Kravis, Heston, and Summers provided comparable price and volume data for gross domestic product and its components for each of 34 countries, varying in affluence from India and Malawi to the United States. The article discusses the data and the way in which they are generated, and considers the use of these data for applied demand analysis.

A firm minimizes cost or maximizes profit subject to the constraint implied by a production function. Demand equations for inputs, formulated in terms of changes over time, are described in two steps. The first is the total input decision, which describes the Divisia input volume index in terms of the change in output (for cost minimization) or price changes (for profit maximization). The second is the input allocation decision, which describes the changes in the demand for the individual inputs in terms of the Divisia input volume index and the input price changes. It is shown that the input allocation decision allows a simple transformation so that (1) the change in the demand for each transformed input is independent of the changes in the relative prices of all others and (2) the log-change in output is the sum of certain components, each representing the contribution of one transformed input, in such a way that the interaction of these inputs is confined to terms of the third order of smallness.

[A value share is defined as the ratio of the expenditure of the ith commodity to total expenditure. Some value shares go up and others go down in successive periods, and in this article the problem is raised whether one can define transitions from the ith share to the jth in an attractive way. The procedure is applied to Dutch data in the period 1921-1963.]

The logarithmic discrepancy of the factor reversal test, to be written DA for (2) and DB for (3), is of the fifth order of smallness (05), but the leading 05 term of DB is closer to zero than that of DA, which is the reason why Sato prefers (3). If the expenditure shares do not change very much, xiyi being close to zero for each i, the discrepancy D is very close to zero for most reasonable choices of f ( ). This applies to the vast majority of annual time series. Tornqvist's choice of the arithmetic average of x and y is then adequate; it has the additional advantage that the sum over i of f(xi, yi) equals 1, which simplifies the index formulas. My own interest in pursuing this topic further was stimulated by C. M. Walsh's proposal to specify f ( ) as the geometric mean of x and y. It can be shown that the leading 03 term of the discrepancy (1) for the geometric mean is equal to minus onehalf of the leading 03 term of that of the arithmetic mean. My immediate reaction was to take the weighted arithmetic mean, with weights equal to 2/3 and 1/3, of the geometric mean and the arithmetic mean so as to eliminate the leading 03 term. This is precisely Sato's choice (3). But I rejected this choice, because D is unbounded under the specification (3) for 0 -O x, y 1. Suppose that x 0, y > 0, so that fB(X, y) 1/6(y) > 0. If this applies to some subscript i, the discrepancy (1) becomes infinitely large. Sato

The article discusses the accounting basis of increased expenditures. The article discusses a situation where, in a business enterprise, a budget of planned expenditures was made before the start of this year, but it appears that many of the actual expenditure figures exceed those of the budget rather substantially. The immediate question is: What are the causes and where in particular are the increases very large? It would be of some help to top management to have at its disposal a tool by which it can "see" the development directly, in the desired degree of detail, even if the picture is one-sided in the sense that it does not take those special circumstances into account which ought to be incorporated when the final verdict is given. Basically, the approach is to take a proportional development of the individual expenditure items as a norm and to formulate measures for the extent to which these items deviate from the proportional development. The approach described in this article is not confined to the analysis of expenditures. It can also be applied to sales by regions and by districts within regions; in fact, to any hierarchy of decompositions of some total into nonnegative parts. The analysis can be further extended to budgets in two dimensions; for example, if one desires to separate wages and salaries from other expenditure categories.

Kravis, Heston, and Summers provided comparable price and volume data for gross domestic product and its components for each of 34 countries, varying in affluence from India and Malawi to the United States. The article discusses the data and the way in which they are generated, and considers the use of these data for applied demand analysis.