imf - whose inflation

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IMF Institute 700 19th Street NW Washington DC 20431 Phone: (+1-202) 623 6107 Fax: (+1-202) 589 6107

Whose Inflation?

A Characterization of the CPI Plutocratic Gap

Eduardo Ley

IMF Institute

Abstract. Prais (1958) showed that the standard CPI computed by most statistical agenciescan be interpreted as a weighted average of household price indexes, where the weight of eachhousehold is determined by its total expenditures. In this paper, we decompose the differencebetween the standard CPI and a democratically weighted index (i.e., the CPI plutocratic gap)as the product of expenditure inequality and the sample covariance between the elementaryindividual price indexes and a parameter which is a function of the expenditure elasticity ofeach good. This decomposition allows us to interpret variations in the size and sign of theplutocratic gap, and also to discuss issues pertaining to group indexes.

Keywords. Consumer price index, plutocratic index, democratic index, group index, aggregation,equivalence scales, inflation.

JEL Classification System. C43, D31, D63

The answer to the question what is the Mean of a given set of magnitudescannot in general be found, unless there is given also the object for the sakeof which a mean value is required. There are as many kinds of averages as

there are purposes; and we may almost say in the matter of prices as manypurposes as writers. Hence much vain controversy between persons who areliterally at cross purposes. To use a metaphor which has been applied tometaphysics, one party makes a good stroke at billiards, and thinks he hasscored off another who is playing chess. [F.Y. Edgeworth (1888), p. 346.]

1. Introduction

The Hong Kong Census and Statistics Department routinely reports three con-sumer price indexes, by income bracket, along with an overall consumer priceindex (CPI).1 CPI-A is based on the expenditure patterns of the bottom 50 per-cent of the population, CPI-B uses the next 30 percent, and CPI-C is for the next

I thank Javier Ruiz-Castillo for numerous conversations and stimulating discussions on thesubject of this paper, and Mario Izquierdo for help with the computations. I received valuablecomments from Ernst Berndt, Alfredo Cuevas, David S. Johnson, Marshall Reinsdorf, RafaRepullo, Hal Varian, Shlomo Yitzhaki, and the participants at seminars at the IMF Institute, theInter-American Development Bank, the Bank of Spain, Universidad Complutense de Madrid,University of Wisconsin-Madison, the XVIII Latin American Econometric Society Meetings(July 2628 2001, Buenos Aires), and at the CRIW/NBER workshop on Output and PriceMeasurement (July 3031 2001, Boston MA).

1 Prior to June 1999, these indexes were computed and reported by Hang Seng Bank, a private

Email: [email protected] Version: February 9, 2003 3:47 P.M.


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10 percent. The composite CPI takes into account the expenditure patterns of allthese households taken together which cover 90 percent of the population. Forthe year 2000, while inflation (deflation) measured by the overall index was 3.7percent, inflation rates of the group indexes were, respectively: 2.8, 3.8, and4.5 percent. Thus, differences in inflation rates by income group can be quitesubstantial. In most countries, however, a single CPI is reported. Even whenmore than one index is made available by the statistical agency, a single one isoften used as an inflation gauge from a macroeconomic policy perspective. Howrepresentative is, in general, the official inflation rate, as measured by the CPI?

It is known since Prais (1958) that the CPI computed by statistical agenciescan be interpreted as a weighted average of household price indexes.2 The weightof each household is given by its total expenditure, hence the term plutocraticindex. Alternatively, we could construct a democratically-weighted index, whereeach household weighs the same. We shall define the CPI plutocratic gap as thedifference between the plutocratic index and the democratic one. Whether price

behavior in a given period hurts relatively more the better-off or the worse-offhouseholds can be expressed in terms of this single scalar (Fry and Pashardes,1985).

This paper investigates the sources of possible discrepancies between pluto-cratic and democratic indexes.3 We show that the plutocratic gap can be ex-pressed as the product of a measure of variation of household expenditures, andthe sample covariance between the elementary individual price indexes and thecorresponding goods expenditure-share regression coefficient on household expen-diture. This coefficient, in turn, is a function of the expenditure elasticity of eachgood. Consequently, because the decomposition is multiplicative, three elementsare required for the gap between plutocratic and democratic indexes to exist.First, there must be some dispersion in the distribution of expenditure acrosshouseholds. Second, there must be differences in behavior among households atdifferent expenditure brackets. And, third, there must be differences in behaviorin prices. This paper ascribes mathematical quantities to these three elements.

entity. Methodological information on the elaboration of CPIs (and other economic data) canbe obtained from the IMF Data Dissemination Standard website, http://dsbb.imf.org/.

2 J.L. Nicholson derived similar results about the same time, which were later published asNicholson (1975). See Diewert (1983) for a systematic exposition of results pertaining cost-of-living indexes.

