I. Consistent aggregation of a multisectoral economy to a two-sector model, 106. — II. The Sato technology frontier, 110. — III. The neoclassical model, 111. — IV. Implications and extensions, 112. — Appendix: a simple empirical test for reswitching, 114.

W ITH the pathbreaking article by Arrow, Al Chenery, Minhas and Solow [1] introducing the constant elasticity of substitution (CES) production function, interest in production theory has multiplied. No longer is the Cobb-Douglas function the workhorse for neoclassical theory; rather, the role of the production function has been examined anew in the theory of the firm, in growth theory, and in the theory of international trade. This re-examination has taken the form (1) of theoretical analysis of the role of the elasticity of substitution, (2) of empirical estimation of the elasticity, and (3) of the introduction of new forms for the production relation, e.g., Zellner and Revankar [14] and Revankar [8]. Unfortunately, knowledge about the appropriate micro-economic or macro-economic production function seems further away now than before 1961, the year of the ACMS article [1]. Nerlove [7, p. 58] reports that even slight variations in the period or concepts tend to produce drastically different estimates of the elasticity [of substitution] and he presents a summary of empirical studies of the CES production function to support his conclusion. In this paper it is first shown that Nerlove's conclusion on the definition of time periods is an inappropriate interpretation of previous estimates (however, part of the difficulty was outside Nerlove's control); that in fact changes in period do not produce significantly different estimates of the elasticity. To demonstrate this contention, estimates of the elasticity from the factor demand equation for labor for two consecutive years are obtained. However, evidence of serial correlation leads to a correction of the labor and wage rate variables for quality variation in the workers over states. Since this correction does not remove the serial correlation, the estimation is then undertaken using the efficient estimation technique of Zellner [12] and the results suggest that use of different time periods does not produce different estimates of the elasticity. Second, the estimates of the elasticity are constrained to be equal for the two years and the efficient estimation techniques are again used with the labor quality correction included. A test on the null hypothesis that the elasticity of substitution equals one for each industry indicates that the elasticity does not in general depart significantly from one. This conclusion from estimates of the labor demand equation supports a similar conclusion of Griliches [3, p. 292] based upon least squares regressions for two-digit manufacturing industries. However, he uses a smaller sample (only 1958 data) than here, and his estimates are biased toward one because labor quality variation was not included in his two-digit industry estimates (he only considers labor quality variation in his estimates for manufacturing as a whole). Third, direct estimates of the CES production function are obtained for each industry using Kmenta's approximation [5]. Again, using efficient estimation and correcting for labor quality differences across states, the elasticity of substitution is not in general significantly different from one. It is incidentally shown that returns to scale can be accepted as being equal to unity for most industries.

There are two primary questions confronting program planners of educational facilities: 1) At what is the minimum cost per pupil achieved, i.e., what is the of school? 5 2) Ascertaining if economies of exist and the magnitude of such economics, i.e., at what rate does expenditure per pupil decrease as increases? Table 1 provides the results of expenditures per pupil regressed on selected independent variables. Use of the regression equation to find the net relationship between cost and is as follows for the different school sizes: 1) Change in cost from 200 to 500 pupils .0503 (500) + .00001121 (500)2 .0503 (200) + .00001121 (200)2 $12.74. 2) Change in cost from 500 to 1000 students is -$16.74. 3) Change in cost from 1,000 to 1,500 students is -$11.14 per pupil. 4) Change in cost from 1,500 to 2,000 students is $5.53. 5) Change in cost from 2,000 to 2,244 is -$.66 per pupil. In other words, the total net effect of increasing school from 200 to 2,244 students would result in a savings of approximately $47 per pupil. Riew, on the other hand, found the total savings would amount to well over $200 per pupil from a school with 200 to 1,675 students. A possible explanation for such a large discrepancy in the rate of decrease in per pupil expenditures is that Riew excluded transportation costs from his expenditure variable. The results of this analysis regarding the net relationship between current expenditures per pupil and of school provides guidelines regarding the added costs incurred if the optimum size school was not provided. Note that most of the economies had been realized as student numbers approached 1500. 5 DY/DX8 =-.0503 + 2 (.00001121) X8. Setting DY/DX8 equal to zero and solving for X8 yields 2,244 students. TABLE 1. REGRESSION RESULTS OF CURRENT PER PUPIL EXPENDITURES REGRESSED ON SELECTED INDEPENDENT VARIABLES MISSOURI SCHOOL SYSTEMS, 1966