BLS WORKING PAPERS

U.S. Department of Labor

U.S. Bureau of Labor Statistics

Office of Prices and Living Conditions

An Alternative Formula for Elementary Producer Price Indexes

Robert Martin, Andy Sadler, Sara Stanley, William Thompson, Jonathan Weinhagen,

U.S. Bureau of Labor Statistics

Working Paper 525

September 2020
1

An Alternative Formula for Elementary Producer Price

Indexes*

Robert S. Martin, Andy Sadler, Sara Stanley, 

William Thompson, Jonathan Weinhagen†

September 2020

Abstract

We re-estimate historical U.S. Producer Price Indexes (PPI) using the geometric Young 

formula at the elementary level. The geometric Young has better axiomatic properties than the 

modified Laspeyres, and may better approximate a feasible economic target. We find in most 

cases, indexes that use the geometric Young escalate between 0.1 and 0.3 percentage points 

less each year than those that use the modified Laspeyres. However, for wholesale and retail 

trade, as well as some other services, the differences are much larger. As a result, using the 

geometric Young at the elementary level lowers the PPI for Final Demand by 0.55 percentage 

points per year during the study period, a magnitude larger than what has been previously

found for the U.S. Consumer Price Index.

Keywords: Inflation; Aggregation; Geometric Young index; Jevons index

JEL Codes: C43, E31

 

* We are grateful to Jeffrey Hill, Thesia Garner, Brian Adams, Rob Cage, Erick Sager, and members of the BLS 

Technical Advisory Committee, in particular Alberto Cavallo and Carol Corrado, for helpful comments. All errors are 

our own.

† Division of Price and Index Number Research (Martin, corresponding author), Division of Price Statistical Methods 

(Sadler), Division of Industrial Prices and Price Indexes (Stanley, Thompson, and Weinhagen), Bureau of Labor 

Statistics, 2 Massachusetts Ave., NE, Washington, DC 20212, USA. Emails: Martin.Robert@bls.gov, 

Sadler.Andy@bls.gov, Stanley. Sara@bls.gov, Thompson.William@bls.gov, Weinhagen.Jonathan@bls.gov.
2

1 Introduction

This paper presents re-estimates of historical U.S. Producer Price Indexes (PPI) using the 

geometric Young formula at the elementary level. The U.S. Bureau of Labor Statistics (BLS) 

currently uses a modified Laspeyres formula (Bureau of Labor Statistics 2015), but statistical 

agencies in Italy, Chile, the Netherlands, among others, use the geometric Young (or something 

similar) for elementary PPIs (OECD 2011). In addition, since 1999, the BLS has used the 

geometric Young for most elementary indexes that comprise the U.S. Consumer Price Index 

(CPI) (Bureau of Labor Statistics 2019). While a number of preceding studies, e.g. Boskin, et al.

(1996), analyzed differences between formulas for consumer prices, relatively little empirical 

research of this nature examines producer prices. PPIs are used widely in adjusting 

procurement contracts and as deflators for other economic time series, so formula choice is of 

broad significance. Unlike with consumer prices, use of the geometric Young may be less

motivated by the issue of substitution bias (Waehrer 2000). However, compared to the 

modified Laspeyres, the geometric Young has better axiomatic properties (IMF 2004) and lower 

formula bias (McClelland 1996, Reinsdorf 1998), which are also relevant factors for PPIs. We 

discuss these considerations further in Section 2.

We use both the modified Laspeyres and geometric Young formulas to calculate 

approximately 7,000 elementary indexes per month covering January 2008 to December 2017. 

We then aggregate these into versions of 1,016 six-digit commodity (product-based) indexes, 

759 six-digit industry indexes, and the headline PPIs for Final Demand and Intermediate 

Demand (FD-ID). For a more detailed description of PPI classifications, see Bureau of Labor 

Statistics (2015). In most cases, six-digit indexes using the geometric Young formula at the 
3

elementary level are between 0.1 and 0.3 percentage points per year lower than those that use 

the modified Laspeyres. However, for wholesale and retail trade, as well as some other 

services, the differences are much larger. As a result, re-estimating the PPI for Final Demand 

using the geometric Young at the elementary level lowers the index by 0.55 percentage points 

per year, a larger magnitude than what has been found previously for the U.S. CPI (Boskin, et al. 

1996).

Formula choice has a greater impact on U.S. PPIs in part because the BLS uses gross 

margins (selling price minus acquisition price) to measure the prices received by firms that 

resell items, such as wholesalers and retailers. Excluding these categories, the geometric Young

lowers the PPI for Final Demand by 0.24 percentage points per year, comparable to Boskin, et 

al.’s finding of 0.25 for the U.S. CPI. Changes in margins tend to be more highly dispersed, 

driving greater differences between index formulas. In addition, the geometric Young is 

sensitive to near-zero margins. However, our results change very little when we impose bounds 

on the most extreme price changes prior to index calculation.

2 Methods

A price index aggregates price changes for many items into a single summary measure.

Producer price aggregation typically occurs in two stages. First, price changes within a narrowly 

defined grouping are combined to form an elementary index. Then, these elementary indexes 

are aggregated into broader measures like the headline PPI for Final Demand.
4

The PPI is currently based on a modified Laspeyres formula (Bureau of Labor Statistics 

2015). More precisely, the target is known as the Lowe index. Let qi

t

and pi

t

denote quantity 

and price, respectively, for an item i in some period t. The Lowe index is then

ILo

t = (

∑ qi

bpi

N t

i=1

∑ qi

bpi

N 0

i=1

) × 100. (1)

The index measures the change in expenditure on a fixed basket {q1

b

, ... , qN

b

} from the reference 

period 0 to the comparison period t. Period b is the base period from which quantity 

information is drawn. If the base and reference periods happen to be the same (b = 0), then 

the index coincides with the well-known Laspeyres formula, which we denote ILasp

t

. If the base 

and comparison periods are the same (b = t), then it coincides with the Paasche formula, IPaas

t

.

However, in practice, it is usually true that period b precedes period 0, particularly at the 

elementary level, where weighting information is not sampled as quickly or frequently as prices. 

Implementation of the Lowe index uses its expenditure share form, given by

ILo

t = (∑si

0b

N

i=1

pi

t

pi

0

) × 100, (2)

where the si

0b = pi0

qi

b ∑ pj0

qj

N b ⁄ j=1

are hybrid expenditure weights using period 0 prices and 

base period quantities, and pi

t

/pi

0

is sometimes referred to as the long-term price relative.

At the upper level of aggregation, the PPI uses something close to Eq. 2. At the 

elementary level, however, shipments data are usually only available in dollar values (the 

products pi

b

qi

b

) rather than quantities (the qi

b

by themselves). As a consequence, the “modified 

Laspeyres” formula actually implemented is closer to the Young index, written
5

IY

t = (∑si

b

N

i=1

pi

t

pi

0

) × 100, (3)

where si

b = pib

qi

b ∑ pjb

qj

N b ⁄ j=1

are the actual expenditure weights from the base period. If b =

0, then the Young, Laspeyres, and Lowe are all equivalent. The axiomatic shortcomings of the 

Young index have been documented in IMF (2004), and include failure of the time-reversal and 

transitivity tests, discussed in the next subsection. The geometric Young index, given in Eq. 4, is 

seen as a superior alternative because it satisfies these tests.

IGY

t = (∏(

pi

t

pi

0

)

si

N b

i=1

) × 100 (4)

This formula has also been called the geometric Lowe, weighted Jevons, and Cobb-Douglas 

price index. It combines the same price and expenditure information, but using a geometric 

mean instead of an arithmetic mean. The BLS uses a version of this formula for the majority of 

elementary CPIs, as do several other countries for their elementary level PPIs. To avoid 

confusion, we use “modified Laspeyres” to refer to the arithmetic form of the Young index 

given in Eq. 3, unless otherwise specified.

A corollary to Jensen’s inequality implies the geometric mean will be less than or equal 

to the arithmetic mean when based on the same weights, and so we should generally expect 

index levels to be lower when using the geometric Young. The quantitative significance, 

however, is an empirical matter. 

2.1 Axiomatic and Statistical Considerations

This subsection outlines how the geometric Young is superior to the modified Laspeyres from 

the axiomatic perspective. As way of background, several different sets of overlapping tests for 
6

index numbers comprise what is known as the axiomatic approaches (IMF, 2004). Historically, 

this approach has favored indexes with some sort of geometric mean, such as the Fisher and 

Tornqvist, both of which are infeasible for the BLS due to the detail and frequency of weight 

information required. IMF (2004) lists a set of 12 tests used to assess modified Laspeyres 

indexes whose weights correspond neither to the reference nor the current period. 

The key axiomatic shortcoming of the modified Laspeyres is that, in general, it fails the 

time reversal test. Formally, a generic index I(0,t) satisfies the time reversal test if the 

following holds:

I(0,t) =

1

I(t, 0)

or I(0,t)I(t, 0) = 1 (5)

The idea of this test is that the index measurement should, in some sense, be independent of 

which period is regarded as the reference and which is regarded as the comparison. For 

example, if the index says the general price level doubled from period 0 to period t, then it 

should also say the price level fell by half if we instead treat t as the base period and 0 as the 

current. The modified Laspeyres fails because IY

(0,t) ≥ 1/IY(t, 0), with equality only in the

unlikely case that all prices change by the same proportion. In other words, the modified 

Laspeyres leads to a higher measurement than if reference and comparison periods were 

reversed, which can be interpreted as an upward bias in the index. On the other hand, it is 

easily verified that the Geometric Young satisfies the time-reversal test.
7

The modified Laspeyres also fails the transitivity (or circularity) test, which says that 

when chained together, indexes over adjacent intervals should equal their direct counterpart.‡

For time periods 0 < s < t, it should be true that:

I(0,t) = I(0, s) × I(s,t). (6)

The degree of failure, which is related to the idea of chain drift, depends on the particular 

patterns of the data. However, unlike time-reversal, intransitivity does not necessarily imply a 

bias in any particular direction, rather a sort of practical user-limitation. Ratios of modified 

Laspeyres indexes might not yield the intended comparison, as failure of Eq. 6 implies 

IY

(0,t)/IY(0, s) ≠ IY(s,t). The Geometric Young satisfies the transitivity test, as will any 

geometric mean of price relatives with time-constant weights summing to one.

Reinsdorf (1998) describes another scenario in which the modified Laspeyres index 

systematically exceeds the Laspeyres indexes, which he refers to as “formula bias.” In a 

statistical model of mean-reversion, meaning prices fluctuate around a common mean, he 

shows how the Young form of the modified Laspeyres index will exceed both the Lowe and 

Laspeyres in expectation. The intuition for his argument is that prices that are temporarily low 

in the reference period will receive excess weight in the Young formula. In a mean-reverting 

model, these are the prices expected to rise the most from 0 to t, leading to an inflation 

estimate that is biased upwards. He gives indirect evidence that this bias was empirically 

relevant for elementary indexes used in the CPI before formula changes that took place in the 

1990s. He also discusses a geometric mean index as a solution. Assuming the elementary price 

 

‡ Note the transitivity test is not part of either the First or the Second Axiomatic Approaches described in IMF 

(2004), but it is included in the 12 tests used to compare the Lowe and arithmetic Young indexes.
8

data in the PPI follow a similar pattern, this argument supports use of the Geometric Young

over the modified Laspeyres.

2.2 Economic Considerations

Despite the geometric Young having superior axiomatic properties to the modified Laspeyres, 

the economic approach to index numbers has led to questions about its appropriateness for 

output PPIs. The economic approach compares a formula against a theoretical target derived 

from a model of an optimizing agent (IMF, 2004). Fisher and Shell (1972) and Archibald (1977) 

propose a class of theoretical output price aggregators called Fixed Input Output Price Indexes

(FIOPIs). A FIOPI measures the firm’s hypothetical change in revenue between 0 and t if it had 

to keep inputs and technology fixed at some level. In this section, we argue that opposition to a 

geometric mean-type index on economic grounds (e.g., from Waehrer, 2000) is too narrowly 

focused on one index (the geometric Laspeyres, which is Eq. 4 with b=0) and one theoretical 

target (the reference-period FIOPI). We offer a plausible scenario in which a geometric Young 

index may better approximate a FIOPI based on average technology and inputs. 

