Dynamic Games in Environmental Economics PhD minicourse Part I: Repeated Games and Self-Enforcing Agreements Bård Harstad UiO December 2017 Bård Harstad (UiO) Repeated Games and SPE December 2017 1 / 48
Content of the Class I. Repeated games, SPEs, Self-enforcing agreements II. Stochastic/dynamic games, MPEs, contracts as legally binding agreements III. Free-riding and coalition formation Bård Harstad (UiO) Repeated Games and SPE December 2017 2 / 48
Limitations: The following will not be covered Empirics Game theory Asymmetric information Continuous time Resources and extraction North vs. South Domestic political economics Bård Harstad (UiO) Repeated Games and SPE December 2017 3 / 48
Literature Fudenberg and Tirole ’96: Game Theory Mailath and Samuelson ’06: Repeated Games and Reputations Bolton and Dewatripont ’05: Contract Theory: Quantities and investments Hart: Theory of the Firm: Hold-up problems and organizational responces Dockner, Jørgensen, Van Long and Sorger ’00: Differentiatial Games in Economics and Management Science Barrett ’03: Environment & Statecraft There is a gap between the game theoretical literature and the more applied one in environmental economics. Filling this gap is the goal of both my research and this class. Bård Harstad (UiO) Repeated Games and SPE December 2017 4 / 48
Outline for Day 1 a. Concepts b. Repeated games and Folk theorem c. Repeated games with emission and pollution d. Repeated games with imperfect public monitoring e. Renegotiation proofness f. Technological spillovers g. Continuous emission levels and policies h. Lessons Bård Harstad (UiO) Repeated Games and SPE December 2017 5 / 48
1. Motivation - Kyoto 1997 The Kyoto Protocol (first commitment period): 35 countries negotiated quotas 5% average emission reduction (from 1990-levels) 5y: 2008-2012 Durban Platform and Doha 2012: EU promised to continue similar commitments, if other countries specify targets by 2015 for 2020 Investments in new technology Importance of technology transfer/develop recognized.. "technology needs must be nationally determined, based on national circumstances and priorities" (§114 in the Cancun Agreement, confirmed in Durban) Bård Harstad (UiO) Repeated Games and SPE December 2017 6 / 48
1. Motivation - Lima 2014 The approach taken by this agreement is quite different from that of the more familiar Kyoto Protocol. There are no formal commitments to reduce carbon emissions by a numerical target during a specific time frame. Instead, each state promises to design its own “nationally determined contribution” that “represent[s] a progression beyond the current undertaking.” There is no clear mechanism to ensure that these national efforts are meaningful nor an obvious way to enforce these self-imposed obligations. (WP) Economist on the Lima Accord: countries must make, before the Paris meeting, their "intended nationally determined contributions" (INDCs) countries "may" (rather than "shall") provide detailed information and a timeframe for their emissions cuts. no formal commitments to reduce carbon emissions by a numerical target during a specific time frame. Bård Harstad (UiO) Repeated Games and SPE December 2017 7 / 48
1. Motivation - Paris 2015 Countries had to suggest, before the Paris meeting, their "intended nationally determined contributions" (INDCs) No formal commitments to reduce carbon emissions by a numerical target during a specific time frame. The agreement calls for the U.N. Framework Convection on Climate Change to publish all national action plans on its Web site and for scientists to calculate the contributions these plans make to curbing emissions. (WP) A Climate Accord Based on "Global Peer Pressure" (NYT) Climate is the ultimate public good International agreements must be self-enforcing There is no explicit sanctions Compliance is the main problem Bård Harstad (UiO) Repeated Games and SPE December 2017 8 / 48
1-a. Important Concepts and Equilibria Refinements Normal form game Nash equilibrium Extensive form game Subgame-perfect equilibrium Repeated game and stage game Renegotiation proofness Stochastic game Markov-perfect equilibrium Bård Harstad (UiO) Repeated Games and SPE December 2017 9 / 48
1-b. The Prisonner Dilemma Game Climate is the ultimate public good Abatements are costly and benefit others The prisonner dilemma game is a reasonable stage game Let g i be the emission of i ∈ { 1 , .., n } , B ( g i ) the benefit of polluting, c the marginal cost of greenhouse gases: n ∑ u i = B ( g i ) − c g i . i = 1 � � If g ∈ g , g , the first-best agreement is simply g = g if: � � < � � B ( g ) − B g g − g cn . But polluting more is a dominant strategy if: � � > � � B ( g ) − B g g − g c . The emission game is a prisonner dilemma game if both holds: � � 1 < B ( g ) − B g � � < n . g − g c Bård Harstad (UiO) Repeated Games and SPE December 2017 10 / 48
1-b. The Repeated Prisonner Dilemma Game Fudenberg and Maskin ’86 : Folk theorem with Nash equilibrium and SPE: Every v ∈ F is possible if v i ≥ v i ≡ min max v i for δ large. In PD, the minmax strategy is simply g = g . With (grim) trigger strategies, cooperation ( g = g ) is an SPE if � � − cng B g B ( g ) − cg − c ( n − 1 ) g + δ B ( g ) − cng ≥ ⇔ 1 − δ 1 − δ � � � � [ δ n + ( 1 − δ )] B ( g ) − B g ≤ c g − g So, as long as the first best requires g = g , cooperation is possible for sufficiently high discount factors: � � � � B ( g ) − B g 1 δ ≥ � � � δ ≡ − 1 < 1 . n − 1 c g − g If δ < � δ , the unique SPE is g = g . Bård Harstad (UiO) Repeated Games and SPE December 2017 11 / 48
1-c. Emissions and Technology Consider next a stage game with both emissions and technology investments ( r i , t ): n ∑ u i , t = B ( g i , t , r i , t ) − c ( r i , t ) g i , t − kr i , t . i = 1 B ( · ) is increasing and concave in both arguments. Examples: "green" technologies: b gr < 0 and c r = 0 "brown" technologies: b gr > 0 and c r = 0 "adaptation" technologies: b gr = 0 and c r < 0 Linear investment-cost k is a normalization Will be added below: Uncertainty, heterogeneity, and stocks Bård Harstad (UiO) Repeated Games and SPE December 2017 12 / 48
1-c. Benchmarks The first-best outcome is ( g ∗ , r ∗ ) satisfying B g ( g ∗ , r ∗ ) nc ( r ∗ ) and = � ∗ � B r ( g ∗ , r ∗ ) − ng ∗ c r r = k . � g b , r b � The business-as-usual outcome is satisfying � g b , r b � � r b � = B g c and � g b , r b � � r b � − ng b c r B r = k . Given g , every country will voluntarily invest optimally in r . Once g has been committed to, there is no need to negotiate r . � � If g ∈ g , g , the first-best agreement is simply g = g . Bård Harstad (UiO) Repeated Games and SPE December 2017 13 / 48
1-c. Problem: Deriving the best SPE The maximization problem is: B ( g , r ) − ngc ( r ) − kr max 1 − δ r , g ∈ { g , g } subject to the two "compliance constraints" (CC-r) and (CC-g): B ( g , r ) − ngc ( r ) − kr B ( g b ( � r ) − [ g b ( � r ) + ( n − 1 ) g b ( r )] c ( � ≥ r ) , � r ) 1 − δ r + δ u b − k � 1 − δ ∀ � r , g + ( n − 1 ) g ] c ( r ) + δ u b B ( g , r ) − ngc ( r ) − δ kr ≥ B ( � g , r ) − [ � 1 − δ ∀ � g . 1 − δ r < 1 and � g < 1 such that the Folk theorem: There exists � δ δ � g � r , � � first-best can be sustained as an SPE iff δ ≥ max δ δ . � g � r , � � Literature says little when δ < max δ δ . Bård Harstad (UiO) Repeated Games and SPE December 2017 14 / 48
1-c. Compliance Constraints Proposition r = 0 ). CC-r never binds if an agreement is beneficial (i.e., � δ CC-g can be written as ( � δ ( r ) can be defined such CC-g binds): � � − ngc − kr − ( 1 / δ − 1 ) � � � − � � � ≥ u b . B g , r B ( g , r ) − B g , r g − g c CC-g is more likely to hold for large δ , n , or c ( r ) . Maximizing rhs of CC-g wrt r gives the best compliance technology � r : � � − ngc r ( � � � − � � B r g , � r r ) − k c � ( � B r ( g , � g , � = r ) − B r r g − g r ) 1 / δ − 1 � � [ B gr − c r ] ⇔ ≈ g − g r ∗ IFF B gr − c r < 0 . � r > Bård Harstad (UiO) Repeated Games and SPE December 2017 15 / 48
Bård Harstad (UiO) Repeated Games and SPE December 2017 16 / 48
1-c. Equilibrium Technology Proposition Let c ( r ) ≡ hf ( r ) . For every r, we have � δ h ( r ) < 0 and � δ n ( r ) < 0 . g ≡ � Suppose δ ≤ � δ ( r ∗ ) . If h , n, or δ decreases, then δ r > r ∗ ↑ for "green" technologies (where B gr < 0 and c r = 0 ) r < r ∗ ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r < r ∗ ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) Bård Harstad (UiO) Repeated Games and SPE December 2017 17 / 48
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