Self‐Control and the Theory of Consumption

To study the behavior of agents who are susceptible to temptation in infinite horizon consumption problems under uncertainty, we define and characterize dynamic self‐control (DSC) preferences. DSC preferences are recursive and separable. In economies with DSC agents, equilibria exist but may be inefficient; in such equilibria, steady state consumption is independent of initial endowments and increases in self‐control. Increasing the preference for commitment while keeping self‐control constant increases the equity premium. Removing nonbinding constraints changes equilibrium allocations and prices. Debt contracts can be sustained even if the only feasible punishment for default is the termination of the contract.     

Leverage and Default in Binomial Economies: A Complete Characterization

Our paper provides a complete characterization of leverage and default in binomial economies with financial assets serving as collateral. Our Binomial No‐Default Theorem states that any equilibrium is equivalent (in real allocations and prices) to another equilibrium in which there is no default. Thus actual default is irrelevant, though the potential for default drives the equilibrium and limits borrowing. This result is valid with arbitrary preferences and endowments, contingent or noncontingent promises, many assets and consumption goods, production, and multiple periods. We also show that only no‐default equilibria would be selected if there were the slightest cost of using collateral or handling default. Our Binomial Leverage Theorem shows that equilibrium Loan to Value (LTV) for noncontingent debt contracts is the ratio of the worst‐case return of the asset to the riskless gross rate of interest. In binomial economies, leverage is determined by down risk and not by volatility.     

If You're so Smart,why Aren't You Rich? Belief Selection in Complete and Incomplete Markets

This paper provides an analysis of the asymptotic properties of Pareto optimal consumption allocations in a stochastic general equilibrium model with heterogeneous consumers. In particular, we investigate the market selection hypothesis that markets favor traders with more accurate beliefs. We show that in any Pareto‐optimal allocation whether each consumer vanishes or survives is determined entirely by discount factors and beliefs. Whereas equilibrium allocations in economies with complete markets are Pareto optimal, our results characterize the limit behavior of these economies. We show that, all else equal, the market selects for consumers who use Bayesian learning with the truth in the support of their prior and selects among Bayesians according to the size of their parameter space. Finally, we show that in economies with incomplete markets, these conclusions may not hold. With incomplete markets, payoff functions can matter for long‐run survival, and the market selection hypothesis fails.     

Uncertainty and Risk in Financial Markets

This paper considers a general equilibrium model in which the distinction between uncertainty and risk is formalized by assuming agents have incomplete preferences over state‐contingent consumption bundles, as in Bewley (1986). Without completeness, individual decision making depends on a set of probability distributions over the state space. A bundle is preferred to another if and only if it has larger expected utility for all probabilities in this set. When preferences are complete this set is a singleton, and the model reduces to standard expected utility. In this setting, we characterize Pareto optima and equilibria, and show that the presence of uncertainty generates robust indeterminacies in equilibrium prices and allocations for any specification of initial endowments. We derive comparative statics results linking the degree of uncertainty with changes in equilibria. Despite the presence of robust indeterminacies, we show that equilibrium prices and allocations vary continuously with underlying fundamentals. Equilibria in a standard risk economy are thus robust to adding small degrees of uncertainty. Finally, we give conditions under which some assets are not traded due to uncertainty aversion.     

Investment and Competitive Matching

We study markets in which agents first make investments and are then matched into potentially productive partnerships. Equilibrium investments and the equilibrium matching will be efficient if agents can simultaneously negotiate investments and matches, but we focus on markets in which agents must first sink their investments before matching. Additional equilibria may arise in this sunk‐investment setting, even though our matching market is competitive. These equilibria exhibit inefficiencies that we can interpret as coordination failures. All allocations satisfying a constrained efficiency property are equilibria, and the converse holds if preferences satisfy a separability condition. We identify sufficient conditions (most notably, quasiconcave utilities) for the investments of matched agents to satisfy an exchange efficiency property as well as sufficient conditions (most notably, a single crossing property) for agents to be matched positive assortatively, with these conditions then forming the core of sufficient conditions for the efficiency of equilibrium allocations.     

Search With Adverse Selection

This paper analyzes a sequential search model with adverse selection. We study information aggregation by the price—how close the equilibrium prices are to the full‐information prices—when search frictions are small. We identify circumstances under which prices fail to aggregate information well even when search frictions are small. We trace this to a strong form of the winner's curse that is present in the sequential search model. The failure of information aggregation may result in inefficient allocations.     

Sentiments and Aggregate Demand Fluctuations

We formalize the Keynesian insight that aggregate demand driven by sentiments can generate output fluctuations under rational expectations. When production decisions must be made under imperfect information about demand, optimal decisions based on sentiments can generate stochastic self‐fulfilling rational expectations equilibria in standard economies without persistent informational frictions, externalities, nonconvexities, or strategic complementarities in production. The models we consider are deliberately simple, but could serve as benchmarks for more complicated equilibrium models with additional features.     

Markov Perfect Industry Dynamics With Many Firms

We propose an approximation method for analyzing Ericson and Pakes (1995)‐style dynamic models of imperfect competition. We define a new equilibrium concept that we call oblivious equilibrium, in which each firm is assumed to make decisions based only on its own state and knowledge of the long‐run average industry state, but where firms ignore current information about competitors' states. The great advantage of oblivious equilibria is that they are much easier to compute than are Markov perfect equilibria. Moreover, we show that, as the market becomes large, if the equilibrium distribution of firm states obeys a certain “light‐tail” condition, then oblivious equilibria closely approximate Markov perfect equilibria. This theorem justifies using oblivious equilibria to analyze Markov perfect industry dynamics in Ericson and Pakes (1995)‐style models with many firms.     

Existence of Equilibria in Economies with Bads

We consider an exchange economy in which there are infinitely many consumers and some commodities are bads, that is, cause disutility to consumers. We give an example of such an economy for which there is no competitive equilibrium or its variants (quasi‐ or pseudo‐equilibrium), and an example of the failure of the so‐called uniform integrability condition of equilibrium allocations of increasingly populous finite economies.     

Virtual Determinacy in Overlapping Generations Models

We reappraise the significance and robustness of indeterminacy in overlapping‐generations models. In any of Gale's example economies with an equilibrium that is not locally unique, for instance, perturbing the economy by judiciously splitting each of Gale's goods into two close substitutes restricts that indeterminacy to each period's allocation of consumption between those substitutes. In particular, prices, interest rates, the commodity value of nominal savings (including money), and utility levels become determinate. Any indeterminacy of equilibrium consumption in the perturbed economy is thus insignificant to consumers, and some forecasting and comparative‐statics policy exercises become possible.     

Efficiency,Equilibrium, and Asset Pricing with Risk of Default

We introduce a new equilibrium concept and study its efficiency and asset pricing implications for the environment analyzed by Kehoe and Levine (1993) and Kocherlakota (1996). Our equilibrium concept has complete markets and endogenous solvency constraints. These solvency constraints prevent default at the cost of reducing risk sharing. We show versions of the welfare theorems. We characterize the preferences and endowments that lead to equilibria with incomplete risk sharing. We compare the resulting pricing kernel with the one for economies without participation constraints: interest rates are lower and risk premia depend on the covariance of the idiosyncratic and aggregate shocks. Additionally, we show that asset prices depend only on the valuation of agents with substantial idiosyncratic risk.     

Bubbles and Self‐Enforcing Debt

We characterize equilibria with endogenous debt constraints for a general equilibrium economy with limited commitment in which the only consequence of default is losing the ability to borrow in future periods. First, we show that equilibrium debt limits must satisfy a simple condition that allows agents to exactly roll over existing debt period by period. Second, we provide an equivalence result, whereby the resulting set of equilibrium allocations with self‐enforcing private debt is equivalent to the allocations that are sustained with unbacked public debt or rational bubbles. In contrast to the classic result by Bulow and Rogoff (1989a), positive levels of debt are sustainable in our environment because the interest rate is sufficiently low to provide repayment incentives.     

Equilibrium in Continuous‐Time Financial Markets: Endogenously Dynamically Complete Markets

We prove existence of equilibrium in a continuous‐time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well‐behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous‐time economy.     

Large Contests

We consider contests with many, possibly heterogeneous, players and prizes, and show that the equilibrium outcomes of such contests are approximated by the outcomes of mechanisms that implement the assortative allocation in an environment with a single agent that has a continuum of possible types. This makes it possible to easily approximate the equilibria of contests whose exact equilibrium characterization is complicated, as well as the equilibria of contests for which there is no existing equilibrium characterization.     

Keeping up with the Neighbors: Social Interaction in a Market Economy

We consider a world in which individuals have private endowments and trade in markets while their utility is negatively affected by the consumption of their neighbors. Our interest is in understanding how the social structure of comparisons, taken together with the familiar fundamentals of the economy (endowments, technology, and preferences), shapes equilibrium prices, allocations, and welfare. We show that equilibrium prices and consumption are a function of a single network statistic: centrality. An individual's “centrality” is given by the weighted sum of paths of different lengths to all others in a social network. In particular, prices are proportional to the sum of centralities, and an individual's consumption depends on how central she is relative to others in the network. Inequalities in wealth and connections reinforce each other in markets: A transfer of resources from less to more central agents raises prices. As segregated communities become integrated, the poor lose while the rich gain in utility! (JEL: D5, D6, D85)     

Existence of Equilibrium in Single and Double Private Value Auctions1

We show existence of equilibria in distributional strategies for a wide class of private value auctions, including the first general existence result for double auctions. The set of equilibria is invariant to the tie‐breaking rule. The model incorporates multiple unit demands, all standard pricing rules, reserve prices, entry costs, and stochastic demand and supply. Valuations can be correlated and asymmetrically distributed. For double auctions, we show further that at least one equilibrium involves a positive volume of trade. The existence proof establishes new connections among existence techniques for discontinuous Bayesian games.     

