Anonymous sequential games: Existence and characterization of equilibria

Summary In this paper we consider Anonymous Sequential Games with Aggregate Uncertainty. We prove existence of equilibrium when there is a general state space representing aggregate uncertainty. When the economy is stationary and the underlying process governing aggregate uncertainty Markov, we provide Markov representations of the equilibria.Table of notation Agents' characteristics space ( ) - A Action space of each agent (aA) - Y Y = x A - Aggregate distribution on agents' characteristics - (X) Space of probability measures onX - C(X) Space of continuous functions onX - _X Family of Borel sets ofX - State space of aggregate uncertainty ( ) - x _t=1 aggregate uncertainty for the infinite game - = (₁,₂,...,_t,...) - ^{t t} (₁, ₂,..., _t) - L₁(^t,C ×A),v _t Normed space of measurable functions from ^t toC( x A) - 8o(^t,( x A)) Space of measurable functions from ^tto( x A) - X^t X^t= x _s=1 ^t X - _X ^t Borel field onX ^t - v Distribution on - v_t Marginal distribution of v on ^t - v(^t)((¦^t)) Conditional distribution on given ^t - v^t(^s)(v_t(¦^s)) Conditional distribution on ^t given ^s (wheres) - _t Periodt distributional strategy - Distributional strategy for all periods =(₁,₂,...,_t,...) - _t Transition process for agents' types - ( _t,^t,y)(P _t+1(, _t , ^t ,y)) Transition function associated with _t - u _t Utility function - V _t (, a, , ^t) Value function for each collection (, a, , ^t ) - W _t (, , ^t ) Value function given optimal action a - C() Consistency correspondence. Distributions consistent with and characteristics transition functions - B() Best response correspondence (which also satisfy consistency) - E Set of equilibrium distributional strategies - x _t=1 ( ^t , (x A)) - S Expanded state space for Markov construction - (, a, ) Value function for Markov construction - P( _t ^* , _t y)(P(, _t ^* , _t , _y )) Invariant characteristics transition function for Markov game We wish to acknowledge very helpful conversations with C. d'Aspremont, B. Lipman, A. McLennan and J-F. Mertens. The financial support of the SSHRCC and the ARC at Queen's University is gratefully acknowledged. This paper was begun while the first author visited CORE. The financial support of CORE and the excellent research environment is gratefully acknowledged. The usual disclaimer applies.     

Equilibria and Pareto optima of markets with adverse selection

Summary This paper examines the efficiency properties of competitive equilibrium in an economy with adverse selection. The agents (firms and households) in this economy exchange contracts, which specify all the relevant aspects of their interaction. Markets are assumed to be complete, in the sense that all possible contracts can, in principle, be traded. Since prices are specified as part of the contract, they cannot be used as free parameters to equate supply and demand in the market for the contract. Instead, equilibrium is achieved by adjusting the probability of trade. If the contract space is sufficiently rich, it can be shown that rationing will not be observed in equilibrium. A further refinement of equilibrium is proposed, restricting agents' beliefs about contracts that are not traded in equilibrium. Incentive-efficient and constrained incentive-efficient allocations are defined to be solutions to appropriately specified mechanism design problems. Constrained incentive efficiency is an artificial construction, obtained by adding the constraint that all contracts yield the same rate of return to firms. Using this notion, analogues of the fundamental theorems of welfare economics can be proved: all refined equilibria are constrained incentive-efficient and all constrained incentive-efficient allocations satisfying some additional conditions can be decentralized as refined equilibria. A constrained incentive-efficient equilibrium is typically not incentive-efficient, however. The source of the inefficiency is the equilibrium condition that forces all firms to earn the same rate of return on each contract.Notation ={ ₁,..., _k } set of outcomes - : ₊ generic contract or lottery - A = () ; - A_o A{, where denotes the null contract or no trade - S={1,...,¦S¦} set of seller types - L(s) number of type-s sellers - M number of buyers - u: × S seller's utility function, which can be extended toA× S by puttingu(, s) ; - v. × S buyer's utility function, which can be extended toA × S by puttingv(, s) ; - f:A ₀ ×S ₊ allocation of sellers - g:A ₀ ×S ₊ allocation of buyers - A ₊ sellers' trading function - :A ×S ₊ buyers' trading function This paper has had a long gestation period, during which I have been influenced by helpful conversations with many persons, by their work, or both. Among those who deserve special mention are Martin Hellwig, Roger Myerson, Edward Prescott, Robert Townsend and Yves Younés. Earlier versions were presented to the NBER/CEME Conference on Decentralization at the University of Toronto and the NBER Conference on General Equilibrium at Brown University. I would like to thank John Geanakoplos, Walter Heller, Andreu Mas Colell, Michael Peters, Michel Poitevin, Lloyd Shapley, John Wooders, Nicholas Yannelis and an anonymous referee for their helpful comments and especially Robert Rosenthal for his careful reading of two drafts. The financial support of the National Science Foundation under Grant No. 912202 is gratefully acknowledged.     

