1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
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1 This documen was generaed a :45 PM 8/8/04 Copyrigh 04 Richard T. Woodward. An inroducion o dynamic opimizaion -- Opimal Conrol and Dynamic Programming AGEC I. Overview of opimizaion Opimizaion is he unifying paradigm in almos all economic analysis. So before we sar le s hink abou opimizaion. The ree below provides a very nice general represenaion of he range of opimizaion problems ha you migh encouner. There are wo hings o ake from his. Firs all opimizaion problems have a grea deal in common: an objecive funcion consrains and choice variables. Second here are los of differen ypes of opimizaion problems and how you solve hem will depend on he branch on which you find yourself. In erms of he enire ree of all opimizaion problems he ones ha could be solved analyically would represen a couple of leaves a bes numerical mehods mus be used o solve he res. Forunaely a grea deal can be learned abou economics by sudying hose problems ha can be solved analyically. Source: The Opimizaion Technology Cener: hp:// In his course we will use boh analyical and numerical mehods o solve a cerain class of opimizaion problems. This class focuses on a se of opimizaion problems ha have wo common feaures: he objecive funcion is a linear aggregaion over ime and a se of variables called he sae variables are consrained across ime. And so we begin
2 - II. Inroducion A simple -period consumpion model Consider he simple consumer's opimizaion problem: max u z z s.. z ( ) pa za + pbzb x [pay aenion o he noaion: z is he vecor of choice variables and x is he consumer's exogenously deermined income.] a Solving he one-period problem should be familiar o you. Wha happens if he consumer lives for wo periods bu has o survive off of he income endowmen provided a he beginning of he firs period? Tha is wha happens if her problem is max U z z z z = U z z s.. z ( ) ( ) p' z + p' z x a b a b where he consrain uses marix noaion wih = [ pa pb ] z [ z z ] a b b p refers o a price vecor and =. We now have a problem of dynamic opimizaion. When we chose z we mus ake ino accoun how i will affec our choices in period. We're going o make a huge (hough common) assumpion and mainain ha assumpion hroughou he course: uiliy is addiively separable across ime : u z = u z + u ( ) ( ) ( ) z Clearly one way o solve his problem would be jus as we would a sandard saic problem: se up a Lagrangian and solve for all opimal choices simulaneously. This may work here where here are only periods bu if we have 00 periods (or even an infinie number of periods) hen his could ge really messy. This course will develop mehods o solve such problems. This is a good poin o inroduce some very imporan erminology: All dynamic opimizaion problems have a ime horizon. In he problem above is discree ={} bu can also be coninuous aking on every value beween 0 and T and we can solve problems where T. x is wha we call a sae variable because i is he sae ha he decision-maker faces in period. Noe ha x is parameric (i.e. i is aken as given) o he decision-maker's problem in and x + is parameric o he choices in period +. However x + is deermined by he choices made in. The sae variables in a problem are he variables upon which a decision maker bases his or her choices in each period. Anoher imporan characerisic of sae variables is ha ypically he choices you make in one period will influence he value of he sae variable in he nex period. A sae equaion defines he ineremporal changes in a sae variable. See Deaon and Muellbauer (37-4) on he negaive implicaions of assuming preferences are addiive
3 - 3 z is he vecor of h period choice variables. Choice variables deermine he (expeced) payoff in he curren period and he (expeced) sae nex period. These variables are also referred o as conrol variables and I will use he erms inerchangeably. p a and p b are parameers of he model. They are held consan or change exogenously and deerminisically over ime. Finally we have wha I call inermediae variables. These are variables ha are really funcions of he sae and conrol variables and he parameers. For example in he problem considered here one-period uiliy migh be carried as an inermediae variable. In firm problems producion or profi migh be oher inermediae variables while produciviy or profiabiliy (a firm s capaciy o generae oupu or profis) could be sae variables. Do you see he difference? This is very imporan (see PS#). When you formulae a problem i is very imporan o disinguish sae variables from inermediae variables. The benefi funcion ells he insananeous or single period ne benefis ha accrue o he planner during he planning horizon. In our problem u(z ) is he benefi funcion. Despie is name he benefi funcion can ake on posiive or negaive values. For example a funcion ha defines he cos in each period can be he benefi funcion. In many problems here are benefis (or coss) ha accrue afer he planning horizon. This is capured in models by including a salvage value which is usually a funcion of he erminal sock. Since he salvage value occurs afer he planning horizon i can no be a funcion of he conrol variables hough i can be a separae opimizaion problem in which choices are made. The sum (or inegral) over he planning horizon plus he salvage value deermines he objecive funcion. We usually use discouning when we sum up over ime. Pay close aenion o his he objecive funcion is no he same as he benefi funcion. All of he problems ha we will sudy in his course fall ino he general caegory of Markov decision processes (MDP). In an MDP he probabiliy disribuion over he saes in he nex period is wholly deermined by he curren sae and curren acions. One imporan implicaion of limiing ourselves o MDPs is ha ypically hisory does no maer i.e. x + depends on z and x irrespecive of he value of x. When hisory is imporan in a problem hen he relevan hisorical variables mus be explicily included as sae variables. In sum he problems ha we will sudy will have he following feaures. In each period or momen in ime he decision maker looks a he sae variables (x ) hen chooses he conrol variables (z ). The combinaion of x and z generaes immediae benefis and coss. They also deermine he probabiliy disribuion over x in he nex period or momen. Insead of using brue force o find he soluions of all he z s in one sep we reformulae he problem. Le x be he endowmen which is available in period and x be he endowmen ha remains in period. Following from he budge consrain we can see
4 - 4 ha x = x p'z wih x 0. In his problem x defines he sae ha he decision maker faces a he sar of period. The equaion which describes he change in he x from period o period x x = p'z is called he sae equaion. This equaion is also someimes referred o as he equaion of moion or he ransiion equaion. We now rewrie our consumer s problem his ime making use of he sae equaion: = ( ) max u z s.. z x + x = p' z = () x + 0 x fixed We now have a nasy lile opimizaion problem wih four consrains wo of hem inequaliy consrains no fun. This course will help you solve and undersand hese kinds of problems. Noe ha his formulaion is quie general in ha you could easily wrie he n-period problem by simply replacing he s in () wih n. III. The OC (opimal conrol) way of solving he problem We will solve dynamic opimizaion problems using wo relaed mehods. The firs of hese is called opimal conrol. Opimal conrol makes use of Ponryagin's maximum principle. To see his approach firs noe ha for mos specificaions economic inuiion ells us ha x >0 and x 3 =0. Hence for = (+=) we can suppress inequaliy consrain in (). We ll use he fac ha x 3 =0 a he very end o solve he problem. Wrie ou he Lagrangian of (): ( ) λ ( + p ' ) () L = u z x + x x z = where we include x in u( ) for compleeness hough u x = 0. More erminology In opimal conrol heory he variable λ is called he cosae variable and following he sandard inerpreaion of Lagrange mulipliers a is opimal value λ is equal o he marginal value of relaxing he consrain. In his case ha means ha λ is equal o he marginal value of he sae variable x. The cosae variable plays a criical role in dynamic opimizaion. The firs order condiions (FOCs) for () are sandard: L z = u z λ p = 0 i = a b = i i
5 - 5 L u λ λ = + = [noe ha x is no a choice variable since i is fixed a he ouse and x 3 is equal o zero] ( p ) L λ = x x + ' z = 0 =. We now use a lile noaional change ha simplifies his problem and adds some inuiion (we'll see how he inuiion arises in laer lecures). Tha is we define a funcion known as he Hamilonian where 0 ( ) λ ( ' ) H = u z x + p z. Some hings o noe abou he Hamilonian: he h Hamilonian only includes curren variables z x and λ unlike in a Lagrangian only he righ-hand side of sae equaion appears in he λ p ' z. erm ( ) In he lef column of able below we presen he well-known FOCs of he Lagrangian. On he righ we presen he derivaive of he Hamilonian wih respec o he same variables. Comparing he wo sides we can see wha we would have o place on he righ-hand side of he derivaives of he Hamilonian o obain he same opimum as when he Lagrangian is used. Lagrangian [ u ( za zb ) + λ ( x x+ ( pa za + pb zb ))] = Hamilonian L = H = u( z x ) + λ ( p ' z ) i i g Sandard FOCs / L u = λ pi = 0 = i=ab u z z z i = λ p z z ( ) L u = λ + λ = 0 x i i ( x ) u z = = = L = x x + p ' z = 0 = i=ab λ λ = p ' z = λ [Fill in he blanks in he righ column before proceeding]
6 - 6 Hence we see ha for he soluion using he Hamilonian o yield he same maximum he following condiions mus hold. = 0 The Hamilonian should be maximized w.r.. he conrol z variable a every poin in ime. The cosae variable changes over ime a a rae equal o. = λ λ for > minus he marginal value of he sae variable o he Hamilonian. 3. = x + x The sae equaion mus always be saisfied. λ When we combine hese wih a 4 h condiion called he ransversaliy condiion (how we ransverse over o he world beyond =) we're able o solve he problem. In his case he condiion ha x 3 =0 (which for now we will assume o hold wihou proof) serves ha purpose. We'll discuss he ransversaliy condiion in more deail in a few lecures. These four condiions are he saring poins for solving mos opimal conrol problems and someimes he FOCs alone are sufficien o undersand he economics of a problem. However if we wan an explici soluion hen we would solve his sysem of equaions. Alhough in his class mos of he OC problems we ll face are in coninuous ime he parallels should be obvious when we ge here. IV. The DP (Dynamic Programming) way of solving he problem The second way ha we will solve dynamic opimizaion problems is using Dynamic Programming. DP is abou backward inducion hinking backwards abou problems. Le's see how his is applied in he conex of he -period consumer's problem. Imagine ha he decision-maker is now in period having already used up par of her endowmen in period leaving x o be spen. In period her problem is simply ( ) = ( ) V x max u z s.. z p ' z x If we solve his problem we can easily obain he funcion V(x ) which ells us he maximum uiliy ha can be obained if she arrives in period wih x dollars remaining. The funcion V( ) is equivalen o he indirec uiliy funcion wih p a and p b suppressed. The period problem can hen be wrien max u z + V x s.. x = x p ' z. (3) z ( ) ( ) The value of having x in period one is he soluion o his problem i.e. ( ) max ( ) ( ) V x = u z + V x. z This equaion is known as he Bellman s equaion and i is he cornersone of dynamic programming.
