<p>As you noticed, positive savings are only beneficial if the next period's consumption is lower. But this can only happen with positive savings if there are positive savings next period. By the same logic, there needs to be positive savings the period after, and so on. So consumption would be nonincreasing over time and <span class="math-container">$c_t&lt;e$</span>. But this is then strictly worse than consuming <span class="math-container">$e$</span> in every period.</p>

CC BY-SA 4.0
2023-11-09T06:48:59.173Z
2023-11-09T06:48:59.173Z
Michael Greinecker
<p>Consider an infinitely-lived agent’s consumption-saving problem. The agent receives <span class="math-container">$e &gt; 0$</span> units of endowment every period, can save via an asset with constant return <span class="math-container">$R$</span>. The agent is endowed with <span class="math-container">$s_0$</span> units of the asset initially. At period <span class="math-container">$t$</span>, he chooses the amount to consume <span class="math-container">$c_t$</span> and to save <span class="math-container">$s_{t+1}$</span>. At <span class="math-container">$t + 1$</span>, his total available resources are endowment e and total returnon saving <span class="math-container">$R s_{t+1}$</span>.
The agent maximizes discounted lifetime utility <span class="math-container">$\sum_{t=0}^\infty \beta^t u(c_t)$</span> with discount factor <span class="math-container">$\beta\in (0, 1)$</span>. He can not borrow, i.e., <span class="math-container">$s_{t+1} \geq 0$</span> for all <span class="math-container">$t ≥ 0$</span>. Assume that <span class="math-container">$R \in [0, \frac{1}{\beta}]$</span>, and u is strictly increasing, strictly concave, and continuously differentiable.</p>
<p>Suppose that <span class="math-container">$s_0=0$</span>. Show that <span class="math-container">$s_{t+1}^*=0$</span> for all <span class="math-container">$t\geq 0$</span> is the unique solution to the sequential problem.</p>
<p>Here is my attempt:</p>
<p>The sequential problem is
<span class="math-container">\begin{align}
\max \sum_{t=0}^\infty \beta^t u(e+Rs_t-s_{t+1})\\
s.t. s_{t+1}\in\Gamma(s_t)\\
s_0\text{ is given}
\end{align}</span></p>
<p>Suppose <span class="math-container">$s_0=0$</span>. If we want to show <span class="math-container">$s_{t+1}^*=0$</span> for all <span class="math-container">$t\geq 0$</span>. By induction hypothesis, let <span class="math-container">$s_t=0$</span>. Thus, the sequential problem at period <span class="math-container">$t$</span> is
<span class="math-container">\begin{align}
\max\sum_{\tau=t}^\infty \beta^t u(c_\tau)\\
s.t. c_t+s_{t+1}=e\\
c_{\tau}+s_{\tau+1}=Rs_{\tau}+e\\
s_{t+1}\geq 0
\end{align}</span></p>
<p>Then the Lagrangian:
<span class="math-container">$$\sum_{\tau=t}^\infty \beta^t u(c_\tau)+\lambda_t (e-s_{t+1}-c_t)+\lambda_{\tau}(Rs_{\tau}+e-c_{\tau}-s_{\tau+1})+\mu s_{t+1}$$</span></p>
<p>First order conditions:
<span class="math-container">\begin{align}
[s_{t+1}]\lambda_t=\lambda_{t+1}R+\mu\\
[c_{t}] \beta^t u'(c_t)=\lambda_t\\
[c_{t+1}]\beta^{t+1}u'(c_{t+1})=\lambda_{t+1}
\end{align}</span></p>
<p>Suppose by contrdiction, <span class="math-container">$s_{t+1}^*\neq 0\implies s_{t+1}^*&gt;0$</span>. By complementary slackness, <span class="math-container">$\mu=0\implies \lambda_t=\lambda_{t+1}R\implies u'(c_t)=\beta u'(c_{t+1})R$</span>. Since <span class="math-container">$\beta R\leq 1$</span>, we have <span class="math-container">$u'(c_t)\leq u'(c_{t+1})\implies c_t\geq c_{t+1}$</span>.</p>
<p>Since <span class="math-container">$s_{t+1}^*&gt;0$</span> and <span class="math-container">$s_t=0$</span>, by <span class="math-container">$c_t+s_{t+1}=Rs_t+e$</span>, we have <span class="math-container">$c_t&lt;e$</span>. Thus, <span class="math-container">$c_{t+1}\leq c_t&lt;e$</span>.</p>
<p>I don't know how to derive a contradiction up. Can someone help?</p>

<dynamic-programming>
How can I show that the optimal savings are 0 for all time periods?
Ludwig Gershwin
# How can I show that the optimal savings are 0 for all time periods?

