This note is intended as a supplement to Downsizing the CPI (the Consumer Price Index): Think Twice! Specifically, it’s the basis of some points made in section 4 (Shaky Ground: Measuring "Equivalent Well-Being"), about the "substitution effect".
When I had written a first draft of that section, I was unsatisfied with my handiwork. It voiced a vague intuition that, since changes in the public’s "market basket" might be forced by adversity, it is not necessarily a "bias" to use the index which paints a less rosy picture.
What made me uneasy was first of all the vagueness of my assertion, but secondly, a technical uncertainty. A cost-of-living index endeavors only to determine the comparative unit price of well-being in two or more periods. If it does that in an independent way, won’t its application to a measure such as GDP then reveal whatever income effects may have been involved? And won’t it measure negative effects as objectively as positive ones?
All three of the indexes under discussion (Laspeyres, geometric mean, and a Tornqvist variant of the latter) are based on price ratios between two periods, weighted in various ways to reflect the importance of the items being priced. The weights differ from one index to another, but in all cases, the weights reflect relative shares of the total expenditure upon a market basket, or upon a combination of two market baskets.[1] This has a nice, objective look about it, not depending upon the absolute level of either prices or item quantities. How can the choice of one index or the other smuggle in an assumption about the causes of substitution?
In its discussion of "quality bias", the Boskin report shows only too clearly how subjective elements enter into the enumeration of the data themselves, especially the price ratios. However, leaving that aside, what about the choice of formulas for combining these expenditure shares and price ratios into a single index? It’s a matter of calculating a representative summand (Laspeyres) versus a representative factor (geometric mean)[2], and of using base-period shares (Laspeyre and the BLS’s experimental geometric mean) versus an average of base-period and end-period shares (Tornqvist). Mightn’t this part of the discussion, at least, be purely technical?
To satisfy my own misgivings, I started fooling around with a model that depicts "income effects" and "substitution effects" in terms of the underlying "utility", and yet combines these with monetary prices in such a way as to study the price indexes and "real incomes" which would be calculated by the various indexes. It’s an amateur effort, and must be taken with a grain of salt, but I think the results are worth sharing.
When I think of comparisons between different bundles of "utility", I think of indifference curves, and this is the tool for which I reached. It is nothing novel to analyze price indexes in the light of indifference curves: Thomas Carroll does exactly that in a standard text on microeconomics.[3] I will make, however, a couple of unauthorized departures, to be noted as we go along.
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The technique I shall use is a little "reverse engineering". A price index starts with observed quantities and prices of things purchased in two periods, and uses one formula or another to calculate an overall price rise (or fall). Its relation to a "cost of living" is then inferred from introspection, since that of which it is the price ("equivalent well-being") is subjective and not directly observable. My reverse engineering will take the opposite tack: if we knew a set of indifference curves (identifying which bundles of goods have equal utility with which other bundles) and could control total expenditure by regulating a fiat money, what sorts of effects might we see in price indexes calculated by the various formulas?
How might each of them be fooled as we move from point to point on the same indifference curve (pure substitution effect), vary the overall money supply but nothing else (pure inflation or deflation), and/or move from points on one indifference curve to another (income effect, which must inherently be accompanied by substitution)?
Step with me, then, into a toy world. (Yes, it’s a toy, but it’s an educational toy. It clarifies what we’re asserting about much more complicated things in the real world.) In this world, there are only two goods: x and y. There’s also money, so that we can talk about prices and price indexes, but it’s a fiat money: the amount of it is set so that total expenditure comes to some stated total. (This helps our calculations no end.) Consumer preferences for a period under study are described by a given set of indifference curves. Dollar prices are perfectly flexible, and simply adjust themselves to make the total expenditures come out right, while conforming to the relative prices (of x versus y) dictated by the indifference curves.[4]
Pure substitution effects, then, will be depicted as movement from one point to another upon a single indifference curve. Pure inflation (or deflation) will be reflected as a change in total expenditures while remaining at the same point on an indifference curve — and thus, an equi-proportional change in dollar prices. "Real income" effects show up as movement from one indifference curve to another. (Inherently, real income effects are accompanied by substitution, if only of "more" for "less" of everything, or vice-versa.)
We’ll indulge in one more flight from reality (in for a penny, in for a pound). Having assumed that x and y are the only goods in the economy, we may as well pretend that the observed quantities are aggregates for the whole economy, and therefore the market basket itself is the Gross National Product.[5] Deflating it by any of the price indexes gives one or another estimate of the economy’s real output.
The discussion will proceed in the form of six examples:
I launched this investigation expecting to find that the Boskin Commission’s allegation of a "substitution bias" in the Laspeyres index could be challenged within the framework of their own assumptions — unchanging (or only slowly changing) consumer preferences, the reasonableness of aggregating different people’s preferences, indifference curves of somewhat the same shape as I’ve employed, and so forth. The above examples, and others which I worked out in the same manner, show the opposite of what I anticipated: the Boskin Commission is right. Under most economically reasonable circumstances, a Tornqvist index (geometric mean with average base-period and end-period weights) turns up the most plausible measure of average price change. It is less confused than the other indexes by a mixture of income and substitution effects, whether real income is going up or going down.
In retrospect, there’s a good reason for this: shots are usually called better after the fact. The Tornqvist index knows everything that the Laspeyres index knows about base-period weights, and it takes into account the end-period weights as well.
By contrast, a base-weighted geometric average can go badly wrong when substitution is extreme. However, such extremes are probably rare, because there is little reason to believe that production possibilities (transformation curves) can furnish such substitutions on short notice. More likely, we’d just see larger income effects.
It doesn’t necessarily follow that the Tornqvist index is a good candidate for compiling the CPI, or any other index which is to determine cost-of-living payments. As a practical matter, it takes longer to collect the necessary data, and more must be collected. The additional resources for doing so, advocated by the Boskin Commission, may not be forthcoming in an era of straitened government budgets. Moreover, if a provisional estimate is published early on, and revised after a year or two, as recommended by the commission, which figure is to be used for the cost-of-living payments? Take-backs in the light of revised estimates are a formula for unending political uproar, while upward revisions will prompt a demand for retroactive supplements.
Finally, my economic model (and I think also the Boskin model) is far too blasé about the income distribution effects of inflation. In my model, the monetary authority somehow puts a stipulated amount of money equally in the hands of everybody, and they don’t care a whit about the absolute level of prices, just about the quantities of x and y that are available for purchase and the trade-off between them. Obviously, it matters intensely to people whether, which way, and for how long they are entangled in contracts that are stated in dollars.
Of course, the ignoring of distribution effects is inherent in any measure which attempts to aggregate the performance of an entire economy — unless we are willing to weight people as well as purchases.