Analytic decision rules and horizon rules are developed for the warehousing problem. A stochastic warehousing problem is defined and solved, first by backwards induction, then by means of a forward working algorithm. The influence of holding costs and discount rates on forecasting horizons is illustrated.
exhibited. We present axiom systems for indexes of several economic variables: inputs, outputs, prices, wages, inflation, and technological change. In addition to conventional smoothness and proportionality conditions, in each case an Invariance Axiom is proposed. For technological change this says, in a sense, that when there is no technological change there is no change in the index. One can prove that in each case there is a unique index satisfying the axioms, and this is a Divisia index.2 Since these continuous indexes are not generally independent of the path, the problem of how often to change index weights may be viewed in the light of a choice between invariance and independence. We show that the unique invariant measure of technological change is a natural generalization of Solow's measure to the case of many commodity types.
The von Neumann-Morgenstern theory of games does not yield determinate solutions (corresponding to unique payoff vectors) for two-person variable-sum games and for n-person games. The present paper outlines a general theory of rational behavior in game situations which does yield determinate solutions for all classes of games. The theory is based on two classes of rationality postulates: those defining rational behavior as'such, and those defining rational expectations concerning the other players' behavior. It is argued that this new approach greatly increases the possibilities for the application of game theory in economics and the other social sciences.
IN RECENT years the Durbin-Watson statistic has been used uncritically to test for serial correlation the of relationships containing lagged endogenous which are estimated by single or simultaneous equations methods. When lagged endogenous are included an equation estimated by ordinary least squares, however, the Durbin-Watson statistic is asymptotically biased towards 2 (the value which it should have if no serial correlation is fact present). is doubtful, therefore, that the statistic should be used either to test for serial correlation the or to provide any indication of the extent of such correlation when the estimated equation contains lagged values of any endogenous variable. The widespread use of the Durbin-Watson statistic inappropriate situations may stem from misinterpretation of a remark one of Durbin's later papers. the original papers setting forth their test, Durbin and Watson stated: It should be emphasized that the tests described this paper apply only to regression models which the independent can be regarded as 'fixed variables'. They do not, therefore, apply to autoregressive schemes and similar models which the lagged values of the dependent variable occur as independent variables [2, p. 159]. a subsequent paper, however, showing that the statistic could be used with some slight modification systems of simultaneous equations, Durbin wrote: In some formulations certain of the x's coincide with lagged values of the y's. The theory becomes much more complicated such cases, and we shall not consider them except to point out that the results obtained later the paper, which are exact for the model specified above, may be expected to hold approximately for models containing lagged dependent variables [1, p. 370]. This later paper discussed the distribution of the Durbin-Watson statistic under the null hypothesis of no serial correlation and did not cover the test's power against alternatives. While the original Durbin and Watson papers showed that the test has high power against Markov alternatives, the asymptotic result, derived from a result obtained by Malinvaud and presented below, indicates that this conclusion does not hold when lagged endogenous are included. a paper which expressions for the asymptotic bias of least squares estimates of regression coefficients various models containing lagged dependent and serially correlated were derived, Griliches commented that in most cases the addition of the lagged dependent variable to the regression will reduce the serial correlation of the residuals and hence increase the Durbin-Watson statistic [3, p. 70]. The asymptotic results were extended to cover the bias the estimated 235