ECON ENV MNGMNT
ECON ENV MNGMNT ESM 204
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Date Created: 10/22/15
Dynamic Programming Under Certainty 5 Marek Kapicka Econ 204b April 13 2009 Review gt The Bellman Operator TVX gym My 1 Has a unique fixed point v in the space of bounded and continuous functions 2 Sequence of functions Tnvo converges to v at rate 3 gt The optimal policy correspondence gX arg maXFXY I3VY y6TX 1 Is upper hemicontinuous and compact valued Today gt We will 1 Look at the properties of the fixed point a Monotonicity b Concavi c Differentiability 2 Look at the example of the optimal growth model again 51 Corollaries to the contraction mapping theorem Corollary Let 5 p be a complete metric space Let T S gt 5 be a contraction mapping that has a fixed point v E 5 Then 1 1 9 is a closed subset ofS and T Q 5 then v E 5 2 Ifin addition T g s then v e s 52 General Approach gt To show that the fixed point has a given property we will gt look at the conditions that guarantee that T maps a set of functions with that property onto itself gt If the set of functions with a given property is closed then by Corollary 1 the fixed point will preserve that property gt If the set of functions with a given property is not closed but Corollary 2 holds then fixed point will preserve the property gt For differentiability the approach fails Corollaries 1 and 2 do not apply gt Other tricks 53 Monotonicity gt Assumption M1 FXy is increasing in X Assumption M2 F is monotone X g X i FX Q FX Theorem If assumptions 12 and M hold then the fixed point v is increasing gt Idea of proof Let 5 be a set of bounded continuous and increasing functions 5 Q 5 and one can show that 5 is closed Under assumptions M T maps increasing bounded and continuous functions onto itself Corollary 1 can be applied 53 Strict Monotonicity Theorem If in addition FXy is strictly increasing in X then the fixed point v is strictly increasing gt Idea of proof Let SJrJr be a set of bounded continuous and strictly increasing functions 5 Q 5 But since SJrJr is not closed one must show that for any v increasing TV is strictly increasing This is guaranteed by the assumption that F is strictly increasing 54 Strict Concavity gt Assumption C1 FXy isjointly strictly concave in w I FAXYY1AX1yl Z AFCCY 1iAFXYyl Assumption C2 F is convex Ify E FX and y E FX then Ay 17Ayl E FAX1AX Theorem If assumptions 12 and C hold then 1 The fixed point v is strictly concave 2 If C1 is strict then the optimal policy function g is single valued and continuous 55 Differentiability gt The usual approach relies on envelope theorems h f X yrgra w gt If f is differentiable then hX ampXgx gt one cannot apply directly since fxy FXy vy and we do not know in advance if vy is differentiable gt One cannot use the same approach as with monotonicity and concavity either gt The space of bounded continuous and differentiable functions is not closed and Corollary 2 does not apply 55 Differentiability BenvenisteSchienkman Theorem Benveniste Schienkman If there is a function WX such that for some X0 1 WX0 vx0 and WX g vx forX in the neighborhood ofxo 2 v and W are both concave 3 W is differentiable Then 1 v is differentiable at X0 2 v x0 WXo 55 Differentiability gt candidate for W WX FX1gXo I3Vgxo gt policy function is fixed for X0 X varies gt Check the conditions of 8 8 WX0 VX0 If gX0 is in the interior of TX0 then gX0 E TX for X in the neighborhood of X0 and so WX g v gt gt U m N If Assumptions C hold F is concave and T is convex then both v and W are concave 5 If F is differentiable in X at X0 then W is differentiable at X0 gt New requirementsF must be differentiable in X and gx0 must be in the interior of FX0 55 Differentiability Theorem If assumptions 12 and C holds F is differentiable in X at X0 and in addition gx0 is in the interior ofFX0 then 1 v is differentiable at X0 2 VXo FxX01gX0 61 Example Optimal Growth Vk Wk 7y l3vy max U 099k 1 For the existence and uniqueness of the fixed point we need 11 3 6 01 12 For boundedness Either U is bounded or there is an upper bound on capital stock R so that k E 0 R This is OK if Iimkw f k 0 13 For TOM U and f are continuous 2 For strict monotonicity of v we need 21 U f are strictly increasing 3 For strict concavity of v we need 31 U f are strictly concave 4 For differentiability of v we need 41 U f are differentiable and strictly concave 42 For interior solution Iim Uc 00 and Iim fk 00 CHO kgt0 61 Example Optimal Growth Optimal Policy Function gt gk