Knowledge that Transforms

Edgeworth series expansions are obtained of the finite sample distributions of the least squares estimator and the associated t ratio test statistic in the context of a first-order noncircular stochastic difference equation. General formulae are given for these expansions up to 0(Th1) where T is the sample size and explicit representations of these in terms of the true parameters are derived up to 0(12). Some numerical comparisons of the approximations and the exact distributions are made in the case of the least squares estimator.

A firm minimizes cost or maximizes profit subject to the constraint implied by a production function. Demand equations for inputs, formulated in terms of changes over time, are described in two steps. The first is the total input decision, which describes the Divisia input volume index in terms of the change in output (for cost minimization) or price changes (for profit maximization). The second is the input allocation decision, which describes the changes in the demand for the individual inputs in terms of the Divisia input volume index and the input price changes. It is shown that the input allocation decision allows a simple transformation so that (1) the change in the demand for each transformed input is independent of the changes in the relative prices of all others and (2) the log-change in output is the sum of certain components, each representing the contribution of one transformed input, in such a way that the interaction of these inputs is confined to terms of the third order of smallness.

["If one is willing to interpret Q̃ [the Goldberger, Nagar, Odeh reduced form coefficient covariance estimate] as a covariance matrix of the random parameter π around the constant extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi\$ extless/tex-math extgreater, rather than as a covariance matrix of the random estimates extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi\$ extless/tex-math extgreater, then using extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi\$ extless/tex-math extgreater for extlesstex-math extgreater$ extbackslashtilde\ extbackslashpi\$ extless/tex-math extgreater and Q̃ for Q extlesstex-math extgreater$[ extbackslashtilde\ extbackslashpi\$ extless/tex-math extgreater and Q are the mean and covariance matrix of the random parameter π] will provide an approximate solution to the evaluation of expectations required in our optimal control problem" [3, p. 641], italics added).]

This paper studies conditions under which aggregate demand behavior will satisfy the usual revealed preference axioms. Assuming a fixed distribution of income and the hypothesis that individual demand is homogeneous in income, it is shown that the weak axiom of revealed preference or the congruence axiom will hold in the aggregate if each individual demand satisfies the corresponding axiom. It is also shown that the hypothesis of homogeneity in income is not necessary for the weak axiom to hold in the aggregate. 1. INTRODUCrION THE PURPOSE OF THIS PAPER is to establish conditions under which aggregate demand behavior will have properties normally associated with individual demand when the distribution of income remains fixed. The best-known result of this type is that if each individual has a homogeneous concave utility function, and the distribution of income is fixed, then the aggregate demand correspondence will be one derived from a homogeneous concave utility function. This was first established by Eisenberg [4], though not in the context of demand theory, who employed duality theory of concave programming. More recently Chipman [1] interpreted Eisenberg's results from the point of view of demand theory and gave a proof of the aggregation theorem based in part on earlier work of Chipman and Moore [2 and 3]. In this paper utility functions will not be employed; instead, a revealed preference approach is taken. Strengthened forms of the weak axiom of revealed preference, the strong axiom of revealed preference, and the congruence axiom are used which are preserved in aggregation, and it is shown that demand correspondences homogeneous of degree one in income which satisfy the regular revealed preference axioms will also satisfy the strengthened versions. One advantage of this approach is that it shows the Eisenberg-Chipman aggregation theorem is a purely algebraic problem and does not require continuity or convexity assumptions. It will also be shown that there are demand functions not homogeneous of degree one in income which satisfy the strengthened form of the

BETWEEN 1940 AND 1950, Kaldor [12], Goodwin [9] and Hicks [11] showed that adequate models of business cycle have to be essentially nonlinear, as only nonlinear systems allow The study of their properties is useful besides the construction of models, for example, the stochastic stability of the system, when shocks and external perturbations occur. This problem, already complex for linear systems, becomes even more complex for nonlinear systems (Kushner [15], Astrom [5]). Klein and Preston [13] and Kosobud and O'Neil [14] obtained interesting results on stochastic stability of nonlinear models of business cycles. Another interesting aspect is the dependence of the business cycle on the parameters characterizing the system. This analysis can be a sort of framework for the control of business cycles. The of stability can provide useful tools to this end. The of stability is a fusion of the two concepts of stability and qualitative behavior in the sense of topological equivalence. Andronov and Pontriagin [3] considered differential equations in two variables in a closed domain. They said that a system X is rough if, by perturbating it slightly (in the Cl-sense), one gets a system Y equivalent to X (in the sense specified in Appendix 1). Later Lefshetz [17] translated rough to structural stable. Thom [24, 25] saw stability, broadly understood, as the preservation of qualitative features under small perturbations. Smale [21, 22], Peixoto [19], and Abraham and Robbin [1] developed the giving fundamental theorems. Sotomayor [23], Andronov et al. [4], Chafee [6], and Sattinger [20] started to give good basis for the so called theory of bifurcations. Points of bifurcation are, in a parameter space, points where the topological structure changes abruptly, that is where stability fails: the creation of limit cycles from a multiple focus (Hopf bifurcation), the creation of a closed trajectory from a multiple limit cycle .... The of bifurcation can provide new criteria to prove the existence of limit cycles, besides the classical ones of Poincare and Bendixon. For example, it is possible to prove the existence of a limit cycle, without resorting to the theorem of Bendixon-Poincare, as done by Chang and Smyth [7] or to the theorem of Levison and Smyth, as done by Ichimura [16] for the Kaldor model.

[This paper examines the estimation of a normal contemporaneous simultaneous equation model in which some of the exogenous variables are measured with error.The theory for asymptotically efficient least squares estimation is developed. The primary result is a structural least squares estimator which offers certain computational advantages relative to the full information maximum likelihood estimator.]

[This paper is concerned with the class of homogeneous programming problems. It is shown, to assure the existence of a saddlepoint for the associated Lagrangian, that some restrictions must be imposed on the degrees of homogeneity of the functions. Some properties of the perturbation function are also demonstrated.]