ON MEASURING INVENTORY CHANGE IN CURRENT AND CONSTANT DOLLARS by Erwin Diewer Deparmen of Economics Universiy of Briish Columbia Augus 2005 Discussion Paper No.: 05-12 DEPARTMENT OF ECONOMICS THE UNIVERSITY OF BRITISH COLUMBIA VANCOUVER, CANADA V6T 1Z1 hp://www.econ.ubc.ca
On Measuring Invenory Change in Curren and Consan Dollars 1 Erwin Diewer, 1 Discussion Paper No. 05-12, Deparmen of Economics, The Universiy of Briish Columbia, Vancouver, Canada, V6T 1Z1. email: diewer@econ.ubc.ca Augus 8, 2005. Absrac In he curren Sysem of Naional Accouns, he reamen of invenory change in real erms is very confusing o users since when nominal invenory change is divided by he corresponding real change, negaive implici prices frequenly occur. This problem is due o he failure of normal index number heory when he value aggregae being deflaed can be of eiher sign in he wo periods under consideraion. The soluion o his problem is sraighforward: he value aggregae should be wrien as he difference beween wo posiive value aggregaes and each of he wo aggregaes should be separaely deflaed. This is analogous o he reamen of he rade balance which is rarely deflaed direcly; raher expors and impors are separaely deflaed and shown as wo separae real aggregaes in he SNA. The paper also considers how invenory change and he user cos of invenories can be joinly derived in a consisen economic framework. Journal of Economic Lieraure Classificaion Numbers C43, D24, D92, M4. Keywords Index number heory, invenory change, he sysem of naional accouns, user coss, economic income, measuremen of capial. 1. Inroducion The curren SNA reamen of invenory change in real erms is very confusing o users. The problem is ha i can happen ha he value of invenory change has a sign ha is opposie o he sign of he corresponding consan dollar invenory change. This means ha he corresponding implici price deflaor is meaningless. In his paper, he naure of he problem is explained and a soluion o he problem is suggesed. In Appendix 1, a heoreical framework ha provides a unified reamen for measuring invenory change 1 My hanks o Charles Aspden, Andrew Baldwin, Peer Harper, Peer Hill, Paul McCarhy, Alice Nakamura and Marshall Reinsdorf for helpful commens on an earlier draf of his noe. The financial assisance of he Bureau of Economic Analysis, he OECD and he SSHRC of Canada is graefully acknowledged. None of he above is responsible for any opinions expressed in his noe.
2 and he user cos of invenories is explained. 2 Appendix 2 gives some background informaion on he origins of he heoreical framework used in Appendix 1. In secion 2, a simple 2 good, 4 period numerical example is inroduced and i is explained how a ypical SNA reamen of invenory change works in he conex of his example. The example illusraes he problem described in he previous paragraph: he curren dollar aggregae invenory change has a sign opposie o he corresponding consan dollar change. In secion 3, he same example is reworked using he mehodological approach suggesed in Appendix 1. The suggesed soluion involves reaing invenory change in a manner ha is symmeric o he curren SNA reamen of expors and impors. Secion 4 concludes. 2. The SNA Treamen of Invenory Change Consider he following daa on he end of period socks of wo invenory iems for hree periods, where p n and q n denoe he price and quaniy of sock n a he end of period : Table 1: Price and Quaniy Daa for Two Invenory Socks p 1 p 2 q 1 q 2 Period 0 1.0 1.0 200 200 Period 1.9 2.0 260 150 Period 2.8 3.0 310 110 Period 3.7 4.0 330 100 Thus he price of he firs sock p 1 is slowly declining while he corresponding end of period sock q 1 grows from 200 o 330 over he hree periods. On he oher hand, he price of he second sock p 2 quadruples over he hree periods while he corresponding end of period sock q 2 seadily falls from 200 o 100 over he hree periods. These price changes are more violen han wha is usually observed over he course of a year bu hey would no necessarily be unusual if we hink of he firs good as compuer chip and he second good as crude oil. The end of period SNA consan dollar sock of invenories, K SNA, using he end of period 0 as he base period, can be defined as he following Laspeyres ype quaniy aggregae: (1) K SNA p 1 0 q 1 + p 2 0 q 2 ; = 0,1,2,3. Noe ha we use he invenory socks q n a he end of period along wih he prices of he socks a he end of period 0, p n 0, in he above definiion of he period consan dollar sock of invenories. Thus for period 0 (he beginning of period 1), he consan dollar 2 This mehodology is based on Diewer and Smih (1994) and Diewer (2004; 36). The iniial accouning mehodology can also be found in Diewer (2005; 21-23). Diewer and Lawrence (2005) used his framework as well.
