Integrating Bottom-Up into Top-Down: A Mixed Complementarity Approach

Transcription

1 Motvaton Mxed Complementarty Integratng Bottom-Up n: A Mxed Complementarty Approach Chrstoph Böhrnger Department of Economcs, Unversty of Oldenburg, Germany Illustraton Thomas F. Rutherford Department of Economcs, Unversty of Wsconsn (Madson), USA Concluson

2 Modelng Energy-Economy Interactons: Fve Approaches, edted by Charles Htch. Publshed by Resources for the Future Energy-Economy Interactons: The Fable of the Elephant and the Rabbt? by Wllam Hogan and Alan S. Manne. 1977

3 Motvaton In many energy polcy studes, the energy sector s approprately vewed n solaton from the remander of the economy. In some stuatons ths may be napproprate, as there may be two way nterdependence between energy markets and the rest of the economy. Even a large change n energy markets may represent a small fracton of aggregate economc output. There may be vrtual one-way lnkages: growth n aggregate GDP nuence energy demand, but not vce versa. If, however, two-way lnkages are mportant, then the analyss of energy market ssues demands an economy-wde perspectve.

4 The Elephant and the Rabbt The energy value share of GDP s typcally on the order of 4-5% n ndustral countres. Ths s somethng lke elephant-rabbt stew. If such a recpe contans just one rabbt (the energy sector) and one elephant (the rest of the economy), doesn't t stll tast very much lke elephant stew? But what f energy prces double, trple or quadruple, and there s sucent tme for the economy to respond? How much wll ths cost the rest of the economy? For large reductons n energy use, the value share of energy n aggregate output need not reman xed. If the value share rses, the metaphor of the elephant and the rabbt may no longer be apprprate.

5 Motvaton Mxed Complementarty Overvew Illustraton Concluson

6 Impact Assessment of Energy Polces Motvaton Complementary (hybrd) modelng framework: Mxed Complementarty Comprehensve coverage of markets: nteractons, dstortons, mperfectons Incorporaton of ncome flows: orgnaton and spendng of ncome (endowments and preferences) Technologcal foundaton: dscrete technologcal optons Illustraton Concluson Top-down (general equlbrum) Bottom-up (partal equlbrum) Synthess of Top-down and Bottom-up

7 Dchotomy of Top-down and Bottom-Up Motvaton Mxed Complementarty Illustraton Concluson Polcy focus and avalablty of solvers mathematcal format Top-down: system of equatons + equlbrum constrants n prces and quanttes actvty analyss, weak nequaltes Excursus: Integrablty Bottom-up: mathematcal programs + actvty analyss, weak nequaltes ntegrablty restrctons Equvalence of frst order condtons wth equlbrum condtons: - concdence of shadow prce of mathematcal programmng constrants wth market prces - restrctve symmetry and effcency propertes of mathematcal programs: - symmetry of (cross-prce) demand elastctes - omsson of multple agents (ncome effects) - effcent allocaton <==> taxes, prce caps, spllover externaltes - sophstcated sequental jont maxmzaton (SJM) technques to overcome non-ntegrabltes n optmzaton approach

8 Framework for Synthess: Mxed Complementarty Problem (MCP) Format (Rutherford 1995, JEDC) Motvaton Mxed Complementarty Illustraton Concluson Mxed Complementarty Problem (MCP): Mxed: Gven : f : R R, l, u R Fnd : z, w, v R N N N s. t.: F( z) w v 0 N l z u, w 0, v 0, T T w ( z l) 0, v ( u z) 0 Mxture of equaltes and nequaltes Complementarty: Complementarty between system varables and system condtons + coverage of system of equatons and mathematcal programs as subcases + equlbrum constrants n prces and quanttes (no ntegrablty restrctons) + actvty analyss, weak nequaltes + avalablty of large-scale robust solvers (PATH)

9 The Arrow-Debreu-Model as MCP Motvaton Mxed Complementarty p := a non-negatve n-vector of prces for all goods and factors ( I={1,...,n} ) y := a non-negatve m-vector of actvty levels for CRTS producton sectors ( J={1,...,m} ) M := a non-negatve k-vector of ncomes ( H={1,...,k} ) Illustraton Concluson Zero proft condton for CRTS producers: - ( p) C ( p) R ( p ) 0 j j j j Market clearance for all goods and factors: j( p) y j bh dh p j h h Budget constrants for households: h pbh M h pd h h d ( p, M ) arg max U ( x) p x M h h h h h

10 Complementarty Features of Economc Equlbra Motvaton Mxed Complementarty Illustraton Walras law ( Non-sataton ) yelds: j j y ( p ) 0 resp. y j j( p) 0 j j( p) p y j bh dh p 0 j p h h j M ( p b p d ) 0 h h h h h h Concluson Ergo: The problem of solvng the economc equlbrum corresponds to a MCP where: z y, p, M resp. ( ) f z ( p),, p b p d h h j h h

11 Economc Equlbrum Problem as MCP Motvaton Mxed Complementarty Illustraton Equvalence of market equlbrum problem wth complementarty problem: n Gven : f : R R Fnd : z R T subject to : f ( z) 0, z 0, z f ( z) 0 l 0, u, z y, p, M, f ( z) ( p),, p b p d h h Lkewse: Mathematcal Programs as a specal case of MCP! n n j h h Concluson From Top-down towards Bottom-up: wrte equatons as weak nequaltes specfy complementarty add actvty analyss/weak nequaltes for energy sectors (replacng smooth producton functon representaton) towards Top-down: re-cast NLP as an MCP add multple markets add ncome constrants

12 The 2x2x1 - Model Motvaton Mxed Complementarty Illustraton Concluson Equlbrum condtons for compettve 2x2x1-economy: Zero proft: Captal demand: Labor demand: Market clearance: Goods markets: Captal market: Income defnton: y,, y p r K r w wl r w 1,2 y p K K r, wy Y r y p L L r, w Y Y w Y X 1, ,2 1,2 X X p, p, M 1,2 2 1 y K r, w Y K M r K w L Numérare: w1 System of 12 nonlnear equatons n 12 varables N.B.: mplct varables K, L, X, M

13 Coeffcent Form versus Calbrated Share Form Motvaton Mxed Complementarty Producton: Cost: CES coeffcent form: y = x 1/ 1/ C w y CES calbrated share form: x y = y x 1/ 1 1/ 1 w y CC w y Illustraton Demand: x p 1 w y y c w x x y c w Concluson Advantage of calbrated share form: No messy nvertng: Drect calbraton from benchmark values

14 Calbraton - The Bascs Motvaton Mxed Complementarty CES functon s determned by: Quanttes (Zeroth order approxmaton - anchor pont) Prces (Frst order approxmaton - slope) Elastcty (Second order approxmaton - curvature) K Illustraton Concluson K w r L L

15 Calbraton - Mcroconsstent Dataset Motvaton Mxed Complementarty Illustraton Benchmark equlbrum: Prce conventon: p 1 = p 2 = r = w =1 Y 1 Y 2 Household Y Y K L Concluson Zero proft: column sum Market clearance: row sum Budget constrant nput-output table Socal Accountng Matrx (SAM)

16 MCP-Implementaton of 2x2x1 - Model Motvaton Equlbrum condtons Varables Complementarty features Zero proft Actvty varables Mxed Complementarty r w p r w p2 y1 0 y r w p1 y r w p2 y2 0 Market clearance M 40 y p 1 Prce varable p1 0 M 1 40 y140 p p1 Illustraton M 40 y p 2 p2 0 M 1 40 y240 p p2 Concluson p y 10 y w p w w 0 p1 p y1 10 y2 w 0 w w p y 30 y r p r r 0 p1 p y1 30 y2 r 0 r r Budget constrant Income varable 30w 50r M 0 M 30w 50r M M 0

