2. Hamilton Method for Dynamic Optimization Theorem A-Z( Special Model I.Letx∈R,u∈R,g:R"x]×R→R R"xRxR→R.For 991). See also Chiang(1992) The following theorem is from Kamien-Schwartz( 1981, p 16) (x4)≡mJ几 Theorem A-6. For H: RXRXR-R, consider problem s.t. i(n=gr().u(t).I x(l0)=xx()≥0 HLu(o),u(oldt define the Hamiltonian as u(to)=lo, I(T) H=f(x,a,1)+A·g(xa where the set of admissible controls is Under certain differentiability conditions, if ur' is a solution, then there exists a fimction A=(continuously differentiable functions u: I'o, T-R A: [o, T]-R such that t'is a solution of If H is continuous w.r. L. its first argument, continuously differentiable w.r. It. its second and H=0 third arguments, then the solution t' must satisfy the euler equation (A.1) If the terminal ralue idT)is free, the transversality condition is (A.12) HA T, u(T)i(n=0 Theorem A-8( Special Model Il.ietr∈R,u∈R,g:R"xRxR→R"and f: RxKXR→R. For problem If the initial value u(to)is free. the transversality condition is H[,'()i’4)=0 A.6) J(xx,4)≡max几x(a"d If the terminal condition is u(T)>0, the transversality conditions are st. x=gx(n), u(o). x()=xxT)20 I'(TH u().i'(T)=0. H:T, u(T),i(T)Iso define the Hamiltonian as Comersely, if H(t, u, i) is concave inu, i ), then any I'E A satisfying the Euler equation H=f(x,a)+A·g(xL,l A.4)and the initial and terminal conditions is a solution of(A3). There is no SOSC, there is only a SONC Under certain differentiability conditions, if ui is a solution, then there exists a finction A: to, T]+IR such that u is a solution of Legendre Condition:Hs【,u(n)i(m)≤0. Example A-3. The mJ中-y(a (A8) with transversality conditie ∫叫y(a24+ mkre=0,XT)≥0. (A.15) Example A-4. Consider Example A-5. Consider consumer's probl (+( J(4)≡ma stx(0>0 st A=rA+y-c-3- 2. Hamilton Method for Dynamic Optimization A good reference for this section is Kamien–Schwartz (1991). See also Chiang (1992). The following theorem is from Kamien-Schwartz (1981, p.16). Theorem A-6. For : k k H × × → , consider problem 0 0 0 max [ ( ) ( )] st ( ) ( ) , T u A t T H t u t u t dt ut u uT u ∈ , , .. = , = ∫  (Α.3) where the set of admissible controls is {continuously differentiable functions [ ] 0 }k A u tT ≡ : , → . If H is continuous w.r.t. its first argument, continuously differentiable w.r.t. its second and third arguments, then the solution u∗ must satisfy the Euler equation: [ ( ) ( )] [ ( ) ( )] u u d H tu t t H tu t t u u dt ∗ ∗ ∗ ∗ ,, =,, .    (Α.4) If the terminal value u T( ) is free, the transversality condition is [ ( ) ( )] 0 H Tu T T u u ∗ ∗ , , =.   (Α.5) If the initial value 0 u t is free, the transversality condition is ( ) 00 0 [ ( ) ( )] 0 Htut t u u ∗ ∗ , , =.   (Α.6) If the terminal condition is u T() 0 ≥ , the transversality conditions are ( ) [ ( ) ( )] 0 [ ( ) ( )] 0 u u u TH Tu T T H Tu T T u u ∗∗ ∗ ∗ ∗ , , =, , , ≤ .     (Α.7) Conversely, if H tuu ( ) , ,  is concave in ( ) u u, ,  then any u A ∗ ∈ satisfying the Euler equation (A.4) and the initial and terminal conditions is a solution of (A.3). There is no SOSC; there is only a SONC: * * Legendre Condition: [ , ( ), ( )] 0. H tu t u t uu   ≤ Example A-3. The principal’s problem is ( ) [ ] [ ] max ( ) ( ) st ( ) ( ) ( ) a s v y s y f y a dy u s y fy a dy ca u , ⋅ − , .. , ≥ + . ∫ ∫ (Α.8) Example A-4. Consider ( ) { () () } ( ) ( ) max st 0 x u x v x dF x θ θ θθ θθ θ θ ⋅ ′ ⎡ ⎤⎡ ⎤ ,+ , ⎣ ⎦⎣ ⎦ . . ≥ . ∫ (Α.9) -4- Theorem A-7 (Special Model I). Let n k nk n xug ∈ , ∈ ,: × × → and n k f :×× → . For problem 0 0 0 0 0 ( ) max [ ( ) ( ) ] st ( ) [ ( ) ( ) ] () () 0 T T u t Jx x t f x t u t t dt x t g xt ut t xt x xT , , ≡ , , .. = , , = , ≥ ∫  define the Hamiltonian as H f = ,, + () () xut λ⋅ g xut ,, . Under certain differentiability conditions, if u∗ is a solution, then there exists a function 0 [ ] n λ : , t T → such that u∗ is a solution of 0 Hu = , (Α.10) λ =−Hx,  (Α.11) with transversality conditions: lim 0 ( ) 0 t T λ λ x T → = , ≥ . (Α.12) Theorem A-8 (Special Model II). Let n k nk n xug ∈ , ∈ ,: × × → and n k f :×× → . For problem 0 0 ( ) 0 0 0 0 ( ) max [ ( ) ( )] st ( ) [ ( ) ( ) ] () () 0 T t t T u t J xxt f x t u t e dt xt gxt ut t xt x xT − − θ , , ≡ , .. = , , = , ≥ ∫  define the Hamiltonian as H = ,+ f () ( ) x u λ⋅ g xut ,, . Under certain differentiability conditions, if u∗ is a solution, then there exists a function 0 [ ] n λ : , t T → such that u∗ is a solution of 0 Hu = , (Α.13) λ θλ = −Hx,  (Α.14) with transversality condition lim 0 ( ) 0 t t T xe T θ λ λ − → = , ≥ . (Α.15) Example A-5. Consider consumer’s problem: 0 0 0 ( ) max ( ) s t (0) t c J A u c e dt A rA y c A A −ρ ≡ .. = + − = . ∫