and Tornqvist indexes were essentially the same—as expected, since the population was
constructed in a manner consistent with the “constant quantities” assumption underlying the
Laspeyres index. The expenditure-weighted population geometric mean indexes also behaved
as expected: G was severely biased down, G2 was similarly biased up, and Gg showed a
moderate downward bias. The equally-weighted geometric mean G, was surprisingly similar
to the Laspeyres, while the Carli and Sauerbeck indexes were predictably biased up.

Table 1. Comparison of Average Laspeyresville Population Indexes

F L P T Aw A Gy Go G.

I 1.45599 | 1.45603 | 1.45595 | 1.45598 | 1.86251 | 1.94884 | 1.09669 | 1.91429 | 1.44750
I-F 0 0.00004 | -0.00004 | -0.00001 | 0.40652 | 0.49285 | -0.35930 | 0.45830 | -0.00849

With respect to composite estimators, Lap, which follows a formula similar to that of the
current CPI, tended to have larger RMSE than Lh, the formula it replaced in the CPI as
of January 1999. Lop had a smaller RMSE only 9/35 times. The composite estimator Gu
performed slightly better than L,,, having a lower RMSE 21/35 times, though these differ-
ences tended to be small. Among all the composite estimators, the unweighted geometric
index Gp,-, which combined Laspeyres sub-indexes, had the lowest RMSE. It was generally
lower than the composite Fisher, exceeding it in only three instances.

Table 2 displays some pure and composite index estimates, along with their RMSE’s and
PMSE’s. Among all sample based indexes, the pure unweighted geometric index G'., though
mildly biased low for the target P, consistently exhibited the lowest RMSE, usually at least
33% lower than the composite Fisher Fy. No other index was as consistently close to the
target population index. The composite Fisher Fy was mildly biased high for the target
F’. For the composite geometric mean indexes, the expenditure weights used at the item
stratum level of aggregation had a more pronounced effect on the aggregate index values
than did the upper level aggregation weights. This is most clearly seen in the differences
between Gea, 1, which is biased up, and Ga, 2, which is biased down. Overall, the method of
lower level aggregation dominated the effect of the method of higher level aggregation. In
Table 2, we see only small differences, for example, between the values of Gr, , and Gr, 2; by
contrast, the differences between Lea and Leo appear quite sharp.

For each of the 50 runs in each experiment, we counted the times certain indexes fell above
or below some others. The pure Laspeyres estimate L exceeded the pure Fisher estimate
about half the time (24.7/50, on average). Similarly, the composite Laspeyres Lx exceeded
the population Fisher index an average of 25.9 out of 50 times. In sharp contrast, Lt lay
above the composite Fisher F, roughly 80% of the time (40.0/50), while the hybrid index
Gia exceeded Fy, only about one third of the time (16.7/50). The difference L,, — Fy, was
generally small—resulting in a low PMSE for Lp—but consistently positive. (In the next
section, we will present a theoretical explanation for this phenomenon.) In general, we found
that the distance to the composite Fisher bore little resemblance to the RMSE. For example,
the PMSE for Ge was not particularly low. And though the RMSE’s for Gz and Ly tended
to be close, the PMSE for Gra was consistently and sharply lower. These findings call into
question the use of distance to Fy, as an appropriate basis for judging substitution behavior,
as has been done in several recent papers on price index estimation.


6. Discussion

In the Introduction we noted four questions, and this paper has indicated some answers,
which we here summarize: (1) Economic theory suggests that a COLI is approximated by
indexes calculated by the Fisher and the Térnqvist formulas. (2) The size and direction of
the gap between these superlative indexes and the Laspeyres, Paasche, and geometric mean
indexes reflect the substitution behavior of consumers, that is, how much they switch from
one set of goods to another merely because of changes in relative prices. These population
indexes are, however, never available in practice. Instead we can at best compare index esti-
mates, which are almost invariably based on data arising from at least two distinct surveys,
elding estimates of sub-indexes, the other yielding weights by which to aggregate the
sub-indexes. (3) The relations between the index estimators are very largely influenced by
formal relations between the estimating formulas—characterized by certain low order Taylor
series expansions—and only indirectly by the substitution behavior of consumers.

