Online Appendix to When o Times of Increasing Uncertainty Call for Centralized Harmonization in International Policy Coordination? Andrzej Baniak Peter Grajzl epartment of Economics, Central European University, Nador u. 9, 1051 Budapest, Hungary. Email: baniaka@ceu.hu (Corresponding author) epartment of Economics, The Williams School of Commerce, Economics, and Politics, Washington and Lee University, Lexington, 4450 VA, USA. Email: grajzlp@wlu.edu
Part A Proof of Claim 1: θ 1 and θ are private information of jurisdictions 1 and, respectively. The appropriate equilibrium concept is thus Bayesian Nash equilibrium, where jurisdictions' equilibrium strategies are functions of their respective fundamentals, or 'types'. Given jurisdiction 's equilibrium policy choice p (θ ), jurisdiction 1 chooses p 1 to maximize γ 1 (θ 1 p 1 ) (1γ 1 )E(p 1 p (θ )), (A1) where E is the expectation operator. Obtaining and re-arranging the corresponding first-order condition gives p 1 =γ 1 θ 1 +(1γ 1 )Ep (θ ). (A) Recognizing that jurisdiction 1's equilibrium strategy is a function of θ 1 and taking expectation with respect to θ 1 on both sides of (A) yields Ep 1 (θ 1 )=(1γ 1 )Ep (θ ). (A3) Following analogous steps as above to characterize jurisdiction 's equilibrium policy choice gives p =γ θ +(1γ )Ep 1 (θ 1 ) (A4) and, eventually, Ep (θ )=(1γ )Ep 1 (θ 1 ). (A5) Given γ 1, γ (0,1), the system of equations (A5) and (A3) has a unique solution: Ep 1 (θ 1 )=0 and Ep (θ )=0. Thus, from (A) and (A4), p 1 =γ 1 θ 1 and p =γ θ. Proof of Claim : The choice of p to maximize E{ω 1 γ 1 (θ 1 p) ω γ (θ p) } yields the firstorder condition ω 1 γ 1 E(θ 1 p)(1)ω γ E(θ p)(1)=0, which, when simplified, gives p[ω 1 γ 1 +ω γ ]=0. Because ω 1 γ 1 +ω γ >0, p H =0. With the objective function concave in p, the second-order sufficient condition for a maximum is satisfied. erivation of expressions (a), (b), (3), (4a), (4b), (5), (6), and (7): Follows from straightforward algebra using the properties of θ 1 and θ : E(θ 1 )=E(θ )=0, E(θ 1 )=σ 1, E(θ )=σ, and, because θ 1 and θ are independent, E(θ 1 θ )=0. Proof of Proposition: Suppose that ν 1 >0. (The proof for the case when ν >0 is analogous and thus omitted.) Then, because ν 1 +ν <0, we know that ν <0. From (7), therefore, d >0 if and only if dσ /dσ 1 >ξ, where ξ=(γ ν )/(γ 1 ν 1 )>0. To prove the properties of ξ, let s=(ω /ω 1 ). Note that ν >0 implies s>1. Then, since ν 1 =ω (1γ 1 )ω 1, ν 1 >0 may be expressed as s(1γ )>1. (A6) Also, ξ can be written as γ[ s (1 γ1)] ξ = γ [ s(1 γ ) 1]. (A7) 1 1
From (A7), it follows that, first, which is negative given that γ 1,γ (0,1). Second, which is negative by (A6) and since s>1. Third, which is positive by (A6) and because s>1. 1 1 4 γ1[ s(1 γ) 1] ξ γ γ [(1 γ )(1 γ ) 1] =, s γγ 1 s γ γ1 = 4 1 1[ s(1 ) 1] ξ [ (1 ) 1][ ( s1)], γ γ γ γ[ (1 γ1)] γ1[ (1 γ) 1] + γ[ (1 γ1)] γ1 = 4 1[ s(1 ) 1] ξ s s s s, γ γ γ (A8) (A9) (A10) Part B This appendix analyzes the model as developed in Sections and 3 of the paper while relaxing the assumption that the means of θ 1 and θ respectively equal to zero. Specifically, assume that now Eθ 1 µ 1 0 and Eθ µ 0, with µ 1 µ. Thus, σ 1 E(θ 1 µ 1 ) = Eθ 1 µ 1 and σ E(θ µ ) = Eθ µ. All other assumptions adopted in Sections and 3 continue to hold. Consider, first, decentralized (anarchic) policy-making. Jurisdiction 1 chooses p 1 to maximize (A1), giving rise to (A). Recognizing that jurisdiction 1's equilibrium strategy is a function of θ 1 and taking expectation with respect to θ 