The Optimal Timing of Transition to New Environmental Technology in Economic Growth
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- Derick Andrews
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1 h Opimal iming of ransiion o Nw Environmnal chnology in Economic Growh h IAEE Europan Confrnc 7- Spmbr, 29 Vinna, Ausria Akira AEDA and akiko NAGAYA yoo Univrsiy Background: Growh and h Environmn Naural rsourc us along w/ conomic growh has bn sudid sinc h 97s. Dasgpa and Hal (974), Sigliz (974), Solow (974), c. In h 98s, h nvironmn bgan o b rcognizd as anohr imporan facor ha drmins h rajcoris of growh. A h sam im, h hory of conomic growh bgan o chang. ndognous chang coms o a cnral issu in h conomic liraur, inc. Romr (99)Lucas (988), c. hs wo srams convrg o a collcion of sudis afr h la 99s. Endognous chnological chang plays cnral rols in susainabl dvlopmn. Barbir (999), ahvonn and Salo (2), Bovnbrg and Smuldrs (995), Schou (2), Cunha--Sá and Ris (27) 2
2 Background: Opimal iming () Cunha--Sá and Ris (27). h Opimal iming of Adopion of a Grn chnology. Environmnal and Rsourc Economics 36. hir focus: As h conomy grows, nw polluion-abaing chnology bcoms indispnsabl; Such a chnology rquirs invsmn ffors a h im of dploymn; h opimal iming will b drmind in an ndognous mannr. hir modl is basd on a ypical Ramsy-yp modl, incorporaing nvironmnal qualiy and clan chnology. h lvl of clan chnology may chang disconinuously onc upgrad happns. Considraion for h n bnfi drmins h opimal iming of upgrad. 3 Background: Opimal iming (2) Cunha--Sá and Ris conribud o h liraur in ha: hy undrlind h significanc of h opimal iming in h framwork of h nvironmn and conomic growh. Som dbaabl issus rmain on h ohr hand: chnological chang in clan chnology is rprsnd as a jump of h lvl. Allowing such a suddn chang of h lvl implis ha h chnology is a kind of flow, no sock. his naur conrass o h fundamnal ida of modrn ndognous growh hory: hir ramn of chnological chang for clan chnology hus sms old-fashiond in his rspc. hir ramn of invsmn-rlad coss sms srang. I may dgrad h clar cu of hir analysis. 4
3 h Purpos is o xamin h choic of iming for chnological chang vis-à-vis nvironmnal qualiy in conomic dvlopmn. Basd on h spiri and mahmaical ramn of Cunha--Sá and Ris, w dvlop an analyical modl ha addrsss h opimal iming of ransiion of nvironmnal chnology from h old on o h nw on. In conras o Cunha--Sá and Ris, w focus upon acclraion of chnological progrss, and w ra coss in a simplr mannr so ha our focus can bcom clarr. 5 Analyical Fram () Ramsy modl w/ closd conomy; consan populaion Y: Producion; : Capial sock; C: Consumpion Q: Environmnal qualiy a: Environmnal chnology lvl Y A d d A C A rprsnaiv conomic agn rprsns h houshold whos im-addiiv uiliy is drmind by no only consumpion (C) bu also nvironmnal qualiy (Q). µ ( C Q ) U < µ <, >, µ ( ) <, µ ( ) < 6
4 Analyical Fram (2) h nvironmnal qualiy (Q) is a funcion of h consumpion (C) and nvironmnal chnology lvl (a) Q ac > h progrss of h nvironmnal chnology lvl islf may occur as h conomy grows. h growh ras of a and ( Y/A) ar proporional o ach ohr: ( da d) a ( d d) Inroducing a posiiv cofficin η : da d d d a h cofficin η may chang disconinuously a im. 7 Analyical Fram (3) Disconinuous chang in h cofficin η: da d d d a $ for < $ for lim, < Coss for h chang ypical coss: R&D and physical invsmns Exising physical capial may bcom obsol, and nd o b scrappd. < lim, < 8
5 h odl J ( ) s.. ax U d { C } $ +, $ µ ( C Q ) U U < µ <, >, d d A C < Q a C a $ a a % % for for > d 9 Solving h Problm Conrol variabls:, { C } [, ] Sa variabls: { } [, ] Firs sp: Opimizing for h valu funcion o a givn capial sock for afr-ransiion. $ ( ) %( ) ax U d { C } Scond sp: Finding h opimal pah unil h ransiion. V ( ) ax { C } $ U hird sp: Solving for h opimal iming. J ( ) axv( ) d + ( )
6 Firs Sp Suppos ha ransiion o h nw chnology occurrd a im. h problm hrafr is: & ( ) s.. ax { C } % µ % ( CQ ) $ ( ) d d d A C Q a C a a FONCs: µ % & µ % $µ % H C $µ a C C ( ) ( ) ( ) ( )( ) µ ( $ ) µ ( & $ ) ( & %µ )( & $ ) ' H ' µ a C % lim $ & & d + A & d Soluion for h Firs Sp C > whr dc d d d C $ ) µ + % (µ > % (µ * % A & ( % ' )( ) µ + % (µ) %( % (µ)( % ' ) (consan) A µ + ( % µ)( A % $ ) [ % ( % µ)( % )]( µ + % µ) > W obain: ( ) a µ ( & )-' +, ( & $µ )( & ) * ( & )( % µ + & $µ ) ( ) & & ' ( ) ( & ) 2
7 Scond Sp Finding h opimal pah o im V FONCs: ( ) s.. ax { C } % d d A C $ ac µ $ ( C Q ) $ $ $ ( ) < Q a a $ d + µ ( % ) & µ ( % ) ( $µ )( % ) ( $µ ) a C H C C µ ( $ ) µ ( & $ ) ( & %µ )( & $ ) ' H ' µ a C $( ) & & d + A & d % 3 Soluion for h Scond Sp C + whr ( ) ( ) $ ) µ + % (µ > % (µ & & ( $ µ )( $ ) C $ C $ $ ( $ µ )( $ ) $ * % A( % ' )( ) µ + % (µ) > % ( % (µ)( % ' ) ( $ µ + ' µ){% ' A( ' )( $ µ + ' µ)} ( $ µ + ' µ){% ' A( ' )( $ µ + ' µ)} ( ' µ )( ' ) ' ( ' µ )( ' ) W obain: A & ( $ ) ' % (* ' ) ) (* ' ) ') ( ' ) + o ') ( + 4
8 h Final Sp Opimizing i w.r.. J( ) axv( ) whr V ( ) ( ) ax U d + { C } $ has alrady bn obaind analyically. hus, w obain: ( )( ) ( ( ( 2µ ( µ ) ( ( 2µ )( ( ) dv ( ) a '. / + $ ( )( ) ' + ( )( ) ( )( ) ( ( ) $ ( 3 ( µ + ( 2µ &, ) & ( ( ( µ ( ( 2µ ( µ ( ( 4 d ( % - * 5 % Examining is sign, w know h propris of h soluion. dv ( ) d * if < * < dv ( ) d * < if * 5 Proposiion For a fini * o xis, ha is, * <, h following condiion is ncssary: Convrsly, if ( $ )( A ) or, quivalnly ( )( A ) < > ' and A & $ < ' and A % $ holds, hn h ransiion of nvironmnal chnology nvr occurs. (i.. * ) 6
9 Inrpraion of Prop. () h xisnc of a fini ransiion im is drmind by: Noic: '( $ ()% A & * + ' $ C A C Y % < & < his rprsns avrag consumpion propnsiy jus bfor h ransiion. σ rprsns h magniud of h lasiciy of marginal uiliy. > ' and A & $ < ' and A % $ I drmins h sign of h cross scond drivaiv of h uiliy 2 funcion: U { µ ( ) } µ ( ) C Q C Q If >σ (<σ), marginal uiliy of nvironmnal qualiy U/ Q is incrasing (dcrasing) in consumpion. ha is, h sociy prfrs mor (lss) nvironmnal qualiy as h consumpion grows. 7 Inrpraion of Prop. (2) For h ransiion o occur in a fini im horizon, h conomy mus b in ihr of h following wo siuaions: h conomy highly prfrs nvironmnal qualiy, and is currn avrag consumpion propnsiy is lowr han a crain valu (/δ). (h conomy has a highr saving ra.) h conomy rahr prfrs consumpion o nvironmnal qualiy, and is currn avrag consumpion propnsiy is highr han a crain valu (/δ). (h conomy has a lowr saving ra.) 8
10 Proposiion 2 Suppos ha h following condiions hold. ( )( A ) > And, ihr or + % * > and < ( $ ) + % * < and > ( $ ) ( % µ ) ( + µ ' {( % ) µ } %( % µ )( % ) & ( % µ ) ( + µ ' {( % ) µ } %( % µ )( % ) & hn hr xiss an opimal ransiion im * such ha < *<, and i is obaind by solving h following quaions: $,, % [ + $ ] o A h capial sock a h im: + % * ( $ ) ( % µ ) ( % µ )( % ) ( + µ ' % % & {( ) µ } 9 Inrpraion of Prop. 2 Givn h xisnc of a fini *, > cas: Capial sock incrass, and rachs h fixd lvl from blow. < cas: Capial sock dclins, and rachs h fixd lvl from abov. > < 2
11 Proposiion 3 ( )( ) Suppos A > Also, suppos h following approximaions hold. µ hn, w hav h following snsiiviis. d d ( ) < > cas: < cas: d d < ( ) d d > ( ) No: da d d d a $ $ for for < 2 Inrpraion of Prop. 3 rprsns h dgr of h innovaion. Inuiivly, i lls how much h ransiion o h nw nvironmnal chnology is bnficial o h conomy. h incras maks h barrir o ransiion lowr. > < 22
12 Conclusion W invsigad h opimal iming of ransiion in nvironmnal chnology from h old on o h nw on along wih conomic growh. Whhr ransiion o nw chnology is ndd or no is drmind by wo facors: h dgr of complmnariy bwn nvironmnal qualiy and consumpion, and h avrag consumpion propnsiy (or saving ra). Givn h ncssiy, ransiions o nw chnology may occur in wo possibl ways. An nvironmnally dvloping conomy would accumula social capial and invs i o h dvlopmn of h nw chnology. A maurd conomy holding a sufficincy of capial and njoying a high consumpion ra would raliz h nd for nvironmnal qualiy improvmn a som im. A highr dgr of innovaion would lad o arlir or lar ransiions, rspcivly. 23
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