PERFORMANCE COMPARISONS OF LASPEYRES INDEXES WITH GEOMETRIC MEAN INDEXES

IN THE U.S. CONSUMER PRICE INDEX

Owen J. Shoemaker, U.S. Bureau of Labor Statistics

 2 Mass Ave., NE, Room 3655, Washington, DC 20212

 KEY WORDS: Quantity weight, price relative

 The official U.S. Consumer Price Index (CPI) is a

Laspeyres index. An alternative form of the index

known as the geometric mean (GeoMeans) index is

scheduled to be published in 1997. In this paper,

rental housing data have been studied with regard to

rent levels, to the average weights attributed to these

levels and to the average rate of rental price change at

these levels. These weights are quantity-based. A twopartition model is constructed, with “low” rents

compared with “higher” rents. Simulations are run

comparing Laspeyres and GeoMeans index results.

The differences between the two indexes are plotted

against the rate of price change in the “low” rent

sector. In the simulations, all the other factors are

treated as fixed. A smooth (quadratic) curve results,

and the GeoMeans indexes are shown to score

consistently higher than the Laspeyres indexes.

1. Background

 When the GeoMeans Analysis Team at BLS

investigated the CPI’s Rent and Rent Equivalence

(REQ) price relatives, they discovered that the

GeoMeans’ Rent/REQ indexes were running

consistently higher than their Test Laspeyres

counterparts. Rent/REQ appeared to be the exception

to the rule that said that GeoMeans indexes will

generally come in lower than Test Laspeyres indexes.

In fact, for nearly all other basic-level indexes in the

CPI (roughly 75% of the Index), the reverse is the

case. There, GeoMeans run consistently lower than

Laspeyres. Providing an explanation for the higher

GeoMeans in Rent/REQ is the subject of this paper.

2. Possible Explanations

 First of all, unless the Laspeyres’ price relative

calculation (PRC) is expressed precisely as a weighted

arithmetic mean, no exact mathematical comparison

can be made between the Laspeyres and its GeoMeans

counterpart. Moreover, only if their respective weights

are equal will the GeoMeans index be guaranteed to

come in lower than a Laspeyres index. Such a

structure is not the case here. (Nor is it the case for

C&S data, that other 75% of the CPI, but that issue is

for a separate investigation.) For Rent/REQ data, the

two BLS production PRCs have the following general

forms:

Laspeyres PRC R

W P

W P

T T

L

i T i

i

i T i

i

= − =

∗

−

∑

∑

,

,

,

,

12

12

 

 

 

 where W Q a quantity weight,

 and is a rent price (

i

,i T,i

=

















= =

= +

∑

∑

−

∗

−

−

•

∗

( )

( )

,

).

,

,

,

,

,

W P

P

P

W P

P P P r

i

i

T i

T i

T i

i T i

i

i

T i

12

12

12

.

GeoMeans PRC R

P

P

W

W

T T

G T i

i T i

i

= =













∑

−

∗

−

, ∏

,

,

,

12

12

 

=









































∗

−

∑

∑

EXP

W

P

P

W

i

T i

i T i

i

i

log

.

,

12,

 Neither of these PRCs exactly duplicates BLS

production, but the two weight structures do duplicate

BLS production. (It should be noted that quantity

weights were used in the experimental GeoMeans

because more appropriate weights were not available.)

Currently, SL = Laspeyres Weights = W ∗ PT-12, while

S

G

 = GeoMeans Weights = W. Thus, the two weight

structures are not the same and there is no

mathematical guarantee that one or the other of these

two PRCs will always be lower than the other.
 (T-12 is the time designation for previous rents,

since that is the time differential that the empirical

results in this paper uses. Actual BLS production uses

six-month rent changes – plus a depreciation

adjustment factor r in the new rents – for its rent

indexes. BLS takes the sixth root of these six-month

relatives to calculate one-month price relatives.)

 The explanation that we will pursue in this paper

focuses on (a) the weighting structure of the rents as it

relates to rent-level, and (b) the relative percentage

increases in the rent-levels. We will use simulations to

compare the relative performances of the two indexes,

by making the difference between the two indexes a

function of the variables outlined in (a) and (b).

