Bellman equation for this dynamic programming problem

Question

For the following problem \begin{equation}\max_{(\tilde{c}_t,\tilde{a}_{t+1+s})}\sum_{s=0}^{\infty}\beta ^su(\tilde{c}_{t+s})\end{equation}

s.t. the following restrictions

$\begin{equation} \begin{split} \tilde{c}_t&=(1-\delta )Y_t+a_t-\frac{\tilde{a}_{t+1}}{R_t}\\ \tilde{c}_{t+1+s}&=(1-\delta )Y_{t+1+s}^{e}+\tilde{a}_{t+1+s}-\frac{\tilde{a}_{t+2+s}}{R_{t+1+s}}, \forall s \geq 0 \end{split} \end{equation}$

where:

$\tilde{c}_t$: Consumption at time $t$

$\tilde{a}_t$: Financial wealth at time $t$

$Y_t$: Income at time $t$

$R_t$: Nominal interest rate at time $t$

$Y_t^e$: Expected income at time $t$

Consider that $u(\tilde{c}_{t+s})$ is defined by the isoelastic utility function: $u(\tilde{c}_{t+s})=\frac{\tilde{c}_{t+s}^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}}$

Find the optimal policy function for $\tilde{c}_t$.

I don't know how to write the Bellman equation because I have two restrictions. What would be the optimal procedure to solve this problem?

Answer 1

The "second" constraint appears redundant and it confuses matters. Re-arrange the first one to obtain

$$\tilde{a}_{t+1} = \big[\tilde a_t+(1-\delta )Y_t-\tilde{c}_t\big]R_t$$

This tells us that wealth in the beginning of next period is fully determined by current-period decisions and known states, without any uncertainty whatsoever: we start with the given wealth at the beginning of the period, our income and the interest rate becomes known, we decide and "set aside" the full amount of consumption (for the whole period, at the beginning of the period), and the rest becomes interest-bearing asset.

Why the second constraint, which looks into the future, confuses matters? Because in order to apply Dynamic Programming/Bellman equation, it has to be the case that our optimization problem can be formulated as a recursive one, meaning that the whole many-periods or even infinite-horizon problem can be broken down to a two-period problem. And we have this condition in the above constraint, we do not need the second one.

If $V(\tilde a_t)$ is the value function, the Bellman equation is then

$$V(\tilde a_t) = \max_{\tilde c_t}\big [u(\tilde c_t) + \beta V(\tilde a_{t+1})\big]$$

and we must find the policy function $h(\tilde a_t)=\tilde c_t$ that satisfies

$$V(\tilde a_t) = u[h(\tilde a_t)] + \beta V\left[\big(\tilde a_t+(1-\delta )Y_t-h(\tilde a_t)\big)\cdot R_t\right]$$