3 On the welfare foundations for aggregate indexes see Pollak (1981) and Fisher (2002). Fisher(2002) considers an infinitesimal change in prices, and looks at the problem of minimizingtotal aggregate expenditure subject to (i) mantaining a given level aggregate welfare, and (ii)preserving the initial distribution of nominal household expenditures. In this framework, heshows that both democratic and plutocratic indexes can only be justified if either (i) exactaggregation is possible, or (ii) the initial distribution of household expenditures is consideredoptimal by the planner. In addition, the democratic index requires this initial distribution tobe egalitarian. In this framework a democratic index would never be justified except when itcoincides with the plutocratic index.

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The gap decomposition allows us to interpret the empirical results obtained onthe size and the sign of the plutocratic gap, and suggests that averaging the gapover long time periods may be misleading.

This paper is organized as follows: Section 2 presents analytical results regard-

ing the plutocratic and democratic budget shares, and relates these new resultswith the approximation in Prais (1958). Section 3 derives a characterization of theCPI plutocratic gap, interprets the empirical evidence under this decomposition,and discusses issues related to group indexes. Section 4 presents an alternativeapproach based on weighting each household proportionately to the number of itsmembers, and section 5 concludes.

2. Plutocratic and Democratic CPI budget shares

The plutocratic budget shares for good i in the aggregate CPI are given by

sPi =

1

Xh

xh

shi , (1)

where xh denotes household h total expenditures, xhi is the expenditure on goodi, so that household h budget share for good i is given by shi = x

hi /x

h. Totalaggregate expenditure is given by X =

xh.4 (See Table 1 for a summary of the

notation used.) The CPI (at time t) is given by5

CP IP =i

sPi Ii, (2)

where Ii = (pit/pi0) are elementary price indexes. Noting that household hindividual index is given by cpih =

i s

hiIi, the CP I

P in (2) may be interpretedas a representative CPI.6 It is natural to ask then what is the household betterrepresented by the CP IP. Muellbauer (1974) searched for the household whose

4 All expenditure magnitudes shall refer to the household-survey performed at the base period,0. We shall assume that the statistical agency observes prices for all goods at 0. See Ruiz-Castilloet al. (2002) for issues regarding modified Laspeyres indexes when prices become available onlysome time after the household survey, as it is the case in most practical situations. See BLS(1997), chapter 17, for a good description about the computation of the U.S. CPI.

5 Prices are typically sampled over J geographical areas, obtaining the elementary price in-dexes, Iij = (pijt/pij0). The official CPI is given by CP I

P = j

i

SijIij , where the

aggregate shares are computed as Sij = (1/X)

hjxhi

. Noting that xhi

= xhshi

, we have that

CP IP =

h

i(xh/X)sh

iIih, where Iih denotes Iij for the region j where household h is

located.

6 It can be established that CP IP =

pitqi/

pi0qi, where qi is the average consumption

of good i. Consequently, at least in this sense, the CP IP is indeed the CPI of an averageconsumer. Similarly, by simply multiplying the average quantities by H, we obtain that theCP IP is also the CPI of an aggregate consumer.

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budget shares were closest to the sPi aggregate weights in the U.K. CPI, andfound it to be at the 71 percentile in the household expenditures distribution.For the U.S. in 1990, Deaton (1998) estimates that this consumer occupies the75 percentile. Thus, the representative consumer embedded in (2) is inclinedtowards upper-expenditure households.

Table 1. Notation

i = 1, . . . ,N Good subscript

h = 1, . . . ,H Household superscript

j = 1, . . . , J Geographical area subscript

xhi

Expenditure on good i by household h at 0

nh Number of people in household h

xh =

ixhi

Total expenditure by household h at 0

x = (x1, . . . , xH) Distribution of household expenditures, xh

X= h

xh Total aggregate expenditure in all goods at 0

x = X/H Average of total aggregate expenditure across households

shi

= xhi/xh Household hs budget share for good i at 0

pit Price of good i at t

pt = (p1t, . . . , pNt ) Vector of prices at t

qhi

= xhi/pit Quantity of good i purchased by household h at 0

qh = (qh1 , . . . , qhN

) Vector of quantities purchased by h at t

pijt Price of good i in geographical area j at t

Ii = pit/pi0 Good i elementary price index between t and 0

Iij = pijt/pij0 Good i elementary price index between t and 0 in area j

(x, y) Sample covariance between xh and yh, h = 1, . . . ,H

2

= (x, x) Sample variance of x

h

, h = 1, . . . ,H = (x, x)/x Inequality measure of household expenditures

hi

= (qhi qi)/(x

h x)(x/qi) Household h expenditure elasticity for good i

i = (si, x)/(x, x) Regression coefficient of shi on xh

i = (qi, x)/(x, x) Regression coefficient of qhi on xh

hi(p, u) Hicksian demand for good i, at prices p and utility u

c(p, u) =

ipihi(p, u) Cost function of achieving u at prices p

(pt,p0, u) = c(pt, u)/c(p0, u) Konus True cost-of-living index with reference utility level u

