Dynamic Games with Applications to Climate Change Treaties

Transcription

1 Dynamic Games with Applications to Climate Change Treaties Prajit K. Dutta Roy Radner MSRI, Berkeley, May 2009 Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

2 Motivation: Dynamic Commons Principal features of climate change problem Global Common - sources of carbon buildup are local but global stock determines warming Near-irreversibility - stock of GHGs depletes slowly so e ect of current emissions is felt into distant future Asymmetry - some regions will su er more than others Nonlinearity & Uncertainty - costs can be very nonlinear and driven by "unknown" & uncertain processes Models need to accommodate these features Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

3 Motivation: Games & Treaties Focus of current research is on treaty-formation Main Question - what agreements or treaties will sovereign nations sign & then carry out? That requires a Strategic Model - Although the players (countries) are relatively numerous, there are some very large players and blocks of like-minded countries "Philosophical Perspective" - Climate Change is an International Issue and hence a Treaty cannot be mandated but has to be Incentive-Compatible. The Treaty needs to be an equilibrium of the game Additional requirement Simplicity in Solution - Desirable for Political Reasons Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

4 Overview Talk in Two Parts Part I - Dynamic (Commons) Games: Set-Up, Strategies, Equilibria, Applicability to Climate Change Treaties Part II - Economics of Climate Change: Recent Research & Results Part I builds on literature going back to Shapley (1953) and Levhari & Mirman (1973). Recent work of relevance includes Abreu-Pearce-Stachetti (1990) and Dutta & Sundaram (2005). Part II builds on recent work by Dutta & Radner (2004, 2006, 2008), Dockner et al (1999, 2006), Barrett (2006) and others. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

5 Dynamic Games: In Words Introduced as "Stochastic Games" by Lloyd Shapley (1953), generalizing Dynamic Programming Multiple Periods - Each period players interact by picking an action. Action interaction take place at a given state which changes as a consequence. Payo received by each player in each period is based on action vector played as well as the state in that period. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

6 Dynamic Games: Notation t = time period (0, 1, 2,...T ). i = players (1,..., I ). s(t) = state at beginning of period t, s(t) 2 S a i (t) = action taken by player i in period, a i (t) 2 A i a(t) = (a 1 (t), a 2 (t),... a I (t)) vector of actions taken in period t. π i (t) = π i (s(t), a(t)) payo of player i in period t. q(t) = q(s(t + 1) j s(t), a(t)) conditional distribution of state at period δ = discount factor, δ 2 [0, 1). Exogenous Variables - Initial value of state, s(0), discount factor δ and game horizon, T ( nite or in nite) Endogenous Variables - Everything else Special Cases - Dynamic Programming (I = 1) & Repeated Games (#S = 1) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

7 Connection to Climate Change Treaties Players = Nations/Blocs; potential signatories State = stock of GHGs - Common State State = technology (coal, oil, nuclear..), capital stock, R&D level, etc. - Private States Action (Common State relevant) = Emissions of GHGs Action (Private State relevant) = switching technologies, investment, R&D expenditures, etc. Remark - Actions "implemented" through national policies, cap & trade, carbon tax, R&D subsidy, contributions to international organizations, foreign aid etc. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

8 Histories & Strategies History at time t, h(t) - list of prior states & action vectors h(t) = s(0), a(0), s(1), a(1),...s(t) Assumption - past actions and states including current state observable Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

9 Histories & Strategies History at time t, h(t) - list of prior states & action vectors h(t) = s(0), a(0), s(1), a(1),...s(t) Assumption - past actions and states including current state observable Strategy for player i at time t, σ i (t) - a complete conditional action plan for every history σ i (t) : fh(t)g! P(A i ) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

10 Histories & Strategies History at time t, h(t) - list of prior states & action vectors h(t) = s(0), a(0), s(1), a(1),...s(t) Assumption - past actions and states including current state observable Strategy for player i at time t, σ i (t) - a complete conditional action plan for every history σ i (t) : fh(t)g! P(A i ) Strategy for entire game, σ i - a list of strategies, one for every period: σ i = σ i (0), σ i (1),...σ i (t),... Let σ = (σ 1, σ 2,...σ I ) denote a vector of strategies, one for each player Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