7 - 7 Noe ha we have implicily assumed an inerior soluion so ha he consrain requiring ha x 3 0 is assumed o hold wih an equaliy and can be suppressed. Once we know he funcional form of V( ) (3) becomes a simple saic opimizaion problem and is soluion is sraighforward. Assume for a momen ha he funcional form of V(x ) has been found. We can hen wrie ou Lagrangian of he firs period problem L = u z + V x + λ x p ' z x. ( ) ( ) ( ) Again we see ha he economic meaning of he cosae variable λ is jus as in he OC seup i.e. i is equal o he marginal value of a uni of x. Of course he problem is ha we do no have an explici funcional form for V( ) and as he problem becomes more complicaed obaining a funcional form becomes more difficul even impossible for many problems. Hence he rick o solving DP problems is o find he funcion V( ). V. Are OC and DP equivalen? Yes. Eiher of he approaches ha we have demonsraed above can be used o solve a dynamic opimizaion problem. In his secion we will quickly show ha he firs-order condiions for a simple problem are equivalen. We will sar wih a dynamic programming se up for a decision maker in which he benefis in are u ( x z ) and x + = x + g ( x z ). The Bellman s equaion for his problem is herefore V x = max u x z + V x + s.. x = x + g x z. or ( ) ( ) ( ) ( ) + + z ( ) + ( ) max ( ) ( ) V x u x z V x g x z z = + +. Since z i a choice variable he opimum will be achieved where he FOC wih respec o z is equal o zero: + u ( x ) z V g ( x z ) + = 0. (4) z z + Bu we know somehing abou he parial derivaive V + + i is he marginal value of x + in period +. A look back a () will confirm ha in he OC formulaion λ is he Lagrange muliplier and he economic inerpreaion of ha is he marginal value o he objecive funcion of relaxing he consrain in period. Hence λ + would be he marginal value of relaxing he sae equaion consrain a ime +. Tha is equivalen o he marginal value of x + so V + + = λ. We can herefore rewrie (4) + ( ) g ( x z ) u x z + λ + = 0. z z which is he firs of he FOCs for he OC specificaion.
8 - 8 Now consider he parial derivaive of he Bellman s equaion wih respec o x. Since x is no a choice variable i is fixed a ime he parial derivaive is no se o zero; i is simply ( ) u ( x * ) ( * z + V x ) V g x z = where we wrie z o clarify ha he derivaive is aken is a he maximum value of z. * Again we can subsiue for λ yielding u ( x * ) ( * z g x z ) λ = + λ + + or u ( x * ) ( * z g x z ) λ λ + = + λ + which is equivalen o he second of he opimizaion condiions for he OC specificaion. Finally i is obvious ha he sae equaion in boh formulaions mus hold. Hence we have shown ha he wo approaches are equivalen. VI. Summary OC problems are solved using he vehicle of he Hamilonian which mus be maximized a each poin in ime. DP is abou backward inducion. Boh echniques are equivalen o sandard Lagrangian echniques and he inerpreaion of he shadow price λ is he same. VII. Reading for nex lecure Leonard and Van Long chaper. VIII. References Deaon Angus and John Muellbauer Economics of Consumer Behavior. New York: Cambridge Universiy Press. To show his for coninuous ime problems replace + wih + and hen ake he limi a 0
1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
This documen was generaed a :37 PM, 1/11/018 Copyrigh 018 Richard T. Woodward 1. An inroducion o dynamic opimiaion -- Opimal Conrol and Dynamic Programming AGEC 64-018 I. Overview of opimiaion Opimiaion
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