RICE and quantum indexes (P, Q) are dual to each other if PQ = E where E is the expenditure index. They satisfy the weak factor reversal test.' If they share an identical weighting formula as weighted averages of price and quantity relatives, they satisfy the strong factor reversal test, that is, they are ideal. The most celebrated ideal economic index is the one associated with the name of Irving Fisher though it was discovered before him. No ideal index as simple as Fisher's has been discovered since. Log-change index numbers have become increasingly popular in recent years, particularly as an approximation to the theoretically desirable Divisia index. Theil (1973) proposed a new log-change index number that alhnost satisfies the strong factor reversal test. I derived several alternative formulas that improve in the degree of approximation (Sato, 1974b). But neither Theil nor I was able to obtain the ideal log-change index. In section II, I report its discovery. Our pessimism has proved premature. Indeed, the formula was self-evident from the very beginning -we simply failed to see it.2 There are dual dualities between economic indexes and homothetic preferences (Samuelson and Swamy, 1974). A price or quantum index is associated with a homothetic indirect or direct preference ordering. If P and Q are dual to each other, so are the direct and indirect preference orderings corresponding to them. If P and Q are ideal, the latter are not only dual but also share an identical mathematical form. They are strictly self-dual as I call them elsewhere.3 An obvious example is the association of Cobb-Douglas indexes and preferences. A less obvious example is the association of Fisher's ideal indexes and quadratic preferences. The association itself was discovered by Konuis and Buscheguence a half century ago in 1926.4 Note that homothetic quadratic preferences are self-dual. Then, what is the selfdual preference ordering that corresponds to our ideal log-change index? We shall show in section III that it is the CES function that has become so popular in the economic literature, originally discussed by Bergson (1936), rediscovered by Solow (1956), and popularized by Arrow et al. (1961). The CES function is known to be self-dual (Samuelson, 1965) and yet the economic index associated with it has eluded discovery until now. Economic indexes are useful because they apply even when underlying preferences are not homothetic.5 We shall show in section IV that the ideal log-change index corresponds to the addilog preference ordering introduced by Houthakker (1960).

A LTHOUGH there are many differences I between planned and market economies, both with respect to the locus of decisionmaking power and the institutions designed to allocate resoturces, household behavior in the two economies may be quite similar. Both Soviet and Western households determine the allocation of their time between work and leisure in response to their preferences and the real wage. Although the Soviet worker does have a limited ability to choose the number of hours he works in his primary place of employment, he can choose to increase or decrease his work effort in response to bonus schemes. He may also get a second job or allocate some time to quasi-legal, free-market activities. Once this choice is made, each household has a certain money income that it may devote to present consumption or save for future consumption. Making the assumption that Soviet households, like their Western counterparts, wish to maximize the utility derived from consumption over their lifetime, we may observe them transferring income from the present to the future, or in the reverse direction, depending upon their present and expected future money income, the current and expected future price of commodities, the interest rate, and their rate of time preference, restricted, of course, by the limited availability of consumer credit. Although the motives for saving may be similar for Soviet and Western households, the significance of saving behavior in the two contexts is quite different. A main concern of those investigating saving behavior in developed market economies has been the relationship between aggregate demand and the level of output and employment. The investigation of saving behavior in developing market economies focuses on the need for increased savings in the household sector to finance ambitious investment programs.1 In the Soviet Union the significance of household saving behavior is different from the two cases above. The resources for the planned level of investment are determined by the central planners independently of household saving decisions, and the remainder of the resources, after subtracting such items as defense and the costs of government administration, are made available for the production of public and private consumer goods. A sales tax on consumer goods is set by the planners to equate the supply of, and demand for, these goods. Thus, in theory the saving decisions of Soviet households do not influence either the total level of output or the share of output devoted to consumption and investment. The above analysis of the Soviet economy, however, is an oversimplification and ignores the impact that Soviet household saving behavior can have on the outcome of the plan. If households are saving at their desired rate, but this rate is greater than anticipated by the planners, a part of the planned output of consumer goods will not be sold and unplanned increases in inventories will result, given that prices are not responsive to excess supply. Assuming the planned level of inventories is considered optimal, these unplanned inventory increases represent a waste of resources, and when they occur they are, and ought to be, of considerable concern to Soviet planners. On the other hand, if money wages are rising rapidly and households find that they are saving at a rate faster than desired, they will attempt to buy more goods and services with the excess savings. To the extent that prices and the supply of consumer goods are completely determined by the planners, and both are unresponsive to consumer demand, households will be unsuccessful in their attempt to purchase goods and to reduce their saving rate, and may choose to reduce their work effort instead. In fact, Soviet planners do not completely determine the price and supply of consumer Received for publication September 30, 1974. Revision accepted for publication June 9, 1975. 1 There is a substantial literature on the effect of rising incomes on savings in developing economies. For a survey of this literature see Mikesell and Zinser (1973

A large number of econometric studies have focused on possibilities for capital-labor in U.S. manufacturing. The evidence, however, indicates substantial disagreement over value of of (cr). Studies based on cross-sectional data provide estimates which are quite close to unity, but time series studies generally report lower estimates. Furthermore, estimates of or seem to vary systematically with choice of functional form: regressions based on marginal product of capital relation generally produce lower estimates of cr than regressions based on marginal product of labor relation. A variety of hypotheses have been advanced to explain diversity of results, including cyclical changes in utilization of factors (Nerlove, 1967), random measurement errors (Leontief, 1964), systematic variation of input prices with product prices (Nerlove, 1967), embodied and disembodied technical change and problems in measurement of inputs (Griliches, 1967a; Hildebrand and Liu, 1965), simultaneous equations bias (Maddala and Kadane, 1966; Nerlove, 1967), serial correlation (Griliches, 1967a), and lagged adjustment (Griliches, 1967a; Lucas, 1969; Jorgenson, 1972). In general, empirical studies attempting to take account of these deficiencies have produced unsatisfactory results. Zvi Griliches, for example, finds that the labor quality variables . . . contribute little in elasticity-of-substitution context (1967a, p. 296), while R. E. Lucas, Jr. concludes that lagged adjustment hypotheses make essentially no contribution to reconciling of time series and cross-sectional evidence of substitution (1969, p. 259). In this paper we report results of a rather successful attempt to reconcile differing estimates of cr. While it may be desirable to consider separately and systematically contribution of each of above hypotheses in reconciling cr estimates, here we limit our concern to two principal areas: (1) data -we attempt to construct time series data on cost of capital services in a more detailed manner than previous researchers have, taking into account real and nominal rates of return, asset prices, depreciation, tax policies, and compositional changes in aggregate capital between equipment and structures; (2) stochastic specification -we estimate cr by a two-stage least squares (2SLS) procedure to circumvent problem of simultaneous equations bias by ordinary least squares (OLS). We then compare estimates of abased on six different functional forms, five alternative measures of rental price of capital services, and two estimation methods. Our most sobering result is that estimates of Cr are extremely sensitive to differences in measurement and data construction. In this respect we concur with Nerlove who finds that even slight variations in period or concepts tend to produce drastically different estimates of elasticity (1967, p. 58). However, with our preferred set of data we obtain OLS and 2SLS time series estimates of owhich exhibit robustness over a variety of functional forms and time periods, and are consistent with cross-sectional evidence

T is widely acknowledged that time spent in school is an important determinant of earnings. There is less agreement about the effects of expenditures per year of schooling on earnings. Most studies of the returns to schooling consider only the extensiveness (time in school) and not the intensiveness (resources invested per year) of investments.' If schools are efficient users of resources, then intensiveness of investment can be called the quality of schooling.2 Several studies have examined the effects on earnings of school quality (see Morgan and Sirageldin, Johnson and Stafford, Wachtel, and Welch) and college quality (Solmon, Wachtel), but no one has examined the trade-off between the two. In this study I provide some strong additional evidence that expenditures at both the school and college level are important determinants of earnings and examine the relative importance of expenditures at each level. Our examination of school expenditures is based on the NBER-TH sample of World War II veterans whose socio-economic background and life cycle behavior have been followed through test data collected by the Army in 1943 and subsequent surveys by Thorndike and Hagen in 1955 and the National Bureau of Economic Research (NBER) in 1969. Among the extensive background data collected by the NBER was information on schools attended and levels attained. Earnings data were collected in both the 1955 and 1969 surveys. These data follow the respondents through a large part of their life cycles as the mean age in 1969 was 47. The respondents chosen for the initial sample were all volunteers for Army pilot or navigator training qualification tests in 1943. Thus, the sample is all male and probably all white and is also drawn from the top half of the population intelligence distribution. Thus, it is not surprising that they attained high levels of education and earnings in the postwar period. School expenditure data for the pre-college level are difficult to obtain. Most previous studies have used state-wide average data which obscure a great deal of the variation in expenditures. Data for individual school districts are available but are incomplete. These data are used even though we are forced to reduce the potential size of the sample.3 About 85 % of the respondents who attended college provided the names of the colleges, which were matched with expenditure data. The sample size for the model estimated is 1,633. The availability of school expenditure data accounts for most of the reduction from an initial sample of 5,084 NBER-TH respondents. The sample was made more homogeneous by eliminating those in poor health in 1969, those with zero or nominal earnings or real earnings in excess of $75,000 in 1955 or 1969, and airplane pilots. An earnings function similar to many estimated before (see, for example, Griliches and Mason) is used to examine the effect of school expenditures. It includes background measures, labor force experience variables, as well as measures of the extensiveness and intensiveness of Received for publication August 16, 1974. Revision accepted for publication July 21, 1975. *This research has been supported by NIE Grant No. OEG 2-71-04798. The author is grateful to Moshe Ben Horim for research assistance. This paper has not undergone the National Bureau of Economic Research review procedures. 1 The terminology is introduced by Leibowitz. Estimates of the rate of return to investments in higher education which measure both theextensive and intensive costs are found in Leibowitz (1974) and Wachtel (1975). 2Although the profit motive is absent, there are incentives for the efficient use of resources by school administrators that suggest that expenditure levels are a reasonable quality index. A related problem is that expenditure differences reflect regional or other variations in the cost of inputs and not differences in the amount of resources used. This is true for some inputs (e.g., land on which schools are built) but it is not true for most resources. 3 About 80% of the respondents to the 1969 survey provided the name and location of their high school. About half of these responses could be matched with available data on expenditures by school district. Missing data are due to incomplete information provided by the respondents and incomplete data available from the Office of Education. In addition, no data were available for those who attended private high schools, some 8% of the sample

Norman D. Aitken, Robert S. Obutelewicz, A Cross-sectional Study of EEC Trade with the Association of African Countries, The Review of Economics and Statistics, Vol. 58, No. 4 (Nov., 1976), pp. 425-433

IN this paper an attempt is made to ascertain the determinants of the demand for international reserves by the monetary authorities of the less developed countries (LDCs).' An important aim of the paper is to improve on the specifications of the demand-for-reserve functions that exist in the literature.2 The model presented improves on existing ones in many respects, including (i) the concepts of actual and optimum reserves are differentiated and the relationship between them rigorously specified; (ii) the determinants of the optimum level of international reserves are obtained by maximizing an intertemporal, stochastic macroeconomic model;3 (iii) an expected export earnings variable is estimated and used as an explanatory variable; and (iv) distributed lag adjustment is introduced. In the next section a modified partial adjustment model is specified. The equation to be estimated has a second-order lag scheme. In section III the estimation methods and data sources are discussed. Estimation of the demand function is done in two steps: first, export earnings functions are estimated for the twentynine LDCs in the sample during 1950-1969. From these, we obtain two variables expected export earnings and an export instability index to use with others in the cross-country equation. Next, a country cross-section regression analysis is carried out for 1970. Steadystate elasticities are calculated and the results are discussed

Direct and Cross Demand Elasticities in a Model with Many Sectors, Econometrica (Apr. 1959). Goldberger, A. S., Econometric Theory (New York: John Wiley & Sons, Inc., 1964). Jorgenson, D. W., Lecture Notes: Economics 241 Part IV: The Multivariate Structural Model (Berkeley, California: Committee on Econometrics and Mathematical Economics, Institute of Business and Economic Research, University of California at Berkeley, 1962). Lemke, C. E., A Method of Solution for Quadratic Programs, Management Science (Aug. 1962), 442-453. Mangasarian, 0. L., Duality in Nonlinear Programming, Quarterly Journal of Applied Mathematics 20 (1962), 300-302.