11 Examples of Strategies Pure Strategy σ i - where σ i (t) is a deterministic choice (from A i ). May however be conditional on history Example - Emission of country i is high if country j had high emissions in previous period but low if j had low previous emissions ("Tit for Tat") Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

12 Examples of Strategies Pure Strategy σ i - where σ i (t) is a deterministic choice (from A i ). May however be conditional on history Example - Emission of country i is high if country j had high emissions in previous period but low if j had low previous emissions ("Tit for Tat") (Stationary) Markovian Strategy f i - When the action map is independendent of history (Markovian) and time (stationary) f i : S! P(A i ) Example - Switch to new technology happens only when capital stock is su ciently "large" (because capital & energy are complements). Switching strategy is non-stationary if it depends on the number of periods left Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

13 Outcomes & Payo s Consider a pure strategy vector σ & suppose that state transition q is also deterministic. Then there is a unique history generated by σ: h(t; σ) = s(0), a(0; σ), s(1; σ), a(1; σ),...s(t; σ) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

14 Outcomes & Payo s Consider a pure strategy vector σ & suppose that state transition q is also deterministic. Then there is a unique history generated by σ: h(t; σ) = s(0), a(0; σ), s(1; σ), a(1; σ),...s(t; σ) where a(τ; σ) = σ(τ; h(τ; σ)) and s(τ + 1; σ) = q(s(τ + 1) j s(τ; σ), a(τ; σ) ) Above called outcome path for σ. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

15 Outcomes & Payo s Consider a pure strategy vector σ & suppose that state transition q is also deterministic. Then there is a unique history generated by σ: h(t; σ) = s(0), a(0; σ), s(1; σ), a(1; σ),...s(t; σ) where a(τ; σ) = σ(τ; h(τ; σ)) and s(τ + 1; σ) = q(s(τ + 1) j s(τ; σ), a(τ; σ) ) Above called outcome path for σ. Associated lifetime payo is R i (σ) = T δ t π i (s(τ; σ), a(τ; σ)) t=0 Generalize when σ not a pure strategy or q not deterministic Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

16 Equilibria Subgame - the game that remains after every history h(t) Restriction of σ to the subgame is denoted σ j h(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

17 Equilibria Subgame - the game that remains after every history h(t) Restriction of σ to the subgame is denoted σ j h(t) Nash Equilibrium (NE) - Strategy vector σ is a NE if R i (σ ) R i (σ i, σ i ), for all i, σ i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

18 Equilibria Subgame - the game that remains after every history h(t) Restriction of σ to the subgame is denoted σ j h(t) Nash Equilibrium (NE) - Strategy vector σ is a NE if R i (σ ) R i (σ i, σ i ), for all i, σ i Subgame Perfect (Nash) Equilibrium (SPE) - σ is a SPE of the game if it is a NE for every subgame R i (σ j h(t)) R i (σ i, σ i j h(t)), for all i, σ i, h(t) Remark - Not all NE are SPE since NE only requires incentive inequality to be satis ed on outcome path generated by σ. But o -equilibrium credible? Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

19 Equilibria Subgame - the game that remains after every history h(t) Restriction of σ to the subgame is denoted σ j h(t) Nash Equilibrium (NE) - Strategy vector σ is a NE if R i (σ ) R i (σ i, σ i ), for all i, σ i Subgame Perfect (Nash) Equilibrium (SPE) - σ is a SPE of the game if it is a NE for every subgame R i (σ j h(t)) R i (σ i, σ i j h(t)), for all i, σ i, h(t) Remark - Not all NE are SPE since NE only requires incentive inequality to be satis ed on outcome path generated by σ. But o -equilibrium credible? Markov Perfect Equilibrium (MPE) - A stationary Markov strategy vector f is a MPE if R i (f ) R i (f i, f i ), for all i, f i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

20 Climate Change - Transition Equation Greenhouse gases form a global common - hence studied by dynamic commons game (DCG) Aggregate E ect of Emissions - (Part of) state space S is a single-dimensional variable with a "commons" structure a ected by aggregate emissions s(t + 1) = q (s(t), A(t)) A(t) denote the global (total) emission during period t; A(t) = I a i (t) i=1 Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

