The Limits to Partial Banking Unions: A Political Economy Approach Dana Foarta. Online Appendix. Appendix A Proofs (For Online Publication)

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1 The Limits to Partial Banking Unions: A Political Economy Approach ana Foarta Online Appendix A Appendix A Proofs For Online Publication A.1 Proof of Proposition 1 Consider the problem for the policymaker in country i with a partial banking union τ, x. Let ζ i {x i, g i, g1, i b i 1} be the policies chosen by policymaker i, with i {, F }. Policymaker solves subject to max ζ uc x, x F + wg + βwg 1, A1 x + g e + βb 1 + τ, A2a x x A2b g1 e b 1, A2c b 1 [ e /β, e ], A2d x θi. A2e According to Assumption 1, constraint A2e does not bind. Policymaker F solves max ζ F uc F x F, x + wg F + βwg F 1, A3 A.1

2 subject to x F + g F e F + βb τ, A4a g1 F e F b F 1, A4b b F 1 [ e F /β, e ] F, A4c x F θi F. A4d According to Assumption 1, constraint A4d binds for policymaker F. By the Envelope Theorem, problem A1 is strictly concave in e τ, and problem A3 is strictly concave in e F and τ. and In choosing τ, x, the supranational authority faces the following maximization problem: max τ,x {η [ uc x, x F + wg + βwg 1 ] +1 η [ uc F x F, x + wg F + βwg F 1 ] } subject to U i x i, x j, g i, g i 1 U i x i0, x j0, g j0, g i0 1, A5 where i, j {, F }, i j, and {x i0, g i0, g i0 1 } denote the solution to policymaker i s maximization problem when τ = 0, x = 0. The supranational authority s objective function is a sum of utilities maximized in A1 and A3, so it is a strictly concave function of τ. Then, any solution to the supranational authority s problem that involves τ > 0 implies a strict increase in the utility of the supranational authority. Given the participation constraints of the two governments, A5, it follows that the utility of households in at least one country must increase. A.2 Proof of Proposition 2 Assumption 1 guarantees that τ e F e F, where e F = θif 1 + β + gf + rf 1 + β, A.2

3 with g F and r F defined in Assumption 1. This means that full recapitalizations are provided in country F x F = θi F even if transfers are made to country. Step 1. The policymakers problem Consider a partial banking union with terms τ and x. Let be the λ, ϑ, and βµ be the Lagrange multipliers on constraints 18a, 18b, and 18c, respectively. The first-order conditions to problem 17 when constraint 18b binds and there is an interior solution lead to 1 γ v r = γ σ Ru c, A6a r + x = x, A6b g = g1 = e + τ x 1 + β. A6c The maximization problem for policymaker F given {τ, x} is to choose ζ F = {r F, x F, g F, g F 1, b F } to solve subject to max 1 γ F vr F + γ [ F uc F x F, x + wg F + βwg1 F ] A7 ζ F r F + x F + g F e F + βb F τ, A8a g1 F e F b F, A8b b F [b F, e F ], A8c x F θi F, A8d where constraint A8d binds. The first-order conditions for an interior solution for r F and g F imply 1 γ F v r F = γ F w g F, A9a g F 1 = g F, A9b x F = θi F. A9c A.3

4 Step 2. The supranational authority s problem The supranational authority sets τ 0 and x 0 in order to maximize 9 given 10 and 11. The minimum reinvestment requirement is r + x x. Setting x at least equal to the policymaker s unconstrained choices is a weakly dominant strategy, so constraint 18b holds with equality for policymaker. The policymaker s utility from rents r and recapitalizations x is concave and additive, so a binding x implies r r 0 and x x 0. Then vr vr 0, and 10 is satisfied as long as U x 0, x F 0, g 0, g 0 1 U x, x F, g, g 1 1 γ γ [ vr vr 0 ]. A10 Step 3. We show that if constraint 10 does not bind for some η C 0, 1, then it does not bind η η C. Assume there exists a value η C 0, 1 at which 10 does not bind. Case A. Corner solution for x. Consider the case in which x = x = θi + r, with r defined implicitly by 1 γ v r = γ σ Ru θi, θi F. In this case, the maximum recapitalization is achieved at η C : x = θi. Let ι denote the Lagrange multiplier on constraint 11. Then, the first order-condition that determines τ is ηw g g = w g F gf +ι1 γ F v r F 1 η + γ F ι rf. A11 Given this condition, applying the Envelope Theorem, an increase in η would increase τ, which is equivalent to increasing e, so U x, x F, g, g1 η > 0, A12 r = r, A13 and the policymaker s utility is also increasing; hence, 10 does not bind A.4

