The Topology of Fear

Abstract

For many years experimental observations have raised questions about the rationality of economic agents – for example, the Allais Paradox or the Equity Premium Puzzle. The problem is a narrow notion of rationality that disregards fear. This article extends the notion of rationality with new axioms of choice under uncertainty and the decision criteria they imply (Chichilnisky, G., 1996a. Anaxiomatic approach to sustainable development. Social Choice and Welfare 13, 257-321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, Vol.1. John Wiley & Sons, Ltd., Chicester). In the absence of catastrophes, the old and the new approach coincide, and both lead to standard expected utility. A sharp difference emerges when facing rare events with important consequences, or catastrophes. Theorem 1 establishes that a classic axiom of choice under uncertainty – Arrow’s Monotone Continuity axiom, or its relatives introduced by De Groot, Villegas, Hernsteinand Milnor – postulate rational behavior that is ‘insensitive’ to rare events as defined in (Chichilnisky, G., 1996a. Anaxiomatic approach to sustainable development. Social Choice and Welfare13, 257-321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, vol. 1. John Wiley & Sons, Ltd., Chicester).Theorem 2 replaces this axiom with another that allows extreme responses to extreme events, and characterizes the implied decision criteria as a combination of expected utility with extremal responses. Theorems 1 and 2 offer a new understanding of rationality consistent with previously unexplained observations about decisions involving rare and catastrophic events, decisions involving fear, the Equity Premium Puzzle, ‘jump diffusion’ processes and ‘heavy tails’, and it agrees with (Debreu, G., 1953. Valuation equilibrium and Pare to optimum. Proceedings of the National Academy of Sciences, 40, 588-592) formulation of market behavior and his pro of of Adam Smith’s Invisible Hand theorem.