INDEX NUMBER THEORY AND MEASUREMENT ECONOMICS. By W.E. Diewert, January, CHAPTER 7: The Use of Annual Weights in a Monthly Index

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1 1 INDEX NUMBER THEORY AND MEASUREMENT ECONOMICS By W.E. Diewert, January, CHAPTER 7: The Use of Annual Weights in a Monthly Index 1. The Lowe Index with Monthly Prices and Annual Base Year Quantities It is now necessary to discuss a major practical prolem with the theory of ilateral indexes that we have een discussing in earlier chapters. Recall that the Lowe (1823) index was defined y equation (19) in chapter 1 as follows: (1) P Lo (p 0,p 1,q) p 1 q / p 0 q. The Lowe index can e written in expenditure share form as follows: (2) P Lo (p 0,p 1,q) n=1 N p n 1 q n / n=1 N p n 0 q n = n=1 N (p n 1 /p n 0 ) s n where the (hypothetical) hyrid expenditure shares s n corresponding to the quantity weights vector q are defined y: 1 (3) s n p n 0 q n / n=1 N p n 0 q n for n = 1,,N. Up to now, it has een assumed that the quantity vector q (q 1,q 2,,q N ) that appeared in the definition of the Lowe index, P Lo (p 0,p 1,q), is either the ase period quantity vector q 0 or the current period quantity vector q 1 or an average of these two quantity vectors. In fact, in terms of actual statistical agency practice, the quantity vector q is usually taken to e an annual quantity vector that refers to a ase year, say, that is prior to the ase period for the prices, period 0. Typically, a statistical agency will produce a Consumer Price Index at a monthly or quarterly frequency ut for the sake of definiteness, a monthly frequency will e assumed in what follows. Thus a typical price index will have the form P Lo (p 0,p t,q ), where p 0 is the price vector pertaining to the ase period month for prices, month 0, p t is the price vector pertaining to the current period month for prices, month t say, and q is a reference asket quantity vector that refers to the ase year, which is equal to or prior to month 0. 2 Note that this Lowe index P Lo (p 0,p t,q ) is not a true Laspeyres index (ecause the annual quantity vector q is not equal to the monthly quantity vector q 0 in general). 3 1 Fisher (1922; 53) used the terminology weighted y a hyrid value while Walsh (1932; 657) used the term hyrid weights. 2 Month 0 is called the price reference period and year is called the weight reference period. 3 Triplett (1981; 12) defined the Lowe index, calling it a Laspeyres index, and calling the index that has the weight reference period equal to the price reference period, a pure Laspeyres index. However, Balk (1980; 69) asserted that although the Lowe index is of the fixed ase type, it is not a Laspeyres price index. Triplett also noted the hyrid share representation for the Lowe index defined y (2) and (3) aove. Triplett noted that the ratio of two Lowe indices using the same quantity weights was also a Lowe index. Baldwin (1990; 255) called the Lowe index an annual asket index.

2 2 The question is: why do statistical agencies not pick the reference quantity vector q in the Lowe formula to e the monthly quantity vector q 0 that pertains to transactions in month 0 (so that the index would reduce to an ordinary Laspeyres price index)? There are two main reasons why this is not done: Most economies are suject to seasonal fluctuations and so picking the quantity vector of month 0 as the reference quantity vector for all months of the year would not e representative of transactions made throughout the year. Monthly household quantity or expenditure weights are usually collected y the statistical agency using a household expenditure survey with a relatively small sample. Hence the resulting weights are usually suject to very large sampling errors and so standard practice is to average these monthly expenditure or quantity weights over an entire year (or in some cases, over several years), in an attempt to reduce these sampling errors. The index numer prolems that are caused y seasonal monthly weights will e studied in more detail in a later chapter. For now, it can e argued that the use of annual weights in a monthly index numer formula is simply a method for dealing with the seasonality prolem. 4 One prolem with using annual weights corresponding to a perhaps distant year in the context of a monthly Consumer Price Index must e noted at this point: if there are systematic (ut divergent) trends in commodity prices and households increase their purchases of commodities that decline (relatively) in price and decrease their purchases of commodities that increase (relatively) in price, then the use of distant quantity weights will tend to lead to an upward ias in this Lowe index compared to one that used more current weights, as will e shown elow. This oservation suggests that statistical agencies should strive to get up to date weights on an ongoing asis. It is useful to explain how the annual quantity vector q could e otained from monthly expenditures on each commodity during the chosen ase year. Let the month m expenditure of the reference population in the ase year for commodity i e v i,m and let the corresponding price and quantity e p i,m and q i,m respectively. Of course, value, price and quantity for each commodity are related y the following equations: (4) v n,m = p n,m q n,m ; n = 1,,N ; m = 1,,12. For each commodity n, the annual total, q n can e otained y price deflating monthly values and summing over months in the ase year as follows: (5) q n = m=1 12 v n,m / p n,m = m=1 12 q n,m ; n = 1,,N 4 In fact, the use of the Lowe index P Lo (p 0,p t,q ) in the context of seasonal commodities corresponds to Bean and Stine s (1924; 31) Type A index numer formula. Bean and Stine made 3 additional suggestions for price indexes in the context of seasonal commodities. Their contriutions will e evaluated in a later chapter.

