# Stochastic growth in continuous time

Literature: See Chang (1988) for theoretical part and Achdou et al. (2015) 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)) \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 $\sigma$? • 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? Jan 29 '16 at 23:53
• 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) Jan 31 '16 at 18:07
• 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) Jan 31 '16 at 18:10
• 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) Jan 31 '16 at 19:20
• 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
Jul 14 '17 at 19:03

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.