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.     

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.     

Equilibria and Indivisibilities: Gross Substitutes and Complements

This paper examines an exchange economy with heterogeneous indivisible objects that can be substitutable or complementary. We show that a competitive equilibrium exists in such economies, provided that all the objects can be partitioned into two groups, and from the viewpoint of each agent, objects in the same group are substitutes and objects across the two groups are complements. This condition generalizes the well‐known Kelso–Crawford gross substitutes condition and is called gross substitutes and complements. We also provide practical and typical examples from which substitutes and complements are both jointly observed.     

Online Manufacturer Referral to Heterogeneous Retailers

Since the development of the Internet, thousands of manufacturers have been referring consumers visiting their websites to some or all of their retailers. Through a model with one manufacturer and two heterogeneous retailers, we investigate whether it is an equilibrium for the manufacturer to refer consumers exclusively to a retailer or nonexclusively to both retailers. Our analysis indicates that nonexclusive referral is the manufacturer's equilibrium choice if the referral segment market size is sufficiently large; otherwise, exclusive referral is the equilibrium choice. In exclusive referral, the manufacturer would refer consumers to the more cost‐efficient and smaller retailer. In the presence of infomediary referral, it is less likely for both exclusive and nonexclusive referrals to be an equilibrium, as the infomediary referral segment grows. We also show our qualitative results are robust even if there were price discrimination among consumers, referral position disparity, local consumers, and asymmetric referral market sizes.     

Stationary Equilibria in Asset‐Pricing Models with Incomplete Markets and Collateral

We consider an infinite‐horizon exchange economy with incomplete markets and collateral constraints. As in the two‐period model of Geanakoplos and Zame (2002), households can default on their liabilities at any time, and financial securities are only traded if the promises associated with these securities are backed by collateral. We examine an economy with a single perishable consumption good, where the only collateral available consists of productive assets. In this model, competitive equilibria always exist and we show that, under the assumption that all exogenous variables follow a Markov chain, there also exist stationary equilibria. These equilibria can be characterized by a mapping from the exogenous shock and the current distribution of financial wealth to prices and portfolio choices. We develop an algorithm to approximate this mapping numerically and discuss ways to implement the algorithm in practice. A computational example demonstrates the performance of the algorithm and shows some quantitative features of equilibria in a model with collateral and default.     

生产能力限制下价格Stackelberg博弈模型

生产能力限制条件下,同质产品市场中两企业以价格为决策变量进行Stackelberg竞争,采取有效配给规则。两企业对称情况下,生产能力较小时模型存在唯一的子博弈精炼纳什均衡,两企业销量达到自身生产能力,得到相同收益;生产能力较大时,追随企业匹配领头企业的价格,博弈存在后动优势。分析了领头企业生产能力大于追随企业的不对称情形,并给出相应均衡。算例分析表明,企业价格竞争的前提是充分大的供给能力,追随企业应在领头企业生产能力较大时进入市场。     

The Retail Space‐Exchange Problem with Pricing and Space Allocation Decisions

We consider retail space‐exchange problems where two retailers exchange shelf space to increase accessibility to more of their consumers in more locations without opening new stores. Using the Hotelling model, we find two retailers’ optimal prices, given their host and guest space in two stores under the space‐exchange strategy. Next, using the optimal space‐dependent prices, we analyze a non‐cooperative game, where each retailer makes a space allocation decision for the retailer's own store. We show that the two retailers will implement such a strategy in the game, if and only if their stores are large enough to serve more than one‐half of their consumers. Nash equilibrium for the game exists, and its value depends on consumers’ utilities and trip costs as well as the total available space in each retailer's store. Moreover, as a result of the space‐exchange strategy, each retailer's prices in two stores are both higher than the retailer's price before the space exchange, but they may or may not be identical.     

竞争市场上保修期与价格的联合质量信号传递作用

虽然顾客重视产品的质量（可靠性），但其在做出购买决策时往往不能完全掌握产品的质量信息。不同质量的产品可能具有不同生产和保修成本，因此保修期和销售价格可以成为传递产品质量的有效信号。考虑市场上存在相互竞争的两个生产商：在位者和新进者，顾客了解在位者的产品质量信息但不了解新进者的质量状况。研究并比较了仅保修期作为单质量信号、保修期和价格同时作为质量信号时，市场实现分离均衡的条件，以及新进者的保修期设计与定价决策。研究表明：在保修期作为单质量信号时，只要高质量新进者的单位时间保修成本小于低质量新进者，即可实现分离均衡；在保修期和价格同时作为质量信号时，当满足某些条件时也可实现分离均衡。相对于保修期作单质量信号的情形，当保修期和价格同作质量信号时，高质量新进者实现分离均衡所需提供的保修期更短、价格更低（信息租金更少），且能获得更高的市场份额和利润。最后，数值分析表明保修期和价格同作质量信号时所需满足的条件是比较宽松的，比保修期单独作质量信号时更容易实现分离均衡。     