5 η η C. Case B: Interior solution for x. The first-order conditions to the supranational authority s maximization problem, in case of an internal solution τ, x, are: [ 1 σ Ru c F 1 η + γ F ι + ησ Ru c ] = ηw g g ηw g g 1 + β = w g F gf 1 + β1 η + γ F ι +ι1 γ F v r F rf From A6a-A6c, 0 < gf < 1 and 0 < rf > 0, g Envelope Theorem, an increase in η implies η 1 + β, A14. A15 g 1 + β = 1, = 1. From A9a, 1+β < 1. Then, from A14 and A15, applying the < 0 and η > 0. So U x, x F, g, g 1 η > 0. A16 From 18b From 18a, V x, x F, g, g 1 V x, x F, g, g 1 0. Then, constraint 10 does not bind for η > η C. A17 > 0. A18 Step 4. We show that if for some η B 0, 1 constraint 10 binds, then it binds η η B. Since x is at least as high as policymaker s policy choices, r r 0. If 10 binds, then τ is inferred implicitly from this constraint as γ 1 + βw e + τ x = 1 γ vr 0 + U x 0, x F 0, g 0, g β 1 γ vr x γ u c x x, θi F. A19 A.5

6 Case A. There is a corner solution for x = x, with x defined in Step 3. A decrease in η would not change the value of x nor the value of τ. Hence, 10 binds η < η B. Case B. If x < x. Constraint 10 binding implies that the first-order conditions A14 and A15 become [ 1 σ Ru c F 1 η B + γ F ι + η B σ Ru c ] η B w g g 1 + β 0, A20 η B w g g + 1 ηw g F gf Then, a decrease in η keeps the constraint 10 binding. 0. A21 Step 5 We show that there exists η B 0, 1 such that constraint 10 binds for η < η B and it does not bind for η > η B. If η = 0, the supranational authority maximizes the utility of the F households only, so τ is minimized and x is maximized given constraint 10. At η = 0, the first-order conditions to the supranational authority s problem are given by A20 and A21, with strict inequality for both. The left-hand side of A20 is strictly decreasing in η, and the left-hand side of condition A21 is strictly increasing in η. By the continuity of the utility functions it then follows that η B > 0 such that A20 and A21 hold with equality. Step 6. We show there exists η 0, 1 such that U x 0, x F 0, g 0, g 0 1 = U x, x F, g, g 1. From Step 5, there exists η B 0, 1 such that constraint 10 binds. Since r r 0, it follows that U x 0, x F 0, g 0, g 0 1 U x, x F, g, g 1. If η = 1, the supranational authority maximizes the utility of the households, so the transfer τ will be at the maximum level at which the participation constraint for the F government is satisfied. It then follows that U x, x F, g, g 1 > U x 0, x F 0, g 0, g 0 1 and vr > vr 0. Given A12 in case A and A16 in case B, and the continuity of U x, x F, g, g 1 A.6

7 it follows that there exists η 0, 1 such that. U x 0, x F 0, g 0, g 0 1 U x η, x F η, g η, g 1 η = 0. A22 A.3 Proof of Corollary 1 The value of η satisfies uc x η, x F βwg η = uc x 0, x F βwg 0. A23 From the supranational authority s first-order condition A15, an internal solution for τ, x implies that η w g η w g F η gf. A24 efine x x x 0 and let g be implicitly given by uc x 0 + x, x F βwg 0 g = uc x 0, x F βwg 0. A25 So, w g η = w g 0 g. When x = θi, x,max θi x 0, and g,max is given implicitly by uc x 0 + x,max, x F βwg 0 g,max = uc x 0, x F βwg 0. A26 A.7

8 If τη > 0, then g F η < g F 0. So w g η w g 0 g,max w g F η gf w g F η gf w g 0 x,max 1+β, A27 w g F 0 where x,max 1+β > g,max given the concavity of u and w. Then, w g 0 x,max w g F 0 1+β = w g 0 + x0 θi 1+β 1+β w e F θif 1+β w e r0 θi 1+β 1+β w e F θif. A28 1+β Then, from A27, w g η w g F η gf w e r0 θi 1+β 1+β w e F θif = Φθ, A29 1+β and so η Φθ. A30 A.4 Proof of Corollary 2 The value η is defined as the value at which uc x η, x F βwg η = uc x 0, x F βwg 0, A31 where x F = x F 0 = θi F, given Assumption 1. The effect of increasing e F Case A: Corner solution with respect to x x = θi. A.8

9 In this case, we have a corner solution with respect to x, so Applying the Envelope Theorem in A31, we obtain = e F = 0. e F = e F 1. A32 From A15, applying the Envelope Theorem, e F > 0 and > 0, so < 0. A33 ef Case B: Internal solution with respect to x x < θi Applying the Envelope Theorem in A31, we obtain e F = Γ Υ, A34 where Γ σ Ru c Υ σ Ru c e + F w g e F e F η + w g η A35 A36 From A14, applying the Envelope Theorem, From A15, applying the Envelope Theorem, e F e F > 0. > 0 and g e F > 0, so e > 0. A37 F ef Similarly, applying the Envelope Theorem in A14 and A15, > 0, and So > 0. Γ > 0, < 0, A38 A39 A.9

