Consider the deterministic consumption-savings problem:
$
V(a_t)
=
\underset{c}{\max}
\int_{\tau =t}^{\tau = \infty} e^{-\rho (\tau - t) } u(c_{\tau}) d\tau
$ w/ $u(c)=\frac{c^{1-\gamma}-1}{1-\gamma}$ and $\gamma, \rho >0$
s.t.
$\frac{da}{d\tau} = \left( r a_{\tau} - c_{\tau} \right) $
Initial condition: $a(0)=a_0$ given
Terminal condition:
$ \underset{t\to\infty}{\lim} e^{-\rho t} \lambda(t) a_{t}
=\underset{t\to\infty}{\lim} e^{-\rho t} u'(c_{t}) a_{t}
=\underset{t\to\infty}{\lim} e^{-\rho t} V_{a}(a_{t}) a_{t}
=0$
Let's call the optimization problem above "Sequential Problem" SP & denote the set of solutions
$\text{Sol}(\textbf{SP}):=
\{V(a_t), c(a_t): \text{solve }\textbf{SP} \}$.
Under routine assumptions $\text{Sol}(\textbf{SP})$ is a set w/ one element.
Using the Hamiltonian we can solve a system of two BV-ODEs in closed form.
Define: $\omega \equiv \left(\frac{r-\rho}{\gamma}\right)$
$c(a_{t}) = \left(r - \omega \right) \times a_{t}$
$a_{t}=a_{0}e^{\omega t}$
$\text{TVC: } \underset{t\to\infty}{\lim} e^{-\rho t} u'(c_{t}) a_{t}
=
\underset{t\to\infty}{\lim}
e^{-\rho t} \left( (r - \omega)a_{t} \right)^{-\gamma} a_{t}
=
\underset{t\to\infty}{\lim}
e^{-\rho t} \left( a_{t} \right)^{1-\gamma}
=
\underset{t\to\infty}{\lim}
e^{-\rho t} \left( a_{0}e^{\omega t} \right)^{1-\gamma}
=
\underset{t\to\infty}{\lim}
e^{-\rho t} e^{(1-\gamma) \omega t}
=
\underset{t\to\infty}{\lim}
e^{-(\rho - (1-\gamma) \omega ) t}
=0
\Leftrightarrow
\rho - (1-\gamma) \omega >0
$
Note: $\rho - (1-\gamma) \omega = r - \omega$
TVC holds $\Leftrightarrow \rho - (1-\gamma) r >0$
$V(a_{t}) = \frac{-1}{(1-\gamma)\rho} +
\frac{1}{\rho - (1-\gamma)\omega}
\frac{\left( \left(r - \omega \right) \times a_{t} \right)^{1-\gamma} }{1-\gamma}$
Note:
$V_{a}=
\frac{\left(r - \omega \right)}{\rho - (1-\gamma)\omega}
\left( \left(r - \omega \right) \times a_{t} \right)^{-\gamma}
=
\left( \left(r - \omega \right) \times a_{t} \right)^{-\gamma}
\Rightarrow
c(a_t) =u'^{-1}(V_a)
$
HJB:
$\rho V(a_t) = \underset{c}{\max}
\{ u(c_{t}) + V_a \times (r a_{t} - c_{t}) \}$
Denote the set of solutions $\text{Sol}(\textbf{HJB})$.
FONC: $
u'(c_{t}) - V_a =0
\Rightarrow
c(a_t) = (V_a)^{-\frac{1}{\gamma}}
\Rightarrow V_a > 0
$
Plugin:
$
\Rightarrow
u(c) - cV_a
=
\frac{\gamma (V_a)^{1-\frac{1}{\gamma}}}{1-\gamma} - \frac{1}{1-\gamma}
$
SOSC: $u''(c_{t})<0
\Rightarrow u''\left( (V_a)^{-\frac{1}{\gamma}} \right)<0
$
We can combine these eqns (maximized HJB & FONC)
$
\left[\begin{array}{l}
\rho V(a_{t}) =
u(c_{t}) + V_{a}(a_{t})\times \left(ra_{t} - c_{t} \right)
\\
c(a_{t}) = (V_{a}(a_{t}))^{-\frac{1}{\gamma}}
\end{array} \right]
\Leftrightarrow
\left[\begin{array}{l}
\rho V(a_{t}) =
\frac{\gamma (V_a)^{1-\frac{1}{\gamma}}}{1-\gamma}
- \frac{1}{1-\gamma}
+ V_a \times r a_{t}
\end{array} \right]
$
DE:
$\rho V(a_t)
=
\frac{\gamma (V_a)^{1-\frac{1}{\gamma}}}{1-\gamma} - \frac{1}{1-\gamma}
+ V_a \times r a_{t} $
Denote the set of solutions $\text{Sol}(\textbf{DE})$.
DE is a non-linear equation (first-order ODE) w/ multiple solutions $V(a_t)$.
- Sol 1: $V(a)=\frac{1}{\rho (\gamma-1)}$, solves DE, if $\gamma>1$
Note: while Sol 1 solves DE it does not solve the $\max$ part of HJB, e.g. adding $V_a >0$ to DE will rule out Sol 1, but this condition isn't enough to pin down the sol to SP. - Sol 2: $V(a)=B_0 + B_1 a$, solves DE, if $r=\rho$ and $\rho B_0 = \frac{\gamma (B_1)^{1-\frac{1}{\gamma}}}{1-\gamma} - \frac{1}{1-\gamma}$.
Note: condition $V_a>0$ is satisfied if $B_1 >0$. - Sol 3: Walde 2010 claims this problem also has a strictly convex solution (I think) in a note I don't fully understand
- Sol 4: $V(a_{t}) = \frac{-1}{(1-\gamma)\rho} + \frac{1}{\rho - (1-\gamma)\omega} \frac{\left( \left(r - \omega \right) \times a_{t} \right)^{1-\gamma} }{1-\gamma}$, solves DE
We have: $\text{Sol}(\textbf{SP}) \subseteq \text{Sol}(\textbf{HJB}) \subseteq \text{Sol}(\textbf{DE}) $
Q1: what conditions does $V(a)$ need to satisfy s.t. the solution to DE is also the solution to the optimization problem SP?
- My understanding of viscosity for dummies is that while DE has many solutions, the optimal solution is the unique "viscosity solution"
Q2 why do the constant & affine solutions above not satisfy the conditions for a viscosity solution?
-the authors of "viscosity for dummies" don't provide a simple example of an HJB w/ multiple closed form solutions & show only the optimal solution satisfies the properties
