Chen Zhen (University of Georgia)

Mary Muth, Shawn Karns (RTI)

Abigail Okrent, and Megan Sweitzer (USDA ERS)

2023 FCSM Research and Policy Conference

Applications of Retail Scanner Data to Calculating Monthly Area Price Measures of Food

October 24─26, 2023

Using Rolling-Window Multilateral Price Indexes to Track 

Food Costs Across Space and Over Time 
Why do we need price indexes?

 Inflation surveillance: 

 price indexes compare costs of identical or similar products. 

 unit values (e.g., $ per pound of fruit) could be literally comparing (the cost 

of) apples to oranges.

 Demand estimation:

 consistency requires the price variables be driven by exogenous market 

conditions. 

 price indexes control for between-product cost differentials.

 unit values register (endogenous) choice over quality—famous unit value 

bias (Cox and Wohlgenant 1986; Deaton 1988). 

Agricultural and Applied Economics
Common price indexes

 Bilateral price indexes: 

 Laspeyres (1871), Paasche (1874), Fisher Ideal (Fisher 1922), and 

Törnqvist (1936). 

 Multilateral price indexes:

 GEKS (Gini 1931; Eltetö and Köves 1964; Szulc 1964) and CCD (Caves, 

Christensen, and Diewert 1982). 

 Rolling-window price indexes

 Rolling-window GEKS (Ivancic, Diewert, and Fox 2011) and CCD (Zhen, 

Finkelstein et al. 2019) indexes. 

Agricultural and Applied Economics
Bilateral price indexes

Laspeyres PPLL

0jj

=

∑vv∈vv0jj ppvv

jj

qqvv

0

∑vv∈vv0jj ppvv

0qqvv0

Key properties: 1) weight barcodes according to their base quantities, 2) hold 

standard of living (i.e., utility) constant between base and jj, and 3) not 

minimum cost at entity jj prices

Paasche PPPP

0jj =

∑vv∈vv0jj ppvv

jj

qqvv

jj

∑vv∈vv0jj ppvv

0qqvv

jj

Key properties: 1) weight barcodes according to their entity jj quantities, 2) 

minimum cost at entity jj prices, and 3) different utility levels at base and jj

Agricultural and Applied Economics
Superlative price indexes

Fisher Ideal PPFF

0jj = PPLL

0jj × PPPP0jj

Key properties:

1) account for substitution between barcodes

2) consistent with the (homothetic) “quadratic mean of order two” cost 

function cc ppjj = ∑kk ∑ll bbkk ppkk

jj

ppll

jj 1⁄2

, where unit cost cc is related to the 

utility level uu and total cost CC by cc = uu

3) no reference utility (i.e., whether it is cost of living at uu0 or uujj is irrelevant)

4) homotheticity implies unit expenditure elasticities for all barcodes 

Agricultural and Applied Economics
Superlative price indexes (continued)

Törnqvist PPTT

0jj = exp 0.5 �

vv∈vv0jj

ssvv

0 + ssvv

jj ln ppvv�jj ppvv

0

Key properties:

1) account for substitution between barcodes

2) consistent with the (homothetic) translog unit cost function 

ln cc ppjj = aa0 + ∑kk aakk ln ppkk

jj + 0.5 ∑kk ∑ll ggkk ln ppkkjj ln pplljj

, and the 

(nonhomothetic) translog total cost function 

ln CC uu, ppjj = aa0 + ∑kk aakk ln ppkk

jj + 0.5 ∑kk ∑ll ggkk ln ppkkjj ln pplljj + bb ln uu + cc ln uu 2

+ ∑kk ddkk ln uu ln ppkk

jj

1) when nonhomothetic, reference utility is uu0uujj

2) theoretically problematic b/c reference utility changes from one entity to 

the next, but empirically fine b/c the role of reference utility is numerically 

trivial 

Agricultural and Applied Economics
Multilateral price indexes 

A price index is transitive if the price ratio between entities ii and jj is the same 

whether ii and jj are compared directly or indirectly through a third entity kk. 

 Transitivity is especially important for spatial price comparison.

 The bilateral Laspeyres, Paasche, Fisher Ideal, and Törnqvist are not 

transitive.

 One solution is to create a MM × MM matrix of bilateral index numbers

 A better solution is to create a MM × 1 column of index numbers using a 

multilateral index 

The GEKS multilateral price index is calculated as

PPGGGGGGGG

0jj = �

ll=0

MM

PPFF

0ll × PPFF

ll 1⁄ MM+1

Agricultural and Applied Economics
Properties of Multilateral price indexes 

 Replacing the Fisher Ideal index in GEKS with the Törnqvist index 

produces the multilateral CCD index 

 Choice of base entity irrelevant (either an artificial base representing 

national average prices and sales or any market-period) 

 Fully transitive

 Violates characteristicity (e.g., I want to compare Atlanta with 

Boston. Why should prices in New York City matter?) 

Agricultural and Applied Economics
The rolling-window GEKS and CCD indexes

 Fixed-basket indexes become less representative over time. Chaining by a 

moving base is prone to a downward chain drift.

 Multilateral indexes avoid chain drift but require revision of published index 

numbers when new data arrive. 

 Solve both by opening a rolling window (RW) to multilateral indexes. 

The RWGEKS index for entity kk in month TT + 1 (TT = end of the base period) 

is 

PPRRRRGGGGGGGG

0kk = PPGGGGGGGG

0jj �

ll∈IITT+1:TT−11

PPFF

jj × PPFF

ll 1⁄MMTT+1:TT−11

PPGGGGGGGG

0jj is link entity jj in TT; IITT+1:TT−11 is the set of all entities between TT − 11 and 

TT + 1 (window = one year); and MMTT+1:TT−11 is the number of entities in the set 

IITT+1:TT−11.

Agricultural and Applied Economics
Properties of RWGEKS and RWCCD indexes

 Replacing PPGGGGGGGG

0jj with PPCCCCCC0jj and PPFF with PPTT gives the RWCCD index 

 Numerically identical to GEKS and CCD for entities in the base period 

 Can be free of chain drift (window length ≥ 1 year to be sure). 

 Not fully transitive for all entities. 

 Transitive between entities of the same period if they share the same link 

entity jj

Agricultural and Applied Economics