Depth versus Rigidity in the Design of International Trade Agreements. Leslie Johns

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1 Depth versus Rigidity in the Design of International Trade Agreements Leslie Johns Supplemental Appendix September 3, 202 Alternative Punishment Mechanisms The one-period utility functions of the home and foreign government W and W, respectively are as follows: W t, τ, a a ut t uτ W t, τ, α α uτ τ ut Losses are: Lτ W t, t B, a W t, τ, a uτ u τ B L t W t B, τ, α W t, τ, α ut u t B Let χ P denote the continuation payoff from the punishment that occurs if at least one player defects. Assume that χ P is not a function of the specific value of the defection tariff. Let χ C denote the continuation payoff if the treaty remains in effect neither player defects. Recall that a, α iid U, A for large A, u > 0, and u < 0.. Optimal Tariffs The home country s expected utility from violating the binding and not paying the fine defection is: A EUD t, a a ut t uτα dhα + σl τ α dhα + δχ P So the optimal defection tariff solves: EUD t, a t a u t 0 u t a t D a u a This violates the binding iff: t D a u > t B a a < u t B a > u t B a B

2 The home country s expected utility from violating the binding and paying the fine settlement is: A EUS t, a a ut t σl t uταdhα + So the optimal settlement tariff solves: EUS t, a t This violates the binding iff: Note that: t S a < t D a for all a. The optimal cooperative tariff is: +H δβχ C + H δχ P a u t σu t 0 u t a σ t Sa u t S a u a σ a > u t B + σ σlταdhα a σ > t B a σ < u t B { td a if a < a t B a B if a B a t B.2 Equilibrium Regions The home country s expected utility from actions C, S, and D given optimal tariff levels are: A EU C t B a, a a u t B a t B a uταdhα + +H δβχ C + H δχ P σlταdhα A EU S t S a, a a u t S a t S a σl t S a uταdhα + +H δβχ C + H δχ P A EU D t D a, a a u t D a t D a uταdhα + To compare utility from actions C and S, define for a: a EU C t B a, a EU S t S a, a Note that t S t B, so 0. Also: a u t B t B a u t S a + t S a + σl t S a σlταdhα σlταdhα + δχ P u t B u t S a a σ u t S a t Sa u t B u t S a < 0 + t Sa 2

3 So S strictly dominates C for all < a. To compare utility from actions S and D, define for a: a EU S t S a, a EU D t D a, a a u t S a t S a σl t S a a u t D a + t D a + δh βχ C χ P a σ u t S a t Sa t Sa + u t S a + t Da au t D a t Da u t D a u t S a u t D a < 0 So D strictly dominates S for sufficiently large values of a. By symmetry, indifference point a D is implicitly defined by: The equilibrium exists iff: > 0. λ a D u t S a D u t D a D + t D a D t S a D σl t S a D + δh a D βχ C χ P 0.3 Continuation Values Let t E a denote equilibrium tariffs when the institution is in place. The continuation payoff for home from the treaty being in effect is: χ C where Ψ A 0 a D au t E a t E a dha σ L t E a dha +σ Ψ δβh a D 2 A L τ E α dhα + δh a D 2 βχ C + δ A H a D 2 χ P a u t E a t E a dha + δ H a D 2 χ P u τ E α dhα.4 Comparative Statics Full Compliance Recall that the binding is not violated if a < u t B + σ. So the probability that the binding is not violated is H. Stability S u t B u t B 2 > 0 and S > 0 3

4 The institution is stable if a < a D. By the implicit function theorem: λ t B and λ σ Then for large A: a D σ u t S a D t S a D t S a D a D u t D a D t D a D + t D a D +u t S a D u t D a D + δh a D βχ C χ P + δh a D βχ C χ P u t S a D u t D a D + δh a D βχ C χ P + δh a D βχ C χ P < 0 λ tb σu t B + δh a D βχ C χ P > 0 λ σ a D σ u t S a D t S a D t S a D u t S a D u t B + δh a D βχ C χ P u t S a D u t B + δh a D βχ C χ P < 0 > 0 and < 0 Depth versus Rigidity Recall that χ C is the expected utility of a state from being a member of the cooperative regime. In equilibrium, λ 0. So for any pair t B, σ: χ C a D u t D a D u t S a D t D a D + t S a D + σl t S a D δβh a D The two first-order conditions on the optimal pair t B, σ are: + χ P β This implies that: dχ C dt B dχ C dσ D So for any pair t B, σ that generates χ C t B, σ χ : 4

5 dt B dσ dχc dσ dχ C dt B D + + D + + D where: δβh a D + σu t B δβh a D + a D u t S a D t S a D χp β δβh a D L t S a D + L t S a D χp + δβh a D β χp β + t S a D + σ u t S a D t S a D dt B dσ L t S a D δβha D + σu t B δβha D + β χp β χp L t S a D + δh a D χ P σu t B δh a D χ P > 0 for small A 2 Asymmetric Type Distributions Assume that home country type, a, is distributed according to distribution function Ha. Denote home continuation payoffs by χ N and χ C. Assume that foreign country type, α, is distributed according to distribution function F α. Denote foreign continuation payoffs χ N and χ C. The one-period utility functions of the home and foreign government W and W, respectively are as follows: W t, τ, a a ut t uτ W t, τ, α α uτ τ ut 5

6 Losses are: Lτ W t, τ B, a W t, τ, a uτ u τ B L t W t B, τ, α W t, τ, α ut u t B 2. Optimal Tariffs Home The home country s expected utility from from violating the binding and not paying compensation defection is: UD t, a a ut t uτα df α + σl τ α df α + δχ N So the optimal defection tariff solves: UD t, a t a u t 0 u t a t D a u a This violates the home binding iff: t D a u > t B a a < u t B a > u t B a B The home country s expected utility from violating the binding and paying compensation settlement is: US t, a a ut t σl t uτα df α + +F δβχ C + F δχ N σlτα df α So the optimal settlement tariff solves: US t, a t a u t σu t 0 u t a σ t Sa u a σ This violates the home binding iff: t S a u a σ a > u t B + σ > t B a σ < u t B 6

7 Note that: t S a < t D a for all a. The optimal cooperative tariff is: { td a if a < a t B a B if a B a t B Foreign The foreign country s expected utility from from violating the binding and not paying compensation defection is: U D τ, α α uτ τ a D uta dha + σl t a dha + δχ N So the optimal defection tariff solves: U D τ, α τ α u τ 0 u τ α τ Dα u α This violates the foreign binding iff: τ D α u > τ B α α < u τ B α > u τ B α B The foreign country s expected utility from violating the binding and paying compensation settlement is: a D U S τ, α α uτ τ σlτ uta dha + σl ta dha +H a D δβχ C + H a D δχ N So the optimal settlement tariff solves: U S τ, α τ α u τ σu τ 0 u τ α σ τ Sα u α σ This violates the foreign binding iff: τ S α u > τ B α σ α > u τ B + σ Note that: τ S α < τ D α for all α. The optimal cooperative tariff is: { τd α if α < α τ B α B if α B α τ B α σ < u τ B 7

8 2.2 Equilibrium Regions Home The home country s expected utility from actions C, S, and D given tariff levels above are: U C t B a, a a u t B a t B a uταdf α + σlταdf α +F δβχ C + F δχ N U S t S a, a a u t S a t S a σl t S a uταdf α + σlταdf α +F δβχ C + F δχ N U D t D a, a a u t D a t D a uταdf α + σlταdf α + δχ N To compare home utility from actions C and S, define for a: a U C t B a, a U S t S a, a a u t B t B a u t S a + t S a + σl t S a Note that t S t B, so 0. Also: u t B u t S a a σ u t S a t Sa u t B u t S a < 0 + t Sa So S strictly dominates C for all < a. To compare home utility from actions S and D, define for a: a U S t S a, a U D t D a, a a u t S a t S a σl t S a a u t D a + t D a + δf βχ C χ N a σ u t S a t Sa t Sa + u t S a + t Da au t D a t Da u t D a u t S a u t D a < 0 So D strictly dominates S for sufficiently large values of a. Indifference point a D is implicitly defined by: λ a D u t S a D u t D a D + t D a D t S a D σl t S a D + δf βχ C χ N 0 8

9 The equilibrium exists iff: > 0. Foreign The foreign country s expected utility from actions C, S, and D given tariff levels above are: U C τ B α, α α u τ B α τ B α a D utadha + σl tadha +H a D δβχ C + H a D δχ N U S τ S α, α α u τ S α τ S α σl τ S α a D utadha + σl tadha +H a D δβχ C + H a D δχ N a D U D τ D α, α α u τ D α τ D α utadha + σl tadha + δχ N To compare foreign utility from actions C and S, define for α: α U C τ B α, α U S τ S α, α α u τ B τ B α u τ S α + τ S α + σl τ S α Note that τ S τ B, so 0. Also: u τ B u τ S α α σ u τ S α τ Sα u τ B u τ S α < 0 + τ Sα So S strictly dominates C for all < α. To compare foreign utility from actions S and D, define for α: α U S τ S α, α U D τ D α, α α u τ S α τ S α σl τ S α α u τ D α + τ D α + δh a D βχ C χ N α σ u τ S α τ Sα τ Sα + u τ Sα + τ Dα αu τ D α τ Dα u τ D α u τ S α u τ D α < 0 So D strictly dominates S for sufficiently large values of α. 9

10 Indifference point is implicitly defined by: λ u τ S u τ D + τ D τ S σl τ S + δh a D βχ C χ N 0 The equilibrium exists iff: > Continuation Values Let t E a and τ E α denote equilibrium tariffs of the home and foreign country, respectively, when the institution is in place. If the institution does not exist, then the home and foreign country choose t D a and τ D α in every time period. This yields anarchy continuation payoffs: { } χ N a u t D a t D a dha u τ D α df α δ { } χ N α u τ D α τ D α df α u t D a dh a δ Home The continuation payoff for home from the treaty being in effect is: a D χ C au t E a t E a dha σ L t E a dha u τ E α df α +σ L τ E α df α + δh a D F βχ C + δ H a D F χ N where Ψ Ψ δβh a D F a D au t E a t E a dha σ L t E a dha +σ L τ E α df α + δ H a D F χ N u τ E α df α Foreign The continuation payoff for foreign from the treaty being in effect is: 0

11 χ C α u τ E α τ E α df α σ L τ E α df α u t E a dha a D +σ L t E a dha + δh a D F βχ C + δ H a D F χ N where Ψ Ψ δβh a D F α u τ E α τ E α df α σ L τ E α df α a D +σ L t E a dha + δ H a D F χ N u t E a dha 2.4 Comparative Statics Full Compliance Recall that the home country does not violate its binding if a u t B + σ. S u t B u t B 2 > 0 and S > 0 Recall that the foreign country does not violate its binding if α u τ B + σ. u τ B τ B u τ B 2 > 0 and > 0 Stability The cutpoints a D, are implicitly defined by the system of equations: By Cramer s Rule: λ a D, 0 λ a D, 0 λ t B λ t B λ a D and λ σ λ σ λ a D

12 τ B λ τ B λ τ B λ and λ σ λ σ λ where: u t S a D u t D a D + δf β D δf β + δf βχ C χ N λ tb σu t B + δf β λ τb δf β τ B λ σ L t S a D + δf β χ λ a D δh a D β C + δh a D βχ C χ N u τ S u τ D + δh a D β χ C D χ λ t B δh a D β C λ τ B σu τ B + δh a D β χ C τ B λ σ L τ S + δh a D β χ C Comparative statics on a D As h a D, H a D, f, F grow small: λ a D u t S a D u t D a D + δf β χ C u τ S u τ D + δh a D β χ C χ δf β + δf βχ C χ N δh a D β C + δh a D βχ C χ N D u t S a D u t D a D u τ S u τ D > 0 2

13 λ t B λ t B λ tb + λ t B σu t B + δf β u τ S u τ D + δh a D β χ C χ + δf β + δf βχ C χ N δh a D β C σu t B u τ S u τ D > 0 λ σ λ σ λ σ + λ σ L t S a D + δf β χ C u τ S u τ D + δh a D β χ C + δf β + δf βχ C χ N L τ S + δh a D β χ C L t S a D u τ S u τ D < 0 λ t B λ t B λ a D σu t B u τ S u τ D u t S a D u t D a D u τ S u τ D > 0 λ σ λ σ λ a D L t S a D u τ S u τ D u t S a D u t D a D u τ S u τ D < 0 Comparative Statics on As h a D, H a D, f, F grow small: λ δf β + δf βχ C χ N δh a D β u t S a D u t D a D + δf β χ C u t S a D u t D a D u τ S u τ D < 0 χ C + δh a D βχ C χ N u τ S u τ D + δh a D β χ C 3

14 λ τ B λ τ B λ τb + λ τ B δf β τ B δh a D β χ C + u t S a D u t D a D + δf β σu τ B u t S a D u t D a D < 0 + δh a D βχ C χ N σu τ B + δh a D β χ C τ B λ σ λ σ λ σ + λ σ L t S a D + δf β + u t S a D u t D a D + δf β L τ S u t S a D u t D a D > 0 χ δh a D β C + δh a D βχ C χ N L τ S + δh a D β χ C τ B λ τ B λ τ B λ σu τ B u t S a D u t D a D u t S a D u t D a D u τ S u τ D > 0 λ σ λ σ λ L τ S u t S a D u t D a D u t S a D u t D a D u τ S u τ D < 0 4