Walter Diewert

Exact and superlative index numbers

Abstract The paper rationalizes certain functional forms for index numbers with functional forms for the underlying aggregator function. An aggregator functional form is said to be ‘flexible’ if it can provide a second order approximation to an arbitrary twice diffentiable linearly homogeneous function. An index number functional form is said to be ‘superlative’ if it is exact (i.e., consistent with) for a ‘flexible’ aggregator functional form. The paper shows that a certain family of index number formulae is exact for the ‘flexible’ quadratic mean of order r aggregator function, (Σ i Σ j a ij x i r 2 x j r 2 ) 1 r , defined by Den and others. For r equals 2, the resulting quantity index is Irving Fishers ideal index. The paper also utilizes the Malmquist quantity index in order to rationalize the Tornqvist-Theil quantity indexin the nonhomothetic case. Finally, the paper attempts to justify the Jorgenson-Griliches productivity measurement technique for the case of discrete (as opposed to continuous) data.

Duality approaches to microeconomic theory

Publisher Summary This chapter develops the duality between cost and production functions. The chapter derives the regularity conditions that a cost function C must have and shows how a production function is constructed from a given cost function. The chapter considers the duality between a (direct) production function F and the corresponding indirect production function G. Under certain regularity conditions, G can also completely describe the technology, and thus there is a duality between direct and indirect production functions. The duality theorems have two interpretations: one in the producer context and the other in the consumer context. The chapter discusses a variety of other duality theorems—that is, other methods for equivalently describing tastes or technology, either locally or globally, in the one-output, N-inputs context. The mathematical theorems presented in the chapter appear to be only theoretical results devoid of practical applications. However, this is not the case. The chapter also surveys some of the applications of the duality theorems developed earlier. These applications fall in two main categories: (1) the measurement of technology or preferences and (2) the derivation of comparative statics results.

Fisher ideal output, input, and productivity indexes revisited

A productivity index for a firm is generally defined as an output index divided by an input index. The first part of the paper uses the test or axiomatic approach to index number theory in order to determine the appropriate functional form for the output and input indexes. It is found that the Fisher ideal index satisfies 21 reasonable tests and is uniquely characterized by a subset of these tests. In the remainder of the paper, the economic approach to productivity indexes introduced by Caves, Christensen, and Diewert is adopted and, again, a strong justification for the Fisher productivity index is provided.

Afriat and Revealed Preference Theory

Suppose that we can observe a number of decisions xi (where xi is a non-negative N dimensional vector for i = 1, 2, ..., I) which some decision-making unit has made and let us further suppose that each vector of decisions xi satisfies a linear constraint of the form pTxi < 1 for i = 1, 2, ..., I where pi is a given N dimensional vector which has positive components.3 Given the above framework, we may ask the following question: is the observed set of decisions {xi} consistent with the hypothesis that the decision-maker chose decision xi (for i = 1, 2, ..., I) because it maximized a real valued function of N variables, ., subject to the constraint pTx < 1? A related question is how may we use the observed data {pi; xi} i = 1, 2, ..., I in order to construct an approximation to the decision-makers true 0, assuming that such a b exists. In Section 3 below, we will give an answer to the above two questions by using the observed data {pi; xi} to construct the coefficients of a linear programming problem [4]. If this linear programme has a positive solution, then it turns out that the observed data does not satisfy the hypothesis of consistency. If on the other hand, the objective function of the linear programme has a solution equal to zero, then we may use the solution to the linear programme to construct a real valued function 0 such that for i = 1, 2, ..., I the vector xi is a solution to the following constrained maximization problem:

A normalized quadratic semiflexible functional form

Estimation of flexible functional forms for large consumer or producer demand systems is often precluded due to computational difficulties or due to a lack of degrees of freedom. We propose the concept of a semiflexible form, which is a special case of a flexible form but which requires fewer free parameters. Our proposed method of estimation allows the researcher to choose the degree of flexibility consistent with feasibility of estimation, while at the same time maintaining the concavity in prices property required by economic theory. In our empirical illustration we estimate a normalized quadratic semiflexible form using Canadian per capita time series data for ten consumer expenditure categories.

