Optimal strategies in linear multisector models: Value function and optimality conditions

In this paper we study an optimal control problem with mixed constraints related to a multisector linear model with endogenous growth. The main aim is to establish a set of necessary and a set of sufficient conditions which are the basis for studying the qualitative properties of optimal trajectories. The presence of possibly degenerate mixed constraints, the unboundedness and non-strict convexity of the Hamiltonian, make the problem difficult to deal with. We develop first the dynamic programming approach, proving that the value function is a bilateral viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation. Then, using our results, we give a set of sufficient and a set of necessary optimality conditions which involve so-called co-state inclusion: this can be interpreted as the existence of a dual path of prices supporting the optimal path.     

Coefficients of ergodicity for Markov chains with uncertain parameters

One of the central considerations in the theory of Markov chains is their convergence to an equilibrium. Coefficients of ergodicity provide an efficient method for such an analysis. Besides giving sufficient and sometimes necessary conditions for convergence, they additionally measure its rate. In this paper we explore coefficients of ergodicity for the case of imprecise Markov chains. The latter provide a convenient way of modelling dynamical systems where parameters are not determined precisely. In such cases a tool for measuring the rate of convergence is even more important than in the case of precisely determined Markov chains, since most of the existing methods of estimating the limit distributions are iterative. We define a new coefficient of ergodicity that provides necessary and sufficient conditions for convergence of the most commonly used class of imprecise Markov chains. This so-called weak coefficient of ergodicity is defined through an endowment of the structure of a metric space to the class of imprecise probabilities. Therefore we first make a detailed analysis of the metric properties of imprecise probabilities.     

Stability of a Cournot-Nash equilibrium: The multiproduct case

This paper examines the stability conditions of Cournot-Nash equilibria in models in which each firm produces multiple outputs in m related markets. Using a discrete adjustment process, we establish sufficient and necessary conditions for stability. We derive further specialized stability conditions for three important cases, namely a multiproduct duopoly, a multiproduct oligopoly with homogeneous products, and a single-product oligopoly with product heterogeneity. The paper shows that the stability conditions of the m markets individually are in general sufficient nor necessary for the stability of the m markets as a group. In addition, an example is given of how the stability conditions can be applied to determine comparative statics for multiproduct oligopoly.     

Weak congestion in output additive data envelopment analysis

Our earlier work [Wei QL, Yan H. Congestion and returns to scale in data envelopment analysis. European Journal of Operational Research 2004;153:641–60] discussed necessary and sufficient conditions for the existence of congestion together with aspects of returns to scale under an output-oriented DEA framework. In line with this work, the current paper investigates the issue of “weak congestion”, wherein congestion occurs when the reduction of selected inputs causes some, rather than all, outputs to increase, without a worsening of others. We define output efficiency for decision making units under a series of typical DEA output additive models. Based on this definition, we offer necessary and sufficient conditions for the existence of weak congestion. Numerical examples are provided for purposes of illustration.     

Existence of a competitive equilibrium in the Lucas (1988) model without physical capital

This paper considers an endogenous growth model with human capital accumulation. It gives sufficient conditions and a necessary condition for the existence of a unique competitive equilibrium with externalities. These conditions are more stringent than those which prevail for the existence of an equilibrium defined as the solution to a fixed-point problem.     

Inconsequential arbitrage

We introduce the concept of inconsequential arbitrage and, in the context of a model allowing short-sales and half-lines in indifference surfaces, prove that inconsequential arbitrage is sufficient for existence of equilibrium. Moreover, with a slightly stronger condition of nonsatiation than that required for existence of equilibrium and with a mild uniformity condition on arbitrage opportunities, we show that inconsequential arbitrage, the existence of a Pareto optimal allocation, and compactness of the set of utility possibilities are equivalent. Thus, when all equilibria are Pareto optimal — for example, when local nonsatiation holds — inconsequential arbitrage is necessary and sufficient for existence of an equilibrium. By further strengthening our nonsatiation condition, we obtain a second welfare theorem for exchange economies allowing short sales.Finally, we compare inconsequential arbitrage to the conditions limiting arbitrage of Hart [Hart, O.D., 1974. J. Econ. Theory 9, 293–311], Werner [Werner, J., 1987. Econometrica 55, abs1403–1418], Dana et al. [Dana, R.A., Le Van, C., Magnien, F., 1999. J. Econ. Theory 87, 169–193] and Allouch [Allouch, N., 1999. Equilibrium and no market arbitrage. CERMSEM, Universite de Paris I]. For example, we show that the condition of Hart (translated to a general equilibrium setting) and the condition of werner are equivalent. We then show that the Hart/Werner conditions imply inconsequential arbitrage. To highlight the extent to which we extend Hart and Werner, we construct an example of an exchange economy in which inconsequential arbitrage holds (and is necessary and sufficient for existence), while the Hart/Werner conditions do not hold.     

Pareto optimality in continuous-time OLG economies

The aim of this paper is the achievement of a complete characterization of the Pareto optimality of competitive equilibria for deterministic, pure exchange, continuous-time economies with a countable number of overlapping generations, where each consumer’s life-span consists in a bounded interval of time. For such an environment, we obtain separate sufficient and necessary conditions for Pareto optimality in the form of the Cass’ criterion, that is, in terms of the equilibrium prices. However, these conditions are not equivalent in general. Therefore, in order to get that equivalence we are compelled to impose certain restrictions, either on consumers’ lifetimes, assuming that all of them have the same longevity, or on the dynamic behaviour of relative intertemporal equilibrium prices. In both cases, we are able to derive a single condition that is sufficient and necessary for efficiency, thus achieving full characterizations.     

The geometry of arbitrage and the existence of competitive equilibrium

We present the basic geometry of arbitrage, and use this basic geometry to shed new light on the relationships between various no-arbitrage conditions found in the literature. For example, under very mild conditions, we show that the no-arbitrage conditions of Hart [Journal of Economic Theory 9 (1974) 293] and Werner [Econometrica 55 (1987) 1403] are equivalent and imply the compactness of the set of utility possibilities. Moreover, we show that if agents’ sets of useless net trades are linearly independent, then the Hart–Werner conditions are equivalent to the stronger condition of no-unbounded-arbitrage due to Page [Journal of Economic theory 41 (1987) 392]—and, in turn, all are equivalent to compactness of the set of rational allocations. We also consider the problem of existence of equilibrium. We show, for example, that under a uniformity condition on preferences weaker than Werner’s uniformity condition, the Hart–Werner no-arbitrage conditions are sufficient for existence. With an additional condition of weak no-half-lines—a condition weaker than Werner’s no-half-lines condition—we show that the Hart–Werner conditions are both necessary and sufficient for existence.     

Causal conditions for loneliness: a set-theoretic analysis on an adult sample in the UK

While age has been identified as a risk factor for loneliness, whether it is a necessary or sufficient condition for loneliness has never been examined. This is the first study that applies fuzzy-set QCA, a special type of set-theoretic method, to discover the necessary and sufficient causal conditions for loneliness, respectively, among adults in the UK, analysing the data collected from the UK sample of Round 6 of the European Social Survey (ESS, 2012, n = 2163). It firstly examines the configurations of five conditions: being female, old age, not living with spouse/partner, bad health, and not being frequently social with others. Gender was found neither a necessary nor a sufficient condition for loneliness, and old age was close to being a necessary condition and became necessary when united with any of the other conditions; the configuration of not living with spouse/partner and not healthy and not frequently social with others is a sufficient condition. Robustness of results was tested with two different conditions (a limiting illness and a confidante), and a separate analysis on the absence of loneliness was conducted. The effect of the unbalanced distribution of cases across different values of the outcome was highlighted as a source of uncertainty, and the results on the absence of loneliness are different from those on its presence.     

