Full disclosure: this problem was part of a final exam that none of our class could really solve definitively. Below the general form is a specific utility function we worked with that I'll try to replicate my work for. Any help on the solution method would be good; specifically, it seems Euler's equations/steady states would be better for this particular form of question rather than using a Bellman, but any guidance on either method would be appreciated.
Suppose a representative agent is solving
$$ \begin{align} & \max_{\left\{ c_t, w_t \right\}^\infty_{t=0}} \sum_{t=0}^\infty \beta^t u(c_t) \\ & \text{s.t.} \\ & w_t = Rw_{t-1} - c_t \\ & c_t \geq 0 \\ & w_t \geq 0 \\ \end{align} $$
For all $t$, and $w_{-1}$. Additionally, $0 < \beta < 1$, $R > 1$.
$c_t$ is consumption at time $t$, and $w_t$ is wealth at the end of time $t$. $\beta$ is a discount factor.
Let $$u(c_t) = \phi \mathrm{e}^ {\theta c_t^p}$$ where $\phi, \theta, p$ are constant parameters.
My first question is what values of the constants $\phi, \theta$, and $p$ are needed to ensure that $u(c_t)$ are strictly increasing and strictly concave in $c_t$, for all values of $c_t$. Later on the problem asks us to assume those above conditions.
So taking the first and second derivative of $u(c_t)$ gives us:
$$ \begin{align} u'(c_t) & = \phi \theta p c_t^{p-1}\mathrm{e}^ {\theta c_t^p} \\ u''(c_t) & = \phi \theta p(p-1) c_t^{p-2}\mathrm{e}^ {\theta c_t^p} + \phi \theta ^2 p^2 c_t^{(p-1)^2}\mathrm{e}^ {\theta c_t^p} \\ & = \phi \theta p c_t^{p-1} \mathrm{e}^{\theta c_t^p}\left[ \frac{p-1}{c_t} + \theta p c_t^{p-1} \right] \\ \end{align}$$
So $u'(c_t) > 0$ in order to satisfy the strictly increasing constraint. $u''(c_t) < 0$ must be true for strict concavity. I suggested that $\phi, \theta > 0$ and $p = 1$ for this to be true. I'm not sure how this ugly second derivative can be simplified to find the appropriate conditions.
After doing this, we had to express Bellman's equation and Euler's equation for the problem.
Bellman:
$$\boxed{V(w) = \max_\tilde{w} \left[ \phi \mathrm{e}^{\theta (Rw - \tilde{w})^p} + \beta V(\tilde{w})\right]}$$
Euler:
$$\boxed{\sum_{t=0}^\infty \beta^t (Rw_{t-1} - w_t)^p = \cdots \beta^t (Rw_{t-1} - w_t)^p + \beta^{t+1}(Rw_t - w_{t+1})^p + \cdots }$$
Finally, we got to say that $\beta \cdot R = 1$. We were then asked to find the value function and optimal decision rules for our choice variable(s). I used the Bellman to try to solve the problem, where I guessed the function form of the value function:
$$V(\tilde{w}) = F \mathrm{e}^{M\tilde{w}} + H$$
where $F, M, H$ are undetermined coefficients. After substituting this into the Bellman and putting in appropriate arguments, taking the derivative to get an optimal choice, substituting the optimal choices back into the Bellman, I got stuck trying to solve for $F, M, H$ in a way that would give me a sensible consumption choice at the end.
The little trick with $\beta R = 1$ is supposed to make consumption constant across all periods, so I assume that the point of getting the Euler's equation is so we can solve it using steady states and make life easier, but my algebra there hasn't borne any fruit either it seems.
For the dynamic programming method, have I set the functional form of the value function correctly? If not, what should be the guess? And finally, how does all the nasty algebra work out?
For the steady states method, what does the path of wealth look like? Does it end up constant as well? (I suppose that question applies for the Bellman approach as well.)
(If it's pedagogically interesting or useful--or if you don't trust that I'm not begging for homework help--I can show my work for both methods.)
Please let me know if anything is unclear.