Volume 7, Issue 1

We extend Milgrom and Weber's affiliated valuations model to the multi-unit case. We show that the discriminatory auction has a unique equilibrium, that corresponds to Milgrom and Weber's first-price equilibrium in the 2-bidder, constant marginal valuations case. This unique equilibrium therefore leads to lower expected prices than the equilibrium of the English auction where the units are bundled together. Hence we show that in an auction of a single object where the object can be divided into k parts and a bidder's valuation for each part is the same, it is not possible to increase revenue by using a multi-unit discriminatory auction. With more than two bidders and constant marginal valuations we show that the first-price equilibrium is an equilibrium of the multi-unit discriminatory auction. Back and Zender show this in the common values case which is a special case of affiliated valuations. We also show that the first-price equilibrium does not hold with decreasing marginal valuations.

Despite their scarcity in the literature, an abundance of globally regular indirect utility functions, involving as many parameters as desired, exist and are easily constructed as a function of simple homothetic component utilities.

In competitive economies with private firm ownership, incomplete markets, and firm shareholders changing over time, several firm objectives have been proposed. Some are useful to understand efficiency of equilibria, and others are explicitly consistent with majority shareholder control or collective choice rules, but it is not always clear if versions of each type are consistent with versions of the other type. This paper shows that ex-post, value maximizing rules, (including those proposed by Dreze, and Grossman and Hart,) are consistent with shareholder preferences in such economies; that is, along the equilibrium path, in every period and state of the world, every coalition of a firm's shareholders in that period and state approves a value maximizing production plan. This result applies to cases when shareholders within a firm and across firms can form coalitions, and when stock trading can be ex-dividend or cum-dividend, and with a combination of both. This result does not resolve the problem of inefficiency of stock market equilibria, or that of ex ante disagreement among shareholders. It can help understand when firm objectives with some desirable properties are consistent with a particular version of shareholder control, and it provides a stability criterion (in terms of robustness to shareholder coalitions) for organizing productive resources in such economies.

This paper presents a model of opportunistic behaviour in decentralized economic exchange and considers the impact of inadequate institutional framework of formal contract enforcement on economic performance. It is shown that (i) when the number of cheating traders is sufficiently large, inadequate institutions result in a loss of decentralized trading contracts, (ii) an adequate institutional framework, while being necessary for the attainment of a Pareto optimal outcome, may not be sufficient if traders perceive it as inadequate; and (iii) sufficiently good formal enforcement provisions help deter contractual breach in environments with corrupt and powerful enforcers.

We study the agenda selection problem faced by a multi-issue committee when there can be (small) delays between the negotiations of different issues. We show that the presence of these delays affects the interplay of the forces in the bargaining game and solves the indeterminacy of equilibria. In particular, a unique SPE can be established among sequential bargaining procedures. In equilibrium, players discuss the most important issue first.

The paper examines under what conditions vertically differentiated duopolists engage in first-degree price discrimination. Each firm decides on a pricing regime at a first stage and sets prices at a second stage. The paper shows that when unit cost is an increasing and convex function of quality, the discriminatory regime is the unique subgame-perfect equilibrium of such two-stage game. In contrast to the case of horizontal differentiation, the discriminatory equilibrium is not necessarily Pareto-dominated by a bilateral commitment to uniform pricing. Also, the quality choices of perfectly discriminating duopolists are welfare maximizing. The paper explains why a threat of entry may elicit price discrimination by an incumbent monopolist.

We analyze credit market equilibrium when banks screen loan applicants. When banks have a convex cost function of screening, a pure strategy equilibrium exists where banks optimally set interest rates at the same level as their competitors. This result complements Broecker's (1990) analysis, where he demonstrates that no pure strategy equilibrium exists when banks have zero screening costs. In our set up we show that interest rate on loans are largely independent of marginal costs, a feature consistent with the extant empirical evidence. In equilibrium, banks make positive profits in our model in spite of the threat of entry by inactive banks. Moreover, an increase in the number of active banks increases credit risk and so does not improve credit market efficiency: this point has important regulatory implications. Finally, we extend our analysis to the case where banks have differing screening abilities.

In a two-stage Cournot oligopoly cost minimization sometimes calls for ex ante identical firms to invest in diverse technologies. It is an implication of this result that it might be optimal to treat identical duopolistic polluters asymmetrically. We discuss the efficiency properties of a per-unit tax and a tradable permit when ex ante identical duopolists should ideally invest asymmetrically. It is argued that a tradable permit is efficient in promoting differential investments.

