The solution of(1. 1)is the solution of the Bellman equation Vi(at= max f(at, ut)+BEV+1(a+1) 9(x Conversely, the solution u*, a* of (3.3) is the solution of (1. 1) if the transversality condition is satisfi lim BEtv+1(x+1)=0. Problem(3.3) imply two equations fa(xt,t)+BE1[Vt+1(xt+1)9a(x;,,e+1)=0 (3.5) v(at=ft,(ct, ut)+BEt Vi+(=+1)gt, (at, Ut, Et+1)I (36) We look for a solution of the form: u*= h(t). By(2. 15)and(2. 2), we can generally find the two functions h and Vt Theorem 3. 1. If ft=f, gt=g and Et is a Markov process, then Vt is time invariant By Theorem 3.1, a solution (ht, Vt) from(2.15) and(2.2) will be time invariant u*=h(at)and V(t=V(t) Example 3.3. Solve the consumer's problem in Example 3. 1. Assuming Rt to be a first-order Markov process. The Euler equation is u'(ct)=B Etla(c+)++1, If u(c)=In c, the optimal consumption has the feedback form: ct=(1-B)At Example 3. 4. Solve the firms problem in Example 3.2. 1.3. Lagrange Method Theorem 3.2.(Kuhn-Tucker). For differentiable f: Rn-R and G: Rn- Rm, letC(A,x)≡f(x)+入G(x).Ifx* is a solution of max f(a) st.G(x)≥0, then there exists AE R such that(Kuhn-Tucker condition)A G(a)=0 and FOC:D2C(A,x)=0.■The solution of (1.1) is the solution of the Bellman equation: Vt(xt) ≡ max ut ft(xt, ut) + βEtVt+1(xt+1) (3.3) s.t. xt+1 = gt(xt, ut, εt+1). Conversely, the solution {u∗ t , x∗ t } of (3.3) is the solution of (1.1) if the transversality condition is satisfied: lim t→∞ βt EtVt+1(x∗ t+1)=0. (3.4) Problem (3.3) imply two equations: ft,u(xt, ut) + βEt  V 0 t+1(xt+1)gt,u(xt, ut, εt+1)  = 0, (3.5) V 0 t (xt) = ft,x(xt, ut) + βEt  V 0 t+1(xt+1)gt,x(xt, ut, εt+1)  . (3.6) We look for a solution of the form: u∗ t = ht(xt). By (2.15) and (2.2), we can generally find the two functions ht and Vt. Theorem 3.1. If ft = f, gt = g and {εt} is a Markov process, then Vt is time invariant.  By Theorem 3.1, a solution (ht, Vt) from (2.15) and (2.2) will be time invariant: u∗ t = h(xt) and Vt(xt) = V (xt). Example 3.3. Solve the consumer’s problem in Example 3.1. Assuming {Rt} to be a first-order Markov process. The Euler equation is: u0 (ct) = β Et[u0 (ct+1)Rt+1], (3.7) If u(c) = ln c, the optimal consumption has the feedback form: ct = (1 − β)At.  Example 3.4. Solve the firm’s problem in Example 3.2. 1.3. Lagrange Method Theorem 3.2. (Kuhn-Tucker). For differentiable f : Rn → R and G : Rn → Rm, let L(λ, x) ≡ f(x) + λ · G(x). If x∗ is a solution of maxx f(x) s.t. G(x) ≥ 0, then there exists λ ∈ Rm + such that (Kuhn-Tucker condition) λ ·G(x∗)=0 and FOC: DxL(λ, x∗ )=0.  3—3