Approximate versus Exact Equilibria in Dynamic Economies

This paper develops theoretical foundations for an error analysis of approximate equilibria in dynamic stochastic general equilibrium models with heterogeneous agents and incomplete financial markets. While there are several algorithms that compute prices and allocations for which agents' first‐order conditions are approximately satisfied (“approximate equilibria”), there are few results on how to interpret the errors in these candidate solutions and how to relate the computed allocations and prices to exact equilibrium allocations and prices. We give a simple example to illustrate that approximate equilibria might be very far from exact equilibria. We then interpret approximate equilibria as equilibria for close‐by economies; that is, for economies with close‐by individual endowments and preferences. We present an error analysis for two models that are commonly used in applications, an overlapping generations (OLG) model with stochastic production and an asset pricing model with infinitely lived agents. We provide sufficient conditions that ensure that approximate equilibria are close to exact equilibria of close‐by economies. Numerical examples illustrate the analysis.     

On the Discrete Version of the Aumann–Shapley Cost‐Sharing Method

Each agent in a finite set requests an integer quantity of an idiosyncratic good; the resulting total cost must be shared among the participating agents. The Aumann–Shapley prices are given by the Shapley value of the game where each unit of each good is regarded as a distinct player. The Aumann–Shapley cost‐sharing method charges to an agent the sum of the prices attached to the units she consumes. We show that this method is characterized by the two standard axioms of Additivity and Dummy, and the property of No Merging or Splitting: agents never find it profitable to split or to merge their consumptions. We offer a variant of this result using the No Reshuffling condition: the total cost share paid by a group of agents who consume perfectly substitutable goods depends only on their aggregate consumption. We extend this characterization to the case where agents are allowed to consume bundles of goods.     

Decentralized Trading With Private Information

The paper studies how asset prices are determined in a decentralized market with asymmetric information about asset values. We consider an economy in which a large number of agents trade two assets in bilateral meetings. A fraction of the agents has private information about the asset values. We show that, over time, uninformed agents can elicit information from their trading partners by making small offers. This form of experimentation allows the uninformed agents to acquire information as long as there are potential gains from trade in the economy. As a consequence, the economy converges to a Pareto efficient allocation.     

Over‐the‐Counter Markets

We study how intermediation and asset prices in over‐the‐counter markets are affected by illiquidity associated with search and bargaining. We compute explicitly the prices at which investors trade with each other, as well as marketmakers' bid and ask prices, in a dynamic model with strategic agents. Bid–ask spreads are lower if investors can more easily find other investors or have easier access to multiple marketmakers. With a monopolistic marketmaker, bid–ask spreads are higher if investors have easier access to the marketmaker. We characterize endogenous search and welfare, and discuss empirical implications.     

A Robust Model of Bubbles With Multidimensional Uncertainty

Observers often interpret boom–bust episodes in asset markets as speculative frenzies where asymmetrically informed investors buy overvalued assets hoping to sell to a greater fool before the crash. Despite its intuitive appeal, however, this notion of speculative bubbles has proven difficult to reconcile with economic theory. Existing models have been criticized on the basis that they assume irrationality, that prices are somewhat unresponsive to sales, or that they depend on fragile, knife‐edge restrictions. To address these issues, I construct a rational version of Abreu and Brunnermeier (2003), where agents invest growing endowments into an asset, fueling appreciation and eventual overvaluation. Riding bubbles is optimal as long as the growth rate of the bubble and the probability of selling before the crash are high enough. This probability increases with the amount of noise in the economy, as random short‐term fluctuations make it difficult for agents to infer information from prices.     

稳健的动态资产组合模型研究

事件风险与参数的不确定性对金融决策有重大的影响,投资者担心股市上极端金融事件的出现,突然改变股票价格和波动率,造成较大的损失.通过引入风险规避的稳健投资者以及模型设定可能存在误差,投资者在最小化模型设定误差的前提下,制定风险资产收益跳跃情况下的动态资产组合战略,最大化投资者的效用.结果表明,当投资者是风险规避和不确定性规避者时,稳健的投资准则会显著降低他们对风险资产的需求.     

