Uncertainty Economics

Abstract

This chapter seeks to reconcile fundamental financial approaches with uncertainty. Uncertainty is defined by the unknown rather than the predictable, counted and accounted for. While financial decisions are reached based on what we know, what we can predict and what we can presume based on experience and the rationalities that financial agents assume. The uncertainty we consider is defined in a limited sense, namely, a partial knowledge of future state preferences and their quantification. There are many approaches to do so such as negligence of the unknown, human intentional rationalities as well behavioral and psychological approaches to confront the unknown. This chapter focuses its attention on the use of entropy for “non-extensive systems” (a term commonly used in physics with its parallel in finance, which we define as “incompleteness”) based on a parametric generalization of the Boltzmann–Gibbs entropy (which assumes extensive systems). Optimization of the Tsallis parametric entropy for non-extensive systems is then used to derive implied power laws and standardized probability distributions that are both asymmetric and have fat tails. This approach provides a parametric definition of the “missing”, namely the tail probabilities not accounted for in selecting an asset future price distribution.

Subsequently, the chapter outlines a number of approaches to robust decision models and ex-post risk management. It concludes with a discussion of risk externalities in financial and environmental regulation and draws a parallel between “banks’ risks” for which they do not assume responsibility for and pollution risks of firms and consumers who consume and who do not assume their pollution consequences. Both cases, call for an efficient regulation and statistical controls which is the topic of Chap. 11.

(*with OREN TAPIERO)

References and Additional Reading

Uncertainty and risks are issues that remain debatable. Brock (1986) and Brock and Dechert (1988) distinguish between random and deterministic systems. In Chap. 5 these issues were both discussed and referred to, each approach seeking some way to formalize our lack of knowledge in some rational and quantified form. In particular, we have used incomplete state preferences (Knight 1921) to define uncertainty. It is however a specific type of uncertainty. Other approaches seek to define alternatives to rational decision making based on a weighting of probabilities. For example, Machina 1982, 1987; Samuelson 1963, 1965 (on the limits of the explanatory power of “numbers”), Machina and Munier 1999; Abdellaoui and Munier 1997, 1999 on the belief and probability assignments to rare events (see also Munier 1986, 1995, 1991; Munier et al. 1999; Rabin 1998 on economics psychology), Simon 1979, 1982 on bounded rationality, Munier 2012 (on volatility and uncertainty supported by an extensive empirical analysis of agricultural commodities), Ahlbrecht and Weber 1997 on the gradual resolution of uncertainty, etc.

The literature in this domain is thus extremely large and applied to a broad set of issues and problems. The references below define various formalizations to decision making with behavioral and psychological factors: Dyer and Jia 1997, Jia and Dyer 1996, Jia et al. 2001 on perceived risks, Eeckoudt and Kimball 1991, Eeckhoudt et al. 1996, Gollier and Pratt 1996 (on background risks), Ellsberg 1961 on ambiguity and Savage Axioms (regret), Evans on Psychological Reasoning and rationality, Gilboa and Schmeidler 1995, Kahneman and Tversky 1979 on Prospect Theory, Thaler et al. 1997; Wakker 1994, 2001, 2010; Wakker and Tversky 1993; Taleb 2007, 2008, 2009.

Implications to economics include seminal papers by Arrow (1963b) on welfare of medical care, Arrow (1982) on the risk perception and psychology in economics, Arrow et al. (1996) on Rational Foundations for decision making, Debreu (1953) on the economics of uncertainty, Friedman and savage (1948, 1952) on utility analysis, Kindleberger (1978) on Mania and panics, Laffont (1989, 1995) on the economics of uncertainty, regulation insurance and environmental risks, Loomes and Sugden (1982, 1987), Mulvey et al. (1991) on scenario based uncertainty and Magill and Quinzii (1996) on markets incompleteness.

Risk externalities are a special form of uncertainty (at least for the public at large, unaware of the actions that parties take with consequences they do not assume). In this sense, Coase (1937, 1960), important contributions discussed in this chapter provide a guideline to assess their effects. For applications, references are given in the text. We refer in particular to Xepapadeas (1994, 1995) on risk externalities and environmental issues.

Chaos, complexity or “the lack of order” are some of the more important sources of uncertainty. Chaos is a field originating in forms of “unpredictability”. For example, a process that cannot converge to any particular end, bifurcating continuously in an unpredictable manner is chaotic. This leads to statements, that a “sneeze” may lead one catch a cold in New York! Namely, that anything and everything are unpredictable and therefore everything may occur. Lorenz (1966) for example, studied and defined chaos in atmospheric data, May (1974) defined dynamic systems in biological systems that are “chaotic”, Gleick (1987) and Peter Edgar (1995) have written popular books on chaos, etc.

In physics, uncertainty in its economic sense, is characterize by non-extensive systems (parallel to incomplete financial markets) and extensive systems (complete financial markets).The characterization of such systems are difficult however. Statistical physics following Gibbs (1961), Boltzmann, Shannon and others, have used entropy as a measure of disorder. The greater the entropy, the greater “disorder”. In this sense entropy has become a “probability” measure to define “disorder” in terms of particular metrics. Stanley et al. (1996) for example, have made immense contributions to such approaches based statistical, economic and financial physics. Numerous attempts have been made to generalize both the measures and the metrics to define and estimate “non-extensiveness”. The results outlined in this chapter were based on such studies. In particular, Jackson (1909) provided an approach to calculate for “jump” derivatives. Jackson’s formulation has shown in this chapter, that Tsallis entropy (Tapiero 1995a, b) and the BG entropy differ only in the metric used to define “disorder”. Sato (2010) uses the Jackson derivative under mean–variance assumptions to obtain a q-Gaussian distributions (based on Tsallis entropy). Other references (not referred to in the text) include Kapur and Kesavan (1992) entropy maximization, Renyi (1961), Dempster et al. (2007) applied empirical copulas to CDO tranche pricing using relative entropy, Akturk et al. (2007) on Sharma-Mittal entropy, Nau (2011) and Naudts (2007) on statistical dynamics and entropy. And Fraundorf (2007) on complexity, Kullback (1959, 1987), Kullback and Leibler (1951), Good (1965, 1968) on information and utility, have provided however a statistical approach to entropy as a metric—both of information and a metric.

Fractal models defined in Chap. 5 (Mandelbrot 1997; Mandelbrot and Wallis 1968 on fractal models for example), provide also another form of uncertainty, modeled by the nonlinear growth of volatility. While information asymmetry and financial intermediation based on contractual agreements, unexpected credit losses, etc. are also sources of uncertainty we have considered in Chaps. 8 and 9 (see also Leland and Pyle 1977 on Riordan 1984; Amato and Remolona 2005).

Finally, networks, the intricate relationships of networked agents, firms, their exchange etc. are a source of uncertainty. Even though, there are some research papers that recognize specific characteristics of networks and networking in economic and financial systems, it remains a challenging topic (Economides 1996). Katz and Shapiro (1985, 1994) provide an outline of networks externalities and their effects on system competition, Liebowitz and Margolis (1994), emphasize as well the uncommon tragedy of network externality.