3. BasicWeights as Related to Rent-Level

 The following table gives summary statistics for

BasicWeights (1987 Revision) as they are distributed

across three rent levels — low, medium, high. The

results confirm that low-rent units do have

significantly higher weights than medium- or high-rent

units. (Three pair-wise comparison tests – H0:

LOW=MEDIUM, H0: LOW=HIGH, and H0:

MEDIUM=HIGH – all give P-values less than 0.0001.)

BasicWeight Averages By Strata Levels

 RENT

 LEVEL

FREQ WEIGHT

 MEAN

WEIGHT

STDERR

 LOW 7728 4842.6 42.33

 MED 13753 4485.4 32.72

 HIGH 5803 3767.5 50.06

 ALL 27284 4433.9 23.12

 It has been suggested that this particular weight

distribution, with the higher weights in the lower rent

levels, is contributory to producing higher GeoMeans

indexes (see internal memoranda between Louise

Campbell and Dave Richardson, September 1996).

However, as it turns out, the weights have little or

nothing to do with the issue at hand. As our

simulations will show, the GeoMeans (Rent/REQ)

index will come in higher than its Test Laspeyres

counterpart whatever the respective weights are.

4. Percentage Increases in Rent Levels

 In the previously referenced memoranda, it was

postulated (1) that low rent units are more likely to

have larger percentage increases than higher rent units

and (2) that the impact of these larger percentage

increases is, in some measure, responsible for

producing a GeoMeans (Rent/REQ) index that tracks

higher than its Test Laspeyres counterpart. Both

statements are true, as we shall demonstrate.

 To test the first supposition, rent data from the

Research Analysis Database (RAD), from 1988

through 1995, were analyzed. New construction units,

rent-control units and special rent-reduction units were

excluded. The idea was to analyze “regular” rental

units in regard to their average percentage increases

across rent levels. The medium and high rents were

collapsed into one higher rent-level and then compared

to low rents. Testing the hypothesis that the percentage

increases for low rents were the same as the increases

for higher rents produced the following results.

Percentage Rent Increases Year to Year

PCT Increase

 (Low rents)

PCT Increase

(Higher

rents)

P-value

1988-89 6.85% 4.47% 0.0005

1989-90 5.78% 4.36% 0.0186

1990-91 5.96% 5.20% 0.1812

1991-92 4.99% 3.66% 0.0233

1992-93 4 81% 3.45% 0.0078

1993-94 5.55% 3.50% 0.0001

1994-95 4.38% 3.54% 0.0927

Low P-values reflect significant differences.

Ana =.05 significance level can be assumed

appropriate here. Variances are assumed to

be unequal, using regular two-sample t-tests.

Sample size for low rents run just under 4000,

with higher rents running around 14000. The

six January-June data panels are used in the

analysis.

 Low rent percentage increases track consistently

higher across the years studied. In two of the yearly

time frames (‘90-’91 and ‘94-’95) the differences are

not significant (at an α=.05 level), but the differences

in the other five yearly time frames are clearly

significant. The question is whether these larger

percentage rate increases for the low rents are affecting

the relative performances of the two indexes. The

answer is they are.

5. The Effect on GeoMeans vs. Test Laspeyres Indexes

 To test the proposition that higher percentage rate

increases for low rents result in the GeoMeans index

coming in higher than Laspeyres, we turn to
simulations, an appropriate paradigm and one general

proof.

 The simulation program need not be very

complicated. We are interested in a mix of higher and

lower percentage rent increases for low rents in tandem

with appropriate weights that may themselves range

higher or lower. This mixture is the essential

paradigm.

 Using appropriate weights and appropriate rents

for the higher rental units, we can simulate these

indexes without too much complication. For one thing,

we can view the varying weights of the low rents as

simple linear combinations of the fixed basic weights

of the higher rents. This allows the basic weights

themselves to cancel out in both indexes.

 We simulate an index consisting of one low rent

unit and one higher rent unit from time t0 to time t1

(think of 12 months as an appropriate time interval).

The higher rent unit and its appended weight will be

treated as fixed, along with the low rent unit’s previous

rent price. The new low rent, along with its weight,

will be allowed to vary considerably. A comparison of

the two index calculations (GeoMeans and Test

Laspeyres) should tell us something useful, one way or

the other.

 The two index formulas applicable here are:

 TLI (Test Laspeyres Index) = IP ∗ RL and

 GMI (GeoMeans Index) = IP ∗ RL ,

 where (WLOG), we can set IP = 1,

 and where Ri = a rent price relative.