10 and ] Υ = [w g σ Ru c + w g = 1 η 1 σ Ru c F + w g η > 0 A40 Then, The effect of increasing γ F < 0. A41 ef Case A: Corner solution with respect to x x = θi. In this case, the corner solution implies = = 0. Applying the γ F Envelope Theorem in A31, we obtain γ F = γ F 1. A42 From A15, applying the Envelope Theorem, γ F where > 0 and > 0, so < 0. A43 γf Case B: Internal solution with respect to x x < θi As above, applying the Envelope Theorem in A31, we obtain Ξ σ Ru c γ F = Ξ Υ, A44 γ + F w g γ. A45 F γ F From A14 and A15, applying the Envelope Theorem, γ F > 0, γ F > 0 A.10

11 and x γ F > 0, which in A44 leads to A.5 Proof of Corollary 3 < 0. A46 γf The value η is defined in A31 above. The effect of a change in α F on σ and σ F is given by σ α F σ F α F α z z F = [α z + 1 α F z F ] 2 = z F σ I, A47 = z F 1 α z [α F z F + 1 α z ] 2 = 1 σf z F I F. A48 Applying the Envelope Theorem in A31, and using the above expressions, we obtain where α F = σ Ru c Ψ σ Ru c 0 zf I x0 + σ Ru c zf I x + g 0 Ψ + w g +1 + βw g 0 α F w g α F α F σ Ru c α F., 0 α F + 1 σf σ θz F α F + 1 σf θz F σ A49 A50 Without a partial banking union, the first-order conditions to the A.11

12 policymaker s problem 17 without a banking union lead to 1 γ v r 0 = γ σ Ru c 0, A51 1 γ v r 0 = γ w g 0. A52 Applying the Envelope Theorem in A51 and A52, g 0 w g 0 α F r 0 1 γ v r 0 α F = σ R 2 u c 0 zf I x σf θz F σ = γ w g 0 α F. 0 α F Ru c 0 σ z F I, A53 g 0 A54 From the budget constraint in country : r 0 α F 0 g β α F + 0 α F = 0. A55 We define Λ β + 1 γ v r 0 γ w g γ σ R 2 u c 0, A56 and from the above conditions, we derive g 0 α F 0 α F = = z F 1 σ F w g 0 Λ 0 σ θ x0 I u c 0 1, A57 σ Ru c 0 I z F 1 σ F σ R 2 u c 0 Λ 0 σ θ x0 I u c 0 1 σ Ru c 0 I + u c 0 z F σ Ru c 0 I + zf x 0 1 σf θi A58 I σ With a partial banking union, when x = θi : When the supranational authority s problem gives the corner solution x = θi, dx d α F = 0 and dc = 0. If d α F α + α F 1, then 1 σ F /σ 1. A.12

13 Since < 0 α F x θi, > 0. Applying the Envelope Theorem to the supranational authority s problem then leads to α F α F > 0. A59 Using A51, A52 and A57, Ψ can be simplified to 1 σ Ψ = σ Ru c 0 F θ w g The upper bound on 1 u c Λ σ σ Ru c 0 I 0. A60 α F α F α F α F A.6 Proof of Proposition 3 is 0, it follows that Ψ 0. The effect of a change in α F on σ θ and c θ for fixed x and τ, and x F = θi F : σ α F = σ F α F = c x F, x α F = R c F x F, x α F = R α z z F [α z + 1 α F z F ] 2 = zf I σ, 1 α z z F [α F z F + 1 α z ] 2 = zf 1 σ F, I F [ ] σ α F x σf α F xf + σ α [ F σ α F x σf α F xf 1 σ α F where, from the first-order conditions to the policymaker s problem, A61 A62 A63 ], A64 α F [ = zf u c + u c R σ x 1 σ F θi ] I Rσ u c + 1. A65 1 γ γ Rσ v r efining ϖ 1 γ γ Rσ 2 v r u c > 0, A66 A.13

14 we can re-write the above as α F = zf σ I ϖ u c Ru c [ σ x 1 σ F θi ], A67 c x F, x = R [σ zf x 1 σ ] F θi + σ I α F I z F α F = R zf I ϖ 1 + ϖ [ σ x 1 σ F θi +σ γ Rσ u c 1 γ v r ] A68 u c c x F, x u c α F = zf I [ ] x 1 σf θi v r + 1 σ v r 1 + ϖ A69 z F I u c c x F, x u c α F = zf 1 1 σ F ϖ x θi v r A70 I 1 + ϖ σ v r Also, [ c F σ = R α F α F x σf α F xf 1 σ α F = R zf I [ σ x 1 σ F θi 1 + ] 1 σ 1 u c σ 1 + ϖ Ru c ] 1 σ σ ϖ A71 A.14

15 and So, if α + α F 1, then also σ + σ F < 1. Since x θi, then z F I u c c x F, x > 0, A72 u c α F c F > 0. A73 αf Consider the first order conditions to the supranational authority s maximization problem when there is an interior solution for x, τ with a binding x for policymaker and an interior solution for { x, r, g } : η [Rσ u c x, x F +1 η e x τ 1 + β ] w [R 1 σ u c F x F, x ] = 0 ; A74 ηw e x τ 1 + β + 1 ηw g F gf = 0. A75 Consider the effect of a change in α F on ησ u c + 1 η 1 σ u c F. A76 = [ ηu c 1 ηu c F ] σ α F α + F ησ u c c α F +1 η 1 σ u c F cf [ α F ] z = ηu c σ F I u c c u c α F 1 ηu c F 1 σ [ ] σ α + u c F c F F u c F α F A77 From A72 and A73, and since c, c F, and x are bounded, it follows that there exist η and η such that: if η < η, then α F < 0, and applying the Envelope Theorem and ignoring A.15

16 second order effects, σ F < 0 and σ < 0; F if η > η, then > 0, and applying the Envelope Theorem and ignoring α F second order effects, σ F A.7 Proof of Proposition 4 > 0 and σ > 0. F Consider the case in which x = θi and x F = θi F corner solutions for recapitalizations. efine φ z z F does not change: and consider a change in φ such that I z F z = α 1 α F, and dφ = zf + z α 1 α F z F 2 = Re-writing σ and σ F, I 1 α F z F 2 = 1 1 σ z. F σ = σ F = α φ α φ + 1 α F, α F α F + 1 α φ. A78 A79 A.16

17 The effect of the change in φ is: σ φ dσ = σ F φ α 1 α F = [α φ + 1 α F ] 2 = σ 1 σ ; A80 φ α I [α φ + 1 α F ] 2 z F 2 = α I ; A81 = αf 1 α 1 σ F ; A82 [α F + 1 α φ] 2 = σf φ 1 σ F dσ F = σf 1 σ F φ 1 σ z F di F = α F α = σf 1 σ z ; A83 1 α F + 1 α = 1 α α F 1 α F. A84 γ Rσ u c = 1 γ v r, A85 dx + dr = dx = 0. A86 γ Ru c α I + γ Rσ u c dc +γ Rσ 2 u c dx + 1 γ v r dx = 0. A87 At full recapitalizations, c = z R, so dc = R. Then, dx = 1 σ u Rz Rz u Rz γ v r γ Rσ 2 u c. A88 If u Rz Rz u Rz < 1, then dx < 0 and then dx > 0, since x = x,max + r,max, γ Rσ u c x,max, x F,MAX = 1 γ v r,max. A89 A90 A.17

18 Applying the Envelope theorem in the first-order condition for τ ignoring second order effects then leads to dτ > 0. If u Rz Rz u Rz > 1, A91 then dx > 0 and then dx < 0. Applying the Envelope theorem in the firstorder condition for τ then leads to dτ < 0. A.8 Proof of Proposition 5 enote by τ, x the equilibrium supranational policy without fiscal rules. Consider introducing a fiscal rule B > b i.e., that binds at τ, x,. enote by τ F R, x F R the optimal policy for the supranational authority when the fiscal rule is B. Also, denote by r, x, g, g1 policymaker s utility maximizing policy choices under τ, x and no fiscal rules, and by g F the public good provision in country F. enote by r, x, g, g 1 the policy choices in country under τ F R, x F R and fiscal rule B, and by g F the public good provision in country F. Without the fiscal rule, the first-order condition A15 implies that with an internal solution for τ, x, ηw g = 1 η w g F gf, A92 g = g1 A93 With a binding fiscal rule B, ηw g = 1 η w g F gf, A94 but w g 1 w g 1, so ηw g 1 1 η w g F gf. A95 A.18

19 Then, reducing B while keeping τ, x, constant increases the utility of the supranational authority. Since this holds true for all B > b and all τ, x,, it implies that the supranational authority is maximized at B = b. A.9 Proof of Proposition 6 We first establish the following lemma: Lemma 1 There exists γ F such that γ F γ F, policymaker F provides full recapitalizations x F = θi F when domestic fiscal rules are in place. Proof. In section A.10 below. The existence of a binding fiscal rule only changes the equilibrium policies { } r, x, g, g1 coming out of policymaker s constrained maximization problem. It does not change the problem for the supranational authority. Step 1. The policymakers problem With a binding debt limit b θ in country and a limit b F θ in country F, policymaker s problem is subject to max 1 γ H vr + γ [ H uc x, x F {x,g,r,b } + wg + βwe b ] A96 r + x + g e + βb + τ, A97 r + x x A98 b b. A99 The first-order conditions with a binding rule x lead to γ Rσ u c x, x F = 1 γ v r, A100 r + x = x, A101 g = e + βb x + τ. A102 A.19

