This is an excerpt from the 4-volume dictionary of economics, a reference book which aims to define the subject of economics today. 1300 subject entries in the complete work cover the broad themes of economic theory. This extract concentrates on utility and probability.

Wealth uncertainty may be resolved before consumption in which case consumption is optimal. But if uncertainty remains at the consumption level, after allocative decisions are made, ex post relative consumption is suboptimal. Therefore, ex ante the consumer has to balance attitudes to risk with underlying preferences. We study this problem in the context of expected utility with preferences over two goods, one of which is subject to additive ex post uncertainty. State preferences determine the utility gambles, and attitudes to risk evaluate these gambles, in order to choose an optimal consumption gamble. We study how risk aversion interacts with preferences to determine the direction and size of comparative statics effects using lattice programming, monotone comparative statics, methods. Comparative statics depend on the evaluation of comparative utility loss along the distribution of uncertainty and lattice theoretic properties allow us to determine how these relate to preferences and consumption. With the global regularity imposed by LSE (quasi-supermodularity) properties in direct value lattices we study the comparative statics of comparative risk aversion, income and underlying risk. We give conditions for a more risk averse consumer to choose a less or more risky consumption bundle and show that the direct LSE property determines comparative statics under first order stochastic dominant changes in ex ante uncertainty. Extensive examples demonstrate.

For several decades, the orthodox economics approach to understanding choice under risk has been to assume that each individual person maximizes some sort of personal utility function defined over purchasing power. This new volume contests that even the best wisdom from the orthodox theory has not yet been able to do better than supposedly naïve models that use rules of thumb, or that focus on the consumption possibilities and economic constraints facing the individual. The authors assert this by first revisiting the origins of orthodox theory. They then recount decades of failed attempts to obtain meaningful empirical validation or calibration of the theory. Estimated shapes and parameters of the "curves" have varied erratically from domain to domain (e.g., individual choice versus aggregate behavior), from context to context, from one elicitation mechanism to another, and even from the same individual at different time periods, sometimes just minutes apart. This book proposes the return to a simpler sort of scientific theory of risky choice, one that focuses not upon unobservable curves but rather upon the potentially observable opportunities and constraints facing decision makers. It argues that such an opportunities-based model offers superior possibilities for scientific advancement. At the very least, linear utility – in the presence of constraints - is a useful bar for the "curved" alternatives to clear.

This dissertation studies a class of path-dependent stochastic control problems with applications to Finance. In particular, we solve the open problem of the continuous time expected utility maximization with addictive consumption habit formation in incomplete markets under two independent scenarios. In the first project, we study the continuous time utility optimization problem with consumption habit formation in general incomplete semimartingale financial markets. Introducing the set of auxiliary state processes and the modified dual space, we embed our original problem into an abstract time-separable utility maximization problem with a shadow random endowment on the product space [mathematic equation] We establish existence and uniqueness of the optimal solution using convex duality by defining the primal value function as depending on two variables, i.e., the initial wealth and the initial standard of living. We also provide market independent sufficient conditions both on the stochastic discounting processes of the habit formation process and on the utility function for the well-posedness of our original optimization problem. Under the same assumptions, we can carefully modify the classical proofs in the approach of convex duality analysis when the auxiliary dual process is not necessarily integrable. In the second project, we examine an example of the optimal investment and consumption problem with both habit-formation and partial observations in incomplete markets driven by Itô processes. The individual investor develops addictive consumption habits gradually while only observing the market stock prices but not the instantaneous rates of return, which follow an Ornstein-Uhlenbeck process. Applying the Kalman-Bucy filtering theorem and Dynamic Programming arguments, we solve the associated Hamilton-Jacobi-Bellman(HJB) equation fully explicitly for this path dependent stochastic control problem in the case of power utility preferences. We provide the optimal investment and consumption policy in explicit feedback form using rigorous verification arguments.

Expected utility provides simple, testable properties of the optimum behavior that should be displayed by risk-averse individuals in risky decisions. Simultaneously, given the existence of paradoxes under the expected utility paradigm, expected utility can only be regarded as an approximation of actual behavior. A more realistic model is needed. This is particularly true when treating attitudes toward small probability events: the standard situation for insurable risks. Non-Expected Utility and Risk Management examines whether the existing results in insurance economics are robust to more general models of behavior under risk.

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