Tables of critical points for the noncentral F are presented with noncentrality equal to 1/2 of numerator degrees of freedom for denominator degrees of freedom of 1-30, 40, 60, 120, 200, 400, and 1,000, and numerator degrees of freedom of 1-30, 40, 60, 120, and 200, and type one errors of 0.05, 0.10, 0.25, and 0.50. These critical points can be used to test the second weak MSE criterion discussed in the companion paper [2]. An approximation is suggested for noncentral F(θ), and accuracy checks are given. An appendix provides a Fortran function for the approximation. The approximation is intended for using the first weak MSE test discussed in the companion paper.

Charles T. Nisbet, Firouz Vakil, Some Estimates of Price and Expenditure Elasticities of Demand for Marijuana Among U.C.L.A. Students, The Review of Economics and Statistics, Vol. 54, No. 4 (Nov., 1972), pp. 473-475

Paul Gayer and Robert Goldfarb (G-G) are certainly correct when they claim that job qualifications are imperfect predictors, not accurate measures of the potential productivity of prospective workers. Their statement that my neglect of this distinction and the learning asymmetry, namely that employers have the opportunity to learn about the true productivity of any employee but not about the productivity of any rejected applicant, are responsible for my conclusion that there is no long-term tradeoff between unemployment and inflation may also be correct. However, G-G confuse the issue by appealing for the relevance of institutional and organizational on wages; this argument is not needed to make their case. In my opinion, they have also overlooked potentially important contributions to our understanding of unemployment, underemployment, and aggregate labor productivity. In this reply I attempt to clarify their argument on the Phillips curve issue and to point out some of these other contributions. To see clearly the implications of the G-G hypothesis for the Phillips curve, we must contrast the equilibrium situation implied by the model analyzed in my paper with that implied by the model as they amended it. For the purpose of the comparison, we explicitly assume that there are no artificial constraints on the eventual adjustment of wages. Hence, in either situation, the wage structure defined as a relationship between wage offers and qualifications will adapt so that all of those willing to work, given the structure, will be able to after an appropriate period of search. This conclusion appears to be at odds with the G-G statement, elaborated in footnote 3, that wages at the lower end of the structure may have to be negative. This appearance of conflict is resolved when one recognizes the fact that participants will decide not to supply their services well before the wage offered falls to zero. In terms of classical supply and demand analysis, it is reasonable to expect, for the reasons given by G-G, that the demand curve for labor whose qualifications are sufficiently poor will intersect the wage axis at a level equal to no less than the wage at which the supply curve intersects the same axis. In other words, no worker with these qualifications is willing to work in return for either his actual net marginal product in my world or his expected net marginal product in the world of Gayer and Goldfarb. Although the characteristics of equilibrium in the two cases can be described in identical terms, the interpretations differ significantly. First, consider the case in which qualifications and productivity are equivalent. Although we may lament for those whose low productivity have forced them to decide to stay home with the kids or engage in some other nonmarket activity and we may recommend training programs and minimum incomes as solutions to their economic plight, we cannot objectively classify them as involuntarily unemployed. However, if qualifications are only related to productivity in a stochastic sense, then there will be some among those who have chosen not to work who are nevertheless productive enough to earn a wage at which they would be willing to work if the true facts were known to the appropriate employer. Their unemployment is hidden behind a veil of imperfect information and will remain so because the existing information exchange mechanisms will not reveal the truth in equilibrium. One is hard pressed to classify the situation of such workers as anything other than involuntary unemployment. What about the Phillips curve? Having * Associate professor, Northwestern University.

"Inl statistical inference proper, the model is never questioned.... The methods of mathe-matical statistics do not provide us with a means of specifying the model. " 1

The empirical literature relating crosssectional variations of per capita public spending to various economic, political and demographic factors is both lengthy and varied.' Differences in expenditures over political units are explained by differences in per capita incomes, urbanization, area, population density, taxable capacity, tax rates, absolute population size, grantsin-aid from higher levels of government, and school-aged population rates. With few exceptions,2 the models employed in these studies are ad hoc constructions with little basis in the theory of choice.3 Our aims are to posit a model of public spending derived from the received theory of collective decision making4 and to test the significance of certain variables assumed by this simple theory to be important determinants of the levels of state and local government expenditures.