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T a given rate of growth in the general A level of personal income, which kind of expenditure will increase its share and which will decrease and, further, what will be the extent of these changes? The requirement for this kind of knowledge has increased in recent years in relation to the theory of economic development as well as for purposes of short-run projection. Prior to the latter half of the 1940's, it had been thought that the above questions might be readily answered by the straightforword application of the generalized Engel's Law. Professors Allen and Bowley I gave the theoretical explanation with a specified form of preference function which was found to be consistent with the empirical cross-section regressions. Thus, it was suggested that if we could assume the invariability of the consumers' preferences both among income groups and over time, it would be possible to predict the consumption of various categories of goods and services and future savings, given income and prices and with the knowledge of the numerical values of the preference parameters. This line of approach had been tried from the time of Professors I. Fisher and R. Frisch.2 It is well known, however, that the validity of the analogy between the cross-section and the time series came to be doubted with the discovery of the long-run stability of the savings ratio in the aggregate time series by Professor Kuznets. Several new theories of consumption appeared with the common target of explaining with a unified theory all of these three kinds of empirical observations: the long-run time series, the short-run time series, and the cross-sectional relations. About ten years ago, the present authors began to try to estimate the consumers' preference functions in a numerical form, making use of continuous time series of the family budget data which were available, at that time, for interwar years in Japan. We undertook this task partly because of pure curiosity and partly because of the practical desire to improve the analysis of index-number problems,3 as well as to make predictions on the simultaneous determination of family expenditures. Starting from the classical model of Professors Allen, Bowley, and A. Wald,4 and after making some tentative computations, we came to see the necessity of introducing some shift elements into the structural equations which were used. First, we tried to adopt Professor Tobin's 5 liquid asset hypothesis in a generalized form which took not only the liquid assets, but also the asset holdings for every individual category of family expenditure into account. The results of the empirical tests, however, did not show a logical consistency with our interpretation of Tobin's hypothesis. This forced us to examine the applicability of Professor Duesenberry's theory of interdependent preferences.6 * The work on this paper was done as part of the Keio Economic Research Project financially supported by the Institute of Industry & Labor Studies, Keio University. The final, considerably revised, version was prepared at Professor W. W. Leontief's Research Seminar during Tsujimura's stay at Harvard as a Fulbright Research Scholar, 1961 to 1962. We are also heavily indebted to Professors J. S. Duesenberry and H. S. Houthakker for valuable suggestions. They, of course, are not responsible for any remaining errors. 'R. G. D. Allen and A. L. Bowley, Family Expenditure: A Study of Its Variation (London, 1935), especially appendix. 2 Ragnar Frisch, New Methods of Measuring Marginal Utility (Tubingen, 1932). It is interesting to see that Frisch's analogy of the spring balance on the dynamic feature of consumer's taste appears to be very close to our present specification of Duesenberry's theory of habit formation. 'H. S. Houthakker, "The Influence of Prices and Incomes on Household Expenditures," Bulletin of the International Statistical Institute (Bruxelles, 1960); W. W. Leontief, "Composite Commodities and the Problem of Index Numbers," Econometrica, iv (1936); and P. A. Samuelson, The Foundations of Economic Analysis (Harvard, 1945). 'A. Wald, "The Approximate Determination of Indifference Surfaces by Means of Engel Curves," Econometrica, viii (April 1940). 'James Tobin, "Relative Income, Absolute Income, and Saving," in, Money Trade, and Economic Growth (Macmillan: New York, 1951). 6 J. S. Duesenberry, Income, Saving, and the Theory of Consumer Behavior (Harvard, 1949).

The traditional method of estimating Engel curve parameters uses either (recorded) income or total expenditure as an independent variable in least squares analysis. Neither of these variables is, however, a satisfactory index of the true economic position of the family. This results in biased estimates of the income elasticities of the various consumption categories. The bias can, however, be eliminated in large samples by using both income and total expenditures in the estimation procedure. This can be accomplished by applying the method of "instrumental variables" to Engel curve analysis, with recorded income serving as the instrumental variable. Having formulated consistent estimation procedures, we use empirical data to investigate and analyze the direction and size of the biases in the traditional estimates of income elasticities of various commodity groups. IT IS COMMON practice to use current income, Y, or perhaps more often, total expenditures, C, as independent variables in least squares analysis of Engel curves. Strong objections, however, have recently been raised against the applications of the least squares procedures to family budget studies. Friedman [3] argued that spending decisions are based on "permanent income," a concept which is generally inadequately represented by Y as obtained from household records. The divergence between the empirical measure of income and its theoretical counterpart leads in Friedman's model to biased estimates of the "true" parameters. The objection to the use of C as regressor has been raised and analysed by Summers [9]. The main point of Summers' contribution is that C and its components (the "dependent variables") are endogenous to the consumer and are determined simultaneously. As is well known, the classical method of least squares leads in the above circumstances to biased estimates of the "true" parameters, i.e., the parameters of the relation between the systematic parts of C and its components. In this paper we shall present a method of obtaining consistent estimates of the "true" parameters of Engel curves when C is used as an explanatory variable. Our method consists of using Y (as recorded in budget studies) as an instrumental variable which eliminates the simultaneous equations bias from the relation between C and its components. Some empirical estimates

EVER SINCE PARETO [11], Slutsky [15], and Hicks and Allen [5] made indifference surfaces pivots of demand analysis, their statistical determination has been a challenging problem. This assumes an overwhelming importance in planned economic growth and also in forecasting, where the aim is either to minimize demand pressure in the market or to have a preview of individual price movements so that repercussions on planning or forecasting may be taken into accouint. If democratic planning is conceived in the sense of Myrdal [iO] as having manipulation of price structure as its chief instrument, the lknowledge of indifference surfaces becomes a must. Most statistical studies short-onircuited the fundamental problem of determining these by concentrating on certain derived functions like demand and Engel curves.' Although they provide very useful tools in themselves, their usefulness is limited in understanding the total consumer reaction to price andi income changes. To be able to have any idea of the shape of indifference suirfaces, it seems necessary that data regarding a sufficient -n.umber of actual choices rnade under varying income and price situations be made available. Ordinarily, family budget data give the experieence of numerous income receivers at various levels of income, but all of them are faced with more or less the same price-complex prevailing at the time of survey. Therefore, although this type of data is eminently suitable for estimating Engel curves, it has not been possible to utilize it for

THIS PAPER PRESENTS, on the assumption of lognormality, a simple graphical method of deriving the Engel elasticities for various items of consumer expenditure from what are known as concentration curves. Usually two types of such curves are distinguished: (a) the Lorenz curve which relates the proportion of total expenditure to the proportion of persons spending up to a given level of total expenditure per capita, and (b) the specific concentration curve which relates the proportion of total consumption of a specific commodity to the proportion of persons spending up to a given level of total expenditure per capita. We shall briefly indicate a method of using these curves to calculate the Engel elasticities and shall present some numerical results for a few important items of consumers' expenditure. These estimates are compared with other estimates obtained by the conventional method of least squares, which requires additional computations.

The problem addressed in this article is that of finding levels of income which typify equivalent levels of poverty for families in different circumstances. An index is constructed which has, as components, descriptions of family circumstances-such as number of persons and geographical location. This index can be used to deflate family income so that it is comparable for all families. It can also be used to provide appropriately differentiated threshold values or "poverty lines" from an initial undifferentiated value, such as the often-cited $3,000 per family used by the Council of Economic Advisers. The solution tentatively proposed uses the share of income devoted to particular categories of consumption as the basis for defining equivalence; e.g., families that, on average, spend an equal fraction on necessities are taken to be equally poor. Given this proposition, one can derive the index from estimated Engel curves. The properties and suitability of alternative forms of the Engel curves are investigated, and the method is applied to data from the 1960 Survey of Consumer Expenditures.

In a recent issue of this Journal,' the author stated that it is not always possible to ' undercompensate ' arbitrary price changes, whereas in fact it is always possible under the conditions employed in that paper.2 The point is of some interest because of the use of undercompensated price changes in elementary taxation theory, in tracinlg out 'inferior Houthakker sequences '3 in revealed preference theory, and in defining lower limits to ' true' price index numbers. Let an individual's demands be q7,...,q, when the corresponding prices are p',...,p? and his total expenditure (income) is E?=0p qq Similarly, let his demands be qi,...ql at prices p'...p,t an-d inicome El =Zpql. Givein the prices, demands depend upon income, thus determining the associated income-consumption curve and Engel curves. The question is whether or not there always exist values of the ql such that

added to the "discretionary spending power" possessed by the great mass of consumers, has increased the pressure to find tools for predicting the volume and direction of consumer expenditures. Under this duress, increased attention has been given of late to the so-called "Engel's Laws." As is known, these "laws" originated nearly ioo years ago from some researches conducted by Ernst Engel, a German statistician. Engel investigated the expenditures of different income groups for broad classes of commodities. His studies led to some rough conclusions as to the effect of income on