(pt,p0,q) = pt q/p0 q Statistical price index with reference bundle q

h Weight of household h in aggregate statistical price index

= (CP I 1) 100 Inflation rate in percentage

G = P

D

plutocratic Gap

Alternatively, we could use democratic budget shares,

sDi =1

H

h

shi , (3)

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where Hdenotes the number of households, to construct a democratically-weightedindex,

CP ID =i

sDi Ii. (4)

(Other possibility, explored later, consists in weighting each household propor-tionately to the number of its members using an equivalence-scale approach.)

From equations (1) and (3), the difference between good-i plutocratic anddemocratic shares in the CPI is given by

(sPi sDi ) =

1

xH

h

(xh x)shi =1

x(x, si) , (5)

where x = X/H is the sample mean of total expenditures, and (x, si) is thesample covariance, across households, of the budget share of good i, shi , and totalexpenditure. We can now rescale the covariance term in (5) and convert it intoa regression coefficient by simply dividing it by the variance of any of the twovariables involved (which would then become the independent variable). Thus,multiplying and dividing the right-hand side of expression (5) by the samplevariance of household total expenditures, 2 = (x, x), we obtain:

(sPi sDi ) = i, (6)

where = (2/x), and i denotes the OLS estimator in the regression given by

(shi sDi ) = i(x

h x) + hi . (7)

Equation (6) indicates that the difference in good is plutocratic and demo-cratic CPI shares depends on the product of: (i) a measure of inequality of

household expenditure, ; and (ii) a measure of how good is budget share varies

with total expenditure in the household sample, i. Since the decompositionis multiplicative, the shares must coincide when there is no inequality in totalexpenditures or when expenditure shares are not affected by those differences.

It is important to note that no distributional or behavioral assumption isneeded to obtain i, because we can always estimate the regression coefficient inequation (7). Of course, if assumptions are made, then different interpretationscould be given to the parameters involved. For now, however, we simply want tostress that the decomposition in (6) holds because of algebraic identities, and doesnot rely on any assumptions on consumer behavior or household-expenditures dis-tribution.

Note that = 2xI2(x), where I2(x) corresponds to the Generalized Entropyinequality measure, Ic(x), for c = 2. The parameter c summarizes the sensitivityof Ic in different parts of the household total expenditures distribution: the morepositive (negative) c is, the more sensitive Ic is to differences at the top (bottom)

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of the distribution (Cowell and Kuga, 1981). Inequality indexes belonging tothe Generalized Entropy family are the only measures of relative inequality thatsatisfy the usual normative properties required for an inequality index and, inaddition, are decomposable by inequality subgroups (Shorrocks, 1984). Finally,

using the fact that i

shi

= 1, it follows from equation (6) that i

= 0 (so that

if, for some i, we have i > 0 we must also have j < 0 for some j).

2.1. Expenditure Elasticity and Prais Results

For any observed variable, yh, we can compute its distance from the populationaverage, yh = (yh y), where y = 1

H

yh is the sample mean ofyh. Assuming

that all households (in the same geographical area) face the same prices, we cancompute a sample analogue of the elasticity of the budget share of good i withrespect to total expenditures, by computing the ratio of percent deviations acrosshouseholds from average quantities:7

shixh

x

si= (hi 1), (8)

where:

hi =qhixh

x

qi, (9)

is a sample analogue of good is expenditure elasticity showing the percentdeviation of household hs consumption of good i from the average consumption,qhi , divided by household hs percent deviation from average expenditure, x

h.

Following Olkin and Yitzhaki (1992), we can rewrite the standard OLS esti-mator in (7) as a weighted average of slopes:

i =h

hshixh

=six

h

h(hi 1), (10)

where the normalized weights are given by8 h =

xh2

/H2. Equation (10)suggests the following estimator of the good-i expenditure elasticity:

i = h

hh

i

=x

qih

hqhi

xh=

x

qii, (11)

7 The differential counterpart can be easily obtained by noting that log(shi

) = log(pi) +

log(qhi

) log(xh) and differentiating with respect to log(xh) in order to obtain the good-ibudget-share expenditure elasticity.

8 Note that this weighting scheme is plutocratic-squared! See Yitzhaki (1996) for a discussionof issues involving regressions on expenditure in the context of welfare economics.

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where i is the OLS coefficient of the regression of qhi on x

h. Equation (11)allows us to rewrite equation (10) as:

i =six

(i 1), (12)

which implies that:i > 0 i > 1. (13)

Thus, the sign ofi is determined by the empirical estimate of good is expenditureelasticity i (equation (11)). Superior goods, displaying an expenditure elasticitylarger than 1, will be associated with positive coefficients. Necessities, on theother hand, will have negative regression coefficients. Using (12), we can nowexpress equation (6) as:

(sPi

sDi

) =si

x(i 1). (14)

Ifhi = i for all households which is guaranteed by constant-elasticity Engelcurves, as assumed in Prais (1958) then, using (14) we find that the percentagedifference between plutocratic and democratic shares is given by:

sPi sDi

sDi=

x(i 1) =

2

x2(i 1), (15)

which is related to the expression obtained by Prais (1958) (his equation (8) on

pages 127 and 134): sPi sDi

sPi 2(i 1). (16)

Note, however, that while (16) is an approximation, equation (15) is exact. Theapproximation in (16) requires assuming that xh is lognormally distributed, andit relies on a first-order Taylor expansion of exponential functions.9 Our analysisshows that no distributional assumption on household expenditures is necessaryto obtain (15).

Finally, note that if exact aggregation is possible, at least locally, because in-direct utility functions take the Gorman generalized polar form (Gorman, 1959),

then shi is identical for all households, and consequently any averaging of individ-

ual shares leads to the same answer. Consequently plutocratic and democraticCPIs coincide.

9 See the Appendix in Prais (1958) for details in his approximation. Lognormality implies2 = log(1 + ), where = (/E(x))2 is the coefficient of variation squared. Using a first-order Taylor expansion, we have that 2 = log(1 + ) , and using (15) we can write:((sP

i sD

i)/sD

i) 2(i 1).

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3. The CPI plutocratic Gap

As discussed before, we shall define the plutocratic gap, G, as:

G P D

100= CP I

P CP ID , (17)where = (CP I 1) 100 is the inflation rate between 0 and t (in percent);using (5) we find:

G =i

(sPi sDi )Ii =

i

iIi = i

i(Ii I), (18)

where I is a simple average, i.e., I = 1N

Ii, which represents simple average

inflation. Equation (18) may be rewritten as:

G = N (,I) , (19)

where (,I) refers to the sample covariance of i and Ii, this time over goods

instead that over households.Equation (19) is our fundamental result. It shows that the plutocratic gap

is determined by the dispersion of household expenditure, measured by , andthe sample covariance between i and Ii. The sign of the plutocratic gap isdetermined by the covariance term. A positive covariance term means that thegoods favored by the richer households experience higher than average inflationand necessities a lower than average inflation. Similarly, a negative covarianceimplies that necessities experience higher than average inflation while superior orluxury goods experience lower than average inflation. These effects are also scaledby the magnitude of the inequality of household expenditures, as measured by .

Ceteris paribus, the higher the dispersion in household expenditures, the higherthe size of the plutocratic gap.

Inspection of equation (19) indicates that three elements are required for theplutocratic gap to be different from zero: (a) there must be some dispersion in

the distribution of household expenditures (reflected by = 0); (b) there mustbe some observed behavioral differences among households with different totalexpenditures (reflected by i = 0 for some i); and (c) there must be some dif-ferences in price behavior across some goods which display behavioral differencesacross households (reflected by Ii = I for some i which has i = 0). These threeconditions are necessary for G = 0, and if they all hold we must have G = 0 i.e.,they are also jointly sufficient.

In practical situations, however, not all households face the same prices. Foreach good i, the statistical agency collects j prices, pijt , one for each geographicalarea j. This can be readily accommodated within the present analysis by thinkingof item i in different areas as different goods. We would expand the good spaceto include N J goods. As a result, shi will be always zero for the goods outsidethe geographical area where household h resides. All the previous analysis applieswithout further changes.

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3.1. Empirical Estimates of the CPI plutocratic Gap

Table 2 summarizes the main findings of various empirical studies that have es-timated the plutocratic gap for different countries during various time periods.10

Care must be applied to cross-country comparisons because the range of publicly-

provided goods (e.g., health care or housing) varies widely across countries. Inaddition, to the extent that age rather than income may determine elegibility forpublic subsidies adds another level of complexity.

