Outline Today s Lecture

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1 Outline Today s Lecture finish Euler Equations and Transversality Condition Principle of Optimality: Bellman s Equation Study of Bellman equation with bounded F contraction mapping and theorem of the maximum Introduction to Dynamic Optimization Nr. 2

2 Infinite Horizon T = subject to, with x 0 given V (x 0 )= sup {x t+1 } t=0 X β t F (x t,x t+1 ) t=0 x t+1 Γ (x t ) (1) sup {} instead of max {} define xt+1ª 0 as a plan t=0 define Π (x 0 ) ª x 0 ª t+1 t=0 x0 t+1 Γ (x 0 t) and x 0 0 = x 0 Introduction to Dynamic Optimization Nr. 3

3 Assumptions A1. Γ (x) is non-empty for all x X A2. lim T P T t=0 βt F (x t,x t+1 ) exists for all x Π (x 0 ) then problem is well defined Introduction to Dynamic Optimization Nr. 4

4 Recursive Formulation: Bellman Equation value function satisfies V (x 0 ) = max {x t+1 } t=0 x t+1 Γ(x t ) = max x 1 Γ(x 0 ) = max x 1 Γ(x 0 ) ( X ) β t F (x t,x t+1 ) t=0 F (x 0,x 1 )+ F (x 0,x 1 )+β max {x t+1 } t=1 x t+1 Γ(x t ) max {x t+1 } t=1 x t+1 Γ(x t ) = max x 1 Γ(x 0 ) {F (x 0,x 1 )+βv (x 1 )} X β t F (x t,x t+1 ) X β t F (x t+1,x t+2 ) t=1 t=0 Introduction to Dynamic Optimization Nr. 5

5 idea: use BE to find value function and V p olicy f unction g [Principle of Optimality] Intro duction to Dynamic Optimization Nr. 5A

6 Outline Today s Lecture housekeeping: ps#1 and recitation day/ theory general / web page finish Principle of Optimality: Sequence Problem (for values and plans) solution to Bellman Equation begin study of Bellman equation with bounded and continuous F tools: contraction mapping and theorem of the maximum Introduction to Dynamic Optimization Nr. 2

7 Sequence Problem vs. Functional Equation Sequence Problem: (SP) X V (x 0 ) = sup β t F (x t,x t+1 ) {x t+1 } t=0 t=0 s.t. x t+1 Γ (x t ) x 0 given... more succinctly V (x 0 )= sup u ( x) x Π(x 0 ) ( SP) functional equation (FE) [this particular FE called Bellman Equation] V (x) = max {F (x, y)+ βv (y)} (FE) y Γ(x) Introduction to Dynamic Optimization Nr. 3

8 Principle of Optimality IDEA: use BE to find value function V and optimal plan x Thm 4.2. V defined by SP V solves FE Thm 4.3. V solves FE and... V = V Thm 4.4. x Π (x 0 ) is optimal V (x t ) = F x t,x t+1 + βv x t+1 Thm 4.5. x Π (x 0 ) satisfies V (x t ) = F x t,x t+1 and... x is optimal + βv x t+1 Introduction to Dynamic Optimization Nr. 4

9 Why isthis Progress? intuition: breaks planning horizon into two: now and then notation: reduces unnecessary notation (esp ecially with uncertainty) analysis: prove existence, uniqueness and properties of optimal policy (e.g. continuity, montonicity, etc...) computation: associated numerical algorithm are powerful for many applications Introduction to Dynamic Optimization Nr. 5

10 Proof oftheorem 4.3 (max case) Assume for any x Π (x 0 ) BE implies Substituting V (x 1 ): lim T βt V (x T )=0. V (x 0 ) F (x 0,x 1 )+ βv (x 1 ), all x 1 Γ (x 0 ) = F (x 0,x 1)+ βv (x 1 ), some x 1 Γ (x 0 ) V (x 0 ) F (x 0,x 1 )+ βf (x 1,x 2 )+ β 2 V (x 2 ), all x Π (x 0 ) = F (x 0,x 1 )+ βf (x 1,x 2)+ β 2 V (x 2), some x Π (x 0 ) Introduction to Dynamic Optimization Nr. 6

