<p>I think the &quot;standard&quot; approach in solving HJB equations like this is to guess a form for <span class="math-container">$V$</span> and then verify. Your first order conditions for <span class="math-container">$c/a$</span> and <span class="math-container">$k/a$</span> are right.</p>
<p>Now, guess that <span class="math-container">$V(a, q) = K\log(a)$</span>. With this you get</p>
<p><span class="math-container">\begin{align}
c &amp;= \frac{a}{K},\\
\frac{k}{a} &amp;= \frac{mq - r}{\delta(q)^2},\\
V'(.) &amp;= \frac{K}{a}.
\end{align}</span></p>
<p>The verification step goes as follows. For notational simplicity, define <span class="math-container">$\xi \equiv \frac{mq - r}{\delta(q)}$</span> as the market price of risk (the Sharpe ratio). Plug in your guess of <span class="math-container">$V$</span> into the HJB and rearrange to get</p>
<p><span class="math-container">$$
\underbrace{\rho K\log(a)}_{\text{LHS of HJB}} = \underbrace{\log(a) - \log K + Kr - 1 + \frac{1}{2}K\xi^2}_{\text{RHS of HJB}}.
$$</span>
Comparing the terms involving <span class="math-container">$a$</span> on the LHS and RHS, we must have <span class="math-container">$\rho K = 1$</span>, which when plugged into the FOC's above gives you the FOC's in the text.</p>

CC BY-SA 4.0
2023-03-20T10:57:34.123Z
2023-03-20T10:57:34.123Z
Wittgenstein's Poker
<blockquote>
<p>Reis (2021) `The constraint on public debt when <span class="math-container">$r &lt; g$</span> but <span class="math-container">$g &lt; m$</span>' :
HJB:
<span class="math-container">$$\rho V(a, q) = \max_{c/a, k/a} [\log c + V'(a,q)[r + (mq-r)\frac{k}{a} - \frac{c}{a}] a + \frac{V''(a,q)}{2} (k/a)^2 \delta^2 a^2 + v \int ((V(a,q') - V(a,q)) dQ(q'|q)]$$</span>
where <span class="math-container">$V'(.) = \partial V(.)/\partial a$</span>. It is standard to derive that at an optimum:
<span class="math-container">$$c = \rho a$$</span>
<span class="math-container">$$\frac{k}{a} = \frac{mq-r}{\delta(q)^2}$$</span>
<span class="math-container">$$V'(.) = \frac1{\rho a}$$</span>
<span class="math-container">$$ \lim_{t\to\infty} e^{-\rho t} V'(.) a_t = 0$$</span></p>
</blockquote>
<p>The author says `standard', but I am having difficulty in deriving these FOC. Using <span class="math-container">$\log c/a + \log a$</span>, I can get <span class="math-container">$c/a = 1/aV'$</span>. Also, it is easy to get <span class="math-container">$k/a = -(mq-r)V'/V'' \delta^2 a$</span>. I guess the next step is to differentiate HJB w.r.t. <span class="math-container">$a$</span> (the envelope condition). This gives me
<span class="math-container">$$\rho V' = \frac1a + V'' \mu_a a + V'\mu_a + \frac{V'''}{2} \sigma_a^2 + V'' (k/a)^2 \delta^2 a + v \int (V'(a, q') - V'(a, q) dQ(q'|q)$$</span>
where <span class="math-container">$\mu_a$</span> is the drift of <span class="math-container">$da$</span> and <span class="math-container">$\sigma_a$</span> is the volatility of <span class="math-container">$da$</span>. For the third FOC in the paper to hold, the terms in right hand side should be cancelled out except <span class="math-container">$1/a$</span>. But, why is that true? Can you give some help?</p>
<p>You can find the <a href="https://personal.lse.ac.uk/reisr/papers/99-mpkrg.pdf" rel="nofollow noreferrer">paper</a>. It is on p.37 (Appendix A).</p>

<mathematical-economics><fiscal-policy><optimal-control>
stochastic optimal control/FOC/Reis(2021)
shk910
# stochastic optimal control/FOC/Reis(2021)

(Asked by: username_0 on 2023-03-19)

> Reis (2021) `The constraint on public debt when $r < g$ but $g < m$' :
> HJB:
> $$\rho V(a, q) = \max\_{c/a, k/a} [\log c + V'(a,q)[r + (mq-r)\frac{k}{a} - \frac{c}{a}] a + \frac{V''(a,q)}{2} (k/a)^2 \delta^2 a^2 + v \int ((V(a,q') - V(a,q)) dQ(q'|q)]$$
> where $V'(.) = \partial V(.)/\partial a$. It is standard to derive that at an optimum:
> $$c = \rho a$$
> $$\frac{k}{a} = \frac{mq-r}{\delta(q)^2}$$
> $$V'(.) = \frac1{\rho a}$$
> $$ \lim\_{t\to\infty} e^{-\rho t} V'(.) a\_t = 0$$

The author says `standard', but I am having difficulty in deriving these FOC. Using $\log c/a + \log a$, I can get $c/a = 1/aV'$. Also, it is easy to get $k/a = -(mq-r)V'/V'' \delta^2 a$. I guess the next step is to differentiate HJB w.r.t. $a$ (the envelope condition). This gives me
$$\rho V' = \frac1a + V'' \mu\_a a + V'\mu\_a + \frac{V'''}{2} \sigma\_a^2 + V'' (k/a)^2 \delta^2 a + v \int (V'(a, q') - V'(a, q) dQ(q'|q)$$
where $\mu\_a$ is the drift of $da$ and $\sigma\_a$ is the volatility of $da$. For the third FOC in the paper to hold, the terms in right hand side should be cancelled out except $1/a$. But, why is that true? Can you give some help?

You can find the [paper](https://personal.lse.ac.uk/reisr/papers/99-mpkrg.pdf). It is on p.37 (Appendix A).

**Tags:** mathematical-economics, fiscal-policy, optimal-control

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## 1 answers

### Answer by username_1 on 2023-03-20. Score: 3 | Accepted answer

I think the "standard" approach in solving HJB equations like this is to guess a form for $V$ and then verify. Your first order conditions for $c/a$ and $k/a$ are right.

Now, guess that $V(a, q) = K\log(a)$. With this you get

\begin{align}
c &= \frac{a}{K},\\
\frac{k}{a} &= \frac{mq - r}{\delta(q)^2},\\
V'(.) &= \frac{K}{a}.
\end{align}

The verification step goes as follows. For notational simplicity, define $\xi \equiv \frac{mq - r}{\delta(q)}$ as the market price of risk (the Sharpe ratio). Plug in your guess of $V$ into the HJB and rearrange to get

$$
\underbrace{\rho K\log(a)}\_{\text{LHS of HJB}} = \underbrace{\log(a) - \log K + Kr - 1 + \frac{1}{2}K\xi^2}\_{\text{RHS of HJB}}.
$$
Comparing the terms involving $a$ on the LHS and RHS, we must have $\rho K = 1$, which when plugged into the FOC's above gives you the FOC's in the text.


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