3 sock coincides wih he curren dollar sock. The value of he curren dollar sock of invenories a he end of period, VK, is defined in he usual fashion as follows: (2) VK p 1 q 1 + p 2 q 2 ; = 0,1,2,3. If we divide VK by K SNA, we obain P SNA, he end of period SNA implici price index for he consan dollar sock of invenories: (3) P SNA VK /K SNA = [p 1 q 1 + p 2 q 2 ]/[p 1 0 q 1 + p 2 0 q 2 ] ; = 0,1,2,3. Noe ha he SNA implici price index for he invenory sock is a Paasche price index beween period and 0. The SNA consan dollar value of invenory change for period, K SNA, can be defined in a sraighforward manner as he difference beween he end of period and beginning of period consan dollar socks defined above by (1): (4) K SNA K 1 SNA K SNA = 1,2,3 = p 0 1 q 1 + p 0 2 q 2 [p 0 1 q 1 1 + p 0 2 q 1 2 ] using (1) 0 = p 1 [q 1 q 1 0 1 ] + p 2 [q 2 q 1 2 ] 0 = p 1 q 0 1 + p 2 q 2 where q n q n q n 1 is he difference in he closing and opening sock of invenory iem n over period. Noe ha he las equaion in (4) shows ha he aggregae change in he consan dollar change in invenories is equal o he sum of he individual iem changes, using he end of period 0 prices as weighs. The (approximae) SNA curren dollar value of invenory change for period, VK SNA, can be defined as he sum of he individual iem changes, q n, weighed by he average of he beginning and end of period prices, (1/2)p n 1 + (1/2)p n : 3 (5) VK SNA [(1/2)p 1 1 + (1/2)p 1 ] q 1 + [(1/2)p 2 1 + (1/2)p 2 ] q 2 ; = 1,2,3. The corresponding implici price index for he SNA invenory change, P SNA, is obained by dividing he value series VK SNA defined by (5) by he consan dollar series K SNA defined by (4): (6) P SNA VK SNA / K SNA = 1,2,3 3 This is no quie he heoreically correc measure of invenory change ha is suggesed in he Sysem of Naional Accouns 1993 on pages 130-131 bu is regarded as an approximaion ha is frequenly used as he following quoaion indicaes: This suggess ha even when prices are changing a good approximaion o he PIM may be obained by aking he difference beween he quaniy of goods held in invenory a he beginning and a he end of he accouning period and valuing he difference a he average prices prevailing wihin he period. This measure, which may be described as he quaniy measure, is widely used in pracice and is someimes misakenly considered o be he heoreically appropriae measure under all circumsances. SNA 1993, page 131. For a more complee discussion of he SNA heoreically correc measure of invenory change, see Hill (2005). However, Hill (2005) noes ha he heoreically correc mehod suffers from he same problems ha arise using he approximae mehod.
= (1/2){[p 1 1 + p 1 ] q 1 + [p 2 1 + p 2 ] q 2 }/{p 1 0 q 1 + p 2 0 q 2 }. 4 The above definiion for he change in socks price index, P SNA, looks a bi srange a firs sigh bu if he weighs q 1 and q 2 are posiive, i can be seen ha i is a perfecly reasonable price index ha compares an average of he beginning and end of period prices wih he base prices (which are he end of period 0 prices for he invenory componens). 4 The above definiions are used o consruc he value, price and quaniy of end of period invenory socks (VK, P SNA and K SNA respecively) for periods 0,1,2 and 3 and he value, price and quaniy of he change in invenory socks ( VK SNA, P SNA and K SNA respecively) for periods 1-3 using he daa in Table 1. The resuls are lised in Table 2 below. Table 2: Values, Prices and Quaniies for Aggregae Invenories a Period 0 Prices VK Period 0 400 1.000 400 Period 1 534 1.302 410 18.0 1.800 10 Period 2 578 1.376 420 57.5 5.750 10 Period 3 631 1.467 430 20.0 2.000 10 P SNA K SNA VK SNA P SNA K SNA A firs glance, he values, prices and quaniies for he aggregae invenory sock look reasonable, wih he values growing fairly quickly due o rapid increases in he price of he second invenory good bu he real socks growing a he much slower rae of 10 unis per year. Turning o he values, prices and quaniies for he changes in he aggregae invenory sock, we see ha he quaniy growh, K SNA, is equal o 10 in periods 1, 2 and 3, which is he difference in he corresponding beginning and end of period socks, K SNA. This is very saisfacory. However, when we calculae he change in he value of invenories a curren prices, VK SNA, a differen picure emerges. Because he price of invenory iem 2 increases much more rapidly han he price of iem 1 declines, he seady decline in he quaniy of iem 2 when valued a curren prices ouweighs he seady increase in he quaniy of iem 1 a curren prices so ha overall, he change in he value of invenories a curren prices urns ou o be srongly negaive ( 18 over he course of period 1, 57.5 over he course of period 2 and 20 over he course of period 3). Thus he corresponding implici price index for invenory change, P SNA, urns ou o be negaive in all hree periods (since he value change is negaive and he corresponding consan dollar quaniy change is posiive). Looking a definiion (6) above, i can be seen ha he roo of he problem is ha he quaniy weighs in he price index number formula, q 1 and q 2, are of opposie signs and he value aggregaes in he numeraor and denominaor of (6) are of opposie signs and fairly small. Index number heory breaks down under hese circumsances and can frequenly give rise o meaningless numbers as is he case in he presen siuaion. 5 4 However, noe ha when we se equal o zero, in general, P 0 SNA will no equal uniy; i.e., he index does no saisfy he ideniy es. 5 Hill (1971) noed his problem wih radiional index number heory many years ago.
5 The exisence of negaive implici prices for an oupu componen of he naional accouns may no creae any grea concepual problems for he compilers of he accouns (since he negaive implici prices are jus a consequence of definiions ha seem reasonable o accounans) bu hey do creae problems for many macroeconomic modelers who base heir models on microeconomic heory: negaive prices creae grea difficulies for his class of user. Hence, in he following secion, a differen heoreical framework (based on microeconomic heory) is suggesed ha will avoid he negaive implici price problem. 6 In addiion o he negaive implici price problem, here is anoher problem wih he above approximae SNA mehodology: namely, i relies on a fixed base Laspeyres ype mehodology. Noe ha he invenory sock aggregae is a fixed base Laspeyres quaniy index which uses he prices of period 0 as he weighs for he individual sock componens. Definiions (7)-(10) below redo definiions (1)-(6) above bu insead of using he prices a he end of period 0 as he base prices, he prices a he end of period 3 are used as he base prices. Definiions (2) and (5) remain unchanged since hey are values bu he counerpars o (1), (3), (4) and (6) are lised below (he new sock and flow aggregaes are denoed by K SNA (3), P SNA (3), K SNA (3) and P SNA (3)): (7) K SNA (3) p 1 3 q 1 + p 2 3 q 2 ; = 0,1,2,3; (8) P SNA (3) VK /K SNA (3) = [p 1 q 1 + p 2 q 2 ]/[p 1 3 q 1 + p 2 3 q 2 ] ; = 0,1,2,3; (9) K SNA (3) K SNA (3) K SNA 1 (3) = 1,2,3 = p 1 3 q 1 + p 2 3 q 2 [p 1 3 q 1 1 + p 2 3 q 2 1 ] using (7) = p 1 3 [q 1 q 1 1 ] + p 2 3 [q 2 q 2 1 ] = p 1 3 q 1 + p 2 3 q 2 ; (10) P SNA (3) VK SNA / K SNA (3) = 1,2,3 = (1/2){[p 1 1 + p 1 ] q 1 + [p 2 1 + p 2 ] q 2 }/{p 1 3 q 1 + p 2 3 q 2 }. The above definiions are used o consruc he price and quaniy of end of period invenory socks (P SNA (3) and K SNA (3) respecively) for periods 0,1,2 and 3 and he price and quaniy of he change in invenory socks ( P SNA (3) and K SNA (3) respecively) for periods 1,2 and 3 using he daa in Table 1. The resuls are lised in Table 3 below. Table 3: Values, Prices and Quaniies for Aggregae Invenories a Period 3 Prices VK P SNA (3) K SNA (3) VK SNA P SNA (3) K SNA (3) Period 0 400 0.4255 940 Period 1 534 0.6829 782 18.0.1139 158 Period 2 578 0.8798 657 57.5.4600 125 Period 2 631 1.0000 631 20.0.7692 26 6 However, here is a cos o he suggesed soluion: he single SNA oupu caegory, change in invenories, is replaced by he difference beween wo oupu caegories: he end of period sock of invenories less he beginning of he period sock of invenories.
