[Experimental studies have shown that the key behavioral assumption of expected utility theory, the so-called "independence axiom," tends to be systematically violated in practice. Such findings would lead us to question the empirical relevance of the large body of literature on the behavior of economic agents under uncertainty which uses expected utility analysis. The first purpose of this paper is to demonstrate that the basic concepts, tools, and results of expected utility analysis do not depend on the independence axiom, but may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions. The second purpose of the paper is to show that this approach may be used to construct a simple model of preferences which ties together a wide body of observed behavior toward risk, including the Friedman-Savage and Markowitz observations, and both the Allais and St. Petersburg Paradoxes.]

Malthusian theory asserts that high population growth in poor countries retards economic development. 2 authors Simon and Pilarski have recently challenged this theory on the basis of their own research; it is their contention that previous studies of population growth and education (1 of the principal modes by which the Malthusian mechanism is assumed to operate) have been flawed. The defect is felt to lie in the assumption of simple rather than partial associations of the 2 variables education and population growth. On the basis of their tightened specifications they have found demographic variables to be nonsignificant in explaining educational expenditures per child once other relevant regressors have been introduced. Similarly little effect of the demographic variable on primary and tertiary school enrollment rates were found although an effect for secondary school enrollment rates was found. On examination of the theoretical framework and the several explanatory variables however the author concludes that a crucial variable which is sure to have a vital bearing on the conclusions reached has been misspecified in the equations. Further the author asserts that a more thorough specification of the tests of the hypothesis (i.e. that high population growth retards educational investment) actually provides additional support for the Malthusian conclusion on educational expenditures despite unpromising initial indications.

URRENT research interest in school finance stems from reforms aimed at narrowing the range of expenditure variations among school districts. These reforms have attempted to compensate for tax base differences by providing state matching aid inversely proportional to tax base.' While expressed in different forms in different states and variously termed Percentage Equalization,' District Power Equalization (DPE), or Guaranteed Tax Base (GTB), these formulas, in their basic form, amount to the state guaranteeing all districts the same tax base per pupil, call it v*. Districts taxing themselves at a tax rate, r (adjusted at the state level to compensate for assessment variations so that r reflects the rate on true market value), are guaranteed revenue equal to rv*. The difference between what is raised locally in a district, rv, and the guarantee at that tax rate is provided through the state aid formula.2 In practice, legislative limitations on aid formulas in most states have led to richer districts maintaining a tax base advantage. For example, districts with a tax base above v* can raise more revenue at the same tax rate than districts at vor below. Also, placing limits on reimbursable expenses (e.g., not to exceed E* = r*v*) means that beyond a certain point, even in districts with tax bases below v*, raising tax rates will not engender any additional state matching aid, as would be required under a strict guarantee. Hence, while the introduction of GTB formulas provided a theoretical improvement, these limits made them equivalent to the foundation aid formulas they were intended to replace, effective improvement coming only when the limits were made more generous.3 In studying these reforms, research has concentrated on models relating current operating expenditures to the following: tax base, some price term reflecting the local share in the matching formula, block grants, and demographic variables. The most popular form has been the single equation log-linear model4 where coefficients are elasticities, which allows for easy comparability of the effects of the independent variables. However, the assumption of constant elasticities and the lack of interaction terms can be misleading in capturing behavioral responses and in making projections. This may be especially true when one is using data from less than equalized systems, where districts are operating in a tax and expenditure range different from what would occur under a fully (tax base) equalized system and when there are different responses to the state aid formula depending upon relative tax burden and educational need. For example, the expenditure response to state aid in poorer districts may be high for small increments in state aid, but, as district expenditures move beyond minimal requirements, increases in state aid may go increasingly into tax relief, depending upon the tax burden. The richer districts, in terms of property, generally face a price of one, i.e., no matching aid at the margin. However, most legislation guarantees some minimal funding for these districts. In order to make aid formulas fully effective, either poorer disReceived for publication October 1, 1981. Revision accepted for publication April 16, 1982. * The American University. This work was supported by National Science Foundation Grant Number SES-8013080. I am indebted to Anthony Boardman for getting me interested in the topic, to Robert Summers, Anita Summers, Janet Pack and Ralph Ginsberg for moral and intellectual support, to numerous people in the Michigan state government, especially Bob Witte and Bob Bosscher, for their expertise and cooperation, and to the School of Public and Urban Policy, University of Pennsylvania, for its encouragement of this research. I Usually property per pupil, which we denote by v. 2 I.e., state aid per pupil is r(v* v). The local share, or net price that the district pays per dollar of educational expenditure, denoted a, is a = v/P*. Under GTB, districts control expenditure levels through their choice of tax rate. 3For a detailed analysis of aid formulas see Reilly (1982). 4 Initially postulated by Feldstein (1975). Park and Carroll of Rand (1979), Black et al. (1979, 1980), and Johnson and Collins (1978, 1979) also use the same model. Other models of local expenditures have been used by Akin and Auten (1976), Barro (1972), Gatti and Tashman (1976, 1978), Grubb and Michelson (1974), Inman (1971, 1978), Ladd (1975), Lovell (1978), Slack (1980), Stem (1973), Welch (1981), and Wentzler (1980).

/-3) + f32X2'+ . . + 8,/Xk k + E (2) can be specified and estimated. On occasion, neither a priori reasoning nor theory clearly dictate the correct functional form (transformation) which an additive model should assume. With the Box-Cox transformation, the functional form is dictated by the parameters, Xi, which are themselves estimated. Note that if X1 = 1 in (2), then y(l) enters the equation linearly; also, y enters (2) as In y, and y(-1) enters (2) as the reciprocal of y. Thus the estimation procedure itself chooses the transformations which best fit the data. Furthermore, hypothesis tests can be made on the estimated Ai in order to determine if alternative functional forms (transformations) are also consistent with the data. (See the appendix for a note on discriminating between functional forms.) Estimation of (2) requires the maximization of a nonlinear likelihood function which can be extremely complicated. Since computer programs for maximizing such complex functions may not be readily available, the estimation of generalized functional forms such as (2) may be impeded. It may not be generally recognized that estimation of the parameters of (2) can be accomplished in at least four different ways. This paper will look at four alternative ways of estimating the parameters f3j, Xi and o-2. Each approach can be made to yield identical parameter estimates, and identical estimates of the covariance matrix of the parameter estimates. In section II, the general problem will be addressed and the likelihood function derived. Section III will look at each estimation approach. Section IV will conclude the paper. Problems of estimation only are dealt with in this paper. Furthermore, the approximate normality of the error terms is assumed throughout. For a discussion of estimation methodology when the error terms are truncated normal, see Poirier (1978). For a discussion of the interpretation of estimated coefficients in Box-Cox models, see Poirier and Melino (1978) or Huang and Kelingos (1979).