11 Continue this way V (x 0 ) = nx β t F (x t,x t+1 )+ β n+1 V (x n+1 ) for all x Π (x 0 ) t=0 nx t=0 β t F x t,x t+1 + β n+1 V x n+1 for some x Π (x 0 ) Since β T V (x T ) 0, taking the limit n on both sides of both expressions we conclude that: V (x 0 ) u ( x) for all x Π (x 0 ) V (x 0 ) = u ( x ) for some x Π (x 0 ) Introduction to Dynamic Optimization Nr. 7

12 Bellman Equation as a Fixed Point define operator T (f)(x) = max {F (x, y)+ βf (y)} y Γ(x) V solution of BE V fixed point of T [i.e. TV = V ] Bounded Returns: if kf k < B and F and Γ are continuos: T maps continuous bounded functions into continuous bounded functions bounded returns T is a Contraction Mapping unique fixed point many other bonuses Introduction to Dynamic Optimization Nr. 8

13 Needed Tools Basic Real Analysis (section 3.1): {vector, metric, noslp, complete} spaces cauchy sequences closed, compact, bounded sets Contraction Mapping Theorem (section 3.2) Theorem of the Maximum: study of RHS of Bellman equation (equivalently of T ) (section 3.3) Introduction to Dynamic Optimization Nr. 9

14 Bellman Equation: Principle of Optimality Principle of Optimality idea: use the functional equation to find V and g V (x) = max {F (x, y)+βv (y)} y Γ(x) note: nuisance subscripts t, t +1, dropped a solution is a function V ( ) thesameonbothsides IF BE has unique solution then V = V more generally the right solution to (BE) delivers V Introduction to Dynamic Optimization Nr. 6

15 Outline Today s Lecture study Functional Equation (Bellman equation) with bounded and continuous F tools: contraction mapping and theorem of the maximum Introduction to Dynamic Optimization Nr. 2

16 Bellman Equation as a Fixed Point define operator T (f)(x) = max {F (x, y)+ βf (y)} y Γ(x) V solution of BE V fixed point of T [i.e. TV = V ] Bounded Returns: if kf k < B and F and Γ are continuous: T maps continuous b ounded functions into continuous b ounded functions bounded returns T is a Contraction Mapping unique fixed point many other bonuses Introduction to Dynamic Optimization Nr. 3

17 Our Favorite Metric Space with S = ½ ¾ f : X R, f is continuous, and kf k sup f (x) < x X ρ (f, g) = kf gk sup f (x) g (x) x X Definition. A linear space S is complete if any Cauchy sequence converges. For a definition of a Cauchy sequence and examples of complete metric spaces see SLP. Theorem. The set of b ounded and continuous functions is Complete. See SLP. Introduction to Dynamic Optimization Nr. 4

18 Contraction Mapping Definition. Let (S, ρ) be a metric space. Let T : S S be an operator. T is a contraction with modulus β (0, 1) for any x, y in S. ρ (Tx, T y) βρ (x, y) Introduction to Dynamic Optimization Nr. 5

19 Contraction Mapping Theorem Theorem (CMThm). If T is a contraction in (S, ρ) with modulus β, then (i) there is a unique fixed point s S, s = Ts and (ii) iterations of T converge to the fixed point ρ (T n s 0,s ) β n ρ (s 0,s ) for any s 0 S, where T n+1 (s) = T (T n (s)). Introduction to Dynamic Optimization Nr. 6

20 CMThm Proof for (i) 1st step: construct fixed point s take any s 0 S define {s n } by s n+1 = Ts n then ρ (s 2,s 1 )= ρ (Ts 1,T s 0 ) βρ (s 1,s 0 ) generalizing ρ (s n+1,s n ) β n ρ (s 1,s 0 ) then, for m >n ρ (s m,s n ) ρ (s m,s m 1 )+ ρ (s m 1,s m 2 ) ρ (s n+1,s n ) β m 1 + β m β n ρ (s 1,s 0 ) β n β m n 1 + β m n ρ (s 1,s 0 ) β n 1 β ρ (s 1,s 0 ) thus {s n } is cauchy. hence s n s Introduction to Dynamic Optimization Nr. 7

