  We consider a generalization of the recursive utility model by adding a new
component that represents utility of investment gains and losses. We also study
the utility process in this generalized model with constant elasticity of
intertemporal substitution and relative risk aversion degree, and with infinite
time horizon. In a specific, finite-state Markovian setting, we prove that the
utility process uniquely exists when the agent derives nonnegative gain-loss
utility, and that it can be non-existent or non-unique otherwise. Moreover, we
prove that the utility process, when it uniquely exists, can be computed by
starting from any initial guess and applying the recursive equation that
defines the utility process repeatedly. We then consider a portfolio selection
problem with gain-loss utility and solve it by proving that the corresponding
dynamic programming equation has a unique solution. Finally, we extend certain
previous results to the case in which the state space is infinite.
