Individual risk and Lebesgue extension without aggregate uncertainty

Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in [Y.N. Sun, The exact law of large numbers via Fubini extension and characterization of insurable risks, J. Econ. Theory 126 (2006) 31-69] to characterize the cancellation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this paper is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk.     

The exact law of large numbers for independent random matching

This paper provides a mathematical foundation for independent random matching of a large population, as widely used in the economics literature. We consider both static and dynamic systems with random mutation, partial matching arising from search, and type changes induced by matching. Under independence assumptions at each randomization step, we show that there is an almost-sure constant cross-sectional distribution of types in a large population, and moreover that the multi-period cross-sectional distribution of types is deterministic and evolves according to the transition matrices of the type process of a given agent. We also show the existence of a joint agent-probability space, and randomized mutation, partial matching and match-induced type-changing functions that satisfy appropriate independence conditions, where the agent space is an extension of the classical Lebesgue unit interval.     

The evolution of exchange

Stochastic stability is applied to the problem of exchange. We analyze the stochastic stability of two dynamic trading processes in a simple housing market. In both models, traders meet in pairs at random and exchange their houses when trade is mutually beneficial, but occasionally they make mistakes. The models differ in the probability of mistakes. When all mistakes are equally likely, the set of stochastically stable allocations contains the set of efficient allocations. When more serious mistakes are less likely, the stochastically stable states are those allocations, always efficient, with the lowest envy level.     

Purification, saturation and the exact law of large numbers

Purification results are important in game theory and statistical decision theory. We prove a new purification theorem that generalizes several earlier results. The key idea of our proof is to make use of the exact law of large numbers. As an application, we show that every mixed strategy in games with finite players, general action spaces and diffused, conditionally independent incomplete information has many strong purifications.     

Competition among banks and the pass-through of monetary policy

This paper introduces monopolistically competitive financial intermediaries into the New Keynesian DSGE setting. Modelling bank market power explicitly contributes to understanding two empirical facts: (i) The short-run transmission of changes in money market rates to bank retail rates is far from complete and heterogeneous. (ii) Stiffer competition among commercial banks implies that loan rates correlate more tightly with the policy rate. In my model, the degree of monopolistic competition in the banking sector has a sizeable impact on the pass-through of changes in the policy rate. In particular, a more competitive market for bank credit amplifies the efficiency of monetary policy.     

A note on the global stability condition of price adjustment processes on the plane

In Mukherji [Global stability condition on the plane: a general law of demand, J. Econ. Theory 134 (2007) 583-592], a set of sufficient conditions for global stability of price adjustment processes on the plane is identified. The aim of this note is to point out that one of the conditions can be dropped.     

Non-marketed options, non-existence of equilibria, and non-linear prices

This paper presents a surprising example that shows that the lattice theoretic properties in Mas-Colell's (1986) seminal work are relevant to the existence of equilibrium even when the commodity space is finite dimensional.The example is a two-period securities model with a three-dimensional portfolio space and two traders. The paper identifies a non-marketed call option that fails to have a minimum cost super-replicating portfolio. Using this option, we construct an economy that satisfies all of Mas-Colell's assumptions, except that the three-dimensional commodity space is not a vector lattice. In this economy, there is no Walrasian equilibrium and the second theorem of welfare economics fails.Our example has important finite- as well as infinite-dimensional implications. It is also an example of a “well behaved” economy in which optimal allocations that are not supported by linear Walrasian prices are decentralized by the non-linear prices studied in Aliprantis-Tourky-Yannelis (2001).     

Global stability condition on the plane: A general law of demand

It is well known that global stability of competitive equilibrium obtains only under quite stringent restrictions. This paper identifies a set of restrictions which may be interpreted as a very weak form of the so-called Law of Demand and shows how this may imply global stability for motion on the plane. It is also of some interest that this condition is satisfied by a perturbed version of the celebrated Scarf example which does not satisfy existing conditions for global stability.     

The existence, uniqueness, and optimality of the terminal wealth depletion time in life-cycle models of saving under uncertain lifetime and borrowing constraint

In life-cycle models of saving under uncertain lifetime and borrowing constraint, the consumer's wealth must be depleted before the maximum lifetime. This paper investigates the existence, uniqueness, and optimality of the terminal wealth depletion time. It is proved that the optimal terminal wealth depletion time, if such exists, must be unique. If the equation that determines the optimal terminal wealth depletion has multiple solutions, then the location of the optimal solution will depend on the configuration of the solutions. An optimality test is developed to verify whether a candidate solution for the terminal wealth depletion time is indeed optimal. The paper introduces a method new to economics, the Dubovitskii-Milyutin adjoint equation, to analyze the properties of the optimal control problem.     