3 3 where (4) was used to derive the second equation in (5). In practice, the aove equations will e evaluated using aggregate expenditures over closely related commodities and the price p n,m will e the month m price index for this elementary commodity group n in year relative to the first month of year. For some purposes, it is also useful to have annual prices y commodity to match up with the annual quantities defined y (5). Following national income accounting conventions, a reasonale 5 price p n to match up with the annual quantity q n is the value of total consumption of commodity n in year divided y q n. Thus we have: (6) p n 12 m=1 v,m n /q n n = 1,,N = 12 m=1 v,m n /[ 12 m=1 v,m n / p,m n ] using (5) = [ 12 m=1 s,m n (p,m n ) 1 ] 1 where the share of annual expenditure on commodity n in month m of the ase year is (7) s n,m v n,m / k=1 12 v n,k ; n = 1,,N ; m = 1,,12. Thus the annual ase year price for commodity n, p n, turns out to e a monthly expenditure weighted harmonic mean of the monthly prices for commodity n in the ase year, p n,1, p n,2,, p n,12. Using the annual commodity prices for the ase year defined y (6), a vector of these prices can e defined as p [p 1,,p N ]. Using this definition, the Lowe index P Lo (p 0,p t,q ) can e expressed as a ratio of two Laspeyres indexes where the price vector p plays the role of ase period prices in each of the two Laspeyres indexes: (8) P Lo (p 0,p t,q ) p t q /p 0 q = [p t q /p q ]/[p 0 q /p q ] = P L (p,p t,q )/P L (p,p 0,q ) = n=1 N (p n t /p n ) s n / n=1 N (p n 0 /p n ) s n where the Laspeyres formula P L was defined in Chapter 1. Thus the aove equation shows that the Lowe monthly price index comparing the prices of month 0 to those of month t using the quantities of ase year as weights, P Lo (p 0,p t,q ), is equal to the Laspeyres index that compares the prices of month t to those of year, P L (p,p t,q ), divided y the Laspeyres index that compares the prices of month 0 to those of year, P L (p,p 0,q ). Note that the Laspeyres index in the numerator can e calculated if the ase 5 Hence these annual commodity prices are essentially unit value prices. Under conditions of high inflation, the annual prices defined y (6) may no longer e reasonale or representative of prices during the entire ase year ecause the expenditures in the final months of the high inflation year will e somewhat artificially lown up y general inflation. Under these conditions, the annual prices and annual commodity expenditure shares should e interpreted with caution. For more on dealing with situations where there is high inflation within a year, see Hill (1996).

4 4 year commodity expenditure shares, s n, are known along with the price ratios that compare the prices of commodity n in month t, p n t, with the corresponding annual average prices in the ase year, p n. The Laspeyres index in the denominator can e calculated if the ase year commodity expenditure shares, s n, are known along with the price ratios that compare the prices of commodity n in month 0, p n 0, with the corresponding annual average prices in the ase year, p n. There is another convenient formula for evaluating the Lowe index, P Lo (p 0,p t,q ), and that is to use the hyrid weights formula, (2). In the present context, the formula ecomes: (9) P Lo (p 0,p t,q ) n=1 N p n t q n / n=1 N p n 0 q n = n=1 N (p n t /p n 0 ) s n 0 where the (hypothetical) hyrid expenditure shares s n 0 corresponding to the quantity weights vector q are defined y: 6 (10) s 0 n p 0 n q n / N n=1 p 0 n q n = p n q n (p 0 n /p n )/ N j=1 p j q j (p 0 j /p j ). for n = 1,,N The second equation in (10) shows how the ase year expenditures on commodity n, p n q n, can e multiplied y the commodity price indexes, p n 0 /p n, in order to calculate the hyrid shares. There is one additional formula for the Lowe index, P Lo (p 0,p t,q ), that will e exhiited. Note that the Laspeyres decomposition of the Lowe index defined y the fourth line in (8) involves the very long term price relatives, p n t /p n, which compare the prices in month t, p n t, with the possily distant ase year prices, p n, and that the hyrid share decomposition of the Lowe index defined y the second line in (9) involves the long term monthly price relatives, p i t /p i 0, which compare the prices in month t, p i t, with the ase month prices, p i 0. Both of these formulae are not satisfactory in practice due to the prolem of sample attrition: each month, a sustantial fraction of commodities disappears from the marketplace and thus it is useful to have a formula for updating the previous month s price index using just month over month price relatives. In other words, long term price relatives disappear at a rate that is too large in practice to ase an index numer formula on their use. The Lowe index for month t+1, P Lo (p 0,p t+1,q ), can e written in terms of the Lowe index for month t, P Lo (p 0,p t,q ), and an updating factor as follows: (11) P Lo (p 0,p t+1,q ) p t+1 q /p 0 q = [p t q /p 0 q ][p t+1 q /p t q ] = P Lo (p 0,p t,q )[p t+1 q /p t q ] = P Lo (p 0,p t,q )[ n=1 N s n t (p n t+1 /p n t )] 6 Fisher (1922; 53) used the terminology weighted y a hyrid value while Walsh (1932; 657) used the term hyrid weights.

5 5 where the hyrid weights s n t are defined y (12) s n t p n t q n / k=1 N p k t q k ; n = 1,,N. Thus the required updating factor, going from month t to month t+1, is the chain link index n=1 N s n t (p n t+1 /p n t ), which uses the hyrid share weights s n t corresponding to month t and ase year. It should e noted that the month t hyrid shares, can e constructed from the previous month s hyrid shares, s n t 1, p n t 1 q n / k=1 N p k t 1 q k, y using the following updating formula: (13) s n t p n t q n / k=1 N p k t q k ; n = 1,,N = p n t 1 q n (p n t /p n t 1 )/ j=1 N p j t q j = [p n t 1 q n (p n t /p n t 1 )/p t 1 q ] /[ j=1 N p j t q j /p t 1 q ] = s n t 1, (p n t /p n t 1 )/[ j=1 N p j t q j /p t 1 q ] = s n t 1, (p n t /p n t 1 )/[ j=1 N (p j t /p j t 1 )p j t 1 q j /p t 1 q ] = s n t 1, (p n t /p n t 1 )/[ j=1 N (p j t /p j t 1 )s j t 1, ] = (p n t /p n t 1 )s n t 1, /[ j=1 N (p j t /p j t 1 )s j t 1, ]. Formula (13) can e used recursively until we get to t = 1, when (13) ecomes: (14) s n 1, p n 1 q n / k=1 N p k 1 q k ; n = 1,,N = (p n 1 /p n 0 )s n 0, /[ j=1 N (p j 1 /p j 0 )s j 0, ]. The hyrid shares, s n 0,, that use the components of the ase year quantity vector q and the ase month price vector p 0, can e constructed from ase year expenditures, p n q n, and the mixed month to year price relatives, (p n 0 /p n ), using formula (10) aove. Thus we have developed a complete set of practical updating formulae. The Lowe index P Lo (p 0,p t,q ) can e regarded as an approximation to the ordinary Laspeyres index, P L (p 0,p t,q 0 ), that compares the prices of the ase month 0, p 0, to those of month t, p t, using the quantity vector of month 0, q 0, as weights. It turns out that there is a relatively simple formula that relates these two indexes. However, efore we present this formula, we digress momentarily and develop a relationship etween the Paasche and Laspeyres price indexes. It turns out that we can adapt this methodology to the prolem of relating the Lowe index to the Laspeyres index. 2. The Bortkiewicz Decomposition etween the Paasche and Laspeyres Indexes In this section, we will develop a relationship etween the ordinary Paasche and Laspeyres price indexes. 7 In order to explain this formula, it is first necessary to make a few definitions. Define the nth price relative etween month 0 and month t as 7 This relationship was originally discovered y Bortkiewicz (1923; ).

