Compute the average income and the average utility I don't know how to compute the average income and the average utility for the second case. It is confused because I have to consider two densities $[0,0.5]$ and $[0.5,1.2]$.Can anybody tell me how to compute this? 

Thank you in advance!

Edit: $\bar {z} =$ the average income, and $z_i$ = the income of an individual $i$. $U(\cdot) =$ utility function for all.
These are not two densities, but one density defined on the interval $[0,1.2]$, taking two different values. The average income is then the integral of the density function, i.e. 

$$
\int_0^{0.5} 1 x \, dx + \int_{0.5}^{1.25} \frac{2}{3} x \, dx = \frac{9}{16} = 0.5625
$$

In order to obtain the average income, you do the same with the Utility and $\bar{z} = \frac{9}{16}$.