Normalized Quadratic Systems of Consumer Demand Functions

Empirically estimated demand systems frequently fail to satisfy the appropriate theoretical curvature conditions. We propose and estimate two demand systems for which these conditions can be imposed globally; the first is derived from a normalized quadratic reciprocal indirect utility function and the second is derived from a normalized quadratic expenditure function. The former is flexible if there are no restrictions on its free parameters, but loses flexibility if the curvature conditions need to be imposed. The latter is flexible, in the class of functions satisfying local money metric scaling, even if the curvature conditions need to be imposed.

Quadratic Spline Models for Producer's Supply and Demand Functions

In this paper, the authors propose and estimate a system of producer output supply and input demand functions that generalizes the standard normalized quadratic form. The generalization adds either linear or quadratic splines in a time (or technical change) variable, yet retains the main attractive property of the normalized quadratic, which is that it can provide a local second order approximation while maintaining the correct curvature globally. However, the generalization has additional desirable approximation properties with respect to the splined variable and, thus, permits a more flexible treatment of technical change than is provided by standard flexible functional forms. Copyright 1992 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

The Theory of the Cost-of-Living Index and the Measurement of Welfare Change

Summary The Consumer Price Index is often regarded as an approximation to a Cost-of-Living Index. This paper reviews the theoretical foundations of the Cost-of-Living Index and the closely related problems involved in measuring changes in economic welfare. The Cost-of-Living Index for a single person is defined as the minimum cost of achieving a certain standard of living during a given period divided by the minimum cost of achieving the same standard of living during a base period. In order to numerically construct an individuals Cost-of-Living Index, it is necessary to know his or her preferences over economic goods. Since these preferences are essentially unobservable, it is necessary to construct approximations to the Cost-of-Living Index. This topic is discussed in Section 2 of the paper. The remaining sections of the paper discuss a number of related topics, including: the closely related problems involved in measuring a group Cost-of-living index and changes in the welfare of a group, the fixed based versus the chain principle, the choice of a functional form for the Cost-of-Living Index, the treatment of durable goods, such as housing and the treatment of taxes and labour supply in a Cost-of-Living Index.

The measurement of the economic benefits of infrastructure services

1. Introduction.- 2. A Simple Producer Benef it Measure.- 3. Willingness to Pav Functions and Marginal Cost Functions.- 4. Approximate Benefit Measures.- 5. Problems with the Producer Benefit Measure.- 5.1. Static versus Dynamic Benefit Measures.- 5.2. The Problem of Endogenous Prices for Local Goods.- 5.3. The Neglect of Consumer Benefits.- 6. Alternative Approaches to Benefit Measurement.- 6.1. The Questionnaire or Sample Survey Approach.- 6.2. Ex Post Accounting Approaches.- 6.3. Engineering and Mathematical Programming Approaches.- 6.4. The Applied General Equilibrium Modelling Approach.- 6.5. The Differential Approach.- 6.6. The Econometr ic Approach.- 7. The Selection of a Functional Form in the Econometric Approach.- 7.1. General Issues.- 7.2. The Translog Restricted Profit Function.- 7.3. The Biquadratic Restricted Profit Function.- 8. The Selection and Measurement of Variables in the Econometric Approach.- 8.1. The Measurement of Outputs.- 8.2. The Measurement of Intermediate Inputs.- 8.3. The Measurement of Labour.- 8.4. The Measurement of Inventories of Goods in Process.- 8.5. The Measurement of Capital Stock Components.- 8.6. Aggregation over Goods.- 8.7. The Measurement of Infrastructure Variables.- 9. The Estimation of Restricted Profit Functions in the Time Series Context.- 10. The Estimation of Restricted Profit Functions in the Cross Sectional Context.- 11. Conclusion.- Appendix 1: Properties of Restricted Profit Functions.- Appendix 2: Properties of Restricted Cost Functions.- Appendix 3: Properties of Restricted Expenditure Functions.- Appendix 4: Proofs of Propositions.- References.

Productivity- and Pareto-Improving Changes in Taxes and Tariffs

The paper investigates the problem of tariff reform in a small open multi-household economy that only has tariffs and domestic commodity taxes as policy instruments. The concept of a productivity improvement in tariffs and taxes is introduced and conditions for its existence are established. We prove that a Pareto-improving change in tariffs and domestic taxes exists if a productivity-improving change in tariffs exists and if the Weymark condition on the matrix of household demands holds. Conditions are established for particular tariff reforms, such as proportional reductions and reductions of extreme rates, to yield Pareto improvements in welfare.