(Asked by: username_0 on 2023-11-09)

Consider an infinitely-lived agent’s consumption-saving problem. The agent receives $e > 0$ units of endowment every period, can save via an asset with constant return $R$. The agent is endowed with $s\_0$ units of the asset initially. At period $t$, he chooses the amount to consume $c\_t$ and to save $s\_{t+1}$. At $t + 1$, his total available resources are endowment e and total returnon saving $R s\_{t+1}$.
The agent maximizes discounted lifetime utility $\sum\_{t=0}^\infty \beta^t u(c\_t)$ with discount factor $\beta\in (0, 1)$. He can not borrow, i.e., $s\_{t+1} \geq 0$ for all $t ≥ 0$. Assume that $R \in [0, \frac{1}{\beta}]$, and u is strictly increasing, strictly concave, and continuously differentiable.

Suppose that $s\_0=0$. Show that $s\_{t+1}^\*=0$ for all $t\geq 0$ is the unique solution to the sequential problem.

Here is my attempt:

The sequential problem is
\begin{align}
\max \sum\_{t=0}^\infty \beta^t u(e+Rs\_t-s\_{t+1})\\
s.t. s\_{t+1}\in\Gamma(s\_t)\\
s\_0\text{ is given}
\end{align}

Suppose $s\_0=0$. If we want to show $s\_{t+1}^\*=0$ for all $t\geq 0$. By induction hypothesis, let $s\_t=0$. Thus, the sequential problem at period $t$ is
\begin{align}
\max\sum\_{\tau=t}^\infty \beta^t u(c\_\tau)\\
s.t. c\_t+s\_{t+1}=e\\
c\_{\tau}+s\_{\tau+1}=Rs\_{\tau}+e\\
s\_{t+1}\geq 0
\end{align}

Then the Lagrangian:
$$\sum\_{\tau=t}^\infty \beta^t u(c\_\tau)+\lambda\_t (e-s\_{t+1}-c\_t)+\lambda\_{\tau}(Rs\_{\tau}+e-c\_{\tau}-s\_{\tau+1})+\mu s\_{t+1}$$

First order conditions:
\begin{align}
[s\_{t+1}]\lambda\_t=\lambda\_{t+1}R+\mu\\
[c\_{t}] \beta^t u'(c\_t)=\lambda\_t\\
[c\_{t+1}]\beta^{t+1}u'(c\_{t+1})=\lambda\_{t+1}
\end{align}

Suppose by contrdiction, $s\_{t+1}^\*\neq 0\implies s\_{t+1}^\*>0$. By complementary slackness, $\mu=0\implies \lambda\_t=\lambda\_{t+1}R\implies u'(c\_t)=\beta u'(c\_{t+1})R$. Since $\beta R\leq 1$, we have $u'(c\_t)\leq u'(c\_{t+1})\implies c\_t\geq c\_{t+1}$.

Since $s\_{t+1}^\*>0$ and $s\_t=0$, by $c\_t+s\_{t+1}=Rs\_t+e$, we have $c\_t. Thus, $c\_{t+1}\leq c\_t.

I don't know how to derive a contradiction up. Can someone help?

**Tags:** dynamic-programming

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## 1 answers

### Answer by username_1 on 2023-11-09. Score: 2 | Accepted answer

As you noticed, positive savings are only beneficial if the next period's consumption is lower. But this can only happen with positive savings if there are positive savings next period. By the same logic, there needs to be positive savings the period after, and so on. So consumption would be nonincreasing over time and $c\_t. But this is then strictly worse than consuming $e$ in every period.


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