Ross risk vulnerability for introductions and changes in background risk

We present two theorems that provide necessary and sufficient conditions for an expected utility maximizer to become more risk averse in the sense of Ross with respect to bearing a foreground risk after the introduction of any independent fair or unfair additive background risk. We call these decision makers Ross risk vulnerable, and show that Ross decreasing absolute risk aversion and Ross decreasing absolute prudence are jointly sufficient for Ross risk vulnerability. Restrictions on utility necessary and sufficient for Ross risk vulnerability with respect to stochastic dominance deteriorations of an existing background risk are also presented. Our analysis concludes with applications of Ross risk vulnerability.     

A sufficiency result for constrained economies

This paper presents a general sufficiency result for constrained economies. The result shows that, when an equilibrium allocation satisfies the first order necessary conditions for hybrid efficiency (a new type of constrained efficiency), there exists a marginal cost pricing equilibrium in an associated economy. If this equilibrium is Pareto efficient in the associated economy, then the original equilibrium allocation is hybrid efficient. Hence. the necessary conditions are also sufficient.     

Poverty Orderings

This paper reviews the literature of partial poverty orderings. Partial poverty orderings require unanimous poverty rankings for a class of poverty measures or a set of poverty lines. The need to consider multiple poverty measures and multiple poverty lines arises inevitably from the arbitrariness inherent in poverty comparisons. In the paper, we first survey the ordering conditions of various individual poverty measures for a range of poverty lines; for some measures necessary and sufficient conditions are identified while for others only some easily verifiable sufficient conditions are established. These ordering conditions are shown to have a close link with the stochastic dominance relations which are based on the comparisons of cumulative distribution functions. We then survey the ordering conditions for various classes of poverty measures with a single or a set of poverty lines; in all cases necessary and sufficient conditions are established. These conditions again rely on the stochastic dominance relations or their transformations. We also extend the relationship between poverty orderings and stochastic dominance to higher orders and explore the possibility and the conditions of increasing the power of poverty orderings beyond the second degree dominance condition.     

Efficient allocations and equilibria with short-selling and incomplete preferences

This paper reconsiders the theory of existence of efficient allocations and equilibria when consumption sets are unbounded below under the assumption that agents have incomplete preferences. Our model is motivated by an example in the theory of assets with short-selling where there is risk and ambiguity. Agents have Bewley’s incomplete preferences. As an inertia principle is assumed in markets, equilibria are individually rational. It is shown that a necessary and sufficient condition for the existence of an individually rational efficient allocation or of an equilibrium is that the relative interiors of the risk adjusted sets of probabilities intersect. The more risk averse, the more ambiguity averse the agents, the more likely is an equilibrium to exist. The paper then turns to incomplete preferences represented by a family of concave utility functions. Several definitions of efficiency and of equilibrium with inertia are considered. Sufficient conditions and necessary and sufficient conditions are given for the existence of efficient allocations and equilibria with inertia.     

Granger causality and path diagrams for multivariate time series

In this paper, we introduce path diagrams for multivariate time series which visualize the dynamic relationships among the variables. In these path diagrams, the vertices represent the components of the time series and are connected by arrows or lines according to the nonvanishing parameters in the autoregressive representation of the time series. We show that these path diagrams provide a framework for the analysis of the dependence structure of the time series. In particular, we give sufficient graphical conditions for Granger-noncausality and Granger-noncausality up to a certain horizon.     

Convergence of optimal harvesting policies to a normal forest

This paper extends the forestry maximum principle of Heaps (1984) to allow the benefits of harvesting to be the utility of the volume of the wood harvested as in Mitra and Wan, 1985, Mitra and Wan, 1986. Unlike those authors, however, time is treated as a continuous rather than as a discrete variable. Existence of an optimal harvesting policy is established. Then necessary conditions are derived for the extended model which are also sufficient. The conditions are used to show that under certain boundedness conditions, sequences of optimal harvesting policies contain subsequences which converge pointwise a.e. and in net present value to an optimal harvesting policy. This result is then used to show that any optimal logging policy must converge in harvesting age to a constant rotation period given by modified Faustmann formula. The associated age class distribution converges to a normal forest.     

