Must the growth rate decline? Baumol s unbalanced growth revisited

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1 Must the growth rate decle? Baumol s ubalaced growth rested Ncholas Oulto Bak of Eglad, Threadeedle Street, Lodo, ECR 8AH. The ews expressed are those of the author, ot ecessarly those of the Bak of Eglad. I should lke to thak wthout mplcatg Wllam Baumol, Alec Chrystal, Roy Cromb, Adrew Gurey, DeAe Julus, Stephe Reddg, Paul Stoema, Joh Vckers, a aoymous referee ad other colleagues at the Bak of Eglad for helpful commets ad suggestos. Issued by the Bak of Eglad, Lodo, ECR 8AH, to whch requests for ddual copes should be addressed: eelopes should be marked for the atteto of Publcatos Group. (Telephoe ). Workg papers are also aalable o the Bak s Iteret ste at Bak of Eglad 999 ISSN

2 Cotets Abstract 5 Itroducto 7. The stagatost argumet ad some facts about serces 7. The pla of the paper.3 A tute argumet 3 The stagatost argumet 5. The Baumol model whe all goods are fal 5. Relate prces ad put shares 8 3 The stagatost argumet whe some products are termedate 3. A alue-added model of productty growth 3. Two cocepts of sectoral TFP growth 3 4 A smple model of edogeous structural chage 8 4. The model 8 4. Dscusso 3 5 TFP growth the Uted Kgdom: a sectoral aalyss Quatfyg the cotrbutos of each sector The effects of structural chage the Uted Kgdom ad the Uted States The effect of measuremet error Outsourcg ad the relabellg of ecoomc acttes 4 6 Cocluso 4 Refereces 44 Appedx A: Based techcal progress 47 Appedx B: The measuremet of real alue-added 50 3

3 Abstract Accordg to Baumol s model of ubalaced growth, f resources are shftg towards dustres where productty s growg relately slowly, the aggregate productty growth rate wll slow dow. Ths cocluso s ofte appled to the adaced dustral ecoomes, where resources are deed shftg towards the relately stagat serce dustres. Howeer, I show that Baumol s cocluso oly follows f the stagat dustres produce fal products. Ths s mportat emprcally, sce the most rapdly expadg serce dustres are those such as facal ad busess serces, whch are large producers of termedate products. Ee f such dustres are stagat, I show that a moemet of resources to them may be assocated wth rsg, ot fallg, aggregate productty growth. JEL codes: O40, O47, D4, O5. Key words: Ubalaced growth, serces, productty. 5

4 Itroducto (). The stagatost argumet ad some facts about serces Suppose that productty s growg at dfferet rates dfferet dustres. Suppose too that resources are gradually shftg towards the dustres where productty s growg more slowly (the techologcally stagat dustres). The the aggregate growth rate of productty wll steadly fall to the rate prealg the stagat dustres. Ths s oe cocluso of the ubalaced growth model frst set out by Baumol (967) ad further deeloped Baumol (985) ad Baumol et al (989). () Ths stagatost argumet s thought to be partcularly applcable to the adaced dustral coutres. Productty growth serces (at least as coetoally measured) s typcally lower tha the rest of the ecoomy. The proportos of output costat prces ad of employmet accouted for by serces hae bee steadly rsg these coutres (Julus ad Butler (998)). So t mght seem that the growth rate of the adaced coutres s fated to decle. The respose to ths gloomy progostcato has usually bee to argue that ot all serce dustres hae low productty growth, that ee f they do ow t may rse the future, ad that ayway measuremet errors mea that growth s uderestmated (Rowthor ad Ramaswamy (997)). It s lkely that errors measurg serce sector output are qute large (see eg Grlches (99) ad (994)). But sce fxg the errors wll ot be easy, the stagatost argumet stll mples that the growth of measured output wll decle, uless ew techology ca be reled o to rase future serce sector growth, somethg t has so far faled to do (Stroh (998)). Whateer the merts of these resposes, they all accept the stagatost argumet as correct. Ths paper argues, to the cotrary, that ths cocluso of the ubalaced growth model may be correct as appled to the adaced coutres. The reaso s that the argumet s logcally correct () Ths paper has bee produced as part of the Bak of Eglad s research programme o the serce sector. The 998 Bak of Eglad Act requres the Bak to hae regard to the deelopmet of dfferet sectors of the ecoomy. Ths paper s offered as a small cotrbuto towards meetg that requremet. Oulto (999b), a compao paper to the preset oe, dscusses some of the same ssues from a growth theoretc ewpot. () The term techologcally stagat ad ts opposte techologcally progresse', are due to Baumol. 7

5 oly f all dustres produce fal goods. Qute a dfferet cocluso results f some of the dustres produce termedate goods. Ad ths could be the releat case practce, sce the serce dustres that hae bee expadg partcularly rapdly are those such as facal ad busess serces, whch are large producers of termedate puts. To set the scee for what follows, Table shows growth rates of output by sector from fe leadg coutres: the Uted Kgdom, the Uted States, Frace, Germay ad Japa. We see mmedately that maufacturg has bee a relately slow-growg sector four out of the fe, the excepto beg Japa. The four serce sectors trasport ad commucatos, the dstrbute trades, face ad busess serces, ad mscellaeous persoal serces hae all bee growg more rapdly tha maufacturg the frst four coutres. I Japa, these sectors hae bee growg at about the same rate as maufacturg. Table shows the correspodg rates of growth of labour productty (alue-added costat prces per hour worked). I the Uted Kgdom, the Uted States ad Frace, labour productty has bee growg more slowly tha maufacturg three out of the four market serces (the excepto s trasport ad commucatos). I Japa, productty has bee growg more slowly two out of the four (face ad busess serces ad o-market serces). I Germay, t has bee growg more slowly the dstrbute trades. What has bee the cosequece of these dsparate growth rates for the allocato of resources betwee sectors? Chart shows the proporto of total hours worked that has bee absorbed by market serces (the aggregate of the four serce sectors Table ) oer the perod I each of the fe coutres, there has bee a strog upward tred. More tha half of total hours worked s ow absorbed by market serces the Uted Kgdom, the Uted States ad Japa, wth somewhat lower proportos Frace ad Germay. I 950, as ow, the Uted States had the hghest share but the t was oly 40%. I the preset cotext, partcular terest attaches to the share of resources deoted to termedate producto. Wth market serces, ths may be proxed by face ad busess serces. (3) Chart shows that there has (3) The share of face ad busess serces s oly a proxy for termedate sales, because some face ad busess serces are sold to fal demad, cludg exports. 8

6 Table Growth rates of output selected sectors, (per cet per aum) Sector UK US Frace Germay Japa Agrculture, forestry, ad fshg Maufacturg Utltes Costructo Trasport & commucatos Dstrbute trades Facal & busess serces Mscellaeous persoal serces Note: Output s alue-added costat prces. Omtted sectors are mg ad o-market serces (health, educato, defece ad publc admstrato). Germay s Wester Germay. Source: Calculated from O Mahoy (999, Table A). Table Growth rates of labour productty selected sectors, (per cet per aum) Sector UK US Frace Germay Japa Agrculture, forestry, ad fshg Maufacturg Utltes Costructo Trasport & commucatos Dstrbute trades Facal & busess serces Mscellaeous persoal serces Note: Labour productty s alue-added costat prces per hour worked. Omtted sectors are mg ad o-market serces (health, educato, defece ad publc admstrato). Germay s Wester Germay. Source: Calculated from O Mahoy (999, Tables A, B ad C). 9

7 Chart Proporto of aggregate hours worked market serces, (per cet) Uted Kgdom Uted States Frace 5. Germay (West) Japa Note: Market serces comprse trasport ad commucatos, the dstrbute trades, face ad busess serces, ad mscellaeous persoal serces Source: Calculated from O Mahoy (999, Tables B ad C). 0

8 Chart Proporto of aggregate hours worked face ad busess serces, (per cet) Uted Kgdom Uted States Frace 3.0 Germay (West) Japa Source: Calculated from O Mahoy (999, Tables B ad C).

9 bee a partcularly sharp crease the share of ths sector total hours worked. The share s ow aroud 3% the Uted States, the Uted Kgdom ad Frace, ad aroud 9% Germay ad Japa. The share was aroud 3%-4% 950, ad has bee rsg steadly for early 50 years. (4). The pla of the paper Secto of the paper sets out the ubalaced growth model whe all goods are fal. Followg Baumol, t shows the codtos uder whch, f output all dustres s growg at the same rate, resources wll shft to the stagat dustres, wth a cosequet slowg of the aggregate productty growth rate. Secto 3 explas how matters are chaged whe some dustres produce termedate products. A geeral result s dered, ad llustrated by meas of a smple, two-dustry example. Cetral to ths secto s the dstcto betwee two cocepts of productty growth at the dustry leel, oe based o gross output, the other o alue-added. The ma result s that, uder the assumpto of perfect competto, a rse the share of resources absorbed by a dustry producg termedate products rases the aggregate growth rate, proded oly that total factor productty (TFP) growth s poste ths dustry. Ths result has a paradoxcal corollary: a shft of resources towards ee a stagat dustry that produces termedate puts rases the growth rate. Secto 4 addresses the questo whe wll a shft resources towards dustres producg termedate products actually occur? A smple, two-dustry model s deeloped, whch the frst dustry supples a put to the secod; perfect competto s assumed. The aswer s show to deped o the elastcty of substtuto the secod dustry betwee the termedate put ad the prmary put. If ths elastcty exceeds oe, resources shft towards the frst dustry. Ths result requres oly that TFP growth the frst dustry be poste; t may be slower tha TFP growth the other dustry. (4) The same source shows that the proporto of aggregate hours worked absorbed by trasport ad commucatos has bee flat or fallg four of the fe coutres (Frace s the excepto). The share absorbed by the dstrbute trades has bee rsg except Japa; the share absorbed by other market serces has bee rsg all fe coutres. The share absorbed by o-market serces has also bee rsg all fe coutres, but output measures are meagless for ths sector.

10 Secto 5 quatfes the effect of structural chage o the UK growth rate oer the perod It demostrates that the shft to face ad busess serces could hae rased the UK growth rate, ee though ths sector has low TFP growth. It also dscusses the effect of measuremet error o the estmates. Secto 6 cocludes..3 A tute argumet Much of the argumet to follow s rather techcal. So to assst tuto, the cetral pot of the paper wll be llustrated by the followg, orgorous argumet. Cosder frst the tuto behd the ubalaced growth model. For purposes of llustrato, suppose there are oly two dustres whch for cocreteess we label cars ad harcuts. Suppose that labour s the oly put. Assume that productty s rsg cars but ot harcuts. Icomes are rsg oer tme because there s productty growth oe sector ee f ot the other. Suppose that people s demad for the two products rses at a equal rate. (We shall see a momet that ths commo growth rate must be declg oer tme.) Assume that total employmet s costat. The sce people wat to hae ther har cut more frequetly as they grow rcher, more hardressers wll be employed. Sce total employmet s fxed, ths meas that fewer car workers wll be employed. Ths s possble sce the productty of car workers s rsg: the growg demad for cars ca be satsfed by progressely fewer car workers. As log as the assumptos cotue to apply, the proporto of the workforce employed hardressg wll go o rsg, approachg oe asymptotcally. Ge that total resources are fxed, the oerall growth rate of the ecoomy must slow dow. Ths s because aggregate productty growth s a weghted aerage of productty growth the two sectors, where the weghts are shares total employmet. (5) We hae already see that the employmet share of harcuts s rsg oer tme. So the sector wth zero productty growth gets a eer-creasg weght ad the oerall productty growth rate must therefore decle. Because total employmet s fxed, the growth rate of aggregate output must decle too. What s happeg to costs ad prces? Assume that wages the two dustres moe step wth each other. The sce t always requres the same amout of labour to cut someoe s har but progressely less labour (5) Ths s true because we are assumg equal growth rates of output the two sectors: see Secto below. 3

11 to produce a car, the relate prce of a harcut must be rsg. It follows that the proporto of cosumers expedture whch falls o harcuts must also be growg, approachg oe asymptotcally. Sce the product whch forms a eer larger share of expedture s subject to zero productty growth, the rate at whch the stadard of lg s rsg must be declg. More precsely, the growth rate of the stadard of lg s fallg asymptotcally to zero. Hag set out the argumet ts most basc form, we ca ow see a smple geeralsato. Suppose that productty growth harcuts s lower tha cars but greater tha zero. The the oerall growth rate of productty ad output wll stll slow dow, approachg ow the low rate foud harcuts. Now mage aother ecoomy where as before oe dustry produces cars but the other dustry supples a termedate put, say busess serces. That s, cars are produced by combg labour ad busess serces. Busess serces requre oly labour. Total employmet s costat as before. Productty growth cars s hgher tha busess serces, but sce the car dustry has two puts we must uderstad productty growth there to mea growth total factor productty. Sce busess serces use oly labour, labour productty ad TFP are detcal that dustry. At frst sght, ths chage of assumpto seems to make o dfferece ad the same argumet as aboe apples. But fact thgs are ery dfferet. At the aggregate leel, we care oly about the output of cars sce ths s the oly product demaded by cosumers. So the ssue s whether a rsg share of employmet busess serces wll be accompaed by a rsg or fallg growth rate of car output. There are two ways whch the ecoomy ca obta more cars, ge that total employmet s fxed. Oe s f TFP rses the car dustry, the other s f TFP rses busess serces. TFP growth the car dustry rases the productty of both the puts, labour ad busess serces: more cars ca be produced for a ge amout of labour drectly employed the car dustry ad drectly employed busess serces. I addto, TFP growth the busess serces dustry rases the productty of labour employed there. Ths meas that more busess serces ca be produced for a ge amout of labour. Hece TFP growth the busess serces dustry causes hgher car output, sce the car dustry buys busess serces. The hgher the proporto of the labour force employed busess serces, the bgger the mpact o car output of TFP growth 4

12 busess serces. Hece ee f productty growth s low busess serces, a shft of resources to ths dustry wll be accompaed by rsg, ot fallg, growth of car output. The reaso s that such a shft wll rase the cotrbuto to the aggregate comg from busess serces wthout reducg the cotrbuto comg from the car dustry. So for the cars/busess serces ecoomy we reach exactly the opposte cocluso to the oe for the cars/harcuts ecoomy. The argumet just stated prodes support for the more geeral proposto, that a shft resources to termedate-producg dustres rases the aggregate growth rate. As the paper wll make clear, the argumet depeds o the shft beg market-duced, as a result of proft-maxmsg behaour by producers, crcumstaces where exteraltes ad other market falures do ot uduly fluece outcomes. The stagatost argumet. The Baumol model whe all goods are fal Assume that all goods ad serces are fal, e there are o termedate puts. Let y deote the gross output of the th product ( =,...,), ad let y deote aggregate output. The growth rate of aggregate output ca be defed as a weghted aerage of the growth rates of the dustry outputs, where the weghts are the alue shares (s ) of the dustres the total alue of output: ˆ = ˆ,, p y y s y s s = () p y = = = Here, the p are the prces ad a hat (^) deotes a growth rate (logarthmc derate wth respect to tme, t). Also = p y py whch mplctly defes p as the aggregate prce dex. Let x be a dex of total put to the th dustry, ad let x be aggregate put the whole ecoomy. All puts are prmary. We ca thk of the x ether as a sgle put, say labour, or as a budle of prmary puts whose composto may well ary across dustres. The growth of aggregate 5

13 put may be defed as a weghted aerage of the growth rates of dustry puts: xˆ = r xˆ, r = = = where r s the proporto of aggregate put employed the th dustry. These proportos wll be defed more precsely below. Defe q y / x as productty the th dustry ad q y / x as aggregate productty. Productty growth the th dustry s the qˆ = yˆ xˆ () The dustry-leel productty growth rates are assumed to be exogeous. The growth rate of aggregate productty s: qˆ = yˆ xˆ = = = s yˆ = = r xˆ = r qˆ + ( s r ) yˆ (3) We see that aggregate productty growth s ot smply a weghted aerage of dustry productty growth rates, the frst term the equato, because of the presece of the secod term. The latter measures the effect of shfts the composto of output. There are two cases where the secod term equato (3) wll be zero. The frst case s the bechmark case cosdered by Baumol et al (989, Appedx to Chapter 6), where the composto of output s assumed costat, e output s growg at the same rate all dustres: ˆ ˆy, Sce the output ad put shares both add to oe, the secod term equato (3) s zero, ad so ths specal case y = all. 6

14 qˆ = r qˆ (4) = The secod case whch (3) reduces to (4) s whe there are costat returs to scale, ad perfect competto preals all markets, cludg the markets for puts. Uder these codtos, the prce of a ge put s the same all dustres, ad measures ts socal margal product. It s the approprate to aggregate the dustry puts usg put prces, e we ca defe the weghts as the alue shares: r w x / = w x, r = = (5) where w s the prce of the prmary put budle dustry. Also, the prce of aggregate put w s mplctly defed by the accoutg relatoshp: wx = w x = Uder perfect competto, log-ru equlbrum, the alue of output equals the cost of the puts (cludg a ormal retur o captal): p y = w x, =,..., py = wx Hece, ddg, s = r, all (6) ad aga (3) reduces to (4). These accoutg relatoshps mply that uder perfect competto the leel of productty curret prces s the same all dustres: p y / w x =, all. So the leels effect the aggregate productty growth equato (3) dsappears, rrespecte of the 7

15 growth rates of dustry outputs. Sce all dustres hae the same productty leel alue terms, there s o ga to reallocato. (6) Suppose that ether of these two cases apples, so that the aggregate growth rate s ge by (4). The f resources are shftg to dustres wth comparately low productty growth, the aggregate growth rate wll clearly decle. We may ote passg that f (4) holds, the growth rate of GDP s ge by yˆ = xˆ + = r qˆ So f the growth rate of aggregate put s take to be costat, fallg productty growth mples fallg GDP growth too. I fact, wth some addtoal assumptos, a stroger proposto apples. We show the ext sub-secto that f output s growg at the same rate all dustres, the the lower a sector s productty growth rate, the faster ts share of total put s rsg. The equato (4) mples that the aggregate productty growth rate wll fall mootocally, coergg o the growth rate of the most stagat sector: ˆ m{ qˆ } as t q Ths s the strog erso of the stagatost result.. Relate prces ad put shares So far derg the stagatost result we hae made two assumptos: frst, that output grows at the same rate all dustres, or alterately, that the ecoomy s compette; ad secod, that the share of resources gog to the stagat dustres s rsg. But f the ecoomy s compette, the secod assumpto ca be dered as a mplcato of the assumpto of equal output growth rates, proded that we make the further smplfyg assumpto that the prce of the prmary put budle s the same all (6) Ee a compette ecoomy, leels effects would stll arse f we chose to measure productty growth usg a fxed weght (eg Laspeyres) dex. But t s better to use a Dsa (cha) dex, as here. Leels effects ca also arse a compette ecoomy, f we are cosderg labour productty ad labour s ot the oly prmary put. 8

16 dustres. Ths last assumpto meas that there s effect oly oe prmary put. (7) Let us assume that w = w, all. The we hae ad r = x / x, r = = x = = x The accoutg relatoshp that the alue of output equals the cost of the puts ow becomes: p y = wx, =,..., (7) Rearragg ths relatoshp, p y x = p q = w, =,..., (8) e the leel of productty curret prces (the curret-prce alue of output per physcal ut of put) s the same all dustres, ee though the growth rate of productty may dffer betwee dustres. Ths s because a compette ecoomy, prces adjust to make ths so, as we ca see by logarthmcally dfferetatg the accoutg relatoshps wth respect to tme: (7) I geeral, the composto of the prmary put budle ares betwee dustres. Hece, ee f put markets are udstorted, so that the prce of a ge prmary put s the same all dustres, the w wll ot geeral be equal across dustres. It ca be show that the w wll be equal oly uder restrcte assumptos: ether f there s oly oe prmary put, or f relate put prces are always the same, or fally f put testes are the same all dustres, at ge put prces. The last codto amouts to assumg that all dustres hae the same producto fucto, up to a multplcate factor. 9

17 pˆ + qˆ = wˆ pˆ + qˆ = wˆ whece pˆ pˆ = qˆ ˆ (9) q e relate to the geeral prce leel, the prce of the th product rses by the dfferece betwee productty growth ad aggregate productty growth. (8) Note that ths result depeds crucally o the assumpto of oly oe (possbly composte) prmary put. If there were more tha oe prmary put, the the eoluto of relate product prces would also be flueced by the eoluto of relate put prces. For example, f the prce of labour s rsg relate to that of captal, the prce of a labour-tese product may be rsg too, ee f the dustry s techologcally progresse (Oulto (999a)). The tme-paths of the output ad resource shares come from logarthmcally dfferetatg the equato defg the output share () wth respect to tme, ad usg (6) ad (8): sˆ = rˆ = qˆ qˆ + yˆ yˆ Now cosder the bechmark case whch output grows at the same rate all dustres (e yˆ = yˆ, all ). The sˆ = rˆ = qˆ qˆ (0) That s, resources shft cotuously towards the relately stagat dustres. The slower productty growth s, the more rapd the shft. The stagatost argumet s therefore stregtheed. Uder the codtos assumed here, f output grows at the same rate all dustres, the aggregate productty growth wll slow dow asymptotcally to that of the (8) The equalty of productty leels dfferet sectors uder competto has bee dscussed by Baumol ad Wolff (984). 0

18 slowest dustry. If output growth s faster more stagat dustres, the slowdow wll be more rapd. 3 The stagatost argumet whe some products are termedate 3. A alue-added model of productty growth Suppose ow that some dustres produce products that are cosumed by other dustres. It mght be thought that ths would make lttle dfferece, f we adopt the usual alue-added approach. Let real alue-added dustry be, ad let omal alue-added be V. The growth of aggregate alue-added () or GDP s a weghted aerage of the growth rates of dustry-leel alue-added, where the weghts are each dustry s share (u ) aggregate omal alue added: ˆ = V ˆ u,, u u V = = = = () We ca defe a dustry-leel prce of alue-added prce of alue-added p from the accoutg relatoshps: p ad a aggregate V = p V = = p Idustry-leel productty growth ca be defed as: qˆ = ˆ xˆ () Note that we use a dfferet symbol here for productty growth from that used Secto. qˆ, ot qˆ sce these two symbols refer to dfferet cocepts. Preously, output was take to be gross output (compare equato ()), ow t s alue-added. We dscuss below the relatoshp betwee these two cocepts of productty growth at the dustry leel.

19 Aggregate productty growth ca ow be wrtte qˆ = ˆ xˆ = = = u ˆ = q + u = r r xˆ = ( r ) ˆ (3) At the aggregate leel, we use the same symbol for productty growth as Secto., qˆ, sce the growth of GDP ewed as a sum of alue-added must be the same prcple as the growth of GDP ewed as a sum of fal expedtures. Now cosder the specal case where alue-added s growg at the same rate all dustres: ˆ = ˆ, all. Now the leels effect s zero ad (3) becomes: qˆ = r qˆ (4) = Alterately, by assumg a compette ecoomy ad employg a exactly aalogous argumet to that of Secto., we ca proe that the output ad resource shares are equal: u = r, all. So (4) apples rrespecte of the alue-added growth rates. Ge competto ad equal alue-added growth rates, we ca also proe that the share of prmary puts deoted to a dustry wll rse faster, the slower ts productty growth rate ( qˆ ): uˆ = rˆ = qˆ qˆ (5) The stagatost argumet would thus seem to go through as before. If the share of puts gog to the stagat dustres s rsg, the the aggregate productty growth rate must fall mootocally to that of the most stagat dustry or at ay rate, so equato (4) seems to be sayg. Howeer, ths argumet cotas a hdde assumpto, amely that TFP growth rates the alue-added sese ( qˆ ) are parameters. I Secto. we assumed that TFP growth rates the gross output sese ( qˆ ) were

20 parameters. We ow show that these two assumptos are cosstet: f the qˆ are costats ad f the share of termedate puts s chagg, the the qˆ must be chagg too. 3. Two cocepts of sectoral TFP growth Let the gross output producto fucto be y = f ( x, m, t) (6) where m s a dex of termedate put (purchases from other dustres) dustry. The accoutg relatoshp s ow p y = w x + p m m m where p s the prce of termedate put. The, assumg competto so that put shares ca be equated to the elastcty of output wth respect to each put, TFP growth the gross output sese dustry s qˆ m w x p m = yˆ xˆ mˆ p y (7) p y We wat to fd the relatoshp betwee qˆ ad qˆ = ˆ xˆ (see equato ()). Assumg that the gross output producto fucto (6) s separable, we ca wrte t the form: where y = f, m ) (7 ) ( = g( x, t) (8) 3

21 ad g ( ) s the alue-added producto fucto. Dfferetatg (7 ) wth respect to tme ad stll assumg competto, we obta the growth rate of real alue-added: (9) ˆ m p y p m = yˆ mˆ w x (9) wx Substtutg ths to equato () ad usg (6), we obta: qˆ p y = qˆ w x (0) Or words, TFP growth gross output sese TFP growth alue-added sese = Share of alue-added gross output Clearly, TFP growth the alue-added sese ca eer be less tha ad wll usually be larger tha TFP growth the gross output sese. (0) Also f we take the qˆ as parameters, the the qˆ become arables, sce the alue-added share s determed by the relate prces of prmary ad termedate puts, ad wll ary oer tme. Hag establshed ths relatoshp betwee the two cocepts of productty growth at the dustry leel, equato (0), we ca ow substtute from ths to the equato for aggregate productty growth (4) to get: (9) Equato (9) s a cotuous-tme, Dsa-dex form of double deflato: see Appedx B for a comparso of double wth sgle deflato. A alterate way of derg (9) s to start wth the defto of omal alue-added dustry : p = p y p m. We ca the obta (9) by dfferetatg ths equato wth respect to tme, whle holdg prces costat. The treatmet the text has the adatage of showg how double deflato s cosstet wth producto theory. (0) The relatoshp betwee these two cocepts of TFP growth was dscussed Oulto ad O Mahoy (994), Chapters ad 6. 4 m

22 = p y qˆ q () = p Ths last result makes use of the fact that r = w x / w x = w x / = p. Equato () exemplfes what has bee called Domar aggregato (Oulto ad O Mahoy (994), Chapter 5, followg Domar (96)). The geeral prcple s that aggregate TFP growth s a weghted sum of the dustry-leel TFP growth rates. The Domar weghts are the rato of gross output each dustry to aggregate alue-added (total fal output). () Note that these weghts sum to more tha oe. Domar aggregato was ge a theoretcal justfcato by Hulte (978): o the assumpto of costat returs to scale ad compette markets, t measures the rate at whch the socal producto possblty froter s shftg out oer tme. The tuto behd Domar aggregato s that productty growth a dustry cotrbutes drectly to aggregate productty growth (a ts fal output), but also drectly whe t supples other dustres. Costs fall the purchasg dustres, ad ths effect s obously bgger the larger such purchases are. It s useful to splt the Domar weght to two, gross output for fal use ad gross output for termedate use, both expressed as a proporto of aggregate fal output: Itermedate sales of Fal sales of Domar weght of sector = + Total fal output Total fal output Note that the sum across dustres of the secod fracto, fal sales of dustry /total fal output, s oe. So f ths fracto rses for oe dustry, t must fall by a correspodg amout for oe or more other dustres. But the same s ot true of a rse the termedate part of the Domar weght. Ths frst fracto ca rse for oe dustry wthout a correspodg fall for ay other sector. For example, suppose that dustry has oly () I the most geeral case, the deomator of the Domar weghts s total fal output. I a closed ecoomy, total fal output equals omal GDP. I a ope ecoomy, t exceeds the latter by the amout of termedate mports, whch should also be cosdered a prmary put (Gollop 983). 5

23 termedate sales. Suppose that there are other dustres that sell oly to fal demad, ad that these ow purchase more of dustry s product, substtutg t for prmary put. The the Domar weght for dustry wll rse wthout ay correspodg fall ay other sector s weght. It follows that the oerall productty growth rate must rse too: see equato (0). To clarfy the argumet, recall that we hae dered two equatos for aggregate productty growth, repeated here for coeece. We also repeat the relatoshp betwee the two cocepts of productty growth: qˆ = r qˆ, r = w x / = p (4) = p y qˆ q () = p qˆ p y = qˆ w x (0) From the frst of these equatos, (4), t appears that a rse the resource share (r ) of a stagat dustry, couterbalaced by a fall the share of a progresse dustry, wll lower the aggregate productty growth rate, rrespecte of whether the stagat dustry supples termedate or fal goods. From the secod, (), a qute dfferet cocluso emerges. If there s a rse the Domar weght of a dustry supplyg a termedate product, the aggregate productty growth rate wll rse, ee f the dustry questo has lower-tha-aerage productty growth. More precsely, aggregate productty growth wll rse proded oly that TFP growth the dustry s poste. The resoluto of ths apparet cotradcto comes from takg accout of the thrd equato, (0). As the dustres supplyg fal goods purchase more from the oe supplyg termedate goods, so the formers alue-added shares decle. Cosequetly, ther TFP growth rates the alue-added sese qˆ ) rse. Ths ga more tha outweghs the loss ( from the reallocato of resources faour of the stagat dustry. 6

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