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.

Abstract
A central issue in price competition with positive network effects is the potential for small price changes to trigger abrupt chain reactions, leading to market tipping, winner-take-all scenarios, and zero-profit equilibria. We show that in a duopoly where consumers are not anonymous but partitioned into at least two groups, a simple group-based network structure can, by itself, generate downward-sloping demand and support profitable shared-market equilibria. These are subgame-perfect pure price equilibria in which both firms earn strictly positive profit. Triggering a bandwagon effect and tipping the market remains possible, but requires aggressive price deviations, or price shocks, that produce demand jumps. However, this is not always profitable, and the fear of bankruptcy can be sufficient to stabilize firms in equilibrium. The result relies on having one group with centripetal influence (stronger impact on peers) and another with centrifugal influence (stronger impact on outsiders). It requires no additional sources of heterogeneity or product differentiation. This mechanism shows that positive network effects - when group structured - can endogenously generate stability in price competition. The analysis reconciles the coexistence of local stability and the potential for tipping, offering a unified explanation of how markets with strong network effects can sustain both competition and profitability. We draw a parallel to Turing's reaction-diffusion patterns and reinterpret Becker's intuition that social influence can produce stable outcomes, even when demand may exhibit upward-sloping segments.