Optimal use of a polluting non renewable resource generating both manageable and catastrophic damages

Size: px
Start display at page:

Download "Optimal use of a polluting non renewable resource generating both manageable and catastrophic damages"

Transcription

1 Opimal ue of a polluing non renewable reoure generaing boh manageable and aarophi damage Jean-Pierre Amigue, Mihel Moreaux and Kaheline Shuber Thi draf: January, 27h 21 We hank Valérie Nowazyk for ehnial aiane Touloue Shool of Eonomi(INRA, LERNA), Manufaure de Taba, 21 allée de Brienne, 31 Touloue, Frane. Touloue Shool of Eonomi (IDEI, IUF and LERNA), Manufaure de Taba, 21 allée de Brienne, 31 Touloue, Frane. Pari Shool of Eonomi and Univeriy Pari 1 Panhéon - Sorbonne, Maion de Siene Eonomique, bd de l Hôpial, Pari.

2 Conen 1 Inroduion 1 2 The model Aumpion The oial planner problem Shadow o of arbon, ariy of he renewable reoure and ompeiivene of he non-renewable one Cae of an abundan olar energy Cae of a rare olar energy Marginal hadow o of he polluion ok along he opimal pah The renewable reoure i abundan Bu i i more expenive han he non-renewable reoure a he eiling And i i heaper han he non-renewable reoure a he eiling 13 4 The renewable reoure i rare And i i more expenive han he non-renewable reoure a he eiling Bu i i heaper han he non-renewable reoure a he eiling Alhough no heap enough o be exploied before he eiling And heap enough o be exploied before he eiling Time profile of he arbon hadow o and omparaive dynami indued by more evere damage Comparaive dynami indued by more harmful manageable damage Comparaive dynami aben he euriy onrain Comparaive dynami under an effeive eiling onrain Comparaive dynami indued by a more ringen euriy onrain Conluion 26 APPENDIX 28

3 1 Inroduion We are now reaonably ure ha greenhoue gaz aumulaion in he amophere i he aue of global warming, and ha he emperaure inreae reae damage o eoyem and eonomi aiviie. However, he preie relaionhip beween arbon onenraion and damage, he damage funion, i no well known, epeially for imporan inreae of emperaure, when irreveribiliie and non-lineariie are likely o our. For inane, he exenive urvey of he available eimaion of damage in he Sern Review (Sern [26]) highligh he exen of our ignorane a far a he damage funion i onerned. So i make ene o found environmenal poliy no only on he value of unerain marginal damage bu alo on he requiremen o mainain amopheri arbon onenraion under a erain eiling, hoen preiely o preven an exeive rie of he emperaure. Previou work were foued eiher on damage due o arbon aumulaion (Ulph and Ulph [1994], Hoel and Kverndokk [1996], Tahvonen [1997]) or on a onenraion arge (Chakravory, Magné and Moreaux [26a], [26b]). Bu here exi for ure damage before he eiling, o we ombine in hi model a onenraion eiling wih a damage funion under he eiling. A he onenraion eiling i impoed preiely o avoid aarophi diurbane of he limae and hene he eonomy, i make ene o aume ha damage remain relaively mall under he eiling, oherwie he eiling would have obviouly been ill-hoen, and ha hey are relaively well known. We onider a model in whih he energy requiremen of oiey an be aified by wo kind of reoure: a non-renewable one, relaively heap bu polluing, le ay oal, and a renewable one, more expenive bu lean, le ay olar. We hooe a imple framework where he damage funion i linear for he mall damage ouring before he eiling, hi lineariy aumpion ogeher wih he lineariy of arbon aumulaion in he arbon ok 1 enabling u o obain lear reul. We are inereed in he ime pah of exraion of he polluing non-renewable reoure, and of peneraion of he renewable one, bu alo in he iniial value and he ime profile of he arbon hadow o, and in he eniviy of hee variable o he ineniy of he marginal damage and he ringeny of he eiling. In a deenralized e-up, he ounerpar of he arbon hadow o would be a uni arbon ax 1 Thi lineariy aumpion, whih mean ha he naural regeneraion rae of he amophere i onan, i learly reriive and i queioned boh by empirial work and he heoreial lieraure, ee e.g. Farzin and Tahvonen [1996] and Toman and Wihagen [2]. We keep i in hi paper beaue naural regeneraion i no a he hear of he queion we wan o adre. 1

4 applied o he non-renewable reoure onumpion in order o implemen he opimal pah. The iniial value and he ime profile of hi arbon ax are a he hear of limae poliy, a well a i eniiviy o marginal damage and o he amopheri arbon onenraion objeive. Moreover, inuiion ugge ha aking ino aoun damage due o he onumpion of oal and impoing a eiling onrain on arbon onenraion hould aelerae he peneraion of lean renewable ubiue o he non-renewable polluing energy. We wan o know if hi inuiion i orre in our framework where here i no endogenou ehnial progre and where he whole oal ok will be ulimaely exraed and onumed. The main reul are he following. Fir, o maer a lo 2. Whaever abundan or rare, if he renewable reoure i more expenive han he non-renewable one a he eiling, i will never be ued before he end of he phae a he eiling, when he nonrenewable reoure beome o are ha he eiling will never be reahed again. On he onrary, if he renewable reoure i le expenive han he non-renewable one a he eiling, i may even be uffiienly heap o be exploied before he eiling, in whih ae fir he non-renewable reoure i exploied alone, eond boh reoure are ued ogeher before, a and afer he eiling, and finally he renewable reoure i exploied alone, afer he exhauion of he non-renewable one. Seond, in all onfiguraion of o, he arbon hadow value i fir inreaing unil he eiling i reahed, hen i dereae during he phae a he eiling, and finally i abilize a a onan value afer he eiling. The iniial arbon hadow o i an inreaing funion of he value of he marginal damage, and a dereaing funion of he eiling: he higher he eiling, ha i he le ringen he onenraion arge, he lower he iniial arbon hadow o. Laly, onrary o he inuiion, higher marginal damage or a more ringen enenraion arge indue a delay in he peneraion of he lean renewable reoure and alo a delay in he omplee raniion oward lean energy. The paper i organized a follow. Seion 2 preen he model. Seion 3 and 4 are devoed o he udy of he ime pah of exraion of he non-renewable reoure and ue of he renewable one in he wo ae of an abundan and a are renewable reoure. Seion 5 preen he ime profile of he arbon hadow o. Seion 6 onlude. 2 Alhough i i very diffiul o give preie figure for he repeive marginal o of he renewable and he non-renewable reoure. 2