Table 2. Empirical Studies of the CPI plutocratic Gap

Range (percentage points per year)

Country Time Period N P G

Carruthers et al. (1980) U.K. 197578 11 8.2 to 24.2 0.1

Fry and Pashardes (1985) id. 197482 95 8.2 to 24.2 negative

Deaton and Muellbauer (1980) id. 197576 10 14.5 2Crawford (1996) id. 197992 74 3.4 to 18.0 +0.16

Newbery (1995) id. 1980s 87 3.4 to 18.0 slightly positive

ibid. Hungary 1980s 87 4.5 to 16.9 slightly positive

Kokoski (1987) U.S. 197280 146 3.3 to 13.5 0.1 to 0.3

Erbas and Sayers (1998) id. 198695 7 1.9 to 5.4 negative

Garner et al. (1999) id. 1980s 207 1.9 to 13.5 slightly negative

Kokoski (2000)a id. 198797 146 2.0 to 5.25 0.28 to +0.56

Lodola et al. (2000) Argentina 198991 9 220 to 10,781 +2.3 to +663.4

ibid. id. 199193 9 11.2 to 20.0 0.66 to 0.78

ibid. id. 199398 9 1.2 to 3.3 0.48 to +0.65

Yahav and Yitzhaki (1991) Israel 196071 10 1.99 to 12.06 0.12 to +0.25ibid. id. 198186 28 19.9 to 373.8 1.7 to +6.3

Ruiz-Castillo et al. (2003) Spain 197381 57 14.54 to 23.02 0.04 to +0.53

ibid. id. 198191 58 4.59 to 9.48 0.19 to +0.30

ibid. id. 199198 2,042 2.49 to 6.99 0.08 to +0.15

Source: Studies cited, IMF Government Financial Statistics and authors calculations.a This paper has a typo in its Table 2: the column headings Democratic and Plutocratic should be switched.

Given a household survey, and the is are then fixed, and any source of

variation in the sign and size of the gap for, e.g., each year must be solely explained10 We are unable to provide estimates sampling variability for the different inflation rates be-cause elementary price data is not publicly available. For Spain, the sampling variability ofhousehold expenditure shares would imply uncertainty well to the right of the third decimal ofannual inflation rates expressed as percentages. However, the variability from price sampling islikely to outweight the variability due to the household shares. For the U.S., the best estimatesfor the standard error of inflation rates (expressed as percentages) is of the order of 0.06 to 0.1;see Leaver and Valiant (1995) and Leaver and Cage (1997).

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by the price behavior reflected by the Iis. The movements in the Iis may cause(,I) to change sign from one year to another (Table 3 below). Thus, as noted,looking at the overall G, simply averaging over a long period may be misleading.

Because of data limitations, most of the results in Table 2 are based on a

substantially smaller number of goods than the number for which prices werecollected by the statistical agencies (column N). In particular, most studiesdo not have information on geographical price variation, they assume that thesame national average CPI prices apply to all households in the sample, andfocus on the effect of expenditure shares variability across households. Ruiz-Castillo et al. (2003), uses 2,042 goods for the 1990s (see below), but only 58and 57 for the 1980s and 1970s.As a result, working with highly aggregated goodscauses an underestimation of the true plutocratic gap for two reasons. First, priceaggregates already embody a plutocratic gap. Second, expenditure elasticitiesrevert to the mean (i.e., to one) as we aggregate goods. As a result, the true sizeof the plutocratic gap is underestimated.11

For Spain during the 1990s, Ruiz-Castillo et al. (2003) estimate that the averageplutocratic gap in Spain amounts to 0.055 percent per year.12 However, as shownin Table 3, annual gaps are typically larger, and price movements significantlychange the sign and magnitude of the annual gap. The results in Table 3 arebased on 2118+3252 = 2, 042 different Iij s; 21 food goods in 18 autonomouscommunities and 32 non-food goods in 52 provinces.13

Thus, as discussed above, the sign and magnitude of the gap may vary sig-nificantly year after year, even when using the same budget survey. As a result,finding the gap small during one particular period has little bearing over its sizeand sign at other time when prices may behave differently. For different household

surveys, not only the price dynamics may change, but also expenditure inequalitymay be different (e.g., was 2% larger for Spain in 198081). As a result, find-ings for one country may have little implications for other countries with largerincome inequality and different price dynamics. For instance, income inequality

11 Algebraically, in equation (18), the first effect implies that the Iis are artificially close to

CP IP. The second effect implies that the is are artificially close to zero. The end result is that

the size of(, I) shrinks towards zero, producing an underestimation of the true plutocraticgap, G.

12 The results in Table 3 are based on the Spanish household budget survey collected by theSpanish statistical agency 199091. This is a household budget survey of 21,155 householdsample points, representative of a population of approximately 11 million households and 38million persons occupying residential housing in all of Spain. The survey was collected fromApril 1990 to March 1991.

13 The statistical agency collects elementary price indexes for a commodity basket consisting of471 items in 130 municipalities spread over the 52 Spanish provinces under the CPI present sys-tem, based in 1992. Approximately 150,000 prices are collected each month from approximately29,000 establishments. Price information at this disaggregated level is not publicly available.Prices are generally collected once a month at each establishment, except for perishable itemswhich are collected three times a month from all establishments.

