文档介绍：Math Review for . Students
ECO 1011H (Fall, 2002)
by Shouyong Shi
Lecture III: Dynamic Optimization
Reading materials:
. Dixit (2nd edition), chapter 10.
Stokey, Lucas with Prescott, chapter 6.
Sargent, chapter 9, pp. 176-196.
1. The Hamiltonian and the maximum principle
(1) An example: A planning problem
8 PT
<> max t=1 U(ct)
. ct + kt+1 F(kt), t = 1; 2; :::; T
:> ¹ · ¹
k1 = k1, kT +1 = kT +1:
The decisions: a sequence of consumption (c ; c ; :::; c ) and a sequence of capital
² 1 2 T
accumulation (k2; k3; :::; kT ).
The initial condition: k = k¹ , given by some unspeci¯ed history.
² 1 1
The terminal condition: k = k¹ , to be bequeathed to the future.
² T +1 T +1
(2) More general speci¯cation:
8 PT
<> max t=1 U(ct; kt; t)
(P) . kt+1 kt F(ct; kt; t), t = 1; 2; :::; T
:> ¹ ¡ · ¹
k1 = k1, kT +1 = kT +1.
Notes:
²
a. Transform the previous problem into the above form.
b. (P) allows the objective function to depend on time directly (., time discounting)
and the constraint function to be time-variant (., if there are exogenous changes
in technology over time).
Stock (state) variables versus °ow (control) variables in a particular period t:
²
a. kt is a stock variable in period t, since it is ¯xed in period t.
b. ct is a °ow variable, since it can be chosen in period t.
c. Intertemporal link: kt+1 is also a °ow variable in period t, but it will be a state
variable next period.
Observation: At the beginning of t = 1, all c's and k's (except k1 and kT +1) are choices.
² This problem can be solved using the Lagrangian method.
(3) Lagrangian Method
XT
L(C; K; ¸) = [U(ct; kt; t) + ¸t[F (ct; kt; t) kt+1 + kt]] ;
t=1 ¡
where C = (c1; c2; :::; cT ) and K = (k1; k2; :::; kT +1).
First-order conditions
²
for ct: Uc(ct; kt; t) + ¸tFc(ct; kt; t) = 0, t = 1; 2; :::; T;
for kt: Uk(ct; kt; t) + ¸tFk(ct; kt; t) = ¸t 1 ¸t, t = 2; :::; T;
¡
for ¸ : F (c ; k ; t) k + k 0; ¸ 0,¡with c:s: