## wp18205 ## Source details **Canonical URL:** [wp18205](https://www.imf.org/-/media/files/publications/wp/2018/wp18205.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp18205.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp18205.pdf.json) --- ### 1. Introduction - Context and motivation - European capital-market integration featured a Northern “core” intermediating debt to a Southern “periphery”; the 2009+ crisis produced a reversal in North-to-South lending driven by downgrades in collateral values (tradable output and nontradable assets such as housing). - The decline in private capital flows into the periphery was severe enough to be classified a “sudden stop” using Calvo et al. (2004) methodology; ECB support and sovereign bailouts partially cushioned the impact. - Model framework and mechanisms - An “imperfect financial union” with politically-sovereign countries, integrated debt markets, and a collateral-based repayment friction; regions N (North/core) and S (South/periphery). - Households face borrowing limit equal to a fraction φ∈(0,1) of period-2 tradable and nontradable outputs; high Southern debt and low nontradable price bind the Southern constraint → region-wide sudden stop. - When the intra-union interest rate is endogenous, sudden stops depress equilibrium r_2 below unconstrained level; falling r_2 hurts lenders (North) and may help or hurt borrowers (South) depending on severity. - Two Pareto-improving supranational interventions (politically feasible given sovereignty) 1. Core-to-periphery governmental loan - Direct public capital flows from core to periphery repaid in period 2; raises r_2 and raises core-to-periphery lending but tightens private borrowing limits via lower p_{S,2}. - Requires periphery governments to have fiscal space to commit to repay. 2. Debt relief today + future budget-neutral fiscal commitment (tax tradable consumption τ_S, subsidize nontradable consumption η_S) - Future commitment raises p_{S,2} and possibly nontradable output, relaxing private borrowing constraint immediately and boosting private capital flows; increases r_2 and benefits core, which can share gains by forgiving debt. - Requires periphery commitment to future fiscal policies but not repayment. - Institutional implications - Regional coalition bargaining internalizing interest-rate effects can exhaust Pareto gains; the existence of ex ante agreement to enable crisis interventions depends on pre-crisis heterogeneity. ### 2. A Simple Financial Union — environment and first results - Environment and key equations (representative expressions retained) - Periods t∈{1,2}; regions i∈{N,S}; common riskless bond r_{i,j,2}=r_2 for all i,j. - Utility: U_{i,j} = (log c_{i,j,T1} + ν log c_{i,j,NT1}) + (log c_{i,j,T2} + ν log c_{i,j,NT2}) (equation (1)). - Borrowing constraint: fraction φ∈(0,1) of period-2 endowments pledgable (equation (3)); when binding, country experiences a sudden stop. - Aggregation assumptions: b_{i,j,1}=b_{i,1}; b_{S,1} = − b_{N,1} ∈ (0, (b_{S,1})_{max}); policies identical within region. - Reduced system characterized by {r_2, p_{S,2}, c_{S,T1}, c_{S,T2}, c_{N,T1}} solving (18)-(23). - Pareto efficiency of laissez-faire - First-best: c_{i,T2}/c_{i,T1} = y_{T2}/y_{T1} for all i (Lemma 1). - Laissez-faire unconstrained: r_2 = y_{T2}/y_{T1}; c_{N,T1}^* = 1/2 ( b_{S,1} + 2 y_{T1} ); c_{S,T1}^* = 1/2 ( − b_{S,1} + 2 y_{T1} ) (equation (25)). - Borrowing constraint threshold: b_{S,1} > b_{S,1}^ = 2 y_{T1} φ(1+ν) / (1 + φ ν) leads to constraint binding; constrained c_{S,C,T1} and c_{S,C,T2} given in (26); (b_{S,1})_{max} specified. - Interest-rate behavior when constraint binds: R_{S,2} > y_{T2}/y_{T1} > r_2 (Proposition 1). - Welfare implications (Proposition 1) - Define ̂ b_{S,1} = 6 y_{T1} φ(1+ν) / (1 + 2 φ + 3 φ ν) (< (b_{S,1})_{max}). - Northern welfare U_N: - = U_N^* for b_{S,1} ∈ [0, b_{S,1}^] - < U_N^* for b_{S,1} > b_{S,1}^ - Southern welfare U_S: - = U_S^* for b_{S,1} ∈ [0, b_{S,1}^] - > U_S^* for b_{S,1} ∈ ( b_{S,1}^, ̂ b_{S,1} ) - < U_S^* for b_{S,1} > ̂ b_{S,1} - Mechanism: binding Southern constraint reduces private North→South capital flows and depresses r_2; North loses from lower r_2; South may gain at moderate debt levels but loses at very high debt. ### 3. Pareto-improving interventions in the simple model - Institutional bargaining setup - Coalitional bargaining chooses {x_S, z_N, τ_S} maximizing Nash product ∏_{i∈{N,S}} ( U_i − U_i(0,0,0) )^{γ_i} subject to equilibrium constraints; γ_N = 1 − γ_S ∈ [0,1] (equation (28)). - Governmental loan ({x_S > 0, z_N > 0, τ_S = 0}) - When Southern constraint binds: c_{S,T2} = (y_{T2} − z_N) (1 − φ) / (1 + φ ν); p_{S,2} y_{NT2} = ν c_{S,T2}. - Small change properties: dz_N > 0 reduces c_{S,T2}, lowers p_{S,2}, reduces b_{S,2}; because R_{S,2} > r_2, increasing r_2 via loans can make both regions better off; ρ_{xz}(γ_N) ∈ [ 1/2 (1 + r_2/R_{S,2}), 1 ] > 0 determines dr_2 response and bargaining outcome. - If tools unlimited, governmental loans can restore first-best frontier; constraints x_S ≤ (x_S)_{max} or z_N ≤ (z_N)_{max} may bind. - Tax-subsidy with debt relief ({x_S > 0, z_N = 0, τ_S > 0}) - When constraint binds: c_{S,T2} = y_{T2} (1 − φ) / (1 + φ (ν + τ_S + ν τ_S)); p_{S,2} y_{NT2} = (ν + τ_S + ν τ_S) c_{S,T2}. - Small-change effects: dτ_S > 0 raises p_{S,2}, increases b_{S,2}, increases private borrowing; because R_{S,2} > r_2, policy can raise r_2 and be Pareto-improving; ρ_{xτ}(γ_N) ∈ [ 1/2 (1 + r_2/R_{S,2}), 1 ] > 0. - Debt-relief coefficient g_{NS} ≡ x_S / τ_S bounded between g_{NS}^{min} and g_{NS}^{max}; for very high b_{S,1}, g_{NS}^{min} < 0 so South may undertake τ_S even without debt relief. - Unlimited tools → bargaining can reach first-best frontier via equation (32); practical bounds may limit full exhaustion of Pareto gains. - Substitutability and tool limits - In the endowment model the two interventions are substitutes: many {x_S, z_N, τ_S} combinations can achieve same Pareto-improving first-best allocations when tools unlimited. - More governmental loan relative to debt relief → higher public capital flows and lower private capital flows. - If one tool constrained, use the other to exhaust remaining Pareto gains. - Mechanisms and extensions - Interventions operate by increasing pledgability of Southern period-2 endowments: - Governmental loan converts partially pledgable tradable endowment y_{T2} into fully pledgable lump-sum repayment revenue z_N. - Tax-subsidy raises the price of pledgable nontradable endowment y_{NT2}, increasing collateral value. - With a non-collateralizable sector, optimal fiscal design implies taxing non-collateralizable nontradables (θ_S > 0 negative subsidy) and using revenues to subsidize collateralizable nontradables; when tools unlimited, θ_S does not change main results; when tools limited, bargaining leads to θ_S < 0. - Transfer-effect: a period-1 transfer dx_S > 0 generates dr_2 > 0 and the provider partially recoups cost via improved intertemporal terms of trade (Lemma 3). - Endogeneity of r_2 is key: future commitments or repayments raise current r_2, enabling Pareto-improving transfers even with limited fiscal space. ### 4. Pre-crisis heterogeneity and institutional design (adding t = 0) - Pre-crisis setup and heterogeneity measure H - Add period t = 0; preferences extend with period-0 utility terms (equation (33)). - Degree of pre-crisis heterogeneity H ≡ (e_N − e_S) / e_T ∈ [0,1] with e_S = e_T − e_N ∈ [0, e_T/2] (equation (42)). - Institutional decision I ∈ {0,1} at start of period 0: I = 1 allows unlimited interventions in crisis; I = 0 forbids them. - Endogenizing b_{S,1} - b_{S,1} = 2 c^{T1}_S − 2 y_{T1} (1 − H) (equation (44)); expectations about I and γ_N affect b_{S,1}. - Constraint binds when H > H̄ = 3 φ (1 + ν) / (1 + φ ν) ⇔ b_{S,1} > b̄_{S,1} (equation (45)). - Lemma 4: an expected constraint reduces debt (b^{C}_{S,1} < b^{*}_{S,1}); an expected intervention increases debt; db_{S,γ_N 1} / dγ_N < 0. - Pre-crisis Pareto design results - Laissez-faire is first-best for H ∈ [0, H̄]; for H ∈ (H̄,1] laissez-faire lies interior to first-best frontier (Proposition 4). - Institutional-design Proposition 5 (thresholds H_1 and H_2, φ sufficiently low): - Institutions I(H) = 0 for H ∈ [0, H_1]; I(H) = 1 for H ∈ (H_1,1]. - Admissible bargaining powers γ_N ∈ [γ^{min}_N(H), γ^{max}_N(H)] for Pareto improvement in period 0: - For H ∈ (H_1, H_2]: γ^{min}_N(H) > 0 and γ^{max}_N(H) = 1. - For H ∈ (H_2,1]: γ^{min}_N(H) > 0 and γ^{max}_N(H) < 1. - Interpretation by heterogeneity regimes: - H ∈ [0, H̄]: union homogeneous; no institutions. - H ∈ (H̄, H_1]: slightly heterogeneous; constraint binds weakly; institutions not Pareto-improving ex ante. - H ∈ (H_1, H_2]: moderately heterogeneous; institutions Pareto-improving if γ_N sufficiently high. - H ∈ (H_2,1]: highly heterogeneous; institutions Pareto-improving only for intermediate γ_N. - Policy-relevant implications - Northern overborrowing fears: setting up crisis institutions can induce Southern pre-crisis overborrowing, worsening outcomes for North when H low. - When H moderate/high, crisis-time Pareto gains can outweigh pre-crisis adverse effects; design bargaining power γ_N to ensure fair sharing and avoid excessive Southern borrowing. ### 5. A More Elaborate Financial Union — production and ROW - Model extensions (periods t∈{1,2}) - Production by firms in tradable and nontradable sectors; capital investment decisions; firms borrow from ROW at rate r; households borrow inside F at rate r2 and from ROW subject to collateral φ_{ROW} ∈ (0,φ). - Capital inflows from ROW to F in period 1 are subject to limits while outflows are not; typically r2 ≥ r and North intermediates ROW funds earning premium r2 − r > 0. - Governments can implement two intervention types: - Governmental loan: x_S funded by Northern tax δ_N on y_{N}^{T1}, repaid via Southern tax ξ_S on y_{S}^{T2} (equation (60)); requires repayment commitment. - Tax-subsidy with debt relief: x_S funded by δ_N; Southern τ_S, η_S in period 2 budget-neutral (equations (61)-(62)); requires commitment to future fiscal actions but not repayment. - Laissez-faire and first-best (Lemma 7, Proposition 6) - First-best: c_{i}^{T2}/c_{i}^{T1} = r_2^*; k_{i}^{T1} = A_2 (α_T / r)^{1/(1−α_T)}; k_{i}^{NT1} = α_{NT} ν c_{i}^{T2}/r. - Laissez-faire unconstrained allocations given in (76); constrained Southern tradable consumption piecewise in (77). - Equilibrium pattern: South borrows from both North and ROW; North intermediates rest-of-world funds and earns premium r2 − r > 0. - Pareto-improving interventions and distortions (Proposition 7) - Interventions remain Pareto-improving starting from constrained equilibrium B_S1 > B_S1, but induce intratemporal distortions due to production and taxation. - Governmental loan (Φ_S^{T2}) reduces period-2 Southern tradable and nontradable output (dy_S^{T2}/dΦ_S^{T2} < 0; dy_S^{NT2}/dΦ_S^{T2} < 0) and raises union current account (dΨ_1/dΦ_S^{T2} > 0). - Tax-subsidy τ_S increases period-2 nontradable output (dy_S^{NT2}/dτ_S > 0), leaves tradable output unaffected, and lowers union current account (dΨ_1/dτ_S < 0). - Optimal policy lies on second-best frontier: interventions used until marginal reduction in intertemporal distortion equals marginal increase in intratemporal distortion. - When both tools available, both are used (ξ_S > 0 and τ_S > 0) to spread distortions; Φ_S^{T2} ≤ (Φ_S^{T2})_{Laffer} never binds; Φ_N^{T1} ≤ (Φ_N^{T1})_{max} may bind. - Pre-crisis period t = 0 and institutional design with production and ROW - Period-0 investment and borrowing determine B_S1; constraint binds when H > H̄ (expression in (92)). - Coalitions choose I ∈ {0,1} and γ_N at period 0; if I = 1, period-1 bargaining uses unlimited {δ_N, ξ_S, τ_S}. - Numerical simulations (illustrative parameterization) - Parameters: φ = 0.05, φ_{ROW} = 0.01, α_T = 0.5, α_{NT} = 0.2, ν = 1, r = 1, e_T = 0.5, A_1 = 2, A_2 = 4 ⇒ Π_1 = 0.5, Π_2 = 1, H = 0.28, r_1^* = 1.96, r_2^* = 1.65. - Simulation findings: - Expectation of binding Southern constraint reduces B_S1. - Expectation of crisis-time intervention increases B_S1 above laissez-faire; larger Southern bargaining power raises post-intervention B_S1. - Post-intervention B_S1 + B_N1 can increase above laissez-faire due to period-1 distortions dominating period-2 effects. - There exist (H, γ_N) configurations where regional crisis-time bargaining is Pareto-improving from pre-crisis perspective despite production and ROW exposure (e.g., with H̄ = 0.28, H_1 = 0.43, H_2 = 0.80 in simulations). ### 6. Conclusion — policy lessons and institutional recommendations - Financial integration of heterogeneous regions generates problems beyond standard optimal currency area concerns: institutions must facilitate efficient core→periphery capital flow and mitigate sudden stops arising from collateral-based frictions and endogenous r_2. - Two interventions—governmental loans and tax-subsidy packages with debt relief—can be Pareto-improving; their implementation feasibility hinges on political-sovereignty constraints and pre-crisis heterogeneity. - Baseline (endowment) model yields closed-form first-best restoration via interventions; more elaborate production model with ROW exposure yields second-best frontier because interventions create intratemporal distortions. - Main policy lesson (both baseline and elaborate models; crisis-time and pre-crisis perspectives): - For sufficiently heterogeneous financial unions, inter-regional lending and debt relief institutions need to be built. *Source: wp18205* ### 1. Introduction ........................................................................................................ ### 1. Introduction ### Context and motivation - European integration after post-WW2 began with greater openness in goods markets and proceeded to liberalization of services, labor, and finance; capital market integration in Europe in the early 2000s featured a Northern “core” intermediating debt purchases to a Southern “periphery.” - As Europe slid into crisis from 2009 onwards, there was a substantial reversal in North-to-South lending driven by downgrades in the value of collateral: tradable output in periphery economies had not kept pace with external debt repayment needs, and prices of collateralizable nontradable assets (such as housing) declined. - The decline in private capital flows into the periphery was severe enough to be classified a “sudden stop” using the Calvo et al. (2004) methodology; structural-model analyses corroborate large private capital-flow reversals. - Crisis-time support from the European Central Bank (low interest rates, Target 2 mechanism, debt-repurchase facilities) and sovereign bailouts (IMF and European supranational institutions) partially cushioned the impact. ### Model framework and key mechanisms - The paper builds a model of an “imperfect financial union”: politically-sovereign countries with integrated debt markets but a collateral-based repayment friction, divided into core (North) and periphery (South) regions. - Households in the union face a borrowing limit equal to a fraction of the value of their future tradable and nontradable outputs; when Southern households issue high debt and the nontradable price is low, the Southern borrowing constraint binds—interpreted as a region-wide sudden stop. - When the intra-union interest rate is endogenous (e.g., the union is financially closed or the North intermediates funds from the rest of the world and earns a premium), sudden stops depress the equilibrium interest rate below the unconstrained level. - For the lender (core), a decline in the interest rate reduces its intertemporal terms of trade and welfare relative to the unconstrained allocation. - For the borrower (periphery), moderate constraint binding can increase welfare, but severe binding makes them worse off. ### Pareto-improving interventions analyzed - Objective: Identify supranational fiscal policies that are Pareto-improving (benefit each country in both core and periphery) and feasible given political sovereignty; financial integration is taken as given (no restrictions on intra-union capital flows). - Two interventions that both raise the interest rate and core-to-periphery lending, but operate differently: 1. Core-to-periphery governmental loan - Directly increases public capital flows from core to periphery and reduces private capital flows by a smaller amount due to tightening of private borrowing limits when the nontradable price falls during loan repayment. - The core is willing to subsidize the governmental loan because it increases the interest rate on private loans, benefiting core households. - Implementation requires the periphery’s governments to have fiscal space (an ability to commit to repay that gets around the private sector’s constraint). 2. Debt relief today combined with future budget-neutral fiscal commitment in the periphery (tax tradable consumption, subsidize nontradable consumption) - The future fiscal commitment raises the future nontradable goods price and possibly nontradable output, immediately relaxing the private borrowing constraint and boosting private core-to-periphery capital flows. - The ensuing increase in the interest rate benefits the core; the core can share welfare gains by offering debt relief to the periphery. - This policy requires no periphery government repayment commitment—only a commitment to implement future fiscal policies. ### Main theoretical results and institutional implications - Section 2 constructs the simplest framework: an endowment-based, two-period financial union (North and South, each a continuum of countries), with a common riskless bond freely traded within a financially closed union. When inherited Southern debt is high, the borrowing constraint binds throughout the South. - In this simple model, both governmental loans and the debt-relief-plus-tax-subsidy package can restore the first-best allocation. - Regional coalition bargaining can exploit all Pareto improvements because coalitions internalize interest-rate effects that individual countries do not. - Section 3 addresses whether crisis-time interventions are Pareto-improving from earlier (pre-crisis) periods when borrowing constraints do not bind: - The ex ante agreement to allow crisis interventions depends on pre-crisis heterogeneity (how much larger Northern endowments are relative to the South). - If heterogeneity is low, the North rules out interventions ex ante because interventions cause higher Southern borrowing and lower Northern consumption pre-crisis. - If heterogeneity is moderate or high, interventions’ ability to increase the interest rate during crises can more than compensate the North for higher pre-crisis Southern borrowing; both regions then agree ex ante to establish institutions enabling governmental loans and debt relief during sudden stops. - Section 4 extends the model to allow borrowing from/lending to the rest of the world and to production (tradable and nontradable) rather than endowments: - If both North and South can issue debt to the rest of the world but face tight limits, the equilibrium resembles Chen et al. (2012): the North intermediates rest-of-world funds to the South and earns a riskless premium; previous results continue to apply because the premium is endogenous. - With production, interventions create intratemporal distortions: - Governmental loans require a tax on Southern tradable output in the repayment period, depressing tradable production. - The tax-and-subsidy component of the debt-relief package causes excessive Southern nontradable production. - Interventions no longer restore the first-best allocation but trace out a second-best frontier: both interventions are used up to the point where the marginal reduction in the constraint-induced intertemporal distortion equals the marginal increase in the policy-induced intratemporal distortion. - Excessive governmental loans yield an excessively large union-wide current account surplus. - Excessive use of the tax-subsidy intervention has the opposite effect. - Institutional design conclusion: inter-regional lending and debt relief institutions are needed when intra-union capital flows are large enough to trigger regional sudden stops; optimal combinations of policies balance intertemporal and intratemporal distortions. ### Relation to literature - Builds on literature on borrowing constraints and sudden stops; closest to Jeanne and Korinek (2013) and Benigno et al. (2013, 2016) in focusing on crisis-time interventions that make crises less painful rather than on crisis-prevention. - Differs from much of the literature by modeling a continuum of countries within a union with an endogenous interest rate, generating novel channels whereby lender regions can gain from transfers because transfers raise the equilibrium interest rate even with identical preferences across countries. - Distinct from optimal currency area literature (Mundell 1961 et al.) by focusing on financial integration and flexible prices rather than sticky-price asymmetric-shock problems; shows exchange-rate-like instruments are less powerful when intra-union capital flows generate regional sudden stops. - Complements literature on European capital flows post-crisis that debates adding impediments to capital flows versus stabilizing policies while accepting higher integration. *Source: wp18205 - 1. Introduction* ### 2. A Simple Financial Union ### 2. A Simple Financial Union ### 2.1. The environment - Union structure - Financial union F contains two regions i∈{N,S} (Northern core N and Southern periphery S), each region composed of a unit measure of countries j∈[0,1]. Rest of world = ROW. - Two periods t∈{1,2}. - Tradable goods are costlessly transported within the union and to ROW; nontradable goods are country-specific. - Financial union: common riskless bond and no government-imposed capital-flow restrictions within the union → r_{i,j,2}=r_2 for all i,j (equation (8)). - Financial and political closure to ROW in this section: - ∑_{i∈{N,S}} ∫_0^1 b_{i,j,t} dj = 0 for all t (equation (9)) - ∑_{i∈{N,S}} ∫_0^1 x_{i,j} dj = ∑_{i∈{N,S}} ∫_0^1 z_{i,j} dj = 0 (equation (10)) - Households and constraints - Representative household utility (log preferences): - U_{i,j} = (log c_{i,j,T1} + ν log c_{i,j,NT1}) + (log c_{i,j,T2} + ν log c_{i,j,NT2}) (equation (1)) - Budget and borrowing constraints (periods 1 and 2) summarized by (2) and (3). - Borrowing constraint parameter: φ∈(0,1) is the fraction of period-2 endowments pledgable as collateral (equation (3)). - When (3) binds for a country, the country is experiencing a sudden stop. - Intratemporal first-order conditions: (5) ν c_{i,j,T1} = p_{i,j,1} c_{i,j,NT1}; (6) (1+τ_{i,j}) ν c_{i,j,T2} = (1−η_{i,j}) p_{i,j,2} c_{i,j,NT2}. Intertemporal Euler: (4) (1+τ_{i,j}) c_{i,j,T2} ≥ r_{i,j,2} c_{i,j,T1}. - Governments and policy tools - Governments benevolent; can impose lump-sum transfers/taxes x_{i,j}, z_{i,j} and consumption taxes/subsidies τ_{i,j}, η_{i,j}. - Budget-neutrality within each country for consumption taxes/subsidies: τ_{i,j} c_{i,j,T2} = η_{i,j} p_{i,j,2} c_{i,j,NT2} (equation (7)). - Two intervention packages: - Governmental loans: lump-sum taxes/transfers {x_{i,j}, z_{i,j}} where recipient governments commit to repay in period 2. - Tax-subsidy with debt relief: lump-sum transfer in period 1 not repaid, followed by budget-neutral τ and η in period 2 (commitment to future fiscal actions but no repayment). - Heterogeneity and symmetry assumptions - Within-region common inherited debt: b_{i,j,1} = b_{i,1} for all j (equation (13)). - Southern periphery is borrower: b_{S,1} = − b_{N,1} ∈ (0, (b_{S,1})_{max}) (equation (14)). - Governments within same region follow identical policies: x_{i,j}=x_i, z_{i,j}=z_i, τ_{i,j}=τ_i, η_{i,j}=η_i (equation (15)). - Aggregation leads to identical equilibrium variables across countries within each region (equation (17)). - Core reduced system (variables to characterize): {r_2, p_{S,2}, c_{S,T1}, c_{S,T2}, c_{N,T1}} as functions of {b_{S,1}, x_S, z_N, τ_S} using equations (18)-(23), including: - 2 c_{N,T1} = b_{S,1} + y_{T1} − x_S + (y_{T2} + z_N) / r_2 (equation (18)) - c_{S,T1} + c_{S,T2} / r_2 = − b_{S,1} + y_{T1} + x_S + (y_{T2} − z_N) / r_2 (equation (19)) - R_{S,2} ≡ c_{S,T2} / c_{S,T1} ≥ r_2 / (1+τ_S) (equation (20)) - b_{S,2} ≡ r_2 (c_{S,T1} + b_{S,1} − y_{T1} − x_S) ≤ φ ( y_{T2} + p_{S,2} y_{NT2} − z_N ) (equation (21)) - p_{S,2} y_{NT2} = (ν + τ_S + ν τ_S) c_{S,T2} (equation (22)) - c_{S,T1} + c_{N,T1} = 2 y_{T1} (equation (23)) - Equations (20) and (21) hold with complementary slackness. ### 2.2. Pareto efficiency of the laissez-faire equilibrium - First best - Pareto weights identical within regions: λ_{i,j} = λ_i (equation (24)). - Lemma 1 (First-best allocation): irrespective of {λ_i}, the first-best allocation features c_{i,T2} / c_{i,T1} = y_{T2} / y_{T1} for all i. - Laissez-faire equilibrium (no interventions: x_S = z_N = τ_S = 0) - Unconstrained (ignoring (21)): r_2 = y_{T2} / y_{T1}, with - c_{N,T1}^* = 1/2 ( b_{S,1} + 2 y_{T1} ) - c_{S,T1}^* = 1/2 ( − b_{S,1} + 2 y_{T1} ) - c_{i,T2}^* = (y_{T2} / y_{T1}) c_{i,T1}^* for all i (equation (25)). - Borrowing constraint lower bound on Southern period-2 consumption: - c_{S,T2} ≥ c_{S,T2} = y_{T2} (1 − φ) / (1 + φ ν). - Threshold debt level where unconstrained c_{S,T2} violates the bound: - b_{S,1} > b_{S,1}^ = 2 y_{T1} φ(1+ν) / (1 + φ ν). - Constrained Southern consumption for b_{S,1} beyond threshold: - c_{S,C,T1} = [ (1 + 3 φ + 4 φ ν) / (1 + 2 φ + 3 φ ν) ] y_{T1} − [ (1 + φ + 2 φ ν) / (1 + 2 φ + 3 φ ν) ] b_{S,1} - c_{S,C,T2} = c_{S,T2} (equation (26)) - Maximum feasible inherited debt (b_{S,1})_{max} = (1 + 3 φ + 4 φ ν) / (1 + φ + 2 φ ν) y_{T1}. - Interest-rate and shadow-rate behavior when constraint binds: - R_{S,2} > y_{T2} / y_{T1} > r_2 (Proposition 1). - Welfare implications (Proposition 1) - Define ̂ b_{S,1} = 6 y_{T1} φ(1+ν) / (1 + 2 φ + 3 φ ν) ( < (b_{S,1})_{max} ). - Northern welfare U_N: - = U_N^* for b_{S,1} ∈ [0, b_{S,1}^] - < U_N^* for b_{S,1} > b_{S,1}^ - Southern welfare U_S: - = U_S^* for b_{S,1} ∈ [0, b_{S,1}^] - > U_S^* for b_{S,1} ∈ ( b_{S,1}^, ̂ b_{S,1} ) - < U_S^* for b_{S,1} > ̂ b_{S,1} - Mechanism: binding Southern constraint reduces North-to-South private capital flows and depresses r_2; North (savers) lose when r_2 falls, South (borrowers) may gain for moderate debt due to improved intertemporal terms but lose at very high debt. - Important characterization: when constraint binds, R_{S,2} rises above y_{T2}/y_{T1} while r_2 falls below y_{T2}/y_{T1}. ### 2.3. Pareto-improving interventions - Institutional set-up for interventions - Union-wide interventions require approval via bargaining among governments organized as regional coalitions. Laissez-faire is the outside option (veto possible). - Objective: choose {x_S, z_N, τ_S} to maximize Nash product ∏_{i∈{N,S}} ( U_i(x_S,z_N,τ_S) − U_i(0,0,0) )^{γ_i} subject to (18)-(23), γ_N = 1 − γ_S ∈ [0,1] (equation (28)). - Governmental loan intervention ({x_S > 0, z_N > 0, τ_S = 0}) - Assumes Southern governments can commit to repay in period 2 (despite private constraint binding). - When constraint binds, Southern period-2 variables: - c_{S,T2} = (y_{T2} − z_N) (1 − φ) / (1 + φ ν) - p_{S,2} y_{NT2} = ν c_{S,T2} - Small change effects: dx_S > 0, dz_N > 0 alter union period-2 tradable consumption and force r_2 to adjust; dr_2 expression given in text. - Welfare differentials for small changes (expressions provided): dU_N and dU_S depend on dx_S, dz_N, dr_2, and R_{S,2} > r_2. - Key result: because R_{S,2} > r_2, a governmental loan that increases r_2 (dr_2 > 0) can be Pareto-improving: North benefits from higher r_2 and can compensate South with period-1 transfers. - Pareto-improving set exists; bargaining (coalitions internalizing r_2 effects) yields dr_2 > 0 with dr_2 formula involving ρ_{xz}(γ_N) ∈ [ 1/2 (1 + r_2/R_{S,2}), 1 ] > 0 and dρ_{xz}/dγ_N < 0. - If loan size unlimited, bargaining can restore the first-best frontier; final allocations characterized by equation (29) and endpoints (equations (30)-(31)). - Effect on Southern private debt: b_{S,2} is lower than in laissez-faire (dz_N reduces South’s available tradables in period 2 and depresses p_{S,2}, tightening the private borrowing constraint in period 1). - Constraints x_S ≤ (x_S)_{max} or z_N ≤ (z_N)_{max} may prevent full exhaustion of Pareto gains; frontier becomes piecewise. - Tax-subsidy with debt relief ({x_S > 0, z_N = 0, τ_S > 0}) - Assumes South can commit in period 1 to undertake budget-neutral τ_S, η_S in period 2 but cannot commit to repay foreigners. - When constraint binds, Southern period-2 variables: - c_{S,T2} = y_{T2} (1 − φ) / (1 + φ (ν + τ_S + ν τ_S)) - p_{S,2} y_{NT2} = (ν + τ_S + ν τ_S) c_{S,T2} - Small change effects: dx_S > 0, dτ_S > 0 alter union period-2 tradable consumption and force r_2 to adjust; dr_2 expression given in text. - Welfare differentials for small changes (expressions provided): dU_N and dU_S depend on dx_S, dτ_S, dr_2, and R_{S,2} > r_2. - Key result: because R_{S,2} > r_2, tax-subsidy plus debt relief can generate dr_2 > 0 and thus Pareto improvements; bounds and coefficients involve ρ_{xτ}(γ_N) ∈ [ 1/2 (1 + r_2/R_{S,2}), 1 ] > 0 and dρ_{xτ}/dγ_N < 0. - Difference vs governmental loan: tax-subsidy with debt relief increases Southern private debt position b_{S,2} relative to laissez-faire (dp_{S,2} > 0 and db_{S,2} > 0). - Debt relief coefficient g_{NS} ≡ x_S / τ_S; g_{NS} bounds g_{NS}^{min} and g_{NS}^{max} derived in text. For very high b_{S,1}, g_{NS}^{min} < 0 meaning South would undertake τ_S even without debt relief and may transfer to North. - Unlimited tools → bargaining yields first-best frontier allocations satisfying equation (32), τ_S ∈ [ τ_{S,γ_N=0}, τ_{S,γ_N=1} ], with τ_{S,γ_N=0} < τ_{S,γ_N=1} and dτ_S/dγ_N > 0. - Constraints x_S ≤ (x_S)_{max} or τ_S ≤ (τ_S)_{max} limit exhaustion of Pareto gains. - Substitutability and combinations - The two interventions are substitutes in this endowment model: many combinations of {x_S, z_N, τ_S} can achieve the same Pareto-improving first-best allocations when tools are unlimited. - More governmental loan relative to debt relief → higher public capital flows and lower private capital flows within the union. - If limits bind on one tool, use the other to exhaust remaining Pareto gains. ### 2.4. Discussion and mechanisms - North-to-South capital flows and role of pledgability - Heterogeneous regions imply North → South capital flows. Pareto improvements hinge on reversing crisis-induced declines in these capital flows. - Interventions operate by increasing pledgability of Southern period-2 endowments: - Governmental loan converts partially pledgable Southern tradable endowment y_{T2} into fully pledgable lump-sum taxation revenue z_N in repayment period. - Tax-subsidy raises the price of pledgable nontradable endowments y_{NT2}, increasing collateral value. - Extended interpretation with non-collateralizable sector {y_{Mt}}: - Augmented budget-neutral condition τ_{i,j} c_{i,j,T2} = η_{i,j} p_{i,j,2} c_{i,j,NT2} + θ_{i,j} q_{i,j,2} c_{i,j,M2}. - Additional FOC: (1 + τ_{i,j}) γ c_{i,j,T2} = (1 − θ_{i,j}) q_{i,j,2} c_{i,j,M2}. - Small-policy comparative stat result: dU_i |_{dU_{−i}=0} ∝ φ(1 + ν) dτ_S − γ φ dθ_S. Result implies governments should tax non-collateralizable nontradables (θ_S > 0 negative subsidy) and use revenues to subsidize collateralizable nontradables. - Lemma 2: when (x_S, τ_S) unlimited, an additional subsidy θ_S > 0 does not change Proposition 3 results; when tools limited, bargaining leads to θ_S < 0. - Transfer effect and intertemporal terms of trade - Pure period-1 transfer dx_S > 0 generates dr_2 > 0 and dU_N(x_S,0,0) = 1 / c_{N,T1} [ − dx_S + (y_{T2} − c_{S,T2}) r_2^2 dr_2 ] ∈ ( − 1 / c_{N,T1} dx_S, 0 ) (Lemma 3). - Unlike classic transfer-effect literature (Keynes, Ohlin), here the provider of transfer partially recoups cost via improved intertemporal terms of trade (higher r_2) when recipient faces binding borrowing constraint. - Endogeneity of the interest rate is key - Promises of future taxes/subsidies or repayments in the South raise r_2 today, benefiting Northern savers and underpinning possibility of Pareto-improving policies even when fiscal space limited. - Contrast with literature assuming exogenous interest rate where borrower unambiguously benefits from tax-subsidy (Benigno et al., 2016). - Coalitional bargaining and institutional implications - Region-based coalitional bargaining internalizes effects on union-wide r_2, enabling exhaustion of Pareto gains; bargaining among individual countries ignoring r_2 effects would fail to achieve full Pareto gains. - Continuum-of-countries assumption supports the importance of regional coalitions and suggests pre-crisis union-wide institutions may be desirable to approve interventions. *Source: IMF Working Paper — "2. A Simple Financial Union" (content unit: wp18205 - 2. A Simple Financial Union)* ### 3. Pre-crisis Heterogeneity and Institutional Design ### 3. Pre-crisis Heterogeneity and Institutional Design ### Pre-crisis environment - Setup: - One additional pre-crisis period t = 0 is added to the model of section 2. - Representative household preferences: - V_{i,j} = (log c^{T,−1}_{i,j} + ν log c^{NT,−1}_{i,j}) + (log c^{T0}_{i,j} + ν log c^{NT0}_{i,j}) + U_{i,j}. (equation (33)) - Household beginning-of-period-1 debt: - b_{i,j 1} ≡ r_{i,j 1} (c^{T0}_{i,j} + p^{0}_{i,j} c^{NT0}_{i,j} − e_{i,j} − p^{0}_{i,j} y^{NT0}). (equation (34)) - Period-0 first-order conditions for consumption: - c^{T1}_{i,j} = r_{i,j 1} c^{T0}_{i,j}. (equation (35)) - ν c^{T0}_{i,j} = p^{0}_{i,j} c^{NT0}_{i,j}. (equation (36)) - Financial union assumptions: - Imperfect financial union: r_{i,j 1} = r_1 for all i,j. (equation (37)) - Financially closed union: c^{NT0}_{i,j} = y^{NT0} for all i,j. (equation (38)) - Tradable market clearing: Σ_{i∈{N,S}} ∫_0^1 c^{T0}_{i,j} dj = e_T. (equation (39)) - Regional heterogeneity: - Within-region common tradable endowment: e_{i,j} = e_i for all i,j. (equation (40)) - Degree of pre-crisis heterogeneity H: - e_S = e_T − e_N ∈ [0, e_T/2]. (equation (41)) - H ≡ (e_N − e_S) / e_T = 1 − 2 e_S / e_T ∈ [0,1]. (equation (42)) - Institutional design decision (at beginning of period 0): - I = {1 if institutions are set up; 0 if no institutions are set up}. (equation (43)) - If I = 0: x_S = z_N = τ_S = 0 during crisis. - If I = 1: unlimited interventions {x_S, z_N, τ_S} allowed during t ∈ {1,2}; governments choose γ_N ∈ [0,1] in period 0 and bargaining in period 1 proceeds with b_{S 1} and γ_N given. ### Endogenizing the debt level b_{S 1} - Households optimally choose c^{T0}_i consistent with c^{T1}_i; equilibrium r_1 and consumption ratios: - r_1 = 2 y_{T1} / e_T - c^{T0}_i = e_T / (2 y_{T1}) c^{T1}_i for all i. - Combining with household budget constraint yields: - b_{S 1} = 2 c^{T1}_S − 2 y_{T1} (1 − H). (equation (44)) - Interpretation: along this upward-sloping relation (indexed by H), household consumption-smoothing determines b_{S 1}; an increase in H (reduction in Southern initial endowment) shifts the line to the right. - Binding constraint / regional sudden stop condition: - H > H̄ = 3 φ (1 + ν) / (1 + φ ν) ⇔ b_{S 1} > b̄_{S 1}. (equation (45)) - Requirement for possibility of binding constraints: H < 1 implies φ < 1/(3 + 2 ν); earlier assumption φ < 1/(2 + ν) is thereby justified. - Lemma 4 (Inherited debt level): - When H > H̄, the Southern debt level is affected by the expectation of {I, γ_N}. - An expected constraint reduces debt: b^{C}_{S 1} (≡ b_{S, I=0 1}) < b^{*}_{S 1}. - An expected intervention increases debt: b_{S, γ_N=0 1} > b_{S, γ_N=1 1} > b^{C}_{S 1}. - db_{S, γ_N 1} / dγ_N < 0. ### Pareto-improving institutional design - First-best and laissez-faire: - Lemma 5 (First-best allocation): irrespective of {λ_i}, first-best features: - c^{T1}_i / c^{T0}_i = 2 y_{T1} / e_T and c^{T2}_i / c^{T1}_i = y_{T2} / y_{T1} for all i. - Proposition 4 (Laissez-faire equilibrium): For φ sufficiently low, φ ∈ [0, φ̂), there exists H ∈ (0,1) such that: - Laissez-faire is first-best for H ∈ [0, H̄], and is interior to first-best frontier for H ∈ (H̄, 1]. - In the interior region: c^{T1}_i / c^{T0}_i = 2 y_{T1} / e_T but c^{T2}_{S,C} = c^{T2}_S and R^S_2 > y_{T2} / y_{T1} > r_2. - There exists Ĥ ∈ (H̄, 1) such that: - V_N = V^*_N for H ∈ [0, H̄] and V_N < V^*_N for H ∈ (H̄, 1]. - V_S = V^*_S for H ∈ [0, H̄]; V_S > V^*_S for H ∈ (H̄, Ĥ); V_S < V^*_S for H ∈ (Ĥ, 1]. - Pre-crisis policy choice with unlimited policies: - Lemma 6: If unlimited policies {x_S, z_N, τ_S} can be selected in period 0 and H > H̄, bargaining between North and South achieves a Pareto-improving allocation on the first-best frontier; b_{S 1} is higher than in constrained equilibrium. - Institutional-design constraint (realistic obstacle): - Only {I, γ_N} chosen in period 0; interventions {x_S, z_N, τ_S} arise from period-1 bargaining. This creates a moral hazard / externality: - Households form expectations based on {I, γ_N} and choose pre-crisis borrowing that determines b_{S 1}, but are too small to internalize the effect of their borrowing on period-1 bargaining outcomes via r_2. - Result: households may borrow too little or too much relative to regional welfare-optimal choices; post-intervention allocations may make regions worse off than laissez-faire from period-0 perspective. - Welfare graphs and critical debt levels (panels III and IV of figure 6): - Southern welfare functions (panel III): - V_{S, γ_N=1} (b_{S 1}) = log{ κ (b_{S 1} + y_{T1} (1 − H)) ( (1 + 3φ + 4φν) / (1 + 2φ + 3φν) − (1 + φ + 2φν) / (1 + 2φ + 3φν) * b_{S 1} / y_{T1} )^{(1−φ)/(1 + φ ν)} }. - V_{S, γ_N=0} (b_{S 1}) = log{ κ (b_{S 1} + y_{T1} (1 − H)) × ( 2 − √( (1 + φ + 2φν)/(1 + 2φ + 3φν) + (1 + φ + 2φν)/(1 + 2φ + 3φν) b_{S 1} / y_{T1} )^{(1 + φ + 2φν)/(1 + φ ν)} )^2 }. - κ = e_T^2 / y_{T2} Π_{t={0,1,2}} y^{NT}_t is a constant. - For γ_N = 1, V_{S, γ_N=1}(b_{S 1}) evaluated at b^{C}_{S 1} has positive slope; it peaks at some (b_{S 1})_M > b^{C}_{S 1}. Southern welfare improves relative to laissez-faire if and only if b_{S, γ_N=1 1} ∈ (b^{C}_{S 1}, (b_{S 1})_M). - V_{S, γ_N=0}(b_{S 1}) lies everywhere above V_{S, γ_N=1}(b_{S 1}). Southern welfare improves under b_{S, γ_N=0 1} iff b_{S, γ_N=0 1} ∈ (b^{C}_{S 1}, (b_{S 1})_N). - Northern welfare functions (panel IV): - V_{N, γ_N=0} (b_{S 1}) = log{ κ (y_{T1} (1 + H) − b_{S 1}) ( (1 + φ + 2φν)/(1 + 2φ + 3φν) + (1 + φ + 2φν)/(1 + 2φ + 3φν) b_{S 1} / y_{T1} )^{(1 + φ + 2φν)/(1 + φ ν)} }. - V_{N, γ_N=1} (b_{S 1}) = log{ κ (y_{T1} (1 + H) − b_{S 1}) × ( 2 − √( (1 + 3φ + 4φν)/(1 + 2φ + 3φν) − (1 + φ + 2φν)/(1 + 2φ + 3φν) b_{S 1} / y_{T1} )^{(1−φ)/(1 + φ ν)} )^2 }. - At b^{C}_{S 1}, slope of V_{N, γ_N=0} is negative; the peak of V_{N, γ_N=0} occurs at debt level lower than b^{C}_{S 1}. - Increase in b_{S 1} induced by expected interventions unambiguously reduces Northern welfare under γ_N = 0. - V_{N, γ_N=1}(b_{S 1}) lies everywhere above V_{N, γ_N=0}(b_{S 1}). The increase b_{S, γ_N=1 1} above b^{C}_{S 1} improves Northern welfare iff b_{S, γ_N=1 1} ∈ (b^{C}_{S 1}, (b_{S 1})_P). - Proposition 5 (Institutional design) — thresholds and admissible γ_N: - Assume φ sufficiently low, φ ∈ [0, φ̃). Define H_1 and H_2 with H_1 ∈ (H̄, 1) and H_2 ∈ (H_1, 1) such that: - (i) Institutions are Pareto-improving in period 0 iff: - I(H) = 0 for H ∈ [0, H_1] - I(H) = 1 for H ∈ (H_1, 1] - (ii) The set of bargaining powers γ_N ∈ [γ^{min}_N(H), γ^{max}_N(H)] consistent with a period-0 Pareto improvement satisfy: - γ^{min}_N(H) > 0 and γ^{max}_N(H) = 1 for H ∈ (H_1, H_2] - γ^{min}_N(H) > 0 and γ^{max}_N(H) < 1 for H ∈ (H_2, 1] - Interpretation across heterogeneity regimes: - H ∈ [0, H̄]: union reasonably homogeneous; Southern borrowing constraint never binds; no institutions (I = 0). - H ∈ (H̄, H_1]: slightly heterogeneous; constraint binds weakly; Pareto gains small; institutions not Pareto-improving from period-0 perspective → I = 0. - H ∈ (H_1, H_2]: moderately heterogeneous; Pareto gains moderately large; institutions Pareto-improving if γ_N sufficiently high (γ_N > γ^{min}_N(H)), and γ^{max}_N(H) = 1. - H ∈ (H_2, 1]: highly heterogeneous; Pareto gains very large; institutions Pareto-improving only for intermediate γ_N: γ_N ∈ [γ^{min}_N(H), γ^{max}_N(H)] with γ^{max}_N(H) < 1. ### Discussion (policy-relevant findings and implications) - Overborrowing concern: - Northern fears that setting up crisis-time bargaining institutions induces Southern overborrowing are rational when union heterogeneity H is low. - Expectation of intervention increases pre-crisis Southern debt b_{S 1}, making sudden stop more severe, reducing union-wide interest rate r_2, and shifting welfare from North to South. - For low H, period-1 Pareto gains are too small to compensate the North even if all Pareto gains are allocated to the North. - When heterogeneity is moderate or high: - Overborrowing worries diminish relative to Pareto gains from crisis-time bargaining. - Pareto gains can be distributed between North and South to offset negative impacts on the North of higher pre-crisis Southern borrowing. - Higher b_{S 1} induced by anticipated interventions can be a positive signal of efficient exploitation of comparative advantages (savers vs borrowers) in union-wide capital market. - Design implication for bargaining power γ_N: - In highly heterogeneous unions, limit Northern bargaining power γ_N so the South receives some Pareto gains during t ∈ {1,2}. - If γ_N is too large, Southern households borrow excessively, lowering the South’s outside option in period-1 bargaining and potentially leaving the South worse off from period-0 perspective. *Source: IMF Working Paper — "3. Pre-crisis Heterogeneity and Institutional Design", wp18205* ### 4. A More Elaborate Financial Union ### 4. A More Elaborate Financial Union ### 4.1 The model environment (periods t ∈ {1,2}) - Structure: - Financial union F with two regions i ∈ {N,S}; each region composed of unit measure of countries j ∈ [0,1]. - Each country contains a unit measure of firms in tradable and nontradable sectors; production by investing in capital. - Households and firms can borrow inside F at interest rate r2 and from the rest of the world ROW at interest rate r. Households face borrowing limits from both sources; firms do not. - Capital inflows from ROW to F in period 1 are subject to borrowing limits whereas outflows from F to ROW are not; hence r2 ≥ r (so households borrow as much as they can from ROW and the remainder from intra-union market; firms borrow everything they need from ROW only). - Households: - Utility (log preferences): U_{i,j} = (log c_{i,j}^{T1} + ν log c_{i,j}^{NT1}) + (log c_{i,j}^{T2} + ν log c_{i,j}^{NT2}) (equation (47)). - Budget and borrowing constraints (equations (48)-(50)) include inherited intra-union debt b_{i,j}1 and ROW debt D_{i,j}t, transfers x_i and z_i, taxes τ_i and subsidies η_i, collateral fractions φ ∈ (0,1) and φ_{ROW} ∈ (0,φ). - First-order conditions (51)-(54) imply: - (1 + τ_i) c_{i,j}^{T2} ≥ r2 c_{i,j}^{T1} (51) - ν c_{i,j}^{T1} = p_{i,j}^1 c_{i,j}^{NT1} (52) - (1 + τ_i) ν c_{i,j}^{T2} = (1 − η_i) p_{i,j}^2 c_{i,j}^{NT2} (53) - D_{i,j}^2 = φ_{ROW} (π_{i,j}^{T2} + z_i) if r2 > r or (49) is binding; otherwise D_{i,j}^2 < φ_{ROW} (π_{i,j}^{T2} + z_i) (54). - Nontradable outputs are pledgable only within F (not to ROW). - Firms: - Profits and production functions (equations (55)-(59)): - π_{i,j}^{T1} = (1 − δ_i) y_{i,j}^{T1} − r k_{i,j}^{T2}, where y_{i,j}^{T1} = A_1^{1−α_T} (k_{i,j}^{T0})^{α_T} (55) - π_{i,j}^{T2} = (1 − ξ_i) y_{i,j}^{T2} − r k_{i,j}^{T1}, where y_{i,j}^{T2} = A_2^{1−α_T} (k_{i,j}^{T1})^{α_T} (56) - π_{i,j}^{NTt} = p_{i,j}^t y_{i,j}^{NTt} − r k_{i,j}^{NT,t−1}, where y_{i,j}^{NTt} = (k_{i,j}^{NT,t−1})^{α_{NT}} (57) - Optimal capital choices: - k_{i,j}^{T1} = A_2 ((1 − ξ_i) α_T / r)^{1/(1−α_T)} = (1 − ξ_i) α_T y_{i,j}^{T2} / r (58) - k_{i,j}^{NT1} = (α_{NT} p_{i,j}^2 / r)^{1/(1−α_{NT})} = α_{NT} p_{i,j}^2 y_{i,j}^{NT2} / r (59) - Government interventions (two types): - Governmental loan {x_i, δ_i, z_i, ξ_i}: - Northern governments provide transfers to Southern governments in period 1 repaid in period 2; funded by production taxes: - x_S = δ_N ∫_0^1 y_{N,j}^{T1} dj > 0; x_N = δ_S = 0 - z_N = ξ_S ∫_0^1 y_{S,j}^{T2} dj > 0; z_S = ξ_N = 0 (60) - Requires governments to commit to repay loans in period 2. - Tax-subsidy with debt relief {x_i, δ_i, τ_i, η_i}: - Northern governments provide transfers x_S = δ_N ∫_0^1 y_{N,j}^{T1} dj > 0; x_N = δ_S = 0 - Southern governments impose budget-neutral combination of taxes and subsidies in period 2: - τ_N = η_N = 0; τ_S, η_S > 0 and τ_S c_{S,j}^{T2} = η_S p_{S,j}^2 c_{S,j}^{NT2} for all j (61)-(62) - Requires commitment to undertake specific fiscal policy actions in period 2, but not to repay. - Resource and market-clearing constraints: - Intra-union loans market clears: b_S1 = − b_N1 ∈ (0, (b_S1)_{max}), and ∑_i ∫_0^1 b_{i,j}^2 dj = 0 (63). - Nontradable market clears within each country: c_{i,j}^{NTt} = y_{i,j}^{NTt} (64). - Union-wide tradable-good resource constraints (65)-(67) and current account Ψ_1 expression capturing principal repayments and new borrowing from ROW. ### 4.2 Pareto efficiency of laissez-faire