Knowledge that Transforms

THE WALRAS CORRESPONDENCE associates the set equilibrium price vectors with an economy. The purpose of this paper is to study some topological properties of the graph of the Walras correspondence such as connectedness and contractibility. This is done once the graph is given the structure of a bundle. The mathematical notations used in this paper are given in Section 2. The bundle structure of the graph is proved in Section 3. The contractibility of the graph is then a straightforward result proved in Section 4. Finally, variable demand functions are introduced in Section 5 and connectivity is then proved.

Many policies use two categories of instruments: a financial incentive (most often a tax) for polluters, and direct undertakings or financing of some restorations or maintenances or improvements of environmental qualities. The financial consequences of such policies, when optimum, are important for considerations of public finance, decentralization (financial autonomy), and equity (must polluters pay?). They turn out essentially to depend upon the mathematical structure of, first, the environment function, i.e., the way in which qualities depend upon both deteriorating and improving activities, and, second, the various constraints of the problem. Constraints which can be expressed by functions homogeneous of any degree are shown to have no direct financial effect. Apart from the constraints' effects, budgetary equilibrium, surplus or deficit are respectively given by functions which present constant, decreasing, or increasing qualitative returns to scale, i.e., weighted homogeneity of degree zero, positive or negative. The opposite polar cases of cleaning and dilution types of improvement technology are presented, with some other mixed simple cases and a few examples of application of the results.

[This paper considers the impact of certain types of policy related voting patterns on the existence and location of equilibrium strategies in the spatial model of two-candidate competition. In contrast to much of the previous literature, this paper makes a distinction between the aggregate level patterns of voting and the individual level variables which bring them about. By so doing the assumptions can focus on objects which have a more direct empirical referent, namely, the aggregate level support function and the distribution of ideal points. Using this approach, sufficient conditions are found for the existence of equilibrium which, although themselves strong, make considerably weaker demands on individuals than have been generally assumed in the literature. Thus, it is not necessary that each voter vote strictly with regard to policy but, rather, it is sufficient that in the electorate as a whole there is a moderate amount of policy related voting. Other results of the analysis are that policy related voting of any type seems to encourage candidates to converge towards the center, with support from extremists only accentuating this tendency. It is the candidate's most loyal supporters who seem to have the least influence over his policy position.]

FIML is shown to be an instrumental variables estimator where the instruments embody all the over-identifying a priori restrictions. FIML is compared to the two alternative estimators 3SLS and full information instrumental variables. 3SLS differs from FIML in not using all a priori restrictions in forming the instruments. The full information instrumental variables estimator when iterated to convergence yields the FIML estimate. For the case of nonlinearity in the parameters a nonlinear 3SLS and a nonlinear full information instrumental variables estimator are proposed. Both estimators are asymptotically efficient. THIS PAPER UNDERTAKES an investigation of asymptotically efficient estimators for linear and nonlinear simultaneous equation econometric models. By using an instrumental variable approach the equivalence of previously proposed linear estimators to full information maximum likelihood (FIML) follows in a straightforward manner, and a class of new estimators which includes a nonlinear three stage least squares estimator (NL3SLS) and nonlinear full information instrumental variables estimator are proposed and shown to be asymptotically equivalent to FIML. First, an instrumental variable interpretation of FIML is developed by investigating the first order conditions for the maximum of the likelihood function without first concentrating the likelihood function. The essential difference between 3SLS and FIML then becomes evident. The difference between the two estimators is first that FIML uses all over-identifying restrictions in forming the instruments while 3SLS ignores some of these restrictions. Also, FIML uses an estimate of the covariance matrix in forming the instruments which is consistent in the sample with the parameter estimates. Thus the instruments used by FIML are mutually consistent with the parameter estimates in the given sample, while for other estimators the instruments are consistent with the parameter estimates only asymptotically. While this difference in forming the instruments is of no importance asymptotically as is known by the earlier results of Sargan [10] and Rothenberg and Leenders [9], in finite samples there seems to be no reason for not using all known prior information. The use of the a priori restrictions gives a more useful criterion than Dhrymes' [3] recent interpretations of a difference in purging the endogenous variables since all other proposed estimators can be shown to be equivalent by simply proving asymptotic equivalence of the instruments used to those instruments used by the FIML estimator. Then using the instrumental variable interpretation, a relation between FIML and the class of estimators recently proposed by Dhrymes [2], Lyttkens [5, 6], and Brundy and Jorgenson [1] is established. The full information instrumental

[This paper examines a notion of coalitional fairness for exchange allocations. An allocation is "c-fair" if no coalition of traders could benefit from achieving the net trade of some other coalition. Properties of c-fair allocations are studied both in exchange economies with a finite number of traders and with an atomless sector.]

THIS NOTE SUGGESTS that the components models employed in [2, 3, 10, 11, 12 and 17] may be usefully analyzed within the state space framework found in the literature of control engineering. Such a formulation has the distinct advantage that a large corpus of filtering and estimation theory may then be brought to bear upon such models and has secondary benefits in the form of a greater lucidity of, and flexibility in, the exposition of the extraction problem. Further, by relating such models to a format that has widespread use in other fields, it is possible to exploit any future computational and theoretical advances therein.

Under the assumption of single peaked preferences, the majority rule equilibrium considered as a correspondence from the voters' preference is shown to be continuous. We also complement the work of Fishburn [6], who first presented a general location theorem for majority rule equilibriums, by dropping the assumptions that the alternative set is finite and that voters' preferences are strict partial orders. SINCE THE WORK of Black [3] on simple majorities and single-peaked preferences, much work has been done in deriving conditions for which some state achieves a majority over all other states. It is also well known that the equilibrium in many cases is the median of the distribution of most preferred states of the voters. (See [3, pp. 14-18].) We present here a result concerning the continuity of the majority rule equilibria. Specifically, we show that in the case of single-peaked preferences, the majority rule equilibrium depends only on the peaks of the voters' preferences and not on the transitivity properties of these preferences. We then show that the equilibrium, viewed as a correspondence of these peaks, is continuous (i.e., both upper and lower semicontinuous). This result is especially important when one tries to prove the existence of an equilibrium in a political-economic system. Most existence theorems are based on fixed point theorems which require at least upper semicontinuity of the correspondences being studied. The theorem proved here shows that the majority rule equilibrium is continuous in voters' peaks. If these are in turn continuous functions of other parameters such as prices, then our result might aid one in deriving general existence theorems for social equilibria. (See [3 or 4].) The only related work seems to be that of Kelly [10] who investigates the existence of a continuous numerical representation of the social preference relation generated by majority rule. He shows that one cannot expect continuity even if the social preference relation has other properties, such as transitivity. Specifically, he presents a case in which a majority of voters are indifferent between two points. In a neighborhood of one of these points, this majority rules between points in the neighborhood and the other point, but a minority (since the majority is indifferent) rules between the two given points. Hence one cannot expect continuity of the

[This paper considers the estimation of a simultaneous equations model in which the dependent variables are qualitative. The model is a simultaneous version of the multivariate logit model, and can be estimated by maximum likelihood. An example is presented, dealing with the prediction of occupation and industry of employment of a worker based on certain demographic variables.]

[In this paper we analyze implications of a singular contemporaneous disturbance covariance matrix for the estimation and hypothesis testing of systems of equations with autoregressive disturbances. We find that this singularity imposes restrictions on the parameters of the autoregressive process. When these restrictions are not imposed, the specification, maximum likelihood estimates, and likelihood ratio test statistics are conditional on the equation deleted. Furthermore, singularity of the contemporaneous disturbance covariance matrix raises issues concerning identification of parameters of the autoregressive process. This identification problem complicates the interpretation of likelihood ratio statistics. The above results are illustrated with an empirical example.]