15.023J / J / ESD.128J Global Climate Change: Economics, Science, and Policy Spring 2008
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1 MIT OpenCourseWare hp://ocw.mi.edu J / J / ESD.128J Global Climae Change: Economics, Science, and Policy Spring 2008 For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.
2 Massachuses Insiue of Technology ESD.128 Climae Change: Economics, Science and Policy Problem Se #2 MODELING ECONOMIC GROWTH AND EMISSIONS Due Monday, 17 March THE ASSIGNMENT In his assignmen you will apply a spreadshee-based model of global CO 2 emissions o prepare 100-year projecions of economic growh and carbon emissions, and o carry ou sudies of emissions resricion. The purpose of he exercise is o gain insigh ino he ways such models are consruced and applied. The model used here is simplified by ignoring he influence of governmen axaion and expendiure, and inernaional rade, and by implemening a simple srucure of producion and inpu pricing. The model is laid ou in Secion 2 along wih parameer values. A emplae wih mos of he equaions already wrien in is provided for download on he course websie. Columns A hrough Q are used in his homework se. You will work wih boh recursive-dynamic (myopic) and forward looking versions. Analyses o be conduced are described in Secion 3. Secion 4 describes how he emplae workshee was se up, and Secion 5 provides some workshee hins. There are nine pars: A1, A2, B1, B2, B3, C1, C2, C3 and D. You may find i convenien o include prin-ous of he solved workshees in your answers. 2. THE EMISSIONS PREDICTION MODEL 1. The Model of Economic Growh and Emissions Naional oupu is modeled as a funcion of hree facors: labor, capial and energy. The equaions for labor and capial are similar o hose discussed in class on 3 March. Carbon dioxide emissions are modeled as proporional o energy consumpion, and he emissionsmiigaion policy insrumen is direc conrol of he level of energy use. Analysis of oher measures ha migh be used in emissions conrol, such as axes and subsidies, is no possible wih his formulaion. Economic variables are saed on an annual basis ($10 12 /yr). The model is solved on a 10-year ime sep.
3 Homework #2 p GDP Growh World GDP is represened by a single good, y, which is modeled as a funcion of inpus of capial, k, labor, l, and energy, e. The unis of y are rillion (10 12 ) $US. The oupu of he economy is represened by a consan-elasiciy-of-subsiuion producion funcion as follows, ρ y = a α k KLE ρ + β l KLE ρ [ + (1 α β) e KLE where he elasiciy of subsiuion, σ KLE, is 1 σ KLE =. 1 ρ KLE The labor force grows a an annual rae λ, 0 (1 + λ 1/ρ KLE (1) l = l ). (2) The capial sock changes over ime as a resul of depreciaion and new invesmen. The magniude of he capial sock is defined as of he firs year of he ime sep, and he level in each subsequen ime sep is a funcion of he depreciaion rae, δ, and he invesmen during he period, i, k ) n + 1 = n i + (1 δ k (3) where n is he number of years in each period of he model s soluion. All-facor produciviy change augmens oupu a an annual rae γ. a = a ), (4) 0 (1 + γ where his shif parameer a has an iniial level a Income, Savings, and Invesmen The oupu of he economy, ne of he coss of supplying energy, becomes he income received by a represenaive global consumer. The relaionship among hese facors is defined by he following accouning ideniy: y = c + s + (pe e ), (5) where pe is he price of energy. Also, all savings are invesed, s = i. (6) ]
4 Homework #2 p. 3 Savings behavior is modeled differenly depending on he version of he model you are working wih. In he recursive-dynamic version, saving is deermined as a fixed fracion, η, of income, s = η (7) y In he forward-looking version, saving is a variable deermined in he muli-period opimizaion, explained below. 1.3 The Cos of Energy Energy use e is measured in exajoules (EJ = J). The marginal cos of energy (and herefore he price) is deermined by he depleion of naural resources. As cumulaive energy use rises, he cos of energy per EJ increases as follows where ν r μ pe = (8) r cume r is a consan represening he fixed amoun of in-ground resources, cume, cumulaive energy consumpion, evolves according o cume = n e + cume cume 0 +1, = 0 (9) μ and ν are esimaed parameers. 1.4 Carbon Accouning The carbon inensiy of energy, ε, is fixed over ime. Emissions of CO 2, denoed m, are saed in gigaons (GT = 10 9 ons) of carbon per year. They are calculaed by muliplying he carbon inensiy of energy by he amoun of energy consumed: m = ε e (10) 1.5 Parameer Values The parameer values are: n = 10 a 0 = 0.8 r = EJ α = 0.25 β = 0.65 δ = 0.05 (number of years per period) (iniial value of echnological shif facor) (energy resource base) (capial share in producion funcion) (labor share in producion funcion) (annual depreciaion rae of capial)