The rationalizability concept was introduced in Bernheim [Ber84] and Pearce [Pea84] to assess what can be inferred by rational players in a non-cooperative game in the presence of common knowledge. However, this notion can be defined in a number of ways that differ in seemingly unimportant minor details. We shed light on these differences, explain their impact, and clarify for which games these definitions coincide.Then we apply the same analysis to explain the differences and similarities between various ways the iterated elimination of strictly dominated strategies was defined in the literature. This allows us to clarify the results of Dufwenberg and Stegeman [DS02] and Chen, Long and Luo [CLL05] and improve upon them. We also consider the extension of these results to strict dominance by a mixed strategy.Our approach is based on a general study of the operators on complete lattices. We allow transfinite iterations of the considered operators and clarify the need for them. The advantage of such a general approach is that a number of results, including order independence for some of the notions of rationalizability and strict dominance, come for free.

This paper analyses comparative statics for first price auctions and all pay auctions with independent private values. In all pay auctions, bidders with low values will respond to a stochastically higher (in the sense of likelihood ratio dominance) distribution of types by playing less aggressively while high value bidders bid more. In the first price auction, a similar change results in all types playing more aggressively. Furthermore, we show that a decrease in dispersion of values, in the sense of a refinement of second order stochastic dominance, although also associated with an increase in competitiveness, may in addition result in less aggressive play by bidders with high values in both auction forms. We also find similar considerations in an oligopoly game with incomplete information: stochastically lower costs can lead to higher prices.

We study a Cournot oligopoly with one low-cost (dominant) firm and one or more high-cost (subordinate) firms. If the equilibrium is interior, with all firms producing positive quantity, a reallocation of production relative to the equilibrium point, such that the dominant firm produces more, while the subordinate firms produce less, can increase consumers' surplus, as well as joint firm profit. We show that a price intervention (either a price floor or a fixed price) may help achieve such an improvement. The result hinges on the dominant firm having sufficiently low cost relative to the subordinate firms.

In this paper we introduce product demand uncertainty in a mixed oligopoly model and reexamine the nature of sub-game perfect Nash equilibrium (SPNE) when firms decide in the first stage whether to lead or follow in the subsequent quantity-setting game. In the non-stochastic setting, Pal (1998) demonstrated that when a public firm competes with a domestic private firm, multiple equilibria exist but the efficient equilibrium outcome is for the public firm to follow. Matsumura (2003a) proved that when the public firm's rival is a foreign private firm, leadership of the public firm is both efficient as well as SPN equilibrium. Our stochastic model shows that when the leader must commit to output before the resolution of uncertainty, multiple SPNE is possible. Whether the equilibrium outcome is public or private leadership hinges upon the degree of privatization and market volatility. More importantly, Pareto-inefficient simultaneous production is a likely SPNE. Our results are driven by the fact that the resolution of uncertainty enhances the profits of the follower firm in a manner that is well known in real option theory.

In many capacity-intensive industries (e.g. electricity, bandwidth), exchanges allow firms, including competitors, to buy and sell wholesale capacity before selling on the retail market. Capacity exchanges allow firms to smooth demand shocks, but do they also facilitate tacit collusion to limit capacity investment? This paper models investment and exchange in a one-shot game and in a repeated game with tacit collusion. It finds that the presence of the exchange does not reduce total capacity investment, and thus does not raise consumer prices. In fact, the exchange may make it more difficult to sustain tacit collusion.

This paper studies the effects of uncoordinated environmental tax policies on firms' incentives to form bilateral R&D collaborations. It is shown that the complete network is pair-wise stable for small differences in the taxation of environmental emissions. Larger tax differentials may induce firms to abandon all their international collaborations.

The paper studies agents' general or specific investment decisions under different ownership structures in a thin, decentralized market where each agent's decision affects the decisions and welfare of other agents mainly through indirect market linkages. It focuses on the roles of both competition and ownership. An investor is more likely to make specific investments as an employee than as an owner. "Excess competition among investors" makes efficient, specific investments more likely. Otherwise, inefficient, general investments and irrelevance of ownership are more likely to result. The problem in which the choice variable is investment level, instead of investment type, yields less contrasting results.

