β nc s.. i k + Ak C ii k k given. here : C k consumpio n capial sock < β < < α < Since he period uiliy funcion is in he HARA class, i is worhwhile o use he guessverify mehod. The Bellman equaion is: ( k { nc + β ( k + { C k, + s.. (i and (ii ( Now guess ha he value fn. is of his form: ( k E F J n k + ( K ( k + E + F nk + L (' where E and F are coefficiens o be deermined. Subsiue consrain (i and (' ino ( o ge: α ( { n[ Ak k + ] + β [ E + F n k + ] k { k + (3 FOC: Ak α k + βf k + 6

19 _ U N TSR Q P O TSR ]\[ Q P Z Z X Y Y ^ k βf (4 + βf α + Ak Now subsiue 4 back ino 3: U βf + βf βf α α α ( k n Ak Ak + βe + BF n Ak O + βf \[] βf X V ( k n A n( + β F + βe + βf n A + βf n + α ( + βf n k + βf (5 Now, comparing equaions ( and (5, we can deermine he coefficien for n k : ( F α + βf α F (6 αβ Similarly, E can be deermined from ( and (5, knowing (6. Since boh he coefficien E and F have been deermined, he iniial guess (equaion has been verified. Now (6 can be subsiued back ino (4 o ge: α k + αβak and using consrain (ii: (7 C α [ αβ ] Ak (8 Given he iniial k equaions (7 - (8 compleely characerize he soluion pahs of C and k. 7

20 b.4. Eample 3: Minimizing Quadraic Coss This is he infinie horizon version of eample. ` β [ + v ] { v Min subjec o i + v + ii given Since his is a linear-quadraic problem, i is possible o use he guess-verify mehod. The Bellman equaion is: ( Min { + v + β ( + { v s.. i + + v ii given ( Now guess ha he value fn. is of his form: ( P ( a ( + + P (' where P is a coefficien o be deermined. Subsiue consrain (i and (', hen he resul ino ( o ge: ( Min { + v + βp[ + v ] { v FOC: v β P[ + v ] + (3 b v [ + βp] βp βp v + βp (4 8

21 hgf ˆ p p nml {z e c d k j i y w k j i i ˆ hgf {z nml e d y s rp w q p k j i Subsiue (4 back ino (3 o ge: ( βp βp + + βp b ( c + βp + βp nml l βp βp + βp + βp + βp o ( uv βp βp + βp + βp + βp (5 P can now be deermined by comparing he equaions ( and (5: P ƒ βp βp + βp + β + β P P ~ (6 The P ha solves (6 is he coefficien ha was o be deermined. For eample, se β, and equaion (6 implies: P 4.4 Equaion (4 becomes : v. 6 (7 Consrain (i becomes: (8 For a given, equaions (7 and (8 compleely characerize he soluion. Noe from equaion 8 ha he sysem will be sable since.38 <. 9

22 Š Ž Œ Ž Œ.5. Eample 4: Life Cycle Consumpion { c β U ( C ( s.. i A R [ A + y C ] + Transiion equaion ( ii A A given (3 here: C y consumpio n known, eogenously given income sream A non - labour wealh beginning in ime R known, eogenous sequence of one-period raes of reurn on A To rule ou he possibiliy of infinie consumpion financed by unbounded borrowing, a presen value budge consrain is imposed: j j + R + k C + j y + R + k y + j j k j k C + A (4 Here, we are no given an eplici funcional form for he period uiliy funcion. e can use mehod 3 in his problem o derive a well-known resul. To use he B-S formula, we need o define he sae and conrol variables o eclude sae variables from he ransiion equaion. Define: Sae Variables: { A, y, R Conrol Variable: A (5 + v R Now we can re-wrie he ransiion equaion as: A R v + (' Noice ha (' and ( imply: C A + y v (6

23 The problem becomes: { v β U ( A + y v (' s.. i A + R v (' Reformulaed problem ii A A given (3 The Bellman equaion is: ( A,, y R { v { U ( A y v + ( A +, y +, R + β s.. 'and 3. Subsiuing (' ino he Bellman equaion yields: ( A, y, R { v { U ( A + y v + β ( R v, y, R + FOC: ' ( A + y v + R ( R v, y, R + U β (7 Now recall ha he B-S formula implies: A (8 + ( R v,y,r U' A + y U' ( C R + Subsiuing (8 ino (7 gives: ( C RU' ( C U' β Euler Equaion (9 + Now, o ge furher resuls, le s impose srucure on he model by specifying a paricular uiliy funcion: Le U ( C n C š U'C ( ( C

