Knowledge that Transforms

such investors can be dichotomized into two steps.'6 First the investor picks the optimal ratios of risky securities in his portfolio. Then, he decides on the optimal amounts of this composite risky security and the certain security. latter problem is just like that already studied by Arrow,17 so it is known that in this case all the risky securities will be inferior or superior 18 as U/U' increases or decreases with increasing wealth. When the utility function is of the form, U= bZa, V is linear homogeneous of degree a. Since, in this case, the slopes of the indifference surfaces along rays from the origin will be constant, all securities (including the certain one) will change in the same proportion as net worth. 16This result has been obtained by Tobin, Sharpe, and Lintner, when the investor's preferences depend only on the mean and standard deviation. For arbitrary probability distributions this means the U must be quadratic. quadratic is a special case of the family given above. Tobin's paper and Lintner's have been cited previously. Sharpe's is to be found in W. Sharpe, Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance (Sept. 1964). 17 See the introduction and K. Arrow, Comment, on James S. Duesenberry, The Portfolio Approach to Demand for Money and Other Assets. 18 Securities are taken to be superior or inferior as the absolute value of their quantity increases or decreases with net worth.

Following the lead of Markowitz,' the portfolio selection problem has usually taken the following form. An investor's utility for his portfolio is assumed to depend only on the mean and variance of its return over the following period. The investor has in mind means and variances for the returns on all the available securities. The problem is to determine the optimal proportions of these securities in his portfolio. Markowitz, himself, was largely concerned with calculating the dominant set of portfolios (i.e., those with minimum variance for given mean and maximum mean for given variance). Later Tobin 2 studied for two securities, one risky and the other riskless, how the optimal proportion of risky security would change with the mean and variance of its return. He also showed that the mean-variance approach of Markowitz would be consistent with the expected utility principle if the investor had a quadratic utility function or if he considered only a two parameter family of probability distributions. Recently Arrow 3 has reformulated the problem studied by Tobin. He lets the investor decide how to allocate a given amount of wealth between a risky and a riskless security so as to maximize his expected utility. Among other things he shows that the amount invested in the risky security will rise or fall with increasing wealth as minus U/U' (U being the investor's utility function) falls or rises with increasing wealth. In particular, for a quadratic U the amount invested in the risky security will necessarily decrease which is unrealistic. Thus, the mean-variance approach appears unduly restrictive for the portfolio problem, unless restrictions are placed on the form of the probability distribution. The restrictions require that the returns on the securities have a multivariate distribution such that a linear combination of the returns has a two parameter distribution. The multivariate normal is an important example of such a multivariate probability distribution. However, it may not be easy to find others when the returns are not independent. In this paper, the problem of portfolio selection is formulated so that it becomes formally identical to the traditional consumer theory for certain commodities. This permits all of the well-known results of this theory to be applied to the portfolio problem. It also permits the direct use of the traditional models for the price determination of certain commodities for determining the prices of securities. Investors are assumed to follow the expected utility principle and to form their own probability beliefs.

UBLIC education is a sector of government where a usable measure of output quality, the achievement score in basic subjects, has existed in refined form for some time. Given the well-known problems in measuring public service outputs, it is surprising that test scores have not been more widely used by economists in educational research. A recent study of school districts in New York State provides information highly useful for relating educational performance to per-pupil expenditure and size of administrative unit, while at the same time making possible a reasonable attempt to isolate the important influences of pupil intelligence and socio-economic background. This paper represents the results of a careful analysis of the New York State study. The findings have been somewhat surprising. Size of school district is negatively related to performance, if at all, and expenditure is related strongly to performance only in larger school districts. Performance in small school districts, here defined as those in which there are fewer than 2,000 pupils in average daily attendance, was found to be highly unpredictable. More detailed findings are presented below after a brief discussion of the data and the simple model of the educational process used as a framework for analysis.