6 6 (15) r n p n t /p n 0 ; n = 1,,N. The ordinary Laspeyres price index, relating the prices of month 0 to those of month t, can e defined as a weighted average of these price relatives as follows: (16) P L (p 0,p t,q 0 ) n=1 N s n 0 (p n t /p n 0 ) = n=1 N s n 0 r n using (15) r* where the month 0 expenditure shares s n 0 are defined as follows: (17) s n 0 p n 0 q n 0 / k=1 N p k 0 q k 0 ; n = 1,,N. Define the nth quantity relative t n as the ratio of the quantity of commodity n used in the month t, q n t, to the quantity used in month 0, q n 0, as follows: (18) t n q n t /q n 0 ; n = 1,,N. The Laspeyres quantity index, Q L (q 0,q t,p 0 ), that compares quantities in month t, q t, to the corresponding quantities in month 0, q 0, using the prices of month 0, p 0, as weights can e defined as a weighted average of the quantity ratios t n as follows: (19) Q L (q 0,q t,p 0 ) p 0 q t /p 0 q 0 = n=1 N s n 0 t n using (17) and (18) t*. Before we compare the Paasche and Laspeyres price indexes, we need to undertake a preliminary computation using the aove definitions of r n and t n : (20) Cov(r,t,s 0 ) n=1 N (r n r*)(t n t*)s n 0 = n=1 N r n t n s n 0 n=1 N r n t*s n 0 n=1 N r*t n s n 0 + n=1 N r*t*s n 0 = n=1 N r n t n s n 0 t* n=1 N r n s n 0 r* n=1 N t n s n 0 + r*t* n=1 N s n 0 = n=1 N r n t n s n 0 t* n=1 N r n s n 0 r* n=1 N t n s n 0 + r*t* using n=1 N s n 0 = 1 = n=1 N r n t n s n 0 t*r* r*t* + r*t* using (16) and (19) = n=1 N r n t n s n 0 t*r*. Note that Cov(r,t,s 0 ) can e interpreted as a weighted covariance etween the vector of price relatives, r [r 1,,r N ], and the vector of quantity relatives, t [t 1,,t N ], using the ase period vector of expenditure shares, s 0 [s 1 0,,s N 0 ], as weights. More explicitly, let r and t e discrete random variales that take on the N values r n and t n respectively. Let s n 0 e the joint proaility that r = r n and t = t n for n = 1,,N and let the joint proaility e 0 if r = r i and t = t j where i j. It can e verified that Cov(r,t,s 0 ) defined in the first

7 7 line of (20) is the covariance etween the price relatives r n and the corresponding quantity relatives t n. This covariance can e converted into a correlation coefficient. 8 Now we are ready to exhiit von Bortkiewicz s formula relating the Paasche and Laspeyres indexes. The Paasche index, P P (p 0,p t,q t ), that compares the prices of the ase month 0, p 0, to those of month t, p t, using the quantity vector of month t, q t, as a weighting vector is defined as follows: (21) P P (p 0,p t,q t ) N n=1 p t n q t n / N n=1 p 0 t n q n = N n=1 r n t n p 0 n q 0 n / N n=1 t n p 0 0 n q n using definitions (15) and (18) = [ N n=1 r n t n p 0 n q 0 n /p 0 q 0 ]/[ N n=1 t n p 0 n q 0 n /p 0 q 0 ] = N n=1 r n t n s 0 n / N 0 n=1 t n s n using definitions (17) = N n=1 r n t n s 0 n /t* using definition (19) = [{ N n=1 (r n r*)(t n t*)s 0 n } + r*t*]/t* using (20) = [ N n=1 (r n r*)(t n t*)s 0 n /t*] + r* = [ N n=1 (r n r*)(t n t*)s 0 n /Q L (q 0,q t,p 0 )] + P L (p 0,p t,q 0 ) using (16) and (19). Sutracting P L (p 0,p t,q 0 ) from oth sides of (21) leads to the following relationship etween the Paasche and Laspeyres price indexes: (22) P P (p 0,p t,q t ) P L (p 0,p t,q 0 ) = n=1 N (r n r*)(t n t*)s n 0 /Q L (q 0,q t,p 0 ) = Cov(r,t,s 0 )/Q L (q 0,q t,p 0 ). Thus the difference etween the Paasche and Laspeyres price indexes relating the prices of period 0 to those of period t is equal to the covariance etween the relative price and relative quantity vectors, Cov(r,t,s 0 ), divided y the Laspeyres quantity index, Q L (q 0,q t,p 0 ). Usually, this covariance will e negative for most value aggregates 9 so that usually the Paasche index will e less than the corresponding Laspeyres index. In the following section, we will develop a similar relationship etween the Lowe and Laspeyres indexes using the same technique as was used y Bortkiewicz. 3. The Relationship etween the Lowe, Laspeyres and Paasche Indexes We shall use the same notation for the long term monthly price relatives r n p n t /p n 0 that was used in the previous section so that (15)-(17) are still used in the present section. However, we shall change the definition of the t n in the previous section in order to relate the ase year annual quantities q n to the ase month quantities q n 0 : (23) t n q n /q n 0 ; n = 1,,N. 8 See Bortkiewicz (1923; ) for the first application of this correlation coefficient decomposition technique. 9 As we shall see later, this corresponds to the situation where demander sustitution effects outweigh supplier sustitution effects.

8 8 We also define a new Laspeyres quantity index Q L (q 0,q,p 0 ), which compares the ase year quantity vector q to the ase month quantity vector q 0, using the price weights of the ase month p 0, as follows: (24) Q L (q 0,q,p 0 ) p 0 q /p 0 q 0 = n=1 N p n 0 q n / n=1 N p n 0 q n 0 = n=1 N p n 0 q n 0 (q n /q n 0 ) / n=1 N p n 0 q n 0 = n=1 N s n 0 (q n /q n 0 ) using definitions (17) = n=1 N s n 0 (t n ) using definitions (23) t*. Using definition (9), the Lowe index comparing the prices in month t to those of month 0, using the quantity weights of the ase year, is equal to: (25) P Lo (p 0,p t,q ) N n=1 p t n q n / N n=1 p 0 n q n = N t n=1 p n t n q 0 n / N 0 0 n=1 p n t n q n using definitions (23) = N 0 n=1 r n p n t n q 0 n / N 0 0 n=1 p n t n q n using definitions (15) = [ N n=1 r n t n p 0 n q 0 n /p 0 q 0 ] / [ N n=1 t n p 0 n q 0 n /p 0 q 0 ] = N n=1 r n t n s 0 n / N 0 n=1 t n s n using definitions (17) = N n=1 r n t n s 0 n / t* using (24) = [{ N n=1 (r n r*)(t n t*)s 0 n } + t*r*]/t* using the identity (20) = [ N n=1 (r n r*)(t n t*)s 0 n /t*] + r* = [ N n=1 (r n r*)(t n t*)s 0 n /t*] + P L (p 0,p t,q 0 ) using definition (16) = [Cov(r,t,s 0 )/Q L (q 0,q,p 0 )] + P L (p 0,p t,q 0 ) where the last equality follows using definitions (20) and (24). Sutracting the Laspeyres price index relating the prices of month t to those of month 0, P L (p 0,p t,q 0 ), from oth sides of (25) leads to the following relationship of this monthly Laspeyres price index to its Lowe counterpart: (26) P Lo (p 0,p t,q ) P L (p 0,p t,q 0 ) = n=1 N (r n r*)(t n t*)s n 0 /Q L (q 0,q,p 0 ) = Cov(r,t,s 0 )/Q L (q 0,q,p 0 ). Thus the difference etween the Lowe and Laspeyres price indexes relating the prices of period 0 to those of period t is equal to the covariance etween the relative price and relative quantity vectors, Cov(r,t,s 0 ), divided y the Laspeyres quantity index, Q L (q 0,q,p 0 ). Thus (26) tells us that the Lowe price index using the quantities of year as weights, P Lo (p 0,p t,q ), is equal to the usual Laspeyres index using the quantities of month 0 as weights, P L (p 0,p t,q 0 ), plus a covariance term N 0 n=1 (r n r*)(t n t*)s n etween the long term monthly price relatives r n p t n /p 0 n and the quantity relatives t n q n /q 0 n (which are equal to the ase year quantities q n divided y the ase month quantities q 0 n ), divided y the Laspeyres quantity index Q L (q 0,q,p 0 ) etween month 0 and ase year.

9 9 Formula (26) shows that the Lowe price index will coincide with the Laspeyres price index if the covariance or correlation etween the month 0 to t price relatives r n p n t /p n 0 and the month 0 to year quantity relatives t n q n /q n 0 is zero. Note that this covariance will e zero under three different sets of conditions: If the month t prices are proportional to the month 0 prices so that all r n equal r*; If the ase year quantities are proportional to the month 0 quantities so that all t n equal t*; If the distriution of the relative prices r n is independent of the distriution of the relative quantities t n. The first two conditions are unlikely to hold empirically ut the third is possile, at least approximately, if consumers do not systematically change their purchasing haits in response to changes in relative prices. If the covariance in (26) is negative, then the Lowe index will e less than the Laspeyres and finally, if the covariance is positive, then the Lowe index will e greater than the Laspeyres index. Although the sign and magnitude of the covariance term, n=1 N (r n r*)(t n t*)s n 0, is ultimately an empirical matter, it is possile to make some reasonale conjectures aout its likely sign. If the ase year precedes the price reference month 0 and there are long term trends in prices, then it is likely that this covariance is positive and hence the Lowe index will exceed the corresponding Laspeyres price index 10 ; i.e., (27) P Lo (p 0,p t,q ) > P L (p 0,p t,q 0 ). To see why this covariance is likely to e positive, suppose that there is a long term upward trend in the price of commodity n so that r n r* (p t n /p 0 n ) r* is positive. With normal consumer sustitution responses 11, q t n /q 0 n less an average quantity change of this type is likely to e negative, or, upon taking reciprocals, q 0 n /q t n less an average quantity change of this (reciprocal) type is likely to e positive. But if the long term upward trend in prices has persisted ack to the ase year, then t n t* (q n /q 0 n ) t* is also likely to e positive. Hence, the covariance will e positive under these circumstances. Moreover, the more distant is the ase year from the ase month 0, the igger the residuals t n t* will likely e and the igger will e the positive covariance. Similarly, the more distant is the current period month t from the ase period month 0, the igger the residuals r n r* will likely e and the igger will e the positive covariance. Thus under the assumptions that there are long term trends in prices and normal consumer 10 It is also necessary to assume that households have normal sustitution effects in response to these long term trends in prices; i.e., if a commodity increases (relatively) in price, its consumption will decline (relatively) and if a commodity decreases relatively in price, its consumption will increase relatively. 11 Walsh (1901; ) was well aware of consumer sustitution effects as can e seen in the following comment which noted the asic prolem with a fixed asket index that uses the quantity weights of a single period: The argument made y the arithmetic averagist supposes that we uy the same quantities of every class at oth periods in spite of the variation in their prices, which we rarely, if ever, do. As a rough proposition, we a community generally spend more on articles that have risen in price and get less of them, and spend less on articles that have fallen in price and get more of them.

