[1]. The index formulas as given above are gathered from a couple of BLS sources, with some adaptation. (We use our own variable names where convenient, and use price ratios in some places where the BLS source substitutes lower-level indexes.)

First, see The Experimental CPI using Geometric Means (CPI-U-XG), published online by the BLS.

Additional detail was found in: Moulton, Brent R., and Stewart, Kenneth J., "An Overview of Experimental U.S. Consumer Price Indexes", paper presented at the Third Meeting of the International Working Group on Price Indexes, Voorburg, Netherlands, April 16-18, 1997 (Washington, D.C., Bureau of Labor Statistics). This is not available online. Besides a wealth of formulas, it recounts some BLS experiments with alternative market baskets reflecting the expenditure patterns elderly and poor consumers.
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[2]. Scaled down to an individual's share, and pro-rated for the length of the prospective stay. [Back to point of reference.]

[3]. Thomas Carroll, Macroeconomic Theory, Concepts and Applications, St. Martin's Press, New York, 1983: pp. 76-79. [Back to point of reference.]