21 Climate Change - Transition Equation Greenhouse gases form a global common - hence studied by dynamic commons game (DCG) Aggregate E ect of Emissions - (Part of) state space S is a single-dimensional variable with a "commons" structure a ected by aggregate emissions s(t + 1) = q (s(t), A(t)) A(t) denote the global (total) emission during period t; A(t) = I a i (t) i=1 Global stock of greenhouse gases (GHGs) at the beginning of period t denoted g(t). Linear Transition - where 0 < σ < 1 g(t + 1) = A(t) + σg(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

22 Climate Change - Payo s Separable E ect of Emissions - Bene ts from emissions & costs from GHG stock h i (a i (t)) c i (g(t)) h i may be thought of as country s GDP. Linked to emissions via energy use. Standard assumptions - concavity, continuity - made. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

23 Climate Change - Payo s Separable E ect of Emissions - Bene ts from emissions & costs from GHG stock h i (a i (t)) c i (g(t)) h i may be thought of as country s GDP. Linked to emissions via energy use. Standard assumptions - concavity, continuity - made. In Dutta & Radner it is assumed that the marginal cost of GHG is constant Linear Cost of GHG c i (g(t)) = c i g(t) where c i is country speci c Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

24 Climate Change - Payo s Separable E ect of Emissions - Bene ts from emissions & costs from GHG stock h i (a i (t)) c i (g(t)) h i may be thought of as country s GDP. Linked to emissions via energy use. Standard assumptions - concavity, continuity - made. In Dutta & Radner it is assumed that the marginal cost of GHG is constant Linear Cost of GHG c i (g(t)) = c i g(t) where c i is country speci c Discussion of Linearity Assumption - Strong assumption that rules out catastrophes. Made for three reasons: - Simple Conclusions - Full Characterization possible, not just qualitative features. Hence calibration of solutions possible - What is the right non-linear cost function? Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

25 More General State Variables Recall, State Variable multi-dimensional including "private" components such as each country s capital stock, technology, R&D level, population etc. Associated action vectors - respectively, investment, technology switching, R&D expenditures, etc. Relevant transition equations - e.g. capital stock grows through savings less destruction through depreciation Relevant payo functions - e.g. switching costs paid when technology is switched Histories, strategies, outcomes and payo s de ned as in general model of Dynamic Games MPE and SPE studied If the actions are unobservable, then strategies and notion of equilibrium needs modi cation. Similarly if there are sources of private information. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

26 Literature Dynamic Games - Shapley (1953), Parthasarathy (1973), Mertens & Parthasarathy (1987) Du e et al (1994), Abreu, Pearce & Stachetti (1990), Dutta (1995) Commons Games - Levhari & Mirman (1973), Benhabib & Radner (1993), Dutta & Sundaram (1993, 1994), Dockner, Long & Sorger (1996), Tornell & Velasco (1992) Climate Change Treaties via Dynamic Games - Dutta and Radner (2004, 2006, 2008), Dockner & Nishimura (1999), Long & Sorger (2006) Climate Change Treaties via Repeated Games - Barrett (2003), Finus (2007) Climate Change Agreements Non-Strategic - IPCC Report (1973), Stern Report (2006), Nordhaus & Yang (1996), Nordhaus & Boyer (2000) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

27 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

28 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor State evolves according to where A(t) is total emissions g(t + 1) = A(t) + σg(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

29 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor State evolves according to g(t + 1) = A(t) + σg(t) where A(t) is total emissions Country i 0 s period t payo, v i (t) given by v i (t) = h i [a i (t)] c i g(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

30 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor State evolves according to g(t + 1) = A(t) + σg(t) where A(t) is total emissions Country i 0 s period t payo, v i (t) given by v i (t) = h i [a i (t)] Payo over the game horizon given by v i = δ t v i (t) t=0 c i g(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

31 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor State evolves according to g(t + 1) = A(t) + σg(t) where A(t) is total emissions Country i 0 s period t payo, v i (t) given by v i (t) = h i [a i (t)] Payo over the game horizon given by Solution concept - SPE v i = δ t v i (t) t=0 c i g(t) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

32 Simple Model Only state variable stock of GHGs, g(t) - constant capital, technology, labor State evolves according to g(t + 1) = A(t) + σg(t) where A(t) is total emissions Country i 0 s period t payo, v i (t) given by v i (t) = h i [a i (t)] Payo over the game horizon given by v i = δ t v i (t) t=0 Solution concept - SPE Example studied for calibration purposes h i [a i (t)] = φ i K γ i i L 1 γ i β i i c i g(t) ai (t) f i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27 βi

33 A Simple MPE - "Business as Usual" There is a simple MPE - dubbed BAU equilibrium - that involves constant emissions a i The level of emissions determined by h 0 i (a i *) = δw i where w i is the lifetime marginal cost w i = c i 1 δσ Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

34 A Simple MPE - "Business as Usual" There is a simple MPE - dubbed BAU equilibrium - that involves constant emissions a i The level of emissions determined by h 0 i (a i *) = δw i c i 1 δσ Same result holds when there are additional state variables such as capital, technology, labor,... (say z i ) There is a "BAU equilibrium" characterized by h i1 (a i *,z i ) = δw i where w i is the lifetime marginal cost w i = Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

35 A Simple MPE - "Business as Usual" There is a simple MPE - dubbed BAU equilibrium - that involves constant emissions a i The level of emissions determined by h 0 i (a i *) = δw i c i 1 δσ Same result holds when there are additional state variables such as capital, technology, labor,... (say z i ) There is a "BAU equilibrium" characterized by h i1 (a i *,z i ) = δw i Result follows directly from the fact that a unit of emission in period t is of size σ in period t + 1, σ 2 in period t + 2, etc. On account of linearity in cost, these surviving units add c i δσ in period t + 1, c i (δσ) 2 in period t + 2, etc.; or over lifetime c i 1 δσ where w i is the lifetime marginal cost w i = Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

36 Global Pareto Optima Let x = (x i ) be a vector of positive numbers, one for each country. A Global Pareto Optimum (GPO) is a solution to Max σ x i v i i For every x there is a GPO that involves constant emissions ba i The level of emissions determined by x i h 0 i (â i ) = δw where w is the lifetime social marginal cost 1 1 δσ i x i c i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

37 Global Pareto Optima Let x = (x i ) be a vector of positive numbers, one for each country. A Global Pareto Optimum (GPO) is a solution to Max σ x i v i i For every x there is a GPO that involves constant emissions ba i The level of emissions determined by x i h 0 i (â i ) = δw where w is the lifetime social marginal cost 1 1 δσ i x i c i Same result holds when there are additional state variables such as capital, technology, labor,... (say z i ) There is a GPO characterized by x i h i1 (a i *,z i ) = δw i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

38 Global Pareto Optima Let x = (x i ) be a vector of positive numbers, one for each country. A Global Pareto Optimum (GPO) is a solution to Max σ x i v i i For every x there is a GPO that involves constant emissions ba i The level of emissions determined by x i h 0 i (â i ) = δw where w is the lifetime social marginal cost 1 1 δσ i x i c i Same result holds when there are additional state variables such as capital, technology, labor,... (say z i ) There is a GPO characterized by x i h i1 (a i *,z i ) = δw i Tragedy of the Common - Since x i c i < j x j c j it follows from the concavity of h i that a i * > â i Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

39 Numerical Results on GPO & BAU Table 1 - Benchmark Case (δ = 0.97, cost = Fankhauser c i, Year = 1998) BAU GPO Region/Emissions BAU (Gtc) GPO (Gtc) % Di erence ( BAU ) United States % Western Europe % Other High Income % Eastern Europe % Middle Income (MI ) % Lower MI % China % Lower Income % Total % Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

40 Numerical Results on GPO & BAU 2 Table 2 - Benchmark Case (δ = 0.995, cost = Fankhauser c i, Year = 1998) BAU GPO Region/Emissions BAU (Gtc) GPO (Gtc) % Di erence ( BAU ) United States % Western Europe % Other High Income % Eastern Europe % Middle Income (MI ) % Lower MI % China % Lower Income % Total % Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