Phillips-Kurve und Bargaining-Theorie

Ohne ZusammenfassungSymbole N Beschäftigung - F (N) Produktionsfunktion (F>0,F <0) - Substitutionselastizität - Arbeitsangebot=Arbeitskräftepotential - Reallohnsatz - w Nominallohnsatz - p Preisniveau - T Zykluslänge - s Kosten der Erhöhung vonv (gemessen in Einheiten der Lohnsumme/Periode) - I Investition (konstant) - s_p (s_w) Sparquote aus Profiten (Löhnen) - Nutzendiskontrate - q Schattenpreis der Erhöhung von     

Market equilibrium with heterogeneous recursive-utility-maximizing agents

Summary The paper by C. Ma [1] contains several errors. First, statement and proof of Theorem 2.1 on the existence of intertemporal recursive utility function as a unique solution to the Koopmans equation must be amended. Several additional technical conditions concerning the consumption domain, measurability of certainty equivalent and utility process need to be assumed for the validity of the theorem. Second, the assumptions for Theorem 3.1 need to be amended to include the Feller's condition that, for any bounded continuous functionf C(S × ⁿ ₊), (f(S_t+1, )¦s_t =s) is bounded and continuous in (s, ). In addition, for Theorem 3.1, the pricep, the endowmente and the dividend rate as functions of the state variables S are assumed to be continuous.The Feller's condition for Theorem 3.1 is to ensure the value function to be well-defined. This condition needs to be assumed even for the expected additive utility functions (See Lucas [2]). It is noticed that, under this condition, the right hand side of equation (3.5) in [1] defines a bounded continuous function ins and. The proof of Theorem 3.1 remains valid with this remark in place.A correct version of Theorem 2.1 in [1] is stated and proved in this corrigendum. Ozaki and Streufert [3] is the first to cast doubt on the validity of this theorem. They point out correctly that additional conditions to ensure the measurability of the utility process need to be assumed. This condition is identified as conditionCE ₄ below. In addition, I notice that, the consumption space is not suitably defined in [1], especially when a unbounded consumption set is assumed. In contrast to what claimed in [3], I show that the uniformly bounded consumption setX and stationary information structure are not necessary for the validity of Theorem 2.1.I would like to thank Hiroyuki Ozaki and Peter Streufert for pointing out correctly some mistakes made in the original article. Comments and suggestions from an anonymous referee are gratefully appreciated. Financially support from SSHRC of Canada is acknowledged.     