Let S denote the production possibilities set associated with a given level of technology 

and inputs. Suppose in periods τ = 0,t, a representative producer facing prices p

τ =

(p1

τ

, ... , pN

τ

) chooses quantities q = (q1, ... , qN) from S to maximize revenue, which maximizes

profit since inputs are fixed. Define the maximized revenue as 

R(p

τ

, S) = max

q∈S ∑ pi

τ

qi

N

i=1

. (7)

The class of FIOPIs is then given by
9

FIOPI(S) =

R(p

t

, S)

R(p0, S)

(8)

for a given S. In general, FIOPIs are infeasible to implement in official statistics. Estimating a 

hypothetical revenue change under fixed inputs and technology would require specifying and 

estimating a model for every industry or commodity group—a challenge because shipments 

data, even if collected frequently enough, do not reflect producers reacting to price change in a 

vacuum.

Economic theory provides some observable bounds, however. Let S

0 be the production 

possibilities set from the reference period, so that under the assumption of profit maximization, 

R(p

0

, S

0

) equals the (in theory) observable revenue level ∑ pi

0

qi

N 0

i=1

. From Archibald (1977), 

FIOPI(S

0

) =

R(p

t

, S

0

)

R(p0, S

0)

≥

∑ pi

t

qi

N 0

i=1

∑ pi

0qi

N 0

i=1

= ILasp

t

. (9)

The result follows because under profit maximization, R(p

t

, S

0

) ≥ ∑ pi

t

qi

N 0

i=1

. Similarly, IPaas

t

is an upper bound for FIOPI(S

t

), where S

t

is the production possibilities set from the 

comparison period. The intuition for these results is that all else equal, the optimizing firm 

would have an incentive to shift production towards items with higher relative prices. The 

geometric Laspeyres index, is necessarily less than or equal to ILasp

t

. Waehrer (2000), therefore, 

opposes the geometric Laspeyres index for producer prices because it has greater bias for 

FIOPI(S

0

) then the arithmetic Laspeyres.

However, this result for the geometric Laspeyres is not applicable to analysis of the 

geometric Young, which does not (to our knowledge) have a known relationship to potential

targets like FIOPI(S

0

) or even FIOPI(S

b

). More broadly, the one-way bounds concerning 
10

ILasp

t

and IPaas

t may be of limited empirical relevance because they relate to different FIOPIs. In 

particular, if ILasp

t ≥ IPaast

, then we have

FIOPI(S

0

) ≥ ILasp

t ≥ IPaast ≥ FIOPI(S

t

). (10)

While we lack data to evaluate the inequality ILasp

t ≥ IPaast

at the elementary level (since both 

indexes are infeasible), Weinhagen (2020) found ILasp

t ≥ IPaast holds for broader industry and 

commodity aggregates. This inequality reflects negative correlation between market prices and 

quantities, implying that even if firms have a ceteris paribus incentive to substitute production 

towards higher relative output prices, this is outweighed in equilibrium by other factors like

consumer demand or technology shocks. If this is the case, then targeting either FIOPI(S

0

) or 

FIOPI(S

t

) might be arbitrary, seeing how the gap between them is at least as large as ILasp

t −

IPaas

t

.

Diewert (1983) offers an alternative FIOPI which, unlike FIOPI(S

0

) or FIOPI(S

t

), has 

two-way observable bounds. Let S

α = αS0 + (1 − α)St

, denote a weighted average of the 

reference and comparison period production possibilities sets. Diewert showed there exists an

α between zero and one such that either ILasp

t ≤ FIOPI(S

α

) ≤ IPaas

t or IPaast ≤ FIOPI(S

α

) ≤

ILasp

t

. If the difference between the ILasp

t

and IPaas

t

is not too great, a symmetric average like

the Fisher index, IFisher

t = √ILasp

t

Ipaas

t

, is a good approximation to the “average” FIOPI(S

α

). 

The Tornqvist index has similar properties (Caves, et. al., 1982). 

