Prais (1958) showed that the standard CPI computed by most statistical agencies can be interpreted as a plutocratic weighted average of household price indexes because the weight of each household in the official CPI is determined by its total expenditures. In this paper, we decompose the difference between the standard CPI and a democratically weighted index as the product of a measure of income inequality and the sample covariance between the elementary individual price indexes and a parameter which is a function of the income elasticity of each good.

Eduardo Ley

English

Research Papers in Economics

On Plutocratic and Democratic CPIs 
Eduardo Ley
IMF Institute
Abstract
Prais (1958) showed that the standard CPI computed by most statistical agencies can be
interpreted as a plutocratic weighted average of household price indexes because the weight
of each household in the official CPI is determined by its total expenditures. In this paper, we
decompose the difference between the standard CPI and a democratically weighted index as
the product of a measure of income inequality and the sample covariance between the
elementary individual price indexes and a parameter which is a function of the income
elasticity of each good.
I thank Javier Ruiz−Castillo for numerous conversations and stimulating discussions on the subject of this paper, and Mario
Izquierdo for most valuable help with the computations. I received very useful comments from Woon Gyu Choi, Nuri Erbas,
Roberto Garcia−Saltos, Josh Greene, David S. Johnson, Mohsin Khan, Marshall Reinsdorf, Rafa Repullo, Hal Varian and
Shlomo Yitzhaki, and the participants at seminars at the IMF Institute, IADB, the XVIII Latin American Econometric Society
Meetings (July 2001, Buenos Aires), and at the CRIW/NBER workshop 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, 2001.  Accepted: January 19, 2002.
URL: http://www.economicsbulletin.com/2002/volume4/EB−01D60003A.pdf1. Introduction
It is known since Prais (1958) that the CPI computed by statistical agencies can be inter-
preted as a weighted average of household price indexes. The weight of each household is
given by its total expenditure, hence the term ‘plutocratic index.’ Alternatively, we could
construct a democratically-weighted index, where each household weights the same. We
shall deﬁne the CPI plutocratic-democratic gap as the diﬀerence between the plutocratic in-
dex and the democratic one. Whether price behavior in a given period hurts relatively more
the better-oﬀ or the worse-oﬀ households can be expressed in terms of this single scalar
(Fry and Pashardes, 1985). Various studies have computed the plutocratic-democratic gap
for diﬀerent countries.1
This paper investigates the sources of possible discrepancies between plutocratic and
democratic indexes. We show that the plutocratic-democratic gap can be expressed as the
product of mean income, a measure of variation of household expenditures, and the sample
covariance between the elementary individual price indexes and the corresponding good’s
expenditure-share regression coeﬃcient on household income. This coeﬃcient, in turn, is
a function of the total expenditures elasticity of each good (Ley, 2001).
2. Plutocratic and Democratic CPI budget shares
Deﬁne the plutocratic budget shares for good i in the aggregate CPI by
˜ sP
i =
1
X
X
h
xhsh
i , (1)
where xh denotes household h total expenditures, xh
i is the expenditure on good i, so that
household h budget share for good i is given by sh
i = xh
i /xh. Total aggregate expenditure
is given by X =
P
xh. The CPI (at time t) is given by CPIP =
P
i ˜ sP
i Ii, where Ii =
(pit/pi0) are elementary price indexes. Noting that household h individual index is given
by cpih =
P
i sh
i Ii, the CPIP may be interpreted as a ‘representative’ CPI. It is natural
to ask then what is the household better represented by the CPIP. Muellbauer (1974)
searched for the household whose budget shares were closest to the ˜ sP
i aggregate weights
in the U.K. CPI, and found 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
the 75 percentile. Thus, the ‘representative’ consumer embedded in the CPIP is biased
towards upper-income households.
Alternatively, we could use democratic budget shares,
˜ sD
i =
1
H
X
h
sh
i , (2)
1 See, e.g., Carruthers et al. (1980), Fry and Pashardes (1985), Deaton and Muellbauer (1980), Crawford
(1996), Newberry (1994), Kokoski (1987, 2000), Erbas and Sayers (1998), Garner et al. (1999), L´ odola et
al. (2000), Yahav and Yitzhaki (1991), Ruiz-Castillo et al. (1999). See also Ley (2001) for a summary of
ﬁndings.where H denotes the number of households, to construct a democratically-weighted index,
CPID =
P
i ˜ sD
i Ii.
From equations (1) and (2), the diﬀerence between good i plutocratic and democratic
shares in the CPI is given by
(˜ sP
i − ˜ sD
i ) =
1
¯ xH
X
h
(xh − ¯ x)sh
i =
1
¯ x
ˆ σ(x,si), (3)
where ¯ x = X/H is the sample mean of total expenditures, and ˆ σ(x,si) is the sample
covariance, across households, of the budget share of good i, sh
i , and total expenditure.
Multiplying and dividing the right-hand side of expression (3) by the sample variance of
household total expenditures, ˆ σ2 = ˆ σ(x,x), we obtain:
(˜ sP
i − ˜ sD
i ) = ˆ ζ ˆ βi, (4)
where ˆ ζ = ˆ σ2/¯ x is a measure of variation of household total expenditures, and ˆ βi =
ˆ σ(x,si)/ˆ σ2 (see Ley (2001) for an interpretation of ˆ βi as a regression coeﬃcient and its
relation the income elasticity of good i).
Equation (4) indicates that the diﬀerence in good i’s plutocratic and democratic CPI
shares depends on the product of: (i) a measure of inequality of household total expendi-
tures,2 ˆ ζ; and (ii) a measure of how good i’s budget share varies with total expenditure in
the household sample, ˆ βi. Since the decomposition is multiplicative, the shares must coin-
cide when there is no inequality or when expenditure shares are not aﬀected by diﬀerences
in total expenditures.
3. The CPI plutocratic-democratic gap
As discussed before, we shall deﬁne the plutocratic-democratic gap, G, as: G ≡ (ΠP −
ΠD)/100 =
 