The Bioeconomics of Marine Sanctuaries

Consider an offshore fishing grounds of size K. Suppose the grounds has been overfished to the point that net revenue has been driven to zero and the fishery is in open access equilibrium at (X, Y). A marine sanctuary, where fishing is prohibited, is then created. Suppose the marine sanctuary is of size K₂ and that fishing is allowed on a smaller grounds, now of size K₁, where K₁ + K₂ = K. In the first, deterministic, model, the present value of net revenue from the grounds-sanctuary system is maximized subject to migration (diffusion) of fish from the sanctuary to the grounds. The size of the sanctuary is varied, the system is re-optimized, and the populations levels, harvest, and value of the fishery is compared to the 'no-sanctuary' optimum, and the open access equilibrium. In the deterministic model, a marine sanctuary reduces the present value of the fishery relative to the 'ideal' of optimal management of the original grounds. In the second model net growth is subject to stochastic fluctuation. Simulation demonstrates the ability of a marine sanctuary to reduce the variation in biomass on the fishing grounds. Variance reduction in fishable biomass is examined for different-sized sanctuaries when net growth on the grounds and in the sanctuary fluctuate independently and when they are perfectly correlated. For the stochastic model of this paper, sanctuaries ranging in size from 60 to 40% of the original grounds (0.6 K₂/K 0.4) had the ability to lower variation in fishable biomass compared to the no sanctuary case. For a sanctuary equal to or greater than 70% of the original grounds (K₂ 0.7K), net revenue would be nonpositive and there would be no incentive to fish.     

On the generic nonconvergence of Bayesian actions and beliefs

Summary SupposeY _n is a sequence of i.i.d. random variables taking values in Y, a complete, separable, non-finite metric space. The probability law indexed by, is unknown to a Bayesian statistician with prior, observing this process. Generalizing Freedman [8], we show that generically (i.e., for a residual family of (,) pairs) the posterior beliefs do not weakly converge to a point-mass at the true. Furthermore, for every open setG , generically, the Bayesian will attach probability arbitrarily close to one toG infinitely often. The above result is applied to a two-armed bandit problem with geometric discounting where armk yields an outcome in a complete, separable metric spaceY _k. If the infimum of the possible rewards from playing armk is less than the infimum from playing armk', then armk is (generically) chosen only finitely often. If the infimum of the rewards are equal, then both arms are played infinitely often.     

The Regulation of Private Schools Serving Low-Income Families in Andhra Pradesh,India

This paper sets out some findings of a research project carried out in private unaided schools in low-income areas of Hyderabad, India. The part of the research project documented here was designed to examine the question: Is the regulatory regime conducive to entrepreneurial action and market discovery with particular reference to the low-income schools in Hyderabad. This paper is narrowly focused, setting out the results of pattern matching empirical data with the Austrian economic concepts of entrepreneurship, rivalry, and market discovery. The research discovered that two regulatory regimes exist, one that is set out on paper in the Education Acts and associated rules, and another that operates in practice. That is, there is a combination of regulations on paper and regulations existing in an extra-legal sector. Generally it was found that the regulations in practice are consistent with market principles. Conversely the regulations set out on paper are not conducive to entrepreneurial innovation and market discovery. Recommendations for potential policy initiatives include the possibility of legitimising the extra-legal sector by introducing self-regulation possibly via self-evaluation systems for the private unaided schools.     

Topologien auf Systemen von Teilmengen des Rn und Konsequenzen für Produktionskorrespondenzen

Zusammenfassung Bei der Beschreibung von Stetigkeitseigenschaften von Produktionskorrespondenzen und insbesondere der zugehörigen inversen Korrespondenzen spielt die Wahl einer geeigneten Topologie auf der Menge der abgeschlossenen Teilmengen des R ⁿ eine wichtige Rolle. Dazu werden die folgenden Topologien eingeführt und verglichen: Hausdorff- und Meyer-Vietoris-Topologie, ferner die topology of closed convergence und die Topologie, die durch eine uniforme Struktur des R ⁿ induziert wird. Insbesondere werden algebraische Eigenschaften dieser Topologien diskutiert. Daraus ergeben sich dann Konsequenzen für die Anwendbarkeit bei der oben skizzierten Fragestellung.