1 on both sides of (A) yields: γµ 1 1+ (1 γ1)e p( θ) = E p1( θ1). (B1) By symmetry, γ µ + (1 γ)e p1( θ1) = E p( θ). (B) Solving the systems of equations (B1) and (B) for Ep 1 (θ 1 ) and Ep (θ ) and plugging the resulting expressions into (B1) and (B) gives γ µ + (1 γ) γµ 1 1 p1 = γθ 1 1+ (1 γ1) (B3) γ1+ γγγ 1 γµ 1 1+ (1 γ1) γµ p = γθ + (1 γ). (B4) γ1+ γγγ 1 Observe that if µ 1 =µ =0, (B3) and (B4) reduce to p 1 =γ 1 θ 1 and p =γ θ, as stipulated in Claim 1 in the paper. Then, E W = E ωγ ( θ p ) ω(1 γ )( p p ) ωγ ( θ p ) ω (1 γ )( p p ), (B5) where { 1 1 1 1 1 1 1 1 } γµ + (1 γ) γµ 1 1 θ1 p1 = (1 γ1) θ1(1 γ1) γ + γ γγ 1 1 (B6)
γµ 1 1+ (1 γ1) γ µ θ p = (1 γ) θ (1 γ) γ + γ γγ p 1 p γθ 1 1 γθ 1 1 1 1 1 1 1 (B7) (1 γ ) γ µ (1 γ ) γ µ = +. (B8) γ + γ γγ Inserting (B6)-(B8) into (B5), taking the expectation, noting that Eθ 1 =σ 1 +µ 1, Eθ =σ +µ, and collecting terms, it follows that E W = ω1(1 γ1) γ1 ω(1 γ) γ 1 σ 1 ω(1 γ) γ ω1(1 γ1) γ + + σ + J, where the term J=J(γ 1,γ,ω 1,ω,µ 1,µ ) is independent of σ 1 and σ. Observe that (B9) differs from the expression (3) in the paper by the term J only. Consider, next, centralized harmonization of policymaking. Acting ex ante, before θ 1 and θ are realized, the supra-jurisdictional authority chooses a uniform policy p to maximize { ωγ 1 1θ1 p ωγθ p } (B9) E ( ) ( ). (B10) Obtaining and re-arranging the resulting first-order condition (second-order condition for a maximum is satisfied) gives ωγ µ p =. (B11) ωγ + ωγ 1 1 Note that if µ 1 =µ =0, (B11) simplifies to p H =0, as noted in Claim in the paper. Then, where { ωγ 1 1θ1 ωγ θ } H H H EW = E ( p ) ( p ), (B1) ωγ µ θ1 p = θ1 ωγ + ω γ 1 1 ωγ µ θ p = θ. ωγ + ω γ 1 1 (B13) (B14) Inserting (B13) and (B14) into (B1), taking the expectation, noting that Eθ 1 =σ 1 +µ 1, Eθ =σ +µ, and collecting terms, it follows that H 1 1 1 ωγ σ E W =ωγσ + K, (B15) where the term K=K(γ 1,γ,ω 1,ω,µ 1,µ ) is independent of σ 1 and σ. Observe that (B15) and the expression (5) in the paper differ only by the term K. Comparison of (B15) and (B9) yields H 1 1 1 νγσ EW E W = ν γ σ + + K J, (B16) where ν 1 ω (1γ )ω 1 and ν ω 1 (1γ 1 )ω (see Section 3 of the paper). Letting dσ 1, dσ >0, 1 1 1 d = γνdσ + γνd σ, (B17) which coincides with expression (7) in the paper. Thus, when µ 1 µ 0, the Proposition stated in Section 3 continues to hold. 3
Part C This appendix shows that under the assumptions of the model as specified in Sections and 3 of the paper, policy harmonization emerges endogenously under the centralized, top-down approach: acting ex ante, the supra-jurisdictional authority chooses a uniform policy p 1 =p =p even if mandating diverse policies in different jurisdictions (i.e. p 1 p ) is a possibility. Consider the following problem: acting ex ante, before θ 1 and θ are realized, the suprajurisdictional authority chooses p 1 and p to maximize E ωγ ( θ p ) ω(1 γ )( p p ) ωγ ( θ p ) ω (1 γ )( p p ) (C1) { 1 1 1 1 1 1 1 1 } Obtaining and re-arranging the resulting first-order necessary conditions (second-order conditions for a maximum are satisfied) gives [ ω + ω (1 γ )] p [ ω (1 γ ) + ω (1 γ )] p = 0 (C) 1 1 1 1 [ ω1(1 γ1) + ω(1 γ)] p1 [ ω + ω1(1 γ1)] p = 0 (C3) Equations (C) and (C3) constitute a linear homogenous system AX=0, where X=[p 1 p ]' and the matrix of coefficients equals ω1+ ω(1 γ) [ ω1(1 γ1) + ω(1 γ)] A = ω1(1 γ1) + ω(1 γ) [ ω + ω1(1 γ1)]. (C4) From (C4), it follows that the determinant of A equals det(a) = ωω 1 ( γ1γ γ1γ) ω(1 γ) γ ω1 (1 γ1) γ1 0. (C5) (C5) implies p 1 =p =0. 4