10 10 sustitution responses, the Lowe index will usually e greater than the corresponding Laspeyres index. Recall relationship (22) in the previous section, which related the difference etween the Paasche and Laspeyres price indexes, P P (p 0,p t,q t ) and P L (p 0,p t,q 0 ), to the covariance term, n=1 N (r n r*)(t n t*)s n 0, where the quantity relatives t n q n t /q n 0 were defined y (18). Although the sign and magnitude of the covariance term in (22), n=1 N (r n r*)(t n t*)s n 0, is again an empirical matter, it is possile to make a reasonale conjecture aout its likely sign. If there are long term trends in prices and consumers respond normally to price changes in their purchases, then it is likely that that this covariance is negative and hence the Paasche index will e less than the corresponding Laspeyres price index; i.e., (28) P P (p 0,p t,q t ) < P L (p 0,p t,q 0 ). To see why this covariance is likely to e negative, suppose that there is a long term upward trend in the price of commodity n 12 so that r n r* (p t n /p 0 n ) r* is positive. With normal consumer sustitution responses, q t n /q 0 n less an average quantity change of this type is likely to e negative. Hence t n t* (q t n /q 0 n ) t* is likely to e negative. Thus, the covariance will e negative under these circumstances. Moreover, the more distant is the ase month 0 from the current month t, the igger in magnitude the residuals t n t* will likely e and the igger in magnitude will e the negative covariance. 13 Similarly, the more distant is the current period month t from the ase period month 0, the igger the residuals r n r* will likely e and the igger in magnitude will e the covariance. Thus under the assumptions that there are long term trends in prices and normal consumer sustitution responses, the Laspeyres index will e greater than the corresponding Paasche index, with the divergence likely growing as month t ecomes more distant from month 0. Putting the arguments in the three previous paragraphs together, it can e seen that under the assumptions that there are long term trends in prices and normal consumer sustitution responses, the Lowe price index etween months 0 and t will exceed the corresponding Laspeyres price index which in turn will exceed the corresponding Paasche price index; i.e., under these hypotheses, (29) P Lo (p 0,p t,q ) > P L (p 0,p t,q 0 ) > P P (p 0,p t,q t ). Thus if the long run target price index is an average of the Laspeyres and Paasche indexes, it can e seen that the Laspeyres index will have an upward ias relative to this target index and the Paasche index will have a downward ias. In addition, if the ase year is prior to the price reference month, month 0, then the Lowe index will also have 12 The reader can carry through the argument if there is a long term relative decline in the price of the ith commodity. The argument required to otain a negative covariance requires that there e some differences in the long term trends in prices; i.e., if all prices grow (or fall) at the same rate, we have price proportionality and the covariance will e zero. 13 However, Q L = t* may also e growing in magnitude so the net effect on the divergence etween P L and P P is amiguous.

11 11 an upward ias relative to the Laspeyres index and hence also to the target index. The previous sentence is not good news for statistical agencies that ase their consumer price index on the Lowe index that uses ase year quantities for a distant year as weights. 5. The Lowe Index and Midyear Indexes The discussion in the previous sections assumed that the ase year for quantities preceded the ase month for prices, month 0. However, if the current period month t is quite distant from the ase month 0, then it is possile to collect expenditure information for a ase year that lies etween months 0 and t. If the year does fall etween months 0 and t, then the Lowe index ecomes a midyear index. 14 It turns out that if the ase year is etween monthly periods 0 and t, then the Lowe midyear index no longer has the upward iases indicated y the inequalities in (29) under the assumption of long term trends in prices and normal sustitution responses y quantities. We now assume that the ase year quantity vector q corresponds to a year that lies etween months 0 and t. Under the assumption of long term trends in prices and normal sustitution effects so that there are also long term trends in quantities (in the opposite direction to the trends in prices so that if the nth commodity price is trending up, then the corresponding nth quantity is trending down), it is likely that the intermediate year quantity vector will lie etween the monthly quantity vectors q 0 and q t. The midyear Lowe index, P Lo (p 0,p t,q ), and the Laspeyres index going from month 0 to t, P L (p 0,p t,q 0 ), will still satisfy the exact relationship given y equation (26) aove. Thus P Lo (p 0,p t,q ) will equal P L (p 0,p t,q 0 ) plus the covariance term [ n=1 N (r n r*)(t n t*)s n 0 ]/Q L (q 0,q,p 0 ), where Q L (q 0,q,p 0 ) is the Laspeyres quantity index going from month 0 to ase year. This covariance term is likely to e negative so that (30) P Lo (p 0,p t,q ) < P L (p 0,p t,q 0 ). To see why this covariance is likely to e negative, suppose that there is a long term upward trend in the price of commodity n so that r n r* (p n t /p n 0 ) r* is positive. With 14 This concept can e traced to Peter Hill (1998; 46): When inflation has to e measured over a specified sequence of years, such as a decade, a pragmatic solution to the prolems raised aove would e to take the middle year as the ase year. This can e justified on the grounds that the asket of goods and services purchased in the middle year is likely to e much more representative of the pattern of consumption over the decade as a whole than askets purchased in either the first or the last years. Moreover, choosing a more representative asket will also tend to reduce, or even eliminate, any ias in the rate of inflation over the decade as a whole as compared with the increase in the CoL index. Thus in addition to introducing the concept of a midyear index, Hill also introduced the terminology representativity ias. Baldwin (1990; ) also introduced the term representativeness: Here representativeness [in an index numer formula] requires that the weights used in any comparison of price levels are related to the volume of purchases in the periods of comparison. However, this asic idea dates ack to Walsh (1901; 104) (1921a; 90). Baldwin (1990; 255) also noted that his concept of representativeness was the same as Drechsler s (1973; 19) concept of characteristicity. For additional material on midyear indexes, see Schultz (1999) and Okamoto (2001). Note that the midyear index concept could e viewed as a close competitor to Walsh s (1901; 431) multiyear fixed asket index where the quantity vector was chosen to e an arithmetic or geometric average of the quantity vectors in the span of periods under consideration.

12 12 normal consumer sustitution responses, q n will tend to decrease relatively over time and since q n is assumed to e etween q n 0 and q n t, q n /q n 0 less an average quantity change of this type is likely to e negative. Hence t n t* (q n /q n 0 ) t* is likely to e negative. Thus, the covariance is likely to e negative under these circumstances. Thus under the assumptions that the quantity ase year falls etween months 0 and t and that there are long term trends in prices and normal consumer sustitution responses, the Laspeyres index will normally e larger than the corresponding Lowe midyear index, with the divergence likely growing as month t ecomes more distant from month 0. It can also e seen that under the aove assumptions, the midyear Lowe index is likely to e greater than the Paasche index etween months 0 and t; i.e., (31) P Lo (p 0,p t,q ) > P P (p 0,p t,q t ). To see why the aove inequality is likely to hold, think of q starting at the month 0 quantity vector q 0 and then trending smoothly to the month t quantity vector q t. When q = q 0, the Lowe index P Lo (p 0,p t,q ) ecomes the Laspeyres index P L (p 0,p t,q 0 ). When q = q t, the Lowe index P Lo (p 0,p t,q ) ecomes the Paasche index P P (p 0,p t,q t ). Under the assumption of trending prices and normal sustitution responses to these trending prices, it was shown earlier that the Paasche index will e less than the corresponding Laspeyres price index; i.e., that P P (p 0,p t,q t ) was less than P L (p 0,p t,q 0 ); recall (22). Thus under the assumption of smoothly trending prices and quantities etween months 0 and t, and assuming that q is etween q 0 and q t, we will have (32) P P (p 0,p t,q t ) < P Lo (p 0,p t,q ) < P L (p 0,p t,q 0 ). Thus if the ase year for the Lowe index is chosen to e in etween the ase month for the prices, month 0, and the current month for prices, month t, and there are trends in prices with corresponding trends in quantities that correspond to normal consumer sustitution effects, then the resulting Lowe index is likely to lie etween the Paasche and Laspeyres indexes going from months 0 to t. If the trends in prices and quantities are smooth, then choosing the ase year half way etween periods 0 and t should give a Lowe index that is approximately half way etween the Paasche and Laspeyres indexes and hence will e very close to an ideal target index etween months 0 and t. This asic idea has een implemented y Okamoto (2001) using Japanese consumer data and he found that the resulting midyear indexes approximated the corresponding Fisher ideal indexes very closely. However, the assumption of smooth trends in prices and quantities is necessary to get this close approximation. All of the inequalities derived in this chapter rest on the assumption of long term trends in prices (and corresponding economic responses in quantities). If there are no systematic long run trends in prices, ut only random fluctuations around a common trend in all prices, then the aove inequalities are not valid and the Lowe index using a prior ase year will proaly provide a perfectly adequate approximation to oth the Paasche and Laspeyres indices. However, there are some reasons for elieving that there are some long run trends in prices. Some of these reasons are:

13 13 The computer chip revolution of the past 40 years has led to strong downward trends in the prices of products that use these chips intensively. As new uses for chips have een developed over the years, the share of products that are chip intensive has grown and this implies that what used to e a relatively minor prolem has ecome a more major prolem. Other major scientific advances have had similar effects. For example, the invention of fier optic cale (and lasers) has led to a downward trend in telecommunications prices as osolete technologies ased on copper wire are gradually replaced. Since the end of World War II, there have een a series of international trade agreements that have dramatically reduced tariffs around the world. These reductions, comined with improvements in transportation technologies, have led to a very rapid growth of international trade and remarkale improvements in international specialization. Manufacturing activities in the more developed economies have gradually een outsourced to lower wage countries, leading to deflation in goods prices in most countries around the world. However, many services cannot e readily outsourced 15 and so on average, the price of services trends upwards while the price of goods trends downwards. At the microeconomic level, there are tremendous differences in growth rates of firms. Successful firms expand their scale, lower their costs, and cause less successful competitors to wither away with their higher prices and lower volumes. This leads to a systematic negative correlation etween changes in item prices and the corresponding changes in item volumes that can e very large indeed. Thus there is some a priori asis for assuming long run divergent trends in prices and hence there is some asis for concern that a Lowe index that uses a distant ase year for quantity weights that is prior to the ase month for prices may e upward iased, compared to a more ideal target index. 6. The Young Index Recall the definitions for the ase year quantities, q n, and the ase year prices, p n, (5) and (6) aove. The ase year expenditure shares s n can e defined in the usual way as follows: (33) s n p n q n / k=1 N p k q k ; n = 1,,N. Define the vector of ase year expenditure shares in the usual way as s [s 1,,s N ]. These ase year expenditure shares were used to provide an alternative formula for the ase year Lowe price index going from month 0 to t defined in (8) as P Lo (p 0,p t,q ) = n=1 N (p n t /p n ) s n / n=1 N (p n 0 /p n ) s n. Rather than using this index as their short run target index, many statistical agencies use the following closely related Young price index: 15 However some services can e internationally outsourced; e.g., call centers, computer programming, airline maintenance, etc.

14 14 (34) P Y (p 0,p t,s ) n=1 N (p n t /p n 0 ) s n. This type of index was first defined y the English economist, Arthur Young (1812). 16 Note that there is a change in focus when the Young index is used compared to the other indexes proposed earlier in this chapter. Up to this point, the indexes proposed have een of the fixed asket type (or averages of such indexes) where a commodity asket that is somehow representative for the two periods eing compared is chosen and then purchased at the prices of the two periods and the index is taken to e the ratio of these two costs. On the other hand, for the Young index, one instead chooses representative expenditure shares that pertain to the two periods under consideration and then uses these shares to calculate the overall index as a share weighted average of the individual price ratios, p t n /p 0 n. Note that this share weighted average of price ratios view of index numer theory is a it different from the view taken at the eginning of this chapter, which viewed the index numer prolem as the prolem of decomposing a value ratio into the product of two terms, one of which expresses the amount of price change etween the two periods and the other which expresses the amount of quantity change. 17 Statistical agencies sometimes regard the Young index defined aove as an approximation to the Laspeyres price index P L (p 0,p t,q 0 ). Hence, it is of interest to see how the two indexes compare. Defining the long term monthly price relatives going from month 0 to t as r n p n t /p n 0 and using definitions (34) and (16): (35) P Y (p 0,p t,s ) P L (p 0,p t,s 0 ) = n=1 N (p n t /p n 0 ) s n n=1 N (p n t /p n 0 ) s n 0 = n=1 N (p n t /p n 0 )[ s n s n 0 ] = n=1 N r n [ s n s n 0 ] using definitions (15) = n=1 N [r n r*][ s n s n 0 ] + r* n=1 N [ s n s n 0 ] = n=1 N [r n r*][ s n s n 0 ] since n=1 N s n = n=1 N s n 0 = 1 and defining r* n=1 N s n 0 r n = P L (p 0,p t,q 0 ). Thus the Young index P Y (p 0,p t,s ) is equal to the Laspeyres index P L (p 0,p t,q 0 ) plus the covariance 16 The attriution of this formula to Young is due to Walsh (1901; 536) (1932; 657). 17 Fisher s 1922 ook is famous for developing the value ratio decomposition approach to index numer theory ut his introductory chapters took the share weighted average point of view: An index numer of prices, then shows the average percentage change of prices from one point of time to another. Irving Fisher (1922; 3). Fisher went on to note the importance of economic weighting: The preceding calculation treats all the commodities as equally important; consequently, the average was called simple. If one commodity is more important than another, we may treat the more important as though it were two or three commodities, thus giving it two or three times as much weight as the other commodity. Irving Fisher (1922; 6). Walsh (1901; ) considered oth approaches: We can either (1) draw some average of the total money values of the classes during an epoch of years, and with weighting so determined employ the geometric average of the price variations [ratios]; or (2) draw some average of the mass quantities of the classes during the epoch, and apply to them Scrope s method. Scrope s method is the same as using the Lowe index. Walsh (1901; 88-90) consistently stressed the importance of weighting price ratios y their economic importance (rather than using equally weighted averages of price relatives). Both the value ratio decomposition approach and the share weighted average approach to index numer theory were studied from the axiomatic perspective in Chapter 3.

15 15 etween the difference in the annual shares pertaining to year and the month 0 shares, s n s n 0, and the deviations of the relative prices from their mean, r n r*. 18 It is no longer possile to guess at what the likely sign of the covariance term is. The question is no longer whether the quantity demanded goes down as the price of commodity n goes up (the answer to this question is usually yes) ut the new question is: does the share of expenditure on commodity n go down as the price of commodity n goes up? The answer to this question depends on the elasticity of demand for the product. However, let us provisionally assume that there are long run trends in commodity prices and if the trend in prices for commodity n is aove the mean, then the expenditure share for the commodity trends down (and vice versa). Thus we are assuming high elasticities or very strong sustitution effects. Assuming also that the ase year is prior to month 0, then under these conditions, suppose that there is a long term upward trend in the price of commodity n so that r n r* (p n t /p n 0 ) r* is positive. With the assumed very elastic consumer sustitution responses, s n will tend to decrease relatively over time and since s n is assumed to e prior to s n 0, s n 0 is expected to e less than s n or s n s n 0 will likely e positive. Thus, the covariance is likely to e positive under these circumstances. Hence with long run trends in prices and very elastic responses of consumers to price changes, the Young index is likely to e greater than the corresponding Laspeyres index. Assume that there are long run trends in commodity prices. Suppose the trend in price for commodity n is aove the mean, and suppose that the expenditure share for the commodity trends up (and vice versa). Thus we are assuming low elasticities or very weak sustitution effects. Assume also that the ase year is prior to month 0 and since we have supposed that there is a long term upward trend in the price of commodity n, then r n r* (p t n /p 0 n ) r* will e positive. With the assumed very inelastic consumer sustitution responses, s n will tend to increase relatively over time and since s n is assumed to e prior to s 0 n, it will e the case that s 0 n is greater than s n or s n s 0 n is negative. Thus, the covariance is likely to e negative under these circumstances. Hence with long run trends in prices and very inelastic responses of consumers to price changes, the Young index is likely to e less than the corresponding Laspeyres index. The previous two paragraphs indicate that a priori, it is not known what the likely difference etween the Young index and the corresponding Laspeyres index will e. If elasticities of sustitution are close to one, then the two sets of expenditure shares, s i and s i 0, will e close to each other and the difference etween the two indices will e close to zero. However, if monthly expenditure shares have strong seasonal components, then the annual shares s i could differ sustantially from the monthly shares s i 0. It is useful to have a formula for updating the previous month s Young price index using just month over month price relatives. The Young index for month t+1, P Y (p 0,p t+1,s ), can e written in terms of the Young index for month t, P Y (p 0,p t,s ), and an updating factor as follows: 18 Strictly speaking, the covariance etween the vectors r and [s s 0 ] is (1/N)[r r*1 N ] [s s 0 ]; i.e., the weighting factor (1/N) is missing in (35).

