1.2. Bellman e quation Let Tt∈Rn,ut∈Rk,ft:Rn×Rk→R,and9t:Rn×Rk×Rm→Rn. Consider E∑(x,) t=0 s t where 0<B< 1, Et E Rm is a random vector unknown until period t, Et is the expectation operator conditional on period t information dt. Here ut is the control and t is the state Example 3.1. Consider a consumer who, at each time t, consumes ct units of goods and has assets At, with an endowment of Ao. The consumer may save at a gross return Rt. The consumer's problem is 64)≡maxE∑a(a) st.A+1=R+1(A-c), given Ao Example 3.2. Consider a firms problem max Eo y =(1+)(=at-s) s.t. 94+1=f(kt, It, Et+1) k+1=(1-0)k+st where St is saving, lt is labor input, kt is capital stock, wt is wage rate, and r is the interest rate Let Vi(t) max EtBs-tfs(rs, us) s.t.as+1=gs(as, us, Es+1), at is given and known1.2. Bellman Equation Let xt ∈ Rn, ut ∈ Rk, ft : Rn × Rk → R, and gt : Rn × Rk × Rm → Rn. Consider V0(x0) ≡ max u0, u1,... E0 [∞ t=0 βt ft(xt, ut) s.t. xt+1 = gt(xt, ut, εt+1), x0 is given and known, (3.1) where 0 < β < 1, t ∈ Rm is a random vector unknown until period t, Et is the expectation operator conditional on period t information Φt. Here ut is the control and xt is the state. Example 3.1. Consider a consumer who, at each time t, consumes ct units of goods and has assets At, with an endowment of A0. The consumer may save at a gross return Rt. The consumer’s problem is V0(A0) ≡ max c0,c1,... E0 [∞ t=0 βt u(ct) s.t. At+1 = Rt+1(At − ct), given A0. Example 3.2. Consider a firm’s problem: max {lt, st} E0 [∞ t=0  1 1 + r t (yt − wtlt − st) s.t. yt+1 = f(kt, lt, εt+1), kt+1 = (1 − δ)kt + st, where st is saving, lt is labor input, kt is capital stock, wt is wage rate, and r is the interest rate.  Let Vt(xt) ≡ max ut, ut+1,... Et [∞ s=t βs−t fs(xs, us) s.t. xs+1 = gs(xs, us, εs+1), xt is given and known. (3.2) 3—2