# 4374

Ql. Consider the dynamic programming total harvest maximization problem from Sec-tion 15 of your notes, with the same conventions. In particular, assume that F(x) is concave, lies above the replacement line y = x if x E (0, K), F(0) = 0, F(K) = K, Su is the smallest positive x such that F'(x) = 1 and recall the equations

F(x), if x < Sm V(x,T —1) = Sm + F(Sm), if x > Sm,

which implied that optimal harvest at the second to last stage is:

(a) Show that

= {0, x — SM,

if x .0 SAf if x > Sm. •

OV(x,T —1) ?ix) 1, if x < Sm Ox 1, ifx>Sm.

(1)

(2)

(3)

(b) Changing the maximization variable from harvest h to the escapement variable s := x — h show that at stage T — 2 the optimality equation (OP) becomes V (x,T —2) = sTit [h v(F(x-h),T —1)] = az [v (F(s),7″ —1)— d+x. (OP) (c) Defining v(s) := [V (F(s),T —1) — s], exploit the equation (3) to show that

Attachments:

Assignment-3.pdf