As stated in the previous subsection, neither the Tornqvist nor the Fisher are feasible for 

elementary PPI. Nevertheless, we describe a plausible scenario under which the geometric 

Young may be preferred to the modified Laspeyres from the standpoint of targeting an average 
11

FIOPI. IMF (2004, Ch. 15, Sec. D.3) describes conditions, reasonable for elementary items, under 

which IY

t

exceeds ILasp

t

. The first is that the base period precedes the reference period, as with 

the U.S. PPI. The second is that prices are trending either up or down over the long-term, which 

is generally true for U.S. PPI data. Next is that changes in market quantities primarily reflect 

purchaser substitutions away from items with higher relative prices. This is true if ILasp

t ≥ IPaast

, 

for which we interpret Weinhagen (2020) as indirect evidence. The last condition is that these 

substitutions are elastic, meaning revenues and prices move in opposite directions. Elastic 

substitution patterns are likely if the elementary product grouping contains highly similar 

varieties, as is the case with the retail sales data studied by Martin (2020).

It is always true that IY

t ≥ IGYt

, and under the scenario described above, we have that 

IY

t ≥ ILaspt ≥ IFisher

t

. This means that switching from the modified Laspeyres to the geometric 

Young moves the index in the direction of the Fisher index, which approximates FIOPI(S

α

). Of 

course, this does not guarantee that the geometric Young will have lower bias for FIOPI(S

α

), 

and the conditions in the preceding paragraph may not be appropriate for all sectors. 

Nevertheless, it is plausible that the geometric Young better approximates a more feasible 

economic target than the modified Laspeyres.

3 Application

3.1 Data

The previous section enumerated reasons for which the geometric Young (Eq. 4) should be

preferred to the modified Laspeyres (Eq. 3) from the standpoint of price index theory. To better 

understand the practical implications of formula choice, we use both formulas with the PPI 
12

microdata to calculate approximately 7,000 elementary indexes per month covering January 

2008 to December 2017. Roughly half of these measure output prices for industries, which are 

organized according to the North American Industry Classification System (NAICS). The other 

half measure prices for commodities (regardless of producing industry) according to an internal 

BLS classification system. 

We then aggregate the elementary indexes to form versions of 1,016 six-digit 

commodity indexes, 759 six-digit industry indexes, and the headline PPIs for Final Demand and 

Intermediate Demand (FD-ID). Because the focus is on differences in elementary calculation, all 

indexes use same the Lowe formula (Eq. 2) at the upper levels. Furthermore, we recalculate 

indexes that use the modified Laspeyres formula at the elementary level, rather than 

comparing to the published PPIs, in order to better hold constant other components of 

methodology such as imputation and item structure changes which are harder to replicate in a 

research environment. In 98.5 percent of observations, monthly percent changes of the reestimated six-digit commodity indexes fall within 0.1 percent of the actual indexes from 

production.

3.2 Results

As stated, we combine each set of elementary indexes into indexes covering 759 six-digit NAICS 

industries and 1,016 six-digit commodity groups. The average annual change across the six-digit 

commodity indexes calculated using the modified Laspeyres is 1.52 percent, versus 1.25

percent for the geometric Young, a difference of 0.27 percentage points. Across industries, the 

modified Laspeyres indexes average 1.70 percent, while the geometric Young indexes average

0.36 points lower at 1.34 percent. There is considerable heterogeneity across commodities and 
13

industries. Figure 1 plots the frequencies of annual percentage point differences for the six-digit 

commodity indexes. About two thirds of commodities show differences in the 0.0 to 0.3 

percentage point range. Frequencies generally decline over higher values, but the right tail is 

long, with 79 commodities having differences exceeding 0.9 percentage points. As expected, 

the modified Laspeyres implies higher inflation than the geometric Young for about 95 percent

of commodities.§ Note that a geometric mean will generally result in lower index levels (i.e., 

reference period to comparison period measurements), but the comparison may not always 

hold for short-term percent changes or when the considered timeframe spans item rotations or 

weight updates.