6 From Table 2, i was seen ha he consan dollar sock of invenories (using he end of period 0 prices as he weighs) grew from 400 o 430 from he end of period 0 o he end of period 3 whereas from Table 3, i appears ha he consan dollar sock of invenories fell from 940 o 631 over he hree periods. Turning o he changes in he consan dollar socks, Table 2 old us ha he change in socks a consan prices was posiive (equal o 10 in each period) while Table 3 ells us ha he change in socks was srongly negaive in each period ( 158, 125 and 26). The corresponding implici prices are all negaive in Table 2 while hey are all posiive in Table 3. The lack of harmony in he wo ses of resuls is due o he large change in relaive prices (and he smaller bu sill significan change in relaive quaniies) over he hree periods and he fac ha a quaniy index is being used ha uses he price weighs of only one of he wo periods being compared. Chaper 16 in he SNA 1993 recommends he use of symmerically weighed index number formulae raher han he asymmerically weighed Laspeyres formula. Thus in he nex secion, he Fisher (1922) price and quaniy index (which is a symmerically weighed formula) will be used in order o consruc invenory sock aggregaes. Since he price and quaniy daa move relaively smoohly over ime, chained Fisher indexes will be used raher han fixed base Fisher indexes. 7 3. A Suggesed Alernaive Treamen of Invenory Change Following he advice given in Chaper 16 of SNA93, he Fisher price and quaniy indexes, P F and K F, are adoped as he measure of he aggregae price and quaniy (or volume) for he end of period socks of invenories. Since he price and quaniy daa have relaively smooh rends over ime, chained Fisher indexes were used. 8 The value of he end of period aggregae invenory sock, VK, is lised in Table 4 below along wih P F and K F. 9 Table 4: Values, Prices and Quaniies for Aggregae Invenories using Chained Fisher Indexes VK Error Period 0 400 1.000 400.0 1.000 Period 1 534 1.374 388.6 15.7 1.374 11.4 46.0 30.3 Period 2 578 1.642 352.1 60.0 1.642 36.5 80.0 20.0 Period 3 631 1.851 340.8 20.8 1.851 11.2 26.0 5.2 P F K F V A Comparing P F and K F in Table 4 wih P SNA and K SNA in Table 2, i can be seen ha he Fisher price index grows more rapidly (from 1 o 1.851) han he SNA price index (from 1 o 1.467) and he corresponding Fisher volume index for he invenory sock grows more slowly. The SNA volume index uses he prices of period 0 as weighs and he decreases in q 2 are jus ouweighed by he increases in q 1. However, when he Fisher chained volume index is used, he decreases in q 2 ge a higher weigh (due o he rapidly P A Q A V I 7 This is consisen wih he advice given in Chaper 16 of he SNA 1993, pages 388-389, where fixed base symmerically weighed indexes are recommended if he daa flucuae or bounce and chained indexes are recommended if he daa have rends. 8 The Bureau of Economic Analysis uses chained Fisher indexes o calculae invenory socks; see Parker and Seskin (1996) and Ehemann (2005). 9 The enries in he final 4 columns of Table 4 will be explained laer.
7 increasing price of q 2 ) han he weigh accorded o he increases in q 1, leading o a decrease in he Fisher volume index compared o he increase in he fixed base SNA ype index. Noe ha he differences are no insignifican. The problem of deermining he price and quaniy of he change in he aggregae invenory socks is now addressed. The mehodology described in Appendix 1 below is used, which provides a consisen heoreical framework based on economic heory for no only he price and quaniy of he change in invenories bu also for he user cos of aggregae invenory socks held a he beginning of each period. According o he model developed in Appendix 1, he heoreically correc period value aggregae for he value of invenory change, 10 V I, is given by (A10), which is rewrien using he noaion in Table 1 as follows: (11) V I j=1 2 P Kj K j = j=1 2 P Kj [K j K j 1 ] = 1,2,3 = j=1 2 p j [q j q j 1 ] = j=1 2 p j q j j=1 2 p j q j 1 = V E V B where (12) V E j=1 2 p j q j is he end of period invenory sock value aggregae and (13) V B j=1 2 p j q j 1 is a (hypoheical) beginning of period invenory sock value aggregae where he beginning of he period socks are valued a he end of period prices. From he second line in (11) above, i would appear ha he heoreical invenory change value aggregae for period, V I, has a sraighforward decomposiion ino prices (he end of period prices for he socks p j ) imes he quaniy changes over period, q j q 1 j. However, because he quaniies in his value aggregae are really quaniy differences and hence can be of eiher sign, index number heory may fail if he prices in he index number formula are aken o be he p j and he quaniies are aken o be he q j q 1 j. The suggesed soluion o his problem is o regard he invenory change value aggregae as he difference beween he end of period value aggregae V E and he hypoheical beginning of period value aggregae V B and hen use normal index number heory o decompose V E ino he produc of he price and quaniy componens P E and Q E respecively and o decompose V B ino he produc of he price and quaniy componens P B and Q B respecively. In oher words, i is suggesed ha he change in invenories value aggregae be reaed in a manner ha is symmeric o he reamen of he curren 10 This is he heoreically correc value aggregae if i is desired o have a formula for he user cos for invenories ha is compleely symmeric o he user cos for reproducible capial. If his symmery propery is no regarded as imporan, hen we need only use he firs 3 columns in Table 4 in order o decompose he beginning and end of period values for he sock of invenories ino Fisher ideal price and quaniy componens. If his second approach is aken, hen i is no necessary o perform he compuaions in he remainder of his secion.
