# Discussion Paper

## Abstract

The most important economic measures are monetary. They have many different names, are derived in different theories and employ different formulas; yet, they all attempt to do basically the same thing: to separate a change in nominal value into a ‘real part’ due to the changes in quantities and an inflation due to the changes in prices. Examples are: real national product and its components, the GNP deflator, the CPI, various measures related to consumer surplus, as well as the large number of formulas for price and quantity indexes that have been proposed.

The theories that have been developed to derive these measures are largely unsatisfactory. The axiomatic theory of indexes does not make clear which economic problem a particular formula can be used to solve. The economic theories are for the most part based on unrealistic assumptions. For example, the theory of the CPI is usually developed for a single consumer with homothetic preferences and then applied to a large aggregate of diverse consumers with non-homothetic preferences.

In this paper I develop a unitary theory that can be used in all situations in which monetary measures have been used. The theory implies a unique optimal measure which turns out to be the Törnqvist index. I review, and partly re-interpret the derivations of this index in the literature and provide several new derivations.

The paper also covers several related topics, particularly the presently unsatisfactory determination of the components of real GDP.

Submitted as Survey and Overview

## JEL Classification

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## Assessment

# Comments and Questions

This is a broad-ranging paper and therefore difficult to comment on. The author is helping us, however, by clearly stating his principal conclusions as

“a. The Törnqvist index is the only one that can be integrated in a realistic and encompassing economic theory. b. The GDP deflator should
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be the CPI in the form of a chained Törnqvist price index. c. The theory of economic measurement should be a core subject for all economists. If economists are uninformed about both theoretical and practical aspects of the data they use, the scientific status of the discipline is in doubt.”

I disagree with conclusion a; I am not convinced by the arguments advanced for conclusion b; but I strongly agree with conclusion c. Hence, I concentrate on a.

What I see as being the main purpose of this paper is to present a generic theory of the decomposition of a value change in a price and a quantity component. The framework is continuous time and Divisia theory (extended in the sense that also difference-type measures are considered). As I have shown elsewhere (Balk 2005), the interesting property of a Divisia index is its path-dependency. One implication is that, given that data materialize only at discrete time-points, there is no unique approximation, because assumptions are needed about the path in the intervening time periods. Hillinger claims at the end of section 4 that this should be “… not a problem as long as the paths are monotone …”, but that seems to be begging the question. The sad fact is that the paths are unknown, and all one can do is to make informed guesses; but there is no uniqueness to be expected.

Though there are good reasons to advocate a Törnvist price or quantity index as a good approximation to a (unknown) Divisia price or quantity index, in the process of approximation several nice properties of Divisia indices get lost. One concerns the so-called Factor Reversal Test: though Divisia price index times Divisia quantity index equals the value ratio, such a relation does not hold for Törnqvist indices. A second concerns Consistency-in-Aggregation: though Divisia indices are consistent-in-aggregation, Törnqvist indices lack this property. Surely, the Geometric Laspeyres and the Geometric Paasche are consistent-in-aggregation, but their geometric mean is not!

Reference

Balk, B. M., 2005, “Divisia price and quantity indices: 80 years after”, Statistica Neerlandica 59, 119-158

Dr. Balk summarizes three conclusions of my paper; he agrees with the third, is skeptical about the second and disagrees with the first. Since the first is the only one he discusses, this will also be the focus of my reply.

At issue is my claim that the Törnqvist
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index is unique in that it can be derived from economically plausible assumptions. I have given three distinct derivations that in my opinion satisfy this criterion; Dr. Balk discusses only one. One of the others involves a reinterpretation of a set of axioms due to Balk and Diewert. The other involves the reinterpretation of the Divisia differentials as total differentials of a function with variable parameters. In that case, the functions corresponding to the differentials are always approximated quadratically by the Törnqvist indexes. Of course, depending on the path, the differential may be very poor; this is true of all approximations.

Before turning to the derivation that is at issue, one preliminary remark: We accept or reject an argument based on its intrinsic merit, but also on the merit of competing arguments. Even a rather weak argument may be accepted if it is the best we can think of. I will therefore not only defend my argument, but also ask what alternatives Dr. Balk is offering.

