Comments and Questions

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 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.