<p>More of a comment:</p>

<p>There should be an expectation operator in the statement of the problem, otherwise problem doesn't make sense.</p>

<p>That "...the deterministic and stochastic value function must be the same..." is not quite right. The value of $\sigma^2$ is crucial in the restriction</p>

<p>\begin{align}
\rho = \left(-n + \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right)(1-\alpha\gamma).
\end{align}</p>

<p>If $\sigma^2 = 0$, then presumably $\rho &lt; 0$ for economically reasonable $\alpha$ and $\gamma$, in which case the deterministic problem may be ill-posed. What is true is that the stochastic value function takes the given form only if the parameter restriction holds.</p>

<p>Factoring out the Ito term $\frac{1}{2} \sigma^2$ from the right hand side</p>

<p>$$
\sigma^2\left(1 - \frac{\alpha\gamma}{2}\right) (1-\alpha\gamma),
$$</p>

<p>the restriction can be written as</p>

<p>$$
\rho + n (1-\alpha\gamma) = \frac{1}{2} \sigma^2 [ (1-\alpha\gamma) - (-\left(1 - \alpha\gamma\right)^2)].
$$</p>

<p>On the right hand side, we have a elasticity of intertemporal substitution term $(1-\alpha\gamma)$ and a risk aversion term $-\left(1 - \alpha\gamma\right)^2$. What the restriction says is that, with a particular choice of $\sigma$, they offset each other, up to time preference $\rho$ and the drift $n(1 - \alpha\gamma)$. Therefore the value function is independent of $\sigma$.</p>

<p>That the value function is independent of $\sigma$ is an artifact of the restriction, and choice of CRRA $u$. Not true in general.</p>

CC BY-SA 3.0
2017-07-27T15:30:57.620Z
2017-07-27T15:30:57.620Z
Michael
<blockquote>
  <p>Literature: See <a href="http://www.jstor.org/stable/1911845?seq=1#page_scan_tab_contents" rel="nofollow noreferrer">Chang (1988)</a> for theoretical part and <a href="http://www.princeton.edu/~moll/HACTproject/HACT_Numerical_Appendix.pdf" rel="nofollow noreferrer">Achdou et
  al. (2015)</a> for numerical part respectively.</p>
</blockquote>

<h2>Model</h2>

<p>Consider the following stochastic optimal growth problem in per capita notation.
\begin{align}
&amp;\max_{c}\int^\infty_0 e^{-\rho t}u(c)dt\\
\text{s.t.}~~~&amp;  dk = [f(k) - (n-\sigma^2) k - c]dt - \sigma kdz\\
&amp;c\in[0,f(k)]\\
&amp;k(0) = k_0
\end{align}
everthing is standard except for $dz$ which is the increment of a standard Wiener process, i.e. $z(t)\sim\mathcal{N}(0,t)$. The population growth rate has mean $n$ and variance $\sigma^2$.</p>

<h2>Analytical Solution</h2>

<p>We presume Cobb-Douglas technology
\begin{align}
f(k) = k^\alpha,\quad \alpha\in(0,1)
\end{align}</p>

<p>and CRRA utility
\begin{align}
u(c) = \frac{c^{1-\gamma}}{1-\gamma},\quad \gamma &gt; 1.
\end{align}
Set up the Hamilton-Jacobi-Bellman equation (HJB-e)
\begin{align}
\rho v(k) = \max_c\left\{\frac{c^{1-\gamma}}{1-\gamma} + v'(k)(k^\alpha - (n - \sigma^2)k - c) + v''(k)\frac{k^2\sigma^2}{2}\right\}
\end{align}</p>

<p>The first order condition (FOC) reads
\begin{align}
c = v'(k)^{-\frac{1}{\gamma}}=:\pi(k)
\end{align}
where $\pi(\cdot)$ denotes the policy function.</p>

<p>Resubstitute FOC into HJB-e
\begin{align}
\rho v(k) = \frac{v'(k)^{\frac{\gamma-1}{\gamma}}}{1-\gamma} + v'(k)k^\alpha - v'(k)(n - \sigma^2)k  -  v'(k)^{\frac{\gamma-1}{\gamma}} + v''(k)\frac{k^2\sigma^2}{2}.
\end{align}</p>

<p>We guess a functional form of $v(k)$ with (<a href="http://www.sciencedirect.com/science/article/pii/S0304407609001456" rel="nofollow noreferrer">Posch (2009, eq. 41)</a>)
\begin{align}
v(k) = \Psi \frac{k^{1-\alpha\gamma}}{1-\alpha\gamma}
\end{align}</p>