5 2 The model 2.1 Aumpion We onider an eonomy in whih he energy need an be aified by exploiing wo kind of primary reoure, a polluing non-renewable reoure and a lean renewable one. Le oal be he diry reoure and olar be he lean one. Denoe by X() he available ok of oal no ye exploied a ime and by X,X >, he oal iniial endowmen: X X(). Le x() be i inananeou exploiaion rae: Ẋ() = x(). I average exploiaion o i aumed o be onan hene equal o i marginal o, and i denoed by x. Burning oal o produe energy generae a flow of new amopheri polluion proporional o he inananeou exploiaion rae. Le ζ be he uniary polluion onen of oal, o ha he flow of newly generaed polluion amoun o ζx(), adding o he amopheri arbon ok denoed by Z(). The amophere ha ome elf regeneraion apaiy we aume o be proporional o he polluion ok Z(). Le α be hi proporional rae of elf regeneraion. The dynami of he arbon ok i given by he differene beween he flow of newly produed polluion and he elf regeneraion effe: Ż() = ζx() αz(). Le y be he available flow of he renewable reoure and y() i exploiaion rae a ime : y y(). I average exploiaion o i aumed o be onan and i denoed by y. The o x and y inlude all he o neeary o deliver energy direly uable by he uer, ha i he exraion, proeing and delivery o, o ha he wo primary reoure may be een a perfe ubiue, and oal energy onumpion q() an be defined a: q() = x() + y(). The ondiion on x, y and y under whih i i opimal o exploi he diry reoure are deailed in ub-eion 2.3 below. The inananeou onumpion q generae a gro urplu u(q) (meaured in moneary uni). The gro urplu funion i aumed o be wie differeniable, rily inreaing, rily onave and aifying he Inada ondiion lim q u (q) = +. We denoe by p he marginal gro urplu u 3

6 and by q d he demand funion, he invere of u. The amopheri arbon onenraion reae damage. Thee damage are of wo kind. Fir he arbon ok generae a eah ime a manageable welfare lo. To implify we aume ha hi welfare lo i proporional o he urren arbon onenraion Z(). Le hz(),h >, be hi inananeou manageable damage. Seond, would he arbon onenraion be higher han ome eiling Z, oiey would experiene a drai hange of he limae regime leading o aarophi damage. Suh an even an be een a an infinie welfare lo. In order o preven uh an ouome, oiey ik o he objeive of keeping he arbon onenraion Z() permanenly a mo equal o he euriy level Z. Z i of oure greaer han he iniial arbon onenraion Z : Z < Z. All he urplu, o and manageable damage are diouned a ome onan poiive oial rae of dioun ρ. 2.2 The oial planner problem The oial planner problem i o deermine he poliy maximizing he um of he diouned ne urplu while aking are of he euriy onrain, and read: W(X,Z ) = max Ẋ() = x() x() X() = X > given Ż() = ζx() αz() e ρ {u(x() + y()) x x() y y() hz()} d Z() = Z > given, Z > Z, Z Z() y > given, y y(), y(). Denoing by λ() he oae variable of X(), by µ() he oppoie of he oae variable of Z(), by ν() he Lagrange muliplier aoiaed o he eiling onrain and by γ x, γ y and γ y he Lagrange muliplier aoiaed o he poiiviy onrain, he urren value Lagrangian i given by: L = u(x() + y()) x x() y y() hz() λ()x() + γ x ()x() µ() (ζx() αz()) + ν() ( Z Z() ) + γ y() (y y()) + γ y ()y() 4

7 Fir order opimaliy ondiion and omplemenariy lakne ondiion read: u (x() + y()) = x + λ() + ζµ() γ x () (1) u (x() + y()) = y + γ y() γ y () (2) γ x (), x(), γ x ()x() = (3) γ y(), y y(), γ y() (y y()) = (4) γ y (), y(), γ y ()y() = (5) λ() = ρλ() (6) and he ranveraliy ondiion are: µ() = (ρ + α)µ() h ν() (7) ν(), Z Z(), ν() ( Z Z() ) = (8) lim e ρ λ()x() = (9) lim e ρ µ()z() =. (1) A uual wih a onan marginal exploiaion o of he non-renewable reoure, he Hoelling rule redue o a ariy ren of oal λ() having o grow a a proporional rae equal o he rae of dioun. From (6) we ge: λ() = λ e ρ where λ λ(). Thu he ranveraliy ondiion (9) may be rewrien a: λ lim X() =. If oal ha ome value, ha i λ >, hen he oal ok mu be exhaued in he long run. Iniially, he arbon onenraion i below he eiling ine Z < Z. Hene on any iniial ime inerval [,) during whih Z() < Z, ν() = o ha equaion (7) beome µ() = (ρ+α)µ() h and an be inegraed ino: where µ µ(). µ() = [ µ h ρ + α ] e (ρ+α) + h, [,), (11) ρ + α Noe ha if he manageable damage effe i aben, ha i if h =, equaion (11) beome µ() = µ e (ρ+α), whih i nohing bu he expreion 5

8 obained in Chakravory e al. [26a], [26b], meaning ha he opimal arbon hadow o grow a a proporional rae equal o he um of he oial dioun rae and he naural aborpion rae when he arbon onenraion i below he eiling. In he preen eing, he arbon hadow o remain onan a he level h/(ρ+α) iff µ = h/(ρ+α), while he gap beween µ and h/(ρ + α) inreae a he rae ρ + α iff µ h/(ρ + α). Le u deermine now he ondiion under whih he diry reoure ha o be exploied, and how ha under hee ondiion µ > h/(ρ + α). 2.3 Shadow o of arbon, ariy of he renewable reoure and ompeiivene of he non-renewable one Aume ha from ome ime x onward he non-renewable reoure i no more exploied, o ha onumpion i aified by he renewable one alone. Define ỹ a he oluion of u (y) = y. Then, from ime x on, he quaniy onumed i min {y,ỹ}. If ỹ y, he renewable reoure i abundan, γ y() =, he ariy ren i nil, γ y () =, and he FOC (2) i aified ogeher wih he omplemenary lakne ondiion (4) and (5). If ỹ > y, he renewable reoure i rare, γ y() = u (y ) y >, he ariy ren i poiive, γ y () =, and he FOC (2) and he omplemenariy lakne ondiion (4) and (5) are aified. We denoe by y he value of min {y,ỹ} and by p he orreponding gro marginal urplu : p = u (y). When he renewable reoure i abundan, p = y and no ren ha ever o be harged for i exploiaion, while when i i rare ome ren γ y() ha o be harged one p() i higher han y. In he former ae p() anno be higher han y, in he laer ae p() i a mo equal o p. Le W(X( x ),Z( x )) be he value funion from x onward: W(X( x ),Z( x )) = x {u(y) y y hz()} e ρ( x) d. I inlude wo omponen, he um of he diouned ne moneary urplu (gro urplu minu o) generaed by he onumpion of y, from whih mu be dedued he um of he diouned manageable damage generaed by he ok of polluion inheried from he pa, Z( x ), auming ha 6

9 Z( x ) Z. Beaue he polluing reoure i no more exploied, hen Ż() = αz() o ha Z() = Z( x )e α( x), < x. Hene: and W(X( x ),Z( x )) = 1 ρ (u(y) yy) dw(x( x ),Z( x )) dz( x ) = h ρ + α. h ρ + α Z( x), (12) Wha i noeworhy i ha he burden of he inheried arbon ok i a linear funion of he ok hank o boh he linear form of he manageable damage funion and he linear form of he arbon ok dynami. The hadow marginal o of he inheried ok in value a ime x and equal o h/(ρ + α), i independen of boh Z( x ) and x. Sine boh Z( x ) and x are arbirary, hi implie, by a raighforward reurive argumen, ha he marginal hadow o of he arbon ok Z() in urren value i alo onan and equal o h/(ρ + α). Aume now ha oal i never exploied. The only upply i he olar energy, and i onumpion i equal o y defined above. Le u examine under wha irumane uh a poliy would no be he opimal one Cae of an abundan olar energy Reall ha hi ae orrepond o y ỹ or equivalenly p = y. Conider ome ime inerval [, 1 ], wih 1 >. Over hi inerval, le u redue a eah dae he onumpion of he renewable reoure by ε, < ε y, and inreae he onumpion of he non-renewable reoure by he ame amoun, o ha x() = ε and y() = y ε. 1 Aume ha ε and are uffiienly mall for he arbon onenraion o remain under he eiling. Sine oal onumpion i held onan a he level y, he balane hee of he ubiuion may be redued o wo omponen: a produion o aving reuling from he onumpion of a le oly reoure provided ha x < y ; 1 Exploiing he non-renewable reoure wihou imulaneouly reduing he exploiaion rae of he renewable reoure would reul in a marginal gro urplu u (y +ε) lower han he marginal o of he renewable reoure y. Then i i eaily heked ha he FOC (2) relaive o he ue of hi reoure and he aoiaed omplemenary lakne ondiion (4) and (5) anno be imulaneouly aified. Thu if he non renewable i o be exploied he exploiaion rae of he renewable mu be redued. 7

10 an inreae in he burden of polluion reuling from he exploiaion of a dirier upply. The value funion a ime along he iniial pah i: W(X( ),Z( )) = {u(y) y y hz()} e ρ( ) d. Beaue he emiion of polluion i nil, Z() = Z( )e α( ), and he value funion i given by equaion (12), wih x =. The value funion a ime on he perurbed pah read: W ε (X( ),Z( )) = wih: 1 {u(y) x ε y (y ε) hz()} e ρ( ) d +e ρ( 1 ) ζε αz(), [, 1 ], Ż() = αz(), > 1, 1 {u(y) y y hz()} e ρ( 1) d, implying ha: ( Z( ) ζε ) e α( ), [, 1 ], α Z() = Z( 1 ) e α( 1), 1, where: Z( 1 ) = (13) ( Z( ) ζε ) e α( 1 ) + ζε α α. (14) Hene he following value funion for he perurbed pah: W ε (X( ),Z( )) = 1 ρ (u(y) yy) + ε ( y x h ζ ) (1 ) e ρ( 1 ) ρ α h ρ + α Z( ) + h ζε ( ) 1 e ρ( 1 ). ρ + α α The differene beween he value funion on he perurbed and he iniial pah amoun o: W ε (X( ),Z( )) W(X( ),Z( )) = ε [ ( y x ) ζh ] (1 ) e ρ. ρ ρ + α 8

11 I i rily poiive, whaever he value of he perurbaion ε and he lengh of he ime inerval, provided ha he eiling onrain i no reahed, if and only if: y x > ζh ρ + α, (15) whih imply mean ha he fir omponen of he balane hee of he perurbaion, he o aving omponen, i larger han he eond one, he addiional damage omponen. Thi ondiion i neeary and uffiien for he exploiaion of he non-renewable reoure. In wha follow we aume ha i i aified. We onlude a follow. When he renewable reoure i abundan, ha i y ỹ, a neeary and uffiien ondiion of exploiaion of he diry nonrenewable reoure i ha i exploiaion o x i uffiienly lower han he exploiaion o of he lean renewable ubiue y, he o direpany being a lea equal o ζh/(ρ + α) Cae of a rare olar energy Reall ha hi ae orrepond o y < ỹ or equivalenly p > y. Exploiing he oal ok doe no imply any more o redue he produion of olar energy. Thu he value funion a ime on he perurbed pah read now: W ε (X( ),Z( )) = 1 {u(y + ε) x ε y y hz()} e ρ( ) d + e ρ( 1 ) {u(y) y y hz()} e ρ( ) d. where Z() i given by: e α < Z() = e [ α Z + ζε α (eα e α ) ] < 1 e [ α Z + ζε α (eα 1 e α ) ] 1. Hene ε an be hoen o ha Z() < Z,. For ε uffiienly mall: u(y + ε) u(y) + u (y)ε = u(y) + p ε, 9

12 o ha: W ε (X( ),Z( )) 1 ρ (u(y) yy) + ε ρ ( (p x ) h ζ ) (1 ) e ρ( 1 ) α h ρ + α Z( ) + h ρ + α ζε α ( 1 e ρ( 1 ) ). The differene beween he value funion on he perurbed pah and he iniial pah i now: W ε (X( ),Z( )) W(X( ),Z( )) ε [ (p x ) ζh ] (1 e ρ ). ρ ρ + α I i rily poiive provided ha: p x > ζh ρ + α. (16) We onlude ha i may happen ha i i opimal o exploi he diry non-renewable reoure even if i average o i higher han he o of i lean renewable ubiue when he renewable reoure i rare. I will be he ae when he upply of he renewable y i uffiienly mall, ha i p i uffiienly larger han x, however low y i Marginal hadow o of he polluion ok along he opimal pah Noie ha ζh/(ρ + α) i he marginal damage aoiaed o any addiional ue of he non-renewable reoure, however mall or large. A poined ou above, i i independen of he addiional ue of he non-renewable reoure ε, of he lengh of ime inerval of addiional ue of he diry reoure and of he polluion ok Z(), provided ha i remain below he eiling. Hene wihou any eiling onrain, he arbon hadow o would be preiely equal o h/(ρ + α) a eah dae. Thu he arbon hadow o, when he eiling onrain i aouned for, mu be higher han h/(ρ + α) before he eiling and when a he eiling, and equal o h/(ρ+a) afer he eiling, when he remaining non-renewable reoure ok i o low ha he eiling an never be reahed again. 2 Noe ha for p y ondiion (16) beome (15). 1

13 3 The renewable reoure i abundan We aume in hi eion ha he renewable reoure i abundan: y > ỹ, or equivalenly p = y, and ha ondiion (15) hold. Le u define x a he maximum flow of he polluing reoure onumpion when he arbon ok i a i eiling Z : x = αz/ζ. We denoe by p he orreponding gro marginal urplu: p = u (x). Two ype of opimal pah may arie, aording o he value of y ompared o p. 3.1 Bu i i more expenive han he non-renewable reoure a he eiling Thi ae, orreponding o y > p or equivalenly ỹ < x, i illuraed in Figure 1. y > p mean ha when a he eiling and he non-renewable reoure i he only reoure whih i exploied hen he lean renewable ubiue i no ompeiive. Thu if here mu exi a phae a he eiling during whih he exploiaion of oal i blokaded a x, hi phae mu be a phae during whih only oal ha o be exploied. Thi i implying ha oal i exploied alone from he beginning of he opimal pah when he eiling i no ye hur, nex when a he eiling and laly afer he eiling when he oal exploiaion doe no emi enough new polluion for he eiling onrain o be effeive again. Finally when p() reahe y, he diry oal ok i exhaued and he lean olar energy i exploied alone. < Figure 1 here > The opimal pah ha four phae. During he fir phae [, ) he nonrenewable reoure i he only exploied one. The prie i equal o: p() = x + λ e ρ + ζµ() < p, (17) wih µ() given by (11) beaue he arbon onenraion i under he eiling Z. In (11) µ i greaer ha h/(ρ + α), o ha he ame hold for µ() ielf. During hi phae Z() inreae: p() < p implie ha x() > x and, ogeher wih Z() < Z, hi implie ha ζx() > αz(). Thi phae end a dae when imulaneouly Z( ) = Z and p( ) = p. The eond phae [, ) i he phae a he eiling, wih he non-renewable reoure aifying he whole demand: p() = p and x() = x. A hi prie, 11

14 he renewable reoure i no ompeiive and anno relax he eiling onrain. During hi phae µ() i dereaing, and i beome preiely equal o h/(ρ + α) a he end of he phae. Beaue he eiling onrain i igh, he aoiaed Lagrange muliplier ν() i rily poiive and he dynami of µ() i given by (7). Beide, equaion (1) yield: µ() = 1 ζ ( p x λ e ρ). (18) Time differeniaing (18) and ubiuing for µ() given by (7) and for µ() given by (18) we obain: ν() = 1 ζ ( (ρ + α) (p x ) αλ e ρ ζh ). The hird phae [, x ) i again a phae during whih he non-renewable reoure i he only one whih i exploied. The prie i now given by: p() = x + λ e ρ + ζh ρ + α. (19) Thu he prie i higher han p, hene x() dereae and he eiling onrain beome unbinding, o ha µ() = h/(ρ+α) a poined ou above 3. A he end of hi phae p() = y and he renewable reoure beome ompeiive and ake he whole marke. Hene he non-renewable reoure mu be exhaued. The la phae [ x, ) i a phae of a definiively lean eonomy: p() = y and y() = ỹ. There are five endogenou variable o be deermined: λ, µ,, and x. They are given a he oluion of he following yem of five equaion: q d (p 1 ())d + ( ) x + x q d (p 3 ())d = X, (2) Z e α + ζq d (p 1 ())e α( ) d = Z, (21) 3 Noe ha x() < x, >, and Z( ) = Z implie ha Z() < Z, >. However, hi doe no imply ha Z() i monoially dereaing from onward. Bu we are ure ha Z() anno hi again he eiling Z. Hene µ() = h/(ρ+α),, o ha he prie pah, for >, i given by p 3 () defined in equaion (24) below. 12

15 p 1 ( ) = p, p 3 ( ) = p, p 3 ( x ) = y, (22) p 1 () and p 3 () being repeively given by: [ p 1 () = x + λ e ρ + ζ µ h ] e (ρ+α) + ζh ρ + α ρ + α, (23) p 3 () = x + λ e ρ + ζh ρ + α. (24) Equaion (2) i he umulaed demand-upply balane equaion of he nonrenewable reoure. Equaion (21) ae he arbon ok oninuiy a. Equaion (22) ae he prie oninuiy repeively a, and x. I i eaily heked ha for any λ, µ,, and x oluion of hee five equaion here exi value of γ x (), γ y(), γ y (y) and ν() uh ha all he opimaliy ondiion (1) (1) are aified by he above prie and quaniy pah. 3.2 And i i heaper han he non-renewable reoure a he eiling Le u now examine he oher ae: y < p or equivalenly x < ỹ. Here, when he arbon ok i a he eiling and he exploiaion of he non-renewable reoure i bounded by x, he demand anno be aified by hi ole reoure, beaue i would imply a prie p > y and he renewable reoure would be ompeiive. Thu he prie y i he maximum prie whih may prevail. The only poibiliy when a he eiling i ha he prie be equal o y and he demand equal o ỹ. The non-renewable reoure being he le oly one, i mu be ued a muh a poible: x() = x and y() = ỹ x, an immediae impliaion of diouning. By le oly i mu be underood ha x + ζh/(ρ + α) < y, ha i ondiion (15) i aified. Noe ha we mu alo have x + λ e ρ + ζh/(ρ + α) y during hi phae a he eiling. The equaliy hold a he end of he phae, whih mean ha he ok of non-renewable reoure mu hen be exhaued. Afer he end of he phae he prie ay a he ame level y. The opimal pah are illuraed in Figure 2. < Figure 2 here > 13

16 The fir phae [, ) i imilar o he fir phae of he preeding enario, p() = p 1 (), exep ha a he end of he phae, when he arbon ok reahe i eiling, he prie mu now be equal o y. The eond phae [, ) a he eiling i he phae we have previouly deribed: p() = y. Sine he non-renewable reoure ok i exhaued a he end of hi phae, we have now = x. The hird and la phae i he phae of exluive exploiaion of he lean reoure. The prie i he ame han during he eond phae: p() = y, bu now q() = y() = ỹ. The yem of equaion deermining he endogenou variable i given in Appendix A. The main differene wih he preeding enario i ha here he ue of he renewable reoure inreae more progreively: y() = during he fir phae, ỹ x during he eond phae, and ỹ, i maximum and definiive level, during he la phae, inead of wihing direly from o ỹ a in he preeding enario. 4 The renewable reoure i rare We aume in hi eion ha he renewable reoure i rare: : y < ỹ or equivalenly y < p, and ha ondiion (16) hold. A in he abundan ae, y may be eiher larger or maller han p, he prie juifying a demand preiely equal o x, he upper bound of he exraion rae of he diry non renewable reoure when a he eiling. 4.1 And i i more expenive han he non-renewable reoure a he eiling In hi ae where p > y, he analyi i a ligh adapaion of he one developed for an abundan and expenive renewable reoure. The opimal pah are five phae pah illuraed in Figure 3. < Figure 3 here > 14

17 The wo fir phae, [, ) and [, ), are he ame han in he abundan ae: a fir phae oward he eiling followed by a phae a he eiling during whih he ole non-renewable reoure i exploied. The hird and fourh phae, [, y ) and [y, x ), are phae during whih he prie pah i he ame: p() = p 3 () defined by equaion (24). Thi i implied by he fa ha he non-renewable reoure i ued during boh phae. During he hird phae [, y ) he non-renewable reoure i ued alone. A ime y he prie reahe y and he renewable reoure beome ompeiive. During he fourh phae [ y, x ) he prie inreae o he level p. Boh reoure are ued now: y() = y and x() = q d (p 3 ()) y. A x, p 3 ( x ) = p and he demand i equal o y. The renewable reoure an aify he whole demand. The ok of non-renewable reoure i exhaued. The fifh and la phae [ x, ) i he phae of exluively lean energy: p() = p, y() = y and x() =. 4.2 Bu i i heaper han he non-renewable reoure a he eiling When y < p, he analyi i lighly more inriae han in he abundan ae. We mu ake are of he fa ha i may happen ha eiher x+y > ỹ or x + y < ỹ. Defining p a he marginal gro urplu generaed by a onumpion rae q = x+y, ha i p = u (x+y ), i may happen equivalenly ha eiher p < y or p > y. Taking for graned ha here mu exi a phae during whih he euriy onrain i aive, when p < y he energy prie mu be equal o he marginal o of he lean renewable ubiue during he phae a he eiling, while when y < p he energy prie mu be equal o p > y during he phae a he eiling. Alhough in boh ae he qualiaive properie of he prie pah are he ame, he mix of reoure are differen. In he fir ae he lean ubiue mu begin o be exploied a he dae a whih he eiling i aained, while in he eond ae i exploiaion mu begin before. Le u examine hee wo ae. 15

18 4.2.1 Alhough no heap enough o be exploied before he eiling In hi ae, p < y 4 Le u how why, when a he eiling, he energy prie mu be preiely equal o y. Aume fir ha, when a he eiling, he energy prie p i lower han y. Then he marginal o of he olar energy y i oo high o be ompeiive ine he marginal gro urplu of energy p = u (y) i lower han y. Thu if uh a prie p were o be he energy prie when a he eiling we hould have y() = and q() = x. Hene p = u (x) = p > y, a onradiion ine we have aumed ha p i lower han y. Nex aume ha when a he eiling he prie p i higher han y bu lower han p, he prie ha prevail when he lean ubiue i he only available upply. For uh a prie p he renewable reoure i ompeiive o ha y() = y. However, ine p < y < p < p, hen q d (p) < ỹ < x + y. Hene a uh prie q d (p) y < x o ha he ue of oal anno be aped by x, meaning ha he eiling onrain i no effeive. Hene again a onradiion. The only remaining poibiliy i ha a he eiling he prie i equal o y. A hi prie, he demand ỹ i aified by a mix of he onrained non-renewable reoure ue x() = x omplemened by a parial exploiaion of i renewable ubiue y() = ỹ x < y. Before he prie y i aained he oal i he only primary reoure having o be exploied. One he prie i higher han y he olar ubiue i fully exploied. We onlude ha he opimal pah are five phae pah, a illuraed in Figure 4. < Figure 4 here > The fir phae [, ) i he phae oward he eiling, wih a prie p() = p 1 () < y, and he non-renewable reoure i he only one o be exploied, ine i i he only one whih i ompeiive a a prie lower han y. A 4 Thu p < y < p. Noe ha p, whih i higher han y, may be eiher lower or higher han p. 16

19 he eiling i aained and imulaneouly he renewable reoure beome ompeiive: p 1 ( ) = y. The eond phae a he eiling i a phae a a prie p() = y during whih boh reoure are exploied. The lea oly reoure, he non-renewable one, mu be exploied a i maximal feaible rae when a he eiling: x() = x. The mo oly, he renewable one, fill he gap beween he demand a prie y, ỹ, and he non-renewable onrained upply x : y() = ỹ x. Sine y() < y, ome par of he renewable reoure poenial y i lef unexploied, explaining why no rariy ren ha o be impued for he exploiaion of he renewable reoure. The nex phae [, x ) i he phae during whih he prie inreae from y o p. Sine he eiling onrain i no more effeive and will never be effeive again, µ() = h/(ρ + α) and p() = p 3 () < p. Sine p() > y hen he renewable reoure ren γ y() = p 3 () y i now poiive and progreively inreae oward i long run level p y. For he ame reaon, ha i p() > y, he whole poenial of he renewable reoure mu be exploied: y() = y. Bu he demand i higher han y ine p() < p, and he omplemen required o balane he demand i upplied by he non-renewable reoure: x() = q d (p 3 ()) y. The exraion of he non-renewable reoure dereae down o zero a he end of hi phae, when p() = p. Then he non-renewable reoure mu be exhaued. The la phae [ x, ) i he lean energy regime. The yem of equaion deermining he endogenou variable i given in Appendix A And heap enough o be exploied before he eiling In hi ae, y < p 5 Now, a he eiling, he prie mu be equal o p. Le u how why. Aume ha a he eiling he prie p i lower han p, hen he onumpion q d (p) would have o be larger han wha a full mobilizaion of he boh reoure, x + y, ould provide ine p < p implie ha q d (p) > q d (p ) = x + y. For prie p higher han p he argumen run a in he preeden paragraph. For p > p hene p > y, he renewable reoure ha o be fully 5 Thu y < p < p. Noe ha p, whih i higher han p, may be eiher lower or higher han p. 17

20 exploied, y = y, while q d (p) < x + y. Thu we hould have x < x and he exploiaion rae of he non-renewable reoure would no be onrained. We onlude ha, when a he eiling, he only poibiliy i ha p = p. The opimal pah are five phae pah, illuraed in Figure 5. < Figure 5 here > The wo fir phae, [, y ) and [ y, ), are he phae oward he eiling. The prie pah i p 1 () beaue he non-renewable reoure i ued during he boh phae. Bu, a y, p 1 ( y ) = y and he renewable reoure beome ompeiive, alhough he eiling i no ye reahed. Thu, while he nonrenewable reoure i he only one o be exploied before y, afer y he upply beome a mix: x() = q d (p 1 ()) y and y() = y. A ime, p 1 ( ) = p and he eiling i reahed. Then begin he hird phae [, ) a he eiling. During hi phae p() = p, x() = x and y() = y. The fourh phae [, x ) i a phae during whih he prie i p3 (), beaue he non-renewable reoure i ill exploied and he eiling onrain will never beome effeive again. Hene he arbon hadow o i h/(ρ+α). A x, p 3 ( x ) = p and he whole demand an now be upplied by he renewable reoure. The non-renewable reoure i exhaued. A hi dae begin he la phae [ x, ) of lean energy. The yem of equaion deermining he endogenou variable i given in Appendix A. 5 Time profile of he arbon hadow o and omparaive dynami indued by more evere damage The qualiaive properie of he ime profile of he arbon hadow o µ() are he ame in all he enarii, he renewable reoure being eiher abundan or rare and eiher expenive or heap. A illuraed in Figure 6, he arbon hadow o iniially inreae up o a maximum level aained when 18

How Much Can Taxes Help Selfish Routing?

How Much Can Taxes Help Selfish Routing? How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU) Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from

More information

6.003 Homework #4 Solutions

6.003 Homework #4 Solutions 6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

More information

Endogenous Growth Models in Open Economies: A Possibility of Permanent Current Account Deficits

Endogenous Growth Models in Open Economies: A Possibility of Permanent Current Account Deficits Endogenou Growh Model in Open Eonomie: Poibili of Permanen Curren oun Defii Taiji HRSHIM Univeri of Tuuba Cabine Offie of Japan pril 5 Verion. bra The paper explore he impa of heerogenei in degree of relaive

More information

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs

Chapter 13. Network Flow III Applications. 13.1 Edge disjoint paths. 13.1.1 Edge-disjoint paths in a directed graphs Chaper 13 Nework Flow III Applicaion CS 573: Algorihm, Fall 014 Ocober 9, 014 13.1 Edge dijoin pah 13.1.1 Edge-dijoin pah in a direced graph 13.1.1.1 Edge dijoin pah queiong: graph (dir/undir)., : verice.

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

How has globalisation affected inflation dynamics in the United Kingdom?

How has globalisation affected inflation dynamics in the United Kingdom? 292 Quarerly Bullein 2008 Q3 How ha globaliaion affeced inflaion dynamic in he Unied Kingdom? By Jennifer Greenlade and Sephen Millard of he Bank Srucural Economic Analyi Diviion and Chri Peacock of he

More information

How to calculate effect sizes from published research: A simplified methodology

How to calculate effect sizes from published research: A simplified methodology WORK-LEARNING RESEARCH How o alulae effe sizes from published researh: A simplified mehodology Will Thalheimer Samanha Cook A Publiaion Copyrigh 2002 by Will Thalheimer All righs are reserved wih one exepion.

More information

Transient Analysis of First Order RC and RL circuits

Transient Analysis of First Order RC and RL circuits Transien Analysis of Firs Order and iruis The irui shown on Figure 1 wih he swih open is haraerized by a pariular operaing ondiion. Sine he swih is open, no urren flows in he irui (i=0) and v=0. The volage

More information

Topic: Applications of Network Flow Date: 9/14/2007

Topic: Applications of Network Flow Date: 9/14/2007 CS787: Advanced Algorihm Scribe: Daniel Wong and Priyananda Shenoy Lecurer: Shuchi Chawla Topic: Applicaion of Nework Flow Dae: 9/4/2007 5. Inroducion and Recap In he la lecure, we analyzed he problem

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Endogenous Growth Practice Questions Course 14.451 Macro I TA: Todd Gormley, tgormley@mit.edu

Endogenous Growth Practice Questions Course 14.451 Macro I TA: Todd Gormley, tgormley@mit.edu Endogenous Grow Praie Quesions Course 4.45 Maro I TA: Todd Gormley, gormley@mi.edu Here are wo example quesions based on e endogenous grow models disussed by Marios in lass on Wednesday, Mar 9, 2005. Tey

More information

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides 7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

More information

The Transport Equation

The Transport Equation The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

More information

Economics Honors Exam 2008 Solutions Question 5

Economics Honors Exam 2008 Solutions Question 5 Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I

More information

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction

On the Connection Between Multiple-Unicast Network Coding and Single-Source Single-Sink Network Error Correction On he Connecion Beween Muliple-Unica ework Coding and Single-Source Single-Sink ework Error Correcion Jörg Kliewer JIT Join work wih Wenao Huang and Michael Langberg ework Error Correcion Problem: Adverary

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

Circuit Types. () i( t) ( )

Circuit Types. () i( t) ( ) Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All

More information

Understanding Sequential Circuit Timing

Understanding Sequential Circuit Timing ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

More information

Solving Equations. PHYSICS Solving Equations. solving equations NOTES. Solving for a Variable. The Rules. The Rules. Grade:«grade»

Solving Equations. PHYSICS Solving Equations. solving equations NOTES. Solving for a Variable. The Rules. The Rules. Grade:«grade» olving equaion NOTES New Jerey ener for Teaching an Learning Progreive Science Iniiaive Thi maerial i mae freely available a www.njcl.org an i inene for he non commercial ue of uen an eacher. Thee maerial

More information

O θ O ' X,X ' Fig. 11. Propagation of a flat wave

O θ O ' X,X ' Fig. 11. Propagation of a flat wave Y Y P 47.4. Doppler s effe Professor Tonnela s mehod The Doppler s effe is manifesed in he hange in he veloi of he signals refleed (radiaed) b moving objes [5] [6] [8] [39] [43] [45]. I lies in he basis

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years.

What is a swap? A swap is a contract between two counter-parties who agree to exchange a stream of payments over an agreed period of several years. Currency swaps Wha is a swap? A swap is a conrac beween wo couner-paries who agree o exchange a sream of paymens over an agreed period of several years. Types of swap equiy swaps (or equiy-index-linked

More information

Experimental Proposal for Determination of One-Way Velocity of Light with One Single Clock. JR Croca

Experimental Proposal for Determination of One-Way Velocity of Light with One Single Clock. JR Croca Experimenal Proposal for Deerminaion of One-Way Veloiy of igh wih One Single Clok JR Croa Deparmen of Physis Fauldade de Ciênias da Universidade de isboa Campo Grande, Ed C8 700 isboa, Porugal email: roa@f.ul.p

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

C Fast-Dealing Property Trading Game C

C Fast-Dealing Property Trading Game C AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Section 7.1 Angles and Their Measure

Section 7.1 Angles and Their Measure Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed

More information

WHAT ARE OPTION CONTRACTS?

WHAT ARE OPTION CONTRACTS? WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

More information

AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 2010 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College

More information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

More information

Option Put-Call Parity Relations When the Underlying Security Pays Dividends

Option Put-Call Parity Relations When the Underlying Security Pays Dividends Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 225-23 Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,

More information

Dividend taxation, share repurchases and the equity trap

Dividend taxation, share repurchases and the equity trap Working Paper 2009:7 Deparmen of Economic Dividend axaion, hare repurchae and he equiy rap Tobia Lindhe and Jan Söderen Deparmen of Economic Working paper 2009:7 Uppala Univeriy May 2009 P.O. Box 53 ISSN

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004 HUT, TUT, LUT, OU, ÅAU / Engineeing depamens Enane examinaion in mahemais May 5, 4 Insuions. Reseve a sepaae page fo eah poblem. Give you soluions in a lea fom inluding inemediae seps. Wie a lean opy of

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 Introduction to Mathematical Finance Lesson 5 26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

More information

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

More information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

The Chase Problem (Part 2) David C. Arney

The Chase Problem (Part 2) David C. Arney The Chae Problem Par David C. Arne Inroducion In he previou ecion, eniled The Chae Problem Par, we dicued a dicree model for a chaing cenario where one hing chae anoher. Some of he applicaion of hi kind

More information

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics

2.4 Network flows. Many direct and indirect applications telecommunication transportation (public, freight, railway, air, ) logistics .4 Nework flow Problem involving he diribuion of a given produc (e.g., waer, ga, daa, ) from a e of producion locaion o a e of uer o a o opimize a given objecive funcion (e.g., amoun of produc, co,...).

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Differential Equations and Linear Superposition

Differential Equations and Linear Superposition Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y

More information

Having an average input power of 1 Watt, compute the S/N ratio at the output of the link, after 4 Km.

Having an average input power of 1 Watt, compute the S/N ratio at the output of the link, after 4 Km. Exerise 1 A miro-oaxial able wih aenuaion A(f) = 9 onnes wo elephone saions. The able arries a PCM binary signal a 4 Mbi/s and roll-off faor of he raised osine filer equal o 0.5. The able is a a noise

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

More information

1. The graph shows the variation with time t of the velocity v of an object.

1. The graph shows the variation with time t of the velocity v of an object. 1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Present Value Methodology

Present Value Methodology Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

More information

11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. Here we always work with relative pressure. That s what everybody always does. 11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

More information

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average

Optimal Stock Selling/Buying Strategy with reference to the Ultimate Average Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July

More information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate Macro Theory II: Notes on Neoclassical Growth Model Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

More information

Market. Recourse. Yan Feng

Market. Recourse. Yan Feng The Value o Sale and Operaion Planning in Oriened Srand Board Indury wih Make-o-Order Manuauring Syem: Cro Funional Inegraion under Deerminii Demand and Spo Marke Reoure Yan Feng Sophie D Amour Rober Beauregard

More information

SAMPLE LESSON PLAN with Commentary from ReadingQuest.org

SAMPLE LESSON PLAN with Commentary from ReadingQuest.org Lesson Plan: Energy Resources ubject: Earth cience Grade: 9 Purpose: students will learn about the energy resources, explore the differences between renewable and nonrenewable resources, evaluate the environmental

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

More information

Lecture III: Finish Discounted Value Formulation

Lecture III: Finish Discounted Value Formulation Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

More information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions to Nonfarm Payroll Employment: 1964 to 2011 Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

When Can Carbon Abatement Policies Increase Welfare? The Fundamental Role of Distorted Factor Markets

When Can Carbon Abatement Policies Increase Welfare? The Fundamental Role of Distorted Factor Markets When Can Carbon Abaemen Poliies Inrease Welfare? The undamenal Role of Disored aor Markes Ian W. H. Parry Roberon C. Williams III awrene H. Goulder Disussion Paper 97-18-REV Revised June 1998 1616 P Sree,

More information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1

Single-machine Scheduling with Periodic Maintenance and both Preemptive and. Non-preemptive jobs in Remanufacturing System 1 Absrac number: 05-0407 Single-machine Scheduling wih Periodic Mainenance and boh Preempive and Non-preempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy

More information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and Decay Functions f (t) = C(1± r) t / K, for t 0, where MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

More information

2.5 Life tables, force of mortality and standard life insurance products

2.5 Life tables, force of mortality and standard life insurance products Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n

More information

Cross-sectional and longitudinal weighting in a rotational household panel: applications to EU-SILC. Vijay Verma, Gianni Betti, Giulio Ghellini

Cross-sectional and longitudinal weighting in a rotational household panel: applications to EU-SILC. Vijay Verma, Gianni Betti, Giulio Ghellini Cro-ecional and longiudinal eighing in a roaional houehold panel: applicaion o EU-SILC Viay Verma, Gianni Bei, Giulio Ghellini Working Paper n. 67, December 006 CROSS-SECTIONAL AND LONGITUDINAL WEIGHTING

More information

AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 2013 Scoring Guidelines AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

More information

Efficient Risk Sharing with Limited Commitment and Hidden Storage

Efficient Risk Sharing with Limited Commitment and Hidden Storage Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing

More information

Distributing Human Resources among Software Development Projects 1

Distributing Human Resources among Software Development Projects 1 Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources

More information

Chapter 6: Business Valuation (Income Approach)

Chapter 6: Business Valuation (Income Approach) Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

I. Basic Concepts (Ch. 1-4)

I. Basic Concepts (Ch. 1-4) (Ch. 1-4) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

1 HALF-LIFE EQUATIONS

1 HALF-LIFE EQUATIONS R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

Motion Along a Straight Line

Motion Along a Straight Line Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

More information

Circle Geometry (Part 3)

Circle Geometry (Part 3) Eam aer 3 ircle Geomery (ar 3) emen andard:.4.(c) yclic uadrilaeral La week we covered u otheorem 3, he idea of a convere and we alied our heory o ome roblem called IE. Okay, o now ono he ne chunk of heory

More information

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton

CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA. R. L. Chambers Department of Social Statistics University of Southampton CHAPTER 11 NONPARAMETRIC REGRESSION WITH COMPLEX SURVEY DATA R. L. Chamber Deparmen of Social Saiic Univeriy of Souhampon A.H. Dorfman Office of Survey Mehod Reearch Bureau of Labor Saiic M.Yu. Sverchkov

More information

Government late payments: the effect on the Italian economy. Research Team. Prof. Franco Fiordelisi (coordinator)

Government late payments: the effect on the Italian economy. Research Team. Prof. Franco Fiordelisi (coordinator) Governmen lae paymens: he effe on he Ialian eonomy Researh Team Prof. Frano Fiordelisi (oordinaor) Universià degli sudi di Roma Tre, Ialy Bangor Business Shool, Bangor Universiy, U.K. Dr. Davide Mare Universiy

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

Two Compartment Body Model and V d Terms by Jeff Stark

Two Compartment Body Model and V d Terms by Jeff Stark Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

More information

CHAPTER FIVE. Solutions for Section 5.1

CHAPTER FIVE. Solutions for Section 5.1 CHAPTER FIVE 5. SOLUTIONS 87 Soluions for Secion 5.. (a) The velociy is 3 miles/hour for he firs hours, 4 miles/hour for he ne / hour, and miles/hour for he las 4 hours. The enire rip lass + / + 4 = 6.5

More information

A Mathematical Description of MOSFET Behavior

A Mathematical Description of MOSFET Behavior 10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

More information

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

Full-wave rectification, bulk capacitor calculations Chris Basso January 2009 ull-wave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal

More information