The Canadian Consumer Price Index Reference Paper
Appendix A – Common Price Index Formulae

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Appendix A – Common Price Index Formulae
Table summary
This table displays the results of Appendix A – Common Price Index Formulae. The information is grouped by Common index formulae for elementary price indices (lower level) (appearing as row headers), (appearing as column headers).
Common index formulae for elementary price indices (lower level)
Name Index formulae Description
Dutot I D , a t 1 : t = i = 1 n 1 n p i t i = 1 n 1 n p i t 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGebGaaiilaiaadggaaeaacaWG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaamaalaaabaGaaG ymaaqaaiaad6gaaaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaaaa aeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaam aaqahabaWaaSaaaeaacaaIXaaabaGaamOBaaaacaWGWbWaa0baaSqa aiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaaaeaacaWGPbGaeyypa0 JaaGymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@55BB@ A price index defined as the ratio of the unweighted arithmetic average of the prices in the current period t to the unweighted arithmetic average of the prices in period t-1. See chapter 6, formula 6.3.
Jevons I J , a t 1 : t = i = 1 n ( p i t ) 1 n i = 1 n ( p i t 1 ) 1 n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGkbGaaiilaiaadggaaeaacaGG0bGaeyOeI0IaaGymaiaa cQdacaWG0baaaOGaeyypa0ZaaSaaaeaadaqeWbqaaiaacIcacaWGWb Waa0baaSqaaiaadMgaaeaacaWG0baaaOGaaiykamaaCaaaleqabaWa aSGaaeaacaaIXaaabaGaamOBaaaaaaaabaGaamyAaiabg2da9iaaig daaeaacaWGUbaaniabg+GivdaakeaadaqeWbqaaiaacIcacaWGWbWa a0baaSqaaiaadMgaaeaacaWG0bGaeyOeI0IaaGymaaaakiaacMcada ahaaWcbeqaamaaliaabaGaaGymaaqaaiaad6gaaaaaaaqaaiaadMga cqGH9aqpcaaIXaaabaGaamOBaaqdcqGHpis1aaaaaaa@58C1@ A price index defined as the ratio of the unweighted geometric average of the prices in the current period t to the unweighted geometric average of the prices in period t-1. See chapter 6, formula 6.2.
Weighted Jevons I W J , a t 1 : t = i = 1 n ( p i t ) w i / i = 1 n w i i = 1 n ( p i t 1 ) w i / i = 1 n w i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGxbGaamOsaiaacYcacaWGHbaabaGaaiiDaiabgkHiTiaa igdacaGG6aGaamiDaaaakiabg2da9maalaaabaWaaebCaeaadaqada qaaiaadchadaqhaaWcbaGaamyAaaqaaiaadshaaaaakiaawIcacaGL PaaadaahaaWcbeqaamaalyaabaGaam4DamaaBaaameaacaWGPbaabe aaaSqaamaaqahabaGaam4DamaaBaaameaacaWGPbaabeaaaeaacaWG PbGaeyypa0JaaGymaaqaaiaad6gaa4GaeyyeIuoaaaaaaaWcbaGaam yAaiabg2da9iaaigdaaeaacaWGUbaaniabg+GivdaakeaadaqeWbqa amaabmaabaGaamiCamaaDaaaleaacaWGPbaabaGaamiDaiabgkHiTi aaigdaaaaakiaawIcacaGLPaaadaahaaWcbeqaamaalyaabaGaam4D amaaBaaameaacaWGPbaabeaaaSqaamaaqahabaGaam4DamaaBaaame aacaWGPbaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaad6gaa4Ga eyyeIuoaaaaaaaWcbaGaamyAaiabg2da9iaaigdaaeaacaWGUbaani abg+Givdaaaaaa@6AE5@ A price index defined as the ratio of the explicitly weighted geometric average of the prices in the current period t to the explicitly weighted geometric average of the prices in period t-1. See chapter 6, formula 6.4.
Common index formulae for aggregate price indices (upper level)
Name Index formulae Description
Fisher I F , A 0 : t = ( I L , A 0 : t × I P , A 0 : t ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaGGgbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaqadaqaaiaadMeadaqhaaWcbaGaamitaiaacYcacaWGbb aabaGaaGimaiaacQdacaWG0baaaOGaey41aqRaamysamaaDaaaleaa caGGqbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaaakiaawI cacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaaikdaaaaa aaaa@4D83@ A price index defined as a geometric average of the Laspeyres price index and the Paasche price index. It is a symmetrically weighted index using quantities of goods and services from both periods 0 and t.
Laspeyres I L , A 0 : t = i = 1 n p i t q i 0 i = 1 n p i 0 q i 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaWcaaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaaba GaamiDaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaaicdaaaaabaGa amyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdaakeaadaaeWb qaaiaadchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0ba aSqaaiaadMgaaeaacaaIWaaaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaaaaa@5402@ A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the base period 0. See chapter 6, formula 6.5.
Lowe I L o , A 0 : t = i = 1 n p i t q i b i = 1 n p i 0 q i b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGmbGaam4BaiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaaiaadchadaqhaaWcbaGaam yAaaqaaiaadshaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWGIbaa aaqaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaGcba WaaabCaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaamyC amaaDaaaleaacaWGPbaabaGaamOyaaaaaeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaa0GaeyyeIuoaaaaaaa@5550@ A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the chosen weight reference period b. See chapter 6, formula 6.6.
Marshall-Edgeworth I M E , A 0 : t = i = 1 n p i t × [ ( q i 0 + q i t ) 2 ] i = 1 n p i 0 × [ ( q i 0 + q i t ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGnbGaamyraiaacYcacaWGbbaabaGaaGimaiaacQdacaWG 0baaaOGaeyypa0ZaaSaaaeaadaaeWbqaaiaadchadaqhaaWcbaGaam yAaaqaaiaadshaaaGccqGHxdaTdaWadaqaamaalaaabaWaaeWaaeaa caWGXbWaa0baaSqaaiaadMgaaeaacaaIWaaaaOGaey4kaSIaamyCam aaDaaaleaacaWGPbaabaGaamiDaaaaaOGaayjkaiaawMcaaaqaaiaa ikdaaaaacaGLBbGaayzxaaaaleaacaWGPbGaeyypa0JaaGymaaqaai aad6gaa0GaeyyeIuoaaOqaamaaqahabaGaamiCamaaDaaaleaacaWG PbaabaGaaGimaaaakiabgEna0oaadmaabaWaaSaaaeaadaqadaqaai aadghadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccqGHRaWkcaWGXbWa a0baaSqaaiaadMgaaeaacaWG0baaaaGccaGLOaGaayzkaaaabaGaaG OmaaaaaiaawUfacaGLDbaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGa amOBaaqdcqGHris5aaaaaaa@69A0@ A price index defined as the ratio of average weighted prices between period 0 and t with weights as the arithmetic average of quantities from both periods 0 and t. It is a symmetrically weighted fixed-basket index.
Paasche I P , A 0 : t = i = 1 n p i t q i t i = 1 n p i 0 q i t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaGGqbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaWcaaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaaba GaamiDaaaakiaadghadaqhaaWcbaGaamyAaaqaaiaadshaaaaabaGa amyAaiabg2da9iaaigdaaeaacaWGUbaaniabggHiLdaakeaadaaeWb qaaiaadchadaqhaaWcbaGaamyAaaqaaiaaicdaaaGccaWGXbWaa0ba aSqaaiaadMgaaeaacaWG0baaaaqaaiaadMgacqGH9aqpcaaIXaaaba GaamOBaaqdcqGHris5aaaaaaa@5483@ A price index defined as an asymmetrically weighted fixed-basket index that uses the quantities of goods and services from the current period t.
Törnqvist-Theil I T , A 0 : t = i = 1 n ( p i t p i 0 ) 1 2 ( s i 0 + s i t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGubGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaqeWbqaamaabmaabaWaaSaaaeaacaWGWbWaa0baaSqaai aadMgaaeaacaWG0baaaaGcbaGaamiCamaaDaaaleaacaWGPbaabaGa aGimaaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaG ymaaqaaiaaikdaaaWaaeWaaeaacaWGZbWaa0baaWqaaiaadMgaaeaa caaIWaaaaSGaey4kaSIaam4CamaaDaaameaacaWGPbaabaGaamiDaa aaaSGaayjkaiaawMcaaaaaaeaacaWGPbGaeyypa0JaaGymaaqaaiaa d6gaa0Gaey4dIunaaaa@541A@


