Abstract
The weighted arithmetic mean is used in a wide variety of applications. An infinite number of possible decompositions of the change in the weighted mean are available, and it is therefore an open question which of the possible decompositions should be applied. In this article, we derive a decomposition of the change in the weighted mean based on a two-stage Bennet decomposition. Our proposed decomposition is easy to employ and interpret, and we show that it satisfies the difference counterpart to the index number time reversal test. We illustrate the framework by decomposing aggregate earnings growth from 2020Q4 to 2021Q4 in Norway and compare it with some of the main decompositions proposed in the literature. We find that the wedge between the identified compositional effects from the proposed two-stage Bennet decomposition and the one-stage Bennet decomposition is substantial, and for some industries, the compositional effects have opposite signs.
References
1.
Baily, M.N., C. Hulten, D. Campbell, T. Bresnehan, and R.E. Caves. 1992. “Productivity dynamics in manufacturing plants.” Brookings Papers on Economic Activity: Microeconomics : 187–267. DOI: https://doi.org/10.2307/2534764.
2.
Balk, B.M. 2021. Productivity: Concepts, Measurement, Aggregation, and Decomposition. Springer.
3.
Benedictow, A., and P. Boug. 2017. “Calculating the real return on a sovereign wealth fund.” Canadian Journal of Economics 50(2): 571–594. DOI: https://doi.org/10.1111/-caje.12270.
4.
Benedictow, A., and P. Boug. 2021. “Exact and inexact decompositions of trade price indices.” Empirical Economics . DOI: https://doi.org/10.1007/s00181-021-02078-4.
5.
Bennet, T.L. 1920. “The Measurement of Changes in the Cost of Living.” Journal of the Royal Statistical Society 83(3): 455–462. DOI: https://doi.org/10.2307/2340777.
6.
Daly, M.C., and B. Hobijn. 2017. “Composition and Aggregate Real Wage Growth.” American Economic Review 107(5): 349–352. DOI: https://doi.org/10.1257/aer.p20171075.
7.
Diewert, W.E. 2002. “The quadratic approximation lemma and decompositions of superlative indexes.” Journal of Economic and Social Measurement 28: 63–88. Available at: https://ip.ios.semcs.net/articles/journal-of-economic-and-social-measurement/jem00200.
8.
Diewert, W.E. 2005. “Index number theory using differences rather than ratios.” American Journal of Economics and Sociology 64(1): 311–360. DOI: https://doi.org/10.1111/j.1536-7150.2005.00365.x.
9.
Diewert, W.E., and K.J. Fox. 2010. “On measuring the contribution of entering and exiting firms to aggregate productivity growth. In Price and Productivity Measurement ”, edited by W.E.
10.
Diewert, B. Balk, D. Fixler, K.J. Fox, and A. Nakamura. 6: 41–66. Trafford Press. Foster, L., J.C. Haltiwanger, and C.J. Krizan. 2001. “Aggregate Productivity Growth: Lessons from Microeconomic Evidence.” In Developments in Productivity Analysis , edited by C.R. Hulten, E.R. Dean, and M.J. Harper: 303–372. University of Chicago Press.
11.
Huerga, J. 2010. “An Application of Index Numbers Theory to Interest Rates.” In Price Indexes in Time and Space: Methods and Practice , edited by L. Biggeri, and G. Ferrari: 239–248. DOI: https://doi.org/10.1007/978-3-7908-2140-6_13.
12.
ILO, IMF, OECD, Eurostat, United Nations, and World Bank. 2004. Consumer price index manual – Theory and practice . I.L. Office, Geneva. Available at: https://www.ilo.org/global/statistics-and-databases/publications/WCMS_331153/lang–en/index.htm.
13.
Kamin, S.B., M. Marazzi, and J.W. Schindler. 2006. “The Impact of Chinese Exports on Global Import Prices.” Review of International Economics 14(2): 179–201. DOI: https://doi.org/10.1111/j.1467-9396.2006.00569.x.
14.
Krueger, A.B. 2017. “Where have all the workers gone? An inquiry into the decline of the U.S. Labor force participation rate.” Brookings Papers on Economic Activity 2017: 1–87. DOI: https://doi.org/10.1353/eca.2017.0012.
15.
OECD. 2018. OECD Compendium of Productivity Indicators 2018 . OECD. DOI: https://doi.org/10.1787/pdtvy-2018-en.
16.
Segnitz, E. 1870. “Über die Berechnung der sogenannten Mittel, sowie deren Anwendung in der Statistik und anderen Erfahrungswissenschaften.” Jahrbücher für Nationalökonomie und Statistik, 14: 183–195.
17.
The Norwegian Tax Administration. 2022. About a-ordningen . Available at: https://www.skatteetaten.no/en/business-and-organisation/employer/the-a-melding/-about-the-a-ordning/about-a-ordningen/ (accessed March 2022).
18.
Theil, H. 1975. Theory and measurement of consumer demand . Vol(1), North-Holland Amsterdam.
19.
Thomas, C.P., and J. Marquez, J. 2009. “Measurement matters for modelling US import prices.” International Journal of Finance and Economics 14(2): 120–138. DOI: https://doi.org/10.1002/ijfe.370.
20.
Von Brasch, T., B. Dapi, and V. Sparrman. 2017. Sammensetningseffekter mellom næringer og veksten i gjennomsnittlig årslønn (Compositional effects and growth in annual earnings): 2017/45. Statistics Norway. Available at: https://www.ssb.no/arbeidog-lonn/artikler-og-publikasjoner/sammensetningseffekter-mellom-naeringer-og-veksten-i-gjennomsnittlig-arslonn.