# Fluctuations in income

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?

• Seems correct to me. – tdm May 8 at 17:19