Equilibrium Investment with Random Risk Aversion: (Non-)uniqueness, Optimality, and Comparative Statics

This paper investigates infinite-dimensional portfolio selection problem under a general distribution of the risk aversion parameter. We provide a complete characterization of all deterministic equilibrium investment strategies. Our results reveal that the solution structure depends critically on the distribution of risk aversion: the equilibrium is unique whenever it exists in the case of finite expected risk aversion, whereas an infinite expectation can lead to infinitely many equilibria or to a unique trivial one (pi equals 0). To address this multiplicity, we introduce three optimality criteria-optimal, uniformly optimal, and uniformly strictly optimal-and explicitly characterize the existence and uniqueness of the corresponding equilibria. Under the same necessary and sufficient condition, the optimal and uniformly optimal equilibria exist uniquely and coincide. Furthermore, by additionally assuming that the market price of risk is non-zero near the terminal time, we show that the optimal (and hence uniformly optimal) equilibrium is also uniformly strictly optimal. Finally, we perform comparative statics to demonstrate that a risk aversion distribution dominating another in the reverse hazard rate order leads to a less aggressive equilibrium strategy.