SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: 0.038/NCLIMATE893 Temporal resoluion and DICE * Supplemenal Informaion Alex L. Maren and Sephen C. Newbold Naional Cener for Environmenal Economics, US Environmenal Proecion Agency 00 Pennsylvania Ave., Washingon, DC 0460, USA Inegraed assessmen models (IAM), along he lines of DICE, are highly simplified represenaions of complex amospheric and economic sysems used o capure imporan (bi-direcional) ineracions beween he sysems ha are relevan for undersanding efficien climae policy design. In choosing he mahemaical represenaions of hese sysems model developers have muliple opions. One approach would be o adop dynamics represened by a sysem of differenial equaions ha are believed o capure he general behaviour of he sysem and whose parameers can be direcly relaed o known characerisics of he underlying coninuous sysem. However, he sysem of coninuous ime differenial equaions necessary o relae he relevan sae variables of he IAM o known characerisics of he underlying sysem may be oo complex o include wihin he IAM (eg, including a global circulaion model wihin an IAM o be opimized over policy variables). Therefore, model developers may choose an alernaive approach hey adop a se of reduced form expressions whose parameers have a more absrac meaning and are calibraed o oupu from more complex models ha do aemp o direcly model he behavior of he sysem based on firs principles. In he firs approach one would represen he behavior of he underlying sysem using a se of differenial equaions dx d f X, N X,, () N N f:, and is a vecor of M parameers relaed o physical characerisics of he underlying sysem. In he likely case ha a closed form soluion does no exis for he sae * The views expressed in his aricle are hose of he auhors and do no necessarily reflec he views or policies of he U.S. Environmenal Proecion Agency. NATURE CLIMATE CHANGE 03 Macmillan Publishers Limied. All righs reserved.

2 variables in () he modeler may implemen a numerical approximaion o he soluion. For simpliciy of exposiion, le us consider a basic explici forward finie difference approximaion such ha X X f X,, () is he size of he ime sep used in he approximaion. Puing aside round-off error in he compuaion of (), he discreizaion error associaed wih he approximaion will be proporional o he sep size. In he second approach he dynamics wihin () are no direcly used, and insead he model developer adops reduced form approximaion Y g Y,, (3) K Y, K K g:, K N, and is a se of L parameers. The dimensionaliy/complexiy of he reduced form approximaion in (3) is chosen such ha i conains he sae variables in X ha are relevan for he IAM and so ha he expression once calibraed is robus enough o approximae he behavior in he more complee sysem in () wih an accepable level of accuracy. In his reduced form expression he parameers in are no longer direcly relaed o he physical characerisics of he underlying sysem as hey were in (). Insead hese parameers,, are calibraed by minimizing he difference beween common sae variables in X and Y for a series of runs by more complex models ha may be using he firs approach o model he sysem in quesion. The value of he calibraed parameers,, will herefore be condiional on he size of he ime sep,, chosen in (3). Therefore, if one were o change he ime sep in his reduced form expression, hey would also be required o recalibrae he parameers for he new ime sep o ensure ha he funcion sill provides an adequae approximaion. According o he model s documenaion (Nordhaus, 007), he climae model in DICE007 was calibraed using his second approach. The carbon cycle is approximaed using a simple hree box sysem M M E, (4) E [G C] represens he oal global carbon emissions in year, T 0 0, is he 3x3 ransiion marix, and he ime sep is se o en years. The firs elemen of M [G C] denoes he amospheric carbon concenraion and he oher wo elemens are used o represen carbon sinks. According o he DICE007 source code obained from he websie of William Nordaus (hp://nordhaus.econ.yale.edu), he ransiion coefficien marix is consrained such ha 03 Macmillan Publishers Limied. All righs reserved.

3 (5) and are parameers o be calibraed., The carbon concenraion in he amosphere is assumed o affec global radiaive forcing, F [W m - ] hrough he relaionship M ln M pre EX F F ln, (6) EX M pre [G C] is he pre-indusrial amospheric carbon concenraion and F [W m - ] are exogenous forcings due o oher climae componens no explicily included in he model. The model considers a an approximaion of he climae describing only he global surface emperaure anomaly, he deep ocean emperaure anomaly, LO T [ o C]. The discree approximaions o he sysem are AT T [ o C], and T T F T T T AT AT AT AT LO T (7) and LO LO AT LO T T 3 T T, (8) T [ o C] is he equilibrium climae sensiiviy, and,, and 3 are parameers o be calibraed. According o Nordhaus (007) he parameers (,,,, and 3 ) were calibraed o provide a good fi beween amospheric carbon concenraion and he surface emperaure anomaly in (4)-(8) using a en year ime sep and oupu from he more complex climae model MAGICC for given emissions scenarios. The parameer esimaes used in DICE007 are presened in he firs column of numbers in Table S Macmillan Publishers Limied. All righs reserved.

4 Table S: Defaul and Recalibraed Climae Model Parameers Parameer DICE Defaul 0 DICE Recalibraed CJL We assume ha he original calibraion was done adequaely such ha he oupu of he simplified climae model in (4)-(8), when using a en year ime sep, provides a close approximaion o he MAGICC oupu for DICE s business as usual, no damage reference emissions scenario. Therefore o ensure ha when swiching o a one year ime sep he oupu from he simplified climae model sill provides a close approximaion o he MAGICC oupu we recalibrae he parameers by minimizing he sum of squared errors beween he wo specificaions a he decadal ime periods over he firs cenury of he simulaion. Specifically we firs esimae recalibraed parameers for he carbon cycle using he same sep-wise emissions pahway as in he defaul DICE007 model wih no carbon price or climae damages. Then, given he updaed carbon cycle parameers we recalibrae he parameers of he emperaure response funcions. The recalibraed parameers are presened in Table S. Conradicory o he DICE documenaion, Cai e al. (0) assume ha he climae sysem expressions in DICE represen a modeling exercise of he firs approach defined above. Tha is hey assume ha (4), (7), and (8) are of he form represened in (). Based on his inerpreaion hey assume ha I, (9), (0) and, () 3 3 I is he ideniy marix and,, and 3 are he se of parameers based on he underlying physical characerisics of he sysem. Using (9)-() Cai e al. (0) recalibrae he parameers of DICE 4 03 Macmillan Publishers Limied. All righs reserved.

5 Temperaure Anomaly ( o C) Difference in Temperaure Anomaly ( o C) by simply adjusing he size of he ime sep. Their updaed parameers for an annual ime sep are presened in Table S. To illusrae he effec of his misinerpreaion, we consider he emperaure anomaly forecas using he defaul DICE007 emissions pahway wih no carbon price or climae damages. Figure S(a) presens he emperaure anomaly projecion for he hree approaches. Noing ha ha he DICE defaul forecas was calibraed o mach he oupu of a more complex climae model. Figure S(b) presens he difference beween he defaul DICE007 projecion using he en year ime sep and he wo alernaive one year ime sep forecass. By recognizing ha he defaul DICE parameers are calibraed condiional on he size of he ime sep, recalibraing he parameers for an annual ime sep mainains he model s abiliy o approximae he oupu of he more complex climae model. However, if one does no recognize ha he parameer esimaes are condiional on he ime sep and insead adjuss he parameers based on (9)-(), he climae projecions derived from he model will no longer be in line wih he original climae oupu ha he model was rying o approximae. The significanly lower emperaure anomaly from he approach of Cai e al. is wha leads o heir significanly lower esimae for he opimal carbon ax pah discussed in he main ex DICE Defaul CJL Approach Recalibraed (a) Temperaure Anomaly Projecion (b) Difference wih Defaul DICE projecion Figure S: Comparison of Climae Model Specificaions This recogniion ha he parameers wihin DICE s climae sysem were calibraed condiional on he en year ime sep does no imply ha he model s resuls are insensiive o he choice of a en year ime sep. Firs, he choice of he ime sep may be relevan o he reduced form climae sysem s abiliy o approximae he oupu of more complex models when emissions scenario is alered. Second, in he economic porion of DICE he parameers appear o be based on he parameers of coninuous ime differenial equaions defining he underlying sysem and are no calibraed condiional on he size 5 03 Macmillan Publishers Limied. All righs reserved.

6 of he ime sep. In hese cases he parameric adjusmen by Cai e al. (0) is in line wih he modeler s approach and discreizaion error of he kind hey refer o is of concern, however numerical analysis presened in he main ex suggess his error has only a modes effec on he opimal policy prescripion. References Cai Y., Judd K.L., Lonzek T.S., 0, Open Science is Necessary, Naure Climae Change (5) 99. Nordhaus W.D., Boyer J., 003, Warming he World: Economic Models of Global Warming, MIT Press. Nordhaus W.D., 007, Accompanying Noes and Documenaion on Developmen of DICE-007 Model: hp://nordhaus.econ.yale.edu/accom_noes_00507.pdf 6 03 Macmillan Publishers Limied. All righs reserved.