21 2nd step: show s = Ts ρ (Ts,s ) ρ (Ts,s n )+ ρ (s,s n ) βρ (s,s n 1 )+ ρ (s,s n ) 0 3nd step: s is unique. Ts 1 = s 1 and s 2 = Ts 2 0 a = ρ (s 1,s 2 ) = ρ (Ts 1,T s 2) βρ (s 1,s 2) = βa only possible if a = 0 s 1 = s 2. Finally, as for (ii): ρ (T n s 0,s )= ρ (T n s 0,T s ) βρ T n 1 s 0,s β n ρ (s 0,s ) Introduction to Dynamic Optimization Nr. 8

22 Corollary. Let S be a complete metric space, let S 0 S and S 0 close. Let T be a contraction on S and let s = Ts. Assume that T (S 0 ) S 0, i.e. if s 0 S, then T (s 0 ) S 0 then s S 0. Moreover, if S 00 S 0 and then s S 00. T (S 0 ) S 00, i.e. if s 0 S 0, then T (s 0 ) S 00 Introduction to Dynamic Optimization Nr. 9

23 Blackwell s sufficient conditions. Let S be the space of bounded functions on X, andk k be given by the sup norm. Let T : S S. Assume that (i) T is monotone, that is, Tf (x) Tg(x) for any x X and g, f such that f (x) g (x) for all x X, and (ii) T discounts, that is, there is a β (0, 1) such that for any a R +, T (f + a)(x) Tf (x)+aβ for all x X. Then T is a contraction. Introduction to Dynamic Optimization Nr. 10

24 Proof. By definition f = g + f g and using the definition of k k, f (x) g (x)+kf gk then by monotonicity i) Tf T (g + kf gk) and by discounting ii) setting a = kf gk Tf T (g)+β kf gk. Reversing the roles of f and g : Tg T (f)+β kf gk ktf Tgk β kf gk Introduction to Dynamic Optimization Nr. 11

25 Bellman equation application (Tv)(x) = max {F (x, y)+βv (y)} y Γ(x) Assume that F is bounded and continuous and that Γ is continuous and has compact range. Theorem. T maps the set of continuous and bounded functions S into itself. Moreover T is a contraction. Introduction to Dynamic Optimization Nr. 12

26 Proof. That T maps the set of continuos and bounded follow from the Theorem of Maximum (we do this next) That T is a contraction follows since T satisfies the Blackwell sufficient conditions. T satisfies the Blackwell sufficient conditions. For monotonicity, notice that for f v Tv(x) = max {F (x, y)+βv (y)} y Γ(x) = F (x, g (x)) + βv (g (x)) {F (x, g (y)) + βf (g (x))} max {F (x, y)+βf (y)} = Tf (x) y Γ(x) A similar argument follows for discounting: for a>0 T (v + a)(x) = max {F (x, y)+β (v (y)+a)} y Γ(x) = max {F (x, y)+βv (y)} + βa = T (v)(x)+βa. y Γ(x) Introduction to Dynamic Optimization Nr. 13

27 Theorem of the Maximum wa n tt to map continuous function into continuous functions (Tv)(x) = max {F (x, y)+βv (y)} y Γ(x) want to learn about optimal policy of RHS of Bellman G (x) =arg max {F (x, y)+βv (y)} y Γ(x) First, continuity concepts for correspondences... then, a few example maximizations... finally, Theorem of the Maximum Introduction to Dynamic Optimization Nr. 14

28 Continuity Notions for Correspondences assume Γ is non-empty and compact valued (the set Γ (x) is non empty and compact for all x X) Upper Hemi Continuity (u.h.c.) at x: for any pair of sequences {x n } and {y n } with x n x and x n Γ (y n ) there exists a subsequence of {y n } that converges to a point y Γ (x). Lower Hemi Continuity (l.h.c.) at x: for any sequence {x n } with x n x and for every y Γ (x) there exists a sequence {y n } with x n Γ (y n ) such that y n y. Continuous at x: if Γ is both upper and lower hemi continuous at x Introduction to Dynamic Optimization Nr. 15

