Interconvertible rules between an aggregative index and a log-change index


Abstract


This paper describes interconvertible rules between an aggregative index like the Laspeyres index and a log-change index like the Tӧrnqvist index. Thus we can compare an aggregative index with a log-change index in the same form. Using these rules, we formulate the logarithmic difference between the Laspeyres price index and the Tӧrnqvist price index. One of the rules may be combined with another. By using these combined rules, we can change from given weights to other weights in an aggregative index (or a log-change index) of which the value is invariable.

DOI Code: 10.1285/i20705948v7n2p394

Keywords: Price index; quantity index; exact index; substitution bias; logarithmic mean.

References


Aizcorbe, A.M., Jackman, P.C. (1993). The commodity substitution effect in CPI data, 1982–91. Monthly Labor Review, 116(12), 25–33.

Balk, B.M. (1996). Consistency-in-aggregation and Stuvel indices. Review of Income and Wealth, 42, 353–363.

Balk, B.M. (1999). On curing the CPI’s substitution and new goods bias. Paper presented at the Fifth Meeting of the International Working Group on Price Indices. Available on the Ottawa Group website.

htttp://www.ottawagroup.org.

Balk, B.M. (2002–3). Ideal indices and indicators for two or more factors. Journal of Economics and Social Measurement, 28, 203–217.

Balk, B.M. (2004). Decompositions of Fisher indexes. Economics Letters, 82, 107–113.

Balk, B.M. (2008). Price and Quantity Index Numbers: Models for Measuring Aggregate Change and Difference. New York: Cambridge University Press.

Bortkiewicz, L.V. (1923). Zweck und struktur einer preisindexzahl. Nordisk Statistisk Tidskrift, 2, 369–408.

Boskin, M.J. (2005). Causes and consequences of bias in the Consumer Price Index as a measure of the cost of living. Atlantic Economic Journal, 38, 1–13.

Cage, R., Greenlees, J., Jackman, R. (2003). Introducing the chained Consumer Price Index. Paper presented at the Seventh Meeting of the International Working Group on Price Indices. Available at

www.bls.gov/cpi/super_paris.pdf.

Carlson, B.C. (1972). The logarithmic mean. American Mathematical Monthly, 79(6), 615–618.

Diewert, W.E. (1976). Exact and superlative index numbers. Journal of Econometrics, 4, 115–145.

Diewert, W.E. (1978). Superlative index numbers and consistency in aggregation. Econometrica, 46, 883–900.

Diewert, W.E. (1981). The economic theory of index numbers: a survey, in Essays in the Theory and Measurement of Consumer Behaviour, ed. A. Deaton, Cambridge: Cambridge University Press, 163–208.

Diewert, W.E. (2002). The quadratic approximation lemma and decompositions of superlative indexes. Journal of Economic and Social Measurement, 28, 63–88.

Diewert, W.E. (2004a). Basic index number theory, in Consumer Price Index Manual: Theory and Practice, eds. International Labour Office et al., Geneva: International Labour Office, 263–288.

Diewert, W.E. (2004b). The economic approach to index number theory: the single-household case, in Consumer Price Index Manual: Theory and Practice, eds. International Labour Office et al., Geneva: International Labour Office, 313–335.

Diewert, W.E. (2004c). Price indices using an artificial data set, in Consumer Price Index Manual: Theory and Practice, eds. International Labour Office et al., Geneva: International Labour Office, 345–354.

Diewert, W.E. (2009). Cost of living indexes and exact index numbers. Discussion paper 09–06. Available at

http://faculty.arts.ubc.ca/ediewert/disc.htm.

Geary, R.C. (1950–51). A note on ‘a constant-utility index of the cost of living.’ Review of Economic Studies, 18, 65–66.

Greenlees, J.S. (2011). Improving the preliminary values of the chained CPI-U. Journal of Economic and Social Measurement, 36, 1–18.

Greenlees, J.S., McClelland, R.B. (2008). Addressing misconceptions about the Consumer Price Index. Monthly Labor Review, 131(8), 3–19.

Greenlees, J.S., Williams, E. (2009). Reconsideration of weighting and updating procedures in the US CPI. BLS Working Papers 431.

Hill, P. (2004). Uses of consumer price indices, in Consumer Price Index Manual: Theory and Practice, eds. International Labour Office et al., Geneva: International Labour Office, 33–37.

Klein, L.R., Rubin, H. (1947–48). A constant-utility index of the cost of living. Review of Economic Studies, 15, 84–87.

Lau, L.J. (1979). On exact index numbers. Review of Economics and Statistics, 61, 73–82.

Lent, J., Dorfman, A.H. (2009). Using a weighted average of base period price indexes to approximate a superlative index. Journal of Official Statistics, 25, 139–149.

Lloyd, P.J. (1975). Substitution effects and biases in nontrue price indices. American Economic Review, 65, 301–313.

Manser, M.E., McDonald, R.J. (1988). An analysis of substitution bias in measuring inflation, 1959–85. Econometrica, 56, 909–930.

McCully, C.P., Moyer, B.C., Stewart, K.J. (2007). Comparing the Consumer Price Index and the Personal Consumption Expenditures Price Index. Survey of Current Business, 87(11), 26–33.

Montgomery, J.K. (1937). The Mathematical Problem of the Price Index. London: P. S. King & Son.

Pittenger, A.O. (1985). The logarithmic mean in n variables. American Mathematical Monthly, 92(2), 99–104.

Reinsdorf, M.B. (1994). A new functional form for price indexes’ elementary aggregates. Journal of Official Statistics, 10, 103–108.

Reinsdorf, M.B., Diewert, W.E., Ehemann, C. (2002). Additive decompositions for Fisher, Tӧrnqvist and geometric mean indexes. Journal of Economic and Social Measurement, 28, 51–61.

Reinsdorf, M., Triplett, J.E. (2009). A review of reviews: ninety years of professional thinking about the Consumer Price Index, in Price Index Concepts and Measurement, eds. W.E. Diewert et al., Chicago: The University of Chicago Press, 17–83.

Samuelson, P.A. (1947–48). Some implications of “linearity.” Review of Economic Studies, 15, 88–90.

Sato, K. (1976). The ideal log-change index number. Review of Economics and Statistics, 58, 223–228.

Shapiro, M.D., Wilcox, D.W. (1997). Alternative strategies for aggregating prices in the CPI. Federal Reserve Bank of St. Louis Review, 79(3), 113–125.

Statistics Bureau, Ministry of Internal Affairs and Communications, Japan (2010). Annual Report on the Family Income and Expenditure Survey, 2009.

Stolarsky, K.B. (1975). Generalizations of the logarithmic mean. Mathematics Magazine, 48(2), 87–92.

Tsuchida, S. (1997). A family of almost ideal log-change index numbers. Japanese Economic Review, 48, 324–342.

Vartia, Y.O. (1976). Ideal log-change index numbers. Scandinavian Journal of Statistics, 3, 121–126.

Zieschang, K. (2004). The system of price statistics, in Consumer Price Index Manual: Theory and Practice, eds. International Labour Office et al., Geneva: International Labour Office, 235–262.


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