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Table 3. Decomposition of the CPI plutocratic Gap: Spain 199397(All values in percentage points)

P D N(,I) G

1993 5.271 5.165 1.27E+06 8.251E-08 0.1051994 4.621 4.701 1.27E+06 -6.286E-08 -0.080

1995 4.079 4.130 1.27E+06 -3.929E-08 -0.050

1996 3.180 3.090 1.27E+06 7.072E-08 0.090

1997 2.494 2.369 1.27E+06 9.823E-08 0.125

x = 2.56E+6, = 1.81E+6, N = 2, 042

in Latin America is very large, IDB (1998) reports that countries in the regionexperience the largest income inequality in the World. It is very likely then that

the CPI plutocratic gap be of a larger significance in Latin America, especiallyin countries with double-digit inflation that may have more differentiated pricedynamics.14

3.2. Group Indexes

The approach presented here may also be used for group indexes i.e., indexesreferring to a group or population of households (Pollak, 1980). Equation (6)implies that the plutocratic shares for the whole population are given by sPi =

si + i, while plutocratic shares for a subgroup G

sGi = sGi +

G Gi , (20)

where the G superscript means that the quantities have been computed for house-hold h G. Then, we have that:

CP IP CP IG =i

(si s

Gi ) + (i

G Gi )

Ii. (21)

The term (si sGi ) picks up differences in average spending patterns. The term

(i G Gi ) reflects differences in total expenditures inequality and behavioral

responses to household total expenditures. If the group is very homogeneous inincome, so that G 0, then:

CP IP

CP IG

=i

(si s

Gi ) + i

Ii = G+

i

(si sGi )Ii. (22)

14 Interestingly, the CPI weights in Ecuador are computed excluding the richer households be-cause of the large dispersion in their consumption patterns see http://www.inec.gov.ec/pry15.htm.(In the U.K., the index weight calculation excludes the top 4% of the population by income andalso pensioners mainly dependent on state benefits.) Nonetheless, in Ecuador, as is typicallydone in many countries (including the U.S.), the household survey on which the CPI is basedis restricted to urban areas.

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Thus, the difference between the overall CPI and the group index would dependon the difference of average spending patterns and the term of the plutocraticgap.

There are two important issues regarding group indexes that fall outside the

framework developed in this paper which implicitly assumes that goods are welldefined. In reality the prices for all existing commodities and services cannotbe sampled, and statistical agencies must choose specific items to represent cate-gories of goods. Thus, a fundamental issue pertains to the selection of items thatrepresent particular goods. Take, for instance, the alcoholic beverages expen-diture category; the statistical agency may chose to follow the price of domesticbeer, imported beer, or, say, champagne (Pollak, 1998). Each choice may be morerelevant for a particular population group, so the decision of which goods to followhas implications for whose inflation is being measured. A related issue concernsthe location where the prices are sampled; as long as distinct population groupsmay shop at different places, again the selection of outlets may implicitly shape

whose inflation is being targeted.While the Hong Kong indexes mentioned in the introduction are obtained

by simple reweighting,15 there are instances where statistical agencies make aneffort to adequately address these two issues. The Indian Ministry of Statisticsand Programme Implementation computes four different CPIs: CPI-IW, CPI-RL,CPI-AL, and CPI-UNME. The CPI-IW covers households headed by industrialworkers in 70 industrial centers following 260 items and sampling approximately160,000 retail price quotes from 16,545 outlets and selected open markets. TheCPI-RL covers all rural households and it is compiled for 20 states and for allIndia, covering 85 to 106 items; 61,005 monthly price quotes are collected fromretail outlets in 600 villages. The CPI-AL, similar to the RL, except that it coversonly households headed by agricultural laborers. Finally, the CPI-UNME covershouseholds headed by non-manual workers in 59 urban centers sampling 1022price quotes on a varying number of items depending on the center, from 146 inImphal to 345 in Delhi. Inflation rates may differ significantly for these indexes.For instance, in January 2001, the annual inflation rates were 3.3, 2.0, 1.6,and 5.8 percent, respectively.

4. Alternative Weighting Schemes

For any family of household weights, h(), parametrized by , we can define

si() =

h

h

()s

h

i , and construct CP I() =

i si()Ii. Defining G() =(CP IP CP I()) leads to the following generalization of equation (19):

G() = N{(,I) () ((),I)} =i

(x, si)

x

((), si)

()

Ii (23)

15 France also publishes a CPI for blue-collar employees which is computed by reweighting.

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where is the inequality index applied to h(), and i() is the OLS estimateof the regression of si() on

h().

As suggested by Nicholson (1975), instead of weighting equally each household,an alternative approach is to consider explicitly the number of members in each

household using an equivalence-scale approach. Equivalence scales are used inempirical studies of consumption behavior to take into account economies of scalein household composition. Following Buhmann et al. (1988) we could adopt anequivalence scale model in which scale economies in consumption depend only onhousehold size. Let nh be the number of members of household h; then h() (nh), with [0, 1], can be interpreted as the number of equivalent adults in ahousehold of size nh. We have h(0) = 1/H, which would weight each householdequally as in our democratic index. At the other extreme, h(1) nh wouldsimply represent the number of members for a super-democratic index.

Table 4.

Alternative Aggregation Schemes: Spain 199397(All values in percentage points)

Equivalence-Scale Weighting: h() (nh)

G()

P = 0 = 0.25 = 0.50 = 0.75 = 1

1993 5.271 0.105 0.098 0.090 0.083 0.077

1994 4.621 -0.080 -0.065 -0.052 -0.042 -0.034

1995 4.079 -0.050 -0.033 -0.018 -0.004 0.009

1996 3.180 0.090 0.088 0.087 0.086 0.085

1997 2.494 0.125 0.121 0.117 0.113 0.110

Equivalent Household Expenditures: h() xh/(nh)

G()P = 0 = 0.25 = 0.50 = 0.75 = 1

1993 5.271 0.000 0.004 0.007 0.010 0.013

1994 4 .621 0.000 -0.009 -0.020 -0.034 -0.050

1995 4 .079 0.000 -0.010 -0.020 -0.032 -0.045

1996 3 .180 0.000 -0.002 -0.004 -0.007 -0.010

1997 2 .494 0.000 -0.002 -0.006 -0.009 -0.014

Muellbauer Homogeneous Social Weights: h() (xh)(1)

G()

P = 0 = 0.25 = 0.50 = 0.75 = 1

1993 5.271 0.000 0.027 0.053 0.080 0.105

1994 4 .621 0.000 -0.015 -0.033 -0.055 -0.080

1995 4 .079 0.000 -0.007 -0.017 -0.032 -0.050

1996 3.180 0.000 0.023 0.045 0.067 0.090

1997 2.494 0.000 0.029 0.060 0.091 0.125

The top panel in table 4 shows G() for different values of , for Spain in

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the 1990s, when h() (nh). The magnitude of the gap decreases with .Moreover, there is even a sign reversal, for 1995, with G(1) > 0 while for all othervalues of the gap, G(), is negative. The results in Table 4 are a consequenceof the fact that, as it is to be expected, for Spain, in that time period, householdsize and total expenditure are correlated. Nonetheless, even for = 1, the size ofthe gaps in Table 4 is not negligible. Another possible weighting scheme could bebased on equivalent household expenditures, h() xh/mh(), which results inthe plutocratic index for = 0 (second panel of table 4).

Finally, Muellbauer (1974) proposes a class ofhomogeneous social price indexesparameterized by a measure of aversion to inequality, [0, 1]. The householdweights are now proportional to (xh)(1), which reduces to democratic weightswhen = 1, and plutocratic weights when = 0. Table 4 shows the gaps forthis aggregating scheme with respect to the plutocratic CPI. It is interesting tofocus on the polar cases. Both indexes coincide with the plutocratic Laspeyreswhen = 0, so the corresponding gap is zero. The last column for the Muelbauer

aggregation scheme corresponds to a democratic index i.e., it coincides withfirst column in the top panel.

5. Concluding Remarks

What is the appropriate inflation gauge from a macroeconomic perspective? Howshould we adjust, e.g., tax brackets, public pensions, or social programs transfersannually?16 BLS (1997, p. 172) warns CPI users should understand that the CPImay not be applicable to all questions about price movements for all populationgroups. Nevertheless, in most places and in most times, these quantities areinvariably revised according to a plutocratic CPI. Thus, a dollar-weight logicprevails over a household-weight logic.

Escalating transfer payments by the plutocratic CPI may result in over- orunder-compensation relative to a democratic index during different time periods.While these deviations may tend to cancel off over longer horizons, there is, how-ever, an important perversity emphasized by Ruiz-Castillo et al. (2003). The plu-tocratic gap in the CPI often accentuates the change in household welfare ratherthan smoothing it. In effect, the worse-off households suffer under-adjustmentswhen inflation is more harmful to them i.e., when they can least afford it.In periods where the plutocratic gap is negative (when prices behave in an waymore detrimental to the poorer households) then social programs, which primarily

benefit the poor, are revised less than what would be the case with a democraticgroup index. Similarly, when price movements are less detrimental to the poorerhouseholds i.e., when the plutocratic gap is positive indexed social transfersgrow more than cost-of-living adjustments would dictate. Thus, plutocratic-CPI

16 See Triplett (1983), Fry and Pashardes (1985), Griliches (1995), Pollak (1998), and Jorgensonand Slesnick (1999).

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adjustments display harmful procyclical features.17

Nonetheless, the plutocratic CPI has its own merits. It naturally arises whencomputing the aggregate Laspeyres price index, and it is consistent with aggregatedeflators arising from the national accounts. It also provides an upper bound for

the theoretical aggregate compensating variation (Hicks, 1940) i.e., by howmuch would monetary national income need increase to compensate for a pricevariation. Plutocratic weights would also arise if we were to draw prices at randomin such a way that each dollar of expenditure had an equal chance of being selected(Theil, 1967; p. 136).