 Then, TLI = R =

B(w F) + c w

b w F) + c w

L

1

0 0

(

B = low rent at time t1

b0 = low rent at time t0

w = base weight for higher rent unit

F = a variable weight factor (>0)

c1 = higher rent at time t1

c0 = higher rent at time t0

 thus, 

 TLI =

B F + c

b F + c

 ,

 

1

0 0

⋅

⋅

 since the weights cancel.

 

GMI = R =

B

b

wF

wF + w c

c

w

wF w

B

b

F

F 1 c

c

1

F 1

G

0

1

0

0

1

0

























+













+ 











+

=

 With these two index formulas, the simulation

program sets up easily enough (using Splus). The

previous low rent (b0) is set at 300. The previous

higher rent (c0) is set at 500. The current higher rent

(c1) is set first at 550 and then at 600. The current low

rent is allowed to vary from 200 to 1000 in increments

of 20. The weight factors are 1⁄4, 1⁄2, 1, 2 and 4.

 Any number of settings can be viewed (with the

same paradigmatic results each time). We can set the

low rents as low as 200 and as high as 500, and set the

higher rents as low as 300 and as high as 800, and still

we will get the same pattern. The paradigm holds

steady and clear in any and all plausible settings. A

(smooth) quadratic curve defines the function. At one

clear-cut mathematical point, both indexes are equal.

At another less definable point, the two indexes are

again equal. Rent price reality exists, in the main,

between these two points — i.e., precisely where

GeoMeans indexes always run higher than its

Laspeyres counterpart. The difference between

GeoMeans and Test Laspeyres (GMI–TLI) is plotted

against the ratio of new to old low-rents (B / b0). The

three fixed numbers in the header refer to c1, c0, and

b0, respectively.

 The simulations point to a clear change of direction

as the two indexes pass their point of intersection, i.e.,

at the point where the two price relatives are equal.

Thus, in the locality just past that intersection point,

GeoMeans runs higher than Test Laspeyres. Only

when B starts to range out of the low rent level does

the inequality begin to reverse itself. Thus, for all

practical purposes, when low rents have higher

percentage increases than their higher rent

counterparts, the GeoMeans index comes in higher

than Test Laspeyres. Moreover, this higher GeoMeans

index begins tracking higher as soon as that

intersection point has been crossed, no matter what the

respective weights are.

 Furthermore, the weights just do not influence the

direction of this paradigm. Lower low-rent weights

make for lower differences between the two indexes,
but the signed differences themselves are never

changed (under the conditions of the paradigm as

denoted above). So long as low rent units on average

exhibit higher percentage rate increases than their

higher rent counterparts, the GeoMeans index will

track higher than Test Laspeyres.

GMI-TLI

1.0 1.5 2.0 2.5 3.0 3.5

-0.06 -0.02 0.0 0.02 0.04

Geo vs Laspeyres with 550, 500, 300

F=.25 

F=.50

F=1

F=2

F=4

 B / b0
GMI-TLI

1.0 1.5 2.0 2.5 3.0 3.5

-0.04 -0.02 0.0 0.02 0.04

Geo vs Laspeyres with 600, 500, 300

F=.25 

F=.50

F=1

F=2

F=4

 B / b0

 The simulations bear out the following results:

 (1) GMI = TLI whenever 

B

b

=

c

c 0

1

0

 ,

 no matter what F is.

 Let 

B

b

=

c

c

=

x

y 0

1

0

 , then,

 

 GMI =

x

y

x

y

 =

x

y

=

x

y

F

F+1

1

F+1

F

F+1

+

1

F+1





































 and likewise, TLI =

x F + x

y F + y

=

x

y

⋅

⋅

.

 (2) GMI > TLI if 

(a) B

b

>

c

c

 and 

(b) 0 < b < B < c < c

0

1

0

0 0 1

,

 for all reasonable

F.

If we let F=1 (i.e., equal weights), we can prove

that GMI > TLI , provided conditions (a) and (b)

are true.

 Given (1) B/A > D/C and (2) 0 < A < B < C <

D,

 Prove: [(B/A)(D/C)]^1/2 > (B+D) / (A+C).

