Knightian uncertainty and poverty trap in a model of economic growth

This paper explores how Knightian uncertainty affects dynamic properties in an economic growth model. The decision-making theory employed in the analysis is the theory of expected utility under a non-additive probability measure, i.e., the Choquet expected utility model of preference. We apply this decision-making theory to an overlapping generations model where producers face “uncertainty” in their technologies. When the producer is averse to uncertainty, the firm's profit function may not be differentiable. Therefore, the firm's decision to invest and hire labor becomes rigid for a certain measurable range of real interest rates. In dynamic equilibrium, the existence of firm-level rigidity causes discontinuity in the wage function; this makes multiple equilibria the more likely outcomes under the log utility and Cobb–Douglas production functions. In this paper, we show that even if aversion to uncertainty is small, the “poverty trap” can arise for a wide range of parameter values.     

Expected utility maximization and demand behavior

This paper extends the non-parametric approach to consumer demand analysis to the case of expected utility maximization. We provide necessary and sufficient conditions for a set of price-quantity pairs to be consistent with expected utility maximization, and investigate their relationship to revealed preference conditions. Extensions to incomplete markets and intertemporal resolution of uncertainty are also considered.     

Aggregation of multiple prior opinions

Experts are asked to provide their advice in a situation of uncertainty. They adopt the decision maker?s utility function, but each has a potentially different set of prior probabilities, and so does the decision maker. The decision maker and the experts maximize the minimal expected utility with respect to their sets of priors. We show that a natural Pareto condition is equivalent to the existence of a set Λ of probability vectors over the experts, interpreted as possible allocations of weights to the experts, such that (i) the decision maker?s set of priors is precisely all the weighted-averages of priors, where an expert?s prior is taken from her set and the weight vector is taken from Λ; (ii) the decision maker?s valuation of an act is the minimal weighted valuation, over all weight vectors in Λ, of the experts? valuations.     

Import-Price Uncertainty, Production Decisions, and Relative Factor Shares

The effects of import-price uncertainty on factor income in Switzerland are estimated. The production-theory approach is used to derive the import demand function from an expected utility maximization problem, treating imports as an input to the technology. The model is also used to test for risk aversion and to assess the impact of uncertainty on the volume of imports and gross output. Evidence is found that, for most years, labor has been relatively more vulnerable to uncertainty than has capital.     

ECONOMIC MODELLING OF SUICIDE UNDER INCOME UNCERTAINTY: FOR BETTER UNDERSTANDING OF MIDDLE‐AGED SUICIDE*

This paper formalises an individual's decision about suicide within a framework of lifetime utility maximisation models. This is in line with the literature on economic modelling of suicide. The novelty of the paper is to take into account income uncertainty. Income uncertainty reduces a risk‐averse individual's expected utility, making them more likely to commit suicide. On the other hand, income uncertainty creates a value to postponing suicide even when their income gets sufficiently low. This is because income uncertainty means that if things go well, they will get higher income in the future. Thus, income uncertainty has two opposite effects on suicidal behaviour. The main objective of this paper is to construct an economic model of suicide for investigating net impacts of income uncertainty on suicidal behaviour. For this purpose, it is assumed that the wage evolves according to a stochastic process. Then, the threshold wage, below which an individual commits suicide, is derived as a function of the parameters of the stochastic process assumed for the wage evolution. Impacts of changes in these parameters on the threshold wage are calculated. With the result, the paper shows how income uncertainty affects suicidal behaviour.     

Relative risk aversion, relative prudence and comparative statics under uncertainty: The case of (μ, σ)‐preferences

From the expected‐utility approach, relative risk aversion being smaller than one and relative prudence being smaller than two emerge as preference restrictions that fully determine the optimal responses of decisions under uncertainty to certain shifts in probability distributions. We characterize the magnitudes of relative risk aversion and relative prudence in terms of the two‐parameter, mean‐standard deviation approach. We demonstrate that this characterization is instrumental in obtaining comparative static results in the two‐parameter setting. We further relate our findings to the results in the expected‐utility framework.     

