[Part I of this paper introduces a general framework for the discussion of discrete production sets and the associated programming problems which arise when a particular endowment of factors is specified. In this part of the paper we shall apply these ideas to integer programming problems with two activities and bring to bear some of the basic considerations of the theory of computational complexity. The numbering of sections, figures, and equations will follow those used in Part I.]

modities (such as various consumer goods and labor), M fixed factors (such as land, natural resources and various types of fixed capital), and a government which taxes commodities and fixed factors in order to finance various govern- ment expenditures. It is well known2 that if the government can raise its required revenue by taxing the fixed factors alone, then the resulting allocation of resources is Pareto optimal-no single household's utility or real income can be increased without decreasing the utility of some other household. Suppose we are at an initial equilibrium where government revenue is being raised by taxing the fixed factors alone. Then the resulting equilibrium can be rationalized by maximizing a certain weighted sum of utility functions subject to various feasibility constraints. Now think of the government replacing the taxes on fixed factors with distortionary commodity taxes. In Section 3, we calculate the second order directional derivative of the above weighted sum of utility functions with respect to any feasible direction of tax change, evaluated at the initial equilibrium which is Pareto optimal. Of course, the first order directional derivatives of the weighted sum of utility functions with respect to feasible directions of tax change are zero evaluated at this initial equilibrium. We obtain a measure of economic due to tax distortions which is virtually identical to that of Boiteux (3, p. 113) and which bears a resemblance to the dead loss of Hotelling (22, p. 254), the consumer's surplus measures of Hicks (19; 20, pp. 330-3), and the deadweight loss measure of Harberger (16, p. 61; 17, p. 788). In Section 4, we calculate a measure of welfare based on Debreu's (4, 5) coefficient of resource utilization (which is a modification of a measure of due to Allais (1, 2)) and we show that under certain conditions, the Hotelling,

[The comparative statics of optimization models which have nonlinear constraints are examined. It is shown that most of the standard results of "linear" comparative statics still apply. However, it is also shown that individual substitution and income effects are systematically affected by the nature and degree of nonlinearity of the constraint. A model of quantity/quality trade-offs, previously examined in the literature, is analyzed, and several new results are developed.]

An important purpose in pooling time-series and cross-section data is to control for individual-specific unobservable effects which may be correlated with other explanatory variables, e.g. latent ability in measuring returns to schooling in earnings equations or managerial ability in measuring returns to scale in firm cost functions. Using instrumental variables and the time-invariant characteristics of the latent variable, we derive: 1. (1) a test for the presence of this effect and for the over-identifying restriction we use; 2. (2) necessary and sufficient conditions for identification of all the parameters in the model; and 3. (3) the asymptotically efficient instrumental variables estimator and conditions under which it differs from the within-groups estimator. We calculate efficient estimates of a wage equation from the Michigan income dynamics data which indicate substantial differences from within-groups and Balestra-Nerlove estimates — particularly a significantly higher estimate of the returns to schooling.

[If independent observations x are drawn from the distribution located at @m, f (x; @m)=c"3 exp[-g(x -@m)], and if g is symmetric and strictly convex, then the maximum likelihood estimate of μ lies between the smallest and largest folded sample observations. If the distribution has fatter tails than a normal distribution, then the maximum likelihood estimate lies between the smallest and largest means of trimmed subsamples. If the distribution is assumed to be symmetric and unimodal, the centers of tight clusters of observations can be maximum likelihood estimates. If observations are not independent, then there is no bound: given any example any number is a maximum likelihood estimate for some sampling distribution. Stationary is not sufficient to bound the estimate between the minimum and maximum observations.]

[This paper investigates the distribution of the least squares estimator of the coefficient α in the model @c"t = @a@c"t -"1 + @?"t where the @?"t where the @?"t are independently distributed N (O, @s extasciicircum2). The exact finite sample and limiting distributions are calculated when α ≥ 1 and finite sample distributions when α extless 1. These distributions are used to compute the power functions of tests of the random walk hypothesis α = 1 as well as the hypotheses.]