Table 1 presents the average annual percent changes for seven broad commodity 

categories. In all but one category (Wholesale and Retail Trade), formulas give average percent 

changes of the same sign, and the average differences mainly fall in the 0.2 to 0.4 percentage 

point range. Notable exceptions include Construction, where the average difference is only 0.05

percentage points, and Wholesale and Retail Trade, where the difference (1.35 percentage 

points) is more than three times that of any other category. The formulas disagree in sign for 

only about 3.3 percent of six-digit commodities overall, but within Trade, they disagree in 32

percent of cases. Similarly, Table 2 gives the average annual percent changes for the six-digit 

industry indexes within broad NAICS categories. As with the commodities, most differences 

average well under one percentage point per year with the exception of Wholesale Trade,

 

§ The distribution across six-digit industries is very similar, so we omit the corresponding histogram. Out of 759 

industries, 520 have index differences in the 0.0 to 0.3 percentage point range, and 64 have differences exceeding 

0.9 percentage points. The modified Laspeyres implies higher inflation than the Geometric Young for about 97 

percent of industries.
14

Retail Trade, and Finance and Insurance, where the average differences are 1.14, 1.71, and 0.72

percentage points per year, respectively.

As a general principle, greater dispersion of the underlying elements (in this case, longterm price relatives) is associated with a greater difference between the arithmetic and 

geometric mean. We should then expect to see greater dispersion in industries like Trade. 

Because of periodic discontinuations, we can only recover the long-term relatives for items that 

are observed during the entire period between sample rotations, a group which we label 

“survivors”. To check representativeness, we construct sets of industry indexes using only this 

subsample and present their average differences in column 4 of Table 3. The full sample 

differences from Table 2 have been copied to column 3 for comparison. Using the survivors 

only, the average differences are slightly greater in magnitude (0.42 versus 0.35 percentage 

points per year), but qualitatively similar to those based on the full-sample. Column 5 shows 

the average coefficients of variation within each NAICS category. Indeed, within the Trade, 

Financial Services, and Insurance industries, the long-term price relatives have coefficients of 

variation of 0.36 on average, versus 0.14 for all other industries.

BLS views firms that resell items as providers of services rather than goods. As such, the 

prices used for Trade are primarily gross margins (selling price minus acquisition price). Gross 

margins for retailers, for example, reflect the value added by the establishment for services 

such as marketing, storing, displaying goods, and making the goods easily available for 

customers to purchase. Several indexes within financial services also use measures that are 

similar to margins, like bid-ask spreads. Margin prices tend to be more volatile than selling 

prices alone. The BLS excludes zero or negative margins from calculation. Geometric mean 
15

indexes are still sensitive to margins that are close to zero, which can cause the long-term and 

month-to-month price relatives to be very small or very large (IMF 2004).

To assess potential sensitivity, we calculate the commodity indexes after imposing 

bounds of 0.05 and 20 on the monthly relatives, which matches the BLS procedure for the CPI. 

For example, if a relative is less than 0.05, we use the value 0.05 in its place. The results change 

very little. For the Trade category, the average percent changes for the modified Laspeyres and

geometric Young increase by 0.025 and 0.04 percentage points, respectively, decreasing the 

gap between them by only 0.015 percentage points. Similar results hold for tighter bounds of 

[0.25, 4], which decrease the gap by an additional 0.026 percentage points. There are still 

influential outliers inside these bounds, but it does not appear that the most extreme price 

relatives are driving the formula differences.

Aggregation of the commodity indexes into the headline PPIs shows the importance of 

formula choice. Table 4 presents average annual percent changes for the FD-ID indexes over 

2010-2017. We also calculate the indexes with and without Trade and Finance, which include 

margin prices to varying degrees. The Final Demand index escalates 0.55 percentage points per 

year less when the elementary indexes use the geometric Young formula. This magnitude is

larger, but near the range found by similar studies of CPI elementary indexes, such as Boskin, et

al. (1996), which found an all-items index difference of 0.25 percentage points, and Reinsdorf 

and Moulton (1996), which found a difference of 0.47 percentage points. In the case of the PPI 

for Final Demand, much of the difference is due to the Trade and Finance sectors, which

collectively show 1.53 percentage points lower inflation using the geometric Young. Excluding 

these, the difference between Final Demand indexes is only 0.24 percentage points per year. 
16

Figures 2 and 3 illustrate the role these industries play in driving the relative evolution of the 

alternative indexes. The Intermediate Demand indexes follow a similar pattern. The Processed 

and Unprocessed Goods indexes differ by 0.25 and 0.15 percentage points per year, 

respectively. As with Final Demand, the indexes covering Services for Intermediate Demand 

differ to a greater degree (0.54 percentage points per year), though this gap narrows 

considerably (to 0.17 percentage points per year) when excluding Trade and Finance.