Thus the inference of substitution behavior from the relations among index estimates, shown
in various studies of data from the U.S. CPI,° is called into question. That the geometric
mean usually falls below the superlative indexes can be taken as a sign that consumers do not
substitute freely across major categories of goods. On the other hand, as indicated by the
“Laspeyresville” study of Section 3 and implied by the Taylor expansions of Section 4, the
fact that the Laspeyres estimator is typically well above the superlative estimator cannot
in itself be taken as evidence that consumers are substituting goods in response to price
change. (4) The relation between a sample weighted aggregated geometric index estimator
and a target index of the same form depends on the bias and variance of each sub-index
and their relation to the aggregating weights. Results 2 and 3 of Section 5 may offer some
guidance for experimentation with various sub-indexes and weights. If nothing else, they
serve as reminders of the distinction between an index and an index estimator.

References

Aizcorbe, A, and Jackman, P. 1993. “Phe Commodity Substitution Effect in CPI Data, 1982-1991; Anatomy
of Price Change”, Monthly Labor Review, December 1993, pp. 25-33. U.S. Government Printing Office

Carli, G. 1764. “Del valore ¢ della proporzione de ’metalli monetati,” reprinted in pp. 297-336 in Scrittori
classici italiani di economia politica, Volume 13, Milano: G.G. Destefanis, 1804. (Reference obtained from
Diewert 1995.)

Dalén, J. 1992. “Computing Elementary Aggregates in the Swedish Consumer Price Index.” Journal of
Official Statistics, Vol.8. No. 2, 1992, pp. 129-147. Statistics Sweden.

Diewert, W. E. 198) ndex numbers.” In The New Palgrave: A Dictionary of Economics, (edited by J
Eatwell, M. Milate, and P. Newman), MacMillan, London

Diewert, W. E. 1995. On the Stochastic Approach to Index Numbers. Discussion paper no. DP 95-31, Sep.
1995, Department of Economics, University of British Columbia, Vancouver, Canada V6T 1Z1.

Fisher, I. 1922. The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability. Sentry
Press, New York, NY 10019.

Greenlees, J. 1998. “Random Errors and Superlative Indexes.” August 9, 1998. Unpublished BLS manuscript.

Shapiro, M. and Wilcox, D. 1997 “Alternative Strategies for Aggregating Prices in the CPI,” Federal Reserve
Bank of St. Louis Review, May/Jun 1997, pp. 113- 125

Turvey, R. et al. 1989. Consumer Price Indexes: An ILO manual. International Labour Office, Geneva.

°For example, Aizcorbe and Jackman (1993), Shapiro and Wilcox (1997)


Thus, to the second order, F, — Tw,,w, © (of — 73) /4, while

Wy We wp
, 1 1 1 2
Fu, aw. 7 Gu, x 9 (v2 —jn)+ 3 (ci _ 02) _ g (2 =yn)’,
and
1

Tray ww — Gun, 5 (Ha ~ fa) + ; (oi — 02) - ; (2 — Iu)”.

These differences may be either positive or negative, depending on the weights and price
ratios. The first term, (j12 — 41) /2, may be expected to dominate the differences. A positive
value for this term indicates that consumers spent more in the second period on items
whose prices had risen between the two periods, i.e., they did not adjust the quantities they
purchased enough to entirely compensate for changes in relative prices. The second term is
indeterminate in sign, while the third is clearly negative but of lower order. Examining the

difference between F'y,,w, and the arithmetic mean index, we have, to the second order,

1 1 9
Aw, — Fw, w R502 — 5 (2 — fu) + (2 — Wn)

yy Wy rte >

ale

Since the first term 03/2 is positive and will likely dominate, we should expect Ay, to exceed

Fu, w,; moreover, the two indexes will diverge as the variability in the rj increases.

