Discrete fixed point theorem reconsidered

The aim of this note is to indicate an example that demonstrates the incorrectness of Iimura’s discrete fixed point theorem [Iimura, T., 2003. A discrete fixed point theorem and its applications. Journal of Mathematical Economics 39, 725–742] and to present a corrected statement using the concept of integrally convex sets.     

On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem when individual preferences are weak orders

We will show that in the case where there are two individuals and three alternatives (or under the assumption of the free-triple property), and individual preferences are weak orders (which may include indifference relations), the Arrow impossibility theorem [Arrow, K.J., 1963. Social Choice and Individual Values, second ed. Yale University Press] that there exists no binary social choice rule which satisfies the conditions of transitivity, Pareto principle, independence of irrelevant alternatives, and non-existence of dictator is equivalent to the Brouwer fixed point theorem on a 2-dimensional ball (circle). Our study is an application of ideas by Chichilnisky [Chichilnisky, G., 1979. On fixed points and social choice paradoxes. Economics Letters 3, 347–351] to a discrete social choice problem, and also it is in line with the work by Baryshnikov [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404–415].     

The joint continuity of the expected payoff functions

Glicksberg [Glicksberg, I.L., 1952. A further generalization of the Kakutani fixed point theorem, with applications to Nash equilibrium points. In: Proceedings of the American Mathematical Society 3, pp. 170–174] generalized the Kakutani fixed point theorem to the setting of locally convex spaces and used it to prove that every k-person strategic game with action sets convex compact subsets of locally convex spaces and continuous payoff functions has a Nash equilibrium. He subsequently used this result to establish the following fundamental theorem of game theory: Every k-person strategic game with action sets metrizable compact topological spaces and continuous payoff functions has a mixed strategies equilibrium. However, in his proof of the latter result, Glicksberg did not show that the expected payoff functions were jointly continuous, something that was required for the existence of a mixed strategies equilibrium.     

一类时滞微分方程的非负周期解的存在性

文章利用Schauder不动点定理,研究了一类时滞微分方程的非负周期解的存在性熏得到了一些新的结果并改进了相应的结论。     

Some results on general assumptions about the existence of economic equilibrium

In the present paper we maximally use the possibilities provided by the Nash approach and the Kakutani fixed point theorem for proving the existence of an economic equilibrium. We obtain a general existence theorem which does not require a special form for income distribution functions and producer's objectives, independence of consumers' tastes, ordered preference and zero degree homogeneous price dependence. The role of the non-satiation assumption becomes more clear.     

Pseudo-likelihood estimation and bootstrap inference for structural discrete Markov decision models

This paper analyzes the higher-order properties of the estimators based on the nested pseudo-likelihood (NPL) algorithm and the practical implementation of such estimators for parametric discrete Markov decision models. We derive the rate at which the NPL algorithm converges to the MLE and provide a theoretical explanation for the simulation results in Aguirregabiria and Mira [Aguirregabiria, V., Mira, P., 2002. Swapping the nested fixed point algorithm: A class of estimators for discrete Markov decision models. Econometrica 70, 1519–1543], in which iterating the NPL algorithm improves the accuracy of the estimator. We then propose a new NPL algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model. We also develop one-step NPL bootstrap procedures for discrete Markov decision models. The Monte Carlo simulation evidence based on a machine replacement model of Rust [Rust, J., 1987. Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica 55, 999–1033] shows that the proposed one-step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations.     

利用有源低通滤波器电路探讨抽样定理与信号恢复

以有源低通滤波器为例,探讨抽样定理与信号恢复原理,深入了解高职高专《信号与系统》的实验教学内容,使学生能够形象直观地观察离散信号频谱,了解其频谱特点。文章通过设计模拟信号源及时针信号源模块、模拟滤波器及抽样定理模块,引导学生全面具体分析抽样定理与信号恢复原理。     

Solvability of perturbation solutions in DSGE models

We prove that the undetermined Taylor series coefficients of local approximations to the policy function of arbitrary order in a wide class of discrete time dynamic stochastic general equilibrium (DSGE) models are solvable by standard DSGE perturbation methods under regularity and saddle point stability assumptions on first order approximations. Extending the approach to nonstationary models, we provide necessary and sufficient conditions for solvability, as well as an example in the neoclassical growth model where solvability fails. Finally, we eliminate the assumption of solvability needed for the local existence theorem of perturbation solutions, complete the proof that the policy function is invariant to first order changes in risk, and attribute the loss of numerical accuracy in progressively higher order terms to the compounding of errors from the first order transition matrix.     

An existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy

We establish an existence theorem for Cournot–Walras equilibria in a monopolistically competitive economy. Instead of the traditional approach which depends on Kakutani’s fixed point theorem, we employ the theories of aggregative games and best reply potential games. We show that, if there exists a representative consumer, under some conditions on preferences and production technologies, the profit maximization game is a (pseudo) best reply potential game. Hence, the existence of the equilibria is proved independently of the well known convex-valued assumption on the best responses. Although our assumptions result in the additive separability on a utility function of a representative consumer, the existence of increasing returns and indivisible productions can be allowed. In our model, it is shown that the game played by firms exhibits strategic substitutes whether the products of firms are substitutes or complements, and this plays an important role for the existence of the equilibria.     

Universally converging adjustment processes—a unifying approach

Both in game theory and in general equilibrium theory, there exists a number of universally converging adjustment processes. In game theory, these processes typically serve the role of selecting a Nash equilibrium. Examples are, the tracing procedure of Harsanyi and Selten or the equilibrium selection procedure proposed by McKelvey and Palfrey. In general equilibrium, the processes are adjustment rules by which an auctioneer can clear all markets. Examples are the processes studied by Smale, Kamiya, van der Laan and Talman, and Herings. The underlying reasons for convergence have remained rather mysterious in the literature, and convergence of different processes has seemed unrelated. This paper shows that convergence of all these processes relies on Browder’s fixed point theorem.     

A topological approach to Wilson’s impossibility theorem

We will present a topological approach to Wilson’s impossibility theorem [Wilson, R.B., 1972. Social choice theory without the Pareto principle. Journal of Economic Theory 5, 478–486] that there exists no non-null binary social choice rule which satisfies transitivity, independence of irrelevant alternatives, non-imposition and has no dictator nor inverse dictator. Our research is in line with the studies of topological approaches to discrete social choice problems initiated by [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404–415]. This paper extends the result about the Arrow impossibility theorem shown in [Tanaka, Y., 2006. A topological approach to the Arrow impossibility theorem when individual preferences are weak orders. Applied Mathematics and Computation 174, 961–981] to Wilson’s theorem.     

Applications of the Likelihood Theory in Finance: Modelling and Pricing

This paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.     

Supply-constrained fixed price equilibria in monetary economies

In this paper we consider an economy with restrictions on the relative prices of non-money commodities. The non-money commodities are partitioned into two groups, index makers and price following commodities. Then two cases are considered, namely that the relative prices of the index makers are fixed, respectively free. In both cases the money prices are flexible, whereas the relative prices of the price following commodities are tied to the prices of the index makers. The existence of a supply-constrained equilibrium is proved with (i) no rationing on the money commodities (stores of value), and (ii) at least one non-money commodity is not rationed. If prices of the index makers are fixed the result strengthens a theorem of Dehez and Drèze, if the prices are free a theorem of Kurz is strengthened. This paper is not only concerned with these existence results, but also with the question whether supply-constrained equilibria should appear more frequently than demand-constrained equilibria.     

巧用不动点求数列通项公式

确定数列的通项公式,对于研究数列的性质起着至关重要的作用。不动点求通项就是其中之一,运用不动点求数列通项既是竞赛数学的内容之一,也是近年来高考命题的热点。文章着重介绍两种运用不动点求数列通项的方法。     

Insiemi invarianti globalmente attrattivi nell'interazione fra il “mercato dei beni” ed il “mercato della moneta”

Si studia il comportamento dinamico locale e globale di un modello macroeconomico che si rappresenta con un sistema nonlineare di due equazioni alle differenze (sistema dinamico discreto, o mappa). Si mostra come l'uso di un nuovo strumento di analisi, le linee critiche, consenta di studiare molte delle proprietà globali (quali i bacini di attrazione di insiemi invarianti) e delle biforcazioni globali, che si verificano in mappe del piano con inversa non unica. Si prova che il punto fisso localmente attrattivo è anche globalmente attrattivo, mentre in regimi di instabilità del punto fisso esistono curve chiuse, o altri insiemi invarianti (regolari o caotici) di attrazione, in aree assorbenti globalmente attrattive.

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Summary The local and global dynamical behaviour of a macroeconomic model, represented by a nonlinear system of two difference equations (discrete dynamical system or map), is studied. It is shown how several properties (as basins of attraction of invariant sets) and global bifurcations occurring in two-dimensional maps with a non-unique inverse can be studied by use of a new analytical tool, the critical curves. It is proven that the locally attractive fixed point is globally attractive, while closed invariant curves or other attractive invariant sets (regular or chaotic) exist in globally attractive absorbing areas, when the fixed point is repulsive.

     

General competetive equilibrium of the spatial economy

The spatial economy is analyzed in the general equilibrium framework by considering the production and the utility functions depending on spatial distribution and sets. The geometric constraint between the location and the extension of goods supplies the equilibrium condition for space. The existence of an equilibrium is demonstrated by extending the Gale-Nikaido theorem to the case under examination. Consequently, the competetive equilibrium exists, under the assumptions of the theorem, although each point of space is heterogeneous with any other point (because of the different location): the existence is allowed by the perfect partibility of space.     

Ungünstige Verteilungen bei einparametrigen Parameterschätzproblemen

Summary One of the commonly used methods for determining minimax point estimators is based on least favorable distributions, because Bayes estimators with respect to a least favorable distribution are frequently minimax point estimators. Therefore it is worthwhile to investigate the structure of least favorable distributions. In the present paper it will be proved that, under certain conditions, each least favorable distribution is finite discrete. We give sufficient conditions for point estimation problems with the property that each least favorable distribution is finite discrete. Some examples are presented.     

Uniqueness conditions for strongly point-rationalizable solutions to games with metrizable strategy sets

This paper introduces the iterative solution concept of strong point-rationalizability as a strengthening of standard point-rationalizability by stipulating that any two players have identical beliefs about the strategy choices of their common opponents. By adopting and generalizing the contraction-property approach of Moulin [Moulin, H., 1984. Dominance solvability and cournot stability. Mathematical Social Sciences 7, 83–102] and Bernheim [Bernheim, B.D., 1984. Rationalizable strategic behavior. Econometrica 52, 1007–1028] we derive sufficient conditions for the existence of unique strongly point-rationalizable strategies in strategic games with best response functions. These uniqueness results are derived under fairly general assumptions and are especially useful for complete and bounded, as well as finite strategy sets. For games with monotonic individual best response functions, equivalence between a unique strongly point-rationalizable and a unique point-rationalizable solution in the standard sense is established. Furthermore, the existence of unique fixed points is proven under conditions that generalize for bounded metric spaces an established fixed point theorem by Bonsall [Bonsall, F.F., 1962. Lectures on Some Fixed Point Theorems of Functional Analysis. Tata Institute of Fundamental Research, Bombay] and Smart [Smart, D.R., 1974. Fixed Point Theorems. Cambridge University Press, Cambridge, London]. Possible extensions of our findings to games with multi-valued best response correspondences are discussed.     

Risk aversion and risk vulnerability in the continuous and discrete case

This paper discusses utility functions for money, where allowable money values are from an arbitrary nonempty closed subset of the real numbers. Thus, the classical case, where this subset is a closed interval (bounded or not) of the real line, is included in the study. The discrete case, where this subset is the set of all integer numbers, is also included. In a sense, the discrete case (which has not been addressed in the literature thus far) is more suitable for real-world applications than the continuous case. In this general setting, the concepts of risk aversion and risk premium are defined, an analogue of Pratt’s fundamental theorem is proved, and temperance, prudence, and risk vulnerability are examined.     

Comparative statics in dynamic programming models with an application to job search

This paper presents a technique for qualitative comparative statics analysis in dynamic programming models. Let the value function υ be the fixed point of a contraction mapping which depends differentiably on some exogenous parameter θ. Ten the derivative of υ with respect to θ exists and is also the fixed point of a contraction mapping. Since this derivative is the fixed point of a contraction mapping, its qualitative properties can be investigated using mathematical induction. This comparative statics methodology is illustrated with an application to a model of job search.