[1] A very illuminating comparison of the Laspeyres index and the BLS version of a geometric mean may be found in The Experimental CPI using Geometric Means (CPI-U-XG), published online by the BLS.The authorship is modestly concealed, but inquiries are directed to stewart_k@bls.gov. A further BLS reference by Brent R. Moulton and Kenneth J. Stewart, unfortunately not available online, is cited in my Appendix 2: Formulas and Meanings of Various Price Indexes.

Note that a geometric mean could also base itself, like the Paasche index, on end-period expenditure shares. The Tornqvist index is just a geometric mean based on the average of the base-period and end-period expenditure shares. The BLS has a preference for base-period weights because the information is available in time to produce the index on schedule. [Back to point of reference.]

[2] For a particularly clear definition of arithmetic and geometric means, see http://www.math.toronto.edu/mathnet/questionCorner/geomean.html. (And this site is aimed at high school students!) [Back to point of reference.]

[3] Microeconomic Theory, by Thomas M. Carroll, St. Martin's Press, New York, 1983. [Back to point of reference.]

[4] Explicit introduction of a fiat money, so managed as to fix overall expenditure at some arbitrarily stipulated level, and of perfectly flexible money prices that adjust to suit that total, is an "unauthorized departure". It provides a sharper way for the model to distinguish "pure inflation" effects from substitution and income effects. [Back to point of reference.]

[5] This, too, is an "unauthorized departure", though it cannot be an unprecedented one, or I'd never have heard of a "transformation curve". Indifference curves are more usually depicted as the preferences of an individual consumer, and the "transformation curve" is replaced with a straight "budget line". The idea is that the budget of an individual consumer is too small to affect the relative prices of things, so the available combinations of x and y present themselves to the consumer as a straight line. When we deal with the aggregate choices of an entire economy, producing more and more x requires the sacrifice of y at an increasing rate, so the "budget line" assumes the somewhat hyperbolic shape that we call a "transformation curve". [Back to point of reference.]

[6] Assigning an algebraic function to the indifference curve is the last of my "unauthorized departures" (I promise). Indifference curves are usually drawn freehand, and nobody claims that they conform to any precise mathematical formulation. They merely look vaguely hyperbolic. It's convenient, though, to impose the algebra, because the calculations which can then be made clarify what's going on, without departing significantly from the freehand versions. [Back to point of reference.]

[7] Bear in mind, I haven't conceded that any "cost of living" index is suitable for stabilizing the value of promises which people make to each other. Here, though, I am exploring the arguments of the Boskin Commission on their own terms.[Back to point of reference.]

[8] (Footnote 8 replaced in online version by direct link to index inequalities expounded by Thomas Carroll.)

[9] It's tempting to suppose that the indexes should show greater reductions of real income towards the bottom of the table, where the situation looks most like an emergency. Remember, though, we postulated an indifference curve according to which the amounts of item y which are substituted for x are satisfactory amends. If they aren't, we should have drawn a different indifference curve. [Back to point of reference.]

[10] For the derivation, see Appendix 3: Checking that Distinct Indifference Curves Don't Cross. [Skip the derivation, take me back to the point of reference.]

[11] Toward a More Accurate Measure of the Cost of Living: Final Report to the Senate Finance Committee from the Advisory Commission to Study the Consumer Price Index, December 4, 1996 Updated Version, Part 4: "Table 1 -- Hypothetical Example of Substitution Bias" and the discussion surrounding it. [Back to point of reference.]

[12] My model equates "inflation", in effect, with changes in total dollar expenditures. (The present example was calculated with an "inflation rate", in that sense, of 0.44.) This handily separates monetary issues from the substitution and real income effects, but only in a toy world where prices are perfectly flexible. It would be a useless definition for a monetary authority charged with keeping prices stable.[Back to point of reference.]

[13] We've pretended, throughout this discussion, that prices Px and Py are known. In reality, the price of x during a period may vary from season to season, from place to place, from store to store, and even from bargain to bargain. Moreover, a single category in the overall index, such as apples, may include several varieties, such as delicious, Macintosh, and Granny Smith. The figures fed into the "upper level" calculation for, say, Px, will thus themselves be indexes. Since different varieties of apples are pretty close substitutes, these are conditions in which the geometric mean is least likely to be led astray. [Back to point of reference.]