关于大众日益青睐的网络虚拟世界的思考

网络的可能性是不可估量的,而大众的要求也是不断提高的,大众的要求是推动网路功能不断前进的动力。透视现实,正是现实与理想的差异空间,为网路虚拟提供了广阔的平台,才能不断吸引大众的目光。也就是,网络能让人们过一种崭新的虚拟人生,它具有强烈的可控性、向往与目标的一致性,能全面满足人们对理想生活的欲望,也就是所说的"‘在线’的乌托邦"。     

Necessary and sufficient conditions for consistency of generalizedM-estimates

GeneralizedM-estimates (minimum contrast estimates) and their asymptotically equivalent approximate versions are considered. A relatively simple condition is found which is equivalent with consistency of all approximateM-estimates under wide assumptions about the model. This condition is applied in several directions. (i) A more easily verifiable condition equivalent with consistency of all approximateM-estimates is derived and illustrated on models with stationary and ergodic observations. (ii) A condition sufficient for inconsistency of all approximateM-estimates is obtained and illustrated on models with i.i.d. observations. (iii) A simple necessary and sufficient condition for consistency of all approximateM-estimates in linear regression with i.i.d. errors is found. This condition is weaker than sufficient conditions for consistency ofM-estimators known from the literature. A linear regression example is presented where theM-estimate is consistent and an approximateM-estimate is incosistent.Supported by CSAS grant N. 17503.     

Some further remarks on a superiority problem in misspecified restricted singular linear models

Razzaghi (1987) examined the difference of covariance matrices of competing estimators in misspecified restricted linear models. Further, Gross, Trenkler and Liski (1998) extended Razzaghi's result by asserting his sufficient condition for nonnegative definiteness of the covariance matrix difference to be also necessary. In this paper, when the covariance matrix of the disturbance vector is nonnegative definite, the necessary and sufficient conditions for nonnegative definiteness of the covariance matrix difference are derived. So above results are strengthened. Received: February 1999     

Asymptotic properties of random extremes under general normalization from nonidentical distributions

In this paper we study the weak convergence of the generally normalized extremes (extremes under nonlinear monotone normalization) of random number of independent (nonidentically distributed) random variables. When the random sample size is assumed to converge in probability and the interrelation between the basic variables and their random size is not restricted, the limit forms as well as the sufficient conditions of convergence are derived. Moreover, when the random sample size is assumed to converge weakly and independent of the basic variables, the necessary and sufficient conditions for the convergence are derived.Acknowledgment. The authors are grateful to a referee for several helpful comments and for pointing out the extensive paper by Galambos (1992) leading to improvement of the representation of the paper.Received January 2002     

Constrained suboptimality when prices are non-competitive

The paper addresses the following question: how efficient is the market system in allocating resources if trade takes place at prices that are not competitive? Even though there are many partial answers to this question, an answer that stands comparison to the rigor by which the first and second welfare theorems are derived is lacking. We first prove a “Folk Theorem” on the generic suboptimality of equilibria at non-competitive prices. The more interesting problem is whether equilibria are constrained optimal, i.e. efficient relative to all allocations that are consistent with prices at which trade takes place. We discuss an optimality notion due to Bénassy, and argue that this notion admits no general conclusions. We then turn to the notion of p-optimality and give a necessary condition, called the separating property, for constrained optimality: each constrained household should be constrained in each constrained market. If the number of commodities is less than or equal to two, the case usually treated in the textbook, then this necessary condition is also sufficient. In that case equilibria are constrained optimal. When there are three or more commodities, two or more constrained households, and two or more constrained markets, this necessary condition is typically not sufficient and equilibria are generically constrained suboptimal.