As a non-Walrasian system tracks through the phase space, the differential equations which govern its motion will typically change as the system crosses certain borders. This increases the complexity of the stability problem considerably. In the present paper we find that some straightforward modifications to Lyapunov's method render the problem tractable. These methods are derived, and their use is illustrated in the case of two different systems which have trading out of equilibrium. 1 . THOUGH ECONOMISTS HAVE BEEN INTERESTED in the stability on non-Walrasian systems at least since Clower's paper' over a decade ago, we have yet to get very far with the inquiry. There may be a number of reasons for this, but the most important seems to be that we have not yet fully appreciated the differences between the methods of analysis suitable for studying the stability of Walrasian and non-Walrasian systems. It is widely known that the primary distinction between the two systems is that quantities actually traded enter as arguments in the non-Walrasian excess demand functions. These quantities will sometimes be demand quantities and sometimes supply quantities, depending upon the overall state of the markets-but then this implies that the excess demand functions themselves will be changing as the system moves through time and that the system itself is not everywhere differentiable. Take, for example, an output supply function which depends upon the actual quantity of labor hired. If the actual quantity hired is the lesser of the quantities supplied and demanded, then under the usual assumptions the partial derivative of output supply with respect to the price of labor will sometimes be positive, sometimes negative, and sometimes non-existent, depending upon whether the demand for labor is greater than, less than, or equal to the supply. What we have in effect is a dynamic system which has its endogenous variables sometimes governed by one set of equations and sometimes by another, with the overall system lacking differentiability at the points of changeover. This much is fairly clear, but the methods which can be used to study such systems have, with few exceptions,2 yet to be seriously explored. In the present paper we are interested in finding modifications to Lyapunov theorems which will render them suitable for the study of non-Walrasian systems. Two such modifications are found and their usefulness in studying non-Walrasian systems is illustrated by means of some relatively simple economic examples.

[The fine structure of earnings is defined by a theoretically meaningful decomposition of the covariance matrix of earnings (or log earnings) time series. A three-element variance components model is proposed for analyzing earnings of young workers. These components are interpreted as the effects of differential on-the-job training (OJT) and differential economic ability. Several properties of these components and relationships between them are deduced from the OJT model. Background noise generated by a nonstationary first-order autoregressive process, with heteroscedastic innovations and time-varying AR parameters is also assumed present in observed earnings. ML estimates are obtained for all parameters of the model for a sample of Swedish males. The results are consistent with the view that the OJT mechanism is an empirically significant phenomenon in determining individual earnings profiles.]

[In his paper, extasciicircum1 Roth has used a three-person game example to illustrate certain difficulties connected with generalizations of the Shapley value for games without side payment. This note argues that cooperative solution concepts in general often give rise to similar difficulties, and that the best way of avoiding them is to analyze cooperative games by means of noncooperative bargaining models.]

IN THE LATE 1960's, Richardson [18] and Sawa [20] derived the exact distribution of the two-stage least squares (2SLS) estimator in a structural equation (of a simultaneous system) that contained two endogenous variables and an arbitrary number of degrees of overidentification. Their results refer to the 2SLS estimator of the coefficient of the endogenous variable included on the right hand side of the equation and were obtained under the classical assumptions (to use the term employed by Sargan [19]) of normally distributed disturbances and nonrandom exogenous variables. Very little exact finite sample theory has been published so far for estimators in structural equations containing more than two endogenous variables. Basmann et al. [4] extract the joint probability density function (p.d.f.) of the 2SLS estimator in a just identified equation containing three endogenous variables. Basmann [3] quotes a result due to Richardson for the same set up but with an 2 arbitrary number of degrees of overidentification . In Basmann's notation, this last result characterizes the subclass

AbstractRecursive equilibrium theory is extended and generalized. Optimality of equilibria and supportability of optima are established in a direct way. Four economic applications are reformulated as recursive competitive equilibria and analyzed.

asymptotic variances of multipliers for nonlinear models. It is used to estimate the uncertainty of the results of eight policy experiments for a particular model. ALTHOUGH MACROECONOMETRIC MOI)ELS are widely used to analyze the effects of alternative government actions on the economy, estimates of the uncertainty of these effects are rarely, if ever, presented. This is, of course, not surprising, since most macroeconometric models are nonlinear. Unlike for linear models, formulas for the asymptotic variances of impact and dynamic multipliers are not known for nonlinear models. ’ It is possible, however, to estimate these variances for nonlinear models by stochastic simulation, and the purpose of this paper is to discuss the method by which this can be done. The method is discussed in Section 2, and results of applying the method to eight policy experiments for the model in Fair [7,10] are presented in Section 3.3 Given the obvious importance of knowing how much confidence to place on the results of any particular policy experiment in a model, it is hoped that this study will stimulate others to obtain uncertainty estimates for their models similar to those presented in Section 3. 2. THE METHOD The. method can be applied to a model that is nonlinear in both variables and coefficients. Let G denote the total number of equations in the model, M the number of stochastic equations, and N the total number of predetermined (both exogenous and lagged endogenous) variables. Assume (for exposition.4 con-venience only) that the model is quarterly, and let the ith equation of the model for quarter t be written: (1) CpdYi,, YGh Zlb, ZN,,

[In this paper we consider the problem of the existence of a well-defined reduced form in the context of piecewise linear models. We give a general theorem which provides necessary and sufficient conditions, called coherency conditions, for such an existence. This result is applied to various kinds of models: self-selectivity models, simultaneous equation probit and tobit models, multimarkets disequilibrium models.]

Consider a closed economy with several deposits of an exhaustible resource, with the marginal cost of extraction differing from deposit to deposit but constant for each deposit. It is widely believed that social optimality requires that deposits be exploited in strict sequence, beginning with the lowest cost deposit. It is shown that, in a general equilibrium context, with Ricardian techniques of extraction, the validity of the proposition depends on what is meant by constancy of cost. It is also believed that if there exists a high-cost substitute for the resource then the resource should be exhausted before production of the substitute is begun. It is shown that this proposition is false.

It was conjectured by Pigou that an increase in real national income, as reckoned in the prices of either the initial or the terminal period, would always correctly indicate an improvement in national welfare provided the increase referred to the aggregate income of a given group of persons with fixed preferences and a fixed proportional distribution of income among them. We show that if the individual preferences are assumed to be homothetic, and if by a welfare improvement one means respectively a potential improvement (in which losers can be compensated by gainers) or an actual improvement (in which all are gainers), then on either of these respective criteria Pigou's conjecture holds true under these conditions if and only if individual preferences are identical.