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 . Bård Harstad (UiO) Repeated Games and SPE February 2019 6 / 44
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 . Bård Harstad (UiO) Repeated Games and SPE February 2019 6 / 44
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 February 2019 6 / 44
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. Bård Harstad (UiO) Repeated Games and SPE February 2019 7 / 44
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 . Bård Harstad (UiO) Repeated Games and SPE February 2019 7 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 7 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 7 / 44
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 February 2019 7 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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: Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 With binary g , we can define � � B gr ≡ B r ( g , � r ) − B r g , � r . g − g Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 With binary g , we can define � � B gr ≡ B r ( g , � r ) − B r g , � r . g − g Linear investment-cost k is a normalization. (Q: Why?) Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
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 With binary g , we can define � � B gr ≡ B r ( g , � r ) − B r g , � r . g − g Linear investment-cost k is a normalization. (Q: Why?) Will be added below: Heterogeneity, continuous g , uncertainty, and stocks Bård Harstad (UiO) Repeated Games and SPE February 2019 8 / 44
c. Benchmarks The first-best outcome is g = g and � ∗ � � ∗ � B r g , r − ngc r r = k . Bård Harstad (UiO) Repeated Games and SPE February 2019 9 / 44
c. Benchmarks The first-best outcome is g = g and � ∗ � � ∗ � B r g , r − ngc r r = k . The business-as-usual outcome is g = g and � g , r b � � r b � − ngc r = k . B r Bård Harstad (UiO) Repeated Games and SPE February 2019 9 / 44
c. Benchmarks The first-best outcome is g = g and � ∗ � � ∗ � B r g , r − ngc r r = k . The business-as-usual outcome is g = g and � g , r b � � r b � − ngc r = k . B r Given g , every country will voluntarily invest optimally in r . Bård Harstad (UiO) Repeated Games and SPE February 2019 9 / 44
c. Benchmarks The first-best outcome is g = g and � ∗ � � ∗ � B r g , r − ngc r r = k . The business-as-usual outcome is g = g and � g , r b � � r b � − ngc r = k . B r Given g , every country will voluntarily invest optimally in r . Once g has been committed to, there is no need to negotiate r . Bård Harstad (UiO) Repeated Games and SPE February 2019 9 / 44
c. Benchmarks The first-best outcome is g = g and � ∗ � � ∗ � B r g , r − ngc r r = k . The business-as-usual outcome is g = g and � g , r b � � r b � − ngc r = k . B r Given g , every country will voluntarily invest optimally in r . Once g has been committed to, there is no need to negotiate r . With such commitments, the first-best agreement is simply g = g . Bård Harstad (UiO) Repeated Games and SPE February 2019 9 / 44
c. Problem: Deriving the best SPE The maximization problem is: B ( g , r ) − ngc ( r ) − kr max 1 − δ r , g ∈ { g , g } Bård Harstad (UiO) Repeated Games and SPE February 2019 10 / 44
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 ): � � − ngc ( r ) − kr B g , r 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 , � � − ngc ( r ) − δ kr � � c ( r ) + δ u b B g , r ≥ B ( g , r ) − g + ( n − 1 ) g 1 − δ . 1 − δ Bård Harstad (UiO) Repeated Games and SPE February 2019 10 / 44
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 ): � � − ngc ( r ) − kr B g , r 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 , � � − ngc ( r ) − δ kr � � c ( r ) + δ u b B g , r ≥ B ( g , r ) − g + ( n − 1 ) g 1 − δ . 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 . Bård Harstad (UiO) Repeated Games and SPE February 2019 10 / 44
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 ): � � − ngc ( r ) − kr B g , r 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 , � � − ngc ( r ) − δ kr � � c ( r ) + δ u b B g , r ≥ B ( g , r ) − g + ( n − 1 ) g 1 − δ . 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 February 2019 10 / 44
c. Compliance Constraints Proposition r = 0 ). CC r never binds if an agreement is beneficial (i.e., � δ Bård Harstad (UiO) Repeated Games and SPE February 2019 11 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 11 / 44
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 ) . Bård Harstad (UiO) Repeated Games and SPE February 2019 11 / 44
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 lhs of CC g wrt r gives the ’best’ compliance technology � r : � � − ngc r ( � � � − � � B r g , � r r ) − k B r ( g , � g , � c r ( � = 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 February 2019 11 / 44
c. Compliance Constraints Bård Harstad (UiO) Repeated Games and SPE February 2019 12 / 44
