3. Consider the following two-period problem. The utility function is

$$u(c1) + \beta \mathbb{E}[u(c2)]$$

with $u' > 0, u'' < 0$, where $\beta \in (0,1)$ is the discount factor, $c_t$ is consumption in period $t, \forall t = 1, 2$. The budget constraints are

$$c1 + a2 = \omega$$

and

$$c2 = x + a2(1 + r),$$

where r is the net real interest rate, $a_2$ is net savings at the end of period 1, $\omega$ is a deterministic endowment, and x is a random variable that takes value $\omega + \sigma$ with probability 0.5 and $\omega - \sigma$ with probability 0.5, with $\omega > \sigma \ge 0$.