4 Conclusion

The geometric Young formula has superior axiomatic properties to the modified Laspeyres, and 

it may better approximate a FIOPI based on an average production possibilities set. Our 

application to U.S. PPI data shows that these theoretical differences have economically 

significant consequences for elementary index aggregation. Using the geometric Young, for 

example, would lower the PPI for Final Demand by 0.55 percentage points per year. The effect 

on most industry and commodity PPIs is smaller—between 0.1 and 0.3 percentage points. For 

services like wholesale and retail trade, however, higher dispersion in margin prices leads to 

differences often exceeding one percentage point. Our main findings are little changed when 

bounding the price relatives, implying the formula differences are not primarily driven by 

outliers, e.g., values close to zero. The issue of margin prices is unique to PPIs and helps explain 

why we find greater differences between formulas than earlier studies found using consumer 

prices.
17

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18

Reinsdorf, Marshall B. 1998. "Formula Bias and Within-Stratum Substitution Bias in the U.S. 

CPI." The Review of Economics and Statistics (MIT Press) 80 (2): 175-187. 

doi:10.1162/003465398557375.

Reinsdorf, Marshall B., and Brent R. Moulton. 1996. "The Construction of Basic Components of 

Cost-of-Living Indexes." In The Economics of New Goods, 397-436. University of Chicago 

Press. http://www.nber.org/chapters/c6073.

Waehrer, Keith. 2000. A Comparison of the Substitution Effects for Input and Output Price 

Indexes. Working Paper 327, Washington, DC: Bureau of Labor Statistics. 

https://www.bls.gov/osmr/research-papers/2000/ec000020.htm.

Weinhagen, Jonathan. 2020. "Measuring the substitution effect in Producer Price Index goods 

data: 2002–16." Monthly Labor Review (Bureau of Labor Statistics). 

https://www.bls.gov/opub/mlr/2020/article/measuring-the-substitution-effect-in-ppidata.htm.
19

Tables

Table 1: Commodity Averages by Category, 2008-17 (Annual Percent Change)

Category Mod. Lasp. Geo. Young Lasp. – Geo.

Food 1.66 1.45 0.21

Energy -0.02 -0.40 0.38

Goods Less Food & Energy 1.68 1.50 0.19

Wholesale and Retail Trade 1.09 -0.26 1.35

Transportation 2.04 1.64 0.40

Services Less Trade & Transp. 1.08 0.64 0.44

Construction 1.60 1.55 0.05

Note: Rows are averages of six-digit commodity indexes within specified category. Mod. 

Lasp. and Geo. Young refer to formulas used for elementary aggregation. Upper-level 

aggregation uses Lowe formula in all cases.

Table 2: Industry Averages by NAICS Category, 2008-17 (Annual Percent Change)

NAICS Description Mod. Lasp. Geo. Young Lasp. – Geo.

11 Ag., Forestry, Fishing and Hunting 1.63 1.47 0.16

21 Mining, Quarrying, and Oil and Gas 2.92 2.46 0.47

22 Utilities 1.17 0.64 0.53

23 Construction 2.10 1.86 0.23

31-33 Manufacturing 1.78 1.55 0.23

42 Wholesale Trade 1.96 0.82 1.14

44-45 Retail Trade 1.05 -0.66 1.71

48-49 Transportation and Warehousing 2.13 1.79 0.34

51 Information -0.57 -1.00 0.43

52 Finance and Insurance 1.83 1.11 0.72

53 Real Estate and Rental and Leasing 1.02 0.67 0.35

54 Prof., Scientific, and Technical Services 1.57 1.40 0.17

56 Admin., Supp., Waste, & Rem. Services 1.10 0.97 0.14

61 Educational Services 1.23 0.96 0.27

62 Health Care and Social Assistance 1.47 1.29 0.19

71 Arts, Entertainment, and Recreation 2.19 1.80 0.39

72 Accommodation and Food Services 0.81 0.61 0.20

81 Other Services (ex. Public Admin.) 2.24 1.85 0.40

Note: Rows are averages of six-digit industry indexes within specified NAICS category. Mod. Lasp. and Geo. Young 

refer to formulas used for elementary aggregation. Upper-level aggregation uses Lowe formula in all cases.
20

Table 3: Industry Differences and Dispersion, 2008-17

NAICS Description

Full Sample 

Lasp. – Geo.

