Knowledge that Transforms

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.

We investigate the structure of optimal policies in general multiperiod multiasset consumption-investment problems in the presence of transfer costs. A number of objectives such as utility of a consumption stream, utility of terminal wealth, and multiattribute utility are encompassed by the formulation. The general problem is first formulated as a stochastic dynamic program. The one-period subproblems are then analyzed using convex duality theory. The principal result is the characterization of a not necessarily convex of no for each period. If in any period the entering asset position is in this set, no transactions are made. Each point of the set is the vertex of a cone such that if the entering asset position is outside the set, the optimal policy is to move to the vertex of the cone in which the entering asset position lies. It is shown that the region of no transactions is a connected set and that it is a cone when the utility function is assumed to be positively homogeneous. In the latter case, the optimal decision policy and induced utility IN THIS PAPER we study a general class of multiperiod, multiasset investmentconsumption problems. Our purpose is to characterize the structure of optimal policies in the presence of transaction costs. Related problems have been studied by several authors, e.g., Constantinides [2, 3], Fama [5], Eppen and Fama [4], Kamin [7], Magill and Constantinides [10], Zabel [17], Hakansson [6], Merton [11], Samuelson [14], and Mukherjee and Zabel [12], and in these papers optimal policies have been characterized for a number of special cases. Our methodology significantly generalizes and sharpens many of the above results. For example, much of the previous work has been limited to the two-asset case or to particular utility functions, whereas our formulation allows any number of assets and general concave utility functions. The principal result of this paper, in the case of proportional transaction costs and concave utility functions, is the characterization of the optimal policy in each period and the set of entering asset positions from which no transactions should be made. This set or of no (RNT) can take on many forms ranging from a simple halfline to a nonconvex set. (See examples in Section 3.)

[Alternative measures of the rate of diffusion of a new innovation are discussed and applied to study the spread of hybrid corn in the United States, in the light of more recent data and improved estimating techniques. An assessment is made of the importance of "profitability" variables in accounting for variations in the rate of diffusion between states.]

This paper presents a parameter covariance matrix estimator which is consistent even when the disturbances of a linear regression model are heteroskedastic. This estimator does not depend on a formal model of the structure of the heteroskedasticity. By comparing the elements of the new estimator to those of the usual covariance estimator, one obtains a direct test for heteroskedasticity, since in the absence of heteroskedasticity, the two estimators will be approximately equal, but will generally diverge otherwise. The test has an appealing least squares interpretation

[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.]

Rational behavior under complete ignorance is described by means of requirements such as invariance of choice with respect to modifications of states of nature. Possible criteria necessarily involve intransitivities of indifference, and are incompatible with the ascribing of personal probabilities to events. Characterization of criteria shows that in first order approximation they take into account only the extremal possible outcomes of each choice; effects linked to events also come into play, although only in the second order, whereas an axiom system like that of Arrow and Hurwicz, which requires transitivity of indifference, excludes their being taken into account at all. IN PERSONAL PROBABILITY THEORIES, there is no such thing as non-probabilizable uncertainty. According to these theories, the choices of a rational decision maker in an uncertain environment are explainable by a mathematical expectation of utility criterion. The decision maker thus assigns, consciously or unconsciously, to any given event, a certain probability determined by the a priori information he possesses or, in the absence of such information, by considerations of symmetry known as the principle of insufficient reason. This principle, in actual fact, does not permit equal treatment of all events; we shall investigate the decision criteria which do permit such treatment. Arrow and Hurwicz [1] have shown that, among such decision criteria, those whose weak preferences are transitive vary only according to the maximum and minimum values of the outcomes of each decision. They thus have the advantage of being easily stated but the drawback of not automatically preferring, over any given decision, any other which weakly dominates it. We shall prove that decision criteria can be made to take weak dominance into account while yet giving up only transitivity of indifference, a property which is, for that matter, seldom genuinely present, even in the case of choice under certainty, since it turns out to be generally noncomparability rather than true indifference. The criteria characterized by the axioms of Arrow and Hurwicz will then prove to be an approximation of those fulfilling ours.