We characterize optimal strategies in a simple herding model where observations have a small cost. We assume that there are two states and two possible signals that each agent may get. The prior distribution is biased towards adopting behavior. That is, ex-ante, adopting gives a higher expected utility than not adopting. Contrary to Kultti & Miettinen (2005) herding does not arise deterministically in this model when the cost of observation is small.

This paper introduces two-player bargaining problems allowing for asymmetric subjective uncertainty about factors that determine whether agreement is achieved, focusing on surprising events, i.e. events believed possible by only one player. A `subjective Nash' solution is proposed, in which a bargaining outcome is defined as immune to all possible appeals given the subjective uncertainty of disagreement. In addition to Pareto optimality and independence of irrelevant alternatives, the main axiom in the solution characterization is `subjective symmetry'. The solution is shown to be equivalent to a specific, non-symmetric, Nash solution, with preference dependent weights that equal the subjective probability players assign to the intersection of their belief supports.

We extend the Baye and Morgan (2001) model to study competition between price comparison sites in the information market on the internet. We identify one symmetric sub-game perfect Nash equilibrium in which (1) price comparison sites set the same advertising fees; (2) the same proportion of consumers subscribe to each site; (3) each firm mixes between advertising on all sites and not advertising; and (4) advertised prices are dispersed. The introduction of additional price comparison sites may reduce social welfare and joint profits of price comparison sites. In the equilibrium with each consumer subscribing to one site, as the number of price comparison sites goes to infinity, the information market approaches that without price comparison sites.

This paper considers ordered vector spaces with arbitrary closed cones and establishes a number of characterization results with applications to monotone comparative statics (Topkis (1978), Topkis (1998), Milgrom and Shannon (1994)). By appealing to the fundamental theorem of calculus for the Henstock-Kurzweil integral, we generalize existing results on increasing differences and supermodularity for C1 or C2 functions. None of the results are based on the assumption that the order is Euclidean. As applications we consider a teamwork game and a monopoly union model.

In this paper, we study a two-period pure exchange economy with idiosyncratic uncertainty, moral hazard and multiple consumption goods. We consider two different market structures: contingent commodity markets on the one hand, and financial plus spot commodity markets on the other hand. We propose a competitive equilibrium concept for each market structure. We first verify that it is possible to decentralize constrained efficient allocations as equilibria with contingent markets. Subsequently, we characterize the conditions which prevent constrained efficient allocations from being decentralized as equilibria with financial markets.

We consider the risk premium demanded by a decision maker in order to be indifferent between obtaining a new level of wealth with certainty, or to participate in a lottery which either results in unchanged wealth or an even higher level than what can be obtained with certainty. We study preferences such that the corresponding relative risk premium is a decreasing function of present wealth, and we determine the set of associated utility functions. We find a new characterization of risk vulnerability and determine a large set of utility functions, closed under summation and composition, which are both risk vulnerable and have decreasing relative risk premium. We finally introduce the notion of partial risk neutral preferences on binary lotteries and show that partial risk neutrality is equivalent to preferences with decreasing relative risk premium.

By combining a theory of herding behavior with the phenomenon of availability heuristic, this paper shows that non-informative advertisements can affect people's choices by influencing their perception of product quality. We present a model in which people can learn about product quality by observing the choices of others. Consumers are, however, not able to fully distinguish between the observations of real people and fictitious characters in advertisements. Even if a person is aware of this limitation and updates his beliefs accordingly, it is still rational for him to choose the product he has observed most often. In equilibrium the most observed product is always most likely to be of the highest quality. The analysis has important policy implications.

This paper develops a model of competition among multiproduct retailers that is consistent with observed pricing regularities, such as the facts that virtually all products have large mass points in their price distributions and that most deviations fall below these mass points. The basis of the model is that, when consumers prefer to buy a bundle of goods from the same retailer, a given discount on any one good in the bundle will have a similar effect on consumers' likelihood of visiting that retailer. Consequently, discounts on goods sold by a retailer are substitute instruments for attracting customers, and factors that influence one goods' price will also affect other goods' prices. Hence, if intertemporal price changes are a means of price discriminating (as the literature suggests), then the impact of these changes will be reflected in the prices of many goods - including those for which discrimination is infeasible.

In this paper we revisit the first price and the second price sealed-bid auctions, but, unlike the standard model, we assume that bidding is conducted by an expert on behalf of the client, and that the client does not completely trust the expert's qualifications. In particular, if the client does not win the auction, but could have won it by submitting a bid below her valuation or won but feels she could have paid less for the object, the client asks the expert to justify the strategy. The objective of this paper is to incorporate the concern for the justifiability into the expert's objective function. We show that under some assumptions about the justification process the requirement of justifiability increases the optimal bid in the first price sealed-bid auction, while bidding the client's true value remains the optimal strategy in the second price auction sealed-bid auction. Hence, the first price auction may raise more revenue than the second price auction and thus it will be preferred by the seller. Both auctions allocate the good to the client with the highest valuation. However, the second price sealed-bid auction is more efficient, since the experts do not incur costs from the failure to justify their strategies.

This paper explores the implications of using multidimensional majorization criteria to derive inequality measures, without taking into consideration the idea behind the Pigou-Dalton principle, in the sense that if a richer person transfers something of at least one attribute to a poorer person the inequality falls. A new and basic criterion proposed by Fleurbaey and Trannoy (2003) which generalizes this idea to the multidimensional framework is explored, and its logical relationships with the dominance criteria that exist in the literature are analyzed. The paper also surveys the existent multidimensional inequality indices in order to see whether they meet this new criterion.

The effect of replacing an agent in a two-person two-state finance economy by a more risk averse agent is studied. It is established under which conditions the other agent benefits or looses in equilibrium from dealing with a more risk averse agent. If one agent becomes more risk averse, then the equilibrium allocation moves towards that agent's certainty line. Whether or not that is beneficial for the other agent depends on the location of the endowment point.

A substantial portion of information goods is sold through upgrades. I model a monopolist offering successive generations of an information good in a dynamic model. In each period, the monopolist offers up to two prices for each generation: a full price to those who have never purchased and a version upgrade price to consumers who own a previous generation. I employ an overlapping generations model with infinite-lived firms and consumers that reflects the effect of future profits on current decisions better than previous two-period models. The model's predictions accord well with data from the PC software industry. The model explains why: 1) firms issued version upgrades with every new generation, 2) firms provided a discount to those upgrading relative to first-time buyers and 3) late adopters commonly purchased the latest version at full price even though some earlier adopters with higher valuations did not upgrade to the latest version.

This paper explores the hold-up problem between two parties (an entrepreneur and an investor) when one of the parties (the entrepreneur) is unable to commit not to repudiate the initial contract. To mitigate hold-up we allow the parties to stage investments over time and derive the optimal investment path in a model that places no restrictions on the growth of collateral. Our model predicts that neither positive wealth of the entrepreneur nor the lack of discounting ensures that all profitable projects proceed. We also derive necessary and sufficient conditions for the project to be financeable when there are no costs of delay.

We characterize the steady state of a market with random matching and bargaining, where the sellers' goods can perish overnight. Generically, the quantity traded is suboptimal, prices are dispersed and there is a dead-weight loss caused by excess supply or demand. In the limit as the cost of staying in the market tends to zero, only the amount of trade tends to the efficient level, the other two non-competitive characteristics remain. We discuss the implications of these findings on the foundations of competitive equilibrium and on the robustness of the results in the literature on durable-good markets.

It is a common practice for governments to offer scrappage subsidies in order to stimulate the early removal of used cars and modify the distribution of vehicle holdings. In this paper, we analyze the market implications of such subsidies when producers have market power and face competition from a secondary used car market. One key result is that, with market power, a subsidy can induce scrappage even if it pays less than the price of a used car in the absence of the subsidy. We provide a full characterization of the effects of scrappage subsidies on primary and secondary markets for the case of a monopoly, and show that the subsidy that maximizes aggregate welfare lowers prices in the used car market. Our results contrast with the predictions derived from a model with perfect competition.

In rational beliefs (RB) models there is an observed empirical distribution for the stochastic process of state variables. Many different weakly asymptotic mean stationary (WAMS) processes could have generated this empirical distribution, i.e. are consistent with it, and each of them are therefore called a rational belief.We provide a general framework for using RB in general equilibrium models. Individual rational beliefs are assumed correlated by means of sunspots which at the aggregate level lead to excess volatility.The application adapts the proof by Duffie et al (1994) of the existence of a stationary ergodic RE equilibrium to the case where agents hold rational beliefs.

