We study the problem of maximizing expected utility from terminal wealth for a not necessarily concave utility function U and for a budget set given by one fixed pricing measure. We prove the existence and several fundamental properties of a maximizer. We analyze the (not necessarily concave) value function (indirect utility) u(x, U). In particular, we show that the concave envelope of u(x, U) is the value function u(x, U c) of the utility maximization problem for the concave envelope U c of the utility function U. The two value functions are shown to coincide if the underlying probability space is atomless. This allows us to characterize the maximizers for several model classes explicitly

The pricing kernel puzzle is the observation that the pricing kernel might be increasing in some range of the market returns. This paper analyzes the pricing kernel in a nancial market equilibrium. If markets are complete and investors are risk-averse and have common and true beliefs, the pricing kernel is a decreasing function of aggregate resources. If at least one of these assumptions is violated, the pricing kernel is not necessarily decreasing. Thus, incomplete markets, riskseeking behaviour and incorrect beliefs can induce increasing parts in the pricing kernel and can be seen as potential solutions for the pricing kernel puzzle. We construct examples to illustrate the three explanations. We verify the robustness of the explanations under aggregation and compare the phenomena with the ndings in the empirical literature. The results are used to reveal strengths and weaknesses of the three solutions. Risk-seeking behaviour is a fragile explanation that can only work in a model with atomic state space. Biased beliefs are robust under aggregation and consistent with the empirical ndings. In incomplete markets, it is easy to nd a pricing kernel with increasing parts. In order to get situations where all pricing kernels have increasing parts, we need extreme assumptions on the wealth distribution.