<p>where $\Psi$ is some constant. The first and second order derivative of $v$ are given by
\begin{align}
v'(k) &amp;= \Psi k^{-\alpha\gamma}\\
v''(k) &amp;= -\alpha\gamma\Psi k^{-1-\alpha\gamma}.
\end{align}</p>

<p>The HJB-e then reads
\begin{align}
&amp;\rho \Psi\frac{k^{1-\alpha\gamma}}{1-\alpha\gamma} = \frac{\Psi^{\frac{\gamma-1}{\gamma}}k^{\alpha(1-\gamma)}}{1-\gamma} + \Psi k^{\alpha(1-\gamma)} - (n-\sigma^2) \Psi k^{1-\alpha\gamma} -  \Psi^{\frac{\gamma-1}{\gamma}} k^{\alpha(1-\gamma)}  - \alpha\gamma\Psi k^{1-\alpha\gamma}\frac{\sigma^2}{2}\\[2mm]
\Longleftrightarrow \quad &amp;
k^{1-\alpha\gamma}\left(\frac{\rho}{1-\alpha\gamma} + n - \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right) = k^{\alpha(1-\gamma)}\left[1+\Psi^{-\frac{1}{\gamma}}\frac{\gamma}{1-\gamma}\right]
\end{align}</p>

<p>The maximized HJB-e is true iff the following conditions hold
\begin{align}
\rho = \left(-n + \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right)(1-\alpha\gamma)\quad \wedge \quad \Psi = \left(\frac{\gamma-1}{\gamma}\right)^{-\gamma}
\end{align}</p>

<p>Resubstitute $\Psi$ into $v$ which finally gives the true value function
\begin{align}
v(k) = \left(\frac{\gamma-1}{\gamma}\right)^{-\gamma} \frac{k^{1-\alpha\gamma}}{1-\alpha\gamma}.
\end{align}</p>

<ul>
<li>How come that $v$ does not depend on $\sigma$? </li>
</ul>

<p>So the deterministic and stochastic value function must be the same. The policy function is then readily given by (use FOC and derivative of value function)</p>

<p>\begin{align}
\pi(k) = \left(1-\frac{1}{\gamma}\right)k^\alpha.
\end{align}</p>

<p>Note that this function does not depend on $\sigma$ either.</p>

<h2>Numerical Approximation</h2>

<p>I solved the HJB-e by an upwind scheme. Error tolerance $\epsilon=1e-10$. In the figure below I plot the policy function for varying $\sigma$. For $\sigma\to 0$ I arrive at the true solution (purple). But for $\sigma&gt;0$ the approximated policy function deviates from the true one. Which should not be the case, since $\pi(k)$ does not depend on $\sigma$, right?  </p>

<ul>
<li>Can anyone confirm that the approximated policy functions should be the same for any $\sigma$, since the true one is independent of $\sigma$?</li>
</ul>

<p><a href="https://i.sstatic.net/ne5BA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ne5BA.png" alt="enter image description here"></a></p>

CC BY-SA 3.0
2016-01-29T23:53:13.273Z
What bothers me here is the first "iff" condition after you write "the maximized  HJB-e is true iff the following conditions hold": this is a very specific _equality_ relation that must hold between _all the parameters of the model_ -preference parameters, population growth, capital productivity and volatility. I wonder:can we really work with guessed functions whose validity depend on such a very narrow condition on the parameters?
Alecos Papadopoulos
CC BY-SA 3.0
2016-01-31T18:07:50.287Z
Well, here I actually fix $\rho = \rho(\alpha, \gamma, n, \sigma)$ as a function of the four remaining parameters. So the equation is always true if in addition, $\rho > 0$ holds. I wonder: is there some rule when guessing a function is not allowed? I mean, we are interested in finding the true solution and under some specific conditions we obtain the true solution. I'm not sure what bothers you here from a theoretical point of view? Sure, it may limits empirical work, but that's not the point here. We are rather interested in solving the HJBe and that can be done. If an empiricist (1/2)
clueless
CC BY-SA 3.0
2016-01-31T18:10:09.307Z
estimates $\{\alpha, \gamma,n,\rho,\sigma\}$ and we find that the condition $\rho = ....$ is violated, then we may reject the model. However, the solution remains true in principle. (2/2)
clueless
CC BY-SA 3.0
2016-01-31T19:20:06.390Z
My concern is not about empirical validity. What I wonder is, to what extent the specific guess about the _functional form_ of the value function is dependent on this relation between the parameters. Without reference to any empirical data, if we assume that the relation does not hold, what then? Should we guess a value function that is not even exponential in $k$, or would it suffice to keep the exponential structure but try different ways to include the parameters in it? (by the way, I am also looking into your main question, since this discussion is probably peripheral)
Alecos Papadopoulos
CC BY-SA 3.0
2017-07-14T19:03:50.527Z
Are you sure the optimization problem is stated correctly? There is no, for example, expectation operated on say, $f(k)$? As it is stated now, $k$ and therefore $f(k)$ likely assume any value given the Wiener process $z$.
Hans
<academic-graduate><continuous-time><optimal-control><numerical-methods>
Stochastic growth in continuous time
clueless
# Stochastic growth in continuous time

