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Consider the following income fluctuation problem. An agent lives for two periods, 0 and 1, and faces stochastic income in period 1. We are given this utility maximization problem: $$\text{max}\ U(c_0)+\beta E_0U(c_1)\ \text{s.t}\ $$ $$c_0+a_1=y_0$$ $$c_1=(1+r)a_t+\tilde{y_1}$$ where $\tilde{y_1}$ is stochastic: $y_1=y_0+e$ with probability $1/2$ and $y_1=y_0-e$ with probability $1/2$. We consider $\beta(1+r)=1$

We are asked to show that if $u'''> 0$ (i.e. $u'$ is convex) and $e > 0$, then $a_1 > 0$. And how can we interpret this result?

In regards to interpretation, the third utility is usually related to prudence. Now what my train of thought was: $$u'(y_0-a_1)=E_0[u'((1+r)a_1+\tilde{y_1})]>u'[E_0((1+r)a_1+\tilde{y_1})]=u'((1+r)a_1+y_0)$$ $$\implies y_0-a_1<(1+r)a_1+y_0$$ Suggesting that $a_1>0$. Meaning that the agent is prone to save?

Is this correct?

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    $\begingroup$ Seems correct to me. $\endgroup$ – tdm May 8 at 17:19

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