3
$\begingroup$

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. It is on p.37 (Appendix A).

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.