(Asked by: username_0 on 2015-09-12)

> Literature: See [Chang (1988)](http://www.jstor.org/stable/1911845?seq=1#page_scan_tab_contents) for theoretical part and [Achdou et
> al. (2015)](http://www.princeton.edu/~moll/HACTproject/HACT_Numerical_Appendix.pdf) for numerical part respectively.

## Model

Consider the following stochastic optimal growth problem in per capita notation.
\begin{align}
&\max\_{c}\int^\infty\_0 e^{-\rho t}u(c)dt\\
\text{s.t.}~~~& dk = [f(k) - (n-\sigma^2) k - c]dt - \sigma kdz\\
&c\in[0,f(k)]\\
&k(0) = k\_0
\end{align}
everthing is standard except for $dz$ which is the increment of a standard Wiener process, i.e. $z(t)\sim\mathcal{N}(0,t)$. The population growth rate has mean $n$ and variance $\sigma^2$.

## Analytical Solution

We presume Cobb-Douglas technology
\begin{align}
f(k) = k^\alpha,\quad \alpha\in(0,1)
\end{align}

and CRRA utility
\begin{align}
u(c) = \frac{c^{1-\gamma}}{1-\gamma},\quad \gamma > 1.
\end{align}
Set up the Hamilton-Jacobi-Bellman equation (HJB-e)
\begin{align}
\rho v(k) = \max\_c\left\{\frac{c^{1-\gamma}}{1-\gamma} + v'(k)(k^\alpha - (n - \sigma^2)k - c) + v''(k)\frac{k^2\sigma^2}{2}\right\}
\end{align}

The first order condition (FOC) reads
\begin{align}
c = v'(k)^{-\frac{1}{\gamma}}=:\pi(k)
\end{align}
where $\pi(\cdot)$ denotes the policy function.

Resubstitute FOC into HJB-e
\begin{align}
\rho v(k) = \frac{v'(k)^{\frac{\gamma-1}{\gamma}}}{1-\gamma} + v'(k)k^\alpha - v'(k)(n - \sigma^2)k - v'(k)^{\frac{\gamma-1}{\gamma}} + v''(k)\frac{k^2\sigma^2}{2}.
\end{align}

We guess a functional form of $v(k)$ with ([Posch (2009, eq. 41)](http://www.sciencedirect.com/science/article/pii/S0304407609001456))
\begin{align}
v(k) = \Psi \frac{k^{1-\alpha\gamma}}{1-\alpha\gamma}
\end{align}

where $\Psi$ is some constant. The first and second order derivative of $v$ are given by
\begin{align}
v'(k) &= \Psi k^{-\alpha\gamma}\\
v''(k) &= -\alpha\gamma\Psi k^{-1-\alpha\gamma}.
\end{align}

The HJB-e then reads
\begin{align}
&\rho \Psi\frac{k^{1-\alpha\gamma}}{1-\alpha\gamma} = \frac{\Psi^{\frac{\gamma-1}{\gamma}}k^{\alpha(1-\gamma)}}{1-\gamma} + \Psi k^{\alpha(1-\gamma)} - (n-\sigma^2) \Psi k^{1-\alpha\gamma} - \Psi^{\frac{\gamma-1}{\gamma}} k^{\alpha(1-\gamma)} - \alpha\gamma\Psi k^{1-\alpha\gamma}\frac{\sigma^2}{2}\\[2mm]
\Longleftrightarrow \quad &
k^{1-\alpha\gamma}\left(\frac{\rho}{1-\alpha\gamma} + n - \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right) = k^{\alpha(1-\gamma)}\left[1+\Psi^{-\frac{1}{\gamma}}\frac{\gamma}{1-\gamma}\right]
\end{align}

The maximized HJB-e is true iff the following conditions hold
\begin{align}
\rho = \left(-n + \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right)(1-\alpha\gamma)\quad \wedge \quad \Psi = \left(\frac{\gamma-1}{\gamma}\right)^{-\gamma}
\end{align}

Resubstitute $\Psi$ into $v$ which finally gives the true value function
\begin{align}
v(k) = \left(\frac{\gamma-1}{\gamma}\right)^{-\gamma} \frac{k^{1-\alpha\gamma}}{1-\alpha\gamma}.
\end{align}

* How come that $v$ does not depend on $\sigma$?