Decision-foundations for properties of nonadditive measures: general state spaces or general outcome spaces

This paper characterizes properties of chance attitudes (nonadditive measures). It does so for decision under uncertainty (unknown probabilities), where it assumes Choquet expected utility, and for decision under risk (known probabilities), where it assumes rank-dependent utility. It analyzes chance attitude independently from utility. All preference conditions concern simple violations of the sure-thing principle. Earlier results along these lines assumed richness of both outcomes and events. This paper generalizes such results to general state spaces as in Schmeidler's model of Choquet expected utility, and to general outcome spaces as in Gilboa's model of Choquet expected utility.     

Uncertainty and the Voluntary Provision of a Pure Public Good in a Two‐Moment Decision Model

In this paper, we explore the potential benefits of uncertainty that may arise in a two‐moment model of the voluntary provision of a pure public good. We find that an increase in a given contributor i’s risk associated with the aggregate contribution level of the other contributors (i.e., an increase in social uncertainty) induces that contributor to increase his own contribution level if and only if the uncertainty's incremental effect on the expected value of his net marginal utility is negative. Contributor i’s welfare likewise increases when a closely related condition is met, namely that the uncertainty's marginal effect on his expected marginal utility value of the public good exceeds its countervailing effect on the numeraire. Further, the corresponding aggregate contribution to the public good increases in the presence of free‐riding if and only if the incremental effect of contributor i’s contribution on the aggregate expected value of all other contributors’ net marginal utilities is small‐enough positive. We derive similar conditions for the case of private uncertainty, where the increase in contributor i’s risk is associated with his own marginal valuation of the public good. A simple example illustrates these conceptual results. Numerical analysis demonstrates that an increase in private uncertainty can have a nonmonotonic impact on contributor i’s welfare.     

Epistemic Conditions for Equilibrium in Beliefs without Independence

Aumann and Brandenburger [Econometrica63(1995), 1161–1180.] provide sufficient conditions on the knowledge of the players in a game for their beliefs to constitute a Nash equilibrium. They assume, among other things, mutual knowledge of rationality. By rationality of a player, it is meant that the action chosen by him maximizes his expected utility, given his beliefs. There is, however, no need to restrict the notion of rationality to expected utility maximization. This paper shows that their result can be generalized to the case where players' preferences over uncertain outcomes belong to a large class of non-expected utility preferences.Journal of Economic LiteratureClassification Numbers: C72, D81.     

Piecewise additivity for non-expected utility

A model of choice under purely subjective uncertainty, Piecewise Additive Choquet Expected utility, is introduced. PACE utility allows for optimism and pessimism simultaneously, but represents a minimal departure from expected utility. It can be seen as a continuous version of NEO-expected utility (Chateauneuf et al. in J Econ Theory 137:538–567, 2007) and, as such, is especially suited for applications with rich state spaces. The main theorem provides a preference foundation for PACE utility in the Savage framework of purely subjective uncertainty with an arbitrary outcome set.     

An axiomatic approach to choice under uncertainty with catastrophic risks

This paper analyses decision under uncertainty with catastrophic risks, and is motivated by problems emerging from global environmental risks. These are typically low-probability events with major irreversible consequences. For such risks, the Von Neumann–Morgenstern (NM) axioms for decision making under uncertainty are not appropriate, since they are shown here to be insensitive to low-probability events. The paper introduces an alternative set of axioms requiring sensitivity to both low- and large-probability events. Through a new representation theorem in functional analysis, the results characterize all the operators whose maximization leads to the fulfillment of these axioms. They involve a convex combination of expected utility and a criterion based on the desire to avoid low probability and potentially catastrophic events. It is shown that the new axioms help resolve the Allais paradox. Open questions about risk aversion, games under uncertainty and calculus of variations are discussed.     

On the stability of the constant relative risk aversion (CRRA) utility under high degrees of uncertainty

Growth models under uncertainty and constant relative risk aversion (CRRA) utility are fragile in explaining consumers’ choice, as equilibrium consumption is dependent on distributional assumptions. We show that, under semi-nonparametric distributions, general equilibrium models are stable, as the existence of expected utility is guaranteed.     

Doubts or variability?