A dual characterization of incentive efficiency

We show that incentive efficient allocations in economies with adverse selection and moral hazard problems can be determined as optimal solutions to a linear programming problem and we use duality theory to obtain a complete characterization of the optima. Our dual analysis identifies welfare effects associated with the incentives of the agents to truthfully reveal their private information. Because these welfare effects may generate non-convexities, incentive efficient allocations may involve randomization. Other properties of incentive efficient allocations are also derived.     

The nucleolus of large majority games

Members of a shareholder meeting or legislative committee have greater or smaller voting power than meets the eye if the nucleolus of the induced majority game differs from the voting weight distribution. We establish a new sufficient condition for the weight and power distributions to be equal, and we characterize the limit behavior of the nucleolus in case all relative weights become small.     

Linear and non-linear price decentralization

Compendious and thorough solutions to the existence of a linear price equilibrium problem, the second welfare theorem, and the limit theorem on the core are provided for exchange economies whose consumption sets are the positive cone of arbitrary ordered Fréchet spaces—dispensing entirely with the assumption that the vector ordering of the commodity space is a lattice. The motivation comes from economic applications showing the need to bring within the scope of equilibrium theory vector orderings that are not lattices, which arise in the typical model of portfolio trading with missing options. The assumptions are on the primitives of the model. They are bounds on the marginals of non-linear prices and for ω-proper economies they are both sufficient and necessary.     

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.     

The dynamics of risk-sensitive allocations

This paper describes Pareto optimal allocations when agents have risk-sensitive preferences as formulated by Hansen and Sargent (IEEE Trans. Automat. Control 40(5) (1995) 968-971). Necessary and sufficient conditions are given for the existence and stability of steady states at which Pareto weights are time-invariant. When all agents are risk-sensitive with the same power reward function there is a unique interior steady state which is stable when the power is positive and unstable when the power is negative. When there is at least one agent with time-additive preferences eventually all risk-sensitive agents have zero Pareto weights.     

A continuum of commitment

We examine a generic three-stage game for two players with alternating moves, where the first player can choose the level of adjustment cost to be paid in the last period to modify the action she announced in the first period. In the resulting continuum of commitment options, convexifying the choice between first-mover and second-mover advantage in pure strategies, we characterize when an intermediate adjustment-cost level is chosen in equilibrium. We show that the wastefulness of the adjustment cost may be in the players’ best interest, improving both of their individual net payoffs over making any fraction of the adjustment cost a side-payment from the first to the second player.     

Uniqueness of Markov equilibrium in stochastic OLG models with nonclassical production

A large class of stochastic OLG economies with nonclassical production is shown to possess a unique Markov Equilibrium (ME) which is also the unique sequential equilibrium. Additional properties such as monotonicity, continuity, and smoothness of the ME are also discussed.     

The communication requirements of efficient allocations and supporting prices

We show that any communication finding a value-maximizing allocation in a private-information economy must also discover supporting prices (in general personalized and nonlinear). In particular, to allocate L indivisible items between two agents, a price must be revealed for each of the 2^L-1 bundles. We prove that all monotonic prices for an agent must be used, hence exponential communication in L is needed. Furthermore, exponential communication is needed just to ensure a higher share of surplus than that realized by auctioning all items as a bundle, or even a higher expected surplus (for some probability distribution over valuations). When the utilities are submodular, efficiency still requires exponential communication (and fully polynomial approximation is impossible). When the items are identical, arbitrarily good approximation is obtained with exponentially less communication than exact efficiency.     

On the first-order approach in principal-agent models with hidden borrowing and lending

We provide sufficient conditions for the validity of the first-order approach for two-period dynamic moral hazard problems where the agent can save and borrow secretly. The first-order approach is valid if the following conditions hold: (i) the agent has non-increasing absolute risk aversion utility (NIARA), (ii) the output technology has monotone likelihood ratios (MLR), and (iii) the distribution function of output is log-convex in effort (LCDF). Moreover, under these three conditions, the optimal contract is monotone in output. We also investigate a few possibilities of relaxing these requirements.