Survivors

Lasp. – Geo.

Survivors

LTR C.V.

11 Ag., Forestry, Fishing and Hunting 0.16 0.48 0.17

21 Mining, Quarrying, and Oil and Gas 0.47 0.42 0.18

22 Utilities 0.53 0.84 0.28

23 Construction 0.23 0.23 0.11

31-33 Manufacturing 0.23 0.30 0.13

42 Wholesale Trade 1.14 1.41 0.47

44-45 Retail Trade 1.71 1.68 0.35

48-49 Transportation and Warehousing 0.34 0.35 0.16

51 Information 0.43 0.40 0.16

52 Finance and Insurance 0.72 1.26 0.33

53 Real Estate and Rental and Leasing 0.35 0.33 0.16

54 Prof., Scientific, and Technical Services 0.17 0.13 0.10

56 Admin., Supp., Waste, & Rem. Services 0.14 0.12 0.10

61 Educational Services 0.27 0.17 0.13

62 Health Care and Social Assistance 0.19 0.28 0.15

71 Arts, Entertainment, and Recreation 0.39 0.53 0.17

72 Accommodation and Food Services 0.20 0.41 0.21

81 Other Services (ex. Public Admin.) 0.40 0.41 0.22

Note: Rows are averages within specified NAICS category. Index differences are expressed as percentage points 

per year. “Survivors” refers to indexes calculated using only those items available during entire sample period. 

“LTR C.V.” denotes coefficient of variation for the long-term relatives. Lasp. and Geo. refer to formulas used for 

elementary aggregation. Upper-level aggregation uses Lowe formula in all cases.

Table 4: PPI for Final and Intermediate Demand, 2010-17 (Annual Percent Change)

Elementary Indexes

Index Mod. Lasp. Geo. Young Lasp. – Geo.

Final Demand 1.56 1.01 0.55

Less Trade and Finance 1.47 1.23 0.24

Trade and Finance 1.74 0.22 1.53

Intermediate Demand -- -- --

Processed Goods 0.98 0.73 0.25

Unprocessed Goods -1.25 -1.40 0.15

Services 2.02 1.48 0.54

Less Trade and Finance 1.38 1.21 0.17

Trade, and Finance 3.05 1.87 1.18

Note: Mod. Lasp. and Geo. Young refer to formulas used for elementary aggregation. Upper-level 

aggregation uses Lowe formula in all cases.
21

Figures

Figure 1: Difference between Modified Laspeyres and Geometric Young for Six-Digit 

Commodities

Note: Observations are differences in annual percent changes for 6-digit commodities.

Modified Laspeyres and Geometric Young refer to formulas used for elementary 

aggregation. Upper-level aggregation uses Lowe formula in all cases.

71

264 258

146

70

48

23 27

15 15

79

0

50

100

150

200

250

300

≤ 0 0.10% 0.20% 0.30% 0.40% 0.50% 0.60% 0.70% 0.80% 0.90% > 0.90%

Frequency
22

Figure 2: PPI for Final Demand (Jan. 2010 = 100)

Note: Mod. Lasp. and Geo. Young refer to formulas used for elementary aggregation. 

Upper-level aggregation uses Lowe formula in all cases.

Figure 3: PPI for Final Demand with and without Trade and Finance (Jan. 2010 = 100)

Note: Mod. Lasp. and Geo. Young refer to formulas used for elementary aggregation. 

Upper-level aggregation uses Lowe formula in all cases.

99

101

103

105

107

109

111

113

115

201001201005201009201101201105201109201201201205201209201301201305201309201401201405201409201501201505201509201601201605201609201701201705201709

Mod. Lasp. Geo. Young

95

100

105

110

115

120

201001201005201009201101201105201109201201201205201209201301201305201309201401201405201409201501201505201509201601201605201609201701201705201709

Mod. Lasp.: FD less Trade & Fin. Geo. Young: FD less Trade & Fin.

Mod. Lasp.: Trade & Fin. Geo. Young: Trade & Fin.