纵向差异下的连接外部性和网络结构分析

在寡头竞争的市场环境里,相互竞争的企业通常会与竞争对手建立连接,从而影响他们在产品市场上竞争的方式。本文我们建立两阶段博弈模型研究企业建立连接的激励以及由此而形成的网络结构形态。在博弈的第一期,具有纵向差异化的企业决定是否与其竞争对手建立连接;企业观察到连接结果后在第二期进行价格竞争。本文的分析显示均衡网络结构与连接效应、连接成本和消费者偏好有关,均衡网络可能仅是一个企业与其它所有企业建立连接的星型网络结构,也可能是没有任何企业建立连接的空网络结构。本文还分析了均衡网络和社会有效网络之间的关系,发现均衡网络和社会有效网络并不总是一致的,从社会计划者的角度看,企业建立连接可能存在激励不足,因此公共政策应该鼓励企业建立连接。     

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.     

Strongly Consistent Self‐Confirming Equilibrium

Fudenberg and Levine (1993a) introduced the notion of self‐confirming equilibrium, which is generally less restrictive than Nash equilibrium. Fudenberg and Levine also defined a concept of consistency, and claimed in their Theorem 4 that with consistency and other conditions on beliefs, a self‐confirming equilibrium has a Nash equilibrium outcome. We provide a counterexample that disproves Theorem 4 and prove an alternative by replacing consistency with a more restrictive concept, which we call strong consistency. In games with observed deviators, self‐confirming equilibria are strongly consistent self‐confirming equilibria. Hence, our alternative theorem ensures that despite the counterexample, the corollary of Theorem 4 is still valid.     

On Forward Induction

A player's pure strategy is called relevant for an outcome of a game in extensive form with perfect recall if there exists a weakly sequential equilibrium with that outcome for which the strategy is an optimal reply at every information set it does not exclude. The outcome satisfies forward induction if it results from a weakly sequential equilibrium in which players' beliefs assign positive probability only to relevant strategies at each information set reached by a profile of relevant strategies. We prove that if there are two players and payoffs are generic, then an outcome satisfies forward induction if every game with the same reduced normal form after eliminating redundant pure strategies has a sequential equilibrium with an equivalent outcome. Thus in this case forward induction is implied by decision‐theoretic criteria.     

ECONOMICS AND CLIMATE CHANGE: INTEGRATED ASSESSMENT IN A MULTI‐REGION WORLD

This paper develops a model that integrates the climate and the global economy—an integrated assessment model—with which different policy scenarios can be analyzed and compared. The model is a dynamic stochastic general‐equilibrium setup with a continuum of regions. Thus, it is a full stochastic general‐equilibrium version of RICE, Nordhaus’s pioneering multi‐region integrated assessment model. Like RICE, our model features traded fossil fuel but otherwise has no markets across regions—there is no insurance nor any intertemporal trade across them. The extreme form of market incompleteness is not fully realistic but arguably not a bad approximation of reality. Its major advantage is that, along with a set of reasonable assumptions on preferences, technology, and nature, it allows a closed‐form model solution. We use the model to assess the welfare consequences of carbon taxes that differ across as well as within oil‐consuming and ‐producing regions. We show that, surprisingly, only taxes on oil producers can improve the climate: taxes on oil consumers have no effect at all. The calibrated model suggests large differences in views on climate policy across regions.     

A Quantitative Theory of Unsecured Consumer Credit with Risk of Default

We study, theoretically and quantitatively, the general equilibrium of an economy in which households smooth consumption by means of both a riskless asset and unsecured loans with the option to default. The default option resembles a bankruptcy filing under Chapter 7 of the U.S. Bankruptcy Code. Competitive financial intermediaries offer a menu of loan sizes and interest rates wherein each loan makes zero profits. We prove the existence of a steady‐state equilibrium and characterize the circumstances under which a household defaults on its loans. We show that our model accounts for the main statistics regarding bankruptcy and unsecured credit while matching key macroeconomic aggregates, and the earnings and wealth distributions. We use this model to address the implications of a recent policy change that introduces a form of “means testing” for households contemplating a Chapter 7 bankruptcy filing. We find that this policy change yields large welfare gains.     

Financial Intermediaries and Markets

A complex financial system comprises both financial markets and financial intermediaries. We distinguish financial intermediaries according to whether they issue complete contingent contracts or incomplete contracts. Intermediaries such as banks that issue incomplete contracts, e.g., demand deposits, are subject to runs, but this does not imply a market failure. A sophisticated financial system—a system with complete markets for aggregate risk and limited market participation—is incentive‐efficient, if the intermediaries issue complete contingent contracts, or else constrained‐efficient, if they issue incomplete contracts. We argue that there may be a role for regulating liquidity provision in an economy in which markets for aggregate risks are incomplete.     

Split‐Award Procurement Auctions—Can Bayesian Equilibrium Strategies Predict Human Bidding Behavior in Multi‐Object Auctions?

