The exact law of large numbers via Fubini extension and characterization of insurable risks

A Fubini extension is formally introduced as a probability space that extends the usual product probability space and retains the Fubini property. Simple measure-theoretic methods are applied to this framework to obtain various versions of the exact law of large numbers and their converses for a continuum of random variables or stochastic processes. A model for a large economy with individual risks is developed; and insurable risks are characterized by essential pairwise independence. The usual continuum product based on the Kolmogorov construction together with the Lebesgue measure as well as the usual finitely additive measure-theoretic framework is shown further to be not suitable for modeling individual risks. Measurable processes with essentially pairwise independent random variables that have any given variety of distributions exist in a rich product probability space that can also be constructed by extending the usual continuum product.     

Default and aggregate income

This paper studies how default varies with aggregate income. We analyze a model in which optimal contracts enable risk sharing of privately observed, idiosyncratic income by allowing for default. Default provisions allow agents with low idiosyncratic income realizations to repay less and thus provide insurance. Default penalties ensure that only these agents default. We show that default can occur under the optimal contract and that default provisions vary with aggregate income. We provide conditions such that both the amount of default and default penalties vary countercyclically with aggregate income and show that the default rate can be discontinuous.     

Dynamic contracting with persistent shocks

In this paper, we develop continuous-time methods for solving dynamic principal-agent problems in which the agent's privately observed productivity shocks are persistent over time. We characterize the optimal contract as the solution to a system of ordinary differential equations and show that, under this contract, the agent's utility converges to its lower bound—immiserization occurs. Unlike under risk-neutrality, the wedge between the marginal rate of transformation and a low-productivity agent's marginal rate of substitution between consumption and leisure will not vanish permanently at her first high-productivity report; also, the wedge increases with the duration of a low-productivity report. We apply the methods to numerically solve the Mirrleesian dynamic taxation model, and find that the wedge is significantly larger than that in the independently and identically distributed (i.i.d.) shock case.     

Informational size and incentive compatibility with aggregate uncertainty

In McLean and Postlewaite (Econometrica 56, 1992, p. 2421), we analyzed pure exchange economies with asymmetrically informed agents. We defined a notion of informational size and showed that, when the aggregate information of all agents resolves nearly all the uncertainty regarding the state of nature, the conflict between incentive compatibility and (ex post) efficiency can be made small if agents have sufficiently small informational size. This paper investigates the relationship between informational size and efficiency for the case in which nontrivial aggregate uncertainty is present, i.e., when significant uncertainty about the world persists even when the information of all agents is known.     

Output and price level effects of monetary uncertainty in a matching model

Monetary uncertainty and information lags are put into a random matching model so that the resulting setting has some meetings in which producers are relatively informed and others in which consumers are relatively informed.For that setting, the ex ante socially optimal way to conduct trade is characterized. The optimum can display a variety of relationships between money and total output and the price level. While the price level is always sticky, even the direction of its response and that of total output depend on the magnitude of the lag and on subtle features of the serial correlation properties of the money supply.     

Endogenous supply of fiat money

We consider whether reputation concerns can discipline the behavior of a long-lived self-interested agent who has a monopoly over the provision of fiat money. We obtain that when this agent can commit to a choice of money supply, there is a monetary equilibrium where it never overissues. We show, however, that monetary equilibria with no overissue do not exist when there is no commitment. This happens because the incentives this agent has to maintain a reputation for providing valuable currency disappear once its reputation is high enough. More generally, we prove that in the absence of commitment overissue happens infinitely often in any monetary equilibrium. We conclude by showing that imperfect memory can restore the positive result obtained with commitment.     

An expository note on individual risk without aggregate uncertainty

In economies where agents bear some risk, the analysis would often be facilitated by the assumption that the risks are i.i.d. and disappear in the aggregate. A formal appeal to the law of large numbers requires the consideration of a sequence of finite economies. In an economy with a continuum of agents there are countable families of sets for which it is impossible for a law of large numbers to be valid. With countably many agents and a finitely additive measure, independence is compatible with the law of large numbers.     

Knightian uncertainty and moral hazard

This paper presents a principal-agent model in which the agent has imprecise beliefs. We model this situation formally by assuming the agent?s preferences are incomplete as in Bewley (1986) [2]. In this setting, incentives must be robust to Knightian uncertainty. We study the implications of robustness for the form of the resulting optimal contracts. We give conditions under which there is a unique optimal contract, and show that it must have a simple flat payment plus bonus structure. That is, output levels are divided into two sets, and the optimal contract pays the same wage for all output levels in each set. We derive this result for the case in which the agent?s utility function is linear and then show it also holds if this utility function has some limited curvature.     

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.     

