Howards policy iteration algorithm Consider the Brock-Mirman problem: to maximize E0 t=0 t ln ct, subject to ct + kt+1 Ak t t

Howard’s policy iteration algorithm Consider the Brock-Mirman problem: to maximize E0

∞

t=0

βt ln ct, subject to ct + kt+1 ≤ Akα

t θt , k0 given, A > 0, 1 >α> 0, where {θt} is an i.i.d.

sequence with ln θt distributed according to a normal distribution with mean zero and variance σ2 .

Consider the following algorithm. Guess at a policy of the form kt+1 = h0(Akα

t θt)

for any constant h0 ∈ (0, 1). Then form J0 (k0, θ0) = E0

∞

t=0

βt ln (Akα

t θt − h0Akα

t θt).

Next choose a new policy h1 by maximizing ln (Akαθ − k

) + βEJ0 (k

, θ

), where k = h1Akαθ . Then form J1 (k0, θ0) = E0

∞

t=0

βt ln (Akα

t θt − h1Akα

t θt).

Continue iterating on this scheme until successive hj have converged.

Show that, for the present example, this algorithm converges to the optimal policy function in one step.