Lecture Notes: Self-enforcing agreements

Transcription

1 Lecture Notes: Self-enforcing agreements Bård Harstad ECON 4910 March 2016 Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

2 1. Motivation Many environmental problems are international By definitition, an "externality" crosses some kind of border Ozone layer, acid rain, climate change There is no world government which can impose taxes or quotas We must rely on agreements and negotiations Distinguish between self-enforcing agreements and legally binding agreements. Must use game theory to better understand the countries strategic actions Many environmental problems are dynamic The relationship is ongoing Pollution stocks accumulate over time Technology are invested in and accumulates over time Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

3 1. Motivation - Kyoto 1997 The Kyoto Protocol (first commitment period): 35 countries negotiated quotas 5% average emission reduction (from 1990-levels) 5y: 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 (ECON 4910) Self-enforcing agreements March / 34

4 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 (ECON 4910) Self-enforcing agreements March / 34

5 Content of the next classes 1: Repeated games, stochastic games, and business as usual 2: Self-enforcing vs. legally binding agreements 3: Short-term vs. long-term agreements 4: Renegotiation and updating 5: Trade, intellectual property rights, and heterogeneity 6: Participation and coalition size (?) Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

6 Outline for today a. Concepts b. Repeated games and Folk theorem c. Repeated games with emission and technology d. Etc. (renegotiation-proofness; stocks) e. Continuous emission levels f. Repeated games with imperfect public monitoring (?) g. Lessons Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

7 1-a. Important Concepts and Equilibria Refinements Normal form game Extensive form game Repeated game and stage game Stochastic/dynamic game Nash equilibrium Subgame-perfect equilibrium Renegotiation proofness Markov-perfect equilibrium Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

8 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: u i = B (g i ) c n 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 ) c ( g g ) < n. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

9 1-b. The Repeated Prisonner Dilemma Game Folk theorem with Nash equilibrium and SPE: Every v F is possible if v i v i min max v i and δ large. In PD, the minmax strategy is simply g = g. With (grim) trigger strategies, cooperation (g = g) is an SPE if B ( g ) cng B (g) cng B (g) cg c (n 1) g + δ 1 δ 1 δ B (g) B ( g ) c ( g g ) [δn + (1 δ)] So, as long as the first best requires g = g, cooperation is possible for suffi ciently high discount factors: [ δ δ 1 ( ) ] B (g) B g n 1 c ( g g ) 1 < 1. If δ < δ, the unique SPE is g = g. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

10 1-c. Emissions and Technology Consider next a stage game with both emissions and technology investments (r i,t ): u i,t = B (g i,t, r i,t ) c (r i,t ) n i=1 g i,t kr i,t. 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 (ECON 4910) Self-enforcing agreements March / 34

11 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. The business-as-usual outcome is ( g b, r b) satisfying B g (g b, r b) ( = c r b) and ( B r g b, r b) ng b c r (r b) = 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 (ECON 4910) Self-enforcing agreements March / 34

12 1-c. Problem: Deriving the best SPE The maximization problem is: B (g, r) ngc (r) kr max r,g {g,g} 1 δ subject to the two "compliance constraints" (CC-r) and (CC-g): B (g, r) ngc (r) kr 1 δ B (g, r) ngc (r) δkr 1 δ B(g b ( r), r) [g b ( r) + (n 1) g b (r)]c ( r) k r + δub 1 δ r, B ( g, r) [ g + (n 1) g] c (r) + δub 1 δ g. Folk theorem: There exists δ r < 1 and δ g < 1 such that the first-best can be sustained as an SPE iff δ max { δ r, δ g }. Literature says little when δ < max { δ r, δ g }. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

13 1-c. Compliance Constraints Proposition CC-r never binds if an agreement is beneficial (i.e., δ r = 0). CC-g can be written as ( δ (r) can be defined such CC-g binds): B ( g, r ) ngc kr (1/δ 1) [ B (g, r) B ( g, r ) ( g g ) c ] u b. 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: ( ) B r g, r ngcr ( r) k ( ) ( ) = B r (g, r) B r g, r g g c ( r) 1/δ 1 ( g g ) [B gr c r ] r > r IFF B gr c r < 0. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

14 1-c. Equilibrium Technology Proposition Let c (r) hf (r). For every r, we have δ h (r) < 0 and δ n (r) < 0. Suppose δ δ g δ (r ). If h, n, or δ decreases, then 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 (ECON 4910) Self-enforcing agreements March / 34

15 1-c. Heterogeneity Proposition CC-g only depends on individual parameters. Suppose δ i δ i (ri ). If h i, δ i, n or i s size decreases, then r i >ri for "green" technologies (where B gr <0 and c r =0) r i <ri for "brown" technologies (where B gr >0 and c r =0) r i < ri for "adaptation" technologies (where B gr =0 and c r <0) Reluctant countries should contribute more! (i.e., invest more in green technologies and less in brown.) True: One problem is to persuade a reluctant country to participate. However, the harder problem is to ensure that they are willing to comply - once they expect others to comply. Reluctant countries should be helped to make such self-commitment, and this can be done with technology! Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

16 1-f. Uncertainty and Imperfect Public Monitoring Let p I be the probability of type I error (punishment despite cooperation) and let p II the probability of type II error (continued cooperation despite more pollution). For example, (i) the individual g i,t s may be unobservable, and (ii) Nature s emission may be θ t with cdf F : g t = n g i,t + θ t. i=1 The probabilities will depend on the threshold ĝ: p I = 1 F ( ĝ ng ) and p II = F ( ĝ (n 1) g g ) Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

17 1-f. Strategy Consider the following trigger strategy with T-period punishment phase: If r i,t = r, reversion to BAU forever If g t > ĝ, reversion to BAU for T periods. When p I > 0, the best SPE may require T <. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

18 1-e. Uncertainty and Imperfect Monitoring: Cooperation Proposition The triplet ( g, r, T ) is an SPE if δ δ (r, T ) where δ T < 0, δ pi > 0, δ pii > 0 and, as before, δ n < 0, δ h < 0 and sign δ r = sign (b gr c r ). Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

19 1-f. Uncertainty and Imperfect Monitoring: Proof Proof: Let V c (r) be the continuation value in the cooperation phase: V c (r) = B ( r, g ) ngc (r) kr + δ [p I V p (r) + (1 p I ) v c (r)], where the continuation value at the start of the punishment phase is: V p (r) = T 1 δ τ v b + δ T V c (r) = 1 δt τ=0 1 δ v b + δ T V c (r), where v b = max B (r, ḡ) nḡc (r) kr. r As before, if the agreement is valuable, CC-r is never binding. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

20 1-f. Uncertainty and Imperfect Monitoring: Proof A country may be tempted to pollute a lot to get V d (r) = B (r, ḡ) [ (n 1) g + ḡ ] c (r) kr + δ [(1 p II ) V p (r) + p II V c (r)] The best equilibrium maximizes V c (r) subject to CC-g: V c (r) V d (r) (CC im) [ ( ) ] V c (r) (1 p II p I ) δ 1 δ T + 1 δ B (r, ḡ) [ ḡ + (n 1) g ] c (r) kr + (1 p II p I ) δ ( 1 δ T ) V b, Let δ (r, T, p II, p I ) be defined such that the inequality holds with identity. Doing comparative static w.r.t. this equation completes the proof. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

21 1-f. Uncertainty and Imperfect Monitoring: r vs. T Proposition Let δ ( r T, T ) = δ. If T decreases or p I or p II increases, then r T >r for "green" technologies (B gr <0 and c r =0) r T <r for "brown" technologies (B gr >0 and c r =0) r T <r for "adaptation" technologies (B gr =0 and c r <0) Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

22 1-f. Uncertainty and Imperfect Monitoring: r vs. T Proposition Let δ (r, T r ) = δ. T r increases in p I and p II and it decreases in r for "green" technologies increases in r for "brown" technologies increases in r for "adaptation" technologies Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

23 1-f. Uncertainty and Imperfect Monitoring: r and T Proposition The cost/effect of increasing T determines r e : B r ( r e, g ) ngc r (r) k B r (r e, ḡ) [ (n 1) g + ḡ ] c r (r e ) k = If δ (r e, T ) = δ holds for some T, r e h = r e n = r e δ = 0 while T e h T e < 0, n p I 1 p II T e < 0 and δ < 0. The continuation value v c is concave in r but linear in p I δ T, while CC-g is linear in (1 p I p II ) δ T. The errors can also be endogenized: The threshold ĝ should maximize p I / (1 p I p II ). Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