20 The maximization problem for policymaker F facing debt limit b F is max 1 γ F vr F + γ [ F uc F x F, x + wg F + βwe b F ] {x F,g F,r F,b F } A103 subject to r F + x F + g F e F + βb F τ, A104a g1 F e F b F, A104b b F b F, A104c x F θi F. A104d Therefore, the above conditions together with Assumption 1 imply 1 γ F v r F = γ F w g F, A105a x F = θi F. A105b In autarky, the policies { x 0, g 0, r 0} satisfy: γ Rσ u c 0 x 0, x F 0 = 1 γ v r 0, A106a Rσ u c 0 x 0, x F 0 = w g 0, A106b The policies { x F 0, g F 0, r F 0} satisfy: x 0 + g 0 + r 0 = e + βb, A106c g 0 1 = e b. A106d γ w g F 0 = 1 γ v r F 0, A107a x F 0 = θi F, A107b x F 0 + g F 0 + r F 0 = e F + βb F, A107c g F 0 1 = e b F. A107d Steps 2-4 are analogous to the proof of Proposition 2. A.20

21 It then follows that there exists η 0, 1 such that. U θ, b, 0, 0 U θ, b, τ, x = 0. A108 For η as defined in Proposition 2. Let b η denote the equilibrium debt at η under the partial banking union with no fiscal rules. Consider a fiscal rule that marginally decreases the debt b : b = b ε, where ε 0. This has no effect on b 0, since b > b 0. So, at b σ Ru c x 0, x F 0 0 At η : b + w g 0 g0 b βw g 0 1 = 0. A109 σ Ru c x, x F + w g βw g 1 = σ Ru c x 0, x F 0 + w g 0 βw g 0 1. A110 So, by the Envelope Theorem, the change in η in response to a change of ε in b is: σ Ru c + w g β + b = b b σ Ru c + w η g b βw g 1. A111 From the first-order conditions to the supranational authority s problem, A14 and A15, < 0, > 0, < 0 and > 0. b b As shown in the proof to Corollary 2, at η, the sign of the denominator in A111 is positive. At b b, we have g1 g, so the numerator of A111 simplifies to σ Ru c w g b + w g b < 0 A112 A.21

22 So b > 0. A113 Then, when b = b ε, η decreases. This then implies η η. A.10 Proof of Lemma 1 Consider the rents r F and public good g F defined implicitly by: w g F = σ F Ru c F θi F, θi, A114a 1 γ F v r F = γ F σ F Ru c F θi F, θi. A114b Let b F γ F = β 1 e F + θi F + r F γ F + g F. Then, by construction, policymaker F s maximization problem yields solutions { r F, g F, θi } F. Assumption 1 guarantees that e F is suffi ciently large to allow for this. The rule b F γ F gives the minimum budget in period 0 needed to obtain x F = θi F when x = θi H. A fiscal limit b F > b F is preferred by the F households if w g F gf b F βw g F 1 0, A115 where g F 1 e F b F γ F. A116 From A114a and A114b, γ F w g F = 1 γ F v r F, A117a g F γ F = 0. A117b and applying the Envelope Theorem in policymaker F s problem, we obtain r F γ = w g F + v r F F 1 γ F v r F < 0. A118 A.22

23 The effect of increasing γ F in A115 is given by w g1 F rf > 0. A119 γf For γ F 1, r F 0, so A115 holds since w g F > w g1 F for any binding fiscal rule. Then, γ F < 1, such that A115 is satisfied γ F > γ F. A.11 Proof of Corollary 4 Let τ, x denote the equilibrium policy chosen by the supranational authority without fiscal rules, and by τ F R, x F R the equilibrium policy chosen by the supranational authority with fiscal rule b in country and fiscal rule b F in country F. Consider country. From the first-order conditions to the government s problem, it follows that g b > 0. A120 Then, a decrease in debt from the non-binding value b to b in first-order condition A15 implies an increase in τ to some τ F R > τ. From condition A14 it follows that x F R < x. Given the government s first-order conditions, then x x F R < x x A121 and g F τ F R, x F R < g F τ, x. A122 Therefore, the utility of the Financing households is given by U F = uc F x F, x x F R + wg F τ F R, x F R +βwg F 1 τ F R, x F R. A123 From policymaker F s problem, u F c F is an increasing function of x, A.23

24 g F = g F 1, and wg F is an increasing function of g F. Then, U F decreases if debt in country is limited to b. A.12 Proof of Proposition 8 enote the country debt limits by b θ, 1 = b, and b θ, 0 = b 0. The supranational authority sets x, τ given b. The minimum debt limit that can be set is b,min = 0. enote the policies in country under the partial banking union by { } r, x, g, g 1. Assume first a partial banking union with τ > 0 and x x 0 + r 0. Consider the utility of policymaker inside the partial banking union: V = 1 γ vr + γ uc x, θi F +γ wg + βγ wg 1. A124 The change in policymaker s payoff as b changes: V = 1 γ v r r b b +γ σ Ru c x, θi F b +γ w g β + b βγ w g b 1. A125 Applying the Envelope Theorem in 10 and 11, < 0, > 0,and b x < 0. b Let b be the fiscal rule at which households in country maximize utility. Then, if b b, b γ σ Ru c x, θi F b + γ w g β + b βγ w g b 1 0, A126 A.24

