<p>Problem:</p>
<p>suppose I have the following maximization problem:
<span class="math-container">\begin{align}
&amp;\max_{(c_t)_{t \geq 0}}E_0\left[\sum^{\infty}_{t=0}\beta^{t}\ln c_t\right]\\[2mm]
\text{s.t.} \quad &amp; k_{t+1}=A^{1-\alpha}_tk^{\alpha}_t - c_t\\[2mm]
&amp; c_t \in [0,A^{1-\alpha}_tk^{\alpha}_t]
\end{align}</span>
with <span class="math-container">$\beta \in (0,1)$</span> and <span class="math-container">$\alpha\in (0,1)$</span>.
I know that I am supposed to “guess” the value function and I have this <span class="math-container">$V(A_t,k_t)=X+Y \ln(A_t)+Z \ln(k_t)$</span>.</p>
<p>Derive the values of the constants.</p>
<p>This value function comes from bellman equation that is not given.</p>
<p>We are given this other bit:</p>
<p><span class="math-container">$\ln A_{t+1}=\rho \ln A_t +\epsilon _{t+1}$</span> <span class="math-container">$\rho \in (0,1)$</span></p>
<p>My question: should I start off this problem by setting up a Lagrangian function and then calculating the values or no?</p>

CC BY-SA 4.0
2020-07-01T13:12:21.420Z
You should start by writing down the proper problem.
clueless
CC BY-SA 4.0
2020-07-01T21:24:56.940Z
Thank you for bringing this to my attention. I have made the necessary edits.
Tony456
CC BY-SA 4.0
2020-07-03T09:07:12.753Z
Maybe this helps http://people.brandeis.edu/~ghall/econ182/_build/html/stochastic_dynamic_programming.html
clueless
CC BY-SA 4.0
2020-07-03T13:13:17.363Z
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.
Tony456
<macroeconomics><bellman-equations><recursive-macroeconomics>
How does one find values of constants in a value function?
Tony456
# How does one find values of constants in a value function?

(Asked by: username_0 on 2020-07-01)

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?

**Tags:** macroeconomics, bellman-equations, recursive-macroeconomics
> **Comments:**
> **username_1**: You should start by writing down the proper problem.  
> **username_0**: Thank you for bringing this to my attention. I have made the necessary edits.  
> **username_1**: Maybe this helps http://people.brandeis.edu/~ghall/econ182/_build/html/stochastic_dynamic_programming.html  
> **username_0**: 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.  

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## 0 answers