17 From Bottom-Up (1) Motvaton Mxed Complementarty Illustraton Least-cost energy supply plannng problem: Margnal Costs mn s.. t x d p cx a x b r d E x := actvty level of technology, c := unt cost coeffcent (Leontef) of technology, a := unt capacty requrement (Leontef) of technology, b := capacty constrant for technology, d := exogenous energy demand p E := shadow prce of energy market constrant r := shadow prce of capacty constrant for technology Concluson Nuclear Coal Natural Gas... Producton

18 From Bottom-Up (2) Motvaton MCP formulaton of supply plannng problem: Mxed Complementarty Equlbrum condtons Varables Complementarty features Zero proft Actvty varable c ar pe, 0 x x c a r p 0 E Illustraton Concluson Market clearance Prce varable x d p E 0 pe x d a x b 0 r r a x b = 0 = 0

19 From Bottom-Up (3) Smplstc CGE extenson: Motvaton Mxed Complementarty addtonal macro-good as endowment (nput to energy producton and fnal consumpton) only energy producton actvtes Cobb-Douglas preferences n energy and the macro-good Economy Illustraton Concluson M b s, x

20 From Bottom-Up (4) Motvaton Mxed Complementarty MCP formulaton of smplstc CGE-extenson: Equlbrum condtons Zero proft Varables Actvty varable cp ar pe, 0 x x c p a r p 0 Complementarty features E Illustraton Market clearance Prce varable x M / p E pe 0 p E x M / pe = 0 a x b 0 c x (1 ) M / p s 0 r r a x b (1 ) / 0 p p cx M p s = 0 Concluson Budget constrant sp r b M Income varable M 0 M sp rb M 0 p := market prce of the macro-good, M := ncome of the representatve agent, s := endowment wth macro-good, := share parameter for energy n Cobb-Douglas utlty functon

21 Benchmark Data of Stylzed Economy (Böhrnger & Rutherford 2008, ENEECO) Motvaton Mxed Complementarty Illustraton Concluson Emboded least-cost energy supply problem: s.t. Here: Supply of demand for energy good j (electrcty) by alternatve technologes t subject to capacty constrants!

22 Technologes for Electrcty Generaton Motvaton Mxed Complementarty Illustraton Concluson

23 Polcy Smulaton: Nuclear Phase-Out Motvaton Mxed Complementarty Gradual reducton n permssble nuclear power capacty: Electrcty supply by technology Illustraton Concluson Nuclear capacty reducton (% vs-à-vs BaU) coal gas nuclear hydro wnd solar bomass

24 Polcy Smulaton: Green Quota Motvaton Subsdzed ncreased of renewable electrcty producton: Mxed Complementarty Illustraton Concluson Green quota n % of overall electrcty supply ev_short Technology-specfc captal ev_long Malleable captal

25 Polcy Smulaton: Envronmental Tax Reform Motvaton Mxed Complementarty Illustraton Concluson Imposton and recyclng of carbon taxes: ntal partal consumpton tax on non-energy commodtes fxed level of publc good provson Welfare changes Carbon emsson reducton (n % vs-à-vs base year) ls Lump-sum recyclng tc Cut of dstortnary consumpton tax

26 Summary Motvaton Mxed Complementarty Illustraton Concluson Perceved Dchotomy: Bottom-up versus Top-Down - specal (restrcted) cases of general equlbrum condtons - polcy focus and avalablty of effcent/robust algorthms MCP framework for synthess (hybrd models) : - economc rchness of top-down (CGE) models - technologcal foundaton of bottom-up models - avalablty of soluton algorthms for large-scale problems

27 Varaton: Decomposton of Large-Scale Hybrd Models (Böhrnger & Rutherford 2009, JEDC) Motvaton Mxed Complementarty TD model determnes prces (p ) and a set of lnear demand curves (D ). BU model s solved as QP takng prces and demand curves as gven. Illustraton Concluson : p TD model s solved as MCP takng net energy supples (e ) and energy sector nputs (x) as gven. BU model determnes net energy supples and energy sector nputs.

28 Outlook: Applcaton To Energy Polcy Scenaros Motvaton Mxed Complementarty Illustraton Concluson