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Topologies on systems of subsets of R ⁿ and the consequences for correspondences of production

     

Strategy-proof and individually rational social choice functions for public good economies

Summary We consider the problem of choosing an allocation in an economy in which there are one private good and one public good. Our purpose is to identify the class of procedures of choosing an allocation which satisfy strategy-proofness, individual rationality, no exploitation and non-bossiness. Any such procedure is a scheme of semi-convex cost sharing determined by the minimum demand principle.I wish to thank Professors Salvador Barbera, Matthew Jackson, Herve Moulin and William Thomson for their helpful suggestions and two anonymous referees for their detailed comments. Conversations with Professors Hideo Konishi, Shinji Oseto Ken-ichi Shimomura and Stephen Ching were helpful. This work is supported by the Japan Economic Research Foundation and Research Grants PB89-0294 and PB89-0075 from the Direcion General de Investigacion Cientifica y Tecnica, Spanish Ministry of Education.     

The supply side in a post-Keynesian two-sector model

Summary In order to avoid missing the wood for trees a brief summary of the results having been obtained above appears to be appropriate.Firstly, we regard it as a result in itself of the present paper to have presented a two-sector model encapsulating a Kaleckian (and Kaldorian) vision of a capitalist economy, a model in which supply conditions of primary products take up a prominent position.Secondly, it belongs to the main results of the paper that it has laid bare, through the model presented, an astoundingly simple pattern in the way economic key-concepts such as activity, employment and distributive shares are affected by on one hand the demand side (which has so far captured an excessive amount of attention in macroeconomic modelling) and on the other hand the largely neglected supply side of the economy.Thirdly, by means of an arbitrary but not implausible numerical example we have attempted to indicate how changes in activity and distributive shares caused by exogeneous changes on the demand and supply side of the economy, respectively, are in themselves crucially dependent on (the assumptions concerning) the supply elasticity of primary products.The notation applied will be as follows C consumption - C ⁰ autonomous element of the consumption function - I investment (gross) - A autonomous expenditure - S savings (gross) - U stock of the primary product - Q real output (not necessarily real income) (gross) - Y income (gross) - W wage bill - L employment - w money wage rate - p price level - mark-up factor - level parameter of the production function pertaining to the primary sector - a labour-input coefficient of the industrial sector - b raw-material input coefficient of the industrial sector - s _w marginal propensity to save out of wages - s marginal propensity to save out of profits - s (weighted) average ofS _w ands - (unit) raw material costs as a proportion of total (unit) prime costs - share of wages in total income - E _{y, x} partial or total elasticity ofY with respect toX. I am most grateful to Søren Gammelgård, Peter Guldager, Erik Strøjer Madsen, Jørgen Ulff-Møller Nielsen, Kurt Pedersen and an anonymous referee for their valuable suggestions and helpful comments on an earlier draft of this paper.     

Convergence and approximation results for non-cooperative Bayesian games: Learning theorems

Summary LetT denote a continuous time horizon and {G ^t :tT} be a net (generalized sequence) of Bayesian games. We show that: (i) if {x ^t : tT} is a net of Bayesian Nash Equilibrium (BNE) strategies for G^t we can extract a subsequence which converges to a limit full information BNE strategy for a one shot limit full information Bayesian game, (ii) If {x ^t : tT} is a net of approximate or _t-BNE strategies for the game G^t we can still extract a subsequence which converges to the one shot limit full information equilibrium BNE strategy, (iii) Given a limit full information BNE strategy of a one shot limit full information Bayesian game, we can find a net of _t-BNE strategies {x ^t : tT} in {G ^t :tT} which converges to the limit full information BNE strategy of the one shot game.We wish to thank Larry Blume, Mark Feldman, Jim Jordan, Charlie Kahn, Stefan Krasa, Gregory Michalopoulos, Wayne Shafer, Bart Taub, and Anne Villamil for several useful discussions. The financial support of the University of Illinois at Urbana-Champaign Campus Research Board is gratefully acknowledged.     

