Population Growth and Technological Change: a Pure Welfarist Approach

Transcription

1 Poplao owh ad Tehologal Chage: a Pe ela ppoah by Thoma Reöm ad La Spaao ba I h pape we haaeze he opmal eady ae ad dyam o a eempoal eoomy peee o edogeo ely. o dog h we popoe a ew veo o Cal Level Ulaam CLU a là Blaoby e al 995 ha we all Relave CLU whh how o be axomaally oded ad allow o me-vayg al level. I pala we how ha de h oal peeee boh he ho ad he log ee o ehologal ho o he opmal poplao gowh ae o-val depedg o he haae o boh peeee ad podo ehology. Howeve we how ha he egave elaohp bewee ely ad ehologal developme emege de a boad la o peeee whh ypally od empal eeah. EL Claao: D63 D9 E 3 O33. Keywod: al level laam poplao ehology ho aoal dyam.. Iodo The e o poplao gowh ha bee exevely aalyzed eoom. e he poeeg de o hola h a Malh ad Rado he laal appoah ha expeeed a dele poplay e he empal evdee eemed oe wh dameal mplao aodg o whh eoom gowh wold lead dvdal o epode a a ae ae ad h wold ae deeag pe apa ood pply ad deeag lvg adad owad bee level. Howeve he ame empal evdee ha obed o a eewed ee he op o poplao eoom e he 96 de o wo appaely oag pheomea: o he oe had he woyg pope o developed oe who have bee degog a log-lag demogaph ao volvg loge aveage expeed leme ad eded ely ae ad o he ohe had he ll aed poplao gowh ae o he dedeveloped ad developg oe. Coeqely h pheomea have ped eveal wo oeg o example he e o opmal poplao gowh ae Samelo 975 Deado 976 ad moe eely aege ad Khle 9 de la Cox e al he elaohp bewee edogeo ely ad he welae ae pala a o oal ey ee o example Cgo ad Roa 99 Zhag ad hma 99 ad 993 Ceme e al 6 Yew ad Zhag 9 Mee ad ede ; ee alo Spaao ad a o deb opmaly ad Spaao ad Reöm o axao aaly peee o edogeo ely. ohe ad o leae ha bee og o he Uvey o Dham UK. Emal:..eom@dham.a.. Dpameo d Eooma e Maageme Uvey o Pa Ialy. Emal: l.paao@e.p..

2 log- elaohp bewee edogeo ely ehologal poge ad eoom gowh ee o example alo ad el eadez-vllavede Hae ad Peo eewood ad Sehad Kaleml-Oza Lagelo 3 Doepe 4 Soae 5 Tama 6 oe ad Tel 6 oe e al. pomg bah o he above meoed leae ha bee pg mh eo povdg a oheely oded oolbox o poplao eoom. I a well ow ha he evalao o aleave pbl pole oe mple he ompao o ae o he wold wh dee poplao ad h a a be poblema whe wela ea ae o be ed ha ea baed o he m o well-beg le o he dvdal who ae alve dee ae o he wold. Depe he elevae o h poblem o opo he heoeal odao o oal evalao wh vaable poplao have eeved lle aeo he leae. Typally wela pple ae adoped h a laal laam whee he objeve he m o he le ove he poplao o aveage laam whee he objeve he welae o a epeeave dvdal o ome mxe o boh. Howeve hee ea poe eveal poblem. o example laal laam ao avod he epga olo RC heeoh; ee Pa Blaoby e al. wheeby ay ae whh eah membe o he poplao ejoy a le above ealy delaed eo o a ae whh eah membe o a lage poplao lve a le wh lowe ly Blaoby e al oe ha model o eoom gowh h a olo ae he om o a ppe oe olo o he poplao gowh ae oey epode a phyal maxmm ae. Thee ae eveal way o deal wh he RC deed ome aho beleve he lae o eally a poblem. See o example Täjö. I a eveal eoom have deded o mve he poblem by eog o pala peeee ee o example Bao-Bee 988 Bee- Bao 989 ad alhogh dee amewo g 986 Ha ad alo ad el. Howeve ypally h olo ae o o a lea oly paally axomaally oded ha hadly ompable wh ome polae ha we beleve oal odeg hold abde ode o be ehally aepable. poble way o o h poblem epeeed by Cal Level Ulaam CLU whh axomaally oded ee Blaoby e al The al level deed a a ly vale o a exa peo ha added o he aeed poplao wold mae oey a well o a who ha peo. CLU ha bee ed by eveal aho: Pa 984 ad Shell 8 age ha CLU ao avod he RC beae a log a aveage ly hghe ha wold be alway oally peeable o ge lage poplao wh lowe ly level ely loe o ; ad h olo aodg o hee aho wold be epga he al level e oo low. Moeove Boome 99 po o ha ae he al level oo hgh he addo o a peo whoe le woh lvg.e. wh a pove ly level wold be peveed. The lae olo wold eplae he oe emmg om he applao o he aveage ly eo. ally g 986 ad he poed o ha CLU volve oeve oal odeg ae he aveage ly lowe ha.e. he o alled ad olo 3. I woh og ha mo ae he ado o poplao eh ha deal wh he RC a evome ee o example Shell 8 whh o deal e poplao e aally eal og dyam mplao. The aemp o haaeze he poblem a dyam amewo povded by Reöm ad Spaao whee a ll haaezao o a eempoal eoomy ealed wh CLU See Ehlh ad L 997 alo 5 ad Doepe 8 o -deph evew o hee op. mog wo g o laa pple ee o example oloov oe ad Tel 7. 3 The ad olo mple ha addg a peo wh egave ly o a poplao may be peeed o a aleave whee all dvdal have pove le alhogh below he al level ee he.

3 peeee aed o. I ha wo how ha he RC avoded oly he al level belog o a pove ope eval h omehow aweg o ome o he m meoed above oeg he hoe o. ollowg he lae pape he pee wo we wa o mae a ep owad he aaly o he elaohp bewee eoom gowh CLU peeee ad ehologal hage. Moe peely we popoe a ew poplao eo Relave Cal Level Ulaam RCLU whee he al level allowed o deped o pa ly level. The ma eao delyg o wo ha we beleve ha a xed al level o all exe oo gd. e all he deo o have hlde ae by pae ad he lae ae lely o hooe he mbe o he opg o he ba o he ow adad o lvg. Hee h wo we ame ha he e al level deped o he age o developme ha a oey ha bee expeeg o a h ha pae wll odo he hoe o ely ompag he ow hlde welae elao o he ow welae. I h o he ae ha he al level amed o be xed by he oey oe o all eveal dawba may ae. o example CLU model a povde lle jao o he hgh ely ae ha eveal a o a oe ae ely expeeg le oe poe o beleve ha h behavo ae ompleely aoal ad h hadly eaable by eoom model. Se we deve o model welae o by eg he e hogh poplao polae o obo epee alo a hoe heoe odao o exg model h a Keme 993 ad Hae ad Peo ha have exogeoly amed a elaohp bewee poplao gowh ae ad dvdal adad o lvg 4. Moeove we alo am a bldg a model ha allow o o-val dyam o poplao gowh a ggeed by eveal empal de ee o example alo 5 ad oe ad Tel 6 Ba ad Lea 5. I a he pee pape we how ha de RCLU boh a eo opmm ad h dyam o poplao gowh emege depedg boh o peeee ad o ehology. ally we avod edogezg ehologal poge beae we wa o eep o aaly a loe a poble o he Ca-Koopma-Ramey appoah. e leave he exeo o edogeo amlao o ehology o e eeah. The pape ogazed a ollow: eo we lay o he model ep eo 3 we povde he olo ad d he exee ad he ably o he log eqlbm. I eo 4 we d he ee o ehologal ho o boh he ho dyam ad log eqlbm. I eo 5 we exploe he ole o peeee dvg he el. ally a ppedx oa he axoma odao o RCLU.. The model. Peeee e oeae o a gle dyay hoehold o a polymae hoog ompo ad poplao gowh ove me o a o maxmze:... [ ] β 4 o a evew o pevo elaed leae o edogeo peeee ee Maovh I a a oe ad Tel ae [ ] depe he exeve empal wo hee ll o olve evdee o he exa elaohp bewee ome ad ely. Qog Hema ad ale 99 «mo eoom wold agee wh ad ad Bz 98 ha e ad e wage ad ome [ ] ae lely deema o ely. Thee lle ageeme beyod h». oe ad Tel 6 p.. 3

4 β he egeeaoal do ao ad he al level ly. I ppedx we how ha h he oly omlao ayg: Idepedee o he Log Dead Saoay Idepedee o Da e eeao Idepedee o Ule o Uoeed Idvdal oymy Sog Paeo ad Relave Cal Level Depedee. oe ha deely om pevo leae we allow h a al vale o be a pove whee he poplao amly ze he aaeo ly o ad o o pevo geeao ly oly a oa h oal odeg wold ode wh he CLU. e all o poplao eo Relave CLU RCLU. Thogho o pape we mae he mplyg ampo ha he al level o lea ha. wh > < ad <. Moeove o aalyal aably we wll wo oo me. e deoe he poplao gowh ae a.e. & wh [ ] U e. The oo me veo o eq. a be we a ee ppedx B.: [ ]d. 3 The egal e oly > ha < he lae ampo mple > whh we ame hogho he pape; moeove gve... Tehology mg a oa e-o-ale CRS podo ehology K L ~ K L wh he paamee epeeg oal ao podvy ad a apal depeao ae o zeo he apal amlao eqao : K K & 4 3. Solo The e-vale Hamloa o he hoehold poblem he ollowg: H [ ] q [ K ] λ ν ϑ 5 The em λ he Hamloa aoaed wh eq. ape he a ha a eah a o me he poplao ze gve ad h a ae vaable ad a oly be oolled by he 4

5 hoe o whh a ool vaable. The law o moo o he poplao ze povded by. Hee λ a be epeed a he hadow vale o poplao. The ode odo o he poblem mply: H H H [ ] q [ ] q 6 λ ν ϑ ϑ ν λ 7 λ [ ] q [ ] λ & λ & λ 8 H K q [ ] q& q& q 9 K eq. 4 ad he avealy odo lm e q K lm e λ. I wha ollow we ame eoy o he olo o 6 h ha boh ν ad ϑ eq. 7 ae zeo ad alog he ao pah & deed we wll how Rema he eeay odo wheeby he RC.e. avoded. Le dee he apal ey K / h ha by explog CRS he podo o ad amg ha L we a we: K L K. Moeove by bg o eq. 6 ad 7 o 8 we ge: [ ][ ] & λ. e oba hee dyam eqao ha ogehe wh he avealy odo lly haaeze o dyam yem we om me bp o he ae o oao: & [ ] & 3 & & K K 4 whee >. Eq. obaed by ag he me devave o eq. 7 ad ombg wh eq. ad eq. 4 by ag he me devave o 6 ad ombg wh eq. 9. ally eq. 3 eq. 4 pe apa em. 6 o oe he eoomy wold behave a he adad Ca-Koopma-Ramey model wh exogeo oa poplao gowh. 5

6 3.. The eady-ae eqlbm om eq. 4 3 ad epevely he eady-ae olo haaezed by he ollowg eqao: * 5 * * * * 6 * * * * *.. 7 Some omme o he olo ae woh dog. Eq. 5 p dow a qe pove vale o eady-ae apal ey * a adad he Ca-Koopma-Ramey model wh exogeo ely. Eq. 6 ae ha a he opmm boh ely * ad ompo * m ay he eoe avalable o he eoomy. ally eq. 7 ae ha a he opmm boh ompo ad ely hold be hoe h a way ha he addo o oal welae o eag he poplao a he mag hold eqal he magal vale ly o wha a ewbo ae o o oey [- ]. I a he e o o a ew dvdal em o eoe he deee bewee wha he ome ad wha he bg he magal vale o labo. I he oal vale o oe moe peo he ame a he o oey o plae dee aleg he poplao ze ad ompo a opmal vale. By g eq. 5 ad 6 ad explog CRS we ge 7 :. 8 Moeove by deg / > a he vee o he elay o ly wh > o > de oavy o he ly o ad by g eq. 5 ad 7 we ge whee ly pove. ally plggg 8 o he deo o he ollowg expeo povdg he mpl olo o eady-ae ompo deed: 4. 9 oe ha he olo o ad.e. eq. 5 ad 9 gve alo he olo o he poplao gowh ae by eq. 8. ow we a gve he ollowg Rema layg he ole o RCLU avodg he RC: Rema : The Repga Colo ao be avoded. Poo: Reall ha he Repga Colo mple ha bewee ad T o oba. Sppoe ow ha. Iegae 7 o ae o oao om ow o we om he a pep o eady-ae vale wheeve ae o ambgy o he eade. 6

7 T d d T λt e λ e [ ]d The a T by explog he avealy odo we have > λ e d de oavy o ly o deed o ay τ ha λ τ > whh by eq. 7 mple ν τ > ad h τ. 3.. Exee ad qee o he eady ae ve ha eqao 9 yeld oly a mpl expeo o eady-ae ompo he Popoo below povde he odo de whh boh exee ad qee o he olo ae gaaeed: Popoo : I he eeay ad e o he exee o a qe eady ae ha >. I > ad / he eeay ad e o he exee o a qe eady ae ha < / o all he lae eqaly beg lea ad /. d Poo: See ppedx B.. I ohe wod he lae odo ae ha ode o hee o be a qe eady ae he vee o he elay o he ly o whh meae he ga em o oal welae emmg om ompo avy o a exa peo hold o be oo eave o ompo hage. he > we have aed he el o /. hle mla el ae obaed o > / we hooe o o o o ha pala ae. 8 The eao ha / a e odo o ably whh we wll ome o he ex eo Sably o he eqlbm e ow how he odo de whh he eqlbm ajeoy addle-pah able. Le leaze he yem gve by eq. -4 aod he eady ae. e ge he ollowg expeo o he aoba max: 8 I ho he el > ad > / he a oave o lea yeld a qe eady ae whle o a ovex </ eeay ad e o exee ad qee. 7

8 Ω Ω Ω Ω Ω eempoal elay o bo o ompo. whee Ω [ σ ] ad > σ he vee o he Hee he haae eqao aoaed wh he aoba max above : Ω λ λ λ he oo o whh ae: Ω λ λ 3 ±. 4 ve ha λ > he eeay ad e odo o addle-pah ably ha λ > ad λ < 3. Se o he lae odo ha Ω >. Howeve e h odo o ealy epeable we povde moe apae e odo below hogh he ollowg Popoo: Popoo : Se o ably o he eady ae Ehe O < </ ad /. Poo: See ppedx B.3. Hee e odo o exee ad qee o he eqlbm povded Popoo alo mply ably o he eady-ae eqlbm. oe ha e a he eady ae he ollowg elaohp hold Ω / ollow ha he Ω >. O he ohe had e / / he Ω < deed ha whe < </ Ω a ae ehe g. e wll explo ad aalye deepe deal h popey he ex eo. ally de he odo povded Popoo eqaly Ω > hold e Taoal dyam e ow o o he popee o he aoal dyam o he model. e a mmaze o dg hogh he ollowg Popoo: 8

9 Popoo 3: log he ao pah <>* he boh ompo ad apal ey eae deeae. o <>* eae deeae Ω > whle deeae eae Ω <. Poo: See ppedx B.4. omme o he el obaed o a woh mag: o model boh pe-apa ompo ad apal behave a he adad Ca-Koopma-Ramey model. Howeve he ae o gowh o poplao a ae ehe dyam pah depedg o he g o Ω. I ode o povde a eoom epeao o h popey o he model we eo o eady-ae elao by deeag eq. 7 o a gve level o whh yeld: d d Ω. The lae eqao epee he magal ae o bo o oey deee ve bewee ompo ad ely ad he g o Ω deeme he ae o he elaohp bewee hee wo good. I a Ω > he good ae ompleme whle Ω < hey ae be. Moeove Ω ogly egave ha he yem able he he deee ve ae la h ha a oe olo o lely o o: ohe wod h ae oey pee o devoe a mh eoe o ompo a poble ad mmze hoe dedaed o ag hlde. 4. The ee o he ehologal hage I h paagaph we ay o ompaave a ad dyam aalye ode o ae he ee o hage oal ao podvy o boh he log ad he aoal pah o he eoom yem. Pelmaly we d he g o he aoba max o o model. The deema o he aoba max eqal o: D Ω. ve ha de ably Ω aoba max egave. > ad ha < > deed ha he deema o he 4. The ee o o eady-ae ompo Ug Came le we a we: d d D De whee he aoba max whh he olm bed o by he devave o eq. o 4 wh egave g wh epe o. 9

10 d D Se > we a mmaze o dg hogh he ollowg Popoo: d De Popoo 4: pove ehologal ho eae he log pe-apa ompo. Poo: See ppedx B.5 4. The ee o o eady-ae apal ey Le ow o he ee o a ehologal ho o he apal ey. I omla we a we he ollowg: d d D De whee he aoba max whh he eod olm bed o by he devave o eq. o 4 wh egave g wh epe o. d D Se > we a povde he ollowg Popoo: d De Popoo 5: pove ehologal ho eae he log pe-apa apal ey Poo: See ppedx B The ee o o eady-ae ely ae Pelmaly le o o he deema o whh he aoba max whh he hd olm bed o by he devave o eq. o 4 wh egave g wh epe o. I o o be: D [ Ω ] K. 3 d TMRS log Reallg ha γ he elay o log d bo o p he eq. 3 a be ewe a ee ppedx B.7: D K 4 { [ ] γ } h ha we a povde he ollowg e odo o he g o Popoo 6: Se o d / d < : ad γ wh oe eqaly o d / d.

11 Ω Se o d / d > : ad γ wh oe eqaly 3 Se o d / d : ad γ. Poo: See ppedx B.7 I he lgh o he aaly aed o o a we a epe o dg a ollow. ehologal mpoveme ede apal moe podve mag ovee o he eoomy o ve hghe amo o eoe apal amlao ad podo o good. Moeove h a eaed level o podvy allow o hghe pe-apa ompo. o ely hg ae moe omplex. I a whe oal podvy eae by eq. 8 wo dee oe dve he hage he poplao gowh ae: o he oe had he hage he ao / ad o he ohe had he hage he ao /. The ome ee deped o he elave elay o ompo ad apal whle he lae a o o he elay o bo o p 9. hh ee wll peval deped o he exa hape o he o ha ae ed o peeee ad podo ehology ha Ω ad γ. To a wh we a povde he ollowg oollay: Coollay : I he podo o ha ay ao elay o bo o example Cobb-Dogla he he g o d/d deped oly o dvdal peeee.e. o he g o. Moe peely a pove ehologal ho deeae eae he ely ae ad oly ><. Poo: I γ / ema oa o ay level o ad he el deed by obevao o eq. 4. e wll aalye he hape o ad o Ω o a boad la o dvdal peeee eo 5 ae ompleg he aaly o he aoal dyam Compleg he aoal dyam I h eo we omplee he haaezao o he aoal dyam o he yem geeaed by a pove ehologal ho. e wll ame ha h a ho ae he yem a ome me whe a eady ae ad he ho ome a a pe. 9 I ode o how h popey eall ha γ. Moeove a he eady ae. I we d / d / g d g / d dee g we ge γ. d d d d

12 o ompo ad apal ey hg ae lea: wheeve eae boh eady-ae ompo ad apal ey ae eaed. Moeove by eallg Popoo 4 a be how ha he al jmp : d d d C d d d whee C λ3 Ω >. 5 d d ee ppedx B.4. Se boh ad ae pove he g o 5 ambgo. I ay d d ae o mae he g o he al jmp o ompo ae he pove ehologal ho ompo ad apal ey eae eadly owad he ew eady-ae vale. o he dyam o he ae o gowh o poplao hg ae moe omplex ha a we have how Popoo 3 ad 6 boh he eady-ae hage ad he dyam pah ollowed by ae he ehologal ho ae ambgo. Howeve we a how ha he al jmp alway egave. e a mmaze o el o he al jmp o ad hogh he ollowg: Popoo 7: he a pove ehologal ho o: d d > < < Ω Ω wh > Ω λ3 >. o he al jmp o he ely ae: d <. d Poo: See ppedx B.8 Le ow ame ha γ o he ae o mply. e a mmaze he behavo o he yem ae a ehologal ho a ollow: ae a eae o oal ao podvy whle apal ey eadly eae owad o ew hghe eady-ae vale a o he ae o gowh o poplao we a have hee poble ae eall ha Ω / eady ae: a > h ae d / d < ad Ω <. e he ehologal ho all a a emedae level bewee he old ad he ew lowe eady-ae vale ad he aodg o Popoo 3 deeae owad ew eady-ae vale. b > h ae d / d < ad Ω >. odg o Popoo 3 whe Ω > eae owad ew lowe eady-ae vale. Hee ae he ehologal ho he ely ae expeee a dehoog eao e aaeoly all below he ew eady-ae vale. < h ae d / d > ad eealy Ω >. The dyam he ame a he pevo ae alhogh he ew eady-ae vale o hghe.

13 ally a o pe-apa ompo ae a hee a pove jmp whle ae b ad he jmp pove egave Ω < > Ω. The ay ae eadly eae owad ew hghe eady-ae vale. o llave ppoe ge we dep o dg o he dyam behavo o ad ae a oeee pove ehologal ho og o he ae whh he p elay o bo γ eqal o. g. a g. b Ω < Ω < Ω > * Ω > Ω > Ω > Ω > * > Ω < ge : Qalave dyam o he poplao gowh ae ad o ompo pe-apa ae a pove ehologal ho o γ. oe ha o eealy he al jmp o qaavely he ame egadle he g o. 5. Example o peeee lae I h eo we exploe he popee o whh al o deemg he eao boh o ely ad ompo o a podvy ho. e o o he hape o Ω eo The HR ly o Le ae a lage la o peeee HR omalzed h a way ha ad : ε ε M ε a ε ε a ε whee a ε > a M>. Reall ha h a o eompae he o-ela ly a ad ε > ad o a > eompae he omalzed logahm ly a ε ε egave expoeal a ε ad M a qada ly ε ad lea ly a 3

14 ad ε. o h la o peeee deed: Lemma : Ude HR peeee ad a>: > o [ ε a ε > lm ε < ε qada ly he ε ε ε a a ad he ollowg Lemma ε ε lm. a Poo: See ppedx B.9. Hee de a boad la o ly o > e ow o ome peal ae o HR whee we oba loed om olo. 5.. CES ly o σ Ude CES ly obaed om HR by eg a ad ε p o σ σ ad h Ω >. The σ we oba he ollowg loed om olo: σ 4. σ σ σ a mlplave oa wh σ we have I he lgh o he el above we a gve he ollowg wo Coollae: Coollay : I he ly o CES he he g o d/d deped oly o he g o γ. Moe peely a ehologal ho deeae eae he eady-ae ely ae ad oly he ao elay o bo lowe hghe ha y. Poo: de CES ly ad by eq. 4 gd/d-g γ. Coollay 3: I ly CES o Cobb-Dogla ad he podo o ha γ he d/d alway zeo. 4

15 5.. HR ly o wh ε ε Th ae woh dog ha allow o ge a loed om olo. Moe peely whe a a. Reall ha h a ly o dplay he ollowg a >. ally he ollowg loed om olo hold: a a popee a a a wh < a.e. <. 5.. o HR peeee Havg how ha o a lage la o peeee we wll ow povde a example o a ly o o whh egave obvoly h wll be a o-hr o. Code he ollowg ly o: σ σ wh σ. oe ha ; a be how ha σ > σ σ σ σ < ad Ω >. I h ao alhogh we do o have a loed om σ σ olo by Popoo 6 d / d a be pove o example γ. 6. The hape o Ω Up o ow we have how ha he dyam o he model ad pala o poplao gowh ggeed by a podvy ho a be vey h depedg boh o ehology ad o peeee. Howeve whe Ω < ad he ao elay o bo eqal o whh eem plable o ame ha ohewe he hae o ome o ehe apal o labo wold h o zeo he log de hghly-eqe ehologal ho he he ee o a pove ho o oal podvy ae voally deemed ha he lae ae a eae o he eady ae vale o boh apal ey ad ompo ad a deeae o he eady-ae vale o he poplao gowh ae. Moeove a o he dyam o he eoom yem we have d d how ha whe Ω < < > ad heeae deeae eae owad he ew d d eady-ae vale. oe ha ha he dyam ag he lae ae eem le wh he empal evdee oeg he egave elaohp bewee ehologal poge ad pe-apa ome o he oe had ad poplao gowh o he ohe had whh aodg o ome hola ha bee haaezg he mo developed oe ove he la wo ee ee o example alo 5 ad oe ad Tel 6. 5

16 Hee h eo we p dow ome e odo o haaezg he g o Ω. Moe peely we povde he odo g ha ha beyod a ea vale o oal ao podvy.e. o > he boh he dyam o he yem ad he al jmp o ad ae he oe meoed above ha Ω <. Popoo 8: Sppoe ha < ae o vale bewee ad. ow </ ad ¼ / ; o ay peeee la whee σ σ > σ whh llled o HR σ peeee ε < ad a> he hee ex a qe level o h ha a he eady ae Ω wh Ω <. Poo: See ppedx B.. Hee we a olde ha whe γ de HR peeee wh ε < ad a> beyod a ea level o ehology a pove ehologal ho wll voally eae ompo ad apal ey ad deeae poplao gowh boh mmedaely ad he log. 7. Colo I h pape we develop a gowh model wh edogeo ely whh a avod he epga olo. e popoe a exeded veo o al level laam CLU whh we ame Relave CLU. I o o odog he al level ed by pae o dedg o gve bh o a exa hld o he ow adad o lvg. e how ha h oal welae eo o vaable poplao ze axomaally oded. Lae we povde a ll haaezao o he exee ably ad dyam popee o he eqlbm. I pala we aalyze he elaohp bewee he model vaable ad ehologal ho ad we how ha he e o epoe o he poplao gowh ae o he lae ho ae qe h depedg o boh peeee ad ehology. o example we how ha whe peeee ae o he HR om b o CES ad he elay o bo o podo ao eqal o oe a pove ho o oal ao podvy eae he eadyae vale o boh pe-apa ompo ad apal ey ad ede he eady-ae ely ae. o he lae he g o he al jmp ad o he gowh ae alog he ao pah o eealy he ame a he g o he hage o he eady-ae vale. Moeove we p dow he odo de whh he model a eplae he egave empal elaohp bewee ehologal hage ad pe-apa ome o he oe had ad poplao gowh o he ohe had veled ome ee wo o he developed oe ove he la wo ee. Thee odo ae ha h a egave elaohp apple beyod a ea level o ehologal developme. ally we how ha he model allow o eve he log- ee o ehologal ho o poplao gowh ha pove ho oal ao podvy a eae poplao gowh ae he log whe peeee ae o o-hr om ad/o he elay o bo o podo ao dee om oe. Reeee he. : mpobly heoem o wela axologe Eoom ad Phloophy 6: 47-66; 6

17 Ba M. ad Leha O. : Demogaph Tao ad Idal Revolo: Maoeoom Ivegao Revew o Eoom Dyam 3: 44-5; Bao R.. ad Bee 988: ely Choe a Model o Eoom owh Eoomea 57: 48-5; Bee.S. ad Bao R.. 989: Reomlao o he Eoom Theoy o ely Qaely oal o Eoom 3: -5; Blaoby C. Boe. ad Doaldo D. 995: Ieempoal poplao eh: allevel laa pple Eoomea 63: 33-; Blaoby C. Boe. ad Doaldo D. 997: Bh-dae depede poplao eh: al-level pple oal o Eoom Theoy 77: 6-84; Blaoby C. Boe. ad Doaldo D. : Cal lelvel poplao pple ad he epga olo Depame o Eoom Uvey o Bh Colmba Do pape -6; Boome. 99: Cog he o o global wamg. he Hoe Pe Lodo; Ca D. 965: Opmm owh a ggegave Model o Capal mlao Revew o Eoom Sde 3: 33-4; Cgo. ad Roa E. 99: The ee o aal mae ad oal ey o avg ad ely behavo Ialy oal o Poplao Eoom 5: 39 4; Ceme H. ahva. ad Peea P. 6: Peo wh edogeo ad oha ely oal o Pbl Eoom 9: 33 ; Deado.V. 976: The Opmm owh Rae o Poplao: Comme Ieaoal Eoom Revew vol. 7: 5-5; de la Cox D. Peea P. ad Pohee. : How powel demogaphy? The Seedpy Theoem eved oal o Poplao Eoom 53: 899-9; Doepe M. 4: og o ely dele dg he ao o gowh oal o Eoom owh 9: ; Doepe M. 8: owh aeo I: Seve. Dla Lawee E. Blme Ed. The ew Palgave Doay o Eoom. Seod ed. Palgave MaMlla Lodo; Ehlh I. ad L. 997: The poblem o poplao ad gowh: evew o he leae om Malh o oempoay model o edogeo poplao ad edogeo gowh oal o Eoom Dyam ad Cool : 5-4; eadez-vllavede.. a Malh gh? Eoom gowh ad poplao dyam PIER og Pape -46; alo O. 5: The demogaph ao ad he emegee o aed eoom gowh oal o he Eopea Eoom oao 3: ; alo O. ad el D.. : Poplao ehology ad gowh: om he Malha egme o he demogaph ao mea Eoom Revew : 86 8; oloov M. oe L.E. ad Tel M. 7: Eey wh Edogeo Poplao owh Eoomea 75: 39-7; eewood. ad Sehad. : The U.S. demogaph ao mea Eoom Revew 9: 53 59; Hae. ad Peo E. : Malh o Solow mea Eoom Revew 9: 5 7; Hema.. ad ale.r. 99: The Relaohp bewee age ad he Tmg ad Spag o Bh: Evdee om Swedh Logdal Daa Eoomea 586: 4-4; Ha T. : Comme o Poplao Pple wh mbe-depede Cal level pblhed map Depame o Phloophy Uvey o Calgay; aege K. ad Khle. 9: The opmm gowh ae o poplao eodeed oal o Poplao Eoom : 3-4; oe L.E. Shoobood. ad Tel M.. ely Theoe: Ca They Expla he egave ely-iome Relaohp? BER Chape : Demogaphy ad he 7

18 Eoomy 43-; oe L.E. Tel M. 6. " Eoom Hoy o ely he U.S.: 86-96" BER og Pape 796; Kaleml-Oza S. : Doe he moaly dele pomoe eoom gowh. oal o Eoom owh 7: 4 39; Koopma T.C. 96: Saoay Odal Uly ad Impaee Eoomea 8: 87-39; Koopma T.C. 965: O he Coep o Opmal Eoom owh Poae ademae Seam Spa Vaa 8: Reped a Cowle odao Pape ; Keme M. 993: Poplao gowh ad ehologal hage: Oe mllo B.C. o 99 Qaely oal o Eoom 8: 68 76; Lagelo. 3: om Malh o mode gowh: The hee egme eved Ieaoal Eoom Revew 44: ; Maovh D.. 998: ely ad he Eael hypohe: aeme o he leae oal o Poplao Eoom : 53-; Mee V. ad ede M. : Peo ely ad edao oal o Peo Eoom ad ae 9: 75-93; g Y.K. 986: Soal Cea o Evalag Poplao Chage: a leave o he Blaoby-Doaldo Ceo oal o Pbl Eoom 9: 375-8; Pa D. 976: O Dog he Be o O Chlde Eh ad Poplao ed. By M. Bayle. Cambdge: Shema; Pa D. 984: Reao ad Peo Oxod/ew Yo: Oxod Uvey Pe; Ramey.P. 98: Mahemaal Theoy o Savg Eoom oal 38: ; Reöm T.I. ad Spaao L : The Opmm owh Rae o Poplao de Cal- Level Ulam ; oal o Poplao Eoom 43: 8-; Samelo P.. 975: The Opmm owh Rae o Poplao Ieaoal Eoom Revew 63: 53-38; Shell L. 8: The Repga Colo ad Ulaam de Doma Reo oal o Pbl Eoom Theoy 6: -3; Soae R.R. 5: Moaly edo edaoal aame ad ely hoe mea Eoom Revew 95 3: 58-6; Spaao L. ad a L. : The Opmal Level o Deb a OL Model wh Edogeo ely ema Eoom Revew 3: Spaao L. ad Reöm T.I. : Opmal axao al-level laam ad eoom gowh oal o Pbl Eoom 96: ; Tama R.. : Hma apal ad he wh om agle o dy oal o Eoom Dyam ad Cool 7: 7 4; Tama R.. 6: Hma apal ad eoom developme oal o Developme Eoom 79: 6-7; Täjö T. : hy e Ogh o ep he Repga Colo Ula 4: ; Yew S.L. ad Zhag. 9: Opmal oal ey a dya model wh hma apal exeale ely ad edogeo gowh oal o Pbl Eoom 933-4: 65-9; Zhag. ad hma K. 99: Pay a yo go pbl peo wh edogeo ely oal o Pbl Eoom 48: 39-58; Zhag. ad hma K. 993: The Old-ge Sey Syem Hypohe Reved. oal o Developme Eoom 4 9-; ad M.P. ad Bz.P. 98: Compleed ely ad I Tmg oal o Polal Eoomy 885: 97-4; 8

19 ppedx ppedx. Relave Cal-Level Ulaam e popoe a poplao eo he p o CLU b whee he jdgme he al level o ly o le woh lvg elave o he exg geeao level o wellbeg. odg o h a eo a oey a low level o ly e a lowe hehold o ly o he ex geeao ad a oey wh hgh lvg adad e a hghe level. So pae had a good le hey eqe he hlde o have a good le a well ad ve vea. e all h a eo Relave Cal Level Ulaam RCLU. I eaoable ha oee yea ago had a eely dee age level o ly o le woh lvg ha oee oday. lo oee ollowg he moe lexble poplao eo wll d eae o adap o lao he exeal evome. o example ood eoe beome ae o a peod ad ly all below a abole al level oey doe o op epodg de RCLU b may ll vew le woh lvg he hlde have a ly elaed o pae ly. Soee ollowg abole CLU may beome ex a hey op epodg whe ly o a mbe o geeao all below he abole hehold level. ollowg he appoah by Koopma 96 we beg by amg a geeal welae o deed ove poplao ad ly aleave. e he ode oe by oe he polae axom o ave a he al welae epeeao. The appoah ha he advaage ha he expoo mple ad a moe aeble ad oe a lealy ee how he welae o developed. o ay me we dee he aleave X a he ly veo { } o geeao ad he ze o he ex geeao.e. X { }. e ame he poplao eo epeeed by a geeal welae o. X X X X e ame om he oe ha he welae o we deeable. e beg wh a mbe o depedee ad depedee polae axom. I eaoable o ame ha he hoe o ly veo ad poplao ze he ely da e depede o he oome o he eale geeao. Tha he peeee ove ad o lage hold be depede o -- ad -. Th he axom o Idepedee o he Log Dead a Blaoby e al To allow he pobly ha poplao jdgme deped o he ly level o he paeal geeao we m allow o X o deped o X. Idepedee o he Log Dead Peeee ove X ae depede o X -- o. Paeal Depedee Peeee ove X deped o X -. depedee o he log dead mple ha ay deee elao bewee ad a me m be depede o he vale o X -. Tha he peeee ove X m be ~ epeeed by ome : I geeao do o ovelap h Depedee o he Reely Dead. 9

20 ~ ~. X X X... ad ay peeee ove e ad e X m be epeeed by ome V.e. he aggegao o Koopma 96: ~ X X X X X....3 V ~ a he he peeee ove X wold be depede o X - ad h volae Paeal elae Depedee. Coeqely Idepedee o he Log Dead ad Paeal elae Depedee mply a wealy epaable welae o o he om.3. oe ha we ao have V X X X X... oe ha ~ poeally dee om howeve he o dee he he peeee ove X a o me gve a pala hoy o X wold be dee om he peeee ove X a o me - whh wold geeae me-oey. e hall heeoe ame me-depedee o me oey: Saoay Tme Idepedee/Tme Coey Deoe a eqee o all e aleave a X τ {X τ X τ }. The o ay hoy X X ad e vale X he peeee ove X hold be he ame a ove X gve hoy X X ad e vale X X X. Saoay mple he o ~ X X X X X X ˆ X X V.4 ~ V.4 m epee he ame peeee ove X whh a oly be he ae V V ad ~ ˆ. Coeqely om.3 we have X X X... V X X X X X....5 Tha a evely epaable oal welae o. Th welae o howeve mple ha geeal he peeee ove ay ad deped o he vale o ad o all e geeao. e lm h depedee a ollow. Idepedee o Da e eeao Peeee ove X ae depede o X o all. e oe ha we ao have depedee o X who ag depedee o X -. Th a eeay oeqee o Paeal Depedee ad Idepedee o he Log Dead depedee o a lea he ex mmedae geeao. The oly poble way o lmg he depedee o eae he degee o epaably o X X X... V X X X X X... φ.6

21 Oe moe eo gve X X X... V X X V X X X. X... φ.7 To ave o oao deoe X X φ X X φ φ.8 φ.8 y deee elao bewee ad gve by d V V φ V φ φ V φ V φ φ V φ φ V φ φ V φ V φ φ V φ φ V φ φ V φ φ φ.9 o h deee elao o be depede o ha o X he ao o he paal devave m be oa: V φ V φ V φ V φ φ V φ V φ φ β Th mple ha he o V m be lea: X X X... X X β[ φ X X β X. X...] O by eo. φ. X X X... φ X X β. Tha a doed m o geeaoal welae o. Havg eablhed he oeqee o Idepedee o he Log Dead Idepedee o Da e eeao ad Paeal Depedee we ow o he pe o he poplao hoe. e ow we he pa X - { - } ad X { } explly. Reall ha a veo o dvdal le { }.e. φ φ. oe ha a be dopped a me a we ae oly odeg wela ea poplao oly mae o he exe bg dvdal le he φ φ.3 e wll ow dee o poplao eo o eah geeao.

22 Relave Cal Level oey dee addg dvdal o a exg poplao eveyhg ele eqal he ly o hoe added eqal a al level o - depedg o he ly o ompo o omeoe he pevo geeao. I dvdal de wh a geeao h level o ompo m be deed. e label h a eeee-level ompo -. The al level ao deped o he poplao ze o he pevo geeao oly he eeee ompo level a a be ee.3 led o by Idepedee o he Log Dead a - wa a deo ae by he - geeao. The al level m alo ay aoymy o be oded lae mplyg m be vaa wh epe o ay pemao eamg he dexe. pemble elave al level he ly o a ao <δ< o he ompo level o he op h dvdal o oho -: δ.4 Th level vaa o addg dvdal o al level ompo dela o oo lage. To ee h ode ompo by eamg dvdal a L L. ddg dvdal o he al level appled o he geeao wll o hage he level o a log a δ. hemoe he al level vaa wh epe o eamg dvdal a he odeg o ompo depede o he ame. I hold be oed ha aveage ompo a eeee ompo doe o have he popey o beg depede o addg dvdal o he al level a wold be delg. The ame alo e o meda ompo. Howeve o all I a de a -be alloao he he aveage he meda ad he op h level ae he ame. Coeqely. : X X... βφ β X. X... φ.5 o gve poplao ze he oal peeee ove dvdal le may deped o poplao ze ad pa le. e hall mpoe ha he oal peeee ove le wh a oho hold be depede o le o oeed dvdal b o eealy he exee ha he ly veo { } peeed o aohe ly veo { } h hold be he ae egadle le o ohe dvdal. e eqe Idepedee o Ule o Uoeed Idvdal Peeee ove { } ad { j } ae depede o { h } o h j ad o e alo eqe :. oymy Peeee ove { } ae depede o dey. Thee ae bh-dae depede aeme a Blaoby e al 997.

23 3 Sog Paeo Pple elae eag eah. Idepedee o he le o oeed dvdal eqe.5 o be wealy epaable { }.e Ψ Π Ψ Π X X β βφ φ o ome o Π ad Ψ ha he veo ee a a o Ψ h ha φ - Ψ. Seod eqe Ψ o be a addve o. To ee h deeae he welae o wh epe o j ad o oba: j j X X Ψ Ψ whh a be depede o h h j oly addve: Ψ ψ. oymy eqe he o ψ o be depede o ad m be a ly eag o de o he Sog Paeo Pple. Deeag wh epe o ad gve: X X Ψ Π Π Ψ ψ φ φ β ψ φ hh a be depede o h h oly φ lea. Coeqely Π Π X X β ψ β ψ.6 ll he eaog p l ow ha bee o xed poplao. e ow o he poplao eo el. Relave Cal Level Ulaam oey dee addg dvdal o a exg poplao eveyhg ele eqal he ly o hoe added eqal a al level o -. Sppoe we add m mbe o dvdal wh le a he al level -.6 he Π Ψ Π X m m m m X β β ψ ψ.7

24 4 oe ha Π ad Ψ peod ae aeed by he poplao ze a oeqee o o deo o eeee ly. The.7 by deo o al level ly m be oa m.e. Π m m m m m dm d ψ ψ ψ Se - ao deped o he le o he ohe dvdal he devave o ψ wh epe o m be zeo oeqely m Π ψ The Π ψ whh bed o.7 gve Popoo. Relave Cal Level Ulaam oal welae o epeeg a oal welae odeg ove poplao ad ompo hoe ha ae Idepedee o he Log Dead Saoay Idepedee o Da e eeao Idepedee o Ule o Uoeed Idvdal oymy Paeo Pple ad Relave Cal Level m ae he om: [ ]... ψ ψ β.8 whee > ψ ψ o ay. oe ha he Relave Cal Level doe o deped o he pa eeee ly he oa ad he poplao pple ede o eealed Cal Level Ulaam a Blaoby e al I he al level zeo ede o eealed Claal Ulaam. he odeg a be alloao wh eah geeao.e. he o ψ edda ad we have [ ]... β.9 Thogho o pape we mae he mplyg ampo ha he al level o lea.

25 ppedx B: Poo ppedx B.: The om o eq. 3 By ag om eq. ad olleg ly em o he ame dae he welae o a be we a: β β By gog a eleva o he plag hozo ad deg β we ge: expeo a be we a ollow:. I oo me by appoxmag he lae U e [ ]d. ppedx B.: Poo o Popoo ay * olvg 8 a eady-ae olo. The gve * oe a d he eadyae a well a he le-had de alo eqal o -. oe ha gve ay we a d a eady-ae apal o ay * ad a aoaed /. The le-had de o 8 egave a ad lealy eag wh oee /. The gh-had de RHS eag ad pove a a ehe > Iada odo hold o ohewe. I oa o deeag o he RHS oa o deeag he hee alway a qe ay * olvg he eqao above. Coeqely he hee alway ex a qe eady ae. I > RHS ao gow a a oo hgh ae o ee exee o *. e e o he ae whe / yeldg RHS ovex o lea /. I ovex o lea he RHS ovex o pobly lea lea ad /. Se o hee beg a olo ha he lope o RHS o geae ha / o ay. Se o h ha he lm a he lope o RHS o geae ha /. eeay o a qe eady ae ha he lope eve geae ha / ohewe hee wold be a eod eeo o ome lage. Coeqely eeay ad e o hee beg a drhs qe eady ae ha lm. d Tag he devave o RHS we ge: drhs o d 4 5

26 6 4 d drhs. The a goe o y e eag we have lm lm d drhs. Th eag ad ovex o lea he eeay ad e o exee o a qe eady ae ha / wh eqaly ad RHS ae lea. I oave he RHS a be ehe ovex o oave o ego o. The e o a qe eady ae ha he lope o he RHS lowe ha / o all. Combg ad we have ha eeay ad e o exee o a qe eady ae ha / wh eqaly lea ad /. The odo alo eeay. Sppoe o he we a p a ovex ad have a oado. ppedx B.3: Poo o Popoo om eq. we ge ha eeay ad e o he ably o he eqlbm > Ω o by g he deo o Ω >. Moeove by g he deo o ad o σ we ge > σ. ex by g eq. 9 we ge > σ. Moeove by g he deo o deed > B. I < / he e o ably o Ug he qada eqao ha gve eq. 8: we ge he e odo o ably a

27 7 The la em pove by he olo o he qada eqao. Coeqely a e odo o ably /. I mple. Ug eq. 9 we ge a e odo: o o by g B. 4 whh alway e. ppedx B.4: Poo o Popoo 3 Le ewe he dyamal yem a ollow: * * * & & & B. o whh he geeal olo e e e I H E D C B 3 * * * λ λ λ. B.3 Se λ > λ > deed ha he oee BDEH ae eqal o zeo ohewe he olo wold be explove ad wold volae avealy odo. Hee by deeag he yem B.3 wh epe o me ad eqag o he oepodg eqao o yem B. we ge: I C I I C I C C λ λ λ

28 Se a be ealy obaed om eq. B.3 by explog he ag odo h ha * he yem bol dow o he ollowg wo eqao : λ3 C 3I [ *] C I λ [ *] 3 The olo o C ad I ae: 3 C 3 λ3 λ ad I λ3 λ3 λ ally by bg om he aoba max we ge ha C λ Ω λ 3 3 ad I Ω λ λ Ω λ Moeove by bg o he expeo o ollow ha: emmg om ad mplyg C λ 3 Ω ad I Ω λ3 Ω. B.4 Se he deomao o C ad I ae pove de ably ad λ 3 < deed ha C>. O he oay he g o I ambgo ad moe peely gig Ω. ally he leazed olo o he dyam yem * Ce λ * e * Ie λ3 [ *] 3 [ *] λ3 [ *] wh C> ad gig Ω. B.5 ppedx B.5: Poo o Popoo 4 Reall ha he deema o he aoba max egave de ably o he eqlbm. Se D Ω whh Oe o he hee eqao edda a a be obaed by g he ohe wo. 8

29 9 Ω Ω Ω K ollow ha: [ ]. De K B.6 I we dee g we oba > g g ad > g K K. Hee > De D d d. ppedx B.6: Poo o Popoo 5 Ω Ω Ω Ω Ω K By explog he eady-ae elaohp K ad we a we he ollowg D K Ω < de ably o he eqlbm. B.7 ppedx B.7. Poo o Popoo 6

30 3 Ω Ω Ω Ω Ω K The Deema Ω D K K By eallg ha K K ad explog CRS we oba: [ ] D K Ω. B.8 Moeove by explog he eady-ae elaohp Ω ad Ω we ge K whee σ >. ally by g he eady-ae elaohp ad olleg em we ge [ ] { } γ D K B.9 whh povde he el. ppedx B.8. Poo o Popoo 7 o ompo by explog eq. 5 we have: De C De d d De. Moeove g eq. B.4 B.6 ad B.7 ad mplyg we ge:

31 d De λ3 d. By explog he expeo o om eq. ollow ha: λ3 d λ3 De d Hee we ge: d > > De Ω Ω. d < < Ω Ω. whee Ω λ3 >. B. d > ally e De < we a olde ha d < o ely eall ha om B.5 [ *] λ3 * Ie whee gig Ω. < Ω Ω. > ow ppoe ha a ome a a pove ehologal ho o. e have ha d d d I * * d g De g[ De I De ]. B. d d d d Explog B.5. B.7 ad B.8 we a we: d De Ω λ3ω. d oe ha he em qae bae egave Ω ; hemoe e De d have < Ω. d To how ha he lae el hold alo o egave Ω e eq. o be o d λ3 De λ3 Ω Ω. d Ω I Ω < he > ad he baeed em above geae ha: Ω < we he o oba: 3

32 λ3 Ω Ω > λ3 Ω > λ3 Ω > Ω > whee eqaly ollow om Ω beg egave eod em om oavy o podo o ad hd om ably. d Hee he baeed em pove ad o d De > whh mple < alo Ω <. d d ppedx B.9. Poo o Lemma ε ε ε a a Ude HR peeee wh a> we ge ha ε ε ε ε ψ ad a whee ε ε ε ε ε ψ ad ψ >. ε a a a a ψ Se lm lm by g lhôpal le we ge ha ψ lm lm. ally e ψ > o > deed ha > o a [ [. Moeove e lm by g lhôpal le we ge ha ε > ε ψ ε ε lm lm. a a ε < ex ε he whle o ε omalzed qada ly deed a a a h ha lm. a l ally ε he l h ha lm ad lm. ppedx B.. Poo o Popoo 8 Reall ha Ω [ σ ] ad by explog eq. 8 ad 9 deed 3

33 Ω 4 σ B. Hee whe σ whh lowe ha he Ω > whle o σ Ω <. Hee Ω hage g a lea oe wh h a eval o σ. I a Ω he σ 4 < <. B.3 Reall ha explog 9 ad he elaohp < </; mple he ollowg eqaly o hold: σ ably odo σ 4 B.4 I ode o gaaee ha he veo o he g o Ω.e. odo B.9 doe o volae ably odo.e. B. e o e ha he RHS o B.4 bgge ha oe ha ae ome maplao. < 8 4. B.5 Se we amed ha < he e odo o B.5 o hold ha >. 4 Moeove de he ampo ha γ he ao oa wh epe o. I ode o gaaee ha Ω hage g oly oe we wll povde e odo wheeby d Ω <. By deeag B. w.. we have: d Ω d Ω d σ d σ d d σ σ d σ d Ω σ d σ Ω Ω σ σ B.6 σ σ σ 33

34 Moeove by evalag he B.6 Ω ad explog he elaohp: ge ha d Ω σ σ d d σ. B.7 σ σ d d Ω Se < he o ha e odo o have < ha σ d σ σ. B.8 σ a o HR peeee wh a> oe a how ha σ σ. Hee a ε a ε eallg ha σ we have ha o h a la o peeee > mple ha σ > σ > >. Moeove a σ > ε <. B.9 a ε ε ally a o eq. B.8 o ha σ σ ε a ε σ ε. B. Howeve by B.9 ε <. σ oe 34