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As will become apparent this model represents a simplified description of the public expenditure process and the application of the Barnett formula. Nevertheless this model retains the key aspect of the system for present purposes: and enables us to draw some useful, and perhaps surprising, conclusions. 3) The government plans its future programme of public expenditure on the basis of a three year rolling programme. Thus, in the CSR2000 exercise, the government set its programmes for the future years of 2001-02, 2002-03 and 2003-04. When the government comes to carry out its 2001 planning round, therefore, the first thing it has to do is to establish a starting, or baseline, figure for the new end year in the three year planning horizon. In this case this will be 2004-05. 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Thus, in the CSR2000 exercise, the government set its programmes for the future years of 2001-02, 2002-03 and 2003-04. When the government comes to carry out its 2001 planning round, therefore, the first thing it has to do is to establish a starting, or baseline, figure for the new end year in the three year planning horizon. In this case this will be 2004-05. The way the government does this is of critical importance for the ap    ўџџџ !"#$%&'ўџџџ)*+4-./012356789:;<=>?@Eџџџџџџџџџџџџџџџџўџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџplication of the Barnett formula. Since 1993, the government has set the new final year base line for each programme as being equal to the previous final year in cash: (source: personal communication from HMT). Prior to 1993, the new final year base line was apparently set as the previous final year figure, uprated for inflation: (see HMT (1997). The government then considers what adjustments are required to the baseline for English departments for each of the three future years. 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These lags will, typically, be five years, four years, and three years successively. The average lag, therefore, between the date of the population estimates used and the year being adjusted is likely to be of the order of four years. 5) The following idealised model of how Barnett works abstracts from these complications. In this model, the Scottish DEL for any given year is adjusted only once by Barnett, when the new baseline for that year is first established. Another simplification is the assumption that public expenditure in England is growing by a constant percentage each year. 6) Specifically, the following notation and assumptions are used:- Let  EMBED Equation.2  denote expenditure in England in year t, and  EMBED Equation.2  expenditure in Scotland. Let  EMBED Equation.2  denote population in England in year t, and  EMBED Equation.2  population in Scotland. Let  EMBED Equation.2  denote the ratio of per capita expenditures between Scotland and England at time t. Let k denote lag, (in years). Suppose