8 rade balance as he difference beween he value of expors less he value of impors. 11 This rade balance aggregae has exacly he same ype of problem as he invenory change aggregae: i could be posiive in one period and negaive in he following period. Index number heory canno decompose his ype of difference value aggregae ino meaningful price and volume componens, unless i is guaraneed ha he value differences will remain well away from zero. Before we illusrae our suggesed reamen of invenory change using he daa in Table 1, we will firs use hese daa o illusrae a simple approach ha is problemaic. An approximae approach o he reamen of invenory change can be implemened as follows. Firs consruc he chained Fisher price and quaniy indexes for he end of period invenory socks, P F and K F respecively. Now define he period approximae price for invenory change, P A, o be he end of period Fisher sock price for invenory componens P F and define he period approximae quaniy or volume of invenory change Q A o be he difference beween he beginning and end of period Fisher quaniy indexes for he invenory socks; i.e., we have he following definiions: 12 (14) P A P F ; = 1,2,3; (15) Q A [K F K 1 F ] ; = 1,2,3. The corresponding approximae value of invenory change in period is V A defined as he produc of he approximae price and quaniy defined above: (16) V A P A Q A ; = 1,2,3. We noe ha definiions (14)-(16) collapse down o he heoreical model presened in Appendix 1, provided ha here is only one invenory iem in he aggregae. Moreover, he use of hese definiions makes he aggregae invenory socks perfecly consisen wih he aggregae value of invenory change; i.e., he sock and flow aggregaes are perfecly consisen. V A, P A and Q A are lised in Table 4 above for periods 1,2 and 3. However, even hough definiions (14)-(16) are perfecly consisen wih he heoreical approach explained in Appendix 1 when here is only one invenory iem in he aggregae, his correspondence does no hold in general when here are wo or more 11 This mehodological approach was suggesed in Diewer (2004; 36). 12 This approximae mehod for he reamen of invenory change is very close o he mehod presenly in use by he BEA o calculae real esimaes of invenory change. The BEA mehod uses he average of he beginning and end of period Fisher sock prices, (1/2)P F 1 + (1/2)P F, in place of P F on he righ hand sides of (14) and (15); see Parker and Seskin (1996) and Ehemann (2005). However, if here is only one invenory iem, hen he use of our (14) and (15) will give he righ answer if we use he user cos framework developed by Diewer and Smih (1994) whereas he BEA procedure will no. Ehemann (2005) developed a varian of he BEA procedure by consrucing Fisher indexes of acquisiions and disposals and aking heir difference, say B F S F using he noaion in he Appendix, in place of he difference in Fisher socks, K F K F 1. In he case of only one invenory iem, he Ehemann mehod will coincide wih he BEA mehod, provided ha U and G in he Appendix equaions (A3) and (A4) are zero in he wo periods being compared. In he many invenory iem case, even if U and G are zero, he BEA and Ehemann mehods will differ due o he differen weighs in he Fisher indexes K F, B F and S F.
9 invenory iems in he aggregae. 13 When here are wo or more invenory iems in he aggregae, i is no necessarily he case ha he value of he approximae change in he value of invenories V A is equal o he heoreically correc value of invenory change, V I, and so here will generally be an aggregaion error beween hese wo value aggregaes defined for period as follows: (17) Error V I V A ; = 1,2,3. The rue values of he invenory change aggregae, V I, and he aggregaion error beween his value and he approximae value V A are lised in he las wo columns of Table 4. I can be seen ha he aggregaion errors are oo large o be ignored in his case. Hence, we conclude ha while under some circumsances, he approximae mehod for calculaing he price and quaniy for invenory change can be saisfacory, in many cases i will no be saisfacory. Recall he end of period value of invenories aggregae, V E defined by (12) and he hypoheical beginning of period value of invenories aggregae V B defined by (13). Using chained Fisher indexes for periods 1 o 3, he resuling price and quaniy decomposiions using he daa in Table 1 are lised in Table 5. Table 5: Values, Prices and Quaniies for Beginning and End of Period Invenory Aggregaes using Chained Fisher Indexes V E P E Q E V B P B Q B Period 1 534 1.0000 534.0 580 1.0000 580.0 Period 2 578 1.1947 483.8 658 1.2707 517.8 Period 3 631 1.3473 468.4 657 1.4769 444.9 Looking a Table 5, i can be seen ha he volume of end of period invenories is decreasing more slowly (from 534.0 o 468.4) han he volume of beginning of period invenories (from 580.0 o 444.9). Hence he difference beween he wo volume aggregaes is increasing. I can also be seen ha he price of end of period invenories increases more slowly (from 1 o 1.3473) han he corresponding price of beginning of period invenories (from 1 o 1.4769). The relaively large discrepancy in hese wo raes of price increase explains why he approximae mehod for dealing wih invenory change does no work well for his example. 14 Since he beginning of period invenory sock ges a negaive weigh when an invenory change aggregae is formed and i has a higher inflaion rae han he end of period sock, i can be expeced ha he price of his ne oupu aggregae will decrease. 15 Alhough his resul is counerinuiive from he 13 If eiher end of period prices of invenory iems vary in sric proporion over ime or he quaniies in he end of period invenory socks vary in sric proporion over ime, hen he approximae approach will be perfecly consisen wih he heoreical approach explained in Appendix 1. This is because he Fisher formula is consisen wih boh Hicks (1946; 312-313) and Leonief s (1936; 54-57) Aggregaion Theorems; see Allen and Diewer (1981). 14 If he wo raes of price increase were equal, hen he aggregaion errors associaed wih he approximae mehod would be zero. 15 The problem is similar o an analogous problem ha occurs when he price of impors increases faser han he price of expors and oher oupu componens of GDP. In his case, he increase in he price of impors will reduce he GDP deflaor. Kohli (1982; 211) (1983) (2004; 91) noiced his problem wih he
