Mathematics for Economists

Dynamic programming

2. Now we move to time periodT 1 .By Bellmanís principle one should
   solve the problem
   VT 1 (w) = max
   c 2 [ 0 ,w]

p

c+VT(k(wc))

=

= max

c 2 [ 0 ,w]

p

c+

q

k(wc)



.

Obviously the goal function is concave as

p

cis concave,k(wc)is

linear hence concave and thex7!

p

xis concave and increasing. So it is a

convex KT-problem.

d

dc

p

c+

q

k(wc)



=

1

2

p

c

+

1

2

p

k(wc)

(k)= 0

p^1

c

=

p

p k

wc

,ck=wc

σT 1 (w)=c=

w

1 +k.