Table 3 gives index estimates, computed from CPI data, using both the index formulae and
the second-order Taylor series approximations (indicated by a superscript 0). In this case,
the “price relatives” rj; are actually sub-indexes, and the weights are estimated expenditure
shares for the corresponding item categories. Clearly the second-order approximations work
well for these estimators of year-to-year change. The final column of the table shows the
estimated values of (fi2 — 11) /2, which, for these data, is the dominant term in the differences
Fo, wy — Gw, and Tw,,w, — Gw,-'The positive values of this term—for all but one year—are
consistent with the superlative index estimates exceeding the geometric mean estimates. The
terms (0? — 03) /2 and — (2 — jx.) /8 contribute little to the total differences.

Table 3. Index Estimates and their Taylor Series Approximations

Year | Aw, | Gu, Mu, | Fopws | Tews | Fuywe | Mayu, || M2 = fa) /2

1.03975 | 1.03806 | 1.03803 | 1.03838 | 1.03833 | 1.03839 | 1.03835 |] 0.00027331
1.04555 | 1.04394 | 1.04385 | 1.04451 | 1.04441 | 1.04447 | 1.04438 || 0.00059465
1.05118 | 1.04942 | 1.04920 | 1.04962 | 1.04940 | 1.04967 | 1.04941 || 0.00019134
1.03919 | 1.03759 | 1.03747 | 1.03786 | 1.03774 | 1.03783 | 1.03773 || 0.00029062
1.02880 | 1.02726 | 1.02722 | 1.02744 | 1.02739 | 1.02743 | 1.02740 |} 0.00017333
1.02764 | 1.02624 | 1.02615 | 1.02674 | 1.02665 | 1.02677 | 1.02667 || 0.00047575
1.02591 | 1.02461 | 1.02456 | 1.02486 | 1.02481 | 1.02483 | 1.02481 || 0.00024865
1.02737 | 1.02600 | 1.02596 | 1.02619 | 1.02615 | 1.02617 | 1.02614 |} 0.00019976
1.02809 | 1.02685 | 1.02683 | 1.02665 | 1.02664 | 1.02656 | 1.02659 || -.00014877
1.01976 | 1.01848 | 1.01850 | 1.01873 | 1.01876 | 1.01867 | 1.01878 || 0.00023259



Table 2. Comparison of Average Laspeyresville Pure and Composite Indexes

F L Ge Fr Ly_| Gear | Geis | Gra | Gre | Lei Lae

L 1.468 | 1.467 | 1.452 | 1.467 | 1.482 | 1.926 | 1.136 | 1.464 | 1.471 1.963
I—F || 0.012 | 0.011 | -0.004 | 0.011 | 0.026 | 0.470 | -0.320 | 0.008 | 0.015 0.507
RMSE || 0.181 | 0.176 | 0.108 | 0.176 | 0.183 | 0.566 | 0.355 | 0.178 | 0.177 0.612
PMSE 0 0.031 | 0.149 0 0.027 | 0.499 | 0.343 | 0.013 | 0.013 0.547

4. Order Relations of Index Estimators
In this section, we establish certain order relations between some index estimators. Following
Dalén (1992), we write the Taylor series expansions’ of the index formulas around the point

~)G
r= 1 in terms of the standardized moments of a set of price ratios r= {rj ye i= {¥,} 1
g

(or, depending on the level of aggregation considered, r; = pj2/pji). Let the weights wj:
represent expenditure shares for time period t € {1,2}, and define the moments

N N N

2 2 3

Me = Ye wyers; = YL wy (rj — pa) 5 and N= Y wie (rj — pu).
j=l j=l j=l

First consider the indexes based on a single set of fixed normalized weights w; = {wy},
The arithmetic mean index is Ay, = py, and, to the third order,

N

4 w 2 a VW
Gu, = [[ rf" & in - of (1 - ) +

j=l “

while

yi ;
PF, =) 2+ Ri — oT (1 - ) 2
7 {=e wy /T; } mee 2 * 2

Clearly, Fw, — Gw, © 71/6 is generally a small positive difference when the price ratios r;
are right skewed (e.g., log-normally distributed). Thus, in practice, we should expect index
estimates based on Fy, and Gw, to closely approximate one another. By contrast,

~ a at Vy
An, ~ Fa, #03 (1-2) - 3B.