c. Equilibrium Technology Proposition Let c ( r ) ≡ hf ( r ) . For every r, we have � δ h ( r ) < 0 and � δ n ( r ) < 0 . Bård Harstad (UiO) Repeated Games and SPE February 2019 13 / 44
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 Bård Harstad (UiO) Repeated Games and SPE February 2019 13 / 44
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 ) Bård Harstad (UiO) Repeated Games and SPE February 2019 13 / 44
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 ) Bård Harstad (UiO) Repeated Games and SPE February 2019 13 / 44
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 February 2019 13 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r i > r ∗ i ↑ for "green" technologies (where B gr < 0 and c r = 0 ) Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r i > r ∗ i ↑ for "green" 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.) Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r i > r ∗ i ↑ for "green" 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 . Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r i > r ∗ i ↑ for "green" 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. Bård Harstad (UiO) Repeated Games and SPE February 2019 14 / 44
c. Heterogeneity Proposition CC g only depends on individual parameters. Suppose δ i ≤ � δ i ( r ∗ i ) . If h i , δ i , n or i’s size decreases, then r i < r ∗ i ↓ for "adaptation" technologies (where B gr = 0 and c r < 0 ) r i < r ∗ i ↓ for "brown" technologies (where B gr > 0 and c r = 0 ) r i > r ∗ i ↑ for "green" 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 (UiO) Repeated Games and SPE February 2019 14 / 44
c. Multiple Technologies g . Suppose δ < � δ Bård Harstad (UiO) Repeated Games and SPE February 2019 15 / 44
c. Multiple Technologies g . Suppose δ < � δ Green technologies and brown technologies are strategic complements : The more countries invest in drilling technologies, the more they must invest in green technologies. Bård Harstad (UiO) Repeated Games and SPE February 2019 15 / 44
c. Multiple Technologies g . Suppose δ < � δ Green technologies and brown technologies are strategic complements : The more countries invest in drilling technologies, the more they must invest in green technologies. Green technologies and adaptation technologies are strategic complements : The more countries adapt, the more they must invest in green technologies. Bård Harstad (UiO) Repeated Games and SPE February 2019 15 / 44
c. Multiple Technologies g . Suppose δ < � δ Green technologies and brown technologies are strategic complements : The more countries invest in drilling technologies, the more they must invest in green technologies. Green technologies and adaptation technologies are strategic complements : The more countries adapt, the more they must invest in green technologies. Brown technologies and adaptation technologies are strategic substitutes : The more countries invest in brown technologies, the less they should adapt. Bård Harstad (UiO) Repeated Games and SPE February 2019 15 / 44
d. Continuous Emission Levels Proposition � r � g , δ (i) The Pareto optimal SPE is first best when δ ≥ max δ ; Bård Harstad (UiO) Repeated Games and SPE February 2019 16 / 44
d. Continuous Emission Levels Proposition � r � g , δ (i) The Pareto optimal SPE is first best when δ ≥ max δ ; � g � r < δ r ( g , r ) , δ g and, when δ ∈ � (ii) If k / b > 1 / 2 , then δ δ , we have: r = r ∗ ( g ∗ ) = r ∗ ( g ) + φ ( δ ) b + k and g = g ∗ + φ ( δ ) > g ∗ with φ ( δ ) > 0 b Bård Harstad (UiO) Repeated Games and SPE February 2019 16 / 44
d. Continuous Emission Levels Proposition � r � g , δ (i) The Pareto optimal SPE is first best when δ ≥ max δ ; � g � r < δ r ( g , r ) , δ g and, when δ ∈ � (ii) If k / b > 1 / 2 , then δ δ , we have: r = r ∗ ( g ∗ ) = r ∗ ( g ) + φ ( δ ) b + k and g = g ∗ + φ ( δ ) > g ∗ with φ ( δ ) > 0 b � r � g ( g , r ) , δ r > δ g and, when δ ∈ � (iii) If k / b < 1 / 2 , then δ δ , we have: r = r ∗ − ψ ( δ ) < r ∗ and g = g ∗ + ψ ( δ ) > g ∗ with ψ ( δ ) > 0 k k Bård Harstad (UiO) Repeated Games and SPE February 2019 16 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . In general, a combination of the two will be optimal. Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . In general, a combination of the two will be optimal. When g i > g ∗ , it is less valuable with a high r i (for green technology). Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . In general, a combination of the two will be optimal. When g i > g ∗ , it is less valuable with a high r i (for green technology). The optimal r ∗ ( g ) is then a decreasing function of g . Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . In general, a combination of the two will be optimal. When g i > g ∗ , it is less valuable with a high r i (for green technology). The optimal r ∗ ( g ) is then a decreasing function of g . There is thus a force pushing r i down when δ is small. Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels If g i ∈ R + , then when δ < � δ either r i is distorted, or g i > g ∗ . In general, a combination of the two will be optimal. When g i > g ∗ , it is less valuable with a high r i (for green technology). The optimal r ∗ ( g ) is then a decreasing function of g . There is thus a force pushing r i down when δ is small. Either effect may be strongest. Bård Harstad (UiO) Repeated Games and SPE February 2019 17 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. If y i is total consumption of energy, g i comes from fossul fuel, while r i comes from renewable energy sources. Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. If y i is total consumption of energy, g i comes from fossul fuel, while r i comes from renewable energy sources. 2 ( Y − y i ) 2 , where y i = g i + r i . Let B ( g , r ) = − b Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. If y i is total consumption of energy, g i comes from fossul fuel, while r i comes from renewable energy sources. 2 ( Y − y i ) 2 , where y i = g i + r i . Let B ( g , r ) = − b So, green technology. Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. If y i is total consumption of energy, g i comes from fossul fuel, while r i comes from renewable energy sources. 2 ( Y − y i ) 2 , where y i = g i + r i . Let B ( g , r ) = − b So, green technology. Let the investment-cost be k 2 r 2 i . Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - Quadratic costs Return to the homogenous setting. If y i is total consumption of energy, g i comes from fossul fuel, while r i comes from renewable energy sources. 2 ( Y − y i ) 2 , where y i = g i + r i . Let B ( g , r ) = − b So, green technology. Let the investment-cost be k 2 r 2 i . We can define d i ≡ Y − y i , so that g i = Y − d i − r i , and B = − b 2 d 2 i . Bård Harstad (UiO) Repeated Games and SPE February 2019 18 / 44
d. Continuous emission levels - First Best The socially optimal decisions are: b ( Y − r − g ) = cn ⇒ g ∗ ( r ) = Y − r − cn = bd b cn = bd = b ( Y − r − g ) ⇒ r ∗ ( g ) = b ( Y − g ) kr = . k + b Bård Harstad (UiO) Repeated Games and SPE February 2019 19 / 44
d. Continuous emission levels - First Best The socially optimal decisions are: b ( Y − r − g ) = cn ⇒ g ∗ ( r ) = Y − r − cn = bd b cn = bd = b ( Y − r − g ) ⇒ r ∗ ( g ) = b ( Y − g ) kr = . k + b Combined, the first-best is g ∗ = Y − cn b − cn k and r ∗ = cn k . Bård Harstad (UiO) Repeated Games and SPE February 2019 19 / 44
d. Continuous emission levels - BAU The Nash equilibrium/BAU of the stage game is: bd = c and kr = c = bd , so g b = Y − c b − c k and r b = c k . Bård Harstad (UiO) Repeated Games and SPE February 2019 20 / 44
d. Continuous emission levels - BAU The Nash equilibrium/BAU of the stage game is: bd = c and kr = c = bd , so g b = Y − c b − c k and r b = c k . This gives the BAU payoff: � � + c 2 � � − cnY c 2 n − 1 n − 1 V b = b 2 k 2 . 1 − δ Bård Harstad (UiO) Repeated Games and SPE February 2019 20 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b g be defined such that CC g binds at the first best. Let δ Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b g be defined such that CC g binds at the first best. Let δ The best deviation at the investment stage is r = c / k , giving CC r : � � r − c ≤ V e − V b . c ( n − 1 ) k Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b g be defined such that CC g binds at the first best. Let δ The best deviation at the investment stage is r = c / k , giving CC r : � � r − c ≤ V e − V b . c ( n − 1 ) k r ensure that CC r binds at the first best. By comparison, Let δ r < δ g iff k / b > 1 / 2. δ Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b g be defined such that CC g binds at the first best. Let δ The best deviation at the investment stage is r = c / k , giving CC r : � � r − c ≤ V e − V b . c ( n − 1 ) k r ensure that CC r binds at the first best. By comparison, Let δ r < δ g iff k / b > 1 / 2. δ � g � r , δ , g e > g ∗ while r e = r ∗ . Then, if δ ∈ δ Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance An equilibrium gives: 2 d 2 − k 2 r 2 − cn ( Y − d − r ) V e = − b . 1 − δ The best deviation at the emission stage is d = c / b , giving the CC g : � � � � � V e − V b � d 2 − c 2 b d − c − c ≤ δ . b 2 2 b g be defined such that CC g binds at the first best. Let δ The best deviation at the investment stage is r = c / k , giving CC r : � � r − c ≤ V e − V b . c ( n − 1 ) k r ensure that CC r binds at the first best. By comparison, Let δ r < δ g iff k / b > 1 / 2. δ � g � r , δ , g e > g ∗ while r e = r ∗ . Then, if δ ∈ δ Thus, r e > r ∗ ( g e ) , and countries over-invest conditional on g . Bård Harstad (UiO) Repeated Games and SPE February 2019 21 / 44
d. Continuous emission levels - Compliance Bård Harstad (UiO) Repeated Games and SPE February 2019 22 / 44
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