5 Homework #2 p. 4 ε = GT/EJ μ = 0.01 ν = 1 σ = 1.25 ρ = (carbon coefficien on energy use) (slope parameer in energy cos funcion) (exponen parameer in energy cos funcion) (elasiciy of subsiuion) ( σ 1) / σ = 0.2 (subsiuion parameer in producion funcion) θ = 0.03 λ = 0.01 γ = 0.01 (annual discoun rae) (annual growh rae of labor supply) (annual growh rae of all facor produciviy change) The iniial-year values of he variables in he model are he following: l 0 = k 0 = 130 e 0 = 600 (10 9 worker-hours) (10 12 $US) (EJ) The ime = 0 is he year 2000, and he spreadshee provided for Pars A, B, and C of he homework se is already se up in 10-year periods. 3. TASKS WITH THE MODEL You are o prepare calculaions wih wo versions of his model, and use i o predic global emissions under various assumpions. For each version, please hand in 1. A copy of each workshee where appropriae. 2. Plos of he ime pahs of consumpion, carbon emissions, and amospheric concenraions. Feel free o plo any oher variables ha you find of ineres. 3. Commens on he soluions as requesed below. Once you have se up he workshee, you may wan o explore he effec on he resuls of alernaive values of key parameers, e.g., γ, η, σ KLE. Noe ha he model may no solve if you depar far from he reference parameers specified above. Par A. Tasks wih he Recursive-Dynamic Version The Excel emplae incorporaes a myopic version of he model, and shows a projecion of global CO 2 emissions up o 2100 (column N). This version assumes ha energy use grows from is year 2000 level of 600 EJ according o an elasiciy relaionship wih GDP, y, wih a one-period lag (column F). Tha is,
6 Homework #2 p. 5 e e 1 = 0.5 = 0.5 χ [ e y ] χ [ e ] 0 y where χ = 0.25 is he elasiciy of energy use o oupu. Solve his model in separae cases for Problem A1. A higher rae of produciviy growh (γ=0.015), and Problem A2. A lower rae of labor force growh (λ=0.005). (Reurn γ o is original value of 0.01 before running his case.) Commen briefly on he way hese changes influence he oher variables in he model. Par B. A Tasks wih he Forward-Looking Version Solve he model assuming ha he level of energy use (he conrol variable) is se over all periods o maximize he presen value of welfare. In any period, he index describing welfare is W = ln(c ). This funcional form, (column K) implemens an assumpion ha he marginal uiliy of consumpion declines as consumpion rises. Thus he funcion o be maximized is 1 max PV ( W ) = ln c (11) 1+ θ where θ is he uiliy discoun rae (rae of ime preference), which is se o θ = 0.03 on an annual basis in he workshee. The model maximizes equaion (11) by choosing he pah of e. The iniial value of his variable is fixed, wih e 0 = 600 (EJ). Problem B1. Prepare a reference ( business as usual ) scenario on he assumpion here is no carbon consrain. (Make sure produciviy growh and labor force growh are rese (γ=0.01; λ=0.01). Problem B2. Prepare a conrol scenario applying an inensiy arge of he form announced by he Bush Adminisraion and some developing counries. Assume he energy inensiy of GDP (m/y) declines by 15% per decade, beginning from he inensiy in he 2000 base year. (Hin: Se up he inensiy consrain in wih P24-P33 <= Q24-Q33 wih he limi se up in he Q column.) Solver is sensiive o saring condiions here. If you ge repeaed errors, reload he emplae and sar again.) Problem B3. Consruc a curve of marginal cos, defined in erms of los (nondiscouned) consumpion, for global emissions reducions of up o 4 GC, for he periods 2010 and This may be done by solving he model for reducions below reference values (from Problem B1) of 1.0, 2.0, 3.0, and 4.0 GC and hen ploing he marginal decrease in consumpion. Commen briefly on he shapes of he curves. Remember ha
7 Homework #2 p. 6 consumpion is in unis of rillions of dollars, and so a 0.1 change in he consumpion level is acually 100 billion dollars. (Noe also ha, because of is simpliciy, his model will yield coss higher han you will see in oher analyses, and a non-zero inercep.) Par C. Sabilizaion of Amospheric CO 2 Concenraion A simple model of CO 2 accumulaion (in GC) in he amosphere is where and n skm = n m skm 1 (12) τ e skm 0 = GC (carbon in amosphere in 2000) τ e = 120 years (e-folding ime of carbon in he amosphere). The amospheric concenraion, in ppmv, is hen conc = skm / The emplae already includes wo columns for he calculaion of amospheric concenraions of CO 2, and a figure o display he resuls is provided in he workshee. Implemen hese equaions and conduc he following exercises using he forward-looking model. Problem C1. Solve for he paern of energy use (and associaed carbon emissions) ha sabilizes amospheric carbon concenraions a or below 550 ppmv for every period of he model. Wha is he yearly carbon upake in 2100 under his scenario? Commen on he model assumpion abou erresrial and ocean carbon upake. Problem C2. Solve he same case as C1, only wih he discoun rae reduced from 0.03 o Commen briefly on he difference his makes o he soluion. (Do no forge o rerun he solver, oherwise he opimizaion doesn happen and all you do is change he ne presen value.) Problem C3. Solve he C1 case again (i.e., <= 550ppmv for all periods), only assume ha because of poliical consrains he level of energy use canno be reduced more rapidly han 10% per decade. Commen briefly on how he soluions differ. Par D. How o Improve he Model Lis hree addiional feaures you would add o he model o make i more realisic, commening briefly on 1. Which you believe are he mos imporan for undersanding he climae issue? 2. Wha problems (daa, analysis mehods) would arise in implemenaion? 3. Are here issues ha are imporan ha his ype of model would never be able o address, even wih significanly more complexiy?
8 Homework #2 p SETUP OF THE SAMPLE WORKSHEET TEMPLATE The problem is se up wih 11 en-year periods, * = 0,...,10. For each period, * he value of he economic variables (y, k, l, c, i) are se a he appropriae annual levels of he firs year in he 10-year calculaion sep. Tha is, y(* = 0) y( = 0) y(* = 1) y( = 10) ec. Thus equaions (3), (9) and (12) assume ha in he firs year of he 10-year ime sep invesmen, energy and emissions (respecively) apply for all years of ha sep, so ha hese (annual) quaniies mus be muliplied by n = 10. Similarly, in equaions (3) and (12) he annual survival facors (1 - δ) and (1-1/τ e ) mus be updaed over an enire decade, which is accomplished by raising hem o he power n. 5. WORKSHEET HINTS 1. Before beginning work save a maser copy of he workshee (o be lef unchanged) and se up a working version. You may need o exi and sar over and his will save you having o go back o he web. 2. This noe is wrien as if he dynamic version of he model is o be solved in Excel using Solver. Equivalen faciliies should be available in oher spreadshee programs. The policies limiing emissions and/or concenraions mus be enered as consrains in Solver. 3. When solving he dynamic model, bounds someimes mus be placed on values of he variables o ensure ha Solver finds an opimum ha in fac makes economic sense. If you run ino problems ry ensuring ha energy use, energy prices, consumpion and invesmen canno be negaive. Rule ou hese possibiliies by adding he consrains e 0, pe 0, c 0 and i 0 for all. 4. When solving he dynamic model under consrain, always re-run he unconsrained version before running wih a differen consrain. Solver does no always move accuraely from one consrained soluion o anoher. Also, someimes Solver ges suck (e.g., when confroned wih a very resricive consrain), and canno reurn o he noconsrain soluion. Therefore, you should always run an unconsrained version of any problem before moving from one consrained soluion o anoher. 5. Someimes Solver reurns a soluion ha is no an opimum. If you run ino his problem, ofen you can recover by limiing he zone of search for he opimal soluion and performing successive solves, using he soluion of one as he iniial poin for he nex. This process should converge o a sensible answer.
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