Reinterpreting most of the market price of risk as a price of model uncertainty eradicates a link between asset prices and measures of the welfare costs of aggregate fluctuations that was proposed by Hansen, Sargent, and Tallarini [17], Tallarini [30], Alvarez and Jermann [1]. Prices of model uncertainty contain information about the benefits of removing model uncertainty, not the consumption fluctuations that Lucas [22] and [23] studied. A max-min expected utility theory lets us reinterpret Tallarini's risk-aversion parameter as measuring a representative consumer's doubts about the model specification. We use model detection instead of risk-aversion experiments to calibrate that parameter. Plausible values of detection error probabilities give prices of model uncertainty that approach the Hansen and Jagannathan [11] bounds. Fixed detection error probabilities give rise to virtually identical asset prices as well as virtually identical costs of model uncertainty for Tallarini's two models of consumption growth.     

Choice under uncertainty with the best and worst in mind: Neo-additive capacities

We develop the simplest generalization of subjective expected utility that can accommodate both optimistic and pessimistic attitudes towards uncertainty—Choquet expected utility with non-extreme-outcome-additive (neo-additive) capacities. A neo-additive capacity can be expressed as the convex combination of a probability and a special capacity, we refer to as a Hurwicz capacity, that only distinguishes between whether an event is impossible, possible or certain. We show that neo-additive capacities can be readily applied in economic problems, and we provide an axiomatization in a framework of purely subjective uncertainty.     

Interim efficient allocations under uncertainty

This paper considers an exchange economy under uncertainty with asymmetric information. Uncertainty is represented by multiple priors and posteriors of agents who have either Bewley's incomplete preferences or Gilboa-Schmeidler's maximin expected utility preferences. The main results characterize interim efficient allocations under uncertainty; that is, they provide conditions on the sets of posteriors, thus implicitly on the way how agents update the sets of priors, for non-existence of a trade which makes all agents better off at any realization of private information. For agents with the incomplete preferences, the condition is necessary and sufficient, but for agents with the maximin expected utility preferences, the condition is sufficient only. A couple of necessary conditions for the latter case are provided.     

Efficient allocations under ambiguity

Important implications of the expected utility hypothesis and risk aversion are that if agents have the same probability belief, then consumption plans in every efficient allocation of resources under uncertainty are comonotone with the aggregate endowment, and if their beliefs are concordant, then the consumption plans are measurable with respect to the aggregate endowment. We study these two properties of efficient allocations for models of preferences that exhibit ambiguity aversion using the concept of conditional beliefs, which we introduce in this paper. We provide characterizations of such conditional beliefs for the standard models of preferences used in applications.     

The impossibility of compromise: some uniqueness properties of expected utility preferences

Summary. We focus on the following uniqueness property of expected utility preferences: Agreement of two preferences on one interior indifference class implies their equality. We show that, besides expected utility preferences under (objective) risk, this uniqueness property holds for subjective expected utility preferences in Anscombe-Aumann's (partially subjective) and Savage's (fully subjective) settings, while it does not hold for subjective expected utility preferences in settings without rich state spaces. Indeed, when it holds the uniqueness property is even stronger than described above, as it needs only agreement on binary acts. The extension of the uniqueness property to the subjective case is possible because beliefs in the mentioned settings are shown to satisfy an analogous property: If two decision makers agree on a likelihood indifference class, they must have identical beliefs. Received: November 15, 1999; revised version: December 29, 1999     

A Definition of Uncertainty Aversion

A definition of uncertainty or ambiguity aversion is proposed. It is argued that the definition is well-suited to modelling within the Savage (as opposed to Anscombe and Aumann) domain of acts. The defined property of uncertainty aversion has intuitive empirical content, behaves well in specific models of preference (multiple-priors and Choquet expected utility) and is tractable. Tractability is established through use of a novel notion of differentiability for utility functions, called eventwise differentiability.     

A simple explanation of some key time preference anomalies

Abstract Individual time discounting behaviour experimentally exhibits important anomalies that are inconsistent with the standard discounted utility model. These include the time preference reversal characteristic of hyperbolic discounting, the magnitude effect and the extreme sign effect. I propose a simple explanation of discounting that accounts for these three anomalies simultaneously, within the context of the expected utility model with uncertainty, risk aversion, and preference for precautionary saving.