So the deterministic and stochastic value function must be the same. The policy function is then readily given by (use FOC and derivative of value function)

\begin{align}
\pi(k) = \left(1-\frac{1}{\gamma}\right)k^\alpha.
\end{align}

Note that this function does not depend on $\sigma$ either.

## Numerical Approximation

I solved the HJB-e by an upwind scheme. Error tolerance $\epsilon=1e-10$. In the figure below I plot the policy function for varying $\sigma$. For $\sigma\to 0$ I arrive at the true solution (purple). But for $\sigma>0$ the approximated policy function deviates from the true one. Which should not be the case, since $\pi(k)$ does not depend on $\sigma$, right?

* Can anyone confirm that the approximated policy functions should be the same for any $\sigma$, since the true one is independent of $\sigma$?

[![enter image description here](https://i.sstatic.net/ne5BA.png)](https://i.sstatic.net/ne5BA.png)

**Tags:** academic-graduate, continuous-time, optimal-control, numerical-methods

> **Comments:**
> **username_1**: What bothers me here is the first "iff" condition after you write "the maximized  HJB-e is true iff the following conditions hold": this is a very specific _equality_ relation that must hold between _all the parameters of the model_ -preference parameters, population growth, capital productivity and volatility. I wonder:can we really work with guessed functions whose validity depend on such a very narrow condition on the parameters?  
> **username_0**: Well, here I actually fix $\rho = \rho(\alpha, \gamma, n, \sigma)$ as a function of the four remaining parameters. So the equation is always true if in addition, $\rho > 0$ holds. I wonder: is there some rule when guessing a function is not allowed? I mean, we are interested in finding the true solution and under some specific conditions we obtain the true solution. I'm not sure what bothers you here from a theoretical point of view? Sure, it may limits empirical work, but that's not the point here. We are rather interested in solving the HJBe and that can be done. If an empiricist (1/2)  
> **username_0**: estimates $\{\alpha, \gamma,n,\rho,\sigma\}$ and we find that the condition $\rho = ....$ is violated, then we may reject the model. However, the solution remains true in principle. (2/2)  
> **username_1**: My concern is not about empirical validity. What I wonder is, to what extent the specific guess about the _functional form_ of the value function is dependent on this relation between the parameters. Without reference to any empirical data, if we assume that the relation does not hold, what then? Should we guess a value function that is not even exponential in $k$, or would it suffice to keep the exponential structure but try different ways to include the parameters in it? (by the way, I am also looking into your main question, since this discussion is probably peripheral)  
> **username_2**: Are you sure the optimization problem is stated correctly? There is no, for example, expectation operated on say, $f(k)$? As it is stated now, $k$ and therefore $f(k)$ likely assume any value given the Wiener process $z$.  

---

## 1 answers

### Answer by username_3 on 2017-07-27. Score: 1

More of a comment:

There should be an expectation operator in the statement of the problem, otherwise problem doesn't make sense.

That "...the deterministic and stochastic value function must be the same..." is not quite right. The value of $\sigma^2$ is crucial in the restriction

\begin{align}
\rho = \left(-n + \sigma^2\left(1 - \frac{\alpha\gamma}{2}\right)\right)(1-\alpha\gamma).
\end{align}

If $\sigma^2 = 0$, then presumably $\rho < 0$ for economically reasonable $\alpha$ and $\gamma$, in which case the deterministic problem may be ill-posed. What is true is that the stochastic value function takes the given form only if the parameter restriction holds.

Factoring out the Ito term $\frac{1}{2} \sigma^2$ from the right hand side

$$
\sigma^2\left(1 - \frac{\alpha\gamma}{2}\right) (1-\alpha\gamma),
$$

the restriction can be written as

$$
\rho + n (1-\alpha\gamma) = \frac{1}{2} \sigma^2 [ (1-\alpha\gamma) - (-\left(1 - \alpha\gamma\right)^2)].
$$

On the right hand side, we have a elasticity of intertemporal substitution term $(1-\alpha\gamma)$ and a risk aversion term $-\left(1 - \alpha\gamma\right)^2$. What the restriction says is that, with a particular choice of $\sigma$, they offset each other, up to time preference $\rho$ and the drift $n(1 - \alpha\gamma)$. Therefore the value function is independent of $\sigma$.

That the value function is independent of $\sigma$ is an artifact of the restriction, and choice of CRRA $u$. Not true in general.


---

