Applied Intertemporal Optimization


 Arnold Greer
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1 . Applied Ineremporal Opimizaion Klaus Wälde Universiy of Mainz CESifo, Universiy of Brisol, UCL Louvain la Neuve
2 These lecure noes can freely be downloaded from A prin version can also be bough a his } auhor = {Klaus Wälde}, year = 2010, ile = {Applied Ineremporal Opimizaion}, publisher = {Mainz Universiy Guenberg Press}, address = {Available a Guenberg Press Copyrigh (c) 2010 by Klaus Wälde ISBN
3 1 Acknowledgmens This book is he resul of many lecures given a various insiuions, including he Bavarian Graduae Program in Economics, he Universiies of Dormund, Dresden, Frankfur, Glasgow, LouvainlaNeuve, Mainz, Munich and Würzburg and he European Commission in Brussels. I would herefore rs like o hank he sudens for heir quesions and discussions. These conversaions revealed he mos imporan aspecs in undersanding model building and in formulaing maximizaion problems. I also pro ed from many discussions wih economiss and mahemaicians from many oher places. The high download gures a repec.org show ha he book is also widely used on an inernaional level. I especially would like o hank Chrisian Bayer and Ken Sennewald for many insighs ino he more suble issues of sochasic coninuous ime processes. I hope I succeeded in incorporaing hese insighs ino his book. I would also like o hank MacKichan for heir repeaed, quick and very useful suppor on ypeseing issues in Scieni c Word.
4 2 Overview The basic srucure of his book is simple o undersand. I covers opimizaion mehods and applicaions in discree ime and in coninuous ime, boh in worlds wih cerainy and worlds wih uncerainy. discree ime coninuous ime deerminisic seup Par I Par II sochasic seup Par III Par IV Table Basic srucure of his book Soluion mehods Pars and chapers Subsiuion Lagrange Ch. 1 Inroducion Par I Deerminisic models in discree ime Ch. 2 Twoperiod models and di erence equaions Ch. 3 Muliperiod models , 3.7 Par II Deerminisic models in coninuous ime Ch. 4 Di erenial equaions Ch. 5 Finie and in nie horizon models 5.2.2, Ch. 6 In nie horizon models again Par III Sochasic models in discree ime Ch. 7 Sochasic di erence equaions and momens Ch. 8 Twoperiod models 8.1.4, 8.2 Ch. 9 Muliperiod models Par IV Sochasic models in coninuous ime Ch. 10 Sochasic di erenial equaions, rules for di erenials and momens Ch. 11 In nie horizon models Ch. 12 Noaion and variables, references and index Table Deailed srucure of his book
5 Each of hese four pars is divided ino chapers. As a quick reference, he able below provides an overview of where o nd he four soluion mehods for maximizaion problems used in his book. They are he subsiuion mehod, Lagrange approach, opimal conrol heory and dynamic programming. Whenever we employ hem, we refer o hem as Solving by and hen eiher subsiuion, he Lagrangian, opimal conrol or dynamic programming. As di erences and comparaive advanages of mehods can mos easily be undersood when applied o he same problem, his able also shows he mos frequenly used examples. Be aware ha hese are no he only examples used in his book. Ineremporal pro maximizaion of rms, capial asse pricing, naural volailiy, maching models of he labour marke, opimal R&D expendiure and many oher applicaions can be found as well. For a more deailed overview, see he index a he end of his book. 3 Applicaions (selecion) opimal Dynamic Uiliy Cenral General Budge conrol programming maximizaion planner equilibrium consrains 2.1, , 3.4, , , 5.3, , , 9.2, , , Table Deailed srucure of his book (coninued)
6 4
7 Conens 1 Inroducion 1 I Deerminisic models in discree ime 7 2 Twoperiod models and di erence equaions Ineremporal uiliy maximizaion The seup Solving by he Lagrangian Examples Opimal consumpion Opimal consumpion wih prices Some useful de niions wih applicaions The idea behind he Lagrangian Where he Lagrangian comes from I Shadow prices An overlapping generaions model Technologies Households Goods marke equilibrium and accumulaion ideniy The reduced form Properies of he reduced form More on di erence equaions Two useful proofs A simple di erence equaion A slighly less bu sill simple di erence equaion Fix poins and sabiliy An example: Deriving a budge consrain Furher reading and exercises Muliperiod models Ineremporal uiliy maximizaion The seup
8 6 Conens Solving by he Lagrangian The envelope heorem The heorem Illusraion An example Solving by dynamic programming The seup Three dynamic programming seps Examples Ineremporal uiliy maximizaion wih a CES uiliy funcion Wha is a sae variable? Opimal R&D e or On budge consrains From ineremporal o dynamic From dynamic o ineremporal Two versions of dynamic budge consrains A decenralized general equilibrium analysis Technologies Firms Households Aggregaion and reduced form Seady sae and ransiional dynamics A cenral planner Opimal facor allocaion Where he Lagrangian comes from II Growh of family size The seup Solving by subsiuion Solving by he Lagrangian Furher reading and exercises II Deerminisic models in coninuous ime 75 4 Di erenial equaions Some de niions and heorems De niions Two heorems Analyzing ODEs hrough phase diagrams Onedimensional sysems Twodimensional sysems I  An example Twodimensional sysems II  The general case Types of phase diagrams and xpoins
9 Conens Mulidimensional sysems Linear di erenial equaions Rules on derivaives Forward and backward soluions of a linear di erenial equaion Di erenial equaions as inegral equaions Examples Backward soluion: A growh model Forward soluion: Budge consrains Forward soluion again: capial markes and uiliy Linear di erenial equaion sysems Furher reading and exercises Finie and in nie horizon models Ineremporal uiliy maximizaion  an inroducory example The seup Solving by opimal conrol Deriving laws of moion The seup Solving by he Lagrangian Hamilonians as a shorcu The in nie horizon Solving by opimal conrol The boundedness condiion Boundary condiions and su cien condiions Free value of he sae variable a he endpoin Fixed value of he sae variable a he endpoin The ransversaliy condiion Su cien condiions Illusraing boundary condiions A rm wih adjusmen coss Free value a he end poin Fixed value a he end poin In nie horizon and ransversaliy condiion Furher examples In nie horizon  opimal consumpion pahs Necessary condiions, soluions and sae variables Opimal growh  he cenral planner and capial accumulaion The maching approach o unemploymen The presen value Hamilonian Problems wihou (or wih implici) discouning Deriving laws of moion The link beween CV and PV
10 8 Conens 5.8 Furher reading and exercises In nie horizon again Ineremporal uiliy maximizaion The seup Solving by dynamic programming Comparing dynamic programming o Hamilonians Dynamic programming wih wo sae variables Nominal and real ineres raes and in aion Firms, he cenral bank and he governmen Households Equilibrium Furher reading and exercises Looking back III Sochasic models in discree ime Sochasic di erence equaions and momens Basics on random variables Some conceps An illusraion Examples for random variables Discree random variables Coninuous random variables Higherdimensional random variables Expeced values, variances, covariances and all ha De niions Some properies of random variables Funcions on random variables Examples of sochasic di erence equaions A rs example A more general case Twoperiod models An overlapping generaions model Technology Timing Firms Ineremporal uiliy maximizaion Aggregaion and he reduced form for he CD case Some analyical resuls CRRA and CARA uiliy funcions
11 Conens Riskaverse and riskneural households Pricing of coningen claims and asses The value of an asse The value of a coningen claim Riskneural valuaion Naural volailiy I The basic idea A simple sochasic model Equilibrium Furher reading and exercises Muliperiod models Ineremporal uiliy maximizaion The seup wih a general budge consrain Solving by dynamic programming The seup wih a household budge consrain Solving by dynamic programming A cenral planner Asse pricing in a oneasse economy The model Opimal behaviour The pricing relaionship More real resuls Endogenous labour supply Solving by subsiuion Ineremporal uiliy maximizaion Capial asse pricing Sicky prices Opimal employmen wih adjusmen coss An explici ime pah for a boundary condiion Furher reading and exercises IV Sochasic models in coninuous ime SDEs, di erenials and momens Sochasic di erenial equaions (SDEs) Sochasic processes Sochasic di erenial equaions The inegral represenaion of sochasic di erenial equaions Di erenials of sochasic processes Why all his? Compuing di erenials for Brownian moion
12 10 Conens Compuing di erenials for Poisson processes Brownian moion and a Poisson process Applicaions Opion pricing Deriving a budge consrain Solving sochasic di erenial equaions Some examples for Brownian moion A general soluion for Brownian moions Di erenial equaions wih Poisson processes Expecaion values The idea Simple resuls Maringales A more general approach o compuing momens Furher reading and exercises In nie horizon models Ineremporal uiliy maximizaion under Poisson uncerainy The seup Solving by dynamic programming The KeynesRamsey rule Opimal consumpion and porfolio choice Oher ways o deermine ~c Expeced growh Maching on he labour marke: where value funcions come from A household The Bellman equaion and value funcions Ineremporal uiliy maximizaion under Brownian moion The seup Solving by dynamic programming The KeynesRamsey rule Capial asse pricing The seup Opimal behaviour Capial asse pricing Naural volailiy II An real business cycle model A naural volailiy model A numerical approach Furher reading and exercises Miscellanea, references and index 299
13 Chaper 1 Inroducion This book provides a oolbox for solving dynamic maximizaion problems and for working wih heir soluions in economic models. Maximizing some objecive funcion is cenral o Economics, i can be undersood as one of he de ning axioms of Economics. When i comes o dynamic maximizaion problems, hey can be formulaed in discree or coninuous ime, under cerainy or uncerainy. Various maximizaion mehods will be used, ranging from he subsiuion mehod, via he Lagrangian and opimal conrol o dynamic programming using he Bellman equaion. Dynamic programming will be used for all environmens, discree, coninuous, cerain and uncerain, he Lagrangian for mos of hem. The subsiuion mehod is also very useful in discree ime seups. The opimal conrol heory, employing he Hamilonian, is used only for deerminisic coninuous ime seups. An overview was given in g on he previous pages. The general philosophy behind he syle of his book says ha wha maers is an easy and fas derivaion of resuls. This implies ha a lo of emphasis will be pu on examples and applicaions of mehods. While he idea behind he general mehods is someimes illusraed, he focus is clearly on providing a soluion mehod and examples of applicaions quickly and easily wih as lile formal background as possible. This is why he book is called applied ineremporal opimizaion. Conens of pars I o IV This book consiss of four pars. In his rs par of he book, we will ge o know he simples and herefore maybe he mos useful srucures o hink abou changes over ime, o hink abou dynamics. Par I deals wih discree ime models under cerainy. The rs chaper inroduces he simples possible ineremporal problem, a woperiod problem. I is solved in a general way and for many funcional forms. The mehods used are he Lagrangian and simple subsiuion. Various conceps like he ime preference rae and he ineremporal elasiciies of subsiuion are inroduced here as well, as hey are widely used in he lieraure and are used frequenly hroughou his book. For hose who wan o undersand he background of he Lagrangian, a chaper is included ha shows he link beween Lagrangians and solving by subsiuion. This will also give us he 1
14 2 Chaper 1. Inroducion opporuniy o explain he concep of shadow prices as hey play an imporan role e.g. when using Hamilonians or dynamic programming. The woperiod opimal consumpion seup will hen be pu ino a decenralized general equilibrium seup. This allows us o undersand general equilibrium srucures in general while, a he same ime, we ge o know he sandard overlapping generaions (OLG) general equilibrium model. This is one of he mos widely used dynamic models in Economics. Chaper 2 concludes by reviewing some aspecs of di erence equaions. Chaper 3 hen covers in nie horizon models. We solve a ypical maximizaion problem rs by using he Lagrangian again and hen by dynamic programming. As dynamic programming regularly uses he envelope heorem, his heorem is rs reviewed in a simple saic seup. Examples for in nie horizon problems, a general equilibrium analysis of a decenralized economy, a ypical cenral planner problem and an analysis of how o rea family or populaion growh in opimizaion problems hen complee his chaper. To complee he range of maximizaion mehods used in his book, he presenaion of hese examples will also use he mehod of solving by insering. Par II covers coninuous ime models under cerainy. Chaper 4 rs looks a differenial equaions as hey are he basis of he descripion and soluion of maximizaion problems in coninuous ime. Firs, some useful de niions and heorems are provided. Second, di erenial equaions and di erenial equaion sysems are analyzed qualiaively by he socalled phasediagram analysis. This simple mehod is exremely useful for undersanding di erenial equaions per se and also for laer purposes for undersanding qualiaive properies of soluions o maximizaion problems and properies of whole economies. Linear di erenial equaions and heir economic applicaions are hen nally analyzed before some words are spen on linear di erenial equaion sysems. Chaper 5 presens a new mehod for solving maximizaion problems  he Hamilonian. As we are now in coninuous ime, woperiod models do no exis. A disincion will be drawn, however, beween nie and in nie horizon models. In pracice, his disincion is no very imporan as, as we will see, opimaliy condiions are very similar for nie and in nie maximizaion problems. Afer an inroducory example on maximizaion in coninuous ime by using he Hamilonian, he simple link beween Hamilonians and he Lagrangian is shown. The soluion o maximizaion problems in coninuous ime will consis of one or several di erenial equaions. As a unique soluion o di erenial equaions requires boundary condiions, we will show how boundary condiions are relaed o he ype of maximizaion problem analyzed. The boundary condiions di er signi canly beween nie and in nie horizon models. For he nie horizon models, here are iniial or erminal condiions. For he in nie horizon models, we will ge o know he ransversaliy condiion and oher relaed condiions like he NoPonzigame condiion. Many examples and a comparison beween he presenvalue and he currenvalue Hamilonian conclude his chaper. Chaper 6 solves he same kind of problems as chaper 5, bu i uses he mehod of dynamic programming. The reason for doing his is o simplify undersanding of dynamic programming in sochasic seups in Par IV. Various aspecs speci c o he use
15 of dynamic programming in coninuous ime, e.g. he srucure of he Bellman equaion, can already be reaed here under cerainy. This chaper will also provide a comparison beween he Hamilonian and dynamic programming and look a a maximizaion problem wih wo sae variables. An example from moneary economics on real and nominal ineres raes concludes he chaper. In par III, he world becomes sochasic. Pars I and II provided many opimizaion mehods for deerminisic seups, boh in discree and coninuous ime. All economic quesions ha were analyzed were viewed as su cienly deerminisic. If here was any uncerainy in he seup of he problem, we simply ignored i or argued ha i is of no imporance for undersanding he basic properies and relaionships of he economic quesion. This is a good approach o many economic quesions. Generally speaking, however, real life has few cerain componens. Deah is cerain, bu when? Taxes are cerain, bu how high are hey? We know ha we all exis  bu don ask philosophers. Par III (and par IV laer) will ake uncerainy in life seriously and incorporae i explicily in he analysis of economic problems. We follow he same disincion as in par I and II  we rs analyse he e ecs of uncerainy on economic behaviour in discree ime seups in par III and hen move o coninuous ime seups in par IV. Chaper 7 and 8 are an exended version of chaper 2. As we are in a sochasic world, however, chaper 7 will rs spend some ime reviewing some basics of random variables, heir momens and disribuions. Chaper 7 also looks a di erence equaions. As hey are now sochasic, hey allow us o undersand how disribuions change over ime and how a disribuion converges  in he example we look a  o a limiing disribuion. The limiing disribuion is he sochasic equivalen o a x poin or seady sae in deerminisic seups. Chaper 8 looks a maximizaion problems in his sochasic framework and focuses on he simples case of woperiod models. A general equilibrium analysis wih an overlapping generaions seup will allow us o look a he new aspecs inroduced by uncerainy for an ineremporal consumpion and saving problem. We will also see how one can easily undersand dynamic behaviour of various variables and derive properies of longrun disribuions in general equilibrium by graphical analysis. One can for example easily obain he range of he longrun disribuion for capial, oupu and consumpion. This increases inuiive undersanding of he processes a hand remendously and helps a lo as a guide o numerical analysis. Furher examples include borrowing and lending beween riskaverse and riskneural households, he pricing of asses in a sochasic world and a rs look a naural volailiy, a view of business cycles which sresses he link beween joinly endogenously deermined shorrun ucuaions and longrun growh. Chaper 9 is hen similar o chaper 3 and looks a muliperiod, i.e. in nie horizon, problems. As in each chaper, we sar wih he classic ineremporal uiliy maximizaion problem. We hen move on o various imporan applicaions. The rs is a cenral planner sochasic growh model, he second is capial asse pricing in general equilibrium and how i relaes o uiliy maximizaion. We coninue wih endogenous labour supply and he 3
16 4 Chaper 1. Inroducion maching model of unemploymen. The nex secion hen covers how many maximizaion problems can be solved wihou using dynamic programming or he Lagrangian. In fac, many problems can be solved simply by insering, despie uncerainy. This will be illusraed wih many furher applicaions. A nal secion on nie horizons concludes. Par IV is he nal par of his book and, logically, analyzes coninuous ime models under uncerainy. The choice beween working in discree or coninuous ime is parly driven by previous choices: If he lieraure is mainly in discree ime, sudens will nd i helpful o work in discree ime as well. The use of discree ime mehods seem o hold for macroeconomics, a leas when i comes o he analysis of business cycles. On he oher hand, when we alk abou economic growh, labour marke analyses and nance, coninuous ime mehods are very prominen. Whaever he radiion in he lieraure, coninuous ime models have he huge advanage ha hey are analyically generally more racable, once some iniial invesmen ino new mehods has been digesed. As an example, some papers in he lieraure have shown ha coninuous ime models wih uncerainy can be analyzed in simple phase diagrams as in deerminisic coninuous ime seups. See ch and ch on furher reading for references from many elds. To faciliae access o he magical world of coninuous ime uncerainy, par IV presens he ools required o work wih uncerainy in coninuous ime models. I is probably he mos innovaive par of his book as many resuls from recen research ow direcly ino i. This par also mos srongly incorporaes he cenral philosophy behind wriing his book: There will be hardly any discussion of formal mahemaical aspecs like probabiliy spaces, measurabiliy and he like. While some will argue ha one can no work wih coninuous ime uncerainy wihou having sudied mahemaics, his chaper and he many applicaions in he lieraure prove he opposie. The objecive here is o clearly make he ools for coninuous ime uncerainy available in a language ha is accessible for anyone wih an ineres in hese ools and some feeling for dynamic models and random variables. The chapers on furher reading will provide links o he more mahemaical lieraure. Maybe his is also a good poin for he auhor of his book o hank all he mahemaicians who helped him gain access o his magical world. I hope hey will forgive me for beraying heir secres o hose who, maybe in heir view, were no appropriaely iniiaed. Chaper 10 provides he background for opimizaion problems. As in par II where we rs looked a di erenial equaions before working wih Hamilonians, here we will rs look a sochasic di erenial equaions. Afer some basics, he mos ineresing aspec of working wih uncerainy in coninuous ime follows: Io s lemma and, more generally, changeofvariable formulas for compuing di erenials will be presened. As an applicaion of Io s lemma, we will ge o know one of he mos famous resuls in Economics  he BlackScholes formula. This chaper also presens mehods for how o solve sochasic di erenial equaions or how o verify soluions and compue momens of random variables described by a sochasic process. Chaper 11 hen looks once more a maximizaion problems. We will ge o know
17 he classic ineremporal uiliy maximizaion problem boh for Poisson uncerainy and for Brownian moion. The chaper also shows he link beween Poisson processes and maching models of he labour marke. This is very useful for working wih exensions of he simple maching model ha allows for savings. Capial asse pricing and naural volailiy conclude he chaper. From simple o complex seups Given ha cerain maximizaion problems are solved many imes  e.g. uiliy maximizaion of a household rs under cerainy in discree and coninuous ime and hen under uncerainy in discree and coninuous ime  and using many mehods, he seps how o compue soluions can be easily undersood: Firs, he discree deerminisic woperiod approach provides he basic inuiion or feeling for a soluion. Nex, in nie horizon problems add one dimension of complexiy by aking away he simple boundary condiion of nie horizon models. In a hird sep, uncerainy adds expecaions operaors and so on. By gradually working hrough increasing seps of sophisicaion and by linking back o simple bu srucurally idenical examples, inuiion for he complex seups is buil up as much as possible. This approach hen allows us o nally undersand he beauy of e.g. KeynesRamsey rules in coninuous ime under Poisson uncerainy or Brownian moion. Even more moivaion for his book Why each a course based on his book? Is i no boring o go hrough all hese mehods? In a way, he answer is yes. We all wan o undersand cerain empirical regulariies or undersand poenial fundamenal relaionships and make exciing new empirically esable predicions. In doing so, we also all need o undersand exising work and evenually presen our own ideas. I is probably much more boring o be hindered in undersanding exising work and be almos cerainly excluded from presening our own ideas if we always spend a long ime rying o undersand how cerain resuls were derived. How did his auhor ge from equaion (1) and (2) o equaion (3)? The major advanage of economic analysis over oher social sciences is is srong foundaion in formal models. These models allow us o discuss quesions much more precisely as expressions like marginal uiliy, ime preference rae or KeynesRamsey rule reveal a lo of informaion in a very shor ime. I is herefore exremely useful o rs spend some ime in geing o know hese mehods and hen o ry o do wha Economics is really abou: undersand he real world. Bu, before we really sar, here is also a second reason  a leas for some economiss  o go hrough all hese mehods: They conain a cerain ype of ruh. A proof is rue or false. The derivaion of some opimal behaviour is rue or false. A predicion of general equilibrium behaviour of an economy is ruh. Unforunaely, i is only ruh in an analyical sense, bu his is a leas some ype of ruh. Beer han none. 5
18 6 Chaper 1. Inroducion The audience for his book Before his book came ou, i had been esed for a leas en years in many courses. There are wo ypical courses which were based on his book. A hird year Bachelor course (for ambiious sudens) can be based on pars I and II, i.e. on maximizaion and applicaions in discree and coninuous ime under cerainy. Such a course ypically ook 14 lecures of 90 minues each plus he same number of uorials. I is also possible o presen he maerial also in 14 lecures of 90 minues each plus only 7 uorials. Presening he maerial wihou uorials requires a lo of energy from he sudens o go hrough he problem ses on heir own. One can, however, discuss some seleced problem ses during lecures. The oher ypical course which was based on his book is a rsyear PhD course. I would review a few chapers of par I and par II (especially he chapers on dynamic programming) and cover in full he sochasic maerial of par III and par IV. I also requires foureen 90 minue sessions and exercise classes help even more, given he more complex maerial. Bu he same ype of arrangemens as discussed for he Bachelor course did work as well. Of course, any oher combinaion is feasible. From my own experience, eaching par I and II in a hird year Bachelor course allows eaching of par III and IV a he Maser level. Of course, Maser courses can be based on any pars of his book, rsyear PhD courses can sar wih par I and II and secondyear eld courses can use maerial of par III or IV. This all depends on he ambiion of he programme, he inenion of he lecurer and he needs of he sudens. Apar from being used in classroom, many PhD sudens and advanced Bachelor or Maser sudens have used various pars of previous versions of his ex for sudying on heir own. Given he deailed sepbysep approach o problems, i urned ou ha i was very useful for undersanding he basic srucure of, say, a maximizaion problem. Once his basic srucure was undersood, many exensions o more complex problems were obained  some of which hen even found heir way ino his book.
19 Par I Deerminisic models in discree ime 7
20
21 This book consiss of four pars. In his rs par of he book, we will ge o know he simples and herefore maybe he mos useful srucures o hink abou changes over ime, o hink abou dynamics. Par I deals wih discree ime models under cerainy. The rs chaper inroduces he simples possible ineremporal problem, a woperiod problem. I is solved in a general way and for many funcional forms. The mehods used are he Lagrangian and simple subsiuion. Various conceps like he ime preference rae and he ineremporal elasiciies of subsiuion are inroduced here as well, as hey are widely used in he lieraure and are used frequenly hroughou his book. For hose who wan o undersand he background of he Lagrangian, a chaper is included ha shows he link beween Lagrangians and solving by subsiuion. This will also give us he opporuniy o explain he concep of shadow prices as hey play an imporan role e.g. when using Hamilonians or dynamic programming. The woperiod opimal consumpion seup will hen be pu ino a decenralized general equilibrium seup. This allows us o undersand general equilibrium srucures in general while, a he same ime, we ge o know he sandard overlapping generaions (OLG) general equilibrium model. This is one of he mos widely used dynamic models in Economics. Chaper 2 concludes by reviewing some aspecs of di erence equaions. Chaper 3 hen covers in nie horizon models. We solve a ypical maximizaion problem rs by using he Lagrangian again and hen by dynamic programming. As dynamic programming regularly uses he envelope heorem, his heorem is rs reviewed in a simple saic seup. Examples for in nie horizon problems, a general equilibrium analysis of a decenralized economy, a ypical cenral planner problem and an analysis of how o rea family or populaion growh in opimizaion problems hen complee his chaper. To complee he range of maximizaion mehods used in his book, he presenaion of hese examples will also use he mehod of solving by insering. 9
22 10
23 Chaper 2 Twoperiod models and di erence equaions Given ha he idea of his book is o sar from simple srucures and exend hem o he more complex ones, his chaper sars wih he simples ineremporal problem, a woperiod decision framework. This simple seup already allows us o illusrae he basic dynamic radeo s. Aggregaing over individual behaviour, assuming an overlappinggeneraions (OLG) srucure, and puing individuals in general equilibrium provides an undersanding of he issues involved in hese seps and in idenical seps in more general seings. Some revision of properies of di erence equaions concludes his chaper. 2.1 Ineremporal uiliy maximizaion The seup Le here be an individual living for wo periods, in he rs she is young, in he second she is old. Le her uiliy funcion be given by U = U c y ; c o +1 U (c ; c +1 ) ; (2.1.1) where consumpion when young and old are denoed by c y and c o +1 or c and c +1 ; respecively, when no ambiguiy arises. The individual earns labour income w in boh periods. I could also be assumed ha she earns labour income only in he rs period (e.g. when reiring in he second) or only in he second period (e.g. when going o school in he rs). Here, wih s denoing savings, her budge consrain in he rs period is c + s = w (2.1.2) and in he second i reads c +1 = w +1 + (1 + r +1 ) s : (2.1.3) 11
24 12 Chaper 2. Twoperiod models and di erence equaions Ineres paid on savings in he second period are given by r +1 : All quaniies are expressed in unis of he consumpion good (i.e. he price of he consumpion good is se equal o one. See ch for an exension where he price of he consumpion good is p :). This budge consrain says somehing abou he assumpions made on he iming of wage paymens and savings. Wha an individual can spend in period wo is principal and ineres on her savings s of he rs period. There are no ineres paymens in period one. This means ha wages are paid and consumpion akes place a he end of period 1 and savings are used for some producive purposes (e.g. rms use i in he form of capial for producion) in period 2. Therefore, reurns r +1 are deermined by economic condiions in period 2 and have he index + 1: Timing is illusraed in he following gure. w s = w c c w +1 c +1 (1 + r +1 )s + 1 Figure Timing in woperiod models These wo budge consrains can be merged ino one ineremporal budge consrain by subsiuing ou savings, w + (1 + r +1 ) 1 w +1 = c + (1 + r +1 ) 1 c +1 : (2.1.4) I should be noed ha by no resricing savings o be nonnegaive in (2.1.2) or by equaing he presen value of income on he lefhand side wih he presen value of consumpion on he righhand side in (2.1.4), we assume perfec capial markes: individuals can save and borrow any amoun hey desire. Equaion (2.2.14) provides a condiion under which individuals save. Adding he behavioural assumpion ha individuals maximize uiliy, he economic behaviour of an individual is described compleely and one can derive her consumpion and saving decisions. The problem can be solved by a Lagrange approach or by simply insering. The laer will be done in ch and in deerminisic seups or exensively in ch for a sochasic framework. Insering ransforms an opimizaion problem wih a consrain ino an unconsrained problem. We will use a Lagrange approach now. The maximizaion problem reads max c; c +1 (2.1.1) subjec o he ineremporal budge consrain (2.1.4). The household s conrol variables are c and c +1 : As hey need o be chosen so ha hey saisfy he budge consrain, hey can no be chosen independenly of each oher. Wages and ineres raes are exogenously given o he household. When
25 2.1. Ineremporal uiliy maximizaion 13 choosing consumpion levels, he reacion of hese quaniies o he decision of our household is assumed o be zero  simply because he household is oo small o have an e ec on economywide variables Solving by he Lagrangian We will solve his maximizaion problem by using he Lagrange funcion. This funcion will now be presened simply and is srucure will be explained in a recipe sense, which is he mos useful one for hose ineresed in quick applicaions. For hose ineresed in more background, ch. 2.3 will show he formal principles behind he Lagrangian. The Lagrangian for our problem reads L = U (c ; c +1 ) + w + (1 + r +1 ) 1 w +1 c (1 + r +1 ) 1 c +1, (2.1.5) where is a parameer called he Lagrange muliplier. The Lagrangian always consiss of wo pars. The rs par is he objecive funcion, he second par is he produc of he Lagrange muliplier and he consrain, expressed as he di erence beween he righhand side and he lefhand side of (2.1.4). Technically speaking, i makes no di erence wheher one subracs he lefhand side from he righhand side as here or vice versa  righhand side minus lefhand side. Reversing he di erence would simply change he sign of he Lagrange muliplier bu no change he nal opimaliy condiions. Economically, however, one would usually wan a posiive sign of he muliplier, as we will see in ch The rsorder condiions are L c = U c (c ; c +1 ) = 0; L c+1 = U c+1 (c ; c +1 ) [1 + r +1 ] 1 = 0; L = w + (1 + r +1 ) 1 w +1 c (1 + r +1 ) 1 c +1 = 0: Clearly, he las rsorder condiion is idenical o he budge consrain. Noe ha here are hree variables o be deermined, consumpion for boh periods and he Lagrange muliplier. Having a leas hree condiions is a necessary, hough no su cien (hey migh, generally speaking, be linearly dependen) condiion for his o be possible. The rs wo rsorder condiions can be combined o give U c (c ; c +1 ) = (1 + r +1 ) U c+1 (c ; c +1 ) = : (2.1.6) Marginal uiliy from consumpion oday on he lefhand side mus equal marginal uiliy omorrow, correced by he ineres rae, on he righhand side. The economic meaning of his correcion can bes be undersood when looking a a version wih nominal budge consrains (see ch ). We learn from his maximizaion ha he modi ed rsorder condiion (2.1.6) gives us a necessary equaion ha needs o hold when behaving opimally. I links consumpion c oday o consumpion c +1 omorrow. This equaion ogeher wih he budge consrain
26 14 Chaper 2. Twoperiod models and di erence equaions (2.1.4) provides a wodimensional sysem: wo equaions in wo unknowns, c and c +1. These equaions herefore allow us in principle o compue hese endogenous variables as a funcion of exogenously given wages and ineres raes. This would hen be he soluion o our maximizaion problem. The nex secion provides an example where his is indeed he case. 2.2 Examples Opimal consumpion The seup This rs example allows us o solve explicily for consumpion levels in boh periods. Le preferences of households be represened by U = ln c + (1 ) ln c +1 : (2.2.1) This uiliy funcion is ofen referred o as CobbDouglas or logarihmic uiliy funcion. Uiliy from consumpion in each period, insananeous uiliy, is given by he logarihm of consumpion. Insananeous uiliy is someimes also referred o as feliciy funcion. As ln c has a posiive rs and negaive second derivaive, higher consumpion increases insananeous uiliy bu a a decreasing rae. Marginal uiliy from consumpion is decreasing in (2.2.1). The parameer capures he imporance of insananeous uiliy in he rs relaive o insananeous uiliy in he second. Overall uiliy U is maximized subjec o he consrain we know from (2.1.4) above, where we denoe he presen value of labour income by W = c + (1 + r +1 ) 1 c +1 ; (2.2.2) W w + (1 + r +1 ) 1 w +1 : (2.2.3) Again, he consumpion good is chosen as numeraire good and is price equals uniy. Wages are herefore expressed in unis of he consumpion good. Solving by he Lagrangian The Lagrangian for his problem reads L = ln c + (1 ) ln c +1 + W c (1 + r +1 ) 1 c +1 : The rsorder condiions are L c = (c ) 1 = 0; L c+1 = (1 ) (c +1 ) 1 [1 + r +1 ] 1 = 0; and he budge consrain (2.2.2) following from L = 0:
27 2.2. Examples 15 The soluion Dividing rsorder condiions gives c = c +1 1 c 1 (1 + r +1) 1 c +1 : = 1 + r +1 and herefore This equaion corresponds o our opimaliy rule (2.1.6) derived above for he more general case. Insering ino he budge consrain (2.2.2) yields W = (1 + r +1 ) 1 c +1 = 1 1 (1 + r +1) 1 c +1 which is equivalen o I follows ha c +1 = (1 ) (1 + r +1 ) W : (2.2.4) c = W : (2.2.5) Apparenly, opimal consumpion decisions imply ha consumpion when young is given by a share of he presen value W of lifeime income a ime of he individual under consideraion. Consumpion when old is given by he remaining share 1 plus ineres paymens, c +1 = (1 + r +1 ) (1 ) W : Equaions (2.2.4) and (2.2.5) are he soluion o our maximizaion problem. These expressions are someimes called closedform soluions. A (closedform) soluion expresses he endogenous variable, consumpion in our case, as a funcion only of exogenous variables. Closedform soluion is a di eren word for closedloop soluion. For furher discussion see ch Assuming preferences as in (2.2.1) makes modelling someimes easier han wih more complex uiliy funcions. A drawback here is ha he share of lifeime income consumed in he rs period and herefore he savings decision is independen of he ineres rae, which appears implausible. A way ou is given by he CES uiliy funcion (see below a (2.2.10)) which also allows for closedform soluions for consumpion (for an example in a sochasic seup, see exercise 6 in ch ). More generally speaking, such a simpli caion should be jusi ed if some aspec of an economy ha is fairly independen of savings is he focus of some analysis. Solving by subsiuion Le us now consider his example and see how his maximizaion problem could have been solved wihou using he Lagrangian. The principle is simply o ransform an opimizaion problem wih consrains ino an opimizaion problem wihou consrains. This is mos simply done in our example by replacing he consumpion levels in he objecive funcion (2.2.1) by he consumpion levels from he consrains (2.1.2) and (2.1.3). The unconsrained maximizaion problem hen reads max U by choosing s ; where U = ln (w s ) + (1 ) ln (w +1 + (1 + r +1 ) s ) :
28 16 Chaper 2. Twoperiod models and di erence equaions This objecive funcion shows he radeo faced by anyone who wans o save nicely. High savings reduce consumpion oday bu increase consumpion omorrow. The rsorder condiion, i.e. he derivaive wih respec o saving s ; is simply 1 = (1 + r +1 ) : w s w +1 + (1 + r +1 ) s When his is solved for savings s ; we obain w +1 + s = (w s ) r +1, w +1 1 = w 1 + r +1 s = (1 ) w w r +1 = w W ; s 1, where W is lifeime income as de ned afer (2.2.2). To see ha his is consisen wih he soluion by Lagrangian, compue rsperiod consumpion and nd c = w s = W  which is he soluion in (2.2.5). Wha have we learned from using his subsiuion mehod? We see ha we do no need sophisicaed ools like he Lagrangian as we can solve a normal unconsrained problem  he ype of problem we migh be more familiar wih from saic maximizaion seups. Bu he seps o obain he nal soluion appear somewha more curvy and less elegan. I herefore appears worhwhile o become more familiar wih he Lagrangian Opimal consumpion wih prices Consider now again he uiliy funcion (2.1.1) and maximize i subjec o he consrains p c + s = w and p +1 c +1 = w +1 + (1 + r +1 ) s : The di erence o he inroducory example in ch. 2.1 consiss in he inroducion of an explici price p for he consumpion good. The rsperiod budge consrain now equaes nominal expendiure for consumpion (p c is measured in, say, Euro, Dollar or Yen) plus nominal savings o nominal wage income. The second period consrain equally equaes nominal quaniies. Wha does an opimal consumpion rule as in (2.1.6) now look like? The Lagrangian is, now using an ineremporal budge consrain wih prices, L = U (c ; c +1 ) + W p c (1 + r +1 ) 1 p +1 c +1 : The rsorder condiions for c and c +1 are L c = U c (c ; c +1 ) p = 0; L c+1 = U c+1 (c ; c +1 ) (1 + r +1 ) 1 p +1 = 0 and he ineremporal budge consrain. Combining hem gives U c (c ; c +1 ) p = U c +1 (c ; c +1 ) p +1 [1 + r +1 ] 1, U c (c ; c +1 ) U c+1 (c ; c +1 ) = p p +1 [1 + r +1 ] 1 : (2.2.6)
29 2.2. Examples 17 This equaion says ha marginal uiliy of consumpion oday relaive o marginal uiliy of consumpion omorrow equals he relaive price of consumpion oday and omorrow. This opimaliy rule is idenical for a saic 2good decision problem where opimaliy requires ha he raio of marginal uiliies equals he relaive price. The relaive price here is expressed in a presen value sense as we compare marginal uiliies a wo poins in ime. The price p is herefore divided by he price omorrow, discouned by he nex period s ineres rae, p +1 [1 + r +1 ] 1 : In conras o wha someimes seems common pracice, we will no call (2.2.6) a soluion o he maximizaion problem. This expression (frequenly referred o as he Euler equaion) is simply an expression resuling from rsorder condiions. Sricly speaking, (2.2.6) is only a necessary condiion for opimal behaviour  and no more. As de ned above, a soluion o a maximizaion problem is a closedform expression as for example in (2.2.4) and (2.2.5). I gives informaion on levels  and no jus on changes as in (2.2.6). Being aware of his imporan di erence, in wha follows, he erm solving a maximizaion problem will neverheless cover boh analyses. Those which sop a he Euler equaion and hose which go all he way owards obaining a closedform soluion Some useful de niions wih applicaions In order o be able o discuss resuls in subsequen secions easily, we review some de niions here ha will be used frequenly in laer pars of his book. We are mainly ineresed in he ineremporal elasiciy of subsiuion and he ime preference rae. While a lo of his maerial can be found in micro exbooks, he noaion used in hese books di ers of course from he one used here. As his book is also inended o be as selfconained as possible, his shor review can serve as a reference for subsequen explanaions. We sar wih he Marginal rae of subsiuion (MRS) Le here be a consumpion bundle (c 1 ; c 2 ; ::::; c n ) : Le uiliy be given by u (c 1 ; c 2 ; ::::; c n ) which we abbreviae o u (:) : The MRS beween good i and good j is hen de ned by MRS ij (:) (:) j : (2.2.7) I gives he increase of consumpion of good j ha is required o keep he uiliy level a u (c 1 ; c 2 ; ::::; c n ) when he amoun of i is decreased marginally. By his de niion, his amoun is posiive if boh goods are normal goods  i.e. if boh parial derivaives in (2.2.7) are posiive. Noe ha de niions used in he lieraure can di er from his one. Some replace decreased by increased (or  which has he same e ec  replace increase by decrease ) and hereby obain a di eren sign. Why his is so can easily be shown: Consider he oal di erenial of u (c 1 ; c 2 ; ::::; c n ) ; keeping all consumpion levels apar from c i and c j x. This yields du (c 1 ; c 2 ; ::::; c n ) i dc i j dc j :
30 18 Chaper 2. Twoperiod models and di erence equaions The overall uiliy level u (c 1 ; c 2 ; ::::; c n ) does no change if du (:) = 0, dc j dc i (:) = MRS ij (:) j Equivalen erms As a reminder, he equivalen erm o he MRS in producion heory is he marginal rae of echnical subsiuion MRT S ij (:) = i j where he uiliy funcion was replaced by a producion funcion and consumpion c k was replaced by facor inpus x k. On a more economywide level, here is he marginal rae of ransformaion MRT ij (:) = i j where he uiliy funcion has now been replaced by a ransformaion funcion G (maybe beer known as producion possibiliy curve) and he y k are oupu of good k. The marginal rae of ransformaion gives he increase in oupu of good j when oupu of good i is marginally decreased. (Ineremporal) elasiciy of subsiuion Though our main ineres is a measure of ineremporal subsiuabiliy, we rs de ne he elasiciy of subsiuion in general. As wih he marginal rae of subsiuion, he de niion implies a cerain sign of he elasiciy. In order o obain a posiive sign (wih normal goods), we de ne he elasiciy of subsiuion as he increase in relaive consumpion c i =c j when he relaive price p i =p j decreases (which is equivalen o an increase of p j =p i ). Formally, we obain for he case of wo consumpion goods ij d(c i=c j ) d(p j =p i ) p j =p i c i =c j. This de niion can be expressed alernaively (see ex. 6 for deails) in a way which is more useful for he examples below. We express he elasiciy of subsiuion by he derivaive of he log of relaive consumpion c i =c j wih respec o he log of he marginal rae of subsiuion beween j and i, ij d ln (c i=c j ) d ln MRS ji : (2.2.8) Insering he marginal rae of subsiuion MRS ji from (2.2.7), i.e. exchanging i and j in (2.2.7), gives ij = d ln (c i=c j ) = u c j =u ci d (c i =c j ) : d ln u cj =u ci c i =c j d u cj =u ci The advanage of an elasiciy when compared o a normal derivaive, such as he MRS, is ha an elasiciy is measureless. I is expressed in percenage changes. (This can be bes seen in he following example and in ex. 6 where he derivaive is muliplied by p j=p i c i =c j :) I can boh be applied o saic uiliy or producion funcions or o ineremporal uiliy funcions.
31 2.2. Examples 19 The ineremporal elasiciy of subsiuion for a uiliy funcion u (c ; c +1 ) is hen simply he elasiciy of subsiuion of consumpion a wo poins in ime, ;+1 u c =u c+1 c +1 =c d (c +1 =c ) d u c =u c+1 : (2.2.9) Here as well, in order o obain a posiive sign, he subscrips in he denominaor have a di eren ordering from he one in he numeraor. The ineremporal elasiciy of subsiuion for logarihmic and CES uiliy funcions For he logarihmic uiliy funcion U = ln c +(1 an ineremporal elasiciy of subsiuion of one, where he las sep used ;+1 = d (c +1 =c ) = d c = 1 c +1 c = 1 c +1 d (c +1 =c ) = 1; c +1 =c d c = 1 c +1 1 d (c +1 =c ) d (c +1 =c ) = 1 ) ln c +1 from (2.2.1), we obain : I is probably worh noing a his poin ha no all exbooks would agree on he resul of plus one. Following some oher de niions, a resul of minus one would be obained. Keeping in mind ha he sign is jus a convenion, depending on increase or decrease in he de niion, his should no lead o confusions. When we consider a uiliy funcion where insananeous uiliy is no logarihmic bu of CES ype U = c 1 + (1 ) c 1 +1 ; (2.2.10) he ineremporal elasiciy of subsiuion becomes ;+1 [1 ] c = (1 ) (1 ) c+1 d (c +1 =c ) c +1 =c d [1 ] c = (1 ) (1 ) c De ning x (c +1 =c ) ; we obain d [1 d (c +1 =c ) ] c = (1 ) (1 ) c +1 = 1 = 1 +1 d (c +1 =c ) d c = 1 1 x 1 1 : =c +1 : (2.2.11) dx 1= Insering his ino (2.2.11) and cancelling erms, he elasiciy of subsiuion urns ou o be ;+1 c =c c+1 = 1 c +1 =c : This is where he CES uiliy funcion (2.2.10) has is name from: The ineremporal elasiciy (E) of subsiuion (S) is consan (C). c dx
32 20 Chaper 2. Twoperiod models and di erence equaions The ime preference rae Inuiively, he ime preference rae is he rae a which fuure insananeous uiliies are discouned. To illusrae, imagine a discouned income sream x r x 1 + x 2 + ::: 1 + r where discouning akes place a he ineres rae r: Replacing income x by insananeous uiliy and he ineres rae by, would be he ime preference rae. Formally, he ime preference rae is he marginal rae of subsiuion of insananeous uiliies (no of consumpion levels) minus one, T P R MRS ;+1 1: As an example, consider he following sandard uiliy funcion which we will use very ofen in laer chapers, U 0 = 1 =0 u (c ) ; 1 ; > 0: (2.2.12) 1 + Le be a posiive parameer and he implied discoun facor, capuring he idea of impaience: By muliplying insananeous uiliy funcions u (c ) by, fuure uiliy is valued less han presen uiliy. This uiliy funcion generalizes (2.2.1) in wo ways: Firs and mos imporanly, here is a much longer planning horizon han jus wo periods. In fac, he individual s overall uiliy U 0 sems from he sum of discouned insananeous uiliy levels u (c ) over periods 0; 1; 2;... up o in niy. The idea behind his objecive funcion is no ha individuals live forever bu ha individuals care abou he wellbeing of subsequen generaions. Second, he insananeous uiliy funcion u (c ) is no logarihmic as in (2.2.1) bu of a more general naure where one would usually assume posiive rs and negaive second derivaives, u 0 > 0; u 00 < 0. The marginal rae of subsiuion is hen MRS ;+1 (:) 0 (:) (c 0 (:) (c +1 ) = (1= (1 + )) +1 = 1 + : (1= (1 + )) The ime preference rae is herefore given by : Now ake for example he uiliy funcion (2.2.1). Compuing he MRS minus one, we have = 1 1 = : (2.2.13) The ime preference rae is posiive if > 0:5: This makes sense for (2.2.1) as one should expec ha fuure uiliy is valued less han presen uiliy. As a side noe, all ineremporal uiliy funcions in his book will use exponenial discouning as in (2.2.12). This is clearly a special case. Models wih nonexponenial or hyperbolic discouning imply fundamenally di eren dynamic behaviour and ime inconsisencies. See furher reading for some references.
33 2.3. The idea behind he Lagrangian 21 Does consumpion increase over ime? This de niion of he ime preference rae allows us o provide a precise answer o he quesion wheher consumpion increases over ime. We simply compue he condiion under which c +1 > c by using (2.2.4) and (2.2.5), c +1 > c, (1 ) (1 + r +1 ) W > W, 1 + r +1 > 1, r +1 > 1 +, r +1 > : 1 Consumpion increases if he ineres rae is higher han he ime preference rae. The ime preference rae of he individual (being represened by ) deermines how o spli he presen value W of oal income ino curren and fuure use. If he ineres rae is su cienly high o overcompensae impaience, i.e. if (1 ) (1 + r) > in he rs line, consumpion rises. Noe ha even hough we compued he condiion for rising consumpion for our special uiliy funcion (2.2.1), he resul ha consumpion increases when he ineres rae exceeds he ime preference rae holds for more general uiliy funcions as well. We will ge o know various examples for his in subsequen chapers. Under wha condiions are savings posiive? Savings are from he budge consrain (2.1.2) and he opimal consumpion resul (2.2.5) given by 1 s = w c = w w + w +1 = w r r +1 where he las equaliy assumed an invarian wage level, w = w +1 w. Savings are posiive if and only if s > 0, 1 >, 1 + r +1 > 1 + r +1 1, r +1 > 2 1 1, r +1 > (2.2.14) This means ha savings are posiive if ineres rae is larger han ime preference rae. Clearly, his resul does no necessarily hold for w +1 > w : 2.3 The idea behind he Lagrangian So far, we simply used he Lagrange funcion wihou asking where i comes from. This chaper will o er a derivaion of he Lagrange funcion and also an economic inerpreaion of he Lagrange muliplier. In maximizaion problems employing a uiliy funcion, he Lagrange muliplier can be undersood as a price measured in uiliy unis. I is ofen called a shadow price.
34 22 Chaper 2. Twoperiod models and di erence equaions Where he Lagrangian comes from I The maximizaion problem Le us consider a maximizaion problem wih some objecive funcion and a consrain, max F (x 1 ; x 2 ) subjec o g(x 1; x 2 ) = b: (2.3.1) x 1 ;x 2 The consrain can be looked a as an implici funcion, i.e. describing x 2 as a funcion of x 1 ; i.e. x 2 = h (x 1 ) : Using he represenaion x 2 = h (x 1 ) of he consrain, he maximizaion problem can be wrien as max x 1 F (x 1; h (x 1 )) : (2.3.2) The derivaives of implici funcions As we will use implici funcions and heir derivaives here and in laer chapers, we brie y illusrae he underlying idea and show how o compue heir derivaives. Consider a funcion f (x 1 ; x 2 ; :::; x n ) = 0: The implici funcion heorem says  saed simply  ha his funcion f (x 1 ; x 2 ; :::; x n ) = 0 implicily de nes (under suiable assumpions concerning he properies of f (:)  see exercise 7) a funcional relaionship of he ype x 2 = h (x 1 ; x 3 ; x 4 ; :::; x n ) : We ofen work wih hese implici funcions in Economics and we are also ofen ineresed in he derivaive of x 2 wih respec o, say, x 1 : In order o obain an expression for his derivaive, consider he oal di erenial of f (x 1 ; x 2 ; :::; x n ) = 0; df (:) (:) dx 1 + dx 2 + ::: + dx n = n When we keep x 3 o x n consan, we can solve his o ge dx 2 dx 1 (:) (:) 2 : (2.3.3) We have hereby obained an expression for he derivaive dx 2 =dx 1 wihou knowing he funcional form of he implici funcion h (x 1 ; x 3 ; x 4 ; :::; x n ) : For illusraion purposes, consider he following gure.
35 2.3. The idea behind he Lagrangian 23 Figure The implici funcion visible a z = 0 The horizonal axes plo x 1 and, o he back, x 2 : The verical axis plos z: The increasing surface depics he graph of he funcion z = g (x 1 ; x 2 ) b: When his surface crosses he horizonal plane a z = 0; a curve is creaed which conains all he poins where z = 0: Looking a his curve illusraes ha he funcion z = 0, g (x 1 ; x 2 ) = b implicily de nes a funcion x 2 = h (x 1 ) : See exercise 7 for an explici analyical derivaion of such an implici funcion. The derivaive dx 2 =dx 1 is hen simply he slope of his curve. The analyical expression for his is  using (2.3.3)  dx 2 =dx 1 = (:) 1 ) = (:) 2 ) : Firsorder condiions of he maximizaion problem The maximizaion problem we obained in (2.3.2) is an example for he subsiuion mehod: The budge consrain was solved for one conrol variable and insered ino he objecive funcion. The resuling maximizaion problem is one wihou consrain. The problem in (2.3.2) now has a sandard rsorder condiion, df dh = 0: (2.3.4) dx 2 dx 1 Taking ino consideraion ha from he implici funcion heorem applied o he consrain, dh = dx 2 1; x 2 1 ; (2.3.5) dx 1 dx 1 ; x 2 2 he opimaliy condiion (2.3.4) can be wrien 1 ;x he Lagrange muliplier 2 2 and obain = 0: Now 1; x 2 1 = 0: (2.3.6)
36 24 Chaper 2. Twoperiod models and di erence equaions As can be easily seen, his is he rsorder condiion of he Lagrangian L = F (x 1; x 2 ) + [b g (x 1; x 2 )] (2.3.7) wih respec o x 1 : Now imagine we wan o underake he same seps for x 2 ; i.e. we sar from he original problem (2.3.1) bu subsiue ou x 1 : We would hen obain an unconsrained problem as in (2.3.2) only ha we maximize wih respec o x 2 : Coninuing as we jus did for x 1 would yield he second 1; x 2 ) = 2 We have hereby shown where he Lagrangian comes from: Wheher one de nes a Lagrangian as in (2.3.7) and compues he rsorder condiion or one compues he rsorder condiion from he unconsrained problem as in (2.3.4) and hen uses he implici funcion heorem and de nes a Lagrange muliplier, one always ends up a (2.3.6). The Lagrangianroue is obviously faser Shadow prices The idea We can now also give an inerpreaion of he meaning of he mulipliers. Saring from he de niion of in (2.3.6), we can rewrie i according o 2 @b : One can undersand ha he rs equaliy can cancel he 2 by looking a de niion of a (parial) derivaive: 1 ;:::;x i = lim 4x i!0 f(x 1 ;:::;x i +4x i ;:::;x n) f(x 1 ;:::;x n) lim 4xi!0 4x i : The second equaliy uses he equaliy of g and b from he consrain in (2.3.1). From hese ransformaions, we see ha equals he change in F as a funcion of b. I is now easy o come up wih examples for F or b: How much does F increase (e.g. your uiliy) when your consrain b (your bank accoun) is relaxed? How much does he social welfare funcion change when he economy has more capial? How much do pro s of rms change when he rm has more workers? This is called shadow price and expresses he value of b in unis of F:
37 2.3. The idea behind he Lagrangian 25 A derivaion A more rigorous derivaion is as follows (cf. Inriligaor, 1971, ch. 3.3). Compue he derivaive of he maximized Lagrangian wih respec o 1 (b) ; x 2 = (F (x 1(b); x 2 (b)) + (b) [b g (x 1(b); x 2 1 = F x + 2 x + 0 (b) [b = (b) g(:)] + (b) 1 g x 1 2 g x The las equaliy resuls from rsorder condiions and he fac ha he budge consrain holds. As L(x 1; x 2) = F (x 1; x 2) due o he budge consrain holding wih equaliy, An example (b) 1; x (x 1; x The Lagrange muliplier is frequenly referred o as shadow price. As we have seen, is uni depends on he uni of he objecive funcion F: One can hink of price in he sense of a price in a currency, for example in Euro, only if he objecive funcion is some nominal expression like pro s or GDP. Oherwise i is a price expressed for example in uiliy erms. This can explicily be seen in he following example. Consider a cenral planner ha maximizes social welfare u (x 1 ; x 2 ) subjec o echnological and resource consrains, max u (x 1 ; x 2 ) subjec o x 1 = f (K 1; L 1 ) ; x 2 = g (K 2; L 2 ) ; K 1 + K 2 = K; L 1 + L 2 = L: Technologies in secors 1 and 2 are given by f (:) and g (:) and facors of producion are capial K and labour L: Using as mulipliers p 1 ; p 2 ; w K and w L ; he Lagrangian reads L = u (x 1 ; x 2 ) + p 1 [f (K 1; L 1 ) x 1 ] + p 2 [g (K 2 ; L 2 ) x 2 ] + w K [K K 1 K 2 ] + w L [L L 1 L 2 ] (2.3.8) and rsorder condiions p 1 = @f = p 1 w K = p 1 w L = = 2 p 2 = = p 2 w K = p 2 w L = 0:
38 26 Chaper 2. Twoperiod models and di erence equaions Here we see ha he rs muliplier p 1 is no a price expressed in some currency bu he derivaive of he uiliy funcion wih respec o good 1, i.e. marginal uiliy. By conras, if we looked a he muliplier w K only in he hird rsorder condiion, w K = p 1 1 ; we would hen conclude ha i is a price. Then insering he rs rsorder condiion, 1 = p 1 ; and using he consrain x 1 = f (K 1; L 1 ) shows however ha i really sands for he increase in uiliy when he capial sock used in producion of good 1 rises, w = p 1 Hence w K and all oher mulipliers are prices in uiliy unis. I is now also easy o see ha all shadow prices are prices expressed in some currency if he objecive funcion is no uiliy bu, for example GDP. Such a maximizaion problem could read max p 1 x 1 + p 2 x 2 subjec o he consrains as above. Finally, reurning o he discussion afer (2.1.5), he rsorder condiions show ha he sign of he Lagrange muliplier should be posiive from an economic perspecive. If p 1 in (2.3.9) is o capure he value aached o x 1 in uiliy unis and x 1 is a normal good (uiliy increases in x 1 ; i.e. 1 > 0), he shadow price should be posiive. If we had represened he consrain in he Lagrangian (2.3.8) as x 1 f (K 1; L 1 ) raher han righhand side minus lefhand side, he rsorder condiion would read 1 + p 1 = 0 and he Lagrange muliplier would have been negaive. If we wan o associae he Lagrange muliplier o he shadow price, he consrains in he Lagrange funcion should be represened such ha he Lagrange muliplier is posiive. 2.4 An overlapping generaions model We will now analyze many households joinly and see how heir consumpion and saving behaviour a ecs he evoluion of he economy as a whole. We will ge o know he Euler heorem and how i is used o sum facor incomes o yield GDP. We will also undersand how he ineres rae in he household s budge consrain is relaed o marginal produciviy of capial and he depreciaion rae. All his joinly yields ime pahs of aggregae consumpion, he capial sock and GDP. We will assume an overlappinggeneraions srucure (OLG). A model in general equilibrium is described by fundamenals of he model and marke and behavioural assumpions. Fundamenals are echnologies of rms, preferences of households and facor endowmens. Adding clearing condiions for markes and behavioural assumpions for agens complees he descripion of he model.
39 2.4. An overlapping generaions model Technologies The rms Le here be many rms who employ capial K and labour L o produce oupu Y according o he echnology Y = Y (K ; L) : (2.4.1) Assume producion of he nal good Y (:) is characerized by consan reurns o scale. We choose Y as our numeraire good and normalize is price o uniy, p = 1: While his is no necessary and we could keep he price p all hrough he model, we would see ha all prices, like for example facor rewards, would be expressed relaive o he price p : Hence, as a shorcu, we se p = 1: We now, however, need o keep in mind ha all prices are henceforh expressed in unis of his nal good. Wih his normalizaion, pro s are given by = Y w K K w L L. Leing rms ac under perfec compeiion, he rsorder condiions from pro maximizaion re ec he fac ha he rm akes all prices as parameric and se marginal produciviies equal o real = w = wl : (2.4.2) In each period hey equae ; he marginal produciviy of capial, o he facor price w K for capial and he marginal produciviy of labour o labour s facor reward w L. Euler s heorem Euler s heorem shows ha for a linearhomogeneous funcion f (x 1 ; x 2 ; :::; x n ) he sum of parial derivaives imes he variables wih respec o which he derivaive was compued equals he original funcion f (:) ; f (x 1 ; x 2 ; :::; x n ) (:) x 1 + x 2 + ::: + x n : n Provided ha he echnology used by rms o produce Y has consan reurns o scale, we obain from his heorem ha Y K Using he opimaliy condiions (2.4.2) of he rm for he applied version of Euler s heorem (2.4.4) yields Y = w K K + w L L: (2.4.5) Toal oupu in his economy, Y ; is idenical o oal facor income. This resul is usually given he economic inerpreaion ha under perfec compeiion all revenue in rms is used o pay facors of producion. As a consequence, pro s of rms are zero.
40 28 Chaper 2. Twoperiod models and di erence equaions Households Individual households Households live again for wo periods. The uiliy funcion is herefore as in (2.2.1) and given by U = ln c y + (1 ) ln c o +1: (2.4.6) I is maximized subjec o he ineremporal budge consrain w L = c y + (1 + r +1 ) 1 c o +1: This consrain di ers slighly from (2.2.2) in ha people work only in he rs period and reire in he second. Hence, here is labour income only in he rs period on he lefhand side. Savings from he rs period are used o nance consumpion in he second period. Given ha he presen value of lifeime wage income is w L ; we can conclude from (2.2.4) and (2.2.5) ha individual consumpion expendiure and savings are given by Aggregaion c y = w L ; c o +1 = (1 ) (1 + r +1 ) w L ; (2.4.7) s = w L c y = (1 ) w L : (2.4.8) We assume ha in each period L individuals are born and die. Hence, he number of young and he number of old is L as well. As all individuals wihin a generaion are idenical, aggregae consumpion wihin one generaion is simply he number of, say, young imes individual consumpion. Aggregae consumpion in is herefore given by C = Lc y + Lc o : Using he expressions for individual consumpion from (2.4.7) and noing he index (and no + 1) for he old yields C = Lc y + Lc o = w L + (1 ) (1 + r ) w L 1 L: Goods marke equilibrium and accumulaion ideniy The goods marke equilibrium requires ha supply equals demand, Y = C + I ; where demand is given by consumpion plus gross invesmen. Nex period s capial sock is  by an accouning ideniy  given by K +1 = I +(1 ) K : Ne invesmen, amouning o he change in he capial sock, K +1 K, is given by gross invesmen I minus depreciaion K, where is he depreciaion rae, K +1 K = I K. Replacing gross invesmen by he goods marke equilibrium, we obain he resource consrain K +1 = (1 ) K + Y C : (2.4.9) For our OLG seup, i is useful o rewrie his consrain slighly, Y + (1 ) K = C + K +1 : (2.4.10)
41 2.4. An overlapping generaions model 29 In his formulaion, i re ecs a broader goods marke equilibrium where he lefhand side shows supply as curren producion plus capial held by he old. The old sell capial as i is of no use for hem, given ha hey will no be able o consume anyhing in he nex period. Demand for he aggregae good is given by aggregae consumpion (i.e. consumpion of he young plus consumpion of he old) plus he capial sock o be held nex period by he currenly young The reduced form For he rs ime in his book we have come o he poin where we need o nd wha will be called a reduced form. Once all maximizaion problems are solved and all consrains and marke equilibria are aken ino accoun, he objecive consiss of undersanding properies of he model, i.e. undersanding is predicions. This is usually done by rs simplifying he srucure of he sysem of equaions coming ou of he model as much as possible. In he end, afer insering and reinsering, a sysem of n equaions in n unknowns resuls. The sysem where n is he smalles possible is wha will be called he reduced form. Ideally, here is only one equaion lef and his equaion gives an explici soluion of he endogenous variable. In saic models, an example would be L X = L; i.e. employmen in secor X is given by a uiliy parameer imes he oal exogenous labour supply L: This would be an explici soluion. If we are lef wih jus one equaion bu we obain on an implici soluion, we would obain somehing like f (L X ; ; L) = 0: Deriving he reduced form We now derive, given he resuls we have obained so far, how large he capial sock in he nex period is. Spliing aggregae consumpion ino consumpion of he young and consumpion of he old and using he oupufacor reward ideniy (2.4.5) for he resource consrain in he OLG case (2.4.10), we obain w K K + w L L + (1 ) K = C y + C o + K +1 : De ning he ineres rae r as he di erence beween facor rewards w K for capial and he depreciaion rae ; r w K ; (2.4.11) we nd r K + w L L + K = C y + C o + K +1 : The ineres rae de niion (2.4.11) shows he ne income of capial owners per uni of capial. They earn he gross facor rewards w K bu, a he same ime, hey experience a loss from depreciaion. Ne income herefore only amouns o r : As he old consume he capial sock plus ineres c o L = (1 + r )K, we obain K +1 = w L L C y = s L: (2.4.12)
42 30 Chaper 2. Twoperiod models and di erence equaions which is he aggregae version of he savings equaion (2.4.8). Hence, we have found ha savings s of young a is he capial sock a + 1: Noe ha equaion (2.4.12) is ofen presen on inuiive grounds. The old in period have no reason o save as hey will no be able o use heir savings in + 1: Hence, only he young will save and he capial sock in + 1; being made up from savings in he previous period, mus be equal o he savings of he young. The onedimensional di erence equaion In our simple dynamic model considered here, we obain he ideal case where we are lef wih only one equaion ha gives us he soluion for one variable, he capial sock. Insering he individual savings equaion (2.4.8) ino (2.4.12) gives wih he rsorder condiion (2.4.2) of he rm K +1 = (1 ) w L L = (1 (K ; L) L: The rs equaliy shows ha a share 1 of labour income urns ino capial in he nex period. Ineresingly, he depreciaion rae does no have an impac on he capial sock in period + 1. Economically speaking, he depreciaion rae a ecs he wealh of he old bu  wih logarihmic uiliy  no he saving of he young Properies of he reduced form Equaion (2.4.13) is a nonlinear di erence equaion in K : All oher quaniies in his equaion are consan. This equaion deermines he enire pah of capial in his dynamic economy, provided we have an iniial condiion K 0. We have herefore indeed solved he maximizaion problem and reduced he general equilibrium model o one single equaion. From he pah of capial, we can compue all oher variables which are of ineres for our economic quesions. Whenever we have reduced a model o is reduced form and have obained one or more di erence equaions (or di erenial equaions in coninuous ime), we would like o undersand he properies of such a dynamic sysem. The procedure is in principle always he same: We rs ask wheher here is some soluion where all variables (K in our case) are consan. This is hen called a seady sae analysis. Once we have undersood he seady sae (if here is one), we wan o undersand how he economy behaves ou of he seady sae, i.e. wha is ransiional dynamics are. Seady sae by In he seady sae, he capial sock is consan, K = K +1 = K, and deermined K = (1 (K ; L) L:
43 2.4. An overlapping generaions model 31 All oher variables like aggregae consumpion, ineres raes, wages ec. are consan as well. Consumpion when young and when old can di er, as in a seup wih nie lifeimes, he ineres rae in he seady sae does no need o equal he ime preference rae of households. Transiional dynamics Dynamics of he capial sock are illusraed in gure The gure plos he capial sock in period on he horizonal axis. The capial sock in he nex period, K +1 ; is ploed on he verical axis. The law of moion for capial from (2.4.13) hen shows up as he curve in his gure. The 45 line equaes K +1 o K : We sar from our iniial condiion K 0. Equaion (2.4.13) or he curve in his gure hen deermines he capial sock K 1 : This capial sock is hen viewed as K so ha, again, he curve gives us K +1 ; which is, given ha we now sared in 1; he capial sock K 2 of period 2: We can coninue doing so and see graphically ha he economy approaches he seady sae K which we compued in (2.4.14). K +1 N K = K K +1 = (1 )w L (K )L 0 K 0 K N Figure Convergence o he seady sae Summary We sared wih a descripion of echnologies in (2.4.1), preferences in (2.4.6) and facor endowmen given by K 0 : Wih behavioural assumpions concerning uiliy and pro maximizaion and perfec compeiion on all markes plus a descripion of markes in (2.4.3) and some juggling of equaions, we ended up wih a onedimensional di erence equaion (2.4.13) which describes he evoluion of he economy over ime and seady sae in he longrun. Given his formal analysis of he model, we could now sar answering economic quesions.
44 32 Chaper 2. Twoperiod models and di erence equaions 2.5 More on di erence equaions The reduced form in (2.4.13) of he general equilibrium model urned ou o be a nonlinear di erence equaion. We derived is properies in a fairly inuiive manner. However, one can approach di erence equaions in a more sysemaic manner, which we will do in his chaper Two useful proofs Before we look a di erence equaions, we provide wo resuls on sums which will be useful in wha follows. As he proof of his resul also has an esheic value, here will be a second proof of anoher resul o be done in he exercises. Lemma For any a 6= 1; Proof. The lef hand side is given by Muliplying his sum by a yields Now subrac (2.5.2) from (2.5.1) and nd T i=1a i = a 1 at 1 a ; T i=0a i = 1 at +1 1 a T i=1a i = a + a 2 + a 3 + : : : + a T 1 + a T : (2.5.1) a T i=1a i = a 2 + a 3 + : : : + a T + a T +1 : (2.5.2) (1 a) T i=1a i = a a T +1, T i=1a i = a 1 at 1 a : (2.5.3) Lemma T i=1ia i = 1 a 1 at 1 a 1 a Proof. The proof is lef as exercise 9. T a T A simple di erence equaion One of he simples di erence equaions is x +1 = ax ; a > 0: (2.5.4) This equaion appears oo simple o be worh analysing. We do i here as we ge o know he sandard seps in analyzing di erence equaions which we will also use for more complex di erence equaions. The objecive here is herefore no his di erence equaion as such bu wha is done wih i.
45 2.5. More on di erence equaions 33 Solving by insering The simples way o nd a soluion o (2.5.4) consiss of insering and reinsering his equaion su cienly ofen. Doing i hree imes gives x 1 = ax 0 ; x 2 = ax 1 = a 2 x 0 ; x 3 = ax 2 = a 3 x 0 : When we look a his soluion for = 3 long enough, we see ha he general soluion is x = a x 0 : (2.5.5) This could formally be proven by eiher inducion or by veri caion. In his conex, we can make use of he following De niion A soluion of a di erence equaion is a funcion of ime which, when insered ino he original di erence equaion, sais es his di erence equaion. Equaion (2.5.5) gives x as a funcion of ime only. Verifying ha i is a soluion indeed jus requires insering i wice ino (2.5.4) o see ha i sais es he original di erence equaion. Examples for soluions The sequence of x given by his soluion, given di eren iniial condiions x 0 ; are shown in he following gure for a > 1: The parameer values chosen are a = 2, x 0 2 f0:5; 1; 2g and runs from 0 o Figure Soluions o a di erence equaion for a > 1
46 34 Chaper 2. Twoperiod models and di erence equaions Longerm behaviour We can now ask wheher x approaches a consan when ime goes o in niy. This gives < 0 = < 0 < a < 1 lim x = x 0 lim!1!1 a = x 0 : ;, a = 1 : : 1 a > 1 Hence, x approaches a consan only when a < 1: For a = 1; i says a is iniial value x 0 : A graphical analysis For more complex di erence equaions, i ofen urns ou o be useful o analyze heir behaviour in a phase diagram. Even hough his simple di erence equaion can be undersood easily analyically, we will illusrae is properies in he following gure. Here as well, his allows us o undersand how analyses of his ype can also be underaken for more complex di erence equaions. N N x +1 x +1 x = x +1 x +1 = ax x +1 = ax 6  x = x +16 x 0 x Figure A phase diagram for a < 1 on he lef and a > 1 in he righ panel x 0 x N N The principle of a phase diagram is simple. The horizonal axis plos x ; he verical axis plos x +1 : There is a 45 line which serves o equae x o x +1 and here is a plo of he di erence equaion we wan o undersand. In our curren example, we plo x +1 as ax ino he gure. Now sar wih some iniial value x 0 and plo his on he horizonal axis as in he lef panel. The value for he nex period, i.e. for period 1; can hen be read o he verical axis by looking a he graph of ax : This value for x 1 is hen copied ono he horizonal axis by using he 45 line. Once on he horizonal axis, we can again use he graph of ax o compue he nex x +1 : Coninuing o do so, he lef panel shows how x evolves over ime, saring a x 0 : In his case of a < 1; we see how x approaches zero. When we graphically illusrae he case of a > 1; he evoluion of x is as shown in he righ panel.
47 2.5. More on di erence equaions A slighly less bu sill simple di erence equaion We now consider a slighly more general di erence equaion. Compared o (2.5.4), we jus add a consan b in each period, Solving by insering x +1 = ax + b; a > 0: (2.5.6) We solve again by insering. In conras o he soluion for (2.5.4), we sar from an iniial value of x : Hence, we imagine we are in ( as oday) and compue wha he level of x will be omorrow and he day afer omorrow ec. We nd for x +2 and x +3 ha x +2 = ax +1 + b = a [ax + b] + b = a 2 x + b [1 + a] ; x +3 = a 3 x + b 1 + a + a 2 : This suggess ha he general soluion is x +n = a n x + b n 1 i=0 ai = a n x + b an 1 a 1 : The las equaliy used he rs lemma from ch N x +1 Limi for n! 1 and a < 1 The limi for n going o in niy and a < 1 is given by A graphical analysis x = x +1?? x +1 = ax + b  b  6 lim x +n = lim n!1 n!1 an x + b an 1 a 1 = b 1 a : (2.5.7) x +1 N x +1 = ax + b x = x +1 x x N x x N Figure Phase diagrams of (2.5.6) for posiive (lef panel) and negaive b (righ panel) and higher a in he righ panel
48 36 Chaper 2. Twoperiod models and di erence equaions The lef panel in g sudies he evoluion of x for he sable case, i.e. where 0 < a < 1 and b > 0: Saring oday in wih x ; we end up in x. As we chose a smaller han one and a posiive b; x is posiive as (2.5.7) shows. We will reurn o he righ panel in a momen Fix poins and sabiliy De niions We can use hese examples o de ne wo conceps ha will also be useful a laer sages. De niion (Fixpoin) A xpoin x of a funcion f (x) is de ned by x = f (x ) : (2.5.8) For di erence equaions of he ype x +1 = f (x ) ; he xpoin x of he funcion f (x ) is also he poin where x says consan, i.e. x +1 = x. This is usually called he longrun equilibrium poin of some economy or is seady or saionary sae. Whenever an economic model, represened by is reduced form, is analyzed, i is generally useful o rs ry and nd ou wheher xpoins exis and wha heir economic properies are. For he di erence equaion from he las secion, we obain x +1 = x x () x = ax + b () x = b 1 a : Once a xpoin has been ideni ed, one can ask wheher i is sable. De niion (Global sabiliy) A xpoin x is globally sable if, saring from an iniial value x 0 6= x ; x converges o x. The concep of global sabiliy usually refers o iniial values x 0 ha are economically meaningful. An iniial physical capial sock ha is negaive would no be considered o be economically meaningful. unsable De niion (Local sabiliy and insabiliy) A xpoin x is if, locally sable diverges from saring from an iniial value x + "; where " is small, x x converges o. For illusraion purposes consider he xpoin x in he lef panel of g i is globally sable. In he righ panel of he same gure, i is unsable. As can easily be seen, he insabiliy follows by simply leing he x +1 line inersec he 45 line from below. In erms of he underlying di erence equaion (2.5.6), his requires b < 0 and a > 1:
49 2.5. More on di erence equaions 37 Clearly, economic sysems can be much more complex and generae several xpoins. Imagine he link beween x +1 and x is no linear as in (2.5.6) bu nonlinear, x +1 = f (x ) : Unforunaely for economic analysis, a nonlinear relaionship is he much more realisic case. The nex gure provides an example for some funcion f (x ) ha implies an unsable x u and a locally sable xpoin x s. x +1 N?   6? x 01 x u x 02 x s x 03 N x Figure A locally sable xpoin x s and an unsable xpoin x u An example: Deriving a budge consrain A frequenly encounered di erence equaion is he budge consrain. We have worked wih budge consrains a various poins before bu we have hardly hough abou heir origin. We more or less simply wroe hem down. Budge consrains, however, are ricky objecs, a leas when we hink abou general equilibrium seups. Wha is he asse we save in? Is here only one asse or are here several? Wha are he prices of hese asses? How does i relae o he price of he consumpion good, i.e. do we express he value of asses in real or nominal erms? This secion will derive a budge consrain. We assume ha here is only one asse. The price of one uni of he asse will be denoed by v : Is relaion o he price p of he consumpion good will be lef unspeci ed, i.e. we will discuss he mos general seup which is possible for he oneasse case. The derivaion of a budge consrain is in principle sraighforward. One de nes he wealh of he household (aking ino accoun which ypes of asses he household can hold for saving purposes and wha heir prices are), compues he di erence beween wealh oday and omorrow (his is where he di erence equaion aspec comes in) and uses an equaion which relaes curren savings o curren changes in he number of asses. In a nal sep, one will naurally nd ou how he ineres rae appearing in budge consrains relaes o more fundamenal quaniies like value marginal producs and depreciaion raes.
50 38 Chaper 2. Twoperiod models and di erence equaions A real budge consrain The budge consrain which resuls depends on he measuremen of wealh. We sar wih he case where we measure wealh in unis of share, or number of machines k : Savings of a household who owns k shares are given by capial income (ne of depreciaion losses) plus labour income minus consumpion expendiure, s w K k v k + w L p c : This is an ideniy resuling from bookkeeping of ows a he household level. Savings in are used for buying new asses in for which he period price v needs o be paid, s v = k +1 k : (2.5.9) We can rewrie his equaion slighly, which will simplify he inerpreaion of subsequen resuls, as k +1 = (1 ) k + wk k + w L p c v : Wealh in he nex period expressed in number of socks (and hence no in nominal erms) is given by wealh which is lef over from he curren period, (1 ) k, plus new acquisiions of socks which amoun o gross capial plus labour income minus consumpion expendiure divided by he price of one sock. Collecing he k erms and de ning an ineres rae r wk v gives a budge consrain for wealh measured by k ; k +1 = 1 + wk v k + wl v p v c = (1 + r ) k + wl v p v c : (2.5.10) This is a di erence equaion in k bu no ye a di erence equaion in nominal wealh a : Rearranging such ha expendiure is on he lef and disposable income on he righhand side yields p c + v k +1 = v k + w K v k + w L : This equaion also lends iself o a simple inerpreaion: On he lefhand side is oal expendiure in period ; consising of consumpion expendiure p c plus expendiure for buying he number of capial goods, k +1, he household wans o hold in + 1: As his expendiure is made in ; oal expendiure for capial goods amouns o v k +1. The righhand side is oal disposable income which splis ino income v k from selling all capial inheried from he previous period, capial income w K v k and labour income w L : This is he form budge consrains are ofen expressed in capial asse pricing models. Noe ha his is in principle also a di erence equaion in k :
51 2.6. Furher reading and exercises 39 A nominal budge consrain In our oneasse case, nominal wealh a of a household is given by he number k of socks he household owns (say he number of machines i owns) imes he price v of one sock (or machine), a = v k : Compuing he rs di erence yields a +1 a = v +1 k +1 v k = v +1 (k +1 k ) + (v +1 v ) k ; (2.5.11) where he second line added v +1 k v +1 k. Wealh changes depend on he acquisiion v +1 (k +1 k ) of new asses and on changes in he value of asses ha are already held, (v +1 v ) k. Insering (2.5.9) ino (2.5.11) yields a +1 a = v +1 s + (v +1 v ) k v = v +1 w K a a + w L v+1 p c + 1 a, v v v a +1 = v wk a + w L p c : (2.5.12) v v Wha does his equaion ell us? Each uni of wealh a (say Euro, Dollar, Yen...) gives 1 unis a he end of he period as % is los due o depreciaion plus dividend paymens w K =v. Wealh is augmened by labour income minus consumpion expendiure. This endofperiod wealh is expressed in wealh a +1 a he beginning of he nex period by dividing i hrough v (which gives he number k of socks a he end of he period) and muliplying i by he price v +1 of socks in he nex period. We have hereby obained a di erence equaion in a : This general budge consrain is fairly complex, however, which implies ha in pracice i is ofen expressed di erenly. One possibiliy consiss of choosing he capial good as he numeraire good and seing v 1 8: This simpli es (2.5.12) o a +1 = (1 + r ) a + w L p c : (2.5.13) The simpli caion in his expression consiss also in he de niion of he ineres rae r as r w K : 2.6 Furher reading and exercises Raes of subsiuion are discussed in many books on Microeconomics; see e.g. MasColell, Whinson and Green (1995) or Varian (1992). The de niion of he ime preference rae is no very explici in he lieraure. An alernaive formulaion implying he same de niion as he one we use here is used by Buier (1981, p. 773). He de nes he pure rae of ime preference as he marginal rae of subsiuion beween consumpion in wo periods
52 40 Chaper 2. Twoperiod models and di erence equaions when equal amouns are consumed in boh periods, minus one. A derivaion of he ime preference rae for a woperiod model is in appendix A.1 of Bossmann, Kleiber and Wälde (2007). The OLG model goes back o Samuelson. For presenaions in exbooks, see e.g. Blanchard and Fischer (1989), Azariadis (1993) or de la Croix and Michel (2002). Applicaions of OLG models are more han numerous. For an example concerning bequess and wealh disribuions, see Bossmann, Kleiber and Wälde (2007). See also Galor and Moav (2006) and Galor and Zeira (1993). The presenaion of he Lagrangian is inspired by Inriligaor (1971, p ). Treamens of shadow prices are available in many oher exbooks (Dixi, 1989, ch. 4; Inriligaor, 1971, ch. 3.3). More exensive reamens of di erence equaions and he implici funcion heorem can be found in many inroducory mahemaics for economiss books. There is an ineresing discussion on he empirical relevance of exponenial discouning. An early analysis of he implicaions of nonexponenial discouning is by Sroz (1955/56). An overview is provided by Frederick e al. (2002). An analysis using sochasic coninuous ime mehods is by Gong e al. (2007).
53 2.6. Furher reading and exercises 41 Exercises chaper 2 Applied Ineremporal Opimizaion Opimal consumpion in woperiod discree ime models 1. Opimal choice of household consumpion Consider he following maximizaion problem, subjec o Solve i by using he Lagrangian. max U = v (c ) + 1 c ;c v (c +1) (2.6.1) w + (1 + r) 1 w +1 = c + (1 + r) 1 c +1 : (a) Wha is he opimal consumpion pah? (b) Under wha condiions does consumpion rise? (c) Show ha he rsorder condiions can be wrien as u 0 (c ) =u 0 (c +1 ) = [1 + r] : Wha does his equaion ell you? 2. Solving by subsiuion Consider he maximizaion problem of secion and solve i by insering. Solve he consrain for one of he conrol variables, inser his ino he objecive funcion and compue rsorder condiions. Show ha he same resuls as in (2.2.4) and (2.2.5) are obained. 3. Capial marke resricions Now consider he following budge consrain. This is a budge consrain ha would be appropriae if you wan o sudy he educaion decisions of households. The parameer b amouns o schooling coss. Inheriance of his individual under consideraion is n. U = ln c + (1 ) ln c +1 subjec o b + n + (1 + r) 1 w +1 = c + (1 + r) 1 c +1 : (a) Wha is he opimal consumpion pro le under no capial marke resricions? (b) Assume loans for nancing educaion are no available, hence savings need o be posiive, s 0. Wha is he consumpion pro le in his case?
54 42 Chaper 2. Twoperiod models and di erence equaions 4. Opimal invesmen Consider a monopolis invesing in is echnology. Technology is capured by marginal coss c. The chief accounan of he rm has provided he manager of he rm wih he following informaion, = 1 + R 2 ; = p (x ) x c x I ; c +1 = c f (I 1 ) : Assume you are he manager. Wha is he opimal invesmen sequence I 1, I 2? 5. A paricular uiliy funcion Consider he uiliy funcion U = c + c +1 ; where 0 < < 1: Maximize U subjec o an arbirary budge consrain of your choice. Derive consumpion in he rs and second period. Wha is srange abou his uiliy funcion? 6. Ineremporal elasiciy of subsiuion Consider he uiliy funcion U = c 1 + c 1 +1 : (a) Wha is he ineremporal elasiciy of subsiuion? (b) How can he de niion in (2.2.8) of he elasiciy of subsiuion be ransformed ino he maybe beer known de niion Wha does ij sand for in words? 7. An implici funcion Consider he consrain x 2 x 1 x 1 = b. ij = d ln (c i=c j ) d ln u cj =u ci = p j=p i c i =c j d (c i =c j ) d (p j =p i )? (a) Convince yourself ha his implicily de nes a funcion x 2 = h (x 1 ) : Can he funcion h (x 1 ) be made explici? (b) Convince yourself ha his implicily de nes a funcion x 1 = k (x 2 ) : Can he funcion k (x 2 ) be made explici? (c) Think of a consrain which does no de ne an implici funcion. 8. General equilibrium Consider he Diamond model for a CobbDouglas producion funcion of he form Y = K L 1 and a logarihmic uiliy funcion u = ln c y + ln c o +1. (a) Derive he di erence equaion for K : (b) Draw a phase diagram. (c) Wha are he seady sae consumpion level and capial sock?
55 2.6. Furher reading and exercises Sums (a) Proof he saemen of he second lemma in ch , T i=1ia i = 1 a 1 at 1 a 1 a The idea is idenical o he rs proof in ch (b) Show ha k 1 s=0c k 1 s 4 s = ck 4 k c 4 : T a T +1 : Boh parameers obey 0 < c 4 < 1 and 0 < v < 1: Hin: Rewrie he sum as c k 4 1 k s=0 1 (=c 4 ) s and observe ha he rs lemma in ch holds for a which are larger or smaller han Di erence equaions Consider he following linear di erence equaion sysem y +1 = a y + b; a < 0 < b; (a) Wha is he xpoin of his equaion? (b) Is his poin sable? (c) Draw a phase diagram.
56 44 Chaper 2. Twoperiod models and di erence equaions
57 Chaper 3 Muliperiod models This chaper looks a decision processes where he ime horizon is longer han wo periods. In mos cases, he planning horizon will be in niy. In such a conex, Bellman s opimaliy principle is very useful. Is i, however, no he only way o solve maximizaion problems wih in nie ime horizon? For comparison purposes, we herefore sar wih he Lagrange approach, as in he las secion. Bellman s principle will be inroduced aferwards when inuiion for he problem and relaionships will have been increased. 3.1 Ineremporal uiliy maximizaion The seup The objecive funcion is given by he uiliy funcion of an individual, where again as in (2.2.12) U = 1 = u (c ) ; (3.1.1) (1 + ) 1 ; > 0 (3.1.2) is he discoun facor and is he posiive ime preference rae. We know his uiliy funcion already from he de niion of he ime preference rae, see (2.2.12). The uiliy funcion is o be maximized subjec o a budge consrain. The di erence o he formulaion in he las secion is ha consumpion does no have o be deermined for wo periods only bu for in niely many. Hence, he individual does no choose one or wo consumpion levels bu an enire pah of consumpion. This pah will be denoed by fc g : As ; fc g is a shor form of fc ; c +1 ; :::g : Noe ha he uiliy funcion is a generalizaion of he one used above in (2.1.1), bu is assumed o be addiively separable. The corresponding wo period uiliy funcion was used in exercise se 1, cf. equaion (2.6.1). The budge consrain can be expressed in he ineremporal version by 1 = (1 + r) ( ) e = a + 1 = (1 + r) ( ) w ; (3.1.3) 45
58 46 Chaper 3. Muliperiod models where e = p c : I saes ha he presen value of expendiure equals curren wealh a plus he presen value of labour income w. Labour income w and he ineres rae r are exogenously given o he household, is wealh level a is given by hisory. The only quaniy ha is lef o be deermined is herefore he pah fc g : Maximizing (3.1.1) subjec o (3.1.3) is a sandard Lagrange problem Solving by he Lagrangian The Lagrangian reads L = 1 = u (c ) + h 1= (1 + r) ( ) e a 1= (1 + r) ( ) w i ; where is he Lagrange muliplier. Firsorder condiions are L c = u 0 (c ) + [1 + r] ( ) p = 0; < 1; (3.1.4) L = 0; (3.1.5) where he laer is, as in he OLG case, he budge consrain. Again, we have as many condiions as variables o be deermined: here are in niely many condiions in (3.1.4), one for each c and one condiion for in (3.1.5). Do hese rsorder condiions ell us somehing? Take he rsorder condiion for period and for period + 1. They read Dividing hem gives u 0 (c ) = [1 + r] +1 u 0 (c +1 ) = [1 + r] ( ) p ; (+1 ) p +1 : 1 u 0 (c ) u 0 (c +1 ) = (1 + r) p p +1 () u0 (c ) u 0 (c +1 ) = p (1 + r) 1 p +1 : (3.1.6) Rearranging allows us o see an inuiive inerpreaion: Comparing he insananeous gain in uiliy u 0 (c ) wih he fuure gain, discouned a he ime preference rae, u 0 (c +1 ) ; mus yield he same raio as he price p ha has o be paid oday relaive o he price ha has o be paid in he fuure, also appropriaely discouned o is presen value price (1 + r) 1 p +1 : This inerpreaion is idenical o he woperiod inerpreaion in (2.2.6) in ch If we normalize prices o uniy, (3.1.6) is jus he expression we obained in he soluion for he woperiod maximizaion problem in (2.6.1). 3.2 The envelope heorem We saw how he Lagrangian can be used o solve opimizaion problems wih many ime periods. In order o undersand how dynamic programming works, i is useful o undersand a heorem which is frequenly used when employing he dynamic programming mehod: he envelope heorem.
59 3.2. The envelope heorem The heorem In general, he envelope heorem says Theorem Le here be a funcion g (x; y) : Choose x such ha g (:) is maximized for a given y: (Assume g (:) is such ha a smooh inerior soluion exiss.) Le f(y) be he resuling funcion of y; f (y) max g (x; y) : x Then he derivaive of f wih respec o y equals he parial derivaive of g wih respec o y, if g is evaluaed a ha x = x(y) ha maximizes g; d f (y) d y g (x; y : x=x(y) Proof. f (y) is consruced g (x; y) = 0: This implies a cerain x = x (y) provided ha second order condiions hold. Hence, f (y) = max x g (x; y) = g (x (y) ; y) : : The rs erm of he rs erm is zero. Then, d f(y) d y x d x(y) d y y Illusraion The plane depics he funcion g (x; y). The maximum of his funcion wih respec o x is shown as max x g (x; y), which is f (y). Given his gure, i is obvious ha he derivaive of f (y) wih respec o y is he same as he parial derivaive of g (:) wih respec o y a he poin where g (:) has is maximum wih respec o x: The is he derivaive when going in he direcion of y. Choosing he highes poin of g (:) wih respec o x, his direcional derivaive mus be he same as df(y) a he back of he dy gure.
60 48 Chaper 3. Muliperiod models f( y) gxy (, ) ( ) max gxy, x x ( ) g( x( y), y) dx g( x( y), y) df y dy = + x dy g( x( y), y = ) y y y Figure Illusraing he envelope heorem An example There is a cenral planner of an economy. The social welfare funcion is given by U (A; B), where A and B are consumpion goods. The echnologies available for producing hese goods are A = A (cl A ) and B = B (L B ) : The amoun of labour used for producing one or he oher good is denoed by L A and L B and c is a produciviy parameer in secor A: The economy s resource consrain is L A + L B = L: The planner is ineresed in maximizing he social welfare level and allocaes labour according o max LA U (A (cl A ) ; B (L L A )) : The opimaliy B0 = 0: (3.2.1) This makes opimal employmen L A a funcion of c, L A = L A (c) : (3.2.2) The cenral planner now asks wha happens o he social welfare level when he echnology parameer c increases and she sill maximizes he social welfare. The laer requires ha (3.2.1) coninues o hold and he maximized social welfare funcion wih (3.2.2) and
61 3.3. Solving by dynamic programming 49 he resource consrain reads U (A (cl A (c)) ; B (L heorem, he answer is L A (c))). Wihou using he envelope d dc U (A (cl A (c)) ; B (L L A (:) A0 [L A (c) + cl (:) A (c)] B0 [ L 0 A (c)] A0 L A (c) > 0; where he las equaliy follows from insering he opimaliy condiion (3.2.1). Economically, his resul means ha he e ec of beer echnology on overall welfare is given by he direc e ec in secor A: The indirec e ec hrough he reallocaion of labour vanishes as, due o he rsorder condiion (3.2.1), he marginal conribuion of each worker is idenical across secors. Clearly, his only holds for a small change in c: If one is ineresed in nding an answer by using he envelope heorem, one would sar by de ning a funcion V (c) max LA U (A (cl A ) ; B (L L A )) : Then, according o he envelope heorem, d dc V U (A (cl A) ; B (L L A )) A0 L A LA =L A (c) = LA =L A A0 L A (c) > 0: Apparenly, boh approaches yield he same answer. Applying he envelope heorem gives he answer faser. 3.3 Solving by dynamic programming The seup We will now ge o know how dynamic programming works. Le us sudy a maximizaion problem which is similar o he one in ch We will use he same uiliy funcion as in (3.1.1), reproduced here for convenience, U = 1 = u (c ). The consrain, however, will be represened in a more general way han in (3.1.3). We sipulae ha here is a variable x which evolves according o x +1 = f (x ; c ) : (3.3.1) This variable x could represen wealh and his consrain could hen represen he budge consrain of he household. This di erence equaion could also be nonlinear, however, as for example in a cenral planner problem where he consrain is a resource consrain as in (3.9.3). In his case, x would sand for capial. Anoher sandard example for x as a sae variable would be environmenal qualiy. Here we will rea he general case rs before we go on o more speci c examples furher below.
62 50 Chaper 3. Muliperiod models The consumpion level c and  more generally speaking  oher variables whose value is direcly chosen by individuals, e.g. invesmen levels or shares of wealh held in di eren asses, are called conrol variables. Variables which are no under he direc conrol of individuals are called sae variables. In many maximizaion problems, sae variables depend indirecly on he behaviour of individuals as in (3.3.1). Sae variables can also be compleely exogenous like for example he TFP level in an exogenous growh model or prices in a household maximizaion problem. Opimal behaviour is de ned by max fc g U subjec o (3.3.1), i.e. he highes value U can reach by choosing a sequence fc g fc ; c +1 ; :::g and by saisfying he consrain (3.3.1). The value of his opimal behaviour or opimal program is denoed by V (x ) max U subjec o x +1 = f (x ; c ) : (3.3.2) fc g V (x ) is called he value funcion. I is a funcion of he sae variable x and no of he conrol variable c. The laer poin is easy o undersand if one akes ino accoun ha he conrol variable c is, when behaving opimally, a funcion of he sae variable x : The value funcion V (:) could also be a funcion of ime (e.g. in problems wih nie horizon) bu we will no discuss his furher as i is of no imporance in he problems we will encouner. Generally speaking, x and c could be vecors and ime could hen be par of he sae vecor x : The value funcion is always a funcion of he saes of he sysem or of he maximizaion problem Three dynamic programming seps Given his descripion of he maximizaion problem, solving by dynamic programming essenially requires us o go hrough hree seps. This hreesep approach will be followed here, laer in coninuous ime, and also in models wih uncerainy. DP1: Bellman equaion and rsorder condiions The rs sep esablishes he Bellman equaion and compues rsorder condiions. The objecive funcion U in (3.1.1) is addiively separable which means ha i can be wrien in he form U = u(c ) + U +1 : (3.3.3) Bellman s idea consiss of rewriing he maximizaion problem in he opimal program (3.3.2) as V (x ) max fu(c ) + V (x +1 )g (3.3.4) c subjec o x +1 = f (x ; c ) : Equaion (3.3.4) is known as he Bellman equaion. In his equaion, he problem wih poenially in niely many conrol variables fc g was broken down in many problems
63 3.3. Solving by dynamic programming 51 wih one conrol variable c. Noe ha here are wo seps involved: Firs, he addiive separabiliy of he objecive funcion is used. Second, more imporanly, U +1 is replaced by V (x +1 ). This says ha we assume ha he opimal problem for omorrow is solved and we should worry abou he maximizaion problem of oday only. We can now compue he rsorder condiion which is of he form u 0 (c ) + V 0 (x +1 (x ; c = 0: (3.3.5) I ells us ha increasing consumpion c has advanages and disadvanages. The advanages consis in higher uiliy oday, which is re eced here by marginal uiliy u 0 (c ) : The disadvanages come from lower overall uiliy  he value funcion V  omorrow. The reducion in overall uiliy amouns o he change in x +1 ; i.e. he (x ; c ) ; imes he marginal value of x +1 ; i.e. V 0 (x +1 ) : As he disadvanage arises only omorrow, his is discouned a he rae : One can alk of a disadvanage of higher consumpion oday on overall uiliy omorrow as he (x ; c ) needs o be negaive, oherwise an inerior soluion as assumed in (3.3.5) would no exis. In principle, his is he soluion of our maximizaion problem. Our conrol variable c is by his expression implicily given as a funcion of he sae variable, c = c (x ) ; as x +1 by he consrain (3.3.1) is a funcion of x and c. As all sae variables in are known, he conrol variable is deermined by his opimaliy condiion. Hence, as V is wellde ned above, we have obained a soluion. As we know very lile abou he properies of V a his sage, however, we need o go hrough wo furher seps in order o eliminae V (o be precise, is derivaive V 0 (x +1 ) ; i.e. he cosae variable of x +1 ) from his rsorder condiion and obain a condiion ha uses only funcions (like e.g. he uiliy funcion or he echnology for producion in laer examples) of which properies like signs of rs and second derivaives are known. We obain more informaion abou he evoluion of his cosae variable in he second dynamic programming sep. DP2: Evoluion of he cosae variable The second sep of he dynamic programming approach sars from he maximized Bellman equaion. The maximized Bellman equaion is obained by replacing he conrol variables in he Bellman equaion, i.e. he c in (3.3.4), in he presen case, wih he opimal conrol variables as given by he rsorder condiion, i.e. by c (x ) resuling from (3.3.5). Logically, he max operaor disappears (as we inser he c (x ) which imply a maximum) and he maximized Bellman equaion reads V (x ) = u (c (x )) + V (f (x ; c(x ))) : The derivaive wih respec o x reads V 0 (x ) = u 0 (c (x )) dc (x ) + V 0 dc (x ) (f (x ; c (x ))) f x + f c : dx dx
64 52 Chaper 3. Muliperiod models This sep shows why i is imporan ha we use he maximized Bellman equaion here: Now conrol variables are a funcion of sae variable and we need o compue he derivaive of c wih respec o x when compuing he derivaive of he value funcion V (x ) : Insering he rsorder condiion simpli es his equaion o V 0 (x ) = V 0 (f (x ; c (x ))) f x = V 0 (x +1 ) f x (3.3.6) This equaion is a di erence equaion for he cosae variable, he derivaive of he value funcion wih respec o he sae variable, V 0 (x ). The cosae variable is also called he shadow price of he sae variable x : If we had more sae variables, here would be a cosae variable for each sae variable. I says how much an addiional uni of he sae variable (say e.g. of wealh) is valued: As V (x ) gives he value of opimal behaviour beween and he end of he planning horizon, V 0 (x ) says by how much his value changes when x is changed marginally. Hence, equaion (3.3.6) describes how he shadow price of he sae variable changes over ime when he agen behaves opimally. If we had used he envelope heorem, we would have immediaely ended up wih (3.3.6) wihou having o inser he rsorder condiion. DP3: Insering rsorder condiions Now inser he rsorder condiion (3.3.5) wice ino (3.3.6) o replace he unknown shadow price and o nd an opimaliy condiion depending on u only. This will hen be he Euler equaion. We do no do his here explicily as many examples will go hrough his sep in deail in wha follows. 3.4 Examples Ineremporal uiliy maximizaion wih a CES uiliy funcion The individual s budge consrain is given in he dynamic formulaion a +1 = (1 + r ) (a + w c ) : (3.4.1) Noe ha his dynamic formulaion corresponds o he ineremporal version in he sense ha (3.1.3) implies (3.4.1) and (3.4.1) wih some limi condiion implies (3.1.3). This will be shown formally in ch The budge consrain (3.4.1) can be found in many papers and also in some exbooks. The iming as implici in (3.4.1) is illusraed in he following gure. All evens ake place a he beginning of he period. Our individual owns a cerain amoun of wealh a a he beginning of and receives here wage income w and spends c on consumpion also a he beginning. Hence, savings s can be used during for producion and ineres is paid on s which in urn gives a +1 a he beginning of period + 1.
65 3.4. Examples 53 a + 1 = 1 + r s c +1 s = a + w c ( ) c +1 Figure The iming in an in nie horizon discree ime model The consisency of (3.4.1) wih echnologies in general equilibrium is no selfeviden. We will encouner more convenional budge consrains of he ype (2.5.13) furher below. As (3.4.1) is widely used, however, we now look a dynamic programming mehods and ake his budge consrain as given. The objecive of he individual is o maximize her uiliy funcion (3.1.1) subjec o he budge consrain by choosing a pah of consumpion levels c ; denoed by fc g ; 2 [; 1] : We will rs solve his wih a general insananeous uiliy funcion and hen inser he CES version of i, i.e. 1 u (c ) = c1 1 : (3.4.2) The value of he opimal program fc g is, given is iniial endowmen wih a, de ned as he maximum which can be obained subjec o he consrain, i.e. V (a ) max U (3.4.3) fc g subjec o (3.4.1). I is called he value funcion. Is only argumen is he sae variable a : See ch for a discussion on sae variables and argumens of value funcions. DP1: Bellman equaion and rsorder condiions We know ha he uiliy funcion can be wrien as U = u (c ) + U +1 : Now assume ha he individual behaves opimally as from +1: Then we can inser he value funcion. The uiliy funcion reads U = u (c ) + V (a +1 ) : Insering his ino he value funcion, we obain he recursive formulaion V (a ) = max c fu (c ) + V (a +1 )g ; (3.4.4) known as he Bellman equaion. Again, his breaks down a manyperiod problem ino a woperiod problem: The objecive of he individual was max fc g (3.1.1) subjec o (3.4.1), as shown by he value funcion in equaion (3.4.3). The Bellman equaion (3.4.4), however, is a wo period decision problem, he radeo beween consumpion oday and more wealh omorrow (under he assumpion ha he funcion V is known). This is wha is known as Bellman s
66 54 Chaper 3. Muliperiod models principle of opimaliy: Whaever he decision oday, subsequen decisions should be made opimally, given he siuaion omorrow. Hisory does no coun, apar from is impac on he sae variable(s). We now derive a rsorder condiion for (3.4.4). I reads d u (c ) + d V (a +1 ) = u 0 (c ) + V 0 (a +1 ) da +1 = 0: dc dc dc Since da +1 =dc = (1 + r ) by he budge consrain (3.4.1), his gives u 0 (c ) (1 + r ) V 0 (a +1 ) = 0: (3.4.5) Again, his equaion makes consumpion a funcion of he sae variable, c = c (a ) : Following he rsorder condiion (3.3.5) in he general example, we wroe c = c (x ) ; i.e. consumpion c changes only when he sae variable x changes. Here, we wrie c = c (a ) ; indicaing ha here can be oher variables which can in uence consumpion oher han wealh a : An example for such an addiional variable in our seup would be he wage rae w or ineres rae r ; which afer all is visible in he rsorder condiion (3.4.5). See ch for a more deailed discussion of sae variables. Economically, (3.4.5) ells us as before in (3.3.5) ha, under opimal behaviour, gains from more consumpion oday are jus balanced by losses from less wealh omorrow. Wealh omorrow falls by 1 + r, his is evaluaed by he shadow price V 0 (a +1 ) and everyhing is discouned by :, DP2: Evoluion of he cosae variable Using he envelope heorem, he derivaive of he maximized Bellman equaion reads V 0 (a ) = V 0 (a +1 : (3.4.6) We compue he parial derivaive of a +1 wih respec o a as he funcional relaionship of c = c (a ) should no (because of he envelope heorem) be aken ino accoun. From he budge consrain we know = 1 + r : Hence, he evoluion of he shadow price/ he cosae variable under opimal behaviour is described by This is he analogon o (3.3.6). DP3: Insering rsorder condiions V 0 (a ) = [1 + r ] V 0 (a +1 ) : Le us now be explici abou how o inser rsorder condiions ino his equaion. We can inser he rsorder condiion (3.4.5) on he righhand side. We can also rewrie he
67 3.4. Examples 55 rsorder condiion (3.4.5), by lagging i by one period, as (1 + r 1 ) V 0 (a ) = u 0 (c 1 ) and can inser his on he lefhand side. This gives u 0 (c 1 ) 1 (1 + r 1 ) 1 = u 0 (c ), u 0 (c ) = [1 + r ] u 0 (c +1 ) : (3.4.7) This is he same resul as he one we obained when we used he Lagrange mehod in equaion (3.1.6). I is also he same resul as for he woperiod saving problem which we found in OLG models  see e.g. (2.2.6) or (2.6.1) in he exercises. This migh be surprising as he planning horizons di er considerably beween a 2 and an in nieperiod decision problem. Apparenly, wheher we plan for wo periods or for many more, he change beween wo periods is always he same when we behave opimally. I should be kep in mind, however, ha consumpion levels (and no changes) do depend on he lengh of he planning horizon. The CES and logarihmic version of he Euler equaion Le us now inser he CES uiliy funcion from (3.4.2) ino (3.4.7). Compuing marginal uiliy gives u 0 (c ) = c and we obain a linear di erence equaion in consumpion, c +1 = ( [1 + r ]) 1= c : (3.4.8) Noe ha he logarihmic uiliy funcion u (c ) = ln c ; known for he woperiod seup from (2.2.1), is a special case of he CES uiliy funcion (3.4.2). Leing approach uniy, we obain c 1 1!1 lim u (c ) = lim!1 1 = ln c where he las sep used L Hôspial s rule: The derivaive of he numeraor wih respec o is Hence, d d c1 1 = d d e(1 ) ln c 1 = e (1 ) ln c ( ln c ) = c 1 ( ln c ) : lim!1 c = lim!1 c 1 ( ln c ) 1 = ln c : (3.4.9) When he logarihmic uiliy funcion is insered ino (3.4.7), one obains an Euler equaion as in (3.4.8) wih se equal o one Wha is a sae variable? Dynamic programming uses he concep of a sae variable. In he general version of ch. 3.3, here is clearly only one sae variable. I is x and is evoluion is described in (3.3.1). In he economic example of ch , he quesion of wha is a sae variable is less obvious.
68 56 Chaper 3. Muliperiod models In a sric formal sense, everyhing is a sae variable. Everyhing means all variables which are no conrol variables are sae variables. This very broad view of sae variables comes from he simple de niion ha everyhing (apar from parameers) which deermines conrol variables is a sae variable. We can undersand his view by looking a he explici soluion for he conrol variables in he woperiod example of ch We reproduce (2.2.3), (2.2.4) and (2.2.5) for ease of reference, W = w + (1 + r +1 ) 1 w +1 ; c +1 = (1 ) (1 + r +1 ) W ; c = W : We did no use he erms conrol and sae variable here bu we could of course solve his woperiod problem by dynamic programming as well. Doing so would allow us o undersand why everyhing is a sae variable. Looking a he soluion for c +1 shows ha i is a funcion of r +1 ; w and w +1 : If we wan o make he saemen ha he conrol variable is a funcion of he sae variables, hen clearly r +1 ; w and w +1 are sae variables. Generalizing his for our muliperiod example from ch , he enire pahs of r and w are sae variables, in addiion o wealh a : As we are in a deerminisic world, we know he evoluion of variables r and w and we can reduce he pah of r and w by he levels of r and w plus he parameers of he process describing heir evoluion. Hence, he broad view for sae variables applied o ch requires us o use r ; w ; a as sae variables. This broad (and ulimaely correc) view of sae variables is he reason why he rsorder condiion (3.4.5) is summarized by c = c (a ) : The index capures all variables which in uence he soluion for c apar from he explici argumen a : In a more pracical sense  as opposed o he sric sense  i is highly recommended o consider only he variable which is indirecly a eced by he conrol variable as (he relevan) sae variable. Wriing he value funcion as V = V (a ; w ; r ) is possible bu highly cumbersome from a noaional poin of view. Wha is more, going hrough he dynamic programming seps does no require more han a as a sae variable as only he shadow price of a is required o obain an Euler equaion and no he shadow price of w or r. To remind us ha more han jus a has an impac on opimal conrols, we should, however, always wrie c = c (a ) as a shorcu for c = c (a ; w ; r ; :::) : The conclusion of all his, however, is more cauious: When encounering a new maximizaion problem and when here is uncerainy abou how o solve i and wha is a sae variable and wha is no, i is always he bes choice o include more raher han less variables as argumens of he value funcion. Dropping some argumens aferwards is simpler han adding addiional ones.
69 3.4. Examples Opimal R&D e or In his second example, a research projec has o be nished a some fuure known poin in ime T. This research projec has a cerain value a poin T and we denoe i by D like disseraion. In order o reach his goal, a pah of a cerain lengh L needs o be compleed. We can hink of L as a cerain number of pages, a cerain number of papers or  probably beer  a cerain qualiy of a xed number of papers. Going hrough his process of walking and wriing is cosly, i requires e or e a each poin in ime : Summing over hese cos  hink of hem as hours worked per day  evenually yields he desired amoun of pages, T =f (e ) = L; (3.4.10) where f(:) is he page of qualiy producion funcion: More e or means more oupu, f 0 (:) > 0; bu any increase in e or implies a lower increase in oupu, f 00 (:) < 0: The objecive funcion of our suden is given by U = T D T = e : (3.4.11) The value of he compleed disseraion is given by D and is presen value is obained by discouning a he discoun facor : Toal uiliy U sems from his presen value minus he presen value of research cos e. The maximizaion problem consiss in maximizing (3.4.11) subjec o he consrain (3.4.10) by choosing an e or pah fe g : The quesion now arises how hese coss are opimally spread over ime. How many hours should be worked per day? An answer can be found by using he Lagrangeapproach wih (3.4.11) as he objecive funcion and (3.4.10) as he consrain. However, her we will use he dynamic programming approach. Before we can apply i, we need o derive a dynamic budge consrain. We herefore de ne M 1 =1f(e ) as he amoun of he pages ha have already been wrien by oday. This hen implies M +1 M = f(e ): (3.4.12) The increase in he number of compleed pages beween oday and omorrow depends on e orinduced oupu f (e ) oday. We can now apply he hree dynamic programming seps. DP1: Bellman equaion and rsorder condiions The value funcion can be de ned by V (M ) max fe g U subjec o he consrain. We follow he approach discussed in ch and explicily use as sae variable M only, he only sae variable relevan for derivaions o come. In oher words, we explicily suppress ime as an argumen of V (:) : The reader can go hrough he derivaions by
70 58 Chaper 3. Muliperiod models using a value funcion speci ed as V (M ; ) and nd ou ha he same resul will obain. The objecive funcion U wrien recursively reads U = T (+1) D 1 T= e = T (+1) D 1 e + T=+1 e = T (+1) D T=+1 (+1) e e = U +1 e : Assuming ha he individual behaves opimally as from omorrow, his reads U = e + V (M +1 ) and he Bellman equaion reads V (M ) = max e f e + V (M +1 )g : (3.4.13) The rsorder condiion is 1 + V 0 (M +1 ) dm +1 de = 0; which, using he dynamic budge consrain, becomes 1 = V 0 (M +1 ) f 0 (e ) : (3.4.14) Again as in (3.3.5), implicily and wih (3.4.12), his equaion de nes a funcional relaionship beween he conrol variable and he sae variable, e = e (M ) : One uni of addiional e or reduces insananeous uiliy by 1 bu increases he presen value of overall uiliy omorrow by V 0 (M +1 ) f 0 (e ) : DP2: Evoluion of he cosae variable To provide some variaion, we will now go hrough he second sep of dynamic programming wihou using he envelope heorem. Consider he maximized Bellman equaion, where we inser e = e (M ) and (3.4.12) ino he Bellman equaion (3.4.13), The derivaive wih respec o M is V (M ) = e (M ) + V (M + f(e (M ))) : V 0 (M ) = e 0 (M ) + V 0 (M + f(e (M ))) d [M + f (e (M ))] dm = e 0 (M ) + V 0 (M +1 ) [1 + f 0 (e (M )) e 0 (M )] : Using he rsorder condiion (3.4.14) simpli es his derivaive o V 0 (M ) = V 0 (M +1 ) : Expressed for + 1 gives V 0 (M +1 ) = V 0 (M +2 ) (3.4.15) DP3: Insering rsorder condiions The nal sep insers he rsorder condiion (3.4.14) wice o replace V 0 (M +1 ) and V 0 (M +2 ) ; 1 (f 0 (e )) 1 = (f 0 (e +1 )) 1, f 0 (e +1 ) = : (3.4.16) f 0 (e ) The inerpreaion of his Euler equaion is now simple. As f 00 (:) < 0 and < 1; e or e increases under opimal behaviour, i.e. e +1 > e : Opimal wriing of a disseraion implies more work every day.
71 3.5. On budge consrains 59 Wha abou levels? The opimaliy condiion in (3.4.16) speci es only how e or e changes over ime, i does no provide informaion on he level of e or required every day. This is a propery of all expressions based on rsorder condiions of ineremporal problems. They only give informaion abou changes of levels, no abou levels hemselves. However, he basic idea for how o obain informaion abou levels can be easily illusraed. Assume f (e ) = e ; wih 0 < < 1: Then (3.4.16) implies (wih being replaced by ) e +1=e 1 1 =, e +1 = 1=(1 ) e : Solving his di erence equaion yields e = ( 1)=(1 ) e 1 ; (3.4.17) where e 1 is he (a his sage sill) unknown iniial e or level. Saring in = 1 on he rs day, insering his soluion ino he ineremporal consrain (3.4.10) yields T =1f ( 1)=(1 ) e 1 = T =1 ( 1)=(1 ) e 1 = L: This gives us he iniial e or level as (he sum can be solved by using he proofs in ch ) 1= L e 1 = : T =1 ( 1)=(1 ) Wih (3.4.17), we have now also compued he level of e or every day. Behind hese simple seps, here is a general principle. Modi ed rsorder condiions resuling from ineremporal problems are di erence equaions, see for example (2.2.6), (3.1.6), (3.4.7) or (3.4.16) (or di erenial equaions when we work in coninuous ime laer). Any di erence (or also di erenial) equaion when solved gives a unique soluion only if an iniial or erminal condiion is provided. Here, we have solved he di erence equaion in (3.4.16) assuming some iniial condiion e 1. The meaningful iniial condiion hen followed from he consrain (3.4.10). Hence, in addiion o he opimaliy rule (3.4.16), we always need some addiional consrain which allows us o compue he level of opimal behaviour. We reurn o his poin when looking a problems in coninuous ime in ch On budge consrains We have encounered wo di eren (bu relaed) ypes of budge consrains so far: dynamic ones and ineremporal ones. Consider he dynamic budge consrain derived in (2.5.13) as an example. Using e p c for simpliciy, i reads a +1 = (1 + r ) a + w e : (3.5.1) This budge consrain is called dynamic as i only akes wha happens beween he wo periods and + 1;ino accoun. In conras, an ineremporal budge consrain akes
72 60 Chaper 3. Muliperiod models wha happens beween any saring period (usually ) and he end of he planning horizon ino accoun. In his sense, he ineremporal budge consrain is more comprehensive and conains more informaion (as we will also see formally below whenever we alk abou he noponzi game condiion). An example for an ineremporal budge consrain was provided in (3.1.3), replicaed here for ease of reference, 1 = (1 + r) ( ) e = a + 1 = (1 + r) ( ) w : (3.5.2) From ineremporal o dynamic We will now ask abou he link beween dynamic and ineremporal budge consrains. Le us choose he simpler link o sar wih, i.e. he link from he ineremporal o he dynamic version. As an example, ake (3.5.2). We will now show ha his ineremporal budge consrain implies which was used before, for example in (3.4.1). Wrie (3.5.2) for he nex period as Express (3.5.2) as a +1 = (1 + r ) (a + w c ) ; (3.5.3) 1 =+1 (1 + r) ( 1) e = a =+1 (1 + r) ( 1) w : (3.5.4) e + 1 =+1 (1 + r) ( ) e = a + w + 1 =+1 (1 + r) ( ) w, e + (1 + r) 1 1 =+1 (1 + r) ( 1) e = a + w + (1 + r) 1 1 =+1 (1 + r) ( 1) w, 1 =+1 (1 + r) ( 1) e = (1 + r) (a + w e ) + 1 =+1 (1 + r) ( 1) w : Inser (3.5.4) and nd he dynamic budge consrain (3.5.3) From dynamic o ineremporal Le us now ask abou he link from he dynamic o he ineremporal budge consrain. How can we obain he ineremporal version of he budge consrain (3.5.1)? Technically speaking, his simply requires us o solve a di erence equaion: In order o solve (3.5.1) recursively, we rewrie i as w a = a +1 + e ; a +i = a +i+1 + e +i w +i : 1 + r 1 + r +i
73 3.5. On budge consrains 61 Insering su cienly ofen yields a = = = a +2 +e +1 w +1 1+r +1 + e w = a +2 + e +1 w r (1 + r +1 ) (1 + r ) + e w 1 + r a +3 +e +2 w +2 1+r +2 + e +1 w +1 (1 + r +1 ) (1 + r ) + e w 1 + r a +3 + e +2 w +2 (1 + r +2 ) (1 + r +1 ) (1 + r ) + e +1 w +1 (1 + r +1 ) (1 + r ) + e w 1 + r a +i = ::: = lim i!1 (1 + r +i 1 ) (1 + r +1 ) (1 + r ) + 1 i=0 (1 + r +i ) (1 + r +1 ) (1 + r ) : The expression in he las line is hopefully insrucive bu somewha cumbersome. We can wrie i in a more concise way as a = lim a +i i!1 i 1 e +i w +i e +i s=0 (1 + r +s ) + 1 i=0 i s=0 (1 + r +s ) where indicaes a produc, i.e. i s=0 (1 + r +i ) = 1 + r for i = 0 and i s=0 (1 + r +i ) = (1 + r +i ) (1 + r ) for i > 0: For i = 1; i s=0 (1 + r +i ) = 1 by de niion. Leing he limi be zero, a sep explained in a second, we obain e +i w +i a = 1 i=0 i s=0 (1 + r +s ) = 1 = e w s=0 (1 + r +s ) 1 = w +i e w R where he las bu one equaliy is subsiued + i by : We can wrie his as 1 e = Wih a consan ineres rae, his reads w = a + 1 = : (3.5.5) R R 1 = (1 + r) ( +1) e = a + 1 = (1 + r) ( +1) w : (3.5.6) Equaion (3.5.5) is he ineremporal budge consrain ha resuls from a dynamic budge consrain as speci ed in (3.5.1) using he addiional condiion ha a lim +i i!1 = 0. i 1 s=0 (1+r +s) Noe ha he assumpion ha his limi is zero has a sandard economic inerpreaion. I is usually called he noponzi game condiion. To undersand he inerpreaion more easily, jus focus on he case of a consan ineres rae. The condiion hen reads lim i!1 a +i = (1 + r) i = 0: The erm a +i = (1 + r) i is he presen value in of wealh a +i held in + i: The condiion says ha his presen value mus be zero. Imagine an individual ha nances expendiure e by increasing deb, i.e. by leing a +i becoming more and more negaive. This condiion simply says ha an individual s longrun deb level, i.e. a +i for i going o in niy mus no increase oo quickly  he presen value mus be zero. Similarly, he condiion also requires ha an individual should no hold posiive wealh in he long run whose presen value is no zero. Noe ha his condiion is ful lled, for example, for any consan deb or wealh level.
74 62 Chaper 3. Muliperiod models Two versions of dynamic budge consrains Noe ha we have also encounered wo subspecies of dynamic budge consrains. The one from (3.5.1) and he one from (3.5.3). The di erence beween hese wo consrains is due o more basic assumpions abou he iming of evens as was illusraed in g.s and These wo dynamic consrains imply wo di eren versions of ineremporal budge consrains. The version from (3.5.1) leads o (3.5.6) and he one from (3.5.3) leads (wih a similar noponzi game condiion) o (3.5.2). Comparing (3.5.6) wih (3.5.2) shows ha he presen values on boh sides of (3.5.6) discouns one ime more han in (3.5.2). The economic di erence again lies in he iming, i.e. wheher we look a values a he beginning or end of a period. The budge consrain (3.5.1) is he naural budge consrain in he sense ha i can be derived easily as above in ch and in he sense ha i easily aggregaes o economy wide resource consrains. We will herefore work wih (3.5.1) and he corresponding ineremporal version (3.5.6) in wha follows. The reason for no working wih hem righ from he beginning is ha he ineremporal version (3.5.2) has some inuiive appeal and ha is dynamic version (3.5.3) is widely used in he lieraure. 3.6 A decenralized general equilibrium analysis We have so far analyzed maximizaion problems of households in parial equilibrium. In woperiod models, we have analyzed how households can be aggregaed and wha we learn abou he evoluion of he economy as a whole. We will now do he same for in nie horizon problems. As we did in ch. 2.4, we will rs specify echnologies. This shows wha is echnologically feasible in his economy. Which goods are produced, which goods can be sored for saving purposes, is here uncerainy in he economy semming from producion processes? Given hese echnologies, rms maximize pro s. Second, household preferences are presened and he budge consrain of households is derived from he echnologies presened before. This is he reason why echnologies should be presened before households are inroduced: budge consrains are endogenous and depend on knowledge of wha households can do. Opimaliy condiions for households are hen derived. Finally, aggregaion over households and an analysis of properies of he model using he reduced form follows Technologies The echnology is a simple CobbDouglas echnology Y = AK L 1 : (3.6.1) Capial K and labour L is used wih a given oal facor produciviy level A o produce oupu Y : This good can be used for consumpion and invesmen and equilibrium on he
75 3.6. A decenralized general equilibrium analysis 63 goods marke requires Y = C + I : (3.6.2) Gross invesmen I is urned ino ne invesmen by aking depreciaion ino accoun, K +1 = (1 ) K + I : Taking hese wo equaions ogeher gives he resource consrain of he economy, K +1 = (1 ) K + Y C : (3.6.3) As his consrain is simply a consequence of echnologies and marke clearing, i is idenical o he one used in he OLG seup in (2.4.9) Firms Firms maximize pro s by employing opimal quaniies of labour and capial, given he echnology in (3.6.1). Firsorder = w = wl (3.6.4) as in (2.4.2), where we have again chosen he consumpion good as numeraire Households Preferences of households are described as in he ineremporal uiliy funcion (3.1.1). As he only way households can ransfer savings from one period o he nex is by buying invesmen goods, an individual s wealh is given by he number of machines k, she owns. Clearly, adding up all individual wealh socks gives he oal capial sock, L k = K : Wealh k increases over ime if he household spends less on consumpion han wha i earns hrough capial plus labour income, correced for he loss in wealh each period caused by depreciaion, k +1 k = w K k k + w L c, k +1 = 1 + w K k + w L c : If we now de ne he ineres rae o be given by we obain our budge consrain r w K ; (3.6.5) k +1 = (1 + r ) k + w L c : (3.6.6) Noe ha he derivaion of his budge consrain was simpli ed in comparison o ch as he price v of an asse is, as we measure i in unis of he consumpion good which is raded on he same nal marke (3.6.2), given by 1. More general budge consrains will become prey complex as soon as he price of he asse is no normalized.
76 64 Chaper 3. Muliperiod models This complexiy is needed when i comes e.g. o capial asse pricing  see furher below in ch Here, however, his simple consrain is perfec for our purposes. Given he preferences and he consrain, he Euler equaion for his maximizaion problem is given by (see exercise 5) u 0 (c ) = [1 + r +1 ] u 0 (c +1 ) : (3.6.7) Srucurally, his is he same expression as in (3.4.7). The ineres rae, however, refers o + 1, due o he change in he budge consrain. Remembering ha = 1= (1 + ), his shows ha consumpion increases as long as r +1 > Aggregaion and reduced form Aggregaion To see ha individual consrains add up o he aggregae resource consrain, we simply need o ake ino accoun ha individual income adds up o oupu, w K K +w L L = Y. Remember ha we are familiar wih he laer from (2.4.4). Now sar from (3.6.6) and use (3.6.5) o obain, K +1 = L k +1 = 1 + w K L k + w L L C = (1 ) K + Y C : The opimal behaviour of all households aken ogeher can be gained from (3.6.7) by summing over all households. This is done analyically correcly by rs applying he inverse funcion of u 0 o his equaion and hen summing individual consumpion levels over all households (see exercise 6 for deails). Applying he inverse funcion again gives where C is aggregae consumpion in : Reduced form u 0 (C ) = [1 + r +1 ] u 0 (C +1 ) ; (3.6.8) We now need o undersand how our economy evolves in general equilibrium. Our rs equaion is (3.6.8), elling us how consumpion evolves over ime. This equaion conains consumpion and he ineres rae as endogenous variables. Our second equaion is herefore he de niion of he ineres rae in (3.6.5) which we combine wih he rsorder condiion of he rm in (3.6.4) o yield : (3.6.9) This equaion conains he ineres rae and he capial sock as endogenous variables.
77 3.6. A decenralized general equilibrium analysis 65 Our nal equaion is he resource consrain (3.6.3), which provides a link beween capial and consumpion. Hence, (3.6.8), (3.6.9) and (3.6.3) give a sysem in hree equaions and hree unknowns. When we inser he ineres rae ino he opimaliy condiion for consumpion, we obain as our reduced form h i u 0 (C ) = u 0 (C +1 ) ; (3.6.10) K +1 = (1 ) K + Y C : This is a wodimensional sysem of nonlinear di erence equaions which gives a unique soluion for he ime pah of capial and consumpion, provided we have wo iniial condiions K 0 and C Seady sae and ransiional dynamics When rying o undersand a sysem like (3.6.10), he same principles can be followed as wih onedimensional di erence equaions. Firs, one ries o idenify a xed poin, i.e. a seady sae, and hen one looks a ransiional dynamics. Seady sae In a seady sae, all variables are consan. Seing K +1 = K = K and C +1 = C = C; we obain 1 = @Y = + ; C = Y where he, sep used he link beween and from (3.1.2). In he seady sae, he marginal produciviy of capial is given by he ime preference rae plus he depreciaion rae. Consumpion equals oupu minus depreciaion, i.e. minus replacemen invesmen. These wo equaions deermine wo variables K and C: he rs deermines K; he second deermines C: Transiional dynamics Undersanding ransiional dynamics is no as sraighforward as undersanding he seady sae. Is analysis follows he same idea as in coninuous ime, however, and we will analyze ransiional dynamics in deail here. Having said his, we should acknowledge he fac ha ransiional dynamics in discree ime can quickly become more complex han in coninuous ime. As an example, chaoic behaviour can occur in onedimensional di erence equaions while one needs a leas a hreedimensional di erenial equaion sysem o obain chaoic properies in coninuous ime. The lieraure on chaos heory and exbooks on di erence equaions provide many examples.
78 66 Chaper 3. Muliperiod models 3.7 A cenral planner Opimal facor allocaion One of he mos solved maximizaion problems in Economics is he cenral planner problem. The choice by a cenral planner given a social welfare funcion and echnological consrains provides informaion abou he rsbes facor allocaion. This is a benchmark for many analyses in normaive economics. We consider he probably mos simple case of opimal facor allocaion in a dynamic seup. The maximizaion problem Le preferences be given by U = 1 = u (C ) ; (3.7.1) where C is he aggregae consumpion of all households a a poin in ime : This funcion is maximized subjec o a resource accumulaion consrain which reads K +1 = Y (K ; L ) + (1 ) K C (3.7.2) for all : The producion echnology is given by a neoclassical producion funcion Y (K ; L ) wih sandard properies. The Lagrangian This seup di ers from he ones we go o know before in ha here is an in nie number of consrains in (3.7.2). This consrain holds for each poin in ime : As a consequence, he Lagrangian is formulaed wih in niely many Lagrangian mulipliers, L = 1 = u (C ) + 1 = f [K +1 Y (K ; L ) (1 ) K + C ]g : (3.7.3) The rs par of he Lagrangian is sandard, 1 = u (C ), i jus copies he objecive funcion. The second par consiss of a sum from o in niy, one consrain for each poin in ime, each muliplied by is own Lagrange muliplier. In order o make he maximizaion procedure clearer, we rewrie he Lagrangian as L = 1 = u (C ) + s 2 = [K +1 Y (K ; L ) (1 ) K + C ] + s 1 [K s Y (K s 1 ; L s 1 ) (1 ) K s 1 + C s 1 ] + s [K s+1 Y (K s ; L s ) (1 ) K s + C s ] + 1 =s+1 [K +1 Y (K ; L ) (1 ) K + C ] ; where we simply explicily wrie ou he sum for s 1 and s: Now maximize he Lagrangian boh wih respec o he conrol variable C s ; he mulipliers s and he sae variables K s. Maximizaion wih respec o K s migh appear
79 3.7. A cenral planner 67 unusual a his sage; we will see a jusi caion for his in he nex chaper. Firsorder condiions are L Cs = s u 0 (C s ) + s = 0 () s = u 0 (C s ) s ; L Ks = s 1 s + 1 = 0 () s 1 = 1 ; s s L s = 0: (3.7.6) u Combining he rs and second rsorder condiion gives 0 (C s 1 ) s 1 u 0 (C s) = 1 s : This is equivalen o u 0 (C s ) u 0 (C s+1 ) = 1 1 : s+1 This expression has he same inerpreaion as (3.1.6) or (3.4.7) for example. When we replace s by ; his equaion wih he consrain (3.7.2) is a wodimensional di erence equaion sysem which allows us o deermine he pahs of capial and consumpion, given wo boundary condiions, which he economy will follow when facor allocaion is opimally chosen. The seady sae of such an economy is found by seing C s = C s+1 and K s = K s+1 in (3.7.7) and (3.7.2). This example also allows us o reurn o he discussion abou he link beween he sign of shadow prices and he Lagrange muliplier a he end of ch Here, he consrains in he Lagrangian are represened as lefhand side minus righhand side. As a consequence, he Lagrange mulipliers are negaive, as he rsorder condiions (3.7.4) show. Apar from he fac ha he Lagrange muliplier here now sands for minus he shadow price, his does no play any role for he nal descripion of opimaliy in (3.7.7) Where he Lagrangian comes from II Le us now see how we can derive he same expression as ha in (3.7.7) wihou using he Lagrangian. This will allow us o give an inuiive explanaion for why we maximized he Lagrangian in he las chaper wih respec o boh he conrol and he sae variable. Maximizaion wihou Lagrange Inser he consrain (3.7.2) ino he objecive funcion (3.7.1) and nd U = 1 = u (Y (K ; L ) + (1 ) K K +1 )! max fk g This is now maximized by choosing a pah fk g for capial. Choosing he sae variable implicily pins down he pah fc g of he conrol variable consumpion and one can herefore hink of his maximizaion problem as one where consumpion is opimally chosen.
80 68 Chaper 3. Muliperiod models Now rewrie he objecive funcion as Choosing K s opimally implies U = s 2 = u (Y (K ; L ) + (1 ) K K +1 ) + s 1 u (Y (K s 1 ; L s 1 ) + (1 ) K s 1 K s ) + s u (Y (K s ; L s ) + (1 ) K s K s+1 ) + 1 =s+1 u (Y (K ; L ) + (1 ) K K +1 ) : s 1 u 0 (Y (K s 1 ; L s 1 ) + (1 ) K s 1 K s + s u 0 (Ks ; L s ) (Y (K s ; L s ) + (1 ) K s K s+1 ) + 1 = s Reinsering he consrain (3.7.2) and rearranging gives u 0 (C s 1 ) = u 0 (C s ) 1 (K s; L s s : This is he sandard opimaliy condiion for consumpion which we obained in (3.7.7). As s can sand for any poin in ime beween and in niy, we could replace s by, or + 1: Back o he Lagrangian When we now go back o he maximizaion procedure where he Lagrangian was used, we see ha he parial derivaive of he Lagrangian wih respec o K in (3.7.5) capures how changes over ime. The simple reason why he Lagrangian is maximized wih respec o K is herefore ha an addiional rsorder condiion is needed as needs o be deermined as well. In saic maximizaion problems wih wo consumpion goods and one consrain, he Lagrangian is maximized by choosing consumpion levels for boh consumpion goods and by choosing he Lagrange muliplier. In he Lagrange seup above in (3.7.4) o (3.7.6), we choose boh endogenous variables K and C plus he muliplier and hereby deermine opimal pahs for all hree variables. Hence, i is a echnical  mahemaical  reason ha K is chosen : deermining hree unknowns simply requires hree rsorder condiions. Economically, however, he conrol variable C is economically chosen while he sae variable K adjuss indirecly as a consequence of he choice of C : 3.8 Growh of family size The seup Le us now consider an exension o he models considered so far. Le us imagine here is a family consising of n members a poin in ime and le consumpion c per individual
81 3.8. Growh of family size 69 family member be opimally chosen by he head of he family. The objecive funcion for his family head consiss of insananeous uiliy u (:) per family member imes he number of members, discouned a he usual discoun facor ; U = 1 = u (c ) n : Le us denoe family wealh by ^a. I is he produc of individual wealh imes he number of family members, ^a n a. The budge consrain of he household is hen given by ^a +1 = (1 + r ) ^a + n w n c Toal labour income is given by n imes he wage w and family consumpion is n c : Solving by subsiuion We solve his maximizaion problem by subsiuion. We rewrie he objecive funcion and inser he consrain wice, s 1 = u (c ) n + s u (c s ) n s + s+1 u (c s+1 ) n s =s+2 u (c ) n (1 + = = s 1 u (c ) n + s rs ) ^a s + n s w s ^a s+1 u n s n s (1 + + s+1 rs+1 ) ^a s+1 + n s+1 w s+1 ^a s+2 u n s =s+2 u (c ) n : n s+1 Now compue he derivaive wih respec o ^a s+1. This gives (1 + u 0 rs ) ^a s + n s w s ^a s+1 ns (1 + = u 0 rs+1 ) ^a s+1 + n s+1 w s+1 ^a s rs+1 n s+1 : n s n s n s+1 n s+1 When we replace he budge consrain by consumpion again and cancel he n s and n s+1, we obain u 0 (c s ) = [1 + r s+1 ] u 0 (c s+1 ) : (3.8.1) The ineresing feaure of his rule is ha being par of a family whose size n can change over ime does no a ec he growh of individual consumpion c s : I follows he same rule as if individuals maximized uiliy independenly of each oher and wih heir personal budge consrains Solving by he Lagrangian The Lagrangian for his seup wih one budge consrain for each poin in ime requires an in nie number of Lagrange mulipliers ; one for each : I reads L = 1 = u (c ) n + [(1 + r ) ^a + n l w n c ^a +1 ] :
82 70 Chaper 3. Muliperiod models We rs compue he rsorder condiions for consumpion and hours worked for one poin in ime s; L cs s u (c s ) s = 0; As discussed in 3.7, we also need o compue he derivaive wih respec o he sae variable. I is imporan o compue he derivaive wih respec o family wealh ^a as his is he rue sae variable of he head of he family. (Compuing he derivaive wih respec o individual wealh a would also work bu would lead o an incorrec resul, i.e. a resul ha di ers from ) This derivaive is L^as = s 1 + s (1 + r s ) = 0, s 1 = (1 + r s ) s : Opimal consumpion hen follows by replacing he Lagrange mulipliers, u 0 (c s 1 ) = [1 + r s ] u 0 (c s ) : This is idenical o he resul we obain by insering in (3.8.1). 3.9 Furher reading and exercises For a much more deailed background on he elasiciy of subsiuion, see Blackorby and Russell (1989). They sudy he case of more han wo inpus and sress ha he Morishima elasiciy is o be preferred o he Allan/ Uzawa elasiciy. The dynamic programming approach was developed by Bellman (1957). Maximizaion using he Lagrange mehod is widely applied by Chow (1997). The example in ch was originally inspired by Grossman and Shapiro (1986).
83 3.9. Furher reading and exercises 71 Exercises chaper 3 Applied Ineremporal Opimizaion Dynamic programming in discree deerminisic ime 1. The envelope heorem I Le he uiliy funcion of an individual be given by U = U (C; L) ; where consumpion C increases uiliy and supply of labour L decreases uiliy. Le he budge consrain of he individual be given by wl = C: Le he individual maximize uiliy wih respec o consumpion and he amoun of labour supplied. (a) Wha is he opimal labour supply funcion (in implici form)? How much does an individual consume? Wha is he indirec uiliy funcion? (b) Under wha condiions does an individual increase labour supply when wages rise (no analyical soluion required)? (c) Assume higher wages lead o increased labour supply. Does disuiliy arising from increased labour supply compensae uiliy from higher consumpion? Does uiliy rise if here is no disuiliy from working? Sar from he indirec uiliy funcion derived in a) and apply he proof of he envelope heorem and he envelope heorem iself. 2. The envelope heorem II (a) Compue he derivaive of he Bellman equaion (3.4.6) wihou using he envelope heorem. Hin: Compue he derivaive wih respec o he sae variable and hen inser he rsorder condiion. (b) Do he same wih (3.4.15) 3. The addiively separable objecive funcion (a) Show ha he objecive funcion can be wrien as in (3.3.3).