24 š Ÿž œ «Subsiuing ino 9 yields: C βr C + now updaing: ( š C R [ βr C ] + β + C + β R + k k C In general: j j + j R + k k C β C ( Subsiuing back ino 4 yields: j j j R R C y + C + β + k + k j k k j j R k + k y + j + A j C + β C " " " ª j j j C β " " " C β " " " C ( β y + R y + A «± j j k + k + j Life Cycle Consumpion Funcion 3. STOCHASTIC INFINITE HORIZON MODELS 3.. Inroducion: As wih deerminisic infinie horizon problems, i is convenien o assume ime separabiliy, saionary, and boundedness of he payoff funcion. However, sochasic models are more general han deerminisic ones because hey allow for some uncerainy.

25 µ The ype of uncerainy ha is usually inroduced ino hese models is of a very specific kind. To preserve he recursive srucure of he models, i is ypically assumed ha he sochasic shocks he sysem eperiences follow a homogeneous firs order Markov process. In he simples erms, his means ha he value of his period's shock depends only on he value of las period's shock, no upon any earlier values. (hie noise shocks are a special case of his: hey do no even depend upon las period's shock. Consider he sochasic problem: ²³ E β U (, v ;,..., v Ω v subjec o: i G(,v ε +, + ii given where { ε is a sequence of random shocks ha ake on values in he inerval [ ε, ε ] and follow a homogeneous firs order Markov process wih he condiional cumulaive densiy funcion ( ε ', ε Pr{ ε ε ' ε ε F (iii + Also, E denoes he mahemaical epecaion, given informaion a ime, I. e assume: The assumed sequence of evens is: ( is observed I (,{ v,{, G(., U (., (. { k k k k ε k k F ( Decision maker chooses v (3 Naure chooses ε + (4 Ne period occurs. The Bellman equaion for his problem is: (, ε { v { U (, v + βe (, ε + + 3

26 As in he deerminisic infinie horizon problem, under regulariy condiions 4 ieraions on (, ε { v { U (, v + βe (, ε saring from any bounded coninuous + (say, + will cause (. o converge as he number of ieraions becomes large. Once again, he ( ha comes ou of his procedure is he unique opimal value funcion for he above problem. Furhermore, associaed wih ( is a unique ime invarian conrol rule v h( which solves he maimizaion problem. 3. How o Use These Resuls: The soluion echniques given in he deerminisic infinie horizon problem sill work in he sochasic infinie horizon problem, and here is no need o repea hem. Perhaps he bes way o illusrae hese resuls is by using an eample. 3.3 Eample 5: A Sochasic Opimal Growh Model wih a Labour-Leisure Trade-off. This eample is chosen no only o illusrae he soluion echniques given above, bu also o inroduce he reader o a paricular ype of model. This modelling approach is used in he "Real Business Cycle" lieraure of auhors such as Kydland and Presco [98] and Long and Plosser [983]. Under he appropriae assumpions made abou preferences and echnology in an economy, he following opimal growh model can be inerpreed as a compeiive equilibrium model. 5 Tha is, his model mimics a simple dynamic sochasic general equilibrium economy. Because of his, he soluion pahs for he endogenous variables generaed by his model can be compared wih acual pahs of hese variables observed in real-world macroeconomies. 4 The same condiions as in Noe, wih he added assumpion of homogeneous firs order Markov shocks. 5 See Sokey e al., (989, chaper 5 for furher discussion on his opic. 4

27 ¼ ½ º Consider he following problem: E ¹ { C,k,n β [ º n C + δ n( n ] subjec o: i C + k y ( ii α α + A + k n ( y (3 iii» n A + ρ» n A + ξ + (4 where n represens unis of labour chosen in period, δ is a posiive parameer, ρ is a parameer which lies in he inerval (-,, and ξ represens a whie noise error process. 6 The endogenous variable y represens oupu in period. The res of he noaion is he same as in eample. These are only wo subsanive differences beween his eample and eample. Firs, a labour-leisure decision is added o he model, represened by he inclusion of he erm ( n in he payoff funcion ( and he inclusion of n in he producion funcion (3. Second, he producion echnology parameer o he Markov process (4. A in equaion 3 is assumed o evolve over ime according Noice ha in his eample C,k, and n are he conrol variables chosen every period by he decision maker, whereas up he Bellman equaion: y and A are he sae variables. To solve his problem, we firs se ( y, A { C, k, n {[ n C + δ¼ n( n ] + βe ( y, A + + (5 Since his is a logarihmic eample, we can use he guess-verify mehod of soluion. The obvious firs guess for he form of he value funcion is: ( y,a D + G½ n y + H n A (6 6 For eample, ξ is normally disribued wih mean and variance σ ξ. 5

28 Â Ë ÊÉÈ Á À Ò Â Ç Æ Å ÑÐÏ Á Ô Ã Ë À ÊÉÈ Ô Î Ì Í À Â Ç Æ Å Á À Ò ÊÉÈ Â Á Ç Æ Á where D, G and H are coefficiens o be deermined. ¾ ( y +,A + D + G n y + + H n A + (6' To verify his guess, firs subsiue (3 ino (6' o ge ( y +,A + D + G[ n A + + α n k + ( α n n ] + H n A + Now subsiue his ino equaion 5: [ y, A ] { C, k, n {[ n C + δ n( n ] + βe [ D + ( G + H n A + Gα n k + G( α n n ] + Using equaion o subsiue ou C in he above yields: [ y, A ] {[ n ( y k + δ n( n ] + βd + βgα n k + βg( α n n β ( G + H E n A Â { k, n (7 + The FOCs are: n ( α δ βg : + n n ( α G( α βg n (8 δ + β k : y k βgα + k αβg (9 + αβg Ä k y Now we can subsiue equaions 8 and 9 back ino 7 o ge: ( y,a δ Ë αβg n y + δ n + βd + αβ n y + αβg δ + βg ( α G( α ( α + αβg Å βg β ( α ( G + H E n A (7' δ + β + G n + β + ρó Recall from equaion 4 ha Ó n A + n A + ξ + where ξ + is a whie noise error erm wih mean zero. Hence: E n A + ρ n A. 6

29 Ö Õ Õ Collecing erms in equaion 7'yields: ( y,a ( + G n y + β ( G + H ρ n A + consans αβ ( Comparing equaions 6 and, i is clear ha G + αβ G ( αβ ( G H H ρβ + Ö ρβ αβ ( ( Similarly, he consan D can be deermined from equaions 6 and, knowing he values of G and H given in equaions and. Since all he coefficiens have been deermined, he iniial guess of he form of he value funcion (equaion 6 has been verified. e can now use equaions and o solve for he opimal pahs of n,k, and C as funcions of he sae y. From equaion 9: k αβy (3 From equaion 8: β ( α ( αβ + β ( α n (4 δ From equaions 3 and : C ( αβ y (5 Noice ha while he opimal k and C are funcions of he sae y, he opimal n is no. Tha is, n is consan. This is a resul ha is peculiar o he funcional forms chosen for he preferences and echnology, and i provides some jusificaion for he absence of labourleisure decisions in growh models wih logarihmic funcional forms. 7

30 REFERENCES Blanchard, O., and Fischer, S., Lecures on Macroeconomics, MIT Press, (989. Chaing, A.C., Fundamenal Mehods of Mahemaical Economics, Third Ediion, McGraw- Hill, New York (984. Kydland, F., and Presco, E. "Time o Build and Aggregae Flucuaions", Economerica 5, pp (98. Long, J., and Plosser, C., "Real Business Cycles", Journal of Poliical Economy 9, pp (983. Manuelli, R., and Sargen, T., Eercises in Dynamic Macroeconomic Theory, Harvard Universiy Press, (987. Sargen, T., Dynamic Macroeconomic Theory, Harvard Universiy Press, (987. Sokey, N., Lucas, R.E., wih Presco, E., Recursive Mehods for Economic Dynamics, Harvard Universiy Press (989. 8