We study the total utility of an agent in a model of narrow framing with constant elasticity of intertemporal substitution and relative risk aversion degree and with infinite time horizon. In a finite-state Markovian setting, we prove that the total utility uniquely exists when the agent derives nonnegative utility of gains and losses incurred by holding risky assets and that the total utility can be non-existent or non-unique otherwise. Moreover, we prove that the utility, when uniquely exists, can be computed by a recursive algorithm with any starting point. We then consider a portfolio selection problem with narrow framing and solve it by proving that the corresponding dynamic programming equation has a unique solution. Finally, we propose a new model of narrow framing in which the agent's total utility uniquely exists in general. Barberis and Huang (2009, J. Econ. Dynam. Control, vol. 33, no. 8, pp. 1555-1576) propose a preference model that allows for narrow framing, and this model has been successfully applied to explain individuals' attitudes toward timeless gambles and high equity premia in the market.

To uniquely define the utility process in this preference model and to yield a unique solution when the model is applied to portfolio selection problems, one needs to impose some restrictions on the model parameters, which are too tight for many financial applications. We propose a modification of Barberis and Huang's model and show that the modified model admits a unique utility process and a unique solution in portfolio selection problems. Moreover, the modified model is more tractable than Barberis and Huang's when applied to portfolio selection and asset pricing.