A law of large numbers for large economies

Summary LetX(i),i[0; 1] be a collection of identically distributed and pairwise uncorrelated random variables with common finite mean and variance ². This paper shows the law of large numbers, i.e. the fact that ₀ ¹ X(i)di=. It does so by interpreting the integral as a Pettis-integral. Studying Riemann sums, the paper first provides a simple proof involving no more than the calculation of variances, and demonstrates, that the measurability problem pointed out by Judd (1985) is avoided by requiring convergence in mean square rather than convergence almost everywhere. We raise the issue of when a random continuum economy is a good abstraction for a large finite economy and give an example in which it is not.I am indebted to Hugo Hopenhayn. Furthermore I would like to thank Dilip Abreu, Glenn Donaldson, Ed Green, Ramon Marimon, Nabil Al-Najjar, Victor Rios-Rull, Timothy van Zandt and the editor for useful comments. The first version of this paper was written in 1987.     

Knowledge Questions: Hayek,Keynes and Beyond

This paper discusses the knowledge problem in terms of both the use and generation of knowledge. This is analyzed in the context of Hayek's failure to respond to the Keynes Challenge—the claim that markets fail to produce relevant knowledge—by suggesting that in the aftermath of The General Theory he was not well-positioned to address that problem. Ironically, his post-World War II work in cognitive psychology, The Sensory Order, offers a theory of the generation of knowledge which can provide a useful analogy for understanding the generation of market-level knowledge.     

Public-Choice Analysis of a New International Organization: The International Sea-Bed Authority

The usefulness of the public-choice approach for a better understanding of international organizations can be demonstrated by applying it to the analysis of the structure and functions of a new international organization, the International Sea-Bed Authority, established in 1994, after two decades of negotiations under the auspices of the United Nations, with the aim to control the oceans' mineral resources beyond the limits of national jurisdictions (which have been proclaimed by the U.N. Assembly common heritage of mankind). First, the reasons for establishing this organization, whose basis is the common heritage of mankind nature of ocean resources, are examined under two aspects: 1) definition and protection of property rights; 2) environmental control of sea-bed mining activities. Secondly, the organization's decisionmaking system is presented, including such features as the assembly, council with chamber voting, finance committee with decision by consensus, features that balance the voting power of members and protect those countries that bear the financial responsibility for the budget. Finally, some comments are offered about more general aspects of the theory of international organizations: the bureaucracy (and diplomacy) of these organizations as well as the interdependence among international organizations, which opens the way to forms of international logrolling and makes it advisable for countries not to exit from these organizations, even when they have no primary interest in them.     

Does Increased Extraction of Natural Gas Reduce Carbon Emissions?

Without an international climate agreement, extraction of more natural gas could reduce emissions of CO₂ as more clean natural gas may drive out dirty coal and oil. Using a computable equilibrium model for the Western European electricity and natural gas markets, we examine whether increased extraction of natural gas in Norway reduces global emissions of CO₂. We find that both in the short run and in the long run total emissions are reduced if the additional quantity of natural gas is used in gas power production in Norway. If instead the additional quantity is exported directly, total emissions increase both in the short run and in the long run. However, if modest CO₂-taxes are imposed, increased extraction of natural gas will reduce CO₂ emissions also when the additional natural gas is exported directed.earlier version of this paper was presented at the 25th Annual IAEE International Conference in Aberdeen June 2002.     

“Buy Domestic” Campaigns and Optimal Tariffs

This paper attempts to analyze the strategic use of optimal tariffs and to examine the effects of national bias on the optimal trade policy and social welfare in a two-country, two-good, price competition model derived from Neven et al. (1991). The major findings are as follows. (1) If all consumers prefer the domestic good, then buy domestic campaigns will decrease the prohibitive tariff rate and increase local welfare. (2) If at least some consumers prefer the foreign good, but not to a great extent, then buy domestic campaigns will not change the optimal tariff rate, but may improve local welfare. (3) When all consumers greatly prefer the foreign good, then promotion of buy domestic decreases the optimal tariff rate, but it cannot improve social welfare. With this framework, we also prove that buy domestic campaigns serve as a substitute for tariffs with respect to a strategic trade policy.     