CPIP − CPID
, where Π = (CPI − 1) × 100 is the inﬂation rate between 0
and t (in percent); using (3) we ﬁnd:
G =
X
i
(˜ sP
i − ˜ sD
i )Ii = ˆ ζ
X
i
ˆ βiIi = ˆ ζ
X
i
ˆ βi(Ii − ¯ I), (5)
where ¯ I is a simple average, i.e., ¯ I = 1
N
P
Ii, and N is the number of goods. Equation
(5) may be rewritten as:
G = ˆ ζ N ˆ σ(ˆ β,I), (6)
where ˆ σ(ˆ β,I) refers to the sample covariance of ˆ βi and Ii, this time over goods instead
that over households.
2 Note that ˆ ζ = 2¯ xI2(x), where I2(x) corresponds to the Generalized Entropy inequality measure, Ic(x),
for c = 2 (Cowell and Kuga, 1981).
2Equation (6) is our fundamental result. It shows that the plutocratic-democratic gap
is determined by the dispersion of household total expenditures, measured by ˆ ζ, and the
sample covariance between ˆ βi and Ii. The sign of the plutocratic-democratic gap is deter-
mined by the covariance term. A positive covariance term means that the goods favored
by the richer households experience higher than average inﬂation and necessities a lower
than average inﬂation. Similarly, a negative covariance implies that necessities experience
higher than average inﬂation while superior or luxury goods experience lower than average
inﬂation. These eﬀects are also scaled by the magnitude of the inequality of household
expenditures, as measured by ˆ ζ.
Inspection of equation (6) indicates that three elements are required for the plutocratic-
democratic gap to be diﬀerent from zero: (a) there must be some dispersion in the distri-
bution of household expenditures (reﬂected by ˆ ζ 6= 0); (b) there must be some observed
behavioral diﬀerences among households with diﬀerent total expenditures (reﬂected by
ˆ βi 6= 0 for some i); and (c) there must be some diﬀerences in price behavior across some
goods which display behavioral diﬀerences across households (reﬂected by Ii 6= ¯ I for some
i which has ˆ βi 6= 0).
Given a household survey, ˆ ζ and the ˆ βi’s are then ﬁxed, and any source of variation
in the sign and size of the gap for, e.g., each year must be solely explained by the price
behavior reﬂected by the Ii’s. The movements in the Ii’s may cause ˆ σ(ˆ β,I) to change sign
from one year to another. Thus, as noted in Ley (2001), looking at the overall G, simply
averaging over a long period may be misleading. Moreover, because of data limitations,
most of the empirical results are based on a smaller number of goods than the number for
which prices were collected by the statistical agencies. In these instances, working with
highly aggregated goods causes an underestimation of the true plutocratic-democratic gap
for two reasons. First, price aggregates already embody a plutocratic-democratic gap.
Second, income elasticities revert to the mean (i.e., to one) as we aggregate goods. As a
result, the true size of the plutocratic-democratic gap is typically underestimated. Finally,
Ley (2001) extends this approach by considering explicitly the number of members in each
household using an equivalence-scale approach.
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The Distributional Impact of Price Changes

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The Plutocratic Bias in the CPI: Evidence from Spain”,

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On Plutocratic and Democratic CPIs

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