Abstract
We consider a continuous lifetime model for investor whose lifetime is a random variable. We assume the investor has an access to the social welfare system, the financial market and the life insurance market. The investor aims to find the optimal strategies that maximize the expected utility obtained from consumption, investing in the financial market, buying life insurance, registering in the social welfare system, the size of his estate in the event of premature death and the size of his fortune at time of retirement if he lives that long. We use dynamic programming techniques to derive a second-order nonlinear partial differential equation whose solution is the maximum objective function. We use special case of discounted constant relative risk aversion utilities to find an explicit solutions for the optimal strategies. Finally, we have shown a numerical solution for the problem under consideration and study some properties for the optimal strategies.

Abstract
We show that in finite settings with identical firms and consumers, asymmetric pure price equilibria with positive profits exist. We consider a price competition duopoly for a homogeneous product. Demand stems from a second-stage consumption game at posted prices, with consumers' behavior impacted by negative network effects. We characterize equilibrium prices and demand. In all subgame-perfect pure price equilibria, both firms have positive profits, and in some, firms charge different prices.