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

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

<p>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.</p>

<p>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 <em>recursive</em> 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.  </p>

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

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

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

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

CC BY-SA 4.0
2019-02-13T11:22:41.717Z
This line is gold: "meaning that the whole many-periods or even infinite-horizon problem can be broken down to a two-period problem"
Jinhua Wang
CC BY-SA 3.0
2017-03-16T19:56:31.987Z
2017-03-16T19:56:31.987Z
Alecos Papadopoulos
<blockquote>
  <p>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}</p>
  
  <p>s.t. the following restrictions</p>
  
  <p>$\begin{equation}
\begin{split}
\tilde{c}_t&amp;=(1-\delta )Y_t+a_t-\frac{\tilde{a}_{t+1}}{R_t}\\
\tilde{c}_{t+1+s}&amp;=(1-\delta )Y_{t+1+s}^{e}+\tilde{a}_{t+1+s}-\frac{\tilde{a}_{t+2+s}}{R_{t+1+s}}, \forall s \geq 0
\end{split}
\end{equation}$</p>
  
  <p>where:</p>
  
  <p>$\tilde{c}_t$: Consumption at time $t$</p>
  
  <p>$\tilde{a}_t$: Financial wealth at time $t$</p>
  
  <p>$Y_t$: Income at time $t$</p>
  
  <p>$R_t$: Nominal interest rate at time $t$</p>
  
  <p>$Y_t^e$: Expected income at time $t$</p>
  
  <p>Consider that $u(\tilde{c}_{t+s})$ is defined by the isoelastic utility function: $u(\tilde{c}_{t+s})=\frac{\tilde{c}_{t+s}^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}}$</p>
  
  <p>Find the optimal policy function for $\tilde{c}_t$.</p>
</blockquote>

<p>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?</p>

CC BY-SA 3.0
2017-03-14T20:05:04.903Z
I would substitute constraint 1 for a(t+1+s) in constraint 2. Then plug constraint 2 into the objective. Then write down the bellman equation.
Tobias
CC BY-SA 3.0
2017-03-14T20:57:50.530Z
Thanks! The results from this is $\tilde{c}_{t+s}=\frac{(1-\delta )Y_{t+1+s}^{e}}{R_{t+s}}+(1-\delta )Y_{t+s}+a_{t+s}-\frac{\tilde{c}_{t+1+s}}{R_{t+s}}-\frac{\tilde{a}_{t+2+s}}{R_{t+1+s}}
$, then you said that I should plug this into the objective function, right? But how could I write the Bellman Equation? I thought the maximization problem always need to have a constraint to write the Bellman Equation. Sorry, I'm just a beginner with dynamic programming.
Renzo GA
<dynamic-programming><bellman-equations>
Bellman equation for this dynamic programming problem
Renzo GA
# Bellman equation for this dynamic programming problem

(Asked by: username_0 on 2017-03-14)

> 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?

**Tags:** dynamic-programming, bellman-equations

> **Comments:**
> **username_1**: I would substitute constraint 1 for a(t+1+s) in constraint 2. Then plug constraint 2 into the objective. Then write down the bellman equation.  
> **username_0**: Thanks! The results from this is $\tilde{c}_{t+s}=\frac{(1-\delta )Y_{t+1+s}^{e}}{R_{t+s}}+(1-\delta )Y_{t+s}+a_{t+s}-\frac{\tilde{c}_{t+1+s}}{R_{t+s}}-\frac{\tilde{a}_{t+2+s}}{R_{t+1+s}} $, then you said that I should plug this into the objective function, right? But how could I write the Bellman Equation? I thought the maximization problem always need to have a constraint to write the Bellman Equation. Sorry, I'm just a beginner with dynamic programming.  

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## 1 answers

### Answer by username_2 on 2017-03-16. Score: 3

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

$$V(\tilde a\_t) = u[h(\tilde a\_t)] + \beta V\left[\big(\tilde a\_t+(1-\delta )Y\_t-h(\tilde a\_t)\big)\cdot R\_t\right]$$

> **Comments:**
> **username_3**: This line is gold: "meaning that the whole many-periods or even infinite-horizon problem can be broken down to a two-period problem"  

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