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Problem:

suppose I have the following maximization problem: \begin{align} &\max_{(c_t)_{t \geq 0}}E_0\left[\sum^{\infty}_{t=0}\beta^{t}\ln c_t\right]\\[2mm] \text{s.t.} \quad & k_{t+1}=A^{1-\alpha}_tk^{\alpha}_t - c_t\\[2mm] & c_t \in [0,A^{1-\alpha}_tk^{\alpha}_t] \end{align} with $\beta \in (0,1)$ and $\alpha\in (0,1)$. I know that I am supposed to “guess” the value function and I have this $V(A_t,k_t)=X+Y \ln(A_t)+Z \ln(k_t)$.

Derive the values of the constants.

This value function comes from bellman equation that is not given.

We are given this other bit:

$\ln A_{t+1}=\rho \ln A_t +\epsilon _{t+1}$ $\rho \in (0,1)$

My question: should I start off this problem by setting up a Lagrangian function and then calculating the values or no?

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  • $\begingroup$ You should start by writing down the proper problem. $\endgroup$
    – clueless
    Jul 1, 2020 at 13:12
  • $\begingroup$ Thank you for bringing this to my attention. I have made the necessary edits. $\endgroup$
    – Tony456
    Jul 1, 2020 at 21:24
  • $\begingroup$ Maybe this helps people.brandeis.edu/~ghall/econ182/_build/html/… $\endgroup$
    – clueless
    Jul 3, 2020 at 9:07
  • $\begingroup$ Thank you for the helpful link. I still need to derive the values of the constants in my value function, but this greatly helped me. $\endgroup$
    – Tony456
    Jul 3, 2020 at 13:13

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