satisfies gt First order condition U fk 00 VEUO gt Envelope condition Wk U fk gkf k gt gk is strictly increasing 71 Dynamics in the Optimal Growth Model gt Is there a steady state gt Is it unique gt Is it stable Does the capital stock converge to the steady state 71 Dynamics in the Optimal Growth Model Existence and Uniqueness of Steady State gt In steady state gk5 kss Hence Ufk 7 kSS VkSS VkSS Ufk 7 kSSfk55 gt Hence fkss gt Steady state exists and is unique for a strictly positive capital stock 71 Dynamics in the Optimal Growth Model Stability of Steady State gt Since v is strictly concave llk 7 v lAltk7 1 g 0 all k I with equality only if k I gt Choose ilt gk Then Mk 7 WWW lk 7 gkl S 0 with equality only in steady state Dynamic Programming Under Certainty 4 Marek Kapicka Econ 204b April 8 2009 Review gt Bellman Equation v as a fixed point v TV of the operator TVX gym My gt Contraction Mapping Theorem If T maps bounded and continuous functions to itself and is a contraction it has a unique fixed point Today gt We will Uquot gtL 39 Look at Sufficient conditions for a contraction Look at the Theorem of Maximum Apply CMT and TOM to the Bellman Operator Have Example Look at the properties of the fixed point 51 Monotonicity 42 Contraction Mapping Blackwell s Sufficient Conditions for a Contraction Theorem Let 5 be a space of bounded functions on X endowed With a sup norm Let T S gt 5 If i T is monotone If fx g gx for allx E X then Tfx g Tgx for allx E X ii T discounts For some 8 E 01 and any a E R Tfax g Tfx a VX E X Where f ax fx a then T is a contraction With modulus 43 Theorem of the Maximum gt We want to make sure that an operator T maps continuous functions into continuous functions gt Assumptions 2a T is nonempty ie TX is nonempty for all X E X 2b T is compact valued ie TX is compact for all X E X 2c T is continuous 77 43 Theorem of the Maximum Continuity of 3 Correspondence gt Two weaker concepts 1 upper hemi continuity quotno dips 2 lower hemi continuity no spikes Definition A correspondence is continuous if it is both uhc and uc 43 Theorem of the Maximum Theorem LetX E R and Y E Rm Define hx max fx y gx arg maxfx y YEUX yerx Suppose that f X X Y gt R is continuous and F X gt Y is nonempty compact valued and continuous Then i h X gt R is continuous and ii g X gt Y is upper hemi continuous and compact valued 44 Bellman Equation Application gt The Bellman Operator TVX yg FX1y l3vy Theorem Let 5 be the space of bounded and continuous functions With a sup norm Suppose that i A1 FX y is bounded and continuous ii 0 lt p lt 1 iii A2 F is nonempty compact valued and continuous Then the Bellman operator T i maps 5 onto itself ii has a unique fixed point v E 5 in MW e vll WW 7 WM iv The optimal policy correspondence gx is compact valued and uhc 44 Bellman Equation Application Proof 1 By TOM T maps continuous functions into continuous functions 2 T is a contraction Blackwell monotonicity obvious discounting Tvax lFXy3vayl lFXy l3vy I33 yg lFX1y3vyl l3a TVX 3983 max yemx max yemx 51 Corollaries to the contraction mapping theorem Corollary Let 5 p be a complete metric space Let T S gt 5 be a contraction mapping that has a fixed point v E 5 Then 1 1 9 is a closed subset ofS and T Q 5 then v E 5 2 Ifin addition T g s then v e s 52 General Approach gt to show that the fixed point has a given property we will gt look at the conditions that guarantee that T maps a set of functions with that property onto itself gt If the set of functions with a given property is closed then by Corollary 1 the fixed point will preserve that property gt If the set of functions with a given property is not closed but Corollary 2 holds then fixed point will preserve the property gt For differentiability the approach fails Corollaries 1 and 2 do not apply gt Other tricks 53 Monotonicity gt Assumption M1 FXy is increasing in X Assumption M2 F is monotone X g X i FX Q FX Theorem If assumptions 12 and M hold then the fixed point v is increasing gt Idea of proof Let 5 be a set of bounded continuous and increasing functions 5 Q 5 and one can show that 5 is closed Under assumptions M T maps increasing bounded and continuous functions onto itself Corollary 1 can be applied Dynamic Programming Under Certainty Contd Marek Kapicka Econ 204b April 1 2009 Review gt Functional Equation and Sequence Problem 5P Vk0 ogk nqgaffk w g tUUUQ 7 kt 0 given FE vltkgt oggga wak e y My gt Today 2 Connection between SP and FE 3 Bellman Equation as a Fixed Point 22a A solution