that a)  EMBED Equation.2  : (i.e., expenditure in England grows at a constant rate.) b)  EMBED Equation.2  for all t, where  EMBED Equation.2 : (i.e., there is a constant relative rate of growth of population in England relative to Scotland). c) In the annual public expenditure planning round, the new final year baseline is determined as being equal to the previous end year figure: and Barnett applies only to that end year, with population shares determined at a lag k. 7) The above model can be solved, to show how the per capita spending relativity between Scotland and England,  EMBED Equation.2 , evolves through time from its initial starting value in year 0. The relevant formula for  EMBED Equation.2  is as follows:-  EMBED Equation.2  (1) The derivation of formula (1) is given in Annex 1. What formula (1) means is that, in the normal circumstance where  EMBED Equation.2  , then the initial per capita spending relativity,  EMBED Equation.2  ,will decay geometrically to the limiting value  EMBED Equation.2  EMBED Equation.2  , which is a function of the expenditure growth rate in England, the rate of relative population growth, and the lag. 8) If  EMBED Equation.2  , then the limiting value  EMBED Equation.2  EMBED Equation.2  collapses to 1, (as is expected), and the rate of convergence is governed by the term  EMBED Equation.2  : that is, the larger  EMBED Equation.2  is, the faster the rate of convergence, which, again, is exactly as expected. However, if  EMBED Equation.2 , then the limiting value will be greater than 1: so the per capita spending ratio will never converge to unity. 9) In order to establish whether formula (1) just represents a curiosity, or whether it has any practical importance, it is necessary to know what range of values of  EMBED Equation.2  and  EMBED Equation.2  is likely to be encountered in practice:- a) The table below shows values of  EMBED Equation.2  back to 1961, for individual years, or averaged over groups of years, with projected values forward to 2021. The projected figures relate to the 1998 based official population projections: the figures on which the table is based are taken from the Annual Abstract of Statistics, 2000. Population Growth in England Relative to Scotland Year EMBED Equation.2 Year EMBED Equation.2 196119961.005619711.005519971.005019811.002019981.004919911.004420011.005419921.002720061.003919931.001420111.003819941.001320211.004019951.0030 EMBED Equation.2  For present purposes, it is relevant to note that, in the 1990s,  EMBED Equation.2 increased from a value of 1.0013 in 1994, to 1.0056 in 1996, and is projected to remain at a value around 1.005 until 2001. b) As regards  EMBED Equation.2 , through the mid 1990s growth in government expenditure was typically in the low single figures: for example, the annual growth in the aggregate control total for the expenditure of government departments was between 1.6% and 4% between 1993-94 and 1997-98. This contrasts with the current situation: for example, in CSR200, total DELs are projected to rise by 10.4%, 8.7%, 8.1%, and 7.1% over the next four years respectively. 10) For illustrative purposes, therefore, a  EMBED Equation.2  value of 1.005, and  EMBED Equation.2  values of 1.03 and 1.09 are considered, (as being typical of values in the mid 1990s, and currently). An illustrative lag of 4 years is considered, as is a starting value of 1.2 for the level of public expenditure per head in Scotland relative to England, (close to the current value). The relevant results are illustrated in Charts 1 to 3. Chart 1 shows the path of the per capita expenditure relativity, when  EMBED Equation.2 =1.03, and  EMBED Equation.2 =1: in this case, of course, the limiting ratio is 1: because of the relatively small growth rate in expenditure in England, convergence is fairly slow. Chart 2 shows what happens when  EMBED Equation.2  = 1.005 : (note that this is the only change from the Chart 1 case.) This time, a radically different picture emerges. The limiting value is now 1.22, ( that is, above the starting value), so that what is observed is slow growth of  EMBED Equation.2  to this level. Finally, Chart 3 shows what happens when  EMBED Equation.2  is increased to 1.09, but still retaining a  EMBED Equation.2 value of 1.005. In this case, the limiting value is 1.08. Although this value is higher than the limiting value of 1 in Chart 1, the values of  EMBED Equation.2  in Chart 3 are actually below those in Chart 1 for about the first 20 years, because the rate of convergence is faster in Chart 3. 11) These results illustrate how levels of relative population growth similar to those currently being experienced can have a very marked effect indeed on the behaviour of per capita spending relativities, when expenditure growth rates are in low single figures. However, this effect is significantly reduced when expenditure growth rates approach double figures. These findings are very relevant to the debate about the likely historical impact of the Barnett squeeze. At times during the 1990s, given the low growth rate in overall public expenditure , and the high relative growth rate of population in England which applies latterly, the model indicates that we would not have expected convergence in per capita spending levels. The situation now, with the high current levels of public expenditure growth, is markedly different. 12) Finally, it is worth noting that the illustrative results quoted here are not very sensitive to variations in the lag parameter, (which, it will be recalled, has been at a value of 4 years, representing the likely average lag in practice.) This is illustrated by the following table, which shows the limiting value of the per capita expenditure ratio, given different choices of the lag parameter, and  EMBED Equation.2 , (and with  EMBED Equation.2 =1.005 in each case). Limiting Values of Per Capita Expenditure Ratio Expenditure growth EMBED Equation.2  1.031.061.0911.2061.0961.064Lag (years)21.2121.1021.06931.2181.1071.07541.2241.1131.08 In other words, the key factor driving the results illustrated in this note is the relative values of  EMBED Equation.2 and  EMBED Equation.2 . REFERENCES HM Treasury (1997) Supplementary Memorandum in the Second Report of the Treasury Committee, “The Barnett Formula”, (London: The Stationery Office). HM Treasury (2000) “Funding the Scottish Parliament, National Assembly for Wales and the Northern Ireland Assembly”. PAGE  PAGE 4 ™ Єƒ.ЅШAІЇЈ Љ Њe considering years 2002-03 to 2004-05. This adjustment will include any required uplift for inflation in the new final year. These adjustments then feed into the Barnett formula to give the required changes to the Scottish DEL for each of the three years. In applying the Barnett formula Scotland is given its population share of changes to English department base lines, based on the most recently available historic population estimates for Scotland and England. Typically, therefore, the population shares used will relate to a date two years before the year in which the spending review is being carried out. Hence for the 2001 CSR, the population estimates would relate to the year 1999. (For details on the operation of Barnett, see HMT (2000)). 4) It will be apparent from the description in the previous paragraph why modelling the operation of Barnett is not simple. 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Native ^џџџџџџџџd(Ole џџџџџџџџџџџџ{PIC `cџџџџyLMETA џџџџџџџџџџџџspCompObjbeџџџџqf6;STYZ[`ghmtuz‚‡Ž”›œЁЈЉЎЕЖЛТУШЯаемнтщъяќw%ќШцШќЫ%т№тwќw%тШцШќЫ%т№тwќw%тШцШќЫ%т№тwќw%тШцШќЫ%т№тwќw%тШцШќЫ%т№тwќw%тШцШќЫ%т№тwќw%тШцШќЫ%ИНњЛ ОnuЁ T (META џџџџџџџџџџџџжCompObjTVџџџџдfObjInfoџџџџWџџџџгEquation Native џџџџџџџџџџџџЯјOle џџџџџџџџџџџџ“PIC loџџџџ‘LMETA џџџџџџџџџџџџŒCompObjnqџџџџŠfObjInfoџџџџџџџџџџџџˆEquation Native pџџџџџџџџ‰(Ole џџџџџџџџџџџџŸPIC ruџџџџLMETA џџџџџџџџџџџџ˜CompObjtwџџџџ–fObjInfoџџџџџџџџџџџџ”Equation Native vџџџџџџџџ•(Ole џџџџџџџџџџџџЋPIC x{џџџџЉLMETA џџџџџџџџџџџџЄCompObjz}џџџџЂfObjInfoџџџџџџџџџџџџ Equation Native |џџџџџџџџЁ(Ole џџџџџџџџџџџџЗPIC ~џџџџЕLMETA џџџџџџџџџџџџАCompObj€ƒџџџџЎfObjInfoџџџџџџџџџџџџЌEquation Native ‚џџџџџџџџ­(Ole џџџџџџџџџџџџУPIC „‡џџџџСLMETA џџџџџџџџџџџџМCompObj†‰џџџџКfObjInfoџџџџџџџџџџџџИEquation Native ˆџџџџџџџџЙ(Ole џџџџџџџџџџџџЯPIC ŠџџџџЭLMETA џџџџџџџџџџџџШCompObjŒџџџџЦfObjInfoџџџџџџџџџџџџФEquation Native ŽџџџџџџџџХ(Ole џџџџџџџџџџџџйPIC “џџџџзLMETA џџџџџџџџџџџџдАCompObj’•џџџџвfObjInfoџџџџџџџџџџџџаEquation Native ”џџџџџџџџб$Ole џџџџџџџџџџџџхPIC –™џџџџуLMETA џџџџџџџџџџџџоCompObj˜›џџџџмfObjInfoџџџџџџџџџџџџкEquation Native šџџџџџџџџл(Ole џџџџџџџџџџџџёPIC œŸџџџџяLMETA џџџџџџџџџџџџъCompObjžЁџџџџшfObjInfoџџџџџџџџџџџџцEquation Native  џџџџџџџџч(Ole џџџџџџџџџџџџ§PIC ЂЅџџџџћLMETA џџџџџџџџџџџџіCompObjЄЇџџџџєfObjInfoџџџџџџџџџџџџђEquation Native Іџџџџџџџџѓ(Ole џџџџџџџџџџџџ PIC ЈЋџџџџLMETA џџџџџџџџџџџџCompObjЊ­џџџџfObjInfoџџџџџџџџџџџџўEquation Native Ќџџџџџџџџџ(Ole џџџџџџџџџџџџPIC ЎБџџџџLMETA џџџџџџџџџџџџCompObjАГџџџџ fObjInfoџџџџџџџџџџџџ Equation Native Вџџџџџџџџ (Ole џџџџџџџџџџџџ$PIC ДЗџџџџ"LMETA џџџџџџџџџџџџдCompObjЖЙџџџџfObjInfoџџџџџџџџџџџџEquation Native Иџџџџџџџџ@Ole џџџџџџџџџџџџ0PIC КНџџџџ.LMETA џџџџџџџџџџџџ)CompObjМПџџџџ'fObjInfoџџџџџџџџџџџџ%Equation Native Оџџџџџџџџ&(Ole џџџџџџџџџџџџ=PIC РУџџџџ;LMETA џџџџџџџџџџџџ5hCompObjТХџџџџ3fObjInfoџџџџџџџџџџџџ1Equation Native Фџџџџџџџџ28Ole џџџџџџџџџџџџPPIC ЦЩџџџџNLMETA џџџџџџџџџџџџCРCompObjШЫџџџџAfObjInfoџџџџџџџџџџџџ>Equation Native Ъџџџџџџџџ?`Ole џџџџџџџџџџџџZPIC ЬЯџџџџXLMETA џџџџџџџџџџџџUАCompObjЮбџџџџSfObjInfoџџџџџџџџџџџџQEquation Native аџџџџџџџџR$Ole џџџџџџџџџџџџhPIC веџџџџfLMETA џџџџџџџџџџџџ_ŒCompObjдзџџџџ]fObjInfoџџџџџџџџџџџџ[Equation Native жџџџџџџџџ\4Ole џџџџџџџџџџџџ{PIC илџџџџyLMETA џџџџџџџџџџџџnРCompObjкнџџџџlfObjInfoџџџџџџџџџџџџiEquation Native мџџџџџџџџj`Ole џџџџџџџџџџџџ…PIC осџџџџƒLMETA џџџџџџџџџџџџ€АCompObjруџџџџ~fObjInfoџџџџџџџџџџџџ|Equation Native тџџџџџџџџ}$Ole џџџџџџџџџџџџ’PIC фчџџџџLMETA џџџџџџџџџџџџŠpCompObjцщџџџџˆfObjInfoџџџџџџџџџџџџ†Equation Native шџџџџџџџџ‡4Ole џџџџџџџџџџџџЂPIC ъэџџџџ LMETA џџџџџџџџџџџџ˜дCompObjьяџџџџ–fObjInfoџџџџџџџџџџџџ“Equation Native юџџџџџџџџ”HOle џџџџџџџџџџџџСPIC №ѓџџџџПLMETA џџџџџџџџџџџџЊCompObjђѕџџџџЈfObjInfoџџџџџџџџџџџџЃEquation Native єџџџџџџџџЄјOle џџџџџџџџџџџџЮPIC іљџџџџЬLMETA џџџџџџџџџџџџЦpCompObjјћџџџџФfObjInfoџџџџџџџџџџџџТEquation Native њџџџџџџџџУ4Ole џџџџџџџџџџџџлPIC ќџџџџџйLMETA џџџџџџџџџџџџгpCompObjўџџџџбfObjInfoџџџџџџџџџџџџЯEquation Native џџџџџџџџа4Ole џџџџџџџџџџџџщPIC џџџџчLMETA џџџџџџџџџџџџрМCompObjџџџџоfObjInfoџџџџџџџџџџџџмEquation Native џџџџџџџџн4Ole џџџџџџџџџџџџџPIC  џџџџ§LMETA џџџџџџџџџџџџ№CompObj  џџџџюfObjInfoџџџџџџџџџџџџъEquation Native  џџџџџџџџыЈOle џџџџџџџџџџџџPIC џџџџLMETA џџџџџџџџџџџџ CompObjџџџџfObjInfoџџџџџџџџџџџџEquation Native џџџџџџџџ`Ole џџџџџџџџџџџџPIC џџџџLMETA џџџџџџџџџџџџpCompObjџџџџfObjInfoџџџџџџџџџџџџEquation Native џџџџџџџџ4Ole џџџџџџџџџџџџ,PIC џџџџ*Lяіїќ , в!г!š#Г$ћ%Ћ'Ќ'№*ё*й,к, - --8-9-:-;-<-A-F-ќ№ќwјw%ќШтШјЫ%ќ№ќwјwќШтШнz Bкџz Vкџz Јкz кџz иџz [иџz иџz Ниz и z иz иџz иz еz вбвбј#%ј#Р#вбвбјбјбИlЛ О ”џ=ц с ѕџИНњЛ ОnuЁ T "META џџџџџџџџџџџџ#„CompObjџџџџ!fObjInfoџџџџџџџџџџџџEquation Native џџџџџџџџDOle џџџџџџџџџџџџ9PIC  #џџџџ7LMETA џџџџџџџџџџџџ1pCompObj"%џџџџ/fObjInfoџџџџџџџџџџџџ-Equation Native $џџџџџџџџ.4Ole џџџџџџџџџџџџHPIC &)џџџџFLMETA џџџџџџџџџџџџ?„CompObj(+џџџџ=fObjInfoџџџџџџџџџџџџ:Equation Native *џџџџџџџџ;DOle џџџџџџџџџџџџUPIC ,/џџџџSLMETA џџџџџџџџџџџџMpCompObj.1џџџџKfяіїќ , в!г!š#Г$ћ%Ћ'Ќ'№*ё*й,к, - --8-9-:-;-<-A-F-ќ№ќwјw%ќШтШјЫ%ќ№ќwјwќШтШнz Bкџz Vкџz Јкz кџz иџz [иџz иџz Ниz и z иz иџz иz еz вбвбј#%ј#Р#вбвбјбјбИlЛ О ”џ=ц с ѕџИНњЛ ОnuЁ T "ˆ‚ƒ„…зџџџџ§џџџ‰и'ŠŽп‘”“т•–ф˜™š›œžŸ ЁЂЃЄЅЇЗЈЉЊЋЌ­ЎЏАБВГДЕЖИѕЙКЛМНОПРСТУФХЦЧЩЪЫЬЭЮЯабв-еŒPд‹йоџџџџџџџџџџџџџџџџчрс’у—хцъшщяыьэюўџџџ№ёђѓєўћїјљњќ §џAF-K-L-M-O-U-[-a-b-n-p-v-|-‚-ƒ-„-†-Œ-’-˜-™-š-œ-Ђ-Ј-­-Ў-Џ-I.J.K.L.M.N.O.Z.ќбшбхбќбќбќбќбшбхбќбќбќбќбшбхбќбќбќбќбшбхбќбќбќбќбшбуz уџz ;уz уz уz уz уz Хz Хz  4€)ИlЛ О”џ=ц 8с #Z.[.я.№.e/f/o/p/q/|/}/~//€/W0^0_00Ј0Њ0н0ј0(1тz тz тz Сz Пz ИЖПИxЖz ПППz ПЂПППŸz ‚z Пz  4€)ИlЛ О”џ=ц 8с `ќџ% Х;§ 4€) 4€)_103847940079mРFtW6fgР@s@єkgР_1038479931џџџџџџџџhРF€"уpggР@s@єkgР_10384804068<aРFРчР‡hgР@s@єkgР_1038480652џџџџџџџџ\РF DqigР@s@єkgР_103847745735…РFрНagР@s@єkgР_1038477617џџџџџџџџРF ЁŠbgР@s@єkgР_1038478207/:yРF`ФЊvcgР@s@єkgР_1038479353џџџџџџџџsРF@ зfgР@s@єkgР_1038471828џџџџџџџџCРF ЖтTgР ЖтTgРOle471236 џџџџџџџџџџџџРF1_1038470057џџџџџџџџ7РF@fк›PgР@fк›PgР_1038471845<41РFоŽ TgРР˜ TgРўџ џџџџРFMicrosoft Equation 2.0 DS Equation Equation.2є9Вq`э ‰ џџџ.1  Р@&џџџџРџџџРџџџ€ & MathType0ћ€ўSymbol- 2 `0qЧ & џџџџћМ"Systemn-№GL`эШшш6жИmJ iJ4nJ R tўџ џџџџРFMicrosoft Equation 2.0 DS Equation Equation.2є9Вq44 Д џџџ.1  &џџџџРџџџРџџџРРOle џџџџџџџџџџџџюPIC X[џџџџЊLMETA џџџџџџџџџџџџ‹ИCompObjZ]џџџџ‰fEquation Native YџџџџџџџџЌјObjInfoџџџџџџџџџџџџ[Ole џџџџџџџџџџџџќPIC ^bџџџџљLMETA џџџџџџџџџџџџёЬEquation Native _џџџџџџџџћ4CompObj`cџџџџяfObjInfoџџџџџџџџџџџџ\1  Р@&џџџџРџџџРџџџ€ & MathType0ћ€ўSymbol- 2 `0qЧ & џџџџћМ"Systemn-№GL`эШшш6жИmJ iJ4nJ R tўџ џџџџРFMicrosoft Equation 2.0 DS Equation Equation.2є9Вq44 Д џџџ.1  &џџџџРџџџРџџџРР & MathTypePћ€ўTimes New Roman- 2 `@RћрўTimes New RomanŽ-№ 2 РltP & џџџџћOle џџџџџџџџџџџџPIC dgџџџџLMETA џџџџџџџџџџџџџpCompObjfiџџџџ§fEquation Native eџџџџџџџџHObjInfoџџџџџџџџџџџџ]Ole џџџџџџџџџџџџBPIC jnџџџџ:LMETA џџџџџџџџџџџџ pEquation Native 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sРF uР›PgР uР›PgР_1038470067'џџџџmРF uР›PgР`Щ›PgР_10384707396gРFР‡HQgРр(PQgР_1038470065џџџџџџџџaРF`Щ›PgР`Щ›PgРObjInfoџџџџџџџџџџџџIEquation Native 0џџџџџџџџJ4CompObjџџџџџџџџџџџџVj_1038477288џџџџџџџџ‹РF /–XagР@s@єkgРK@ёџNormala c"A@ђџЁ"Default Paragraph Font)`Ђё Page Number ` Footer 9r It follows, on summing the resulting geometric series in  EMBED Equation.2 , that  EMBED Equation.2  (A3) and  EMBED Equation.2  (A4) It follows from (A3), when  EMBED Equation.2 , that  EMBED Equation.2  which simplifies to , which is the required expression. It follows from (A4), when  EMBED Equation.2 , that,  EMBED Equation.2  so in this special, (and unlikely), case,  EMBED Equation.2  grows linearly. 1™ Єƒ.ЅШAІЇЈ Љ Њn+?AБГШмоbv()captures the, planningthe 5t0 3 Spending Review 2000. (SR2000)SRSR2000 to the limiting value 5 H.M.TreasuryH.M.Treasury figures ees figuresH.M.Treasurysimplified 5R)+n‚„#%-ACт і ј !!!к!ю!№!к"ю"№"+#?#A#o#F-K-L-M-O-U-[-a-b-n-p-v-|-‚-ƒ-„-†-Œ-’-˜-™-š-œ-Ђ-Ј-­-Ў-Џ-I.J.K.L.M.N.O.Z.ќбшбхбќбќбќбќбшбхбќбќбќбќбшбхбќбќбќбќбшбхбќбќбќбќбшбуz уџz ;уz уz уz уz уz Хz Хz  4€)ИlЛ О”џ=ц 8с #Z.[.я.№.e/f/o/p/q/|/}/~//€/W0^0_00Ј0Њ0н0ј0(1тz тz тz Сz Пz ИЖПИxЖz ПППz ПЂПППŸz ‚z Пz  4€)ИlЛ О”џ=ц 8с `ќџ% Х;§ 4€) 4€)K@ёџNormala c"A@ђџЁ"Default Paragraph Font)`Ђё Page Number ` Footer 9r It follows, on summing the resulting geometric series in  EMBED Equation.2 , that  EMBED Equation.2  (A3) and  EMBED Equation.2  (A4) It follows from (A3), when  EMBED Equation.2 , that  EMBED Equation.2  which simplifies to , which is the required expression. 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