10 perspecive of measuring general inflaion, i is sensible from he perspecive of producion heory: he increase in he beginning of period price of invenories acs like an increase in he price of an inermediae inpu and so he ne reurn o he producer of producing a uni of gross oupu less a uni of he inermediae has decreased; i.e., he price of ne oupu has decreased. To indicae how furher sages of aggregaion migh proceed, an invesmen aggregae is inroduced, which has price p 5 and quaniy q 5 in period. I is assumed ha he price and quaniy of his invesmen aggregae is consan during periods 1 o 3 and in paricular, i is assumed ha: (18) p 5 = 1; q 5 = 1000 ; = 1,2,3. The ask now is o consruc chained Fisher aggregae prices and quaniies for each year, P and Q (wih corresponding value V P Q ), ha aggregae over end of period socks, q 1 and q 2 (wih corresponding prices p 1 and p 2 ), beginning of year hypoheical socks indexed wih negaive signs, q 3 q 1 1 and q 4 q 1 2 (wih corresponding prices p 3 p 1 and p 4 p 2 ) and oher invesmen flows, q 5 (wih corresponding price p 5 ). Thus here are 5 commodiies in all ha are being aggregaed. The resuls for his invesmen plus change in invenories Fisher aggregae, V, P and Q, are lised in he firs hree columns of Table 6. Table 6: Values, Prices and Quaniies for Aggregae Invesmen plus Invenory Change using Chained Fisher Indexes V P Q P 2S Q 2S P AA Q AA V AA Period 1 954 1.0000 954.0 1.0000 954.0 1.0000 984.3 984.3 Period 2 920 0.9473 971.2 0.9484 970.1 0.9933 946.4 940.0 Period 3 974 0.9182 1060.8 0.9217 1056.7 0.9881 991.0 979.2 As expeced, he price of he invesmen plus invenory change aggregae, P, decreases over ime (in a sensible manner) and he corresponding quaniy or volume, Q, seadily increases. The columns in Table 5 ha decompose he wo invenory aggregaes plus he firs 3 columns in Table 6 are he core of he new suggesed presenaion of aggregae invenory change. The key is o decompose he invenory change ino wo aggregaes, show he price and quaniy deail for hose wo aggregaes and hen move o he nex sage of aggregaion where he wo invenory aggregaes are aggregaed wih oher flow aggregaes. All of he columns in Table 5 and he firs 3 columns in Table 6 show GDP deflaor many years ago: Acually, i can easily be seen ha any erms of rade change away from he base period price raio resuls in a fall in real naional produc. This clearly reveals he weakness of his measure of real value added, he drawbacks of direc index numbers, and he dangers of aggregaing posiive wih negaive quaniies. Ulrich Kohli (1983; 142). An example of his anomalous behavior of he GDP deflaor jus occurred in he advance release of gross domesic produc for he hird quarer of 2001 for he US naional income and produc accouns: he chain ype price indexes for C, I, X and M decreased (a annual raes) over he previous quarer by 0.4%, 0.2%, 1.4% and 17.4% respecively bu ye he overall GDP deflaor increased by 2.1%. Thus here was general deflaion in all secors of he economy bu ye he overall GDP deflaor increased. See Table 4 in he Bureau of Economic Analysis (2001).
11 sensible prices, quaniies and values, which is no he case wih he exising SNA mehod for dealing wih invenory change. In he firs 3 columns of Table 6, we consruced he aggregae P and Q by using he Fisher chained formula over he 5 mos finely disaggregaed prices and quaniies in he model. 16 I is also possible o consruc his aggregae price and quaniy in wo sages. In he firs sage, he end of period Fisher chained invenory aggregae price and quaniies, P E and Q E, and he beginning of period hypoheical invenory aggregae price and quaniies, P B and Q B are consruced: see he enries in Table 5. In he second sage of aggregaion, chained Fisher indexes are calculaed using P E, P B and p 5 as he period prices and Q E, Q B and q 5 as he corresponding period quaniies. The resuls of his wo sage aggregaion procedure are lised in Table 6 under he columns wih he headings P 2S and Q 2S (he corresponding wo sage value aggregae equals V and so i is no lised). I can be seen ha hese wo sage esimaes are reasonably close o heir one sage counerpars, P and Q. 17 Finally, he approximae price and quaniy for invenory change lised in Table 4, P A and Q A, can be used, along wih p 5 and q 5, in order o consruc approximae invesmen and invenory change aggregae price, quaniy and value for period, P AA, Q AA and V AA respecively, using he Fisher chain formula. The resuls are lised in he las 3 columns of Table 6. I can be seen ha for his paricular numerical example, he approximae mehod is no accepable. 18 The errors in values, prices and quaniies are large compared o he heoreically preferred measures, V, P and Q. 4. Conclusion The SNA mehod for reaing changes in invenories suffers from wo major problems: Aggregae real invenory socks and changes in socks are evaluaed a consan base period prices which leads o difficulies if he relaive prices of invenory componens are changing over ime (and his mehod is no consisen wih he use of symmerically weighed or superlaive indexes which is recommended in SNA93); The SNA implici prices for invenory change can be negaive and are exremely difficul for users o inerpre. Since he Canberra II Group has recommended ha user coss for reproducible capial socks be added o he SNA producion accouns as a recommended decomposiion of gross operaing surplus and since he Group also recommended ha invenory socks be 16 Diewer and Lawrence (2005) used his sraegy o consruc invesmen plus invenory change aggregaes in heir empirical work for Ausralia. 17 This is in accordance wih he experience of he U.S. Bureau of Economic Analysis in consrucing wo sage chained Fisher aggregaes. Diewer (1978; 888) derived a heoreical resul ha showed ha normally, he single sage and wo sage esimaes should approximae each oher fairly closely. 18 Lasky (1998; 106) and Ehemann (2005) essenially used his mehodology o evaluae he adequacy of he BEA mehod for esimaing invenory change, excep ha hey used all componens of GDP (C+G+I+X M) in place of our use of jus I as he ouside commodiy in he nex sage of aggregaion. Boh Lasky and Ehemann found relaively large aggregaion errors in using he BEA approximae mehod for esimaing invenory change. Thus he problem ha we are describing is no jus a hypoheical one.
