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When creating indices intended for use in cash settlement of futures contracts (or perpetual claims or options, or swaps, or other over‐the‐counter forward contracts or retail insurance contracts), it is critical that that each index represents value associated with a standard claim on future income (or services). The contract settlement must reflect the price of claims on income streams, so that the market can be used to hedge the risk associated with the claims, but the problem is that the available observations on prices or incomes may apply to dissimilar claims, and that standardization in the indices used to settle contracts is essential to liquidity in these markets. This chapter first reviews some existing index number methods, and then extends these methods to deal with the problems described. Chain index and hedonic index number methods are reviewed, and ordinary repeated‐measures indices (like the repeat sales indices) are shown to be in a sense a special case of these, and to have strong parallels to some existing indices used to settle contracts. The last part of the chapter introduces the hedonic repeated‐measures index to allow for control of changing price of quality variables, while retaining the repeated‐measures design.

Most published economic indices are revised after they are first published—information does not come in all at once, and timely publication dictates that the preliminary index numbers be later revised. The repeated‐measures indices developed in the preceding chapters are vulnerable to revisions after especially long intervals of time, since they have the property that, unless the repeated measures come sequentially (which they do not necessarily), there will be revisions in the indices after the index numbers are first produced, even if the raw data used then were perfectly accurate and complete. There are other index number construction methods, such as ordinary‐least‐squares regression‐per‐period hedonic regressions, that do not normally produce revisions; this would at first seem to be an advantage, but an index number construction method that does not produce revisions is not a virtue if new information tends to arrive that implies revisions and is just ignored. This chapter addresses the whole problem of index number revisions in the following sections: Variance components in regression‐per‐period hedonics; Interval‐linked indices; Indices that are derived by conditioning on lagged index values. The final section of the chapter draws some sort of interpretation of what has gone before.

This chapter addresses the fact that creating index numbers for settlement of contracts requires some judgement, and that no single method is likely to be applicable to all circumstances—there are trade‐offs, and choices have to be made with limited information. Before applying a repeated‐measures method like the ones defined in the preceding chapter, a decision has to be made as to whether there are enough repeated measures to ensure that the standard errors are not going to be too high, and whether there is enough unmeasured quality variation across subjects to warrant the increase in error variances caused by the addition of many subject dummies. A choice has to be made as to which kinds of hedonic variables, if any, to include in the analysis, and not all quality measures are appropriate for index number construction, so a choice needs to be made as to whether these variables or the subject dummies are to be constrained in any of various ways. Prior information of an imprecise nature may be used to put probabilistic, rather than rigid, restrictions on the regression coefficients. There are also some fundamentally different variants of the hedonic repeated‐measures regression methods that could be considered, methods in which quality is inferred as an observed factor associated with each subject (factor analytic methods), and methods in which a separate selection equation is used to correct for possible selection bias in the mechanism by which it is determined which subjects are to be measured (selection bias correction methods).

This chapter examines the relatively neglected issue of estimating sample variance in the context of constructing exact index numbers. Published index numbers are rarely accompanied by an indicator of variability, and it is thus difficult to assess whether a new estimate is significantly different from the previous one. The chapter applies the analysis to the specific case of total factor productivity (TFP) indexes, and examines the question of whether TFP growth in Singapore has been negative or positive, which has become an issue of considerable controversy.

This chapter reviews old and recent arguments for and against monetary aggregates as social welfare indicators. It is organized as follows. Sections 3.1 and 3.2 examine two different revealed-preference arguments that can be used to link the sign of welfare variations with the evolution of total consumption at market prices. Section 3.3 reviews the theory of index numbers, which has the more ambitious goal of providing cardinal measures of welfare. Section 3.4 focuses on the more modest, but perhaps more promising idea of decomposing social welfare into factors separately representing efficiency and equity. Section 3.5 introduces the problem of imputing prices for nonmarketed goods and computing a “full income.” The chapter concludes that economic theory does not provide much support for relying on the market value of total consumption as a proxy for social welfare.

The price index, a pervasive and old institution of economics, is a number issued by the Statistical Office that should tell anyone the ratio of costs for maintaining a given standard of living in two periods where prices differ. This picturesque indication amounts to a definition that is the basis for our approach. An immediate consequence by simple argument is the ‘New Formula’, where price indices are determined as ratios of numbers with the role of price levels that are solutions of a system of inequalities. Work on the approach continues and the system of inequalities is a reminder of a half century of experience including the present chapter.

The proposal that demand is governed by utility leads to the question of how to arrive at the utility, from evidence provided by the demand. For while demand is, in principle, directly observable, utility is not; it is just discussed, usually for its part in an explanation of demand. The question became a basis for mathematical questions making an enlargement of ‘mathematical economics’ where a host have contributed a variety of fragments. This chapter has to deal specifically with two construction topics, involving a demand function, or a finite demand correspondence. Pareto (1901) touched the utility construction question, dealing with a demand function, and remarked it could be done by solving certain partial differential equations. These referred to the inverse of the function. Volterra (1906) pointed out, in a review of Pareto, that the existence of a solution was not always assured but required certain ‘integrability conditions’, such as had been provided by a theorem of Frobenius. Thereafter the question became known as the ‘integrability problem’. Antonelli (1886) provided equivalent conditions, stated in the form of a symmetry. Slutsky (1915) did the same, by differentiation of first order Lagrange conditions, and brought into view the further necessary (in fact, more than necessary) second-order conditions. The approach gained a currency after rediscovery by Hicks and Allen (1934). Besides symmetry, the Slutsky matrix was required to have a negativity condition, intermediate between its being non-positive and negative definite and different from both. That this negativity requirement found by these discoverers had to be spurious is demonstrated by ‘The Case of the Vanishing Slutsky Matrix’ found by Afriat (1972), where there is a continuously differentiable demand function that has a utility but the Slutsky coefficients all vanish identically, as would be impossible under the negativity condition of Slutsky and the others. Lionel McKenzie (1957) made an elegant departure when he identified the Slutsky coefficient matrix with the matrix of second derivatives of a utility-cost function, so necessarily both symmetric and non-positive definite. The issue of sufficiency that had always remained completely without mention was taken up and settled by Afriat (1980). Afriat (1977, 1980) dealt with the same question when the demand function is replaced by a finite demand correspondence (1956, 1960), with the result now usually referred to as ‘Afriat's Theorem’.

A first step in the examination of an act is to view it as a choice. With a choice there is the problem of accounting for the singularity of what is done which differentiates it in the variety of what might have been done. The account should involve a principle that applies uniformly to the possibilities, yet singles out the particular act, in terms of its cause, the motive, or objective. Value is a term serving that principle, with preference as a concept for its expression. Whatever the scope of the concept of choice and systematization of choice by preference, here to have attention, it remains just a model, a form of representation, a scheme that can always be applied to behaviour in some fashion, even without assured significance. The model of a choice system based on a preference order is a habit in particular for economics; even in a more general perspective, order appears as an almost invariable component in measurement, though there may be some puzzle about that. Principles concerning choice and preference are to be considered and illustrated, in particular in regard to the analysis of consumption. An approach to the theory of consumption is developed by considering a scheme of consumption data, and then questions that might be asked of that data.