to SP satisfies FE Vk0 Theorem max 03kt13fkto maX OSk1Sfko i WW 7 km max ogkw k io t0 Ufko 7 k1 15 Z H lUWr W t1 max Ufk0 7 1 7 osklgmo i1 l3t 1Ufkt 7 mo max 03kt13fkt 1 Ufko k13Vk1 v satisfies FE 22b A solution to FE satisfies SP vow O gg koufk0 k1 vk1 og f ko w kowkl wos f h um k2 I5Vk2 E l3tUfkt 7 kt1l32Vk2 max ngf1 fkto t0 T BtUfkt kt1 T1vkT1 max 03k1 fko t0 22b A solution to FE satisfies SP harder Theorem Tum 15mm 0 1 for all kTH such that 0 g kTH g fkT then v satisfies 5P Proof If the condition is satisfied then the last term vanishes T vk Iim max thkik T1vk lt 0 ngmgw f i lt lt t m 15 lt no t U f k 7 k oskni gikniafgf t 1 21contd The second Example revisited gt By Theorem 2 it must be true that the condition fails for some feasible sequence of capital gt Consider the following feasible sequence 1 k1 3kg 1 k2 Elm gt Then 1 T 1 T 1 Tllnw VkT1 Tllnw l T1 kOl k0 3A 0 gt Hence Condition 1 is not satisfied for vk k gt Note that Condition 1 is also not satisfied for the right solutionll gt Condition 1 is sufficient but not necessary 23 A Relationship between optimal policies in SP and FE Theorem Ifkt1 attains the maximum of SP then k gkt all t Z 0 Theorem Iim BT1vkT1g 0 Taco then gk0ggk0 attains the maximum of SP 3 Bellman Equation as a Fixed Point gt Let 5 be a space of functions Define a Bellman operator T 5 a 5 by Tvk Wk 7y l3vy max U 099k gt T maps a family of functions to a family of functions gt The solution to FE v is a fixed point of the operator T Tv v gt fwe find a fixed point of the operator T we have solved FE and provided the boundedness condition 2 is satisfied SP 3 Bellman Equation as a Fixed Point gt We will show gt under certain conditions there exists a unique fixed point gt A sequence V51 defined by Vs1 TVS converges to the fixed point useful computationally gt other properties of the fixed point monotonicity concavity etc 31 An Example gt uc In c fk k full depreciation 1 STEP 1 Set v0 0 7 Dc 7 Tv0k fogrnyagzml k y Solution gok 0 Tv0k nk txnk 31 An Example 2 STEP 2 Set v1 Tv0txnk 7 xi Tv1kiogmyagzmlnk ytx ny Solution 113 ac 1tx k Tv1k tx1 tx In k In 100 1 043 1tx tx ln 1tx 31 An Example 3 STEP 3 Set V2 Tvl txltx In k In 11a5 5 I 1545 I Tv2k ogmyagka nk 7y tx 1 tx Iny 0C5 n 1tx 3 1a m 2n Solution WNW3V km 1txl30 l32 TVQka1 043 ix3 In k W 015 0952N I3 2 043 n1tx 00 04 042 H quot1vc5oc 2 l niltx 31 An Example 00 THE LIMIT v IimwvS f S39megs 0c 1 043 vk lnk17 n1itx litx lntx gk 5km Hence 6k k 7gk 1 im dx gt Consume a fraction 17 oc of the resources save tx for the future 31 An Example Results gt By iteration on the value function we have found 1 The value function v that solves FE 2 The optimal policy function gk gt Note The speed of convergence oclnk 0c1 18 In k 0c1 a3 ix3 In k will hold more generally 4 A General Approach gt Consider the following more general setup X E X 50 mao Z 39BtFXt n1 Xt1o 130 st Xt1 E FXt X0 given 5P vx FE vx max FXy 8vy y6TX gt F is a correspondence Assigns a set FX to each X 41 A General Approach Mathematical Preliminaries gt Assumption FXy is bounded and continuous gt The assumption suggests that the value function might also be bounded and continuous gt To study FE in general we need some mathematical tools gt Contraction Mapping Theorem Will tell us under what conditions will an operator mapping the set of bounded and continuous functions onto itself have a unique fixed point gt Theorem of Maximum Will tell us under what conditions will the Bellman operator map the set of bounded and continuous functions onto itself 41 A General Approach Mathematical Preliminaries gt Consider a set of bounded and continuous functions with a SLIP norm 5 f X gt R f is continuous ll fllSUPlfXl lt00 xeX gt Define a metric Mfg ll fig ll lfx gX Definition A metric space Sp is complete if every Cauchy sequence in 5 converges to an element in 5 41 A General Approach Mathematical Preliminaries Definition A sequence Xn is a Cauchy sequence if for all s gt 0 ENE such that pxmxn lt s for all m n 2 NE Theorem The set of bounded and continuous functions is complete Example A set of functions that is not complete 41 Example A set of functions that is not complete gt Let SJrJr be a set of bounded continuous functions that are strictly increasing gt Consider a sequence 1 1 n gt fn is a sequence of functions fnx 1 X X E 01 gt fn is a Cauchy sequence for any 8 gt 0 1 1 fmfn sup 1 Xiliix p Xdovl 1m 1n 1 1 i 7 1m 1n 1 1minmn lt8 for mn 2 Ngwhere NE 71
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