Dr. Balk wrote:

As I have shown elsewhere (Balk 2005), the interesting property of a Divisia index is its path-dependency. One implication is that, given that data materialize only at discrete time-points, there is no unique approximation, because assumptions are needed about the path in the intervening time periods. Hillinger claims at the end of section 4 that this should be “… not a problem as long as the paths are monotone …”, but that seems to be begging the question. The sad fact is that the paths are unknown, and all one can do is to make informed guesses; but there is no uniqueness to be expected.

The above description is a caricature of my argument. The statement in quotation marks is not in Section 4, which has a different topic, nor anywhere else in the paper (I made a search). The argument that I actually made is given below:

Before proceeding to a formal analysis, I will give here a verbal discussion of how I propose to deal with the conceptual problems that have bedeviled the analysis of Divisia integrals. I use a combination of economic and mathematical arguments. The economic argument is that the values of the derived price and quantity indexes should depend solely on prices and quantities at the end points of the interval. This is the standard assumption that has always been made in index theory. It should be noted that an influence of the path on the outcome is by no means excluded. The assumption is only that whatever outcome is reached, the price/quantity data of the initial and final situations are all that is needed for a comparison. The mathematical result is that the Divisia integral is approximated quadratically if prices and quantities grow exponentially or more generally monotonically, over the interval being considered. These arguments together provide a strong, though not the only justification for accepting the Törnqvist index.

Next I ask if Dr. Balk has any better arguments: As discussed in his monograph, one can assume various exotic paths of prices and quantities and derive formulas that approximate, or replicate the Divisia indexes under these assumptions. No one would argue that these paths can ever be expected to be close to the actual paths of the variables. So what is the point?

Do we in general even wish to replicate the Divisia integrals as Dr. Balk implies. Consider the following: Assume a cyclical path of the variables such that initial and final positions are the same, but such that the price integral signifies a huge inflation of several thousand percent. Supposing that we could compute the value of the integral, would we want to do so and proclaim a huge inflation even though nothing has changed? I think not! NIPA statisticians have long ago given a sensible answer to a similar problem: they do not consider seasonal fluctuations in computing secular trends.

Finally, I disagree with the criticisms of the Törnqvist index that Dr. Balk makes in his final paragraph. Törnqvist indexes do not aggregate directly, but, as Dr. Balk notes, the component geometrical indexes do. The aggregate geometrical indexes then combine to give the aggregate Törnqvist. I can see no problem here. What I call the duality property, that the product of price and quantity indexes should give the proportional change in expenditure, holds for the Törnqvist only to a quadratic approximation. I am not aware of any practical problem that would be caused thereby.

I regret that Dr. Balk chose not to comment on other aspects of my paper.

I thank the referee for his careful and knowledgable comments that will certainly serve to improve the paper.

see attached file

The referee suggests that I write a short “technical” paper. I suppose that by this he means the straightforward derivation of some mathematical results from stated assumptions. I could do this, for example by writing a two page note giving my novel derivations of the Törnqvist index. I will not ...[more]

... do this because it would not accomplish my purpose. .

My concern is with economics as an empirical science. In my view, this requires an economic theory of measurement based on plausible assumptions. If the assumptions are implausible, the measurements will not reflect a relevant aspect of reality. If the assumptions are not economic, for example if they are of a purely mathematical nature, then it will not be clear what inferences can be drawn from the measure. In the paper I discuss the various derivations of index formulas that have appeared in the literature and show that they do not satisfy the twin criteria of being based on assumptions that are both economic and plausible. Moreover, different sets of assumptions that have been made lead to different formulas. The situation is thus altogether unsatisfactory.

I believe that my paper demonstrates that a theory satisfying the twin requirements can be given and that it leads to a unique index, namely the Törnqvist

This sort of conceptual analysis does not please the referee. That is unfortunate. In my view of empirical science, conceptual analysis is required to bridge the gap between a formal theory and reality.