45 Stock-Flow problem
Lagrangean
factor out y_t in Lagrangean
take derivative with respect to y_t
where $Y_{t+1} - Y_{t} = Q_{t}$.
Hamiltonian
For each period , maximize H by choosing Z_t Y_t in peroid t.
47 Maximum principle
- maximum principle
p150 Transform dynamic problem into a equence of static problems that are connectec by inter temporal euqations.
(i) p150 interpretation of Hamiltonian
“Thus the Hamiltonina offeres a simple way of altering the one-period objective function F(y,z, t) to take into account the future consequences of the choice of controls z at t.”
“zt will change F and y_t+1. “
- F term
- $\pi Q(y, z, t) $
- Intertemporal connector for $\pi$. “$\pi$ connector “
Or by the Envolope thereom: only direct effect of theta, induced effect zero.
Since
- “$Y$ connector “
And
where $Y_{t+1} - Y_{t} = Q_{t}$.
Then
- Hamiltonian FOC.
a.
“The effect of z_t on y_t+1, equals its effects on Q(y,z,t), and the resulting chagne in the objective function is found by multiplying this by the shadow price $\pi_{t+1}$ of y_t+1”
“Thus the Hamiltonina offeres a simple way of altering the one-period objective function F(y,z, t) to take into account the future consequences of the choice of controls z at t.”
b.
” These can be thought of as a dividend. The change in price $ \pi_{t+1} -\pi_{t} $ is like a capital gain, except that the prices are in present-value terms, so $\pi_{t+1} $ contains an extra discount factor that captures the usual interest or opportunity cost of carrying y_t for one period. “
“When y_t is optimum, the overall marginal return ,or the sum of these components, should be zero. That is just what (10.8) express when written as “
“This is an intertemporal no-arbitrage condition”
c.
Solve
-
clculate a. b. c. for each period
-
start in T
-
solve backwards,
48 Example
P 156
example 10.2 optimum growth