29 Max Examples h (x) = max f (x, y) y Γ(x) G (x) = arg max f (x, y) y Γ(x) ex 1: f (x, y) =xy; X =[ 1, 1] ; Γ (x) =X. G (x) = h (x) = x { 1} x<0 [ 1, 1] x =0 {1} x>0 Introduction to Dynamic Optimization Nr. 16

30 ex 2: f (x,y ) =xy 2 ;X = [ 1, 1] ; Γ (x) =X G (x) = h (x) = max{0,x} {0} x<0 [ 1, 1] x =0 { 1, 1} x>0 Intro duction to Dynamic Optimization Nr. 16A

31 Theorem of the Maximum Define: h (x) = max f (x, y) y Γ(x) G (x) = arg max f (x, y) y Γ(x) = {y Γ (x) : h (x) =f (x, y)} Theorem. (Berge) Let X R l and Y R m. Let f : X Y R be continuous and Γ : X Y be compact-valued and continuous. Then h : X R is continuous and G : X Y is non-empty, compact valued, and u.h.c. Introduction to Dynamic Optimization Nr. 17

32 lim max max lim Theorem. Suppose {f n (x, y)} and f (x, y) are concave in y and f n f in the sup-norm (uniformly). Define g n (x) = arg max n (x, y) y Γ(x) g (x) = arg max f (x, y) y Γ(x) then g n (x) g (x) for all x (pointwise convergence); if X is compact then the convergence is uniform. Introduction to Dynamic Optimization Nr. 18

33 UsesofCorollaryofCMThm Monotonicity of v Theorem. Assume that F (,y) is increasing, that Γ is increasing, i.e. Γ (x) Γ (x 0 ) for x x 0. Then, the unique fixed point v satisfying v = Tv is increasing. If F (,y) is strictly increasing, so is v. Introduction to Dynamic Optimization Nr. 19

34 Proof BythecorollaryoftheCMThm,itsuffices to show Tf is increasing if f is increasing. Let x x 0 : Tf (x) = max {F (x, y)+βf (y)} y Γ(x) = F (x, y )+βf (y ) for some y Γ (x) F (x 0,y )+βf (y ) since y Γ (x) Γ (x 0 ) If F (,y) is strictly increasing max y Γ(x 0 ) {F (x, y)+βf (y)} = Tf (x0 ) F (x, y )+βf (y ) <F(x 0,y )+βf (y ). Introduction to Dynamic Optimization Nr. 20

35 Concavity (or strict) concavity of v Theorem. Assume that X is convex, Γ is concave, i.e. y Γ (x), y 0 Γ (x 0 ) implies that y θ θy 0 +(1 θ) y Γ (θx 0 +(1 θ) x) Γ x θ for any x, x 0 X and θ (0, 1). Finally assume that F is concave in (x, y). Then, the fixed point v satisfying v = Tv is concave in x. Moreover, if F (,y) is strictly concave, so is v. Introduction to Dynamic Optimization Nr. 21

36 Differentiability can t use same strategy:space of differentiable functions is not closed many envelope theorems Formula: if h (x) is differentiable and y is interior then h 0 (x) =f x (x, y) right value... but is h differentiable? one answer (Demand Theory) relies on f.o.c. and assuming twice differentiability of f won t work for us since f = F (x, y)+βv (y) and we don t even know if f is once differentiable! going in circles Introduction to Dynamic Optimization Nr. 22

37 Benveniste and Sheinkman First a Lemma... Lemma. Suppose v (x) is concave and that there exists w (x) such that w (x) v (x) and v (x 0 )=w(x 0 ) in some neighborhood D of x 0 and w is differentiable at x 0 (w 0 (x 0 ) exists) then v is differentiable at x 0 and v 0 (x 0 )= w 0 (x 0 ). Proof. Since v is concave it has at least one subgradient p at x 0 : w (x) w (x 0 ) v (x) v (x 0 ) p (x x 0 ) Thus a subgradient of v is also a subgradient of w. But w has a unique subgradient equal to w 0 (x 0 ). Introduction to Dynamic Optimization Nr. 23

38 Benveniste and Sheinkman Now a Theorem Theorem. Suppose F is strictly concave and Γ is convex. If x 0 int (X) and g (x 0 ) int (Γ (x 0 )) then the fixed point of T, V, is differentiable at x and V 0 (x) =F x (x, g (x)) Proof. We know V is concave. Since x 0 int (X ) and g (x 0 ) int (Γ (x 0 )) then g (x 0 ) int (Γ (x)) for x D a neighborhood of x 0 then W (x) =F (x, g (x 0 )) + βv (g (x 0 )) and t henw (x) V (x) and W (x 0 )=V (x 0 ) and W 0 (x 0 )=F x (x 0,g(x 0 )) so the r esult f ollows from the lemma. Introduction to Dynamic Optimization Nr. 24

39 Recursive Methods Introduction to Dynamic Optimization Nr. 1

40 Outline Today s Lecture finish off: theorem of the maximum Bellman equation with bounded and continuous F differentiability of value function application: neoclassical growth model homogenous and unbounded returns, more applications Introduction to Dynamic Optimization Nr. 2

41 Our Favorite Metric Space with S = ½ ¾ f : X R, f is continuous, and kf k sup f (x) < x X ρ (f, g) = kf gk sup f (x) g (x) x X (Tv)(x) = max {F (x, y)+ βv (y)} y Γ(x) Assume that F is bounded and continuous and that Γ is continuous and has compact range. Theorem 4.6. T maps the set of continuous and bounded functions S into itself. Moreover T is a contraction. Introduction to Dynamic Optimization Nr. 3

42 Proo f. That T maps the s et of continuous a nd bo unded f ollo w f rom the Theorem of Maximum (we do this next) That T is a contraction Blackwell sufficient conditions monotonicity, notice that for f v Tv (x) = max {F (x, y)+ βv (y)} y Γ(x) = F (x, g (x)) + βv (g (x)) {F (x, g (y)) + βf (g (x))} discounting: for a > 0 max {F (x, y)+ βf (y)} = Tf (x) y Γ(x) T (v + a)(x) = max {F (x, y)+ β (v (y)+ a)} y Γ(x) = m ax y Γ(x) {F (x, y)+ βv (y)} + βa = T (v)(x)+ βa. Introduction to Dynamic Optimization Nr. 4

43 Theorem of the Maximum wan t T to map c ontinuous f unction into continuous f unctions (Tv)(x) = max {F (x, y)+ βv (y)} y Γ(x) want to learn about optimal policy of RHS of Bellman G (x) = arg max {F (x, y)+ βv (y)} y Γ(x) First, continuity concepts for correspondences... then, a few example maximizations... finally, Theorem of the Maximum Introduction to Dynamic Optimization Nr. 5

44 Continuity Notions for Correspondences assume Γ is non-empty and compact valued (the set Γ (x) is non empty and compact for all x X) Upper Hemi Continuity (u.h.c.) at x: for any pair of sequences {x n } and {y n } with x n x and x n Γ (y n ) there exists a subsequence of {y n } that converges to a point y Γ (x). Lower Hemi Continuity (l.h.c.) at x: for any sequence {x n } with x n x and for every y Γ (x) there exists a sequence {y n } with x n Γ (y n ) such that y n y. Continuous at x: if Γ is both upper and lower hemi continuous at x Introduction to Dynamic Optimization Nr. 6

45 Max Examples h (x) = max f (x, y) y Γ(x) G (x) = arg max f (x, y) y Γ(x) ex 1: f (x, y) = xy; X = [ 1, 1] ; Γ (x) = X. G (x) = h (x) = x { 1} x< 0 [ 1, 1] x = 0 {1} x> 0 Introd uction to Dynamic O ptimization N r. 7