 Proof 1.1: BC > AD [ from (1) ]

 CD > AB [ from (2) ]

 Then BC – AD > 0

 CD – AB > 0

 (BC – AD)(CD – AB) > 0

 BDC2 – ACB2 – ACD2 + BDA2 > 0

 BDA2 + BDC2 > ACB2 + ACD2

 BDA2 + 2ABCD + BDC2 >

 ACB2 + 2ABCD + ACD2

 BD⋅( A2 + 2AC + C2 ) >

 AC⋅( B2 + 2BD + D2 )

 BD / AC > ( B2 + 2BD + D2 ) /

 ( A2 + 2AC + C2 )

 BD / AC > (B + D)2 / (A + C)2
 [(BD) / (AC)]^1/2 > (B+D) / (A+C)

 In the extended multi-quote setting, the following

condtion should be sufficient (albeit as yet unproven)

to insure the claim that (2) is still true.

 

 where 

m = n, if (m + n) is even.

m = n -1, if (m + n) is odd.

m + n = # of quotes.

 and 

 

B

b

c

c

b b b c c c

B b B B

i

i

m

j

j

m

m

i

i

n

j

j

n

n

m n

k i j

∏

∏

∏

∏





























>







































< < < < < < <

∀ =

0

1

1

0

1

0 1 0 2 0 0 1 0 2 0

0

,

,

,

, , , , , ,

,

,

( / ) ,

 ∀ ( / ) , = .

, ,

c c c c 1 0 k 1 i 1 j

 The extended multi-quote version of (2),

where GMI > TLI would then be

B

b

c

c

B c

b c

i

i i

m m

i

i

n i

i

m

i

i

n

i

i

m

i

i

n

i

n

0

1

1

0

1

1

0 0

,

,

,

,

. ,























 >

+

+

∏

∑ ∑

∑ ∑

∏ .

 Since it has been demonstrated that the weights

matter little in regard to the direction of this

inequality, this version is quite close to production

reality. I.e., when the percentage rate of increase is

greater across the lower rents than it is across the

higher rents, then this inequality holds and holds

strong.

6. Conclusion

 The somewhat simplistic two-partition model that

we have used may not always reflect an actual Rent

or REQ index, but it is not purely hypothetical

either. If the average low rent percentage increase is

higher than its higher rent counterpart, then the

inequality as expressed in (2) will almost always

hold. (An average low-rent percentage increase is

directly analogous to the left-hand side of the multiquote condition above.)

 BLS statistician, Sylvia Leaver, working

independently of this paper, calculated 6-month

price changes in rents over this same 1988-1995

period and found that the GeoMeans index came in

higher in three out the four major geographic regions

of the CPI. Only in the South was the direction

reversed, with GeoMeans there coming in lower than

Test Laspeyres. When I then separated out my own

rent numbers by region and fitted these averages

into the above paradigm, I found that only in the

South did the first “average” rent price relative come

in lower than its higher rent counterpart, thus

corroborating the paradigm.

7. Acknowledgements

 The author would like to thank Dave Richardson

and Louise Campbell for their extensive analyses on

performance comparisons of the Laspeyres and

GeoMeans indexes, and in particular for their raising

the main issue of this paper. The author would also

like to thank Janet Williams and especially Robert

Baskin for their help and encouragement in

developing these issues. The author also wishes to

thank Dave Swanson, Shawn Jacobson, Sylvia

Leaver, and Janet Williams for their careful reading

of this paper and for their helpful comments.
8. References

Bureau of Labor Statistics, BLS Handbook of

Methods (1992), Washington, DC. U.S.

Government Printing Office, 180-181, 190-191,

196-198.

Campbell, L. and Richardson, D. (1996), “The

Comparison of the Geometric Mean Index (GMI)

with the Test Laspeyres Index (TLI) on the Basis of

1990-1991 Data”, BLS Memorandum, Bureau of

Labor Statistics, Washington, DC.

Leaver, S.G. (1997), “Variance Estimates for

Alternative Estimators of Price Change for Rent and

Owners’ Equivalent Rent, for U.S. All Cities and

Regions, 1990-1995”, BLS Memorandum, Bureau

of Labor Statistics, Washington, DC.

Stewart, K., Smedley, K. (1997), “The

Experimental CPI using Geometric Means (CPI-UXG)”, BLS Memorandum, Bureau of Labor

Statistics, Washington, DC.