12 included as asses ha should have user coss in hese opional accouns, i is necessary o carefully specify he links beween he user cos of invenories and he reamen of he change in invenories in he producion accouns. Appendix 1 o his paper presens a coheren heoreical framework for he reamen of invenory change and for he consrucion of user coss for invenory iems. In addiion o suggesing a consisen accouning framework for he user cos of invenories and he reamen of invenory change, he oher main mehodological suggesion in his noe is o rea invenory in a manner ha is symmeric o he reamen of he curren rade balance as he difference beween he value of expors less he value of impors. Alhough his suggesed reamen of invenory leads o sensible price and volume esimaes, i has he downside of being somewha differen han he curren SNA reamen of invenory change, which is well esablished. Hence users may find our suggesed soluion o he problems associaed wih he curren SNA reamen of invenory change o be a bi srange a firs. 19 However, if i is explained o users ha he suggesed reamen of invenory change is enirely analogous o he curren SNA reamen of inernaional rade, he suggesed new reamen will evenually be regarded as being quie accepable. Appendix 1: A Theoreical Treamen of Invenory Change A heoreical framework is needed o measure he conribuion of he change invenory sock over a period o producion. I is also necessary o work ou he user cos of he beginning of he period sock of invenories. A framework o answer hese quesions is oulined, aken from Diewer and Smih (1994). Firs consider he heory for a single invenory sock iem. Consider a firm ha perhaps produces a noninvenory oupu during period, Y, uses a noninvenory inpu X, sells he amoun S of an invenory iem during period and makes purchases of he invenory iem during period in he amoun B. Suppose ha he average prices during period of Y, X, S and B are P Y, P X, P S and P B respecively. Then neglecing balance shee iems, he firm s period cash flow is: 20 (A1) CF P Y Y P X X + P S S P B B. Le he firm s beginning of period sock of invenory be K 1 and le is end of period sock of invenory be K. These invenory socks are valued a he balance shee prices prevailing a he beginning and end of period, P K 1 and P K respecively. Noe ha all 4 prices involving invenory iems, P S, P B, P K 1 and P K can be differen. 19 The problem is ha mos users are no aware ha normal index number heory fails specacularly as a value aggregae approaches zero. 20 Noe ha his framework is flexible enough o allow he firm o eiher purchase or produce inernally invenory iems. Noe also ha firm purchases of invenory iems from oher domesic firms would appear in he naional accouns as inermediae inpu purchases and purchases from foreign suppliers would appear as impors. On he oher hand, sales of invenory iems by he firm o domesic producers, households or foreigners would appear in he naional accouns as gross oupus, final household consumpion or expors respecively.
13 The firm s period economic income or ne profi is defined as is cash flow plus he value of is end of period sock of invenory iems less (1+r ) imes he value of is beginning of period sock of invenory iems: (A2) EI CF + P K K (1+r ) P K 1 K 1 where r is he nominal cos of capial ha he firm faces a he beginning of period. Thus in definiion (A2), i is assumed ha he firm has o borrow financial capial or raise equiy capial a he cos r in order o finance is iniial holdings of invenory iems. This cos could be real (in he case of a firm whose iniial capial is funded by bonds) or i could be an opporuniy cos (in he case of a firm enirely funded by equiy capial). The end of period sock of invenory is relaed o he beginning of he period sock by he following equaion: (A3) K = K 1 + B S U where U denoes invenory iems ha are los, spoiled, damaged or are used inernally by he firm. In he case of livesock invenories, here is a naural growh rae of invenories over he period so equaion (3) is replaced by: (A4) K = K 1 + B S + G where G denoes he naural growh of he sock over period. 21 Define he change in invenory socks over period as: (A5) K K K 1. Using (A5), boh (A3) and (A4) can be wrien as: (A6) K = K 1 + K. Now subsiue (A6) ino he definiion of economic income (A2) and he following expression is obained: (A7) EI CF + P K [K 1 + K ] (1+r ) P K 1 K 1 = CF + P K K [r P K 1 (P K P K 1 ) ]K 1. Thus economic income is equal o cash flow plus he value of he change in invenory (valued a end of period balance shee prices) minus he user cos of invenories imes he saring socks of invenories where his period user cos is defined as (A8) P U r P K 1 (P K P K 1 ). 21 If he firm is consrucing invenory iems eiher for direc sale or as an inermediae sep in is producion processes, hen hese produced addiions o he sock would be included in he erm G.
Noe ha he above algebra works for boh livesock and ordinary invenory iems. 14 Of course, here can be wo versions of he user cos: An ex pos version where he acual end of period balance shee price of invenories is used or An ex ane version where a he beginning of period, we esimae a prediced value for he end of period balance shee price. For he producion accouns in he SNA, he ex ane version is he appropriae version, which means he naional income accounan has some leeway in forming esimaes of he end of period balance shee price for he invenory iem. Looking a (A7), i is imporan o noe ha he change in invenories ha occurred over period, K, should be valued a he end of period price for he invenory iem, P K. 22 If he firm is using or selling many invenory iems, say J iems, hen equaion (A7) becomes: (A9) EI CF + j=1 J P Kj K j j=1 J [r P Kj 1 (P Kj P Kj 1 ) ]K j 1 where he noaion is obvious. The erms involving he value of he change in invenories over he period are he following ones: (A10) j=1 J P Kj K j = j=1 J P Kj [K j K j 1 ] (A11) = j=1 J P Kj K j j=1 J P Kj K j 1. Looking a (A10), i would appear ha normal index number heory could be applied o he sum of erms in he value aggregae on he righ hand side, wih prices defined as he end of period balance shee prices P Kj and corresponding quaniies defined as he invenory changes K j K 1 j over period. However, his value aggregae is no necessarily of one sign over ime: i could be posiive, negaive or zero. Normal index number heory breaks down for value aggregaes ha can be eiher posiive or negaive over ime. 23 Thus index number heory should no be applied o he value aggregae on he righ hand side of (A10). Insead, i is recommended ha index number heory be applied separaely o he wo value aggregaes on he righ hand side of (A11). 24 Thus J j=1 P Kj K j should be decomposed (using normal index number heory) ino P KE K E where P KE is he scalar end of period aggregae price of invenories and K E is he 22 However, he curren SNA mehodology requires ha invenory change over he producion period be evaluaed a he average prices of he period. This requiremen could be accommodaed in our framework by replacing he end of period price of he invenory iem, P K, by an appropriae average invenory price for period. If his is done, and if he acual end of period price of he invenory iem is used for balance shee purposes, hen a reconciliaion enry will be required in he Revaluaion Accouns. 23 To see why his breakdown occurs, consider a siuaion where he value aggregae jus happens o be zero in he base period. Laspeyres price and quaniy indexes will be undefined under hese circumsances and nonsensical numbers will be obained if he value aggregae is very close o zero in he base period. However, if he Laspeyres, Paasche or Fisher formula is used in forming a larger aggregae ha is bounded well away from zero, hen he righ hand side of (A10) can be used when forming his larger aggregae and he same resuls will be obained as using he righ hand side of (A11) in forming he larger aggregae. 24 This soluion o he aggregaion problem was suggesed by Diewer (2004; 36).
15 J 1 corresponding end of period aggregae sock and j=1 P Kj K j should be decomposed ino P KB K B where P KB is he scalar beginning of period aggregae price of invenories and K B is he corresponding beginning of period aggregae sock. Then in place of he curren single aggregae for invenory change ha is repored in he curren Sysem of Naional Accouns, i is recommended ha invenory change be reaed in a manner ha is symmeric o he reamen of aggregae expors and impors in he accouns; i.e., he end of period aggregaes P KE and K E (he counerpars o he aggregae price of expors and he aggregae quaniy of expors) and he beginning of period aggregaes P KB and K B would be repored separaely jus as expors and impors are repored separaely in he curren SNA. There is anoher reamen of invenory change ha could be used by saisical agencies ha is much more sraighforward. The definiion of economic income, (A2) above, can be rewrien as follows: (A12) EI CF + P K K P K 1 K 1 r P K 1 K 1. Using (A12), he value of invenory change for period is simply defined as he end of period value of he sock, VK, 25 less he beginning of period value of he sock, VK 1 : (A13) VK VK 1 = P K K P K 1 K 1. Using his decomposiion of economic income, he user cos value aggregae is defined as he las erm on he righ hand side of (A12) and so he new user cos of invenories is: (A14) P U * r P K 1. The new user cos of invenories, P * U defined by (A14), can be compared o he iniial user cos of invenories, P U defined by (A8), and he new value of invenory change defined by (A13) can be compared o he earlier expression for he value of invenory change defined by (A11). Boh he old and he new decomposiion of economic income are heoreically valid. However, noe ha a nominal ineres rae r appears in (A14) whereas a ype of real ineres rae appeared in (A8). Hence for a counry experiencing high inflaion, he new user cos of invenories will be higher han he old user cos and similarly, he new value of invenory change defined by (A13) will be higher han he old value of invenory change defined by (A11). 26 Thus nominal GDP will end o be higher using he new decomposiion compared o he iniial one and i will be subsanially higher under condiions of high inflaion. There are advanages and disadvanages of using he second decomposiion of economic income compared o he firs: 25 See he firs 3 columns of Table 4 for he Fisher chain decomposiion of he end of period value of he socks VK ino price and quaniy componens for he numerical example. 26 If he iniial decomposiion of economic income is used, hen he beginning of he period invenory socks are valued a he higher end of period prices bu since his value aggregae is given a minus sign, his will reduce nominal GDP.
