<p>As far as I know, the only method that works is to <em>guess and verify</em>: You guess the functional form of the value function <span class="math-container">$J(x)$</span> and verify that it indeed satisfies the Bellman equation.</p>
<p>Of course, depending on how lucky you are, guessing might be a rather time consuming approach.</p>
<ul>
<li><p>In some cases, you can make a more educated guess by solving several finite horizon versions of the problem, i.e. solve,
<span class="math-container">\begin{align*}
J_T(x) = \max \sum_{t = 0}^T u(c_t) \text{ s.t. } &amp;p_t x_{t+1} = (p_t + d_t)x_t-c_t.\\
&amp;x_0 \text{ given }
\end{align*}</span>
for values <span class="math-container">$T = 1,2,\dots$</span>. If you are able to get a closed form solution for <span class="math-container">$J_T(x)$</span>, you could try a similar functional form for the general problem <span class="math-container">$J(x)$</span>.</p>
</li>
<li><p>Sometimes, it is also possible to get <em>shape restrictions</em> on the function <span class="math-container">$J(x)$</span>. For example, you might be able to show that <span class="math-container">$J$</span> is monotone, or concave, or homogeneous. This sometimes restricts the class of possible solutions.</p>
</li>
<li><p>Finally, you could solve the problem numerically and then try to fit a particular shape (e.g. linear in <span class="math-container">$\ln(x)$</span>). This can guide you to a more particular functional form.</p>
</li>
</ul>

CC BY-SA 4.0
2023-08-28T11:11:26.267Z
Thanks a lot for your answer - I'll try out the finite horizon approach soon. Do you already know if it is possible to solve for the value function in the above example model?
Eddie
CC BY-SA 4.0
2023-08-28T05:55:53.517Z
2023-08-28T05:55:53.517Z
tdm
<p>Given the simplest form of a Lucas model, i.e., a Bellman equation given by
<span class="math-container">\begin{align}
    J(x_t) &amp; = \max_{c_t, x_{t+1}} \{ u(c_t) + \beta E_{\pi} [ J(x_{t+1})] \} \\
    &amp; \textrm{ s.t. } p_{t}x_{t+1} = (p_t + d_t)x_t - c_t, \nonumber 
\end{align}</span>
with a FOC given by
<span class="math-container">\begin{align}
     u'(c_t)p_t = \beta E_{\pi} [u'(c_{t+1}) (p_{t+1}+d_{t+1})],
\end{align}</span>
is there any way to calculate the value function <span class="math-container">$J(x_t)$</span> explicitly(i.e. without numerical solution algorithms)?</p>
<p>More generally, when is it possible to calculate the value function from a Bellman set-up explicitly, and what are the standard approaches to do this?</p>
<p>Thanks a lot in advance!</p>
<hr />
<p>Just to explain the above variables:</p>
<p><span class="math-container">$c_t$</span> : consumption,
<span class="math-container">$x_t$</span> : risky asset holdings,
<span class="math-container">$d_t$</span> : dividends,
<span class="math-container">$\pi$</span> : probability law for <span class="math-container">$d_{t+1}$</span>,
<span class="math-container">$u(\cdot)$</span> : standard utility function</p>

<mathematical-economics><asset-pricing><bellman-equations>
When could value functions in Bellman equations be calculated explicitly?
Eddie
# When could value functions in Bellman equations be calculated explicitly?

(Asked by: username_0 on 2023-08-26)

Given the simplest form of a Lucas model, i.e., a Bellman equation given by
\begin{align}
J(x\_t) & = \max\_{c\_t, x\_{t+1}} \{ u(c\_t) + \beta E\_{\pi} [ J(x\_{t+1})] \} \\
& \textrm{ s.t. } p\_{t}x\_{t+1} = (p\_t + d\_t)x\_t - c\_t, \nonumber
\end{align}
with a FOC given by
\begin{align}
u'(c\_t)p\_t = \beta E\_{\pi} [u'(c\_{t+1}) (p\_{t+1}+d\_{t+1})],
\end{align}
is there any way to calculate the value function $J(x\_t)$ explicitly(i.e. without numerical solution algorithms)?

More generally, when is it possible to calculate the value function from a Bellman set-up explicitly, and what are the standard approaches to do this?

Thanks a lot in advance!

---

Just to explain the above variables:

$c\_t$ : consumption,
$x\_t$ : risky asset holdings,
$d\_t$ : dividends,
$\pi$ : probability law for $d\_{t+1}$,
$u(\cdot)$ : standard utility function

**Tags:** mathematical-economics, asset-pricing, bellman-equations

---

## 1 answers

### Answer by username_1 on 2023-08-28. Score: 2

As far as I know, the only method that works is to *guess and verify*: You guess the functional form of the value function $J(x)$ and verify that it indeed satisfies the Bellman equation.

Of course, depending on how lucky you are, guessing might be a rather time consuming approach.

* In some cases, you can make a more educated guess by solving several finite horizon versions of the problem, i.e. solve,
  \begin{align\*}
  J\_T(x) = \max \sum\_{t = 0}^T u(c\_t) \text{ s.t. } &p\_t x\_{t+1} = (p\_t + d\_t)x\_t-c\_t.\\
  &x\_0 \text{ given }
  \end{align\*}
  for values $T = 1,2,\dots$. If you are able to get a closed form solution for $J\_T(x)$, you could try a similar functional form for the general problem $J(x)$.
* Sometimes, it is also possible to get *shape restrictions* on the function $J(x)$. For example, you might be able to show that $J$ is monotone, or concave, or homogeneous. This sometimes restricts the class of possible solutions.
* Finally, you could solve the problem numerically and then try to fit a particular shape (e.g. linear in $\ln(x)$). This can guide you to a more particular functional form.

> **Comments:**
> **username_0**: Thanks a lot for your answer - I'll try out the finite horizon approach soon. Do you already know if it is possible to solve for the value function in the above example model?  

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