Where

s i 0 p i 0 q i 0 i = 1 n p i 0 q i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaaGimaaaakiabggMi6oaalaaabaGaamiCamaa DaaaleaacaWGPbaabaGaaGimaaaakiaadghadaqhaaWcbaGaamyAaa qaaiaaicdaaaaakeaadaaeWbqaaiaadchadaqhaaWcbaGaamyAaaqa aiaaicdaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaaIWaaaaaqaai aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaaaaa@4BC8@

s i t p i t q i t i = 1 n p i t q i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaDa aaleaacaWGPbaabaGaamiDaaaakiabggMi6oaalaaabaGaamiCamaa DaaaleaacaWGPbaabaGaamiDaaaakiaadghadaqhaaWcbaGaamyAaa qaaiaadshaaaaakeaadaaeWbqaaiaadchadaqhaaWcbaGaamyAaaqa aiaadshaaaGccaWGXbWaa0baaSqaaiaadMgaaeaacaWG0baaaaqaai aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaaaaa@4D03@
A price index defined as a geometric average of price relatives weighted by the average expenditure shares in both periods 0 and t. It is a symmetrically weighted index.
Walsh I W , A 0 : t = i = 1 n p i t q i t q i 0 i = 1 n p i 0 q i t q i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysamaaDa aaleaacaWGxbGaaiilaiaadgeaaeaacaaIWaGaaiOoaiaadshaaaGc cqGH9aqpdaWcaaqaamaaqahabaGaamiCamaaDaaaleaacaWGPbaaba GaamiDaaaakmaakaaabaGaamyCamaaDaaaleaacaWGPbaabaGaamiD aaaakiaadghadaqhaaWcbaGaamyAaaqaaiaaicdaaaaabeaaaeaaca WGPbGaeyypa0JaaGymaaqaaiaad6gaa0GaeyyeIuoaaOqaamaaqaha baGaamiCamaaDaaaleaacaWGPbaabaGaaGimaaaakmaakaaabaGaam yCamaaDaaaleaacaWGPbaabaGaamiDaaaakiaadghadaqhaaWcbaGa amyAaaqaaiaaicdaaaaabeaaaeaacaWGPbGaeyypa0JaaGymaaqaai aad6gaa0GaeyyeIuoaaaaaaa@5A56@ A price index defined as the ratio of average weighted prices between period 0 and t with weights as the geometric average of quantities from both periods 0 and t. It is a symmetrically weighted fixed-basket index.

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