84 72 Chaper 3. Muliperiod models (b) Find ou wheher (3.3.3) implies he objecive funcion. (I does no.) 4. Ineremporal and dynamic budge consrains (a) Show ha he ineremporal budge consrain T = 1 1 k= e = a + T = 1 + r k implies he dynamic budge consrain 1 k= r k i (3.9.1) a +1 = (a + i e ) (1 + r ) : (3.9.2) (b) Under which condiions does he dynamic budge consrain imply he ineremporal budge consrain? (c) Now consider a +1 = (1 + r ) a +w does i imply? e : Wha ineremporal budge consrain 5. The sandard saving problem Consider he objecive funcion from (3.1.1), U = 1 = u (c ), and maximize i by choosing a consumpion pah fc g subjec o he consrain (3.6.6), k +1 = (1 + r ) k + w L c : The resul is given by (3.6.7). (a) Solve his problem by dynamic programming mehods. (b) Solve his by using he Lagrange approach. Choose a muliplier for an in nie sequence of consrains. 6. Aggregaion of opimal consumpion rules Consider he opimaliy condiion u 0 (c ) = (1 + r +1 ) u 0 (c +1 ) in (3.6.7) and derive he aggregae version (3.6.8). Find he assumpions required for he uiliy funcion for hese seps o be possible. 7. A benevolen cenral planner You are he omniscien omnipoen benevolen cenral planner of an economy. You wan o maximize a social welfare funcion U = 1 = u (C ) for your economy by choosing a pah of aggregae consumpion levels fc g subjec o a resource consrain K +1 K = Y (K ; L ) K C (3.9.3) (a) Solve his problem by dynamic programming mehods.
85 3.9. Furher reading and exercises 73 (b) Discuss how he cenral planner resul is relaed o he decenralized resul from exercise 5. (c) Wha does he resul look like for a uiliy funcion which is logarihmic and for one which has consan elasiciy of subsiuion, u (C ()) = ln C () and u (C ()) = C ()1 1? (3.9.4) 1 8. Environmenal economics Imagine you are an economis only ineresed in maximizing he presen value of your endowmen. You own a renewable resource, for example a piece of fores. The amoun of wood in your fores a a poin in ime is given by x. Trees grow a b(x ) and you harves a he quaniy c. (a) Wha is he law of moion for x? (b) Wha is your objecive funcion if prices a per uni of wood is given by p, your horizon is in niy and you have perfec informaion? (c) How much should you harves per period when he ineres rae is consan? Does his change when he ineres rae is imevariable? 9. The 10k run  Formulaing and solving a maximizaion problem You consider paricipaion in a 10k run or a marahon. The even will ake place in M monhs. You know ha your ness needs o be improved and ha his will be cosly: i requires e or a 0 which reduces uiliy u (:) : A he same ime, you enjoy being fas, i.e. uiliy increases he shorer your nish ime l: The higher your e or, he shorer your nish ime. (a) Formulae a maximizaion problem wih 2 periods. E or a ecs he nish ime in M monhs. Specify a uiliy funcion and discuss a reasonable funcional form which capures he link beween nish ime l and e or a 0 : (b) Solve his maximizaion problem by providing and discussing he rsorder condiion. 10. A cenral planner Consider he objecive funcion of a cenral planner, The consrain is given by U 0 = 1 =0 u (C ) : (3.9.5) K +1 = Y (K ; L ) + (1 ) K C : (3.9.6) (a) Explain in words he meaning of he objecive funcion and of he consrain.
86 74 Chaper 3. Muliperiod models (b) Solve he maximizaion problem by rs insering (3.9.6) ino (3.9.5) and hen u by opimally choosing K. Show ha he resul is 0 (C ) = (K +1;L +1 ) u 0 (C and discuss his in words. (c) Discuss why using he Lagrangian also requires maximizing wih respec o K even hough K is a sae variable.
87 Par II Deerminisic models in coninuous ime 75
88
89 Par II covers coninuous ime models under cerainy. Chaper 4 rs looks a differenial equaions as hey are he basis of he descripion and soluion of maximizaion problems in coninuous ime. Firs, some useful de niions and heorems are provided. Second, di erenial equaions and di erenial equaion sysems are analyzed qualiaively by he socalled phasediagram analysis. This simple mehod is exremely useful for undersanding di erenial equaions per se and also for laer purposes for undersanding qualiaive properies of soluions o maximizaion problems and properies of whole economies. Linear di erenial equaions and heir economic applicaions are hen nally analyzed before some words are spen on linear di erenial equaion sysems. Chaper 5 presens a new mehod for solving maximizaion problems  he Hamilonian. As we are now in coninuous ime, woperiod models do no exis. A disincion will be drawn, however, beween nie and in nie horizon models. In pracice, his disincion is no very imporan as, as we will see, opimaliy condiions are very similar for nie and in nie maximizaion problems. Afer an inroducory example on maximizaion in coninuous ime by using he Hamilonian, he simple link beween Hamilonians and he Lagrangian is shown. The soluion o maximizaion problems in coninuous ime will consis of one or several di erenial equaions. As a unique soluion o di erenial equaions requires boundary condiions, we will show how boundary condiions are relaed o he ype of maximizaion problem analyzed. The boundary condiions di er signi canly beween nie and in nie horizon models. For he nie horizon models, here are iniial or erminal condiions. For he in nie horizon models, we will ge o know he ransversaliy condiion and oher relaed condiions like he NoPonzigame condiion. Many examples and a comparison beween he presenvalue and he currenvalue Hamilonian conclude his chaper. Chaper 6 solves he same kind of problems as chaper 5, bu i uses he mehod of dynamic programming. The reason for doing his is o simplify undersanding of dynamic programming in sochasic seups in Par IV. Various aspecs speci c o he use of dynamic programming in coninuous ime, e.g. he srucure of he Bellman equaion, can already be reaed here under cerainy. This chaper will also provide a comparison beween he Hamilonian and dynamic programming and look a a maximizaion problem wih wo sae variables. An example from moneary economics on real and nominal ineres raes concludes he chaper. 77
90 78
91 Chaper 4 Di erenial equaions There are many excellen exbooks on di erenial equaions. This chaper will herefore be relaively shor. Is objecive is more o recap basic conceps augh in oher courses and o serve as a background for laer applicaions. 4.1 Some de niions and heorems De niions The following de niions are sandard and follow Brock and Malliaris (1989). De niion An ordinary di erenial equaion sysem (ODE sysem) is of he ype dx () d _x () = f (; x ()) ; (4.1.1) where lies beween some saring poin and in niy, 2 [ 0 ; 1[ ; x can be a vecor, x 2 R n and f maps from R n+1 ino R n : When x is no a vecor, i.e. for n = 1; (4.1.1) obviously is an ordinary di erenial equaion. An auonomous di erenial equaion is an ordinary di erenial equaion where f(:) is independen of ime ; dx () _x () = f (x ()) : (4.1.2) d The di erence beween a di erenial equaion and a normal algebraic equaion obviously lies in he fac ha di erenial equaions conain derivaives of variables like _x (). An example of a di erenial equaion which is no an ODE is he parial di erenial equaion. A linear example is a (x; (x; + b (x; (x; = c (x; ) ; where a (:) ; b (:) ; c (:) and p (:) are funcions wih nice properies. While in an ODE, here is one derivaive (ofen wih respec o ime), a parial di erenial equaion conains 79
92 80 Chaper 4. Di erenial equaions derivaives wih respec o several variables. Parial di erenial equaions can describe e.g. a densiy and how i changes over ime. Oher ypes of di erenial equaions include sochasic di erenial equaions (see ch. 9), implici di erenial equaions (which are of he ype g ( _x ()) = f (; x ())), delay di erenial equaions ( _x () = f (x ( ))) and many oher more. De niion An iniial value problem is described by where x 0 is he iniial condiion. A erminal value problem is of he form where x T is he erminal condiion. _x = f(; x); x ( 0 ) = x 0; 2 [ 0 ; T ] ; _x = f (; x) ; x (T ) = x T ; 2 [ 0 ; T ] ; Two heorems Theorem Exisence (Brock and Malliaris, 1989) If f (; x) is a coninuous funcion on recangle L = f(; x)j j 0 j a; jx x 0 j bg hen here exiss a coninuous di ereniable soluion x () on inerval j 0 j a ha solves iniial value problem _x = f (; x) ; x ( 0 ) = x 0 : (4.1.3) This heorem only proves ha a soluion exiss. I is sill possible ha here are many soluions. Theorem Uniqueness If f and are coninuous funcions on L; he iniial value problem (4.1.3) has a unique soluion for 2 0 ; 0 + min a; b max jf (; x)j If his condiion is me, an ODE wih an iniial or erminal condiion has a unique soluion. More generally speaking, a di erenial equaion sysem consising of n ODEs ha saisfy hese condiions (which are me in he economic problems we encouner here) has a unique soluion provided ha here are n boundary condiions. Knowing abou a unique soluion is useful as one knows ha changes in parameers imply unambiguous predicions abou changes in endogenous variables. If he governmen changes some ax, we can unambiguously predic wheher employmen goes up or down.
93 4.2. Analyzing ODEs hrough phase diagrams Analyzing ODEs hrough phase diagrams This secion will presen ools ha allow us o deermine properies of soluions of differenial equaions and di erenial equaion sysems. The analysis will be qualiaive in his chaper as mos economic sysems are oo nonlinear o allow for an explici general analyic soluion. Explici soluions for linear di erenial equaions will be reaed in ch Onedimensional sysems We sar wih a onedimensional di erenial equaion _x () = f (x ()), where x 2 R and > 0: This will also allow us o review he conceps of xpoins, local and global sabiliy and insabiliy as used already when analyzing di erence equaions in ch Unique xpoin Le f (x) be represened by he graph in he following gure, wih x () being shown on he horizonal axis. As f (x) gives he change of x () over ime, _x () is ploed on he verical axis. N _x() N N x N N x() N Figure Qualiaive analysis of a di erenial equaion As in he analysis of di erence equaions in ch , we rs look for he xpoin of he underlying di erenial equaion. A xpoin is de ned similarly in spiri bu  hinking now in coninuous ime  di erenly in deail.
94 82 Chaper 4. Di erenial equaions De niion A xpoin x is a poin where x () does no change. In coninuous ime, his means _x () = 0 which, from he de niion (4.1.2) of he di erenial equaion, requires f (x ) = 0: The requiremen ha x () does no change is he similariy in spiri o he de niion in discree ime. The requiremen ha f (x ) = 0 is he di erence in deail: in discree ime as in (2.5.8) we required f (x ) = x. Looking a he above graph of f (x) ; we nd x a he poin where f (x) crosses he horizonal line. We hen inquire he sabiliy of he xpoin. When x is o he lef of x ; f (x) > 0 and herefore x increases, _x () > 0: This increase of x is represened in his gure by he arrows on he horizonal axis. Similarly, for x > x ; f (x) < 0 and x () decreases. We have herefore found ha he xpoin x is globally sable and have also obained a feeling for he behaviour of x () ; given some iniial condiions. We can now qualiaively plo he soluions wih ime on he horizonal axis. As he discussion has jus shown, he soluion x () depends on he iniial value from which we sar, i.e. on x (0) : For x (0) > x ; x () decreases, for x (0) < x ; x () increases: any changes over ime are monoonic. There is one soluion for each iniial condiion. The following gure shows hree soluions of _x () = f (x ()), given hree di eren iniial condiions. Figure Qualiaive soluions of an ODE for hree di eren iniial condiions Muliple xpoins and equilibria Of course, more sophisicaed funcions han f (x) can be imagined. Now consider a di erenial equaion _x () = g (x ()) where g (x) is nonmonoonic as ploed in he nex gure.
95 4.2. Analyzing ODEs hrough phase diagrams 83 Figure Muliple equilibria As his gure shows, here are four xpoins. Looking a wheher g (x) is posiive or negaive, we know wheher x () increases or decreases over ime. This allows us o plo arrows on he horizonal axis as in he previous example. The di erence o before consiss of he fac ha some xpoins are unsable and some are sable. De niion showed us ha he concep of a xpoin in coninuous ime is slighly di eren from discree ime. However, he de niions of sabiliy as hey were inroduced in discree ime can be direcly applied here as well. Looking a x 1; any small deviaion of x from x 1 implies an increase or decrease of x: The xpoin x 1 is herefore unsable, given he de niion in ch Any small deviaion x 2; however, implies ha x moves back o x 2: Hence, x 2 is (locally) sable. The xpoin x 3 is also unsable, while x 4 is again locally sable: While x converges o x 4 for any x > x 4 (in his sense x 4 could be called globally sable from he righ), x converges o x 4 from he lef only if x is no smaller han or equal o x 3: Figure Qualiaive soluions of an ODE for di eren iniial condiions II
96 84 Chaper 4. Di erenial equaions If an economy can be represened by such a di erenial equaion _x () = g (x ()), one would call xpoins longrun equilibria. There are sable equilibria and unsable equilibria and i depends on he underlying sysem (he assumpions ha implied he di erenial equaion _x = g (x)) which equilibrium would be considered o be he economically relevan one. As in he sysem wih one xpoin, we can qualiaively plo soluions of x () over ime, given di eren iniial values for x (0). This is shown in g which again highlighs he sabiliy properies of xpoins x 1 o x 4: Twodimensional sysems I  An example We now exend our qualiaive analysis of di erenial equaions o wodimensional sysems. This laer case allows for an analysis of more complex sysems han simple onedimensional di erenial equaions. In almos all economic models wih opimal saving decisions, a reduced form consising of a leas wo di erenial equaions will resul. We sar here wih an example before we analyse wodimensional sysems more generally in he nex chaper. The sysem Consider he following di erenial equaion sysem, _x 1 = x 1 x 2 ; _x 2 = b + x 1 1 x 2 ; 0 < < 1 < b: Assume ha for economic reasons we are ineresed in properies for x i > 0: Fixpoin The rs quesion is as always wheher here is a xpoin a all. In a wodimensional sysem, a xpoin x = (x 1; x 2) is wodimensional as well. The xpoin is de ned such ha boh variables do no change over ime, i.e. _x 1 = _x 2 = 0: If such a poin exiss, i mus saisfy _x 1 = _x 2 = 0, (x 1) = x 2; x 2 = b + (x 1) 1 : By insering he second equaion ino he rs, x 1 is deermined by (x 1) = b+(x 1) 1 and x 2 follows from x 2 = (x 1) : Analyzing he properies of he equaion (x 1) = b + (x 1) 1 would hen show ha x 1 is unique: The lefhand side increases monoonically from 0 o in niy for x 1 2 [0; 1[ while he righhand side decreases monoonically from in niy o b: Hence, here mus be an inersecion poin and here can be only one as funcions are monoonic. As x 1 is unique, so is x 2 = (x 1).
97 4.2. Analyzing ODEs hrough phase diagrams 85 Zeromoion lines and pairs of arrows Having derived he xpoin, we now need o undersand he behaviour of he sysem more generally. Wha happens o x 1 and x 2 when (x 1 ; x 2 ) 6= (x 1; x 2)? To answer his quesion, he concep of zeromoion lines is very useful. A zeromoion line is a line for a variable x i which marks he poins for which he variable x i does no change, i.e. _x i = 0: For our wodimensional di erenial equaion sysem, we obain wo zeromoion lines, _x 1 0, x 2 x 1 ; _x 2 0, x 2 b + x 1 1 : (4.2.1) In addiion o he equaliy sign, we also analyse here a he same ime for which values x i rises. Why his is useful will soon become clear. We can now plo he curves where _x i = 0 in a diagram. In conras o he onedimensional graphs in he previous chaper, we now have he variables x 1 and x 2 on he axes (and no he change of one variable on he verical axis). The inersecion poin of he wo zeromoion lines gives he x poin x = (x 1; x 2) which we derived analyically above. Figure Firs seps owards a phase diagram In addiion o showing where variables do no change, he zeromoion lines also delimi regions where variables do change. Looking a (4.2.1) again shows (why we used he and no he = sign and) ha he variable x 1 increases whenever x 2 < x 1. Similarly, he variable x 2 increases, whenever x 2 < b + x1 1 : The direcions in which variables change can hen be ploed ino his diagram by using arrows. In his diagram, here is a pair of arrows per region as wo direcions (one for x 1 ; one for x 2 ) need o be indicaed. This is in principle idenical o he arrows we used in he analysis of he onedimensional sysems. If he sysem nds iself in one of hese four regions, we know qualiaively, how variables change over ime: Variables move o he souheas in region I, o he norheas in region II, o he norhwes in region III and o he souhwes in region IV.
98 86 Chaper 4. Di erenial equaions Trajecories Given he zeromoion lines, he xpoin and he pairs of arrows, we are now able o draw rajecories ino his phase diagram. We will do so and analyse he implicaions of pairs of arrows furher once we have generalized he derivaion of a phase diagram Twodimensional sysems II  The general case Afer his speci c example, we will now look a a more general di erenial equaion sysem and will analyse i by using a phase diagram. The sysem Consider wo di erenial equaions where funcions f (:) and g (:) are coninuous and di ereniable, _x 1 = f (x 1 ; x 2 ) ; _x 2 = g (x 1 ; x 2 ) : (4.2.2) For he following analysis, we will need four assumpions on parial derivaives; all of hem are posiive apar from f x1 (:) ; f x1 (:) < 0; f x2 (:) > 0; g x1 (:) > 0; g x2 (:) > 0: (4.2.3) Noe ha, provided we are willing o make he assumpions required by he heorems in ch , we know ha here is a unique soluion o his di erenial equaion sysem, i.e. x 1 () and x 2 () are unambiguously deermined given wo boundary condiions. Fixpoin The rs quesion o be ackled is wheher here is an equilibrium a all. Is here a xpoin x such ha _x 1 = _x 2 = 0? To his end, se f (x 1 ; x 2 ) = 0 and g (x 1 ; x 2 ) = 0 and plo he implicily de ned funcions in a graph. x 2 * x 2 f ( x x ) 0 1, 2 = g ( x x ) 0 1, 2 = * x 1 x 1 Figure Zero moion lines wih a unique seady sae
99 4.2. Analyzing ODEs hrough phase diagrams 87 By he implici funcion heorem  see (2.3.3)  and he assumpions made in (4.2.3), one zero moion line is increasing and one is decreasing. If we are furher willing o assume ha funcions are no monoonically approaching an upper and lower bound, we know ha here is a unique xpoin (x 1; x 2) x : General evoluion Now we ask again wha happens if he sae of he sysem di ers from x ; i.e. if eiher x 1 or x 2 or boh di er from heir seady sae values. To nd an answer, we have o deermine he sign of f (x 1 ; x 2 ) and g (x 1; x 2 ) for some (x 1 ; x 2 ) : Given (4.2.2); x 1 would increase for a posiive f (:) and x 2 would increase for a posiive g (:) : For any known funcions f (x 1 ; x 2 ) and g (x 1; x 2 ) ; one can simply plo a 3dimensional gure wih x 1 and x 2 on he axes in he plane and wih ime derivaives on he verical axis. Figure A hreedimensional illusraion of wo di erenial equaions and heir zeromoion lines The whie area in his gure is he horizonal plane, i.e. where _x 1 and _x 2 are zero. The dark surface illusraes he law of moion for x 1 as does he grey surface for x 2 : The inersecion of he dark surface wih he horizonal plane gives he loci on which x 1 does no change. The same is rue for he grey surface and x 2 : Clearly, a he inersecion poin of hese zeromoion lines we nd he seady sae x : When working wih wodimensional gures and wihou he aid of compuers, we sar from he zeromoion line for, say, x 1 and plo i ino a normal gure.
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