Question: How would I go about deriving the values for a Bellman Equation’s value function?
i.e. suppose I have the following maximization problem:
$\max_{c_t\geq 0, k_{t+1}\geq 0}$$E_0$$[\sum^{\infty}_{t=0}\beta^{t}\ln c_t]$ s.t. $k_{t+1}+c_t=A^{1-\alpha}_tk^{\alpha}_t$ $\beta \in (0,1)$; $\alpha\in (0,1)$
I know that I am supposed to “guess” the value function and I have this $V(A_t,k_t)=X+Y \ln(A_t)+Z \ln(k_t)$.
This is where the problem shows up. I have no idea as to how I would begin to derive the values of X,Y and Z to confirm the values. Can anyone “point” me in the right direction as it where?
I assume I probably have to rewrite my bellman, but I have yet to learn how to properly rewrite a bellman out of the sigma notation form.