24 1-f. Hetereogeneity and imperfect monitoring With heterogeneity, T should be set so low that all CC-r holds Some CC-r may bind before others But reducing T further, one may violate the strongest CC-r Reducing the coalition size, in this way, may be beneficial since it reduces the need for penalties. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

25 1-e. Continuous emission levels If g R +, then when δ < δ either r is distorted, or g > g. In general, a combination of the two will be optimal. When g > g, it is less valuable with a high r (for green technology). The optimal r (g) is then a decreasing function of g. There is thus a force pushing r down when δ is small. Either effect may be strongest. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

26 1-e. Continuous emission levels - Quadratic costs Return to the homogenous setting. Let B (g, r) = b (y y i ) 2 /2, where y i = g + r. So, green technology. Let the investment-cost be kr 2 /2. We may write B = bd 2 i /2 if d i = y y i and g i = y d i r i. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

27 1-e. Continuous emission levels - First Best The socially optimal decisions are: kr = b (y y i ) = b (y g r) r (g) = b (y y i ) = cn g i (r i ) = y cn b r i b (y g) k + b. Combined, the first-best is g f = y cn B cn K and r = cn K. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

28 1-e. Continuous emission levels - BAU The Nash equilibrium of the stage game is: kr b = b (y y i ) = b (y g r) r b (g) = r b b (y g) (g) = k + b. b (y y i ) = c g b i This gives the BAU payoff: (r i ) = y c b r i, so g b = y c b c K and r b = c K. V b = c 2 b ( ) ( ) n c 2 k n 1 2 cny 1 δ. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

29 1-e. Continuous emission levels - Compliance An equilibrium gives: V e = b 2 d 2 k 2 r 2 cn (y d r). 1 δ The best deviation at the emission stage is d = c/b, giving the CC-g: b ( d 2 c ) ( c d c ) ( δ V e V b). 2 b b Lets δ g ensure that CC-g binds. The best deviation at the investment stage is r = c/k, giving CC-r: ) k (r 2 c2 ( 2 k 2 c r c ) + b ( d 2 c ) ( cn d c ) δ (V e V b) k 2 b b Lets δ r ensure that CC-r binds. By comparison, δ r < δ g if k/b > 1/2. Then, if δ ( δ r, δ g ), g e > g while r e = r. Thus, r e > r (g e ), and countries over-invest conditional on g. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

30 1-e. Continuous emission levels - Taxes and Subsidies Cooperating on r and g, or emission tax τ and investment subsidy φ are equivalent. Consumers pollute until bd = τ, while investors invest until kr φ = bd = b (y g r). Thus, for any given g, φ (g) = 0. But when δ ( δ r, δ g ), d and thus τ cannot be set at the socially optimal level. Thus, τ < cn. The smaller is δ ( δ r, δ g ), the smaller is the equlibrium d and thus τ. To ensure that kr = cn, φ = cn τ > 0 decreases in δ ( δ r, δ g ). Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

31 1-d. Renegotiation-proofness Grim trigger is not (weakly) renegotiation proof, or collectively rational If instead only the deviator abates after polluting a lot, the others may benefit and, then, the penalty can be renegotiation proof Example: Getting even strategies This requires a larger discount factor. Thus, r should increase (if green investments) Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

32 1-d. Stocks We can reformulate the model to allow for stocks Consider a pollution stock G t = q G G t 1 + g i,t and if just c C / (1 δq G ). ũ i,t = B (g i,t, r i,t ) CG t kr i,t, Similarly, the technology r i,t can be a stock: r i,t = q R r i,t 1 + s i,t, where the investment s i,t has the marginal cost K, if just k K (1 δq R ). Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

33 1-g. Lessons Bård Harstad (ECON 4910) Self-enforcing agreements March / 34

34 1-g. Lessons For climate change, it is reasonable to permit both emissions and technology investments Folk theorems: First-best possible as an SPE if δ δ. If δ < δ: Distort investments. Even with no technological spillovers, countries should cooperate also on technology, to motivate compliance. For example, compliance requires more in green; less in brown and less in adaptation technologies. Particularly if small harm, few participants, large uncertainty. Strategic technologies reduce punishment phase. Model can allow for continuous emission levels. Bård Harstad (ECON 4910) Self-enforcing agreements March / 34