On Plutocratic and Democratic CPIs. Abstract

Transcription

1 On Plutocratic and Democratic CPIs Eduardo Ley IMF Institute Abstract Prais (1958) sowed tat te standard CPI computed by most statistical agencies can be interpreted as a plutocratic weigted average of ouseold price indexes because te weigt of eac ouseold in te official CPI is determined by its total expenditures. In tis paper, we decompose te difference between te standard CPI and a democratically weigted index as te product of a measure of income inequality and te sample covariance between te elementary individual price indexes and a parameter wic is a function of te income elasticity of eac good. I tank Javier Ruiz Castillo for numerous conversations and stimulating discussions on te subject of tis paper, and Mario Izquierdo for most valuable elp wit te computations. I received very useful comments from Woon Gyu Coi, Nuri Erbas, Roberto Garcia Saltos, Jos Greene, David S. Jonson, Mosin Kan, Marsall Reinsdorf, Rafa Repullo, Hal Varian and Slomo Yitzaki, and te participants at seminars at te IMF Institute, IADB, te XVIII Latin American Econometric Society Meetings (July 2001, Buenos Aires), and at te CRIW/NBER worksop on Productivity, Prices and Output (July 2001, Boston MA). Citation: Ley, Eduardo, (2002) "On Plutocratic and Democratic CPIs." Economics Bulletin, Vol. 4, No. 3 pp. 1 5 Submitted: October 4, Accepted: January 19, URL: ttp:// 01D60003A.pdf

2 1. Introduction It is known since Prais (1958) tat te CPI computed by statistical agencies can be interpreted as a weigted average of ouseold price indexes. Te weigt of eac ouseold is given by its total expenditure, ence te term plutocratic index. Alternatively, we could construct a democratically-weigted index, were eac ouseold weigts te same. We sall define te CPI plutocratic-democratic gap as te difference between te plutocratic index and te democratic one. Weter price beavior in a given period urts relatively more te better-off or te worse-off ouseolds can be expressed in terms of tis single scalar (Fry and Pasardes, 1985). Various studies ave computed te plutocratic-democratic gap for different countries. 1 Tis paper investigates te sources of possible discrepancies between plutocratic and democratic indexes. We sow tat te plutocratic-democratic gap can be expressed as te product of mean income, a measure of variation of ouseold expenditures, and te sample covariance between te elementary individual price indexes and te corresponding good s expenditure-sare regression coefficient on ouseold income. Tis coefficient, in turn, is a function of te total expenditures elasticity of eac good (Ley, 2001). 2. Plutocratic and Democratic CPI budget sares Define te plutocratic budget sares for good i in te aggregate CPI by s P i = 1 x s i, (1) X were x denotes ouseold total expenditures, x i is te expenditure on good i, so tat ouseold budget sare for good i is given by s i = x i /x. Total aggregate expenditure is given by X = x. Te CPI (at time t) is given by CP I P = i sp i I i, were I i = (p it /p i0 ) are elementary price indexes. Noting tat ouseold individual index is given by cpi = i s i I i, te CP I P may be interpreted as a representative CPI. It is natural to ask ten wat is te ouseold better represented by te CP I P. Muellbauer (1974) searced for te ouseold wose budget sares were closest to te s P i aggregate weigts in te U.K. CPI, and found it to be at te 71 percentile in te ouseold expenditures distribution. For te U.S. in 1990, Deaton (1998) estimates tat tis consumer occupies te 75 percentile. Tus, te representative consumer embedded in te CP I P is biased towards upper-income ouseolds. Alternatively, we could use democratic budget sares, s D i = 1 s i, (2) H 1 See, e.g., Carruters et al. (1980), Fry and Pasardes (1985), Deaton and Muellbauer (1980), Crawford (1996), Newberry (1994), Kokoski (1987, 2000), Erbas and Sayers (1998), Garner et al. (1999), Lódola et al. (2000), Yaav and Yitzaki (1991), Ruiz-Castillo et al. (1999). See also Ley (2001) for a summary of findings.

3 were H denotes te number of ouseolds, to construct a democratically-weigted index, CP I D = i sd i I i. From equations (1) and (2), te difference between good i plutocratic and democratic sares in te CPI is given by ( s P i s D i ) = 1 xh (x x)s i = 1 x ˆσ(x, s i), (3) were x = X/H is te sample mean of total expenditures, and ˆσ(x, s i ) is te sample covariance, across ouseolds, of te budget sare of good i, s i, and total expenditure. Multiplying and dividing te rigt-and side of expression (3) by te sample variance of ouseold total expenditures, ˆσ 2 = ˆσ(x, x), we obtain: ( s P i s D i ) = ˆζ ˆβ i, (4) were ˆζ = ˆσ 2 / x is a measure of variation of ouseold total expenditures, and ˆβ i = ˆσ(x, s i )/ˆσ 2 (see Ley (2001) for an interpretation of ˆβ i as a regression coefficient and its relation te income elasticity of good i). Equation (4) indicates tat te difference in good i s plutocratic and democratic CPI sares depends on te product of: (i) a measure of inequality of ouseold total expenditures, 2 ˆζ; and (ii) a measure of ow good i s budget sare varies wit total expenditure in te ouseold sample, ˆβ i. Since te decomposition is multiplicative, te sares must coincide wen tere is no inequality or wen expenditure sares are not affected by differences in total expenditures. 3. Te CPI plutocratic-democratic gap As discussed before, we sall define te plutocratic-democratic gap, G, as: G (Π P Π D )/100 = ( CP I P CP I D), were Π = (CP I 1) 100 is te inflation rate between 0 and t (in percent); using (3) we find: G = i ( s P i s D i )I i = ˆζ i ˆβ i I i = ˆζ i ˆβ i (I i Ī), (5) were Ī is a simple average, i.e., Ī = 1 N Ii, and N is te number of goods. Equation (5) may be rewritten as: G = ˆζ N ˆσ( ˆβ, I), (6) were ˆσ( ˆβ, I) refers to te sample covariance of ˆβ i and I i, tis time over goods instead tat over ouseolds. 2 Note tat ˆζ = 2 xi2 (x), were I 2 (x) corresponds to te Generalized Entropy inequality measure, I c (x), for c = 2 (Cowell and Kuga, 1981). 2

