Knowledge that Transforms

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

Maurice Wilkinson, European Migration to the United States: An Econometric Analysis of Aggregate Labor Supply and Demand, The Review of Economics and Statistics, Vol. 52, No. 3 (Aug., 1970), pp. 272-279

SOME OF THE FACTORS WHICH INFLUENCE THE COMMUTER'S CHOICE OF MODE ARE EXPLORED BY PROVIDING QUANTITATIVE ESTIMATES OF THE DEGREE OF TRANSIT IMPROVEMENT WHICH WILL BE NECESSARY TO ATTRACT COMMUTERS. THERE IS DEVELOPED AND ESTIMATED A BEHAVIORALLY-ORIENTED MODEL OR MODAL CHOICE. THE TWO MAIN RESULTS WERE: ESTIMATE OF THE VALUE OF TRAVEL TIME TO COMMUTERS (WHICH WAS ESTIMATED AS 42 PER CENT OF THE COMMUTER'S WAGE RATE); AND ESTIMATE OF THE TIME AND COST ELASTICITIES OF CHOICE BETWEEN MODES (WHICH TURNED OUT TO BE RELATIVELY SMALL). UNLESS COMFORT TURNS OUT TO BE A MUCH MORE IMPORTANT FACTOR THAN EITHER TIME OR COST, THE POSSIBILITY FOR ANY SUBSTANTIAL DIVERSION OF AUTO USERS ONTO THE PROPOSED RAPID TRANSIT SYSTEMS DOES NOT APPEAR TO BE VERY GOOD. /AUTHOR/

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.

In studies of factor productivity in the Soviet economy, as in many other economic investigations, it has been necessary to make some assumption about the degree to which capital and labor are substitutable for each other. In many, if not most, the investigators have implicitly, and often explicitly, assumed either unitary elasticity of factor substitution, i.e., the renowned Cobb-Douglas function, or infinite elasticity of substitution, the arithmetic function [2-5] [7] [10] [12] [13]. Recent analyses of the American and other economies [1] [6] [8] [9] suggest, however, that a numerical value less than one, perhaps on the order of 0.5, may be more nearly consistent with the statistical evidence. Since the elasticity of substitution may be regarded as an index of diminishing returns, and since inputs of capital and labor have been growing at highly disparate, perhaps unprecedentedly disparate, rates in the USSR, it is of particular interest to ask if the traditional assumptions are indeed suitable. Would a closer approximation to the truth be given by the estimates for other countries? In the current state of our knowledge on the Soviet Union can we shed any light on the appropriate value for the elasticity of substitution? If it could very well have been less than unity, as in fact our calculations suggest, what are the implications for quantitative estimates of the sources of Soviet growth? Recently published data on capital and labor inputs and on synthetic factor shares in income [2] [10] have made it possible to suggest some tentative answers to these questions, which is the purpose of this paper. We shall proceed as follows. In section II we analyze data on growth of labor and capital and on synthetic factor shares to obtain a range of implied values for the elasticity of substitution. In section III we use these values to obtain further implications, those pertaining to the part of growth in output explained by combined input of capital and labor and the part left to be explained by other factors. II Factor Inputs, Synthetic Factor Shares, and the Elasticity of Substitution

T HIS paper considers the major influences on the level of current expenditures on public primary and secondary education. Its purpose is to clarify their interrelation, directing attention toward those determinants that can be shown to be of major importance overall in both cross section and time series data. There are a number of empirical studies of determinants of public primary and secondary education in the literature. Among the most recent and thorough are those by Hirsch [14], Shapiro [29], Miner [23], Burkhead [7], James [18, 19] and Pryor [26] . There is, however, the lack of a structural theory in all of these studies, and this lack of a sufficiently explicit theoretical framework somewhat limits the capacity to interpret the economic meaning of the statistical results.1 A neat separation of economic and noneconomic factors is hardly possible. But it is possible to distinguish the economic framework of the problem which interrelates influences on the demand for public education, costs of producing it, and tax behavior (via tax handles or other revenue sources). It is also possible to bring to bear on the analysis of resources for education some of the developments in the broader context of public expenditure theory (e.g., Musgrave [25] and others) and the recent developments in consumption theory (e.g., Ando-Modigliani [1], Houthakker [16], and others).2 Although this is not a normative study of efficient allocation (for which see Bowles [4] for example), it can make use of certain advantages offered by education as a case study in public expenditure theory. For most school administrative units are single purpose units making quasi-independent expenditure decisions. Importance also derives from the fact that education is an extremely large industry, engaged in the furthering'of growth through human capital formation, and in the reduction of inequality. Part I considers the demand, production cost, and tax behavior structural equations. Attention is then turned to their joint solution and the reduced form public expenditure functions that are the result. The reduced forms maintain the degree of comparability to other studies that we desire for the large ones have used single equatioii methods (e.g., Miner [23], James [18, 19]) with data for a sample of individual districts. Others have focused on the relation of the aggregate expenditure-income ratio to per capita income (e.g., Musgrave [25], Pryor [26 pp. 182-226].) Part II considers empirical estimates of these reduced forms for cross-section data among states for 1955-1956, the mid-point of the postwar period, and for time series data for 1946-1968.

of reserves should grow at approximately the same rate as global wealth. A lower growth rate of reserves would impose undesirable policy constraints, while a higher growth rate would imply an inflationary bias to the world economy. Judging from the data used in this study, rates of increase of national wealth varied approximately 2.5 per cent and 7.5 per cent, with a cluster around the 5 per cent figure. If the behavior of the governmental authorities follows the pattern found in this study, we might assume that they will want to increase their stock of international reserves by approximately 5 per cent annually. Table 2 pre-

Transfers from Developed to Underdeveloped Areas, Study Week on the Econometric Approach to Development Planning, Pontificiae Academie Scientiarum Scripts Varia (Amsterdam: NorthHolland Publishing Company, 1965). [4] Hinrichs, H. H., Determinants of Government Revehue Share Among Less Developed Countries, Economic Journal (Sept. 1965), 546-556. [5] Lall, S., Note on Government Expenditure in Developing Countries, Economic Journal (June 1969), 413-416. [6] Rahman, Md. Anisur, Foreign Capital and Domestic Savings: A Test of Haavelmo's Hypothesis with Cross-Country Data, this REVIEW (Feb. 1968), 137-138. [7] Roe, A. R., The Government-Revenue Share in Poorer African Countries -A Comment, Economic Journal (June 1968), 479-481.

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A PREVIOUS paper [9] developed a model of the structure of equilibrium prices for risk assets in a purely competitive in which a set of individually risk averse investors optimize their respective portfolios of risk assets in terms of common expectations and risk assessments with respect to a common horizon. When there is a riskless asset available for holding or borrowing at a fixed interest return and all probability assessments are normal (Gaussian) it was shown that in equilibrium a purely competitive will place an aggregate value on all the outstanding stock of any company V0j equal to the discounted value at the riskless rate r* of the certainty equivalent of -the distribution of its uncertain end-of-period aggregate value. This in turn is less than the statistical expectations V1* by the product of the market price of dollar