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)     

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.     

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.     

Market Good Flexibility in Capacity Auctions

We consider the allocation of limited production capacity among several competing agents through auctions. Our focus is on the contribution of flexibility in market good design to effective capacity allocation. The application studied is a capacity allocation problem involving several agents, each with a job, and a facility owner. Each agent generates revenue by purchasing capacity and scheduling its job at the facility. Ascending auctions with various market good designs are compared. We introduce a new market good that provides greater flexibility than those previously considered in the literature. We allow ask prices to depend both on agents’ utility functions and on the number of bids at the previous round of the auction, in order to model and resolve resource conflicts. We develop both optimal and heuristic solution procedures for the winner determination problem. Our computational study shows that flexibility in market good design typically increases system value within auctions. A further increase is achieved if each agent is allowed to bid for multiple market goods at each round. On average, the multiple flexible market goods auction provides over 95% of the system value found by centralized planning.     

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.     

双边市场中用户满意度与平台战略的选择

目前,有关双边市场理论的研究多在单期博弈模型中展开,并未在多期的情况下考虑平台的动态定价问题.显然,这与两边用户总是在平台中展开多次交易,平台总是面临动态定价的客观现实相违背.基于此,本文构建了一个双寡头两期动态博弈模型,通过引入用户满意度,研究了双边平台在不同定价策略(统一定价策略和歧视定价策略)和不同战略(短期战略和长期战略)情形下的最优定价,比较了策略间和战略间的差异性和有效性.研究发现: 第一期获得较大市场份额并不能保证平台在第二期竞争中占据优势地位,也不能保证获得较高重复购买率;歧视定价策略比统一定价策略更有利于平台获得较高利润,而不利于两边用户效用和社会福利水平的提升;(歧视定价策略,歧视定价策略)是帕累托上策均衡解;两期社会福利总水平仅仅与平台的定价策略有关,而与平台的战略无关.     

Incentives,Contracts, and Markets: A General Equilibrium Theory of Firms1

This paper takes steps toward integrating firm theory in the spirit of Alchian and Demsetz (1972) and Grossman and Hart (1986), contract theory in the spirit of Holmstrom (1979), and general equilibrium theory in the spirit of Arrow and Debreu (1954) and McKenzie (1959). In the model presented here, the set of firms that form and the contractual arrangements that appear, the assignments of agents to firms, the prices faced by firms for inputs and outputs, and the incentives to agents are all determined endogenously at equilibrium. Agents choose consumption—but they also choose which firms to join, which roles to occupy in those firms, and which actions to take in those roles. Agents interact anonymously with the (large) market, but strategically within the (small) firms they join. The model accommodates moral hazard, adverse selection, signaling, and insurance. Equilibria may be Pareto ranked.     

A Dynamic Equilibrium Model of International Portfolio Holdings

This paper develops a continuous‐time equilibrium model of a two‐country exchange economy with heterogeneous agents and nontraded goods. Nontraded goods play the role of state variables that shift the marginal utility of traded goods. This affects prices and generates dynamic hedging demands that explain the well documented home bias puzzle in international equity portfolios. When calibrated to both consumption and production data, the model is able to generate significative home bias in equity portfolios. A new methodology, based on Malliavin calculus, is presented to solve for the portfolio policies along the equilibrium path. This methodology allows one to reduce the determination of equilibrium portfolio holdings to the solution of a linear algebraic system, rather than a partial differential equation.     