4 Equation (6) is our fundamental result. It sows tat te plutocratic-democratic gap is determined by te dispersion of ouseold total expenditures, measured by ˆζ, and te sample covariance between ˆβ i and I i. Te sign of te plutocratic-democratic gap is determined by te covariance term. A positive covariance term means tat te goods favored by te ricer ouseolds experience iger tan average inflation and necessities a lower tan average inflation. Similarly, a negative covariance implies tat necessities experience iger tan average inflation wile superior or luxury goods experience lower tan average inflation. Tese effects are also scaled by te magnitude of te inequality of ouseold expenditures, as measured by ˆζ. Inspection of equation (6) indicates tat tree elements are required for te plutocraticdemocratic gap to be different from zero: (a) tere must be some dispersion in te distribution of ouseold expenditures (reflected by ˆζ 0); (b) tere must be some observed beavioral differences among ouseolds wit different total expenditures (reflected by ˆβ i 0 for some i); and (c) tere must be some differences in price beavior across some goods wic display beavioral differences across ouseolds (reflected by I i Ī for some i wic as ˆβ i 0). Given a ouseold survey, ˆζ and te ˆβ i s are ten fixed, and any source of variation in te sign and size of te gap for, e.g., eac year must be solely explained by te price beavior reflected by te I i s. Te movements in te I i s may cause ˆσ( ˆβ, I) to cange sign from one year to anoter. Tus, as noted in Ley (2001), looking at te overall G, simply averaging over a long period may be misleading. Moreover, because of data limitations, most of te empirical results are based on a smaller number of goods tan te number for wic prices were collected by te statistical agencies. In tese instances, working wit igly aggregated goods causes an underestimation of te true plutocratic-democratic gap for two reasons. First, price aggregates already embody a plutocratic-democratic gap. Second, income elasticities revert to te mean (i.e., to one) as we aggregate goods. As a result, te true size of te plutocratic-democratic gap is typically underestimated. Finally, Ley (2001) extends tis approac by considering explicitly te number of members in eac ouseold using an equivalence-scale approac. References Bumann, B., L. Rainwater, G. Scmauss and T. Smeeding (1988), Equivalence Scales, Well-Being, Inequality and Poverty: Sensitivity Estimates Across Ten Countries Using te Luxembourg Income Study Database, Review of Income and Wealt, 34: Carruters, A., D. Sellwood and P. Ward (1980), Recent Developments in te Retail Price Index, Te Statistician, 29:

5 Crawford, Ian (1996), UK Houseold Cost-of-Living indexes, , in J. Hills (ed), New Inequalities: te Canging Distribution of Income and Wealt in te United Kingdom, Cambridge: Cambridge University Press. Cowell, F.A. and K. Kuga (1981), Additivity and te Entropy Concept: An Axiomatic Approac to Inequality Measurement, Journal of Economic Teory, 25, Deaton, A. (1998), Getting Prices Rigt: Wat Sould Be Done?, Journal of Economic Perspectives, 12: , and J. Muellbauer (1980), Economics and Consumer Beavior, New York: Cambridge University Press. Erbas, N. and C.L. Sayers (1998), Is te United States CPI biased across income and age groups?, Wasington DC: IMF Working Paper ttp:// Fry, V. And P. Pasardes (1985), Te RPI and te Cost of Living, Report Series No. 22, London: Institute for Fiscal Studies. Garner, T, J. Ruiz-Castillo and M. Sastre (1999), Te Influence of Demograpics and Houseold Specific Price indexes on Expenditure Based inequality and Welfare: A Comparison of Spain and te United States, Working Paper 9963 Economic Series 25, Universidad Carlos III, Madrid. ttp://eco.uc3m.es Kokoski, M.F. (1987), Consumer Price indexes by Demograpic Group, BLS Working Paper 167 (April)., (2000), Alternative CPI Aggregations: two approaces, BLS Montly Labor Review, vol. 123, Ley, Eduardo (2001), Wose Inflation? A Caracterization of te CPI Plutocratic- Democratic Gap, Working Paper, ttp:// Lódola, A., M. Busso and F. Cerimedo (2000), Sesgos en el índice de precios al consumidor: el sesgo plutocrático en Argentina, working paper, Universidad Nacional de La Plata. Muellbauer, J. (1974), Te Political Economy of Price indices, Birbeck Discussion Paper no 22. Newberry, David M. (1994), Te Distributional Impact of Price Canges in Hungary and te UK, University of Cambridge Discussion Papers on Economic Transition: DPET Prais, S. (1958), Wose Cost of Living?, Review of Economic Studies, 26: Ruiz-Castillo, J., E. Ley and M. Izquierdo (1999), Te Plutocratic Bias in te CPI: Evidence from Spain, Madrid: FEDEA Working Paper 99/15. ttp:// Sorrocks, A.F. (1984), Inequality Decomposition by Population Subgroups, Econometrica, 52(6): Yaav, J. and S. Yitzaki (1991), On te Appropriate Index for Cost-of-Living Adjustment, in E. Helpman and Y. Natan (eds) Studies in Israel s Economy, 1989, Jerusalem: Te Israel Economic Association. 4