How do I find out which form of utility function is being used? Can someone tell my how i can figure out which type of utility function the following maximization problem has. It is for an overlapping generations model.

$$\underset{x'}{\max} x'\big(E_t (P_{t+1}+δ_{t+1})-(1+r^f)P_t\big) - \tfrac{\gamma}{2} x'Ωx $$
Where:
$x'=$ portfolio of shares 

$E_t (P_{t+1}+δ_{t+1}) =$ expected future payoff 

$\gamma =$ agent i risk aversion

$Ω=$ covariance matrix
I would call this a mean-variance utility function. The agent likes higher mean values, which is the first term, but trades that off against higher variance, which is the second term.

If the random variable of interest is normally distributed with mean $P_{t+1} + \delta_{t+1} - (1+r^f)P_t$ and covariance matrix $\Omega$, and if the agent has constant absolute risk aversion utility, such as $u(w) = 1-e^{-\gamma w}$, then maximizing that utility is equivalent to maximizing mean-variance utility. Here are some details I found with a quick google search.