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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 $c_t<e$. Thus, $c_{t+1}\leq c_t<e$.

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

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1 Answer 1

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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 $c_t<e$. But this is then strictly worse than consuming $e$ in every period.

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