25 and so V > 0. A127 b Therefore, the utility of policymaker is lowest at b,min. Consider the case in which households in country set fiscal rule b,min inside the partial banking union and debt limit b 0 outside the partial banking union so no binding debt limit outside the { partial banking union. The policies } with fiscal rule b,min are denoted by r,min, x,min, g,min, g,min 1 Then, policymaker s participation constraint is given by 1 γ vr,min + γ uc x,min, θi F +γ wg,min + βγ wg,min 1 1 γ vr 0 + γ uc x 0, θi F + γ wg 0 + βγ wg 0 1. A128 The lowest value of γ at which the participation constraint is satisfied for the policymaker is where γ = Φ. A129 Φ U θ, b 0, 0, 0 U θ, b,min, τ, x. A130 vr,min vr 0 If a partial banking union is formed with fiscal rule b,min, then r,min > r 0, so vr,min > vr 0. Also, if U θ, b,min, τ, x < U θ, b 0, τ, x, then Φ > 0 and γ 0, 1; otherwise γ = 0. For γ > γ, households can implement a fiscal rule above b,min inside the partial banking union and a fiscal rule below b 0 outside the partial banking union. Notice that if the partial banking union with τ > 0 is not formed so τ = 0, then the result follows immediately. A.25

26 B Appendix B Alternative Fiscal Rules For Online Publication B.1 omestic Fiscal Rules That o Not Anticipate the Partial Banking Union It is worth noting how the results of the model differ if households were to myopically choose the fiscal rule without anticipating the partial banking union. This is relevant, since several domestic restrictions on government borrowing pre-date the increase in financial integration that would make crosscountry transfers and recapitalizations a concern. 1 The debt limit b θ is chosen as the solution to the following problem: max {b } uc x 0, x F 0 + wg 0 + βwe b 0 B1,x 0,g 0,b 0,r 0 subject to γ Rσ u c x 0, x F 0 = 1 γ v r 0, B2a Rσ u c x 0, x F 0 = w g 0, B2b r 0 + x 0 + g 0 e + βb 0, B2c b 0 b. B2d The problem for the F country is analogous. The supranational authority must propose the transfer τ and reinvestment requirement x taking into account the debt limits b and b F in each country. The problem it faces is max τ,x ηu θ, b, τ, x + 1 ηu F θ, b F, τ, x B3 1 See Budina et al B.1

27 subject to 1 γ vr + γ U θ, b, τ, x 1 γ vr 0 +γ U θ, b, 0, 0, B4a 1 γ F vr F + γ F U F θ, b F, τ, x 1 γ F vr F 0 +γ F U F θ, b F, 0, 0. B4b Constraints B4a and B4b represent the participation constraints for the and F governments, respectively. The participation constraints make it clear that the fiscal rules are set outside of the partial banking union, and therefore they remain in place even if the partial banking union is not accepted. Analogous results to the case in which the partial banking union is anticipated are immediately obtained; however, one important difference emerges. Once the partial banking union is not anticipated, the strategic effect of choosing fiscal rules disappears. This may lead to higher welfare losses to country households. The following result compares the loss in welfare from joining a partial banking union when fiscal rules are in place to the loss in welfare from joining a partial banking union when no fiscal rules are in place. Corollary 1 Consider a partial banking union that achieves full recapitalizations x = θi. Then, there exists η 0, η such that η < η, having domestic fiscal rules in country increases the welfare losses to households from joining a partial banking union. Proof. In section B.3.1. Fiscal rules may increase household welfare compared to having no fiscal rules, both with and without a banking union. What Corollary 1 shows is that the drop in welfare going into a partial banking union is higher when fiscal rules are in place. The result emerges because a low value of η means that the supranational authority allocates a high share of the bailout costs to country. With a limited ability to borrow due to the fiscal rule, the government must finance the spending on the banking sector by significantly reducing public good provision in period 0. This lowers the utility of households, B.2

28 and the welfare loss is higher than in the alternative scenario in which there are no fiscal rules. 2 This case highlights the pitfall of domestic fiscal rules that do not anticipate a partial banking union: if the country carries a low weight at the supranational level, domestic constraints on spending increase the cost of implementing the agreement. The benefit of fiscal rules in terms of reducing rents is offset by the supranational transfers, which allow rents to increase. This creates a situation in which policymaker still derives a higher relative benefit from the supranational agreement due to rent seeking, while the households face higher relative costs. B.2 omestic Fiscal Rules Non-contingent on θ Consider the case in which Θ = [θ, θ] and domestic fiscal rules cannot be made contingent on the value of θ, and they are set without the anticipation of the partial banking union. The debt limit b for country is set so as to maximize expected household utility: subject to max E θ [uc x θ, x F θ, θ + wg θ + βwe b θ] B5 b γ σ Ru c x, x F, θ = 1 γ v r θ, B6a σ Ru c x, x F, θ = w g θ, B6b w g θ = 1 {b <b } w e b θ, B6c r θ + x θ + g θ e + βb θ, B6d b θ b. B6e 2 Corollary 6 discusses a comparison between a partial banking union and no banking union, with domestic fiscal rules in place in both cases. It can still be the case that household welfare in country is higher in a banking union with fiscal rules compared to a banking union without fiscal rules. B.3

29 Problem B5 can be simplified by noticing that if b binds for some θ Θ, then it binds for all θ > θ. For Θ = [θ, θ] R, problem B5 can then be expressed as: max E b θ θb [uc x θ, x F θ, θ + wg θ + βwe b ] + E θ< θb [uc x θ, x F θ, θ + wg θ + βwe b θ], B7 subject to B6a-B6e. In order to ensure that the objective in program B7 is concave in b, we make the following assumption about the government s utility from rent seeking: Assumption 1 For any set of feasible policies {x, g, r } and θ Θ that satisfy γ σ Ru c x, x F, θ = 1 γ v r, B8 γ w g = 1 γ v r, B9 x + g + r e 1 + β, B10 the following conditions are also satisfied: u c σ R u c 2 w g w g 2 γ v r 1 γ v r, 2 γ v r 1 γ v r, 2 B11 B12 where u c, w g, and v r denote the third derivatives of the utility functions. We proceed to analyze the problem by establishing the following lemmas. Lemma 2 The objective function B7 is strictly concave in b and the maximization problem has a unique solution b [ e /β, e ]. Proof. In Section B.3.2. B.4

30 Lemma 3 There exists θ G Θ, θ G < θ such that the debt limit imposed by the domestic fiscal rule is binding for policymaker if θ θ G. Proof. In Section B.3.3. The above lemmas establish that the solution to problem B7 is unique, and that the fiscal rule is binding for a subset of the possible realizations of θ. This setup captures the main trade-off of non-contingent fiscal rules: on the one hand, they limit the government s ability to engage in excessive spending in the first period; since part of first-period spending goes towards rents, the debt limit is beneficial to households because it reduces rents; on the other hand, fiscal rules limit government s ability to borrow in order to recapitalize banks in period 0. For country F, we assume the analogous decision problem to B7, such that debt limit b F e F is set. The following Lemma ensures that θ, full recapitalization of F banks are performed even under the fiscal rule x F θ = θi F. Lemma 4 There exists γ F such that γ F γ F, policymaker F provides full recapitalizations x F = θi F when domestic fiscal rules are in place. Proof. In Section B.3.4. Lemma 4 gives the equivalent result to that of Lemma 1. Consider the supranational authority s problem with debt limits as described above. Analyzing the equivalent problem to problem 27, we obtain the following results. Proposition 1 For γ F γ F, there exists a threshold η such that a partial banking union under domestic fiscal rules achieves a Pareto improvement compared to no banking union whenever η > η. Proof. Analogous to the proof to Proposition 6. The fiscal rules change the cost of funding recapitalizations, but they do not change the trade-off faced by the supranational authority between increasing B.5

31 recapitalizations and reducing public good provision. Therefore, the intuition from the case without fiscal rules carries over to the case with fiscal rules. We can also derive the equivalents of Corollaries 4 and 6. Corollary 2 omestic fiscal rules in country decrease the welfare of F households in the partial banking union, compared to the case without fiscal rules in country. Proof. Same as the proof to Corollary 4. Finally, the equivalent of Corollary 6 holds even if the fiscal rule is not made contingent on θ. The proof is analogous to the proof to Corollary 6. B.3 Proofs B.3.1 Proof of Corollary 6 Consider a fiscal rule that sets a binding debt limit of b, lower than b 0, the debt chosen by the government in the equilibrium without fiscal rules. enote by r, x, g, g1 the policies chosen by the government given τ, x and no fiscal rules, by r 0, x 0, g 0, g1 0 are the policies chosen by the government without a banking union and without fiscal rules, by r, x, g, g1 the policies chosen by the government given policies τ F R, x F R and fiscal rules b, b F, and by r 0, x 0, g 0, g1 0 the policies chosen by the government without a banking union, but under fiscal rule b in country. From the proof to Proposition 2, Step 5, without fiscal rules, η B < η such that the participation constraint for policymaker binds η < η B. Given proof to Proposition 6, which is analogous to that of Proposition 2, η B < η, such that the participation constraint for policymaker binds η < η B. Let η B = min{η B, η B }. Then, η < η B, 1 γ vr + γ U x, x F, g, g 1 = 1 γ vr 0 +γ U x 0, x F 0, g 0, g 0 1, B13 B.6

32 and 1 γ vr + γ U x, x F, g, g 1 = 1 γ vr 0 +γ U x 0, x F 0, g 0, g 0 1, B14 where by Assumption 1 and the condition that γ F γ F described in Lemma 1, x F = x F 0 = x F = x F 0 = θi F. A binding fiscal rule decreases the outside option for the government, since for b < b 0, 1 γ vr 0 + γ U x 0, x F 0, g 0, g 0 1 > 1 γ vr 0 + γ U x 0, x F 0, g 0, g 0 1. B15 Moreover, since the fiscal rules maximize household utility, U x 0, x F 0, g 0, g 0 1 > U x 0, x F 0, g 0, g 0 1. B16 Consider the case in which x F R = x = θi + r, with r defined implicitly by 1 γ v r = γ σ Ru θi, θi F. In this case, the maximum recapitalization is achieved, so x = x = θi. Conditions B13 and B14, together with vr = vr lead to U x 0, x F 0, g 0, g 0 1 U x, x F, g, g 1 > U x 0, x F 0, g 0, g 0 1 U x, x F, g, g 1. B17 B.3.2 Proof of Lemma 2 enote by U 0 θ the value of the household utility given the solution {r 0 θ, x 0 θ, g 0 θ, g 0 1 θ, b 0 θ} to policymaker s maximization problem without the partial banking union and without the fiscal rule. Also, denote by U 0 θ, b the value of household utility given the solution to policymaker s maximization problem without a partial banking union, but with a fiscal rule b. Finally, let θb denote the value of θ at which b 0 = b so B.7

33 g 0 1 θ = e b. Given fθ the p.d.f. for θ over Θ = [θ, θ], the function maximized by problem B7 is EUb = θb θ U 0 θfθdθ + θ θb U 0 θ, b fθdθ. B18 The function U 0 θ, b is a continuous and differentiable function of b, since uc, wg and vr are continuously differentiable. Also, θb is differentiable since it is a continuous function of u, w and v, derived from the solution b to policymaker s problem. Taking the first-derivative with respect to b, we obtain EUb = U 0 θb f θ θb b b + θ U 0 θb, b f θ θb Notice that for θ = θ, we have U 0 θ = U 0 θ, b, so Then, U 0 θ, b fθ dθ θb b b. B19 EUb θ U = 0 θ, b fθ dθ. B20 b θb b 2 EUb θ = 2U 0 θ, b fθ dθ U 0 θ, b f θ b 2 θb b 2 b θb b. B21 But U 0 θ,b f θ = 0 since any increase in b would make the debt constraint b θ b slack. b Therefore, 2 EUb θ 2 U = 0 θ, b fθ dθ. B22 b 2 θb b 2 B.8

34 Then 2 U 0 θ, b fθ b 2 < 0 2 EUb < 0. B23 b 2 The change in household utility due to the change in the binding debt limit b is given by U 0 θ, b b Then, = σ Ru c b + w g g b βw e b. B24 2 U 0 θ, b b 2 = [ σ R 2 u c x, x F 2, θ b g + w g 2 + βw e b ] b +w g 2 r b 2 B25 The first-order conditions to the Home government s problem give Then, γ σ Ru c x, x F, θ = 1 γ v r, B26a γ w g = 1 γ v r. B26b and γ σ R 2 u c x, x F, θ γ w g g b = 1 γ v r r b, b = 1 γ v r r b, b + r b + g b = β. B27a B27b B28 B.9

35 Combining the above conditions, So r b = β [ r r = b 2 b By Assumption 1, 1 γ v r γ σ R 2 u c γ β γ 1 γ v r v r 2 u c σ R u c w g 3 w g 3 2 r ] 1 1 γ v r. B29 γ w g 2 γ v r 1 σ R 2 u c + 1 w g. B30 b 2 0. B31 This, together with the concave increasing functions uc, wg implies and 2 U 0 θ, b b 2 < 0 B32 2 EUb b 2 < 0. B33 Given the strict concavity of the objective function, it follows that the maximization problem has a unique solution b [ e /β, e ]. B.3.3 Proof of Lemma 3 From the proof to Lemma 2, the first-order condition for the household expected utility maximization problem is given by θ U 0 θ, b fθ dθ = 0. B34 θb b B.10

36 Claim 1 Consider the case in which b < b θ, where b θ is the level of debt at which the utility of households is maximized when θ = θ max θ Θ. Proof. Let b < b θ. Then, θ < θ, U 0 θ,b < 0, due to the concavity of b U 0 θ, b. Since it is set to maximize Home household utility, b is lower than the level of debt that maximizes policymaker s utility when θ = θ.,it follows that θb < θ. So, for all nondegenerate probability distribution functions fθ, we have θ U 0 θ, b θb b fθdθ < 0. Then, b = b θ cannot be the solution to B34. B35 Since b < b θ, it follows that θ G Θ such that θ θ G, V 0 θ, b < V 0 θ. B.3.4 Proof of Lemma 4 From Lemma 1, θ Θ, there exists γ F θ such that x F = θi F γ F γ F θ. Then, it follows that γ F = max θ {γ F θ}. B.11