Suppose that the coefficients of an input-output matrix, A, are random variables but that we have ascertained their expected values, EA. What will be the relation of the Leontief inverse of EA, (I EA) ', to the expected value of the inverse, E(I A) ? Will one or the other be uniformly greater? We will show that if all coefficients of A are independent, then the expected value of the inverse is uniformly greater than or equal to the inverse of the expected value. If, on the other hand, the column and row sums of the coefficient matrix are fixed, and smaller than one, so that the variables are not independent, then, in the two-by-two case, the opposite is true of the off-diagonal elements.

[The empirical content of the paper is the relationship between personal attributes and job performance, as measured by rate of promotion, in a large corporation. It focuses, however, on a behavioral model and method of estimation which provide a natural way of dealing with this and similar phenomena.]

[The power function of the Durbin-Watson test for first-order serial correlation is examined. The power function depends upon the regression vectors, but useful upper and lower bounds for the power are established. The bounds are obtained from inequalities on the characteristic roots of real non-definite symmetric matrices developed in this article.]

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 is a study on optimal consumption over time under the assumption of uncertainty about prices and interest rates. A stochastic model has been developed in which present wealth is either consumed or invested in a future commodity and loans. The possibility of lending and borrowing allows the model to capture some of the aspects of a monetary economy, such as the effects of actual or anticipated inflation on optimal plans. Future prices follow a Markovian process. Conditions for the existence of optimal policies are given for concave utility functions in general, but constant elasticity utility functions are studied in greater detail. The case in which the expected value of the discounted stream of utilities is infinite, is also investigated.]

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

Linear decision rules for controlling complex systems are often obtained by matrix inversion, but transform methods offer an alternative approach that yields insights into the structure of the decision problem of maximizing expected payoffs under constraints.