Differences in estimates based on Ay, and Fy, may therefore be substantial and should
increase with o?. Now consider the corresponding indexes based on two sets of weights
wy = {wa}, and wy = {wy}, To the second order, we have

N
Paya, = {5

ws 2
we Dj=1 wia/T;

1/2
1 1, 1
} & Pat + 2) — <0} -— 3 (2 — ju)’.

1 Ary

2

Similarly,

1
(fa - mm)’.

N ( yp
wy twye
1 = II" ue

5 (Ha + pa) — i (oi + 03) -

whe

*The Constant Elasticity of Substitution (CES) index (see Shapiro and Wilcox 1997) may be expanded
and analyzed in similar fashion.


Thus 1 + b, is the multiplicative bias, and v, is the squared coefficient of variation, of /,.
Then E (7) fw [P97 H (wg, bg, 0g) , Where
H (wy, bg, 0g) = 1 + wy Ag + w,B,

2

where A, = {, -4} [2 + v,}} and By = § {, +g [i - (+, - #) }. Then Ie is biased
low for [79 if and only if

aN eo] bey

* 1 {b2 + vy [1 — (by — b02)| Bi + ey

w

For 6, = 0, the bias of Ive is negative (this generalizes a result of Greenlees (1998)). Thus
the positive bias we noted above for Laspeyres sub-indexes may be advantageous if we use
a Térnqvist aggregator. Of course, the above result does not guarantee that the upper- and
lower-level aggregation biases will offset each other, since the relative sizes of the expenditure
weights to the biases and coefficients of variation must be assayed, and this is a non-trivial
prospect. Some bias can be added as a result of sampling error in constructing the weights,
but the following result suggests it will not as a rule be significant.

Result 3 Weighted geometric mean, weights estimated unbiasedly. Let

T, = TI (4) and r= TI (i,)"".
9

g

Assume t, = wy + €y + €, where E (ey) = 0, var (€,) = dy, and €= % Dy €g. Also, assume
B(L,) =1,(1+,), var (f,) = B+ by)’ vy,

as in Result 2. Then

- 1 1
B (Py) & TLL TL A (wg, by, 0) ( +5 Yew) x Ty ( + Du, Hy +5 Yew) .
g “9 g “9

g

where H} = A, + w,By, and
J, = 2B, — AP +N 2 N+0,

with A, = log (L,) + Ay + 2w, By and \ = fa X Ay: |

If the w, are estimated reasonably well, say with coefficient of variation less than 0.2, then the
squared coefficient of variation Gy/w3 will be quite small. Then a comparison of components
of 4D, oyJ, and Y\, w, HF suggests that the former will be relatively small, to the point of
being negligible. In other words, estimating the weights, if done reasonably precisely and
without bias, should have relatively little impact on the bias of the estimator. No work has
been done on determining the variance of T; in the situation corresponding to the above two
results. Intuitively, the less noise there is in estimating [, and t,, the smaller the variance is

likely to be, but the relative impact of the various components awaits further investigation.


SOME OBSERVATIONS ON PRICE INDEX ESTIMATORS
Alan Dorfman, Sylvia Leaver, and Janice Lent
U.S. Bureau of Labor Statistics

1. Introduction

Few statistics produced by governmental agencies have greater impact than price indexes.
Statisticians, however, seldom study these indexes. This is not because index construction
is straightforward. Indeed, in terms of the controversies surrounding price indexes and the
complexity of the data integration they require, they might make a claim to be the most
complicated of statistics. A price index is a measure of change from one time period to
another of the purchasing power of a given population’s monetary unit. The term price
index is also used to refer to the formula—chosen from among a wealth of rival formulas—
used to calculate the measure of change. Economists have developed a body of theory
which they use to compare price index formulas, applied at the population level, to a cost
of living index (COLI), the ratio of minimal costs needed in the two time periods to achieve
a given standard of living. In practice, government agencies apply a given price index
formula to a sample from the target population, yielding an index estimator of the selected
population index. The estimator itself, however, is also often referred to as a price index,
and the resulting ambiguity of the term tends to blur the distinctions between the following
questions: (1) What is the relation of the (population) price index, calculated by a given
formula, to the ideal COLI? (2) What relations exist between various population indexes?
(3) What relations exist between various index estimators? (4) What is the relation of a
given index estimator to a particular target index? These questions form the substrate of
this paper. We focus on the consumer price index, but the proposed answers apply to other
sorts of price indexes as well.

2. Background

The purpose of a consumer price index is said to be “to measure changes over time in the
general level of prices of goods and services that a reference population acquire, use or pay
for consumption.”! A measure of relative change in the price of a particular item j between
time periods | and 2 is the price ratio pj2/pj1, where pj; represents the price of item 7 at
time ¢ € {1,2}. One may therefore obtain a simple price index for a given population just
by averaging these price ratios over the collection of N items purchased by the population:

j=l Pil

The resulting arithmetic mean index A is called the Carli index.? Assigning weights to the
price ratios in A produces the Sauerbeck index, Ay = 0%, w;pja/pjr, where
We shall see that the Carli and Sauerbeck indexes both possess properties that economists
consider undesirable; for decades, in fact, these indexes have been explicitly rejected by
economists such as Irving Fisher (1922). From a statistical standpoint, the deficiency of the

1 p=

'Turvey (1989) p. 4. (This quote is from the “Resolution concerning consumer price indices” of the
Fourteenth International Conference of Labour Statisticians.)
2Carli, G. (1764)


5. Relations of Index Estimators to Population Indexes

Having described the relations of several forms of index estimator to each other, we now
turn to the question of the relation of the estimators to the population indexes they target.®
In the U.S. CPI, the sub-indexes targeted are now predominantly in the form of a geometric
mean. Theory and experience show the sample geometric mean biased up for the population
geometric mean. For suppose we aim at a population unweighted geometric mean G, =

1. r/Na and, by the usual additivity properties of estimators, log(G,,) = 2, aj! logr,;
is unbiased for log(G,) (where xj! are the sample weights); then, exponentiating, we see
that E(G,,) > G,. Such biases decrease with increasing sample size, but the bias of the
aggregated index will also depend on the method of aggregation. In recent years there
has been considerable interest in employing alternate modes of aggregation, in the hope of
achieving an index that is closer to a cost of living index. Since the Fisher and Tornqvist
closely approximate each other, both as estimators (see above) and as population indexes
(Diewert 1987), it is enough to study one of these. We focus primarily on the Térnqvist,

but we first mention one result of interest regarding the Fisher.

Result 1 Fisher Estimator vis-a-vis Fisher. Let

with S,¢ a consistent estimator of

D2 UgitPinet
iS,
DY QitPine

9 i€Sg

Sgt =

and L, a consistent estimator (based on prices survey) of L,. Then

~ -1)1/2
Feo Ft = {r, Syily (< sy2L5') }
1) V2
4 {Epssls(ZysnPot) fo =P

Under typical economic conditions L, > P,, and F'* willexceed F.

We turn to the Tornqvist, that is, to a geometric mean having the average of first and second
period group expenditure shares for weights. More generally, we consider:

Result 2 Weighted geometric mean, weights assumed known. Let

T, =] (y)"” and T; =T] (1) :
g

g

Consider the factors. Suppose

B(f,) =1,(1+0,), var (f,) = B+ by) vy.

5Related work has been carried out by Greenlees (1998).


function of expenditure shares estimated from household survey data; and (3) use the formula
X, with raw ingredients Y, and t,, to get [ = X¢. The result may be termed a “composite
estimator.”

3. Findings from a Simulation Study—‘“Laspeyresville”

We conducted a series of simulation experiments to assess (1) the behaviors of various pop-
ulation indexes and their sample estimators in relation to each other, and (2) the extent
to which the sample estimators approximated their population counterparts. We first con-
structed an artificial population of households, items, and outlets. The economic behavior of
our artificial population was of the Laspeyres type: quantities of items purchased by house-
holds were held constant from period to period, except for random variation, irrespective of
changes in relative prices. The population of “Laspeyresville” consisted of 2000 households
purchasing goods from 100 outlets. Commodities were divided into three item strata, each
comprising 10 item types. All items within a stratum shared the same inflation rate, but
their initial prices and quantities differed.

Next, in a manner similar to the sampling methods of the Consumer Expenditure and CPI
price surveys—though considerably simplified—we drew repeated samples of households,
items, and outlets from the Laspeyresville population. Though we did not attempt to repli-
cate the complexities of CPI sampling (e.g., frame construction based on a Point of Purchase
Survey), we constructed the samples in such a way that the reliability of the simulated sam-
ple estimators of expenditure shares and subindexes approximated that of the corresponding
estimators used in the CPI. We ran 35 experiments, each consisting of one population con-
struction from which 50 samples (“runs”) were drawn. The variances of the population
prices and quantities, as well as the sample sizes for households, outlets, and items, varied
slightly across experiments but were the same for all samples within an experiment. For each
run, samples of households, outlets, and items within outlets were drawn for three pricing
periods: periods 1 and 2 (the periods between which price change was to be estimated) and a
base period B preceding the two pricing periods. Estimates of expenditure weights for each
item stratum were constructed from expenditures associated with the sampled households.

Period 1 to period 2 price indexes were computed for each stratum for the following indexes,
using both population and sample data: Laspeyres, Paasche, Fisher, geometric mean indexes
(with expenditure shares either constant or based in periods 1, 2, or B), Tornqvist, Carli,
and Sauerbeck. Since outlet expenditure data were available, we constructed “pure” index
estimators of these population indexes based on price and expenditure data from only the
sample outlets. We also computed composite index estimators, based on both household and
establishment data, as described in Section 2. We tabulated the distributional characteristics
of the population indexes and their pure and composite estimators, and we compared the
sample estimators with their population targets via means, standard deviations, root mean
square error (RMSE) and predictive mean square error (PMSE) for the pure and composite
Fisher index estimators. The RMSE was computed with reference to the population Fisher
index. For pure indexes, the PSME was computed with respect to the pure Fisher estimate;
for composite indexes, it was computed with respect to the composite Fisher estimate.

Table 1 shows the values of various population indexes (J), averaged over the experiments.
We found that for all experiments, values of the population Laspeyres, Paasche, Fisher


where wjig = (wj1 + wj2) /2. The Fisher and Tornqvist indexes are known as superlative
indexes, and economic theory suggests that they approximate a COLI (see, for example,
Diewert 1987). Intuitively, we can see that using information on buying patterns in both
reference periods allows us to relax the assumption of constant quantities underlying the
Laspeyres and Paasche indexes. Differences between values of the superlative indexes and
the corresponding values of the Laspeyres, Paasche, and geometric mean indexes are referred
to as substitution effects; i.c., these differences reflect the extent to which consumers alter
their buying patterns in response to changes in relative prices.

The Test Approach to Comparing Price Indexes

One approach used by economists to compare price indexes is the test approach, in which
several “tests” of reasonableness are defined. The proportionality test, for example, specifies
that if, for every item j, pj2 = cpji for some constant c, the index J,2, measuring price
change between periods 1 and 2, must equal c. Indexes based on price ratios—including the
Carli index—clearly pass this test. The time reversal test specifies that the “backward”
price index J21, calculated by interchanging the price information from periods 1 and 2,
must equal the multiplicative inverse of [,.2. For practical purposes, this means that chained
indexes behave reasonably when prices change and then revert to their former values. If we
have three time periods, 1, 2, and 3, and pj; = pjs for all j, then [,.2/2.3 = [1,3 = 1, regardless
of the values of the pj2. The superlative indexes F and T pass the time reversal test, as does
the geometric mean index. The Laspeyres and Sauerbeck indexes, however, fail this test;
the Sauerbeck, in particular, is upwardly biased in the presence of price/time reversal.? The
test approach originated with Fisher (1922); for additional tests, see, for example, Diewert
(1987) and Dalén (1992).

