1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
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1 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 I. Overview of opimiaion Opimiaion is a unifying paradigm in almos all economic analysis. So before we sar, le s hink abou opimiaion. The ree below provides a very nice general represenaion of he range of opimiaion problems ha you migh encouner. There are wo hings o ake from his. Firs, all opimiaion problems have a grea deal in common: an objecive funcion, consrains, and choice variables. Second, here are los of differen ypes of opimiaion problems and how you solve hem will depend on he branch on which you find yourself. In erms of he enire ree of all opimiaion 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 Opimiaion Technology Cener: hp:// In his course we will use boh analyical and numerical mehods o solve dynamic opimiaion problems, 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 1 - II. Inroducion A simple -period consumpion model Consider he simple consumer's opimiaion problem: max u, s.. ( ) pa a + pbb x [pay aenion o he noaion: 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,,, = U, s.. p ' + p ' x, ( ) ( ) b 1a 1b a b where he consrain uses marix noaion wih = [ pa, pb ] [, ] 1 1a 1b p refers o a price vecor and =. We now have a problem of dynamic opimiaion. When we chose 1, 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 1 : u = u + u. ( ) ( ) ( ) 1 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, when here are only periods, bu if we have 100 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 opimiaion problems have a ime horion. In he problem above is discree, ={1,}, 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 decisionmaker's problem in, and x+1 is parameric o he choices in period +1. However, x +1 is affeced by he choices made in. The sae variables in a problem are hose ha a decision maker akes as given when making his or her choices in each period. A sae equaion defines he ineremporal changes in a sae variable. This equaion is someimes referred o as he equaion of moion or he ransiion equaion. 1 See Deaon and Muellbauer (137-14) on he negaive implicaions of assuming preferences are addiive
3 1-3 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 or acion variables and I will use all hese erms inerchangeably. To disinguish sae & conrol variables, I like o say, You wake up in he morning, look a your sae variables, make decisions abou your conrol variables, hen go back o sleep. pa and pb 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? When you formulae a problem i is very imporan, bu ofen difficul, o disinguish sae variables from inermediae variables (see PS#1). The benefi funcion ells he insananeous or single period ne benefis ha accrue o he planner during he planning horion. In our problem u( ) is he benefi funcion. Despie is name, he benefi funcion can be posiive or negaive. 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 horion. 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 horion, i canno be a funcion of he conrol variables, hough i can be a separae opimiaion problem in which choices are made. The sum (or inegral) over he planning horion 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+1 depends on and x, irrespecive of he value of x 1. When hisory is imporan in a problem, hen he relevan hisorical variables mus be explicily included as sae variables. A Formal Saemen of he Opimiaion Problem is a se of mahemaical expressions including he objecive funcion and all he consrains. The consrains include he sae equaion, any condiions ha mus be saisfied a he beginning and end of he ime horion, and any consrains ha resric choices beween he beginning and end. A a minimum, dynamic opimiaion problems mus include he objecive funcion, he sae equaion(s) and iniial condiions for he sae variables.
4 1-4 In sum, he problems ha we will sudy will have he following feaures. In each period or momen in ime he decision maker akes as given he sae variables and parameers, hen makes opimal choices for he conrol variables aking ino accoun he objecive funcion and sae equaions. The combinaion of x and generaes immediae benefis and coss and deermines he probabiliy disribuion over x in he nex period he rae of change in x. Insead of using brue force o find he soluions of all he s in one sep, we reformulae he problem. Le x 1 be he endowmen which is available in period 1, and x be he endowmen ha remains in period. Following from he budge consrain, we can see ha x = x 1 p' 1, 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 1 o period, x x 1 = p' 1, is he sae equaion. We now rewrie our consumer s problem, his ime making use of he sae equaion: = 1 ( ) max u s.. x + 1 x = p' = 1, (1) x x1 fixed We now have a nasy lile opimiaion 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 (1) wih n. III. The OC (opimal conrol) way of solving he problem We will solve dynamic opimiaion problems using wo relaed mehods. The firs of hese is called opimal conrol. Opimal conrol makes use of Ponryagin's maximum principle. Firs noe ha for mos specificaions, economic inuiion ells us ha x >0 and x 3 =0. Hence, for =1 (+1=), we can suppress inequaliy consrain in (1). We ll use he fac ha x 3 =0 a he very end o solve he problem. Wrie ou he Lagrangian of (1): (, ) λ ( + 1 p ' ) () L = u x + x x = 1 where we include x in u( ) for compleeness, hough in his case u x = 0.