46 ex 2: f (x, y) = xy 2 ; X =[ 1, 1] ; Γ (x) = X {0} x< 0 G (x) = [ 1, 1] x = 0 { 1, 1} x> 0 h (x) = max {0,x} Introd uction to Dynamic O ptimization Nr. 7A

47 Theorem of the Maximum Define: h (x) = max f (x, y) y Γ(x) G (x) = arg max f (x, y) y Γ(x) = {y Γ (x) : h (x) = f (x, y)} Theorem 3.6. (Berge) Let X R l and Y R m. Let f : X Y R be continuous and Γ : X Y be compact-valued and continuous. Then h : X R is continuous and G : X Y is non-empty, compact valued, and u.h.c. Introduction to Dynamic Optimization Nr. 8

48 lim max max lim Theorem 3.8. Suppose {f n (x, y)} and f (x, y) are concave in y that and Γ is convex and compact valued. Then if f n f in the sup-norm (uniformly). Define g n (x) = arg max y Γ(x) f n (x, y) g (x) = arg max f (x, y) y Γ(x) then g n (x) g (x) for all x (pointwise convergence); if X is compact then the convergence is uniform. Introduction to Dynamic Optimization Nr. 9

49 Uses of Corollary ofcmthm Monotonicity of v Theorem 4.7. Assume that F (,y) is increasing, that Γ is increasing, i.e. Γ (x) Γ (x 0 ) for x x 0. Then, the unique fixed point v satisfying v = Tv is increasing. If F (,y) is strictly increasing, so is v. Introduction to Dynamic Optimization Nr. 10

50 Proof By the corollary of the CMThm, it suffices to show Tf is increasing if f is increasing. Let x x 0 : Tf (x) = max {F (x, y)+ βf (y)} y Γ(x) since y Γ (x) Γ (x 0 ) If F (,y) is strictly increasing = F (x, y )+ βf (y ) for some y Γ (x) F (x 0,y )+ βf (y ) max y Γ(x 0 ) {F (x, y)+ βf (y)} = Tf (x 0 ) F (x, y )+ βf (y ) < F (x 0,y )+ βf (y ). Introduction to Dynamic Optimization Nr. 11

51 Concavity (or strict) concavity of v Theorem 4.8. Assume that X is convex, Γ is concave, i.e. y Γ (x), y 0 Γ (x 0 ) implies that y θ θy 0 +(1 θ) y Γ (θx 0 +(1 θ) x) Γ x θ for any x, x 0 X and θ (0, 1). Finally assume that F is concave in (x, y). Then, the fixed point v satisfying v = Tv is concave in x. Moreover, if F (,y) is strictly concave, so is v. Introduction to Dynamic Optimization Nr. 12

52 convergence of policy functions with concavity of F and convexity of Γ optimal policy correspondence G (x) is actually a continuous function g (x) since v n v uniformly g n g (Theorem 4.8) we can use this to derive comparative statics Introduction to Dynamic Optimization Nr. 13

53 Differentiability can t use same strategy as with monotonicty or concavity: space of differentiable functions is not closed many envelope theorems, imply differentiability of h h (x) = max f (x, y) y Γ(x) always if formula: if h (x) is differentiable and there exists a y int (Γ (x)) then h 0 (x) = f x (x, y)...but is h differentiable? Introd uction to Dynamic O ptimization N r. 14

54 one a pp roach ( e.g. Demand Theo ry) r elies o n s moothness of Γ and f (twice differentiability) use f.o.c. and implicit function theorem won t w ork f or u s s in ce f (x, y) = F (x, y)+ βv (y) don t k no w i f f is once differentiable yet! going in circles...