This paper characterizes a set of Nash equilibria in a first-price sealed-bid repeated auction with the right of first refusal using two bidders and asymmetric information regarding the bidders' value distributions. When contract value is constant from one auction to the next and winners' values are publicized, agents retain the value of incumbency and bids are identical to one-shot auctions. When each agent's contract values are random across auctions, agents choose to bid away the full expected value of incumbency, providing a measure of the value of information in this context.

Recently, it has been claimed that full-information multiple equilibria in games with strategic complementarities are not robust, because generalizing to allow slightly heterogeneous information implies uniqueness. This paper argues that this "global games" uniqueness result is itself not robust. If we generalize by allowing most agents to observe a few previous actions before choosing, instead of forcing players to move exactly simultaneously, then multiplicity of outcomes is restored. Only a small sample of observations is needed to make our herding equilibrium behave like a full-information sunspot equilibrium instead of a global games equilibrium.

We consider a model of bribery in an asymmetric procurement auction. In return for a bribe from the dishonest supplier, the auctioneer has the discretion to allow this supplier to revise his bid downward to match the low bid of the honest supplier. The dishonest supplier can also win the contract outright without paying a bribe by bidding below the honest supplier. We investigate the effect of the bribe share and the cost distributions on the bidding functions, the allocative distortion, and the expected price paid by the buyer. The dishonest supplier bids more aggressively to win the contract outright when the auctioneer takes a larger bribe share. Bribery and the implied right of first refusal introduce a new allocative distortion in favor of the dishonest supplier. Finally, we use the power family of cost distributions to examine the expected price paid by the buyer. When the dishonest supplier has a more favorable cost distribution, there exist bribe shares sufficiently large such that the expected price paid by the buyer can actually decline as a result of bribery.

This paper investigates an oligopolistic market with vertical product differentiation, where qualities are represented by a uni-dimensional interval. We examine the Nash equilibrium in the oligopoly game with a finite number of consumers, where each firm uses its technology and can offer a range of quality-price pairs, and firms' profits are determined by the consumers' selection choice. We show that under the standard conditions on consumers' preferences at most one firm with a positive profit can be active in equilibrium. In contrast, we present an example of a duopoly game with a continuum of consumers, in which both firms yield positive profits in an equilibrium. Finally, we analyze in the linear case the question of existence.

In two-sided matching markets, stable mechanisms are vulnerable to various kinds of manipulations. This paper investigates conditions for the student-optimal stable mechanism (SOSM) and the college-optimal stable mechanism (COSM) to be immune to manipulations via capacities and pre-arranged matches. For SOSM, we find that strongly monotone preferences in populations and weakly maximin preferences are the maximal domains of college preferences that guarantee immunity to manipulations via capacities and pre-arranged matches, respectively. In contrast, COSM is susceptible to both manipulations whenever colleges have multiple positions.

In this paper, we formulate a model prescribing optimal policy for cash disbursements and seasoned equity offerings taking into account the principal-agent problems inherent in these decisions. In order to discipline managers, stockholders demand that excess free cash flow be disbursed either as cash dividends or through stock repurchases. Managers resist stockholders in this regard since they prefer to retain excess free cash flow in order to pursue personal interests and reduce the probability that the company will experience financial distress in the future. However, as a consequence of withholding cash disbursements, managers incur disutility due to the possibility that their control of the firm could be threatened by the market for corporate control. We model this situation as a stochastic impulse control problem, and succeed in finding an analytical solution. We derive several testable implications, some of which have not been fully addressed in the corporate finance literature.

The paper investigates the impact of examinations on incentives and decision-making in bureaucracies and similar organisations. When one amongst a group of bureaucrats can be appointed to give policy advice whose outcome affects all parties, with advisory ability increasing in personal effort, a free-riding problem is generated if preferences are aligned, leading to an ex ante inefficiency. Free-riding may be mitigated by an examination with a pass-mark, i.e., a minimum ability requirement as a necessary criterion for advisory appointment. By collectively punishing all experts when maximal ability is low, it raises private incentive to enhance ability, and improves decision quality.

We apply the von Neumann-Morgenstern stable set to the n-player cake division problem. Only time-preferences á la Rubinstein (1982) are assumed. The stable set is defined with respect to the following dominance relation: x dominates y if there is a player who prefers x over y even with one period lag. The Nash bargaining solution is characterized in the language of stable sets. Through the characterization, we establish the existence and uniqueness of the Nash solution.

