The theory of precautionary saving is shown in this paper to be isomorphic to the Arrow-Pratt theory of risk aversion, making possible the application of a large body of knowledge about risk aversion to precautionary saving, and more generally, to the theory of optimal choice under risk In particular, a measure of the strength of precautionary saving motive analogous to the Arrow-Pratt measure of risk aversion is used to establish a number of new propositions about precautionary saving, and to give a new interpretation of the Oreze-Modigliani substitution effect

Precautionary Saving in the Small and in the Large

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

/pdf/a-smooth-model-of-decision-making-under-ambiguity-2qvm7ggyao.pdf

A Smooth Model of Decision Making under Ambiguity

We propose and axiomatize a model of preferences over acts such that the decision maker
prefers act f to act g if and only if Eμφ (Eπu ◦ f) ≥ Eμφ (Eπu ◦ g), where E is the
expectation operator, u is a vN-M utility function, φ is an increasing transformation,
and μ is a subjective probability over the set Π of probability measures π that the
decision maker thinks are relevant given his subjective information. A key feature of our
model is that it achieves a separation between ambiguity, identified as a characteristic
of the decision maker’s subjective information, and ambiguity attitude, a characteristic
of the decision maker’s tastes. We show that attitudes towards risk are characterized
by the shape of u, as usual, while attitudes towards ambiguity are characterized by the
shape of φ. We also derive φ (x) = −1
α
e−αx as the special case of constant ambiguity
aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by
properties of the subjective set of measures Π. This characterization of ambiguity is
formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang
(2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the welldeveloped
machinery for dealing with risk attitudes can be applied as well to ambiguity
attitudes. The model is also distinct from many in the literature on ambiguity in that
allows smooth, rather than kinked, indifference curves. This leads to different behavior
and improved tractability, while still sharing the main features (e.g., Ellsberg’s Paradox,
etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of
measures may be seen as an extreme case of our model with infinite ambiguity aversion.
Two illustrative applications to portfolio choice are offered.

/pdf/a-smooth-model-of-decision-making-under-ambiguity-yoi825dq9u.pdf

A Smooth Model of Decision Making Under Ambiguity

Many economic problems are naturally modeled as a game of incomplete information, where a player’s payoff depends on his own action, the actions of others, and some unknown economic fundamentals. For example, many accounts of currency attacks, bank runs, and liquidity crises give a central role to players’ uncertainty about other players’ actions. Because other players’ actions in such situations are motivated by their beliefs, the decision maker must take account of the beliefs held by other players. We know from the classic contribution of Harsanyi (1967–1968) that rational behavior in such environments not only depends on economic agents’ beliefs about economic fundamentals, but also depends on beliefs of higher-order – i.e., players’ beliefs about other players’ beliefs, players’ beliefs about other players’ beliefs about other players’ beliefs, and so on. Indeed, Mertens and Zamir (1985) have shown how one can give a complete description of the “type” of a player in an incomplete information game in terms of a full hierarchy of beliefs at all levels. In principle, optimal strategic behavior should be analyzed in the space of all possible infinite hierarchies of beliefs; however, such analysis is highly complex for players and analysts alike and is likely to prove intractable in general. It is therefore useful to identify strategic environments with incomplete information that are rich enough to capture the important role of higher-order beliefs in economic settings, but simple enough to allow tractable analysis. Global games, first studied by Carlsson and van Damme (1993a), represent one such environment. Uncertain economic fundamentals are summarized by a state θ and each player observes a different signal of the state with a small amount of noise. Assuming that the noise technology is common knowledge among the players, each player’s signal generates beliefs about fundamentals, beliefs about other players’ beliefs about fundamentals, and so on. Our purpose in this paper is to describe how such models work, how global game reasoning can be applied to economic problems, and how this analysis relates to more general analysis of higher-order beliefs in strategic settings.

/pdf/global-games-theory-and-applications-1lch3df46c.pdf

Global Games: Theory and Applications

A Bayesian consumer who is uncertain about the quality of an information source will infer that the source is of higher quality when its reports conform to the consumer's prior expectations. We use this fact to build a model of media bias in which firms slant their reports toward the prior beliefs of their customers in order to build a reputation for quality. Bias emerges in our model even though it can make all market participants worse off. The model predicts that bias will be less severe when consumers receive independent evidence on the true state of the world, and that competition between independently owned news outlets can reduce bias. We present a variety of empirical evidence consistent with these predictions.

Media Bias and Reputation

A number of papers have shown that a strict Nash equilibrium action profile of a game may never be played if there is a small amount of incomplete information (see, for example, Carlsson and van Damme (1993a)). We present a general approach to analyzing the robustness of equilibria to a small amount of incomplete information. A Nash equilibrium of a complete information game is said to be robust to incomplete information if every incomplete information game with payoffs almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. We show that many games with strict equilibria have no robust equilibrium and examine why we get such different results from existing refinements. If a game has a unique correlated equilibrium, it is robust. A natural many-player many-action generalization of risk dominance is a sufficient condition for robustness.

The robustness of equilibria to incomplete information

Intrinsic Preference for Information

In this paper we re-examine generic constrained suboptimality of equilibrium allocations with incomplete numeraire asset markets. We provide a general framework which is capable of resolving some issues left open by the previous literature, and encompasses many kinds of intervention in partially controlled market economies. In particular, we establish generic constrained suboptimality, as studied by Geanakoplos and Polemarchakis, even without an upper bound on the number of households. Moreover, we consider the case where asset markets are left open, and the planner can make lump-sum transfers in a limited number of goods. We show that such a perfectly anticipated wealth redistribution policy, though consistent with the assumed incomplete financial structure, is typically effective.

Constrained suboptimality in incomplete markets: a general approach and two applications

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player "updates" his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players' views on uncertainty about the environment affect the strategic outcomes.

Atsushi Kajii

Papers

The robustness of equilibria to incomplete information

Intrinsic Preference for Information

Constrained suboptimality in incomplete markets: a general approach and two applications

Incomplete Information Games with Multiple Priors

A generalization of Scarf's theorem: An α-core existence theorem without transitivity or completeness