16 The main advanage of he second decomposiion is ha i is much more sraighforward and will be easier o explain o users. Also, i is much easier o reconcile quarerly changes in invenories o annual changes using he second decomposiion. The main disadvanage of he second decomposiion is ha he resuling user cos of invenories is differen from he user cos formula for reproducible capial and so an awkward asymmery would be inroduced ino he SNA if a user cos approach o reproducible capial were inroduced. 27 Boh decomposiions of economic income involve a difference in wo value aggregaes where he sign of he difference canno be bounded away from zero. Hence for boh decomposiions, i is recommended ha he beginning and end of period values be separaely deflaed and shown as wo iems in he real accouns in a manner ha is analogous o he presen reamen of expors less impors. Appendix 2: The Underlying Model of Producion In his Appendix, he moivaion for he model of producion ha was defined by equaions (A1) and (A2) in he previous Appendix is explained. This model of is based on a well esablished model of producion ha is used boh by economiss and houghful accounans as he following wo quoaions will show: We mus look a he producion process during a period of ime, wih a beginning and an end. I sars, a he commencemen of he Period, wih an Iniial Capial Sock; o his here is applied a Flow Inpu of labour, and from i here emerges a Flow Oupu called Consumpion; hen here is a Closing Sock of Capial lef over a he end. If Inpus are he hings ha are pu in, he Oupus are he hings ha are go ou, and he producion of he Period is considered in isolaion, hen he Iniial Capial Sock is an Inpu. A Sock Inpu o he Flow Inpu of labour; and furher (wha is less well recognized in he radiion, bu is equally clear when we are sric wih ranslaion), he Closing Capial Sock is an Oupu, a Sock Oupu o mach he Flow Oupu of Consumpion Goods. Boh inpu and oupu have sock and flow componens; capial appears boh as inpu and as oupu John R. Hicks (1961; 23). The business firm can be viewed as a recepacle ino which facors of producion, or inpus, flow and ou of which oupus flow...the oal of he inpus wih which he firm can work wihin he ime period specified includes hose inheried from he previous period and hose acquired during he curren period. The oal of he oupus of he business firm in he same period includes he amouns of oupus currenly sold and he amouns of inpus which are bequeahed o he firm in is succeeding period of aciviy. Edgar O. Edwards and Philip W. Bell (1961; 71-72). Hicks and Edwards and Bell obviously had he same model of producion in mind: in each accouning period, he business uni combines he capial socks and goods in process ha i has inheried from he previous period wih flow inpus purchased in he curren period (such as labour, maerials, services and addiional durable inpus) o produce curren period flow oupus as well as end of he period depreciaed capial sock componens which are regarded as oupus from he perspecive of he curren 27 The ex ane user cos for a reproducible capial asse conains an anicipaed asse inflaion rae in i similar o (A8), which offses he nominal ineres rae erm. The ex ane user cos concep should be close o an acual renal or leasing price for he asse since i based on he same consideraions ha an owner would consider in seing a renal price. Hence, i seems desirable o have he user cos of invenories aligned wih he user cos of reproducible capial.
17 period (bu will be regarded as inpus from he perspecive of he nex period). 28 The model could be viewed as an Ausrian model of producion in honour of he Ausrian economis Böhm-Bawerk (1891) who viewed producion as an aciviy which used raw maerials and labour o furher process parly finished goods ino finally demanded goods. Now relae his heoreical model of producion o equaions (A1) and (A2) in he previous Appendix. All of he flow inpus ha are purchased during he period and all of he flow oupus ha are sold during he period are he inpus and oupus ha appear in he definiion of cash flow in definiion (A1). These are he flow inpus and oupus ha are very familiar o naional income accounans. Bu his is no he end of he sory: he firm inheris an endowmen of asses a he beginning of he producion period and a he end of he period, he firm will have he ne profi or loss ha has occurred due o is sales of oupus and is purchases of inpus during he period. As well, i will have a sock of asses ha i can use when i sars producion in he following period. Hence i seems clear ha jus focusing on he flow ransacions ha occur wihin he producion period will no give a complee picure of he firm s producive aciviies. Naional income accounans are aware of his when hey make allowance for work in progress ; i.e., producion ha akes place during he period bu wihou any visible sales because i akes muliple periods o produce a saleable uni. Hence, o ge a complee picure of he firm s producion over he course of a period, i is necessary o add he value of he closing sock of asses less he beginning of he period sock of asses o he cash flow ha accrued o he firm from is sales and purchases of marke goods and services during he accouning period. Using he noaion explained in he previous appendix, his leads o he following definiion of he firm s period gross income or gross profi, defined as is cash flow plus he value of is end of period sock of invenory iems less he value of is beginning of period sock of invenory iems: (A15) GI CF + P K K P K 1 K 1. The gross income or profi approach does no explicily recognize ineres as a cos of producion. However, in order o induce invesors in he firm o hold he saring socks of capial iems for producive purposes (insead of immediaely selling hem), i is necessary o pay ineres. Thus i is necessary o subrac ineres imes he beginning of he period value of he capial sock from gross income in order o ge he economic income or ne profi EI defined earlier by (A2), which is repeaed here for convenience: (A16) EI CF + P K K (1+r ) P K 1 K 1. There are wo versions of economic income ha could be considered for naional income accouning purposes: An ex pos version ha uses he acual end of period price as he price P K in (A13) or 28 For more on his model of producion and addiional references o he lieraure, see he Appendices in Diewer (1977) (1980).
18 An ex ane version ha uses an anicipaed end of period price as he price P K in (A13). Diewer (1980; 476) and Hill and Hill (2003) endorsed he ex ane version for mos purposes, since i will end o be smooher han he ex pos version and i will generally be closer o a renal or leasing price for he asse. However, here are several pracical measuremen issues ha will make i difficul o implemen he ex ane version of ne income: 29 There may be difficulies in esimaing he beginning of he period values of he various socks held by firms since by definiion, hese socks are being held (and no sold immediaely) and so here are no unambiguous marke prices o value hese socks. There may be difficulies in deermining he righ opporuniy cos of financial capial r. I will be difficul o provide reproducible esimaes of he anicipaed end of period prices for he capial socks being held by firms. References Allen, R.C. and W.E. Diewer (1981), Direc versus Implici Superlaive Index Number Formulae, The Review of Economics and Saisics 63, 430-435. Böhm-Bawerk, E. von (1891), The Posiive Theory of Capial, ranslaed by W. Smar, New York: G. E. Secher (originally published in German in 1888). Bureau of Economic Analysis (2001), BEA News Release: Gross domesic Produc, Ocober 31, 2001, BEA, U.S. Deparmen of Commerce, hp://www.bea.doc.gov/newsrel/gdp301a.hm Diewer, W.E. (1977), Walras' Theory of Capial Formaion and he Exisence of a Temporary Equilibrium, pp. 73-126 in Equilibrium and Disequilibrium in Economic Theory, E. Schwödiauer (ed.), Reidel Publishing Co. Diewer, W.E. (1980), Aggregaion Problems in he Measuremen of Capial, pp.433-528 in The Measuremen of Capial, edied by D. Usher, Sudies in Income and Wealh, Vol. 45, Naional Bureau of Economics Research, Universiy of Chicago Press, Chicago. Diewer, W.E. (1978), Superlaive Index Numbers and Consisency in Aggregaion, Economerica 46, 883-900. 29 For noninvenory asses, here will be difficulies in deermining appropriae depreciaion raes as well as he difficulies lised below.