A great deal of research has been done in recent years on the effects of capital inflows on real investment, saving and output growth in developing countries.' It appears that in many instances these inflows have, in effect, been used for financing an expansion of consumption rather than investment. This note attempts to reinterpret the new research in the broader context of intertemporal resource allocation and intertemporal consumption opportunities. The acceptance of foreign capital, of course, augments the total amount of resources potentially available for current domestic real expenditure (consumption and investment). The invested portion of the foreign capital receipts adds to future potential net domestic product, but part of the domestic product increase may be absorbed by the associated debt service, income remittances and capital repatriations. If the domestic rate of return exceeds the cost of foreign capital, any one of the following three types of capital inflow response is possible: 1) both current and future consumption increases; 2) current consumption increases, while future consumption decreases, or 3) current consumption decreases, while future consumption increases. Crude empirical estimates of the true responses, based on the new research, are now possible.

THE recent focus of radical economists such as Thomas Weisskopf (1972) and Keith Griffin (1970) on the possible reduction in domestic savings caused by aid inflow has raised an important issue: How much does this matter?' A characteristic answer by orthodox economists would be that increased current consumption is also welfare-improving. Hence, the evidence of reduced domestic savings following on the influx of foreign capital -or, what is the same thing, the evidence that aid is only partially used for investment -may be dismissed as interesting but irrelevant to the discussion of the benefits of aid programmes.2 But, while the radical economists have not systematically spelled out an argument to sustain the thesis that a reduction of domestic savings by foreign capital is harmful, it is clear that their concern arises from the notion of dependence: in particular, that reduced savings would somehow increase on the aid-giving country (which is likely to be part of the imperialist or the social-revisionist bloc). This dependence argument can indeed be formalised. Take a typical Harrod-Domar type model and define the capital-recipient country's objective as reaching a Millikan-RosensteinRodan-Rostow self-reliant growth rate by raising its domestic savings rate to a target level. The radical case can then be constructed by examining whether, in reducing domestic savings, an influx of foreign capital postpones, or renders infeasible, the reaching of self-reliance. (Needless to say, dependence can only be one dimension out of many that would enter a complete social welfare function; but it is certainly one that has to be formalised, as here, before it can be usefully discussed.) Two things are clear as soon as the problem of dependence is defined in this way. First, whether capital inflow creates will depend on the assumed parameters of the model as well as the targeted level of the savings rate and the time by which it must be achieved. Contrary to the radical notions, an aid programme may achieve a targeted increase in the savings rate earlier than in the absence of aid, or may make an infeasible target a feasible one. Second, it is therefore useful to take estimates of the parameters involved and to examine simulation runs to see whether the radical concerns are worth bothering about. It is our intention to derive the logical implications of such savings behavior in a dynamic framework to see where it can lead. This paper constructs a simple version of the Harrod-Domar model and discusses the simulation runs of savings and the savings ratio, with and without aid, for a number of less developed countries (LDCs). These countries are those for which Weisskopf (1972) has fitted savings functions, using time series analysis, so that we have had to add only plausible Received for publication March 19, 1975. Revision accepted for publication November 10, 1975. * The research underlying this paper was financed by the National Science Foundation. The facilities provided by the Institute for International Economic Studies, Stockholm, are also gratefully acknowledged. A companion paper by Bhagwati and Grinols (1975) which examines a different argument linking foreign capital inflow to and hence to the feasibility of transition to socialism has been published separately in the Journal of Development Economics. 1 The precise line of arguments developed below may be considered to be implicit in the concerns and writings of the radical economists, though we have not seen them carefully developed and stated. The focus in many of the radical writings is rather on the inadequacy of the early aid-requirements estimates, where the analyst assumed a Harrod-Domar model and a realistic capital-output ratio, fixed a target rate of growth to get a target rate of investment, then made a Keynesian savings assumption and came out with the estimate of aid or capital inflow required to fill the gap between the required investment and the available domestic savings. This method, along with alternative approaches, is reviewed in Bhagwati (1971). If the aid inflow itself affects domestic savings, the model is clearly specified incorrectly. To be fair, however, to the economists (such as Rosenstein-Rodan) who used the approach being criticised, they thought of the Keynesian domestic savings function as one which the economy would adhere to (via tax effort, for example) as part of its matching effort while receiving the capital inflow, so that it was a policy function rather than a behavioral function as implied by the radical writings. 2 Following this line of argument, the radicals should focus on the distribution of benefits from additional consumption instead of on whether such additional consumption follows on the capital inflow