5 1-5 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 opimiaion. The firs order condiions (FOCs) for () are sandard: L = u λ p = 0, i = a, b, = 1, i i i L u λ λ = 1 + = [noe ha x1 is no a choice variable since i is fixed a he ouse and x 3 is equal o ero] ( ) L λ = x x + 1 p ' = 0, =1,. 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 H, x, λ = u, x + λ p '. ( ) ( ) ( ) Some hings o noe abou he Hamilonian: he h Hamilonian only includes curren variables:, x and λ, unlike in a Lagrangian, only he righ-hand side of sae equaion appears afer λ. 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. [Fill in he blanks in he righ column before proceeding] Lagrangian Hamilonian [ u ( a, b ) + λ ( x x+ 1 ( pa a + pb b ))] L = H = u(, x ) + λ ( p ' ) = 1 i i 0 g Sandard FOCs / L u u = λ pi = 0, =1,, i=a,b i = λ p ( ) L u = λ1 + λ = 0 x i i (, x ) u = = = L = x x + 1 p ' = 0, =1,, i=a,b λ = p ' =
6 1-6 Hence, we see ha for he soluion using he Hamilonian o yield he same maximum he following condiions mus hold 1. = 0 The Hamilonian should be maximied w.r.. he conrol variable a every poin in ime. The cosae variable changes over ime a a rae equal o. = λ 1 λ for >1 minus he marginal value of he sae variable o he Hamilonian. 3. = x + 1 x The sae equaion mus always be saisfied. When we combine hese wih a 4 h condiion, called he ransversaliy condiion (how we ransverse over o he world beyond =1,) 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. In his class mos of he OC problems we ll face are in coninuous ime. The parallels beween he discree ime case presened here and he coninuous ime case 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 opimiaion 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 1, leaving x o be spen. In period, her problem is simply ( ) = ( ) V x max u s.. p ' 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 pa and pb suppressed. The period 1 problem can hen be wrien max u + V x s.. x = x p '. (3) 1 ( ) ( ) The value of having x 1 in period one is he soluion o his problem, i.e. ( ) max ( ) ( ) V x = u + V x This equaion is known as he Bellman s equaion and i is he cornersone of dynamic programming.
7 1-7 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 opimiaion problem and is soluion is sraighforward. If he funcional form of V(x ) has been found, hen we can wrie ou Lagrangian of he firs period problem, L = u + V x + λ x p ' x. ( ) ( ) ( ) We see ha he economic meaning of he cosae variable, λ 1, is jus as in he OC seup, i.e., i is equal o he marginal value of a uni of x 1. A major challenge 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. As we will see hroughou his course, eiher of hese approaches can be used o solve a dynamic opimiaion problem. In his secion we will quickly show ha he firs-order condiions for a simple problem are equivalen. Consider he coninuous-ime dynamic opimiaion problem, = 0 ( ) = ( ) max u x, d s.. xɺ f x,, T where, as we will discuss in Lecure, xɺ. The discree-ime analog of his problem is j= 0 ( j j ) + = + ( ) max u x, s.. x x f x,, j T where is some fracion of a period. For example, if =0.5, hen here are wo incremens per period ha go from j=0 o j=t=t/0.5. The Hamilonian and he value funcion for hese wo problems are: H x,, λ = u x, + λ f x,, and ( ) ( ) ( ) ( ) = ( ) + ( + ) = + ( ) V x, u x, V x, where x x f x,. + + Firs, we show he equivalence of he FOCs w.r. he conrol variable as 0. The firs order condiion for he Hamilonian is, as above, = u + λ f = 0. For he value funcion, we know ha a he opimum value for, V = 0, i.e.
8 (, + ) V V x + = u + f = 0. Dividing by and aking he limi as 0 so ha + (, ) V x +, we have u + f = 0. Finally, since V lim = = 0. 0 (, ) V x = λ, i follows ha Nex, we can show he equivalence of he FOC w.r. he sae variable, x, as 0. Again, we saed above ha he FOC of he Hamilonian for he sae variable is u f = + λ = λ λ +. (4) If we divide he middle and las par of his equaliy by and hen ake he limi as 0 his becomes u f + λ =. (5) For he Bellman s equaion, since x is no a choice variable i is fixed a ime he parial derivaive is no se o ero; i is simply V ( x, ) u V ( x +, + ) + = +. (6) + Noice ha here is some nice inuiion in (6): he marginal value of he sae variable is equal o he sum of wha you ge ou of i in he firs period of lengh, ux, plus wha + V + you ge in he fuure because you have more x: = λ + ( 1+ f x ). Hence, (6) can be rewrien u ( 1 f ) λ = x + λ + + x, subracing λ+ from boh sides and dividing λ λ + by we obain = ux + λ + ( f x ). Again we ake he limi a 0, which in his case gives us on he LHS, o obain = ux + λ f x, which is he same as FOC for he Hamilonian, (5). Finally, i is obvious ha he sae equaion in boh formulaions mus hold, regardless of he lengh of. 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 maximied 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
9 1-9 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.
1. An introduction to dynamic optimization -- Optimal Control and Dynamic Programming AGEC
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 637-04 I. Overview of opimizaion Opimizaion is
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