Quadratic Concavity and Determinacy of Equilibrium

One of the central features of classical models of competitive markets is the generic determinacy of competitive equilibria. For smooth economies with a finite number of commodities and a finite number of consumers, almost all initial endowments admit only a finite number of competitive equilibria, and these equilibria vary (locally) smoothly with endowments; thus equilibrium comparative statics are locally determinate. This paper establishes parallel results for economies with finitely many consumers and infinitely many commodities. The most important new condition we introduce, quadratic concavity, rules out preferences in which goods are perfect substitutes globally, locally, or asymptotically. Our framework is sufficiently general to encompass many of the models that have proved important in the study of continuous‐time trading in financial markets, trading over an infinite time horizon, and trading of finely differentiated commodities.     

COUNTRY PORTFOLIOS IN OPEN ECONOMY MACRO‐MODELS

This paper develops a simple approximation method for computing equilibrium portfolios in dynamic general equilibrium open economy macro‐models. The method is widely applicable, simple to implement, and gives analytical solutions for equilibrium portfolio positions in any combination or types of asset. It can be used in models with any number of assets, whether markets are complete or incomplete, and can be applied to stochastic dynamic general equilibrium models of any dimension, so long as the model is amenable to a solution using standard approximation methods. We first illustrate the approach using a simple two‐asset endowment economy model, and then show how the results extend to the case of any number of assets and general economic structure.     

Reverse Auctions with Multiple Reinforcement Learning Agents*

Reverse auctions in business‐to‐business (B2B) exchanges provide numerous benefits to participants. Arguably the most notable benefit is that of lowered prices driven by increased competition in such auctions. The competition between sellers in reverse auctions has been analyzed using a game‐theoretic framework and equilibria have been established for several scenarios. One finding of note is that, in a setting in which sellers can meet total demand with the highest‐bidding seller being able to sell only a fraction of the total capacity, the sellers resort to a mixed‐strategy equilibrium. Although price randomization in industrial bidding is an accepted norm, one might argue that in reality managers do not utilize advanced game theory calculations in placing bids. More likely, managers adopt simple learning strategies. In this situation, it remains an open question as to whether the bid prices converge to the theoretical equilibrium over time. To address this question, we model reverse‐auction bidding behavior by artificial agents as both two‐player and n‐player games in a simulation environment. The agents begin the game with a minimal understanding of the environment but over time analyze wins and losses for use in determining future bids. To test for convergence, the agents explore the price space and exploit prices where profits are higher, given varying cost and capacity scenarios. In the two‐player case, the agents do indeed converge toward the theoretical equilibrium. The n‐player case provides results that reinforce our understanding of the theoretical equilibria. These results are promising enough to further consider the use of artificial learning mechanisms in reverse auctions and other electronic market transactions, especially as more sophisticated mechanisms are developed to tackle real‐life complexities. We also develop the analytical results when one agent does not behave strategically while the other agent does and show that our simulations for this environment also result in convergence toward the theoretical equilibrium. Because the nature of the best response in the new setting is very different (pure strategy as opposed to mixed), it indicates the robustness of the devised algorithm. The use of artificial agents can also overcome the limitations in rationality demonstrated by human managers. The results thus have interesting implications for designing artificial agents in automating bid responses for large numbers of bids where human intervention might not always be possible.     

Dynamic Matching,Two‐Sided Incomplete Information,and Participation Costs: Existence and Convergence to Perfect Competition

Consider a decentralized, dynamic market with an infinite horizon and participation costs in which both buyers and sellers have private information concerning their values for the indivisible traded good. Time is discrete, each period has length δ, and, each unit of time, continuums of new buyers and sellers consider entry. Traders whose expected utility is negative choose not to enter. Within a period each buyer is matched anonymously with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, both the seller and the winning buyer exit the market with their realized utility. Traders who fail to trade continue in the market to be rematched. We characterize the steady‐state equilibria that are perfect Bayesian. We show that, as δ converges to zero, equilibrium prices at which trades occur converge to the Walrasian price and the realized allocations converge to the competitive allocation. We also show the existence of equilibria for δ sufficiently small, provided the discount rate is small relative to the participation costs.