Uninsured idiosyncratic risk, liquidity constraints and aggregate fluctuations

Summary. I study the role played by uninsured idiosyncratic risk and liquidity constraints in the propagation of aggregate fluctuations. To this purpose, I compare the aggregate fluctuations of two model economies that differ in their insurance technologies only. In one of these model economies liquidity constrained households vary their holdings of a nominally denominated asset in order to buffer an uninsured idiosyncratic shock to their individual production opportunities. In the other economy every idiosyncratic component of risk can be costlessly insured. I find that the limited insurance technology implies fluctuations in output that are 20% larger, fluctuations in hours relative to output that are 9% larger, fluctuations in consumption relative to output that are 18% smaller, and a correlation of hours and productivity that is 15% smaller than those that obtain under the full insurance technology. Received: March 6, 1996; revised version August 15, 1996     

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.     

Policy biases with voters’ uncertainty about the economy and the government

We model policy choices where parties competing for election are better informed than voters about the state of the economy. In addition voters are uncertain about the incumbent's preferences and the median voter's vote potentially depends on the state of the economy. The incumbent may bias his policy in order to convey information to the voters and raise his chance of reelection. The bias depends crucially on which kind of uncertainty is the most important. Uncertainty about preferences leads to moderate policies, while uncertainty about the state of the economy leads to extreme policies.     

Anonymity and individual risk

Adverse selection economies with private information are generally studied under the assumption that contracts are exclusive. That is, retrading is prohibited. An alternative market mechanism, the anonymous mechanism, is studied here. Risk averse agents trade contingent claims directly and side markets are in equilibrium. The result is the anonymous equilibrium. The anonymous equilibrium results in a set of endogenous transfers and subsidies.     

Corrigendum to “Reserve price signaling” [J. Econ. Theory 135 (1) (2007) 253-268]

We find an error in [H. Cai, J. Riley, L. Ye, Reserve price signaling, J. Econ. Theory 135 (1) (2007) 253-268]. The existence of the separating signaling equilibrium is not challenged, but the related comparative statics, while valid in the case without informational externalities between bidders, do not generalize to the case with informational externalities as claimed by the authors. General results on the influence of the number of bidders are obtained in a linear specification of the informational externalities.     

Strategic argumentation

I analyze a game between an uninformed decision maker and a possibly biased expert. The expert receives a set of arguments, and each argument favors one of two alternatives. He can disclose each argument credibly, but cannot prove whether he has disclosed everything. In all equilibria, the biased expert sends messages containing arguments both for and against his preferred alternative. However, the decision maker is not influenced by the unfavorable arguments revealed by the biased expert. The latter is able to convince the decision maker to choose the biased expert?s preferred alternative only if he reveals sufficiently many favorable arguments.     

The Shill Bidding Effect versus the Linkage Principle

The analysis of second price auctions with externalities is utterly modified if the seller is unable to commit not to participate in the mechanism. For the General Symmetric Model introduced by Milgrom and Weber [P. Milgrom, R. Weber, A theory of auctions and competitive bidding, Econometrica 50 (1982) 1089-1122] we characterize the full set of separating equilibria that are symmetric among buyers and with a strategic seller being able to bid in the same way as any buyer through a so-called shill bidding activity. The revenue ranking between first and second price auctions is different from the one arising in Milgrom and Weber: the benefits from the highlighted ‘Linkage Principle’ are counterbalanced by the ‘Shill Bidding Effect.’     

Reserve price signaling

In a general auction model in which bidders’ signals are affiliated, we characterize the unique separating equilibrium in which the seller can use reserve prices to credibly signal her private information. When the buyers’ signals are independent, the optimal reserve price is shown to be increasing in the number of bidders under certain conditions. We also demonstrate that the probability that the item is sold at the reserve price can increase as the number of bidders increases, which indicates a more central role for reserve prices than perceived in the standard auction models.     

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514-540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039-1053].     

Asymmetric first-price menu auctions under intricate uncertainty

This paper studies asymmetric first-price menu auctions in the procurement environment where the buyer does not commit to a decision rule and asymmetric sellers have interdependent costs and statistically affiliated signals. Sellers compete in bidding a menu of contracts, where a contract specifies a vector of characteristics and a payment required from the buyer for delivering these characteristics. The buyer does not commit ex-ante to a decision rule but rather upon observing all the menus offered by sellers chooses the best contract. This paper establishes the existence of a continuum of separating monotone equilibria in this game bounded above by the jointly ex-post efficient outcome and below by the jointly interim efficient outcome. It shows that the jointly ex-post efficient equilibrium outcome is the only ex-post renegotiation-proof outcome and it is also ex-ante robust to all continuation equilibria.     

On the nonexistence of universal information structures

This paper extends the nonexistence result of Heifetz and Samet (Games Econ. Behav. 22 (1998) 260-273). They have shown that there exists no universal knowledge space to which every knowledge space can be mapped in a knowledge-preserving manner. We show that an analogous nonexistence result holds in the more general context of information structures. These structures can be viewed as generalizations of knowledge spaces that describe non-probabilistic beliefs.