The paper studies the concept of ``ex post equilibrium" that has recently become popular in literature on auctions, mechanism design and implementation. We ask how one should define ex post equilibrium if one wants to consider extensive games, such as dynamic auctions, and if one wants to include sequential rationality in the equilibrium definition. As it turns out, this raises some conceptual questions that are not present in the static setting. We are lead to introduce three different definitions - all variations of what we call ``information-invariant equilibrium." One of these three definitions is equivalent to ``ex post equilibrium." In static games the three definitions coincide. In extensive games they do not - if we impose sequential rationality. Our main purpose is to make a methodological contribution to game theory, but we illustrate the relevance of this contribution by applying our concepts to several auction games.

We study two-player all-pay contests in which there is a positive probability of a tied outcome. We show that the players' efforts in equilibrium do not depend on the expected prize in the case of a tie given that this prize is smaller than the prize for winning. The implications of this result are twofold. First, in symmetric one-stage contests, the designer who wishes to maximize the expected total effort should not award a prize in the case of a tie which is larger than one-third of the prize for winning. Second, in multi-stage contests, the designer should not limit the number of stages (tie-breaks) but should allow the contest to continue until a winner is decided.

This paper presents a model of local network effects in which agents connected in a social network each value the adoption of a product by a heterogeneous subset of other agents in their neighborhood, and have incomplete information about the structure and strength of adoption complementarities between all other agents. I show that the symmetric Bayes-Nash equilibria of this network game are in monotone strategies, can be strictly Pareto-ranked based on a scalar neighbor-adoption probability value, and that the greatest such equilibrium is uniquely coalition-proof. Each Bayes-Nash equilibrium has a corresponding fulfilled-expectations equilibrium under which agents form local adoption expectations. Examples illustrate cases in which the social network is an instance of a Poisson random graph, when it is a complete graph, a standard model of network effects, and when it is a generalized random graph. A generating function describing the structure of networks of adopting agents is characterized as a function of the Bayes-Nash equilibrium they play, and empirical implications of this characterization are discussed.

We examine the spread of a disease or behavior through a social network. In particular, we analyze how infection rates depend on the distribution of degrees (numbers of links) among the nodes in the network. We introduce new techniques using first- and second order stochastic dominance relationships of the degree distribution in order to compare infection rates across different social networks.

Ability of managers and other nonproduction professionals is key for the productivity of firms. Hence, the assignment of heterogeneous nonproduction workers across firms determines the distribution of productivity. In turn, the transmission of productivity differences into profit differences -- resulting from product market competition -- determines firms' willingness to pay for higher managerial skills. This paper explores the equilibrium assignment of nonproduction workers across ex ante identical firms which results from this interaction between product market and the market for nonproduction skills. The analysis suggests that, typically, large and productive firms coexist with small, low-productivity firms. Consistent with empirical evidence, a skewed distribution of firm size tends to arise. Moreover, the model predicts a positive relationship of firm size to productivity, manager quality, and manager remuneration. Finally, according to comparative-static analysis, higher intensity of product market competition can account for increases in the compensation at the top of the wage distribution.

People reason about real-estate prices both in terms of general rules and in terms of analogies to similar cases. We propose to examine empirically which mode of reasoning fits the data better. To this end, we develop the statistical techniques required for the estimation of the case-based model. It is hypothesized that case-based reasoning will have relatively more explanatory power in databases of rental apartments, whereas rule-based reasoning will have a relative advantage in sales data. We motivate this hypothesis on theoretical grounds, and find empirical support for it by comparing the two statistical models (rule-based and case-based) on two databases (rentals and sales).

We study a market-entry game with a second-mover advantage. In the symmetric equilibrium, there can be a non-monotonic relationship between the probability with which a player will invest (entry) and the length of time until the deadline. Moreover, the probability of investment can move chaotically as the horizon is extended. In the limit when the period length goes to zero chaotic trajectories arise when the efficiency effect does not hold -- that is, when the one-period monopoly profit is less than the total of the one-period duopoly profits. We also show that the presence of chaotic trajectories is associated with a smaller expected delay in entry.

According to a common opinion in economic literature, the National Income Test provides a necessary condition for potential Pareto dominance. This paper demonstrates that this statement is true in pure-exchange economies, but, in general, false in production economies.