POLICY WITH DISPERSED INFORMATION

Information regarding economic fundamentals is widely dispersed in society, is only imperfectly aggregated through prices or other indicators of aggregate activity, and cannot be centralized by the government or any other institution. In this paper we seek to identify policies that can improve the decentralized use of such dispersed information without requiring the government to observe this information. We show that this can be achieved by appropriately designing the contingency of taxation on ex post public information regarding the realized fundamentals and aggregate activity. When information is common (as in the Ramsey literature) or when agents have private information only about idiosyncratic shocks (as in the Mirrlees literature), the contingency on fundamentals alone suffices for efficiency. When instead agents have private information about aggregate shocks, the contingency on aggregate activity is crucial. An appropriate combination of the two contingencies permits the government to: (i) dampen the impact of noise and hence reduce non-fundamental volatility, without also dampening the impact of fundamentals; (ii) induce agents to internalize informational externalities, and hence improve the speed of social learning; (iii) restore a certain form of constrained efficiency in the decentralized use of information; and (iv) guarantee that welfare increases with the provision of any additional information. (JEL: C72, D62, D82)     

Wirkungsweise von Combined-Currency Prices – Eine theoretische und empirisch experimentelle Analyse

Over the last few years, programs to enhance customer retention have strongly increased in popularity. Thus, a growing number of artificial currencies such as bonus points or miles have been created. In consequence, many products and services today can be paid by using a combination of real (e.g. Euro) and artificial currencies. These combined-currency prices offer marketers many possibilities to design prices. Yet, research dealing with the effects of such combined-currency prices is scarce. Therefore, our study addresses the effect of combined-currency prices on consumers' perception of prices purchase behaviour. Moreover, we analysed how such effects can be explained by taking into account important concepts of the field of price psychology. Our experimental results demonstrate that combined-currency prices strongly influence consumers' perception of price fairness. Particularly if planning to increase prices or when designing prices for bonus programs, marketers should therefore favour combined-currency prices instead of one-dimensional prices.     

Adaptive Tactical Pricing in Multi‐Agent Supply Chain Markets Using Economic Regimes

In today's complex and dynamic supply chain markets, information systems are essential for effective supply chain management. Complex decision making processes on strategic, tactical, and operational levels require substantial timely support in order to contribute to organizations' agility. Consequently, there is a need for sophisticated dynamic product pricing mechanisms that can adapt quickly to changing market conditions and competitors' strategies. We propose a two‐layered machine learning approach to compute tactical pricing decisions in real time. The first layer estimates prevailing economic conditions—economic regimes—identifying and predicting current and future market conditions. In the second layer, we train a neural network for each regime to estimate price distributions in real time using available information. The neural networks compute offer acceptance probabilities from a tactical perspective to meet desired sales quotas. We validate our approach in the trading agent competition for supply chain management. When competing against the world's leading agents, the performance of our system significantly improves compared to using only economic regimes to predict prices. Profits increase significantly even though the prices and sales volume do not change significantly. Instead, tactical pricing results in a more efficient sales strategy by reducing both finished goods and components inventory costs.     

中国权证市场认购权证的价格偏误研究

使用Black-Scholes模型对中国权证市场上发行的认购权证进行了研究。研究表明,总体上权证的市场价格高于其模型价格80.38%(历史波动率参数的B/S模型)和140.50%(E-GARCH波动率参数的B/S模型);另一方面,随着时间的推移,市场价格和模型价格之间的偏误有减小的趋势。此外,研究还发现,2006年底至今,部分认购权证的市场价格不仅低于其模型价格,而且低于其价格下限。这似乎违反了无套利原则,但中国市场交易机制的限制(股票卖空限制、股票T 1交易机制)使得这一现象得以较长时间存在。当权证的市场价格低于其价格下限时,在特定的情况下,套利交易可以进行。     

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.     

Toward a Strategic Foundation for Rational Expectations Equilibrium

A step toward a strategic foundation for rational expectations equilibrium is taken by considering a double auction with n buyers and m sellers with interdependent values and affiliated private information. If there are sufficiently many buyers and sellers, and their bids are restricted to a sufficiently fine discrete set of prices, then, generically, there is an equilibrium in nondecreasing bidding functions that is arbitrarily close to the unique fully revealing rational expectations equilibrium of the limit market with unrestricted bids and a continuum of agents. In particular, the large double‐auction equilibrium is almost efficient and almost fully aggregates the agents' information.