We analyze if and when symmetric Bayes Nash equilibrium predictions can explain human bidding behavior in multi‐object auctions. We focus on two sealed‐bid split‐award auctions with ex ante split decisions as they can be regularly found in procurement practice. These auction formats are straightforward multi‐object extensions of the first‐price sealed‐bid auction. We derive the risk‐neutral symmetric Bayes Nash equilibrium strategies and find that, although the two auction mechanisms yield the same expected costs to the buyer, other aspects of the two models, including the equilibrium bidding strategies, differ significantly. The strategic considerations in these auction formats are more involved than in single‐lot first‐price sealed‐bid auctions, and it is questionable whether expected utility maximization can explain human bidding behavior in such multi‐object auctions. Therefore, we analyzed the predictive accuracy of our equilibrium strategies in the laboratory. In human subject experiments we found underbidding, which is in line with earlier experiments on single‐lot first‐price sealed‐bid auctions. To control for regret, we organize experiments against computerized bidders, who play the equilibrium strategy. In computerized experiments where bid functions are only used in a single auction, we found significant underbidding on low‐cost draws. In experiments where the bid function is reused in 100 auctions, we could also control effectively for risk aversion, and there is no significant difference of the average bidding behavior and the risk‐neutral Bayes Nash equilibrium bid function. The results suggest that strategic complexity does not serve as an explanation for underbidding in split‐award procurement auctions, but risk aversion does have a significant impact.     

Supplier–Buyer Negotiation Games: Equilibrium Conditions and Supply Chain Efficiency

In a decentralized supply chain, supplier–buyer negotiations have a dynamic aspect that requires both players to consider the impact of their decisions on future decisions made by their counterpart. The interaction generally couples strongly the price decision of the supplier and the quantity decision of the buyer. We propose a basic model for a repeated supplier–buyer interaction, during several rounds. In each round, the supplier first quotes a price, and the buyer places an order at that price. We find conditions for existence and uniqueness of a well‐behaved subgame‐perfect equilibrium in the dynamic game. When costs are stationary and there are no holding costs, we identify some demand distributions for which these conditions are met, examine the efficiency of the equilibrium, and show that, as the number of rounds increases, the profits of the supply chain increase towards the supply chain optimum. In contrast, when costs vary over time or holding costs are present, the benefit from multi‐period interactions is reduced and after a finite number of time periods, supply chain profits stay constant even when the number of rounds increases.     

Intersectoral Labor Mobility and the Growth of the Service Sector

One of the most striking changes in the U.S. economy over the past 50 years has been the growth in the service sector. Between 1950 and 2000, service‐sector employment grew from 57 to 75 percent of total employment. However, over this time, the real hourly wage in the service sector grew only slightly faster than in the goods sector. In this paper, we assess whether or not the essential constancy of the relative wage implies that individuals face small costs of switching sectors, and we quantify the relative importance of labor supply and demand factors in the growth of the service sector. We specify and estimate a two‐sector labor market equilibrium model that allows us to address these empirical issues in a unified framework. Our estimates imply that there are large mobility costs: output in both sectors would have been double their current levels if these mobility costs had been zero. In addition, we find that demand‐side factors, that is, technological change and movements in product and capital prices, were responsible for the growth of the service sector.     

A Theory of the Stakeholder Corporation

There is a widely held view within the general public that large corporations should act in the interests of a broader group of agents than just their shareholders (the stakeholder view). This paper presents a framework where this idea can be justified. The point of departure is the observation that a large firm typically faces endogenous risks that may have a significant impact on the workers it employs and the consumers it serves. These risks generate externalities on these stakeholders which are not internalized by shareholders. As a result, in the competitive equilibrium, there is under‐investment in the prevention of these risks. We suggest that this under‐investment problem can be alleviated if firms are instructed to maximize the total welfare of their stakeholders rather than shareholder value alone (stakeholder equilibrium). The stakeholder equilibrium can be implemented by introducing new property rights (employee rights and consumer rights) and instructing managers to maximize the total value of the firm (the value of these rights plus shareholder value). If there is only one firm, the stakeholder equilibrium is Pareto optimal. However, this is not true with more than one firm and/or heterogeneous agents, which illustrates some of the limits of the stakeholder model.     

Repeated Games Where the Payoffs and Monitoring Structure Are Unknown

This paper studies repeated games with imperfect public monitoring where the players are uncertain both about the payoff functions and about the relationship between the distribution of signals and the actions played. We introduce the concept of perfect public ex post equilibrium (PPXE), and show that it can be characterized with an extension of the techniques used to study perfect public equilibria. We develop identifiability conditions that are sufficient for a folk theorem; these conditions imply that there are PPXE in which the payoffs are approximately the same as if the monitoring structure and payoff functions were known. Finally, we define perfect type‐contingently public ex post equilibria (PTXE), which allows players to condition their actions on their initial private information, and we provide its linear programming characterization.