While different indexes could be easily computed for different uses, Prais (1958,p. 131) asked: Can more than one index numbers be tolerated without confusion?There is a crucial tradeoff between the simplicity of the current prevailing one-size-fits-all approach and the conceptual superiority of a piecemeal-menu approachto index numbers. The best resolution may well vary in different places andat different times. This paper shows that the larger the income (expenditure)

inequality, the more different the consumption patterns by income group, andthe larger the variance in individual price behavior, the less appealing is a singleplutocratic CPI as the only policy adjuster. Finally, if a single index number isto be computed, then as Prais (1958, p. 126) asked: Whose cost of living shouldone have in mind?

6. References

BLS (1997), Handbook of Methods, Washington DC: U.S. Department of Labor.

Buhmann, B., L. Rainwater, G. Schmauss and T. Smeeding (1988), Equivalence

Scales, Well-Being, Inequality and Poverty: Sensitivity Estimates Across TenCountries Using the Luxembourg Income Study Database, Review of Incomeand Wealth, 34: 115142.

Carruthers, A., D. Sellwood and P. Ward (1980), Recent Developments in theRetail Price Index, The Statistician, 29: 132.

Crawford, Ian (1996), UK Household Cost-of-Living indexes, 1979-92, in J.Hills (ed), New Inequalities: the Changing Distribution of Income and Wealthin the United Kingdom, Cambridge: Cambridge University Press.

Cowell, F.A. and K. Kuga (1981), Additivity and the Entropy Concept: An Ax-iomatic Approach to Inequality Measurement, Journal of Economic Theory,

25, 131143.17 For the richer households, since the plutocratic index is always closer to their true indexthan a democratic one, this problem is unlikely to be too important. However, CPI adjustmentsalso display procyclicality. Possibly the most important CPI adjustment involving the richerhouseholds involves the revision of income-tax brackets. In this case, when inflation is moredetrimental to the richer households, the plutocratic CPI will be below the true inflation of therich, and they would pay too much in taxes. Conversely, they will pay too little when inflationis less detrimental to them.

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Deaton, A. (1998), Getting Prices Right: What Should Be Done?, Journal ofEconomic Perspectives, 12: 37-46.

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Edgeworth, Francis Y. (1888), New Methods of Measuring Variation in GeneralPrices, Journal of the Royal Statistical Society, 51:2, 346368.

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Consumer Price Index, Senate Hearing 104-69, Washington D.C.: U.S. Print-ing Office.

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IDB (1998), Facing Up to Inequality in Latin America. Economic and SocialProgress in Latin America, 199899 Report; Washington DC: Inter-AmericanDevelopment Bank.

Jorgenson, D.W. and D.T. Slesnick (1999), Indexing Government Programs forChanges in the Cost of Living, Journal of Business and Economic Statistics,16:2, 170181.

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Labor Review, vol. 123, 3139.

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Leaver, S. and R. Valiant (1995), Statistical Problems in Estimating the U.S.Consumer Price Index, in Business Survey Methods, B. Cox et al. (eds) New

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York: Wiley.

Leaver, S. and R. Cage (1997), Estimating the Sampling Variance for AlternativeEstimators of the U.S. CPI, Proceedings of the Government Statistics Section,American Statistical Association, Alexandria, VA: ASA.

Lodola, A., M. Busso and F. Cerimedo (2000), Sesgos en el ndice de precios alconsumidor: el sesgo plutocratico en Argentina, working paper, UniversidadNacional de La Plata.

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Nicholson, J.L. (1975), Whose Cost of Living?, Journal of the Royal StatisticalSociety, Series A, 138:4, 5402.

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, (1981), The Social Cost-of Living Index, Journal of Public Eco-nomics, 15:3, 31136.

, (1998), The Consumer Price Index: A Research Agenda and ThreeProposals, Journal of Economic Perspectives, 12: 69-78.

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Spanish CPI, forthcoming Applied Economics., (2003), The Plutocratic Gap in the CPI: Evidence from Spain, forth-

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Shorrocks, A.F. (1984), Inequality Decomposition by Population Subgroups,Econometrica, 52(6): 136985.

Theil, Henri (1967), Economics and Information Theory, Amsterdam: North-Holland.

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