Likelihood of Environmental Coalitions --- An Empirical Testing of Shrinking Core Hypothesis in an Economy with Externality

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1 Lkelhood of Envronmental Coaltons --- An Emprcal Testng of Shrnkng Core Hypothess n an Economy wth Externalty Zl Yang CORE UCL and SUNY Bnghamton

2 IEA: lterature Barrett (1994,2003) Carraro and Snscalco (1993) Chander and Tulkens (1995, 1997) Fnus (2001) Yang (2008) and many others

3 Core concepts In general equlbrum theory In the presence of externalty (envronmental problems) Core and stablty of IEA

4 Shrnkng core hypothess In Arrow-Debreu economy: the core shrnks to the Walrasan equllbrum (Debreu and Scarf, 1963) In the economy wth externalty: the core does not necessarly shrnk to the Lndahl equlbrum (Muench( Muench,, 1972) The queston remans: shrnkng or not?

5 Core and coalton Core solutons are the broadest soluton concept of a cooperatve game Core allocaton s necessary to voluntary IEA The scope of core: a metaphor of randomly thrown dart

6 What does the core look lke? Edgeworth box Hgher dmenson? Dynamc system? Methodologcal prncple here: mappng the core of a complcated game onto a smple metrc space

7 Problem of effcent provson of stock externalty N N E δ t Max V = ϕ W = ( ( ), ( )),. { ( )} ϕu x t B t e dt ϕ = N x t = 1 = 1 0 = 1 N (1) s. t. F ( x ( t), b ( t)) = 0, = 1, 2,..., N. (2) B& ( t) N = = 1 b ( t) σb( t), σ > 0. (3) B ( 0) = B0, x (0) = x, 0, (4) U x U U U >, < 0, < 0, 2 B x B 2 < 0, F b / F x > 0 (5)

8 Problem of non-cooperatve Nash equlbrum M δ t Max W = U ( x ( t), B( t)) e dt, = 1, 2,..., { x ( t)} 0 N. (6) s. t. F ( x ( t), b ( t)) = 0, = 1,2,..., N. (7) N B& ( t) = b ( t) + b ( t) σb( t), = 1,2,..., N. σ > 0. j j (8)

9 Descrbng the cooperatve game of provdng stock externaltes Trplet V(φ, x (t), W C ) for = 1, 2,..., N Frst stage: bargan { φ } on smplex S = {{φ } φ = N}. Second stage: all regons follow the soluton paths x (t) ) gven by (1) under {φ{ } and receve payoff W C The nature of the game: barganng ntal emsson/mtgaton quota then actng cooperatvely and effcently The role of φ : crucal.

10 The core allocatons w.r.t.. Nash equlbrum Core allocaton of (1) or cooperatve game soluton of (1): the grand coalton under {φ{ } cannot be blocked by any sub-coaltons, ncludng the Nash equlbrum (ndvdual ratonalty) Sub-coaltons: cooperate nternally under the same {φ{ } and nteract wth outsders strategcally n the Nash way ( hybrd( hybrd Nash equlbrum or P.A.N.E. n Chander & Tulkens) Core allocatons are determned by ntal endowments (Bergstrom, 1976) or by the unque Nash equlbrum

11 Agenda of ths research Probng the sze of the core constructvely and emprcally Studyng the relatonshp b/w the sze of the core and number of regons (agents) Answerng shrnkng core hypothess Polcy mplcatons of core sze

12 RICE-m m model RICE model wth m regons, m = 2, 3, 4, 5, 6 Smlar (and smpler?) to FEEM s WITCH model Obtanng the scope of the core allocatons for each m Constructng comparable metrcs for the scope of the core

13 Breakng down of regons RICE-6: the Unted States (USA), European Unon (EU), other hgh-ncome ncome countres (OHI), Chna (CHN), Eastern European countres and former Sovet Unon (EEC), and the rest of world (ROW) RICE-5: USA, [EU+ OHI], CHN, EEC, ROW RICE-4: [USA+EU+ OHI], CHN, EEC, ROW RICE-3: [USA+EU+OHI], [CHN+EEC], ROW RICE-2: [USA+EU+OHI], [CHN+EEC+ROW]

14 Methodologes (1) Identfyng the core allocaton (cooperatve game soluton): Incentve checkng on all hybrd Nash equlbrums by searchng proper {φ } on smplex S = {{φ } φ = N}. Two checkng tables for each RICE-m open fle: Suffcent condtons of core propertes based on these two tables:

15 Inner ponts of the core n RICE-m φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ m = m = m = m = m =

16 Under those socal welfare weghts, the two ncentve checkng tables have sweepng homogenous sgn.

17 (2) Probng the borders of core propertes: Increase or decrease of sngle φ from the nner pont dentfed n (1) whle keep other φ j fxed on the smplex untl core propertes are volated. (the process requres solvng hybrd Nash equlbrums repeatedly). RICE-m has 2m such border ponts on the smplex. They are called vertex (a clean geometrc term).

18 Vertexes of the core n RICE-4 φ 1 φ 2 φ 3 φ 4 Inner P up down up down up down up down

19 (3) Calculatng the volumes of core: Span a convex hull wth 2m vertexes of border ponts wth core propertes. All ponts nsde ths convex hull have core propertes (or nsde the core).

20 The approxmate sze of the core: volume rato b/w convex hull and smplex. CS(m) ) = Vol(H(m))/Vol(S(m)) Defnton of volume: Vol(S) --- smplex + orgn Vol(H) --- convex hull + orgn or, pyramd vs. thn prsm cone

21 Volumes of the cores n RICE-m m = 2 m = 3 m = 4 m = 5 m = 6 V(H) E E E E-9 V(S) E E E E-3 CS(m) E E E E E-6

22 The core s shrnkng as m ncrease! The core s very small

23 Polcy mplcatons Formng voluntary IEA s not easy (based on dart throwng metaphor) Some polcy nterventons may be necessary to boost opportuntes of IEA (enlargng the sze of the core) Any connectons b/w shrnkng core phenomena and coalton sze vs. coalton stablty examned by Carraro? (stll thnkng)

24 Thanks. Comments and suggestons are welcome.