16 16 (36) P Y (p 0,p t+1,s ) n=1 N (p n t+1 /p n 0 ) s n = P Y (p 0,p t,s ) n=1 N (p n t+1 /p n 0 ) s n / n=1 N (p n t /p n 0 ) s n = P Y (p 0,p t,s ) n=1 N (p n t+1 /p n t )(p n t /p n 0 ) s n / n=1 N (p n t /p n 0 ) s n = P Y (p 0,p t,s ) n=1 N (p n t+1 /p n t ) s n 0t where the hyrid weights s n 0t are defined as (37) s n 0t (p n t /p n 0 ) s n / n=1 N (p n t /p n 0 ) s n ; n = 1,..,N. Thus the hyrid weights s 0t n can e otained from the ase year expenditure shares s n y updating them; i.e., y multiplying them y the price relatives, (or indexes at higher levels of aggregation), p t n /p 0 n. Thus the required updating factor, going from month t to month t+1, is the chain link index, N 0t n=1 s n (p t+1 n /p t n ), which uses the hyrid share weights s 0t n defined y (37). Even if the Young index provides a close approximation to the corresponding Laspeyres index, it is difficult to recommend the use of the Young index as a final estimate of the change in prices going from period 0 to t, just as it was difficult to recommend the use of the Laspeyres index as the final estimate of inflation going from period 0 to t. Recall that the prolem with the Laspeyres index was its lack of symmetry in the treatment of the two periods under consideration; i.e., using the justification for the Laspeyres index as a good fixed asket index, there was an identical justification for the use of the Paasche index as an equally good fixed asket index to compare periods 0 and t. The Young index suffers from a similar lack of symmetry with respect to the treatment of the ase period. The prolem can e explained as follows. The Young index, P Y (p 0,p t,s ) defined y (34) calculates the price change etween months 0 and t treating month 0 as the ase. But there is no particular reason to treat month 0 as the ase month other than convention. Hence, if we treat month t as the ase and use the same formula to measure the price change from month t ack to month 0, the index P Y (p t,p 0,s ) = n=1 N s n (p n 0 /p n t ) would e appropriate. This estimate of price change can then e made comparale to the original Young index y taking its reciprocal, leading to the following reased Young index 19, P Y *(p 0,p t,s ), defined as follows: (38) P Y *(p 0,p t,s ) 1/ n=1 N (p n 0 /p n t ) s n = [ n=1 N s n (p n t /p n 0 ) 1 ] 1. Thus the reased Young index, P Y *(p 0,p t,s ), that uses the current month as the initial ase period is a share weighted harmonic mean of the price relatives going from month 0 to month t, whereas the original Young index, P Y (p 0,p t,s ), is a share weighted arithmetic mean of the same price relatives. Fisher argued as follows that an index numer formula should give the same answer no matter which period was chosen as the ase: 19 Using Fisher s (1922; 118) terminology, P Y *(p 0,p t,s ) 1/[P Y (p t,p 0,s )] is the time antithesis of the original Young index, P Y (p 0,p t,s ).

17 17 Either one of the two times may e taken as the ase. Will it make a difference which is chosen? Certainly, it ought not and our Test 1 demands that it shall not. More fully expressed, the test is that the formula for calculating an index numer should e such that it will give the same ratio etween one point of comparison and the other point, no matter which of the two is taken as the ase. Irving Fisher (1922; 64). Prolem 1. Show that the Young index and its reased counterpart satisfy the following inequality: (39) P Y *(p 0,p t,s ) P Y (p 0,p t,s ) with a strict inequality provided that the period t price vector p t is not proportional to the period 0 price vector p Thus a statistical agency that uses the direct Young index P Y (p 0,p t,s ) will generally show a higher inflation rate than a statistical agency that uses the same raw data ut uses the reased Young index, P Y *(p 0,p t,s ). The inequality (39) does not tell us y how much the Young index will exceed its reased time antithesis. However, it can e shown that to the accuracy of a certain second order Taylor series approximation, the following relationship holds etween the direct Young index and its time antithesis: (40) P Y (p 0,p t,s ) P Y *(p 0,p t,s ) + P Y (p 0,p t,s ) Var e where Var e is defined as (41) Var e n=1 N s n [e n e*] 2. The deviations e n are defined y (42) 1+e n r n /r*; n = 1,,N where the r n and their weighted mean r* are defined y (43) and (44) elow: (43) r n p n t /p n 0 ; n = 1,,N; (44) r* n=1 N s n r n 20 Walsh (1901; ) explicitly noted the inequality (39) and also noted that the corresponding geometric average would fall etween the harmonic and arithmetic averages. Walsh (1901; 432) computed some numerical examples of the Young index and found ig differences etween it and his est indexes, even using weights that were representative for the periods eing compared. Recall that the Lowe index ecomes the Walsh index when geometric mean quantity weights are chosen and so the Lowe index can perform well when representative weights are used. This is not necessarily the case for the Young index, even using representative weights. Walsh (1901; 433) summed up his numerical experiments with the Young index as follows: In fact, Young s method, in every form, has een found to e ad.

18 18 which turns out to equal the direct Young index, P Y (p 0,p t,s ). The weighted mean of the e n is defined as (45) e* n=1 N s n e n. Prolem 2. Show that e* = 0. Looking at (40), we see that the more dispersion there is in the price relatives p n t /p n 0, to the accuracy of a second order approximation, the more the direct Young index will exceed its counterpart that uses month t as the initial ase period rather than month 0. We indicate how the result (40) can e estalished. The direct Young index, P Y (p 0,p t,s ), and its time antithesis, P Y *(p 0,p t,s ), can e written as functions of r*, the weights s n and the deviations of the price relatives e n as follows: (46) P Y (p 0,p t,s ) = n=1 N s n r n = r*; (47) P Y *(p 0,p t,s ) [ n=1 N s n (p n t /p n 0 ) 1 ] 1 = [ n=1 N s n (r n ) 1 ] 1 using (43) = r*[ n=1 N s n (1+e n ) 1 ] 1 using (42) r* f(e 1,e 2,,e N ). Prolem 3. Calculate the second order Taylor series approximation to f(e) defined in (47) around the point e = 0 N and show that it simplifies to 1 Var e. Hence we otain the approximate equality P Y *(p 0,p t,s ) r*(1 Var e), which is equivalent to the approximate equality (40). 21 Given two a priori equally plausile index numer formula that give different answers, such as the Young index and its time antithesis, Fisher (1922; 136) generally suggested taking the geometric average of the two indexes 22 and a enefit of this averaging is that the resulting formula will satisfy the time reversal test. Thus rather than using either the ase period 0 Young index, P Y (p 0,p t,s ), or the current period t Young index, 21 This type of second order approximation is due to Dalén (1992; 143) for the case r* =1 and to Diewert (1995; 29) for the case of a general r*. 22 We now come to a third use of these tests, namely, to rectify formulae, i.e., to derive from any given formula which does not satisfy a test another formula which does satisfy it;. This is easily done y crossing, that is, y averaging antitheses. If a given formula fails to satisfy Test 1 [the time reversal test], its time antithesis will also fail to satisfy it; ut the two will fail, as it were, in opposite ways, so that a cross etween them (otained y geometrical averaging) will give the golden mean which does satisfy. Irving Fisher (1922; 136). Actually the asic idea ehind Fisher s rectification procedure was suggested y Walsh, who was a discussant for Fisher (1921), where Fisher gave a preview of his 1922 ook: We merely have to take any index numer, find its antithesis in the way prescried y Professor Fisher, and then draw the geometric mean etween the two. Correa Moylan Walsh (1921; 542).