Composite and “Hybrid” Price Indexes

It is natural to separate the vast array of consumer goods into groups characterized by

item category, the place where items are bought, or both, since distinctions of item-type

and of location may give rise to different rates of price change. Correspondingly, we may

write the basic indexes in “composite form,” for example, the Laspeyres as L = YwyiLg,
g

where wg; = ba oitPait! Ny ba QgitPgit is the expenditure share for the gth group, and

= (= dyiPoe) / (= dP) is the “sub-index” for group g. The computation of sub-

indexes is called “lower-level aggregation,” while the process of combining sub-indexes into
an overall index is called “upper-level aggregation.” The formulae used for upper- and
lower-level aggregation need not be the same; we can compute “hybrid” indexes, e.g., Le, =
Y wy1Gyw—a Laspeyres aggregation of geometric mean sub-indexes.

g

For general notation, we write J = Xy where X and Y are the formulas used for upper-
and lower-level aggregation, respectively. Price index estimation then comprises three steps:
(1) estimate sub-indexes Y,, typically using establishment survey data (and perhaps some
household data for weighting purposes); (2) estimate the stratum weights w,, typically a

3Under most economic conditions, the Laspeyres index also suffers from an upward bias in the presence
of price/time reversal. When quantities remain constant across time, however, the Laspeyres index passes
the time reversal test.


arithmetic mean indexes may be seen to stem from the actual probability distributions of the
prices and hence of the price relatives pj2/pj1 : empirical evidence suggests that they follow
an approximate log-normal distribution, the skewness of which renders A upwardly biased
relative to the median of the distribution. A more attractive method of aggregation may be
to average the logarithms of the price relatives, which are more likely to be approximately
normally distributed, and exponentiate the result. This leads to the geometric mean index

N Dia N /piy\”
Gu = exp 4 > w; ln (%*) =TJ (%) .
j=l Pr ja \PA

Also known as the Jevons index, G (or G, in the case of equal weights) preserves the
distribution of the price ratios, i.e., Gy is log-normally distributed when the price ratios are
mutually independent, log-normal random variables.

Rather than focusing on the price ratios of individual items, we may wish to consider the
percentage change in the total price of a bundle of goods and services. This concept leads
to the Laspeyres index

N N
[a Seiet diPie wa (“*) ,
Lys WP ja Py

where qj: denotes the quantity of item j in the bundle at time ¢, and wj: = pjiqje/ we PkeCkt-
The weight w,; represents the expenditure share for item j in period ¢ € {1,2}. For the
Laspeyres index, the quantities considered are based on consumer buying habits during the
initial time period. When buying habits remain unchanged across time (q;2 = qj1 for all j),
the Laspeyres index is considered a true COLI. That is, it reflects the change in the cost
of achieving a given living standard. In order to achieve a given standard, however, some
consumers may be able to shift their purchases toward equivalent lower-cost items as relative
prices change, and the Laspeyres index may exaggerate the increase in their actual cost of
living. Similarly, the Paasche index,

TM aiepie _ 1

P= = =
oh Gj2Pj1 a wy (Pja/pj) |

which is based on quantities for period 2, may underestimate the true change in the cost of
living unless quantities remain constant across the two periods.

To achieve a compromise between the Laspeyres and Paasche indexes, we may take their
geometric mean. This leads to Fisher’s ideal index, advocated by Irving Fisher (1922) and
defined as F = VLP. The Fisher index incorporates quantity information from both peri-
ods 1 and 2. In practice, data on quantities of items purchased are generally not available;
estimates of quantity may be derived indirectly from estimated expenditure shares. Alter-
natively, expenditure shares may be used as weights in the index formula. When estimated
expenditure shares for both periods 1 and 2 are available, we may average the shares across
the two periods. The geometric mean index based on these average share weights is called
the Tornqvist index: