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.     

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.     

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.     

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.     

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。     

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.     

On lags and chaos in economic dynamic models

This paper discusses the role of lags in dynamic economic models. Applications of discrete dynamical systems to economics are considered and the shortcomings of their (implicit) treatment of lags are criticized. An abstract, probabilistic view of lags is then provided, within which fixed delay lags are shown to be a special case. A basic equivalence is proved between: (i) a system with an indefinitely large number of agents, reacting to inputs with randomly gamma distributed, discrete lags, and (ii) a system with one single ‘representative‘ agent reacting to inputs with a continuous, multiple exponential lag. Finally, the paper analyses a single-loop feedback system cuopling a multiple exponential lag and a non-linearity of the one-hump type. The dynamic behavior of the system is studied by means of analytical and numerical methods, and the conditions for periodic and chaotic solutions are investigated.     

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.     

Labour adjustment costs: Estimation of a dynamic discrete choice model using panel data for Greek manufacturing firms

In this paper we estimate a dynamic structural model of employment at firm level. Our dataset consists of a balanced panel of 2790 Greek manufacturing firms. The empirical evidence of this dataset stresses three important stylized facts: (a) there are periods in which firms decide not to change their labour input, (b) there are periods of large employment changes (lumpy nature of labour adjustment) and (c) the commonality is employment spikes to be followed by smooth and low employment growth periods. Following Cooper and Haltiwanger [Cooper, R.W. and Haltiwanger, J. “On the Nature of Capital Adjustment Costs”, Review of Economic Studies, 2006; 73(3); 611–633], we consider a dynamic discrete choice model of a general specification of adjustment costs including convex, non-convex and “disruption of production” components. We use a method of simulated moments procedure to estimate the structural parameters. Our results indicate considerable fixed costs in the Greek employment adjustment.     

A new marked point process model for the federal funds rate target: Methodology and forecast evaluation

Forecasts of key interest rates set by central banks are of paramount concern for investors and policy makers. Recently it has been shown that forecasts of the federal funds rate target, the most anticipated indicator of the Federal Reserve Bank's monetary policy stance, can be improved considerably when its evolution is modeled as a marked point process (MPP). This is due to the fact that target changes occur in discrete time with discrete increments, have an autoregressive nature and are usually in the same direction. We propose a model which is able to account for these dynamic features of the data. In particular, we combine Hamilton and Jordà's [2002. A model for the federal funds rate target. Journal of Political Economy 110(5), 1135–1167] autoregressive conditional hazard (ACH) and Russell and Engle's [2005. A discrete-state continuous-time model of financial transactions prices and times: the autoregressive conditional multinomial-autoregressive conditional duration model. Journal of Business and Economic Statistics 23(2), 166 – 180] autoregressive conditional multinomial (ACM) model. The paper also puts forth a methodology to evaluate probability function forecasts of MPP models. By improving goodness of fit and point forecasts of the target, the ACH–ACM qualifies as a sensible modeling framework. Furthermore, our results show that MPP models deliver useful probability function forecasts at short and medium term horizons.     

On cores and indivisibility

An economic model of trading in commodities that are inherently indivisible, like houses, is investigated from a game-theoretic point of view. The concepts of balanced game and core are developed, and a general theorem of Scarf's is applied to prove that the market in question has a nonempty core, that is, at least one outcome that no subset of traders can improve upon. A number of examples are discussed, and the final section reviews a series of other models involving indivisible commodities, with references to the literature.     

Procuring complex performance in construction: London Heathrow Terminal 5 and a Private Finance Initiative hospital

This paper takes as its starting point the fact that complex projects, interpreted as multiple dependent interactions between many stakeholders over time, challenge traditional procurement practices based on the serial purchase of discrete components. The paper examines how the procurement management of such projects – procuring complex performance – can be conducted. The paper utilises two contrasting case study examples of high-profile UK construction project procurement. The findings suggest that the choice of mechanisms or interfaces for the governance of upstream supply relationships critically relates to subsequent performance. The theoretical contribution is a fusion of procurement literature with the influential CoPS literature.     

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.     

Full fixed cost recovery lot sizing with quantity discounts

Purchasers must often make lot sizing decisions when facing price schedules of price-quantity discounts. It is important to determine the supplier's pricing philosophy when establishing a solution procedure.One approach is to evaluate total costs at all of the appropriate break points. This offers limited information: a lot size and a set of total costs. This is especially true in the case of full fixed cost recovery pricing. In actual practice price schedules can be extremely lengthy: indeed, it may be in the supplier's best interests to offer comprehensive discount schedules. This situation complicates the purchaser's decision making process.An efficient alternative, which solves the price-quantity discount problem when the supplier insists on a full fixed cost recovery schedule, is presented. Computations are reduced to a few simple steps; the result is a least total cost lot size for a simple linear package price model given parameters obtained by an appropriate analysis of the supplier's price-quantity discount schedule. A starting point is determined and the choice of the lot size is made using a simple criterion. Rapid convergence is assured, given a reasonably well-behaved schedule.     

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。     

Equilibrium in abstract economies without ordered preferences

We extend a theorem of G. Debreu on the existence of equilibrium in a generalized N-person game (‘A Social Equilibrium Existence Theorem’). Applying standard techniques the result can be used to prove the existence of equilibrium in economies with interdependent preferences, price-dependent preferences and preferences which may be both non-transitive and non-complete. This generalizes a recent theorem of A. Mas-Colell.     

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.     

Approximation of contractible valued correspondences by functions

Let X and Y be absolute neighborhood retracts (this is a large class of spaces) with X compact, and let F:X→Y be an upper hemicontinuous correspondence whose values are compact and contractible. It is shown that any neighborhood of the graph of F contains the graph of a continuous function f:X→Y. The relevance of this result to fixed point theory is indicated. It is also shown that if X is ‘locally infinite’, then F can be approximated in the stronger sense of the graph of f being close to the graph of F and every point in the graph of F being close to some point in the graph of f. A conjectured generalization of the main result is stated.     

On disequilibrium adjustment processes

Here, we study a class of disequilibrium processes defined by differential inequalities, where prices are adjusted according to the agent's short-term demands, in both pure exchange and monetary settings. We establish convergence theorem and accessibility theorem without convexity hypotheses on the utility functions.