55 Benveniste and Sheinkman First a Lemma... Lemma. Suppose v (x) is concave and that there exists w (x) such that w (x) v (x) and v (x 0 )= w (x 0 ) in some neighborhood D of x 0 and w is differentiable at x 0 (w 0 (x 0 ) exists) then v is differentiable at x 0 and v 0 (x 0 )= w 0 (x 0 ). Proo f. Since v is concave i t h as at least o ne subgradient p at x 0 : w (x) w (x 0 ) v (x) v (x 0 ) p (x x 0 ) Thus a subgradient of v is also a subgradient of w. But w has a unique subgradient equal to w 0 (x 0 ). Introduction to Dynamic Optimization Nr. 15

56 Benveniste and Sheinkman Now a Theorem Theorem. Suppose F is strictly concave and Γ is convex. If x 0 int (X) and g (x 0 ) int (Γ (x 0 )) then the fixed point of T, V, is differentiable at x and V 0 (x) = F x (x, g (x)) Proof. We know V is concave. Since x 0 int (X) and g (x 0) int (Γ (x 0)) then g (x 0 ) int (Γ (x )) fo r x D a n e i g h bo rh ood o f x 0 then W (x) = F (x, g (x 0 )) + βv (g (x 0 )) and then W (x ) V (x ) and W (x 0 )= V (x 0 ) and W 0 (x 0 )= F x (x 0,g (x 0 )) so the r esult follo ws from the l emma. Introduction to Dynamic Optimization Nr. 16

57 Recursive Methods Recursive Methods Nr. 1

58 Outline Today s Lecture Anything goes : Boldrin Montrucchio Global Stability: Liapunov functions Linear Dynamics Local Stability: Linear Approximation of Euler Equations Recursive Methods Nr. 2

59 Anything Goes treat X = [0, 1] R case for simplicity take any g (x) :[0, 1] [0, 1] that is twice continuously differentiable on [0, 1] g 0 (x) and g 00 (x) exists and are bounded define W (x, y) = 1 2 y 2 + yg (x) L 2 x 2 Lemma: W is strictly concave for large enough L Recursive Methods Nr. 3

60 Proof W (x, y) = 1 2 y2 + yg (x) L 2 x2 W 1 W 2 = yg 0 (x) Lx = y + g (x) W 11 W 22 W 12 = yg 00 (x) L = 1 = g 0 (x) thus W 22 < 0; W 11 < 0 is satisfied if L max x g 00 (x) W 11 W 22 W 12 W 21 = yg 00 (x)+ L g 0 (x) 2 > 0 L>g 0 (x) 2 + yg 00 (x) then L > [max x g 0 (x) ] 2 +max x g 00 (x) will do. Recursive Methods Nr. 4

61 Decomposing W (in a concave way) define V (x) = W (x, g (x)) and F so that W (x, y) = F (x, y)+ βv (y) that is F (x, y) = W (x, y) βv (y). Lemma: V is strictly concave Pro of: immediate since W is concave and X is convex. Computing the second derivative is useful anyway: V 00 (x) = g 00 (x) g (x)+ g 0 (x) 2 L since g [0, 1] then clearly our bound on L implies V 00 (x) < 0. Recursive Methods Nr. 5

62 Concavity of F Lemma: F is concave for β h 0, β i for some β > 0 F 11 (x, y) = W 11 (x, y) = yg 00 (x) L F 12 (x, y) = W 12 (x, y) = 1 F 22 (x, y) = W 22 βv 22 = 1 β h i g 00 (x) g (x)+ g 0 (x) 2 L (yg 00 (x) L) F 11 F 22 F 12 2 > 0 ³ h i 1 β g 00 (x) g (x)+ g 0 (x) 2 L g 0 (x) 2 > 0 Recursive Methods Nr. 6

63 ... concavity of F Let η 1 (β) = min η 2 (β) = min x,y x,y ( F 22) F11 F 22 F 12 2 η (β) = min {H 1 (β),h 2 (β)} 0 for β =0 η (β) > 0. η is continuous (Theorem of the Maximum) exists β h > 0 such that H (β) 0 for all β 0, β i. Recursive Methods Nr. 7

64 Monotonicity Use W (x, y) = 1 2 y 2 + yg (x) L 1 2 x 2 + L 2 x L 2 does not affect second derivatives claim: F is monotone for large enough L 2 Recursive Methods Nr. 8