41 Other Equilibria Could there be other - better - equilibria than the BAU? Are they simple? Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

42 Other Equilibria Could there be other - better - equilibria than the BAU? Are they simple? Are there treaties - SPE - sustained by the BAU itself? "Countries pledge to cut emissions z% from the BAU and if someone doesn t then we revert to the BAU" Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

43 Other Equilibria Could there be other - better - equilibria than the BAU? Are they simple? Are there treaties - SPE - sustained by the BAU itself? "Countries pledge to cut emissions z% from the BAU and if someone doesn t then we revert to the BAU" Proposition. There is a continuum of emission cuts that are possible as SPE. There is a (strictly positive) maximum emission aggregate cut sustained by the BAU threat Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

44 Other Equilibria Could there be other - better - equilibria than the BAU? Are they simple? Are there treaties - SPE - sustained by the BAU itself? "Countries pledge to cut emissions z% from the BAU and if someone doesn t then we revert to the BAU" Proposition. There is a continuum of emission cuts that are possible as SPE. There is a (strictly positive) maximum emission aggregate cut sustained by the BAU threat The "best" SPE sustainable by the BAU threat need not be the one that implements the maximal cuts E ect of Asymmetry in Costs - can be shown that the greater the cost asymmetry the lower are aggregate payo s in the "best" SPE Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

45 Maximal Emission Cuts Table 3 - Minimum Sustainable Emissions in Benchmark Case (δ = 0.97, cost = Fankhauser c i, Year = 1998) MIN GPO Country/Emissions BAU (%) UnitedStates 9% Brazil 31% China 21% India 9% Korea DPR +22% Poland +5% Russia +17% Ukraine +21% Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

46 All SPE Can there be better threats than the BAU? Are they simple? is the best such equilibrium? What Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

47 All SPE Can there be better threats than the BAU? Are they simple? is the best such equilibrium? Can one characterize the set of all SPE? What Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

48 All SPE Can there be better threats than the BAU? Are they simple? is the best such equilibrium? Can one characterize the set of all SPE? What Answer - The SPE payo correspondence has a surprising simplicity; the set of equilibrium payo s at a level g is a simple linear translate of the set of equilibrium payo s from some benchmark level, say, g = 0. Consequently, the set of emission levels that can arise in equilibrium is state-independent. Though in a particular equilibrium, emission levels may vary with g. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

49 All SPE Can there be better threats than the BAU? Are they simple? is the best such equilibrium? Can one characterize the set of all SPE? What Answer - The SPE payo correspondence has a surprising simplicity; the set of equilibrium payo s at a level g is a simple linear translate of the set of equilibrium payo s from some benchmark level, say, g = 0. Consequently, the set of emission levels that can arise in equilibrium is state-independent. Though in a particular equilibrium, emission levels may vary with g. Proposition. The equilibrium payo correspondence Ξ is linear; there is a compact set U < I such that for every initial state g Ξ(g) = U fw 1 g, w 2 g,...w I gg Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

50 All SPE Can there be better threats than the BAU? Are they simple? is the best such equilibrium? Can one characterize the set of all SPE? What Answer - The SPE payo correspondence has a surprising simplicity; the set of equilibrium payo s at a level g is a simple linear translate of the set of equilibrium payo s from some benchmark level, say, g = 0. Consequently, the set of emission levels that can arise in equilibrium is state-independent. Though in a particular equilibrium, emission levels may vary with g. Proposition. The equilibrium payo correspondence Ξ is linear; there is a compact set U < I such that for every initial state g Ξ(g) = U fw 1 g, w 2 g,...w I gg Bootstrapping ideas - the Abreu-Pearce-Stachetti operator - employed in proof Computing the equilibrium value set U not an easy task - unlike Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

51 Second-Best Equilibrium Second-Best Problem - To maximize a weightem sum of equilibrium payo s max Second-Best Equilibrium I i=1 x i V i (g), V (g) 2 Ξ(g) Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

52 Second-Best Equilibrium Second-Best Problem - To maximize a weightem sum of equilibrium payo s max Second-Best Equilibrium I i=1 x i V i (g), V (g) 2 Ξ(g) Proposition. There exists a constant emission level a a 1, a 2,...a I - such that no matter what the initial level of GHG, the second-best policy is to emit at the constant rate a. In the event of a deviation from this constant emissions policy by country i, play proceeds to i 0 s worst equilibrium. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

53 Second-Best Equilibrium Second-Best Problem - To maximize a weightem sum of equilibrium payo s max Second-Best Equilibrium I i=1 x i V i (g), V (g) 2 Ξ(g) Proposition. There exists a constant emission level a a 1, a 2,...a I - such that no matter what the initial level of GHG, the second-best policy is to emit at the constant rate a. In the event of a deviation from this constant emissions policy by country i, play proceeds to i 0 s worst equilibrium. Furthermore, the second-best emission rate is always strictly lower than the BAU rate, i.e., a < a. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

54 Second-Best Equilibrium Second-Best Problem - To maximize a weightem sum of equilibrium payo s max Second-Best Equilibrium I i=1 x i V i (g), V (g) 2 Ξ(g) Proposition. There exists a constant emission level a a 1, a 2,...a I - such that no matter what the initial level of GHG, the second-best policy is to emit at the constant rate a. In the event of a deviation from this constant emissions policy by country i, play proceeds to i 0 s worst equilibrium. Furthermore, the second-best emission rate is always strictly lower than the BAU rate, i.e., a < a. Above a critical discount factor (less than 1), the second-best rate coincides with the GPO emission rate ba. Second-Best Simple, Implementable by an across the board cut Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

55 Worst Equilibrium Consider second-best solution in which x i = 0. Denote that emission level a(x i ), the i less second-best Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

56 Worst Equilibrium Consider second-best solution in which x i = 0. Denote that emission level a(x i ), the i less second-best Worst Equilibrium for Country i Is in two parts: Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

57 Worst Equilibrium Consider second-best solution in which x i = 0. Denote that emission level a(x i ), the i less second-best Worst Equilibrium for Country i Is in two parts: 1. There exists a high emission level a(i) (with j6=i a j (i) > j6=i aj ) s.t. each country emits at rate a j (i) for one period (no matter what g is), j = 1,..I. 2. From the second period onwards, each country emits at the constant rate a j (x i ), j = 1,..I. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

58 Worst Equilibrium Consider second-best solution in which x i = 0. Denote that emission level a(x i ), the i less second-best Worst Equilibrium for Country i Is in two parts: 1. There exists a high emission level a(i) (with j6=i a j (i) > j6=i aj ) s.t. each country emits at rate a j (i) for one period (no matter what g is), j = 1,..I. 2. From the second period onwards, each country emits at the constant rate a j (x i ), j = 1,..I. So a sanction is made up of two emission rates, a(i) and a(x i ). The rst imposes immediate costs on country i by increasing the emission levels of countries j 6= i. For the punishing countries, however, the resultant cost increase is o set by a subsequent permanent change, the switch to the emission vector a(x i ), which permanently increases their quota at the expense of country i 0 s. Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

59 Generalizations & Extensions Technology Choice - Switching to cleaner technologies With linear costs to switching, the solution is "bang-bang" - Tragedy obtains Capital Growth - does reducing emissions inhibit growth? Ongoing research suggesting that treaty formation incentives can be very di erent for countries with varying capital stocks Foreign Aid -can aid be Pareto improving for donor & recipient? Yes in some circumstances Other Issues - Treaty negotiation issues. Who joins rst? How many countries are enough? Should climate change treaties be "linked" to trade treaties? Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27

60 Conclusions Climate change treaties can be modeled via dynamic commons games Tragedy of the Commons obtains in the simple BAU equilibrium - on emissions, technology switching etc. BAU can be strictly improved by using it as a threat to reduce emissions Best equilibria may be relatively simple to implement Asymmetry makes treaties harder to implement but foreign aid can help Prajit K. Dutta, Roy Radner () Games & Climate Change MSRI, Berkeley, May / 27