---
tags: macro
---

# Simple Bellman

## The Question

Maximize the lifetime log utility with Cobb-Douglas production function and fully depreciation.

\begin{align*}
\max_{\{c_t, k_t\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t \log(c_t)\\
\text{s.t. }c_t +k_t = Ak_{t-1}^\alpha
\end{align*}

The corresponding Bellman equation is

\begin{align*}
v (k) =\max_{c, k'} \{ \log(c) + \beta v(k') \}\\
\text{s.t. }c = Ak^\alpha - k'
\end{align*}

## Guess and Verify

Substitute the budget constraint into the objective function, we have

\begin{align*}
v (k) =\max_{k'} \{ \log(Ak^\alpha - k') + \beta v(k') \} \\
\end{align*}

Guess $v(k) = a + b \log k$ and plug it into the Bellman equation,

$$
v (k) =\max_{k'} \{ \log(Ak^\alpha - k') + \beta (a + b\log k') \} \\
$$

The first-order condition of the right-hand side is,

$$
\frac{1}{Ak^\alpha - k'} = \frac{\beta b}{k'} \implies k' = \frac{\beta b }{1 + \beta b} Ak^\alpha,
$$

which implies the agent will invest/save a constant portion of their income. Plug it into the Bellman equation,

\begin{align*}
v(k) &= \log \left( \frac{1}{1+\beta b} Ak^\alpha\right) + \beta [a + b \log (k')] \\
&= \log \left( \frac{1}{1+\beta b} Ak^\alpha \right)+ \beta [a + b \log (\frac{\beta b }{1 + \beta b} Ak^\alpha)] \\
&=a + b \log k
\end{align*}

Let's match the coefficients. For the coefficient of $\log k$

$$
b = \alpha + \beta b \alpha \implies b = \frac{\alpha}{1-\alpha \beta}.
$$

$a$ is also solvable, which is a mess. The more important is the policy function,

$$
k' = \frac{\beta b }{1 + \beta b} Ak^\alpha = \alpha \beta Ak^\alpha.
$$