[It is shown that single-peaked, continuous utility functions do not, in general,aggregate in majority voting to a continuous social utility function. Conditions on individual preference orderings to guarantee a continuous social utility function are presented for a simple case.]

where y(t) and x(t) are vectors of G components, z(t) has K components, B is a G x G matrix, and F is a G x K matrix.' The components of z(t) are predetermined at time t and the vector x(t) is not serially correlated, so that (among other requirements) we have

A theory of identification is developed for general stochastic model whose probability law is determined by finite number of parameters. It is shown under weak regularity conditions that local identifiability of the unknown parameter vector is equivalent to nonsingularity of the information matrix. The use of reduced-form parameters to establish identifiability is also analyzed. The general results are applied to the familiar problem of determining whether the coefficients of system of linear simultaneous equations are identifiable. THE IDENTIFICATION PROBLEM concerns the possibility of drawing inferences from observed samples to an underlying theoretical An important part of econometric theory involves the derivation of conditions under which given structure will be identifiable. The basic results for linear simultaneous equation systems under linear parameter constraints were given by Koopmans and Rubin [10] in 1950. Extensions to nonlinear systems and nonlinear constraints were made by Wald [15], Fisher [4, 5, 6], and others. A summary of these results can be found in Fisher's comprehensive study [7]. The identification problem has also been thoroughly analyzed in the context of the classical single-equation errors-in-variables model. The basic papers here are by Neyman [12] and Reiers0l [13]. Most of this previous work on the identification problem has emphasized the special features of the particular model being examined. This has tended to obscure the fact that the problem of structural identification is very general one. It is not restricted to simultaneous-equation or errors-in-variables models. As Koopmans and Reiers0l [9] emphasize, the identification problem is a general and fundamental problem arising, in many fields of inquiry, as concomitant of the scientific procedure that postulates the existence of structure. In their important paper Koopmans and Reiers0l define the basic characteristics of the general identification problem. In the present paper we shall, in the case of general parametric model, derive some identifiability criteria. These criteria include the standard rank conditions for linear models as special cases. Our approach is based in part on the information matrix of classical mathematical statistics. Since this matrix is measure of the amount of information about the unknown parameters available in the sample, it is not surprising that it should be related to identification. For lack of identification is simply the lack of sufficient information to distinguish between alternative structures. The following results make this relationship more precise.2

linear production function, that for some utility functions the optimal initial consumption in the random case decreases for all values of initial wealth as compared with the initial consumption in the deterministic case. For other utility functions the optimal consumption always increases. Hence it seems, from these examples, that two divergent forces are at work. The first is the desire to consume more initially as a hedge against the uncertain future. The second force is the desire to consume less initially so as to increase the future consumption prospects. (It is assumed, of course, that increased inputs increase outputs for all possible random events, or states of the world). The relative strength of each of these forces, as implied by the utility function, is the key to the relationship between random consumption and deterministic consumption in this model. The major conclusion of this paper is that the qualitative difference between optimal consumption decisions in the two different models is very strongly influenced by the shape of the utility function. In particular the third derivative of the utility function plays a rather large role. It is this derivative that determines the attitude toward the skewness of a distribution in the theory of portfolio choices, as may be seen from the analysis of Pratt [7] and Tobin [10]. Even in these models, however, the third derivative cannot be ignored, since ignoring skewness distorts the results. Moreover, there does not seem to be any intuitive economic reason to make any assumptions concerning the third derivative of the utility function. The extent to which the utility function influences savings and consumption decisions is exhibited in a precise manner. It may be shown that the qualitative relationship between random and deterministic consumption depends in general on the initial wealth. It is not true, as one would infer from the papers cited above, that random consumption is always either greater than or less than deterministic consumption independently of the initial wealth. In other words, for many utility functions the initial wealth turns out to be a decisive factor in the qualitative relationship between the random and deterministic case. Naturally this relationship will also normally depend on -the probabilistic structure of the model. The key result of this paper is a theorem which gives a necessary and sufficient condition for determining the qualitative relationship between random consumption and deterministic consumption. This condition, which is both necessary and sufficient, is in a particularly simple form in that it depends only on the known parameters of the model (i.e., the production function, the utility function, and the distribution of the random variable) and also on the optimal deterministic policy which, in general, is much simpler to exhibit than its counterpart in the random case.

In two stage least squares estimation, when the matrix Z of predetermined variables has less than full column rank, Z'Z is singular. It is shown that in this situation 2SLS still can be used and that Y-the result of the first stage-is still uniquely determined; also shown is that 2SLS will often be equivalent to OLS. IN A LARGE SYSTEM containing many equations and many predetermined variables, it is a common occurrence to find that the second moment matrix involving all of the predetermined variables in the system is singular. It seems to be a widely held opinion among econometricians that such singularity prevents estimation of a structural equation by standard methods employing all of the predetermined variables. For example, with specific reference to the method of two stage least squares, we find in two outstanding texts the following statements: