Seminar 5 Sustainability

Transcription

1 Seminar 5 Susainabiliy Soluions Quesion : Hyperbolic Discouning -. Suppose a faher inheris a family forune of 0 million NOK an he wans o use some of i for himself (o be precise, he share ) bu also o beques some of i o his chil (he share 2 ) an granchil (he share 3 ). He hinks abou he problem in he following way: m x U = ln + β(ln 2 + ln 3 )., 2, 3 Hence he Lagrangian is: L = ln + β(ln 2 + ln 3 ) + λ( ) Involving he following coniions for opimaliy: = β 2 2 = 3 = So ha 2 = β an hence = + 2β hus = ( + 2β), so ha = ( + 2β) The iscoun facor which recovers he sharing scheme 4 0, 3 0, 3 is hen β = The son oes no respec he las will of his ol man bu ivies he remaining family forune beween himself an his chil accoring o he following valuaion: U = ln 2 + β ln 3 The uiliy funcion of he faher has no been explici in quesion, an wihou i he problem couln really have been solve. I am very sorry abou ha.

2 Involving he following coniions for opimaliy: 2 = β = So ha 3 = β 2 an hence 0.6 = 2 + β 2 which, wih β = 0.75 implies: 2 = 2 35 = = 9 35 = We are looking for he subgame-perfec soluion of he game beween he firs wo generaions. As he faher akes he acion of his son ino accoun we nee o solve U ( 2 ) = βu ( 2 ) for 2 = BR( ). In our specific case his is: = = 4 7 ( ) Subsiue his ino he original problem of he faher: m x U( ) + β[u(br( )) + U( BR( ))] wih he necessary coniion: U ( ) = β [U ( BR( )) U (BR( ))]BR ( ) + U ( BR( )). Which, in our specific case yiels: = ( ) 4 7 ( ) 4 which can be solve for he following shares of, 2 an 3 : = 3 = = 8 2 = = ( ) 28 =

3 Quesion 2: Opimal growh wih a naural resource consrain Aggregae proucion funcion wih echnological progress M : Y = ƒ (, R ) = K α (M R ) α () Noe ha echnological progress here is resource augmening an ha he proucion funcion has consan reurns o scale in an M R. 2- Show ha he ineres rae in he economy, i.e. he marginal prouciviy of capial, is consan provie MR is consan.: K Y M = αk α (M K R ) α R α = α = r (2) Since α is a consan parameer, his is consan (r = r) if M R is consan over ime. Noe ha he coniion can be rewrien in he following way ha comes in hany laer: Y = α Y = r (3) 2-2 Show ha when he ineres rae is consan, he resource price, i.e. he marginal prouciviy of he resource, is proporional o M.: Y = ( α)k α R M α R α = ( α) K M R α M = p (4) is proporional o M if r consan over ime, i.e. if M R is consan. In ha case, every increase in M over ime has he same effec on he marginal prouciviy of he resource, irrespecive of. 2-3 Explain wha is mean by ineremporal efficiency. Wih regars o he resource, ineremporal efficiency requires ha lim S = 0 an he Hoelling rule hols: ṗ p = r Y R = r (5) Y R Ṁ Ṙ α + ( α) α = r (6) M R In microeconomic erms, he Hoelling rule was an arbirage rule: he resource ren ha o grow a a rae equal o he ousie opion growh rae he rae of ineres. In 3

4 macroeconomic erms, ha implies ha he growh rae in he marginal prouciviy of he resource mus be equal o he marginal prouciviy of capial. Noe ha he oher ineremporal efficiency coniion you have encounere (an will soon mee again), he Ramsey rule, oes no apply here: in his ype of moel, we are given an exogenous savings rae an hence he Ramsey rule (a rule for he opimal ineremporal allocaion of capial, i.e. he savings rae) oes no apply. 2-4 Assume ha M grows a a consan rae m > 0. Show ha in his case i is possible o have an ineremporally efficien growh pah along which he ineres rae is consan an oupu grows a a consan rae g. The approach here is o ask: Given ha M grows a a consan rae, he ineres rae is consan an oupu grows a a consan rae g, is i possible o have an ineremporally efficien growh pah? Firs, as we have seen a consan ineres rae implies ha M R is consan. From he proucion funcion () i hen follows ha if oupu grows a a consan rae g, so oes : M Y = K α (M R ) α R α = Seconly, we see from (3) ha if echnology grows a a consan rae m, hen so oes he marginal prouciviy of he resource Y R. As a hir sep, noice ha he Hoelling rule in (5) hen implies ha r = m. Explicily wriing ou he iffereniaion as one in (6) an insering wha we foun so far we hen ge he following: Ṙ αg + ( α)m α = r = m R Ṙ αg αm = α R Ṙ g m = (7) R Noe ha, by he Hoelling rule, resource exracion is eclining. Denoe Ṙ = which R is consan. For o be negaive, we mus have m > g. Also we see ha resource exracion formally nees o go on forever. 2-5 Wha is he ineres rae along he pah escribe in (2-4)? As we have seen above an ineremporally efficien growh pah in his paricular moel requires r = m. In wors: he consan marginal prouciviy of capial equals he exogenous growh rae in he echnology level. 4

5 2-6 Wha is he relaionship beween he growh rae g an he saving rae s = K Y along he pah escribe in (2-4)? We have: g = = Y Y = s Y Making use of (3) an m = r: g = s r α = s m α (8) In sanar growh moels wihou a naural resource, he balance growh pah is inepenen of he saving rae. This is no he case wih an exhausible resource. Raher we ge ha he higher he saving rae, he higher he growh in oupu. This can also be seen by urning (8) aroun: he growh rae of he economy is exogenous an consan, as is he savings rae an he capial share of proucion. s = g α r = g (9) Y 2-7 (a) Derive he coniions for he opimal oucome. Wha is ifferen now is ha we o no have an exogenous saving rae bu ha we eermine he opimal saving/invesmen in he moel. Also, we are no only ou afer some ineremporally efficien pah bu he socially opimal pah. The opimizaion problem is: m x W = C,R 0 (C )e ρ. subjec o: = ƒ (, R ) C. an Ṡ = R an S 0 = 0 R (0) The curren value Hamilonian for his problem is: H = (C ) + μ[ƒ (, R ) C ] λr. 5

6 Necessary coniions for an opimum inclue: (C ) μ 0 (= 0 for C > 0), () μƒ R (, R ) λ 0 (= 0 for R > 0), (2) μ = μρ μƒ K (, R ) (3) λ = λρ. (4) Then, as in class, iffereniae () wih respec o : μ = (C) an inser his an () ino (3): (C) = (C)ρ (C)ƒ (K, R). This can be re-wrien as Ramsey rule: K r = ρ + η C (5) A his poin, noe ha he consumpion growh rae along he balance growh pah C equals g. This can be seen from he following ransformaions: Y = K + C C = Y K an hence = Ẏ K. As K K = g K = gk we have K = ġk + g K = g K. Puing hings ogeher yiels: C = Ẏ g K Y K (Y K) C = Ẏ g K = gy g K = g(y K) C = g In aiion, he efficiency coniions of he Hoelling rule (5) an lim S = 0 mus hol. 2-7 (b) Show uner wha coniions a growh pah of he ype escribe in (2-4) is opimal (assume ha η = C is consan). When such a growh pah exiss, how oes he growh rae epen on he parameers in he welfare funcion? Ineremporal efficiency is a preconiion. From all he ineremporally efficien growh pahs, we now choose one. We sill have ha r = m. Then rewrie he Ramsey rule as: m = ρ + ηg g = m ρ η Compare his o he coniion erive in (7): g m = 6

7 Where we foun ha i mus be rue ha m > g. To have he same in he socially opimal soluion, in he Ramsey rule ha implies: m > g an g = m ρ m > m ρ η mη > m ρ ρ > m( η) η Since ρ 0 by assumpion, his is always rue if η >. If η <, hen i is only rue if: ρ m < ( η) 7

8 Quesion 3: Discouning an Susainabiliy Suppose uiliy epens on an aggregae consumpion goo C an an aggregae environmenal goo E, an i is given by he specific form (C, E) = ln C γ E γ 3-. Show ha he consumpion iscoun rae r C is given by r C = ρ + g C. where wih r C = ρ C(C, E) C (C, E) C = CC + CEĖ C C = CCC C C + CEE Ė C E C = g C Ė E = g E C = ( γ)c γ E γ C γ E γ CC = γ C 2 CE = 0 = γ C so ha C ( γ)c = C C 2 r C = ρ + g C C γ g C 3-2. Wha is he iscoun rae for he environmenal goo r E? Similar o above we have r E = ρ E(C, E) E (C, E) 8

9 where E = EEĖ + EC E E = EEE E Ė E + ECC E C wih C = g C Ė E = g E E = γc γ E γ C γ E γ = γ E EE = γ E 2 EC = 0 so ha E = γe E E E 2 γ g E r E = ρ + g E 3-3. Which of hese wo raes is likely o be larger? Discuss. r C > r E whenever g C > g E, which is he mos plausible scenario. In oher wors, we fin ha environmenal goos have o be iscoune a a lower han average rae an consumpion goos have o be iscoune a a higher han average rae. In oher wors, he environmen s conribuion o fuure uiliy is becoming relaively more imporan. 2 2 Recall ha he quesion is how much of goo o I nee o have omorrow in orer o accep a one uni reucion oay. If r C is higher han r E ha means ha you have o give me unis of consumpion goo +r C omorrow (e.g. 8/0 when r C = 0.25) an ha you have o give me unis of he environmenal goo +r E omorrow (e.g. 9/0 when r E = 0.). 9