A Supply and Demand Exposition of a Constitutional Tax Loophole: The Case of Tariff Symmetry

The U.S. Constitution permits import tariffs but bans export duties. Yet import taxes are de facto export taxes, just as export taxes are de facto import taxes. Access to this symmetry proposition has been limited by its illustration being in daunting analytics largely restricted to international economics. This is unfortunate. Tariff symmetry exposes a tax loophole of constitutional proportions, a case where economics trumps the intentions of America's Founding Fathers. Moreover, tariff revenue was the U.S. government's pivotal revenue source from 1789 until the 1913 constitutional sanctioning of the income tax. Because U.S. exports were heavily agricultural, tariff symmetry implies that federal taxation had an export dimension with disparate economic and regional consequences. By making tariff symmetry more accessible, this paper lowers the cost of examining important issues.     

Are Uniform Solutions Focal? – The Case of International Environmental Agreements

The application of uniform solutions has several drawbacks, notably their lack of cost efficiency and their inability to guarantee individual rationality. A proper specification of uniform solutions, however, reveals that uniform solutions that satisfy individual rationality always exist. When all countries hold private information about their own reduction costs, there only exists one solution that always satisfies individually rationality without use of side payments and the requirement of dominant strategy implementation: The solution that selects the smallest individually preferred uniform reduction which also give a theoretical explanation of the lowest common denominator effect.     

The greenhouse effect: Damages,costs and abatement

The buildup of so-called greenhouse gases in the atmosphere — CO₂ in particular-appears to be having an adverse impact on the global climate. This paper briefly reviews current expectations with regard to physical and biological effects, their potential costs to society, and likely costs of abatement. For a worst case scenario it is impossible to assess, in economic terms, the full range of possible non-linear synergistic effects. In the most favorable (although not necessarily likely) case (of slow-paced climate change), however, it seems likely that the impacts are within the affordable range, at least in the industrialized countries of the world. In the third world the notion of affordability is of doubtful relevance, making the problem of quantitative evaluation almost impossible. We tentatively assess the lower limit of quantifiable climate-induced damages at $30 to $35 per ton of CO₂ equivalent, worldwide, with the major damages being concentrated in regions most adversely affected by sea-level rise. The non-quantifiable environmental damages are also significant and should by no means be disregarded.The costs and benefits of (1) reducing CFC use and (2) reducing fossil fuel consumption, as a means of abatement, are considered in some detail. This strategy has remarkably high indirect benefits in terms of reduced air pollution damage and even direct cost savings to consumers. The indirect benefits of reduced air pollution and its associated health and environmental effects from fossil-fuel combustion in the industrialized countries range from $20 to $60 per ton of CO₂ eliminated. In addition, there is good evidence that modest (e.g. 25%) reductions in CO₂ emissions may be achievable by the U.S. (and, by implication, for other countries) by a combination of increased energy efficiency and restructuring that would permit simultaneous direct economic benefits (savings) to energy consumers of the order of $50 per ton of CO₂ saved. A higher level of overall emissions reduction — possibly approaching 50% — could probably be achieved, at little or not net cost, by taking advantage of these savings.We suggest the use of taxes on fossil fuel extraction (or a carbon tax) as a reasonable way of inducing the structural changes that would be required to achieve significant reduction in energy use and CO₂ emissions. To minimize the economic burden (and create a political constituency in support of the approach) we suggest the substitution of resource-based taxes in general for other types of taxes (on labor, income, real estate, or trade) that are now the main sources of government revenue. While it is conceded that it would be difficult to calculate the optimal tax on extractive resources, we do not think this is a necessary prerequisite to policy-making. In fact, we note that the existing tax system has never been optimized according to theoretical principles, and is far from optimal by any reasonable criteria.During the academic year 1989–90 Dr. Ayres was at the International Institute for Applied Systems Analysis (IIASA), Laxenburg, Austria.During the summer of 1989 Mr. Walter was a member of the Young Scientists' Summer Program at IIASA.     

Wilhelm Weber—50 Jahre

Ohne ZusammenfassungFerner zahlreicheBuchbesprechungen, insbesondere in der Zeitschrift für Nationalökonomie, und Zeitungsartikel.