question

Problem 2 (Growth Model): Recall the growth model that we discussed in class. We expressed the sequence problem as v(ko) = sup B' In (K; - Kil) subject to the constraint KHI E [0,K; ] = [(K.). Consider the associated Bellman equation v(k) = sup In(k" - y) + Bv(v). Finally, note that 0 x* where, x* = (expp) (1 - [1 - exp(-2p)]1/2). Hint: use a similar conceptual approach to the one that we used in class when w = 0.Problem 1 (A simple consumption problem). Consider the following sequence problem: Find v(x) such that v(ro) = sup > B' In(c.) folio 1=0 subject to a E [0, ",], Tit1 = R(x, - G) with To given. The associated Bellman Equation is given by: v(x) = sup {In(c) + Bv(R(x - c)) } Vr. CE[0,x] The associated Functional Equation is given by: (Bw)(x) = sup {In(c) + Bw(R(x - c))} Vx. CE[0,x] We also be interested in the Finite Horizon Sequence Problem (SP): Find v(x) such that 20(To) = sup B* In(C) {allo t=0 subject to G E [0, r], It+1 = R(It - G) with To given. We also have the associated finite horizon Bellman Equation: u(x) = sup {In(c) + Butti(R(x - c))} Vx CE[0,x] Note that this finite horizon set-up is the same as the infinite-horizon set-up, except that the value functions are subscripted: {up(.), ", (.), v2(.), ..., ur(-)}- Let's adopt the following notation: . ur(x) = Inr . Let vr-1(x) = (Bur)(x) . More generally, let vr-t-1(I) = (Bur-.)(x) . This notation emphasizes connection between - iterating the functional operator B - backwards induction . These two procedures are identical if the initial value function is the value function that applies in the last period of the finite-horizon game (i.e., the initial value function is In r for the current application). a. Using backward induction (starting with vr(x) = Ina) show that: CT-1= 148 AT-18 I CT-2 = 1+ 3 + 81 = AT-21 b. Using an induction argument, show that: where (MPC)Problem 1 (A simple consumption problem). Consider the following sequence problem: Find v(r) such that v(To) = sup B' In(c) folio t=0 subject to o E [0, "], Tit1 = R(r - G) with To given. The associated Bellman Equation is given by: v(x) = sup {In(c) + Bv(R(x - c))} Vr. CE[0,x] The associated Functional Equation is given by: (Bu)(x) = sup {In(c) + Bw(R(x - c))} Vr. CE[0,x] We also be interested in the Finite Horizon Sequence Problem (SP): Find v(x) such that Uo(To) = sup B' In(c) fali=o t=0 subject to q E [0, r], It41 = R(r, - G) with To given. We also have the associated finite horizon Bellman Equation: u(x) = sup {In(c) + Butti(R(x - c))} Vx CE[0,=] Note that this finite horizon set-up is the same as the infinite-horizon set-up, except that the value functions are subscripted: {vo(.), " (.), v2(-), ..., vr(.)}. Let's adopt the following notation: . Up(x) = Inr . Let Ur-1(x) = (Bur)(x) . More generally, let vr-1-1(1) = (Bur-.)(x) . This notation emphasizes connection between - iterating the functional operator B - backwards induction . These two procedures are identical if the initial value function is the value function that applies in the last period of the finite-horizon game (i.e., the initial value function is In r for the current application). a. Using backward induction (starting with vr(r) = Ina) show that: CT-1= 1+BAT-12 CT-2 = 1+3 + 8 = AT-21 b. Using an induction argument, show that: where B (MPC)