[Very often, an index number used in an economic model has been constructed in two or more stages. If the two stage procedure gives the same answer as a single stage procedure, then Vartia calls the index number formula "consistent in aggregation." Paasche and Laspeyres indexes have this consistency in aggregation property, but these index number formulae are consistent only with very restrictive functional forms for the underlying aggregator (i.e., utility or production) function. The present paper shows that the class of superlative index number formulae has an approximate consistency in aggregation property, where a superlative index number formula is one which is consistent with a flexible functional form for the underlying aggregator function. The paper also contains some empirical examples which both illustrate the main theorem and also indicate that the chain principle for constructing index numbers is preferable to the fixed base method. Finally, the paper proves some theorems about the class of pseudo-superlative index numbers.]

[For the two sample linear heteroscedastic regression model, moments of a popular two stage Aitken estimator are derived analytically. Even for small samples and/or near homoscedastic errors, the two stage procedure is surprisingly efficient relative to both unweighted least squares and the Gauss-Markov estimator. These exact results are compared with the author's previous calculations derived from Nagar approximations.]

[A choice function C on X identifies, for each subset Y of X, a set C(Y) of "best" alternatives in Y. A. computationally viable choice function is one for which a member of the choice set C(Y) can be located by a computational procedure which does not waste time, generates reasonably satisfactory intermediate alternatives, and adjusts easily as new alternatives become available. Computational viability is defined precisely and the class of computationally viable choice functions is neatly characterized. In addition, the various types of binary choice functions are distinguished in terms of computational criteria.]

THIS PAPER IS A THEORETICAL examination of the stochastic behavior of equilibrium asset prices in a one-good, pure exchange economy with identical consumers. The single good in this economy is (costlessly) produced in a number of different productive units; an asset is a claim to all or part of the output of one of these units. Productivity in each unit fluctuates stochastically through time, so that equilibrium asset prices will fluctuate as well. Our objective will be to understand the relationship between these exogenously determined productivity changes and market determined movements in asset prices. Most of our attention will be focused on the derivation and application of a functional equation in the vector of equilibrium asset prices, which is solved for price as a function of the physical state of the economy. This equation is a generalization of the Martingale property of stochastic price sequences, which serves in practice as the defining characteristic of market efficiency, as that term is used by Fama [7] and others. The model thus serves as a simple context for examining the conditions under which a price series' failure to possess the Martingale property can be viewed as evidence of non-competitive or irrational behavior. The analysis is conducted under the assumption that, in Fama's terms, prices fully reflect all available an hypothesis which Muth [13] had earlier termed rationality of expectations. As Muth made clear, this hypothesis (like utility maximization) is not behavioral: it does not describe the way agents think about their environment, how they learn, process information, and so forth. It is rather a property likely to be (approximately) possessed by the outcome of this unspecified process of learning and adapting. One would feel more comfortable, then, with rational expectations equilibria if these equilibria were accompanied by some form of stability theory which illuminated the forces which move an economy toward equilibrium. The present paper also offers a convenient context for discussing this issue. The conclusions of this paper with respect to the Martingale property precisely replicate those reached earlier by LeRoy (in [10] and [11]), and not surprisingly, since the economic reasoning in [10] and the present paper is the same. The

[A gap is filled in a proof of a central limit theorem, for regression in a time series context, which has been used in some econometric theory.]

[This paper considers procedures for testing for autocorrelation when there are missing observations on both the dependent and explanatory variables. These procedures include Durbin-Watson type tests given the vector of residuals, tests based on a set of uncorrelated residuals, and large sample likelihood ratio and Wald tests.]

[It has recently been shown that the utility of playing a game with side payments depends on a parameter called strategic risk posture. The Shapley value is the risk neutral utility function for games with side payments. In this paper, utility functions are derived for bargaining games without side payments, and it is shown that these functions are also determined by the strategic risk posture. The Nash solution is the risk neutral utility function for bargaining games without side payments.]

It has recently been shown that the utility of playing a game with side payments depends on a parameter called strategic risk posture. The Shapley value is the risk neutral utility function for games with side payments. In this paper, utility functions are derived for bargaining games without side payments, and it is shown that these functions are also determined by the strategic risk posture. The Nash solution is the risk neutral utility function for bargaining games without side payments. RECENT WORK HAS SHOWN that the Shapley value for a game with side payments is a cardinal utility function which reflects the desirability of playing different positions in a game, or in different games (cf. Shapley [14], Roth [9]). A player's utility for playing some position in a game is determined in part by his assessment of the payoff he will receive in a class of games with side payments called bargaining games. Given a player's evaluation of these bargaining games, his utility for playing a position in any game with side payments can be determined (cf. Roth [11]). It is desirable to extend these results to games without side payments, since the assumption that side payments can be made is not appropriate in many situations. In this paper we will derive a class of utility functions for playing bargaining games without side payments. Games of this sort are studied by Nash [7], who developed a solution to bargaining games which is an extension of the Shapley value for games with side payments. That is, the Nash solution coincides with the Shapley value for bargaining games with side payments. Somewhat surprisingly, the utility of playing a bargaining game without side payments is determined by the same considerations which determine the utility of playing a game with side payments. Given a player's evaluation of bargaining games with side payments, his utility for bargaining without side payments is determined.

IF ONE OF THE EXPLANATORY VARIABLES in a linear regression model is measured with error, the ordinary least squares estimator is known to be biased and inconsistent. Given suitable assumptions, an instrumental variables estimator is known to be consistent. In a large sample, the instrumental variables estimator is thus unambiguously preferred, but the choice of an estimator in a small sample remains a puzzle. The method of maximum likelihood sheds light on this puzzle. It will be shown below that the instrumental variables estimate is the maximum likelihood estimate if, and only if, it lies between the ordinary least squares estimate and the reverse least squares estimate, that is, if and only if it satisfies the bounds implied by the simple errors in variables model. The letters Y, x, and z will indicate, respectively, the vector of observations of the dependent variable, the vector of error-ridden measurements of the explanatory variable and the vector of observations of an instrumental variable, each measured around its mean. The ordinary least squares estimate is then

In this paper we are concerned with the following question: in any economy with several public goods, what are the conditions under which the conventional optimality rule of equality between the sum of marginal rates of substitution and the marginal rate of transformation still holds even in the presence of distortionary taxation?Two cases are considered.In the first case, the taxes may be arbitrary.In the second case, the taxes are optimally chosen.