and first best - First-best allocation (Lemma 7): - Irrespective of Pareto weights {λ_i}, the first-best features: - c_{i}^{T2} / c_{i}^{T1} = r_2^* - k_{i}^{T1} = A_2 (α_T / r)^{1/(1−α_T)} - k_{i}^{NT1} = α_{NT} ν c_{i}^{T2} / r for all i - Union-wide condition (67) holds with equality. - Laissez-faire equilibrium: - Given by equations (68)-(75) with δ_N = ξ_S = τ_S = 0. - Unconstrained allocations (equation (76)): - c_{i}^{*T1} = − 1/(2 + α_{NT} ν) ( B_i1 + (1 + α_{NT} ν) (B_N1 + B_S1)/2 ) + φ_{ROW} / r Π_2 - c_{i}^{*T2} = r_2^* c_{i}^{*T1} - Constrained Southern tradable consumption c_{S,C}^{T1} piecewise given by (77) with breakpoints B_S1 and B_{S,r}1, constants {ω_k} > 0, ω_1 ∈ (1/(2 + α_{NT} ν), 1), and c_{S}^{T2} = (1 − φ_{ROW} − φ) Π_2 / (1 + (α_{NT} + φ(1 − α_{NT})) ν). - Definitions and restrictions: - B_i1 = b_i1 + D_i1 + r k_{i}^{T0} + r k_{i}^{NT0} − y_i^{T1} - Π_t = (1 − α_T) A_t (α_T / r)^{α_T / (1−α_T)} - Φ_{i}^{T1} ≡ δ_i y_i^{T1}, Φ_{i}^{T2} ≡ ξ_i y_i^{T2}, Δ_S^{T2} = Π_2 [1 − (1 − ξ_S)^{1/(1−α_T)}] - (Φ_N^{T1})_{max} = y_N^{T1}; (Φ_S^{T2})_{Laffer} = α^{α_T/(1−α_T)} Π_2 (75). - Unconstrained intra-union interest rate r_2^* = 2 (1 − φ_{ROW}) Π_2 / ((1 + α_{NT} ν) ( − (B_N1 + B_S1) + 2 φ_{ROW} / r Π_2)) > r. - Equilibrium pattern: South borrows from both North and ROW; North intermediates funds from ROW and earns premium (r_2 − r) > 0. - Proposition 6 (Laissez-faire equilibrium) — key points: - Laissez-faire is first-best for some Pareto weights when B_S1 ∈ [0, B_S1]; lies interior to first-best frontier when B_S1 > B_S1. - In latter region c_{S,C}^{T2} = c_S^{T2} and R_S2 > r_2^* > r. - Ψ_1 flat for B_S1 ∈ [0, B_{S,r}1], but dΨ_1 / dB_S1 = 1 / (2 + α_{NT} ν) > 0 when B_S1 > B_{S,r}1. - Welfare: there exists \hat{B}_S1 < (B_S1)_{max} with U_N = U_N^* for B_S1 ∈ [0, B_S1] and U_N < U_N^* for B_S1 > B_S1; U_S = U_S^* for B_S1 ∈ [0, B_S1], U_S > U_S^* for B_S1 ∈ (B_S1, \hat{B}_S1), and U_S < U_S^* for B_S1 > \hat{B}_S1. - Define counterfactual U_{i,CF} fixing D_N2 = φ_{ROW} (Π + Φ_S^{T2}) even if it implies infeasible r_2 < r. For B_S1 > B_{S,r}1, U_N > U_{N,CF} and U_S < U_{S,CF}. ### 4.3 Pareto-improving interventions (periods 1–2) - Objective: start at constrained equilibrium (B_S1 > B_S1) and choose {δ_N, ξ_S, τ_S} to maximize Nash product subject to (68)-(75), bargaining power γ_N = 1 − γ_S ∈ [0,1]. - Key analytical expression (78) gives Pareto gain condition dU_i | dU_{−i} = 0 in terms of dΦ_S^{T2}, dτ_S and parameters including r2, R_S2, φ, φ_{ROW}, α_{NT}, ν, c_{i}^{T1}, c_S^{T2}, ξ_S. - At laissez-faire R_S2 > r2 and δ_N = ξ_S = τ_S = 0, so positive Pareto gains from regional bargaining. - Distortions and trade-offs: - Interventions are distortive; they reduce intertemporal distortion but create intratemporal distortions (tax-induced reductions in period-2 tradable output or increases in period-2 nontradable output). - Governmental loan (Φ_S^{T2}): - Reduces intertemporal distortion but lowers period-2 Southern tradable and nontradable output: dy_S^{T2} / dΦ_S^{T2} < 0 and dy_S^{NT2} / dΦ_S^{T2} < 0. - Raises union current account (dΨ_1 / dΦ_S^{T2} > 0) because it reduces period-1 borrowing. - Tax-subsidy with debt relief (τ_S): - Reduces intertemporal distortion but increases period-2 Southern nontradable output: dy_S^{T2} / dτ_S = 0 and dy_S^{NT2} / dτ_S > 0. - Lowers union current account (dΨ_1 / dτ_S < 0) because higher p_S^2 spurs nontradable-sector investment. - Outcome: - Interventions chosen until marginal reduction in intertemporal distortion equals marginal increase in intratemporal distortion — hence second-best frontier rather than first-best. - Proposition 7 (Second-best frontier): - When B_S1 > B_S1, each intervention achieves Pareto improvement relative to laissez-faire but with different effects on {y_S^{T2}, y_S^{NT2}, Ψ_1}. - Bargaining achieves allocation on second-best frontier defined by dU_i | dU_{−i} = 0 for i ∈ {N,S}. If both interventions available, both are used (ξ_S > 0 and τ_S > 0). - Condition Φ_S^{T2} ≤ (Φ_S^{T2})_{Laffer} never binds; Φ_N^{T1} ≤ (Φ_N^{T1})_{max} may bind. - Combining interventions: if both available, use both to spread intratemporal distortions across economy; deviating from second-best combination reduces Pareto gains (excessive Φ_S^{T2} causes too large fall in y_S^{T2} and current account surplus; excessive τ_S causes excessive y_S^{NT2} and current account deficit). ### 4.4 Pre-crisis environment (period t = 0) - Add pre-crisis period t = 0 with unit measure of firms in tradable and nontradable sectors investing in period 0 to produce in period 1. - Loans available: intra-union at r1 and ROW at r ≤ r1. Households face borrowing constraints; firms do not. - Households: - Utility V_{i,j} = (log c_{i,j}^{T,−1} + ν log c_{i,j}^{NT,−1}) + (log c_{i,j}^{T0} + ν log c_{i,j}^{NT0}) + U_{i,j} as in (79). - Period-0 debt and constraint (80)-(81): b_{i,j}1 ≡ r1 (c_{i,j}^{T0} + p_{i,j}^0 c_{i,j}^{NT0} − e_{i,j} − p_{i,j}^0 y_{NT0} − D_{i,j}1); D_{i,j}1 ≤ φ_{ROW} (π_{i,j}^{T1} + x_{i,j}). - First-order conditions for period 0: c_{i,j}^{T1} = r_{i,j}^1 c_{i,j}^{T0} (82); ν c_{i,j}^{T0} = p_{i,j}^0 c_{i,j}^{NT0} (83), and (81) binds. - Firms: - Period-0 investment decisions (84)-(85): - k_{i,j}^{T0} = A_1 ((1 − δ_i) α_T / r)^{1/(1−α_T)} = (1 − δ_i) α_T y_{i,j}^{T1} / r; y_{i,j}^{T1} = A_1 ((1 − δ_i) α_T / r)^{α_T/(1−α_T)} - k_{i,j}^{NT0} = (α_{NT} p_{i,j}^1 / r)^{1/(1−α_{NT})} = α_{NT} p_{i,j}^1 y_{i,j}^{NT1} / r - Market-clearing for period 0 (86)-(88): - c_{i,j}^{NT0} = y_{NT0}; ∑_i ∫_0^1 c_{i,j}^{T0} dj = e_T + ∑_i ∫_0^1 D_{i,j}1 / r dj; ∑_i ∫_0^1 D_{i,j}1 dj ≤ ∑_i ∫_0^1 φ_{ROW} (A_1 (k_{i,j}^{T0})^{α_T} − r k_{i,j}^{T0}) dj. - Competitive equilibrium extension includes {r_0, p_{i,j}^0, c_{i,j}^{T0}, c_{i,j}^{NT0}, k_{i,j}^{T0}, D_{i,j}1}. - Reduced expressions (regional symmetry suppressed): - c_i^{T0} = c_i^{T1} / r1 for all i - b_S1 = c_S^{T1} − r1 [ e_T / 2 (1 − H) + φ_{ROW} / r (Π_1 + Φ_N^{T1}) ] = − b_N1 - Derived formulas for B_S1 and B_S1 + B_N1 (89)-(90) with r1 = (c_S^{T1} + c_N^{T1}) / (e_T + φ_{ROW} / r (2 Π_1 − Δ_N^{T1} + Φ_N^{T1})) and Φ_N^{T1} ≤ (Φ_N^{T1})_{Laffer} = α^{α_T/(1−α_T)} Π_1 (91). - Binding condition for Southern borrowing constraint in pre-crisis environment: - Constraint binds when H > H̄ where H̄ is given in (92): - H̄ = (3 + 2 α_{NT} ν) (1 + φ_{ROW} / r 2 Π_1 / e_T) / ( φ (1 + (α_{NT} + (1 − φ_{ROW}) (1 − α_{NT})) ν) (1 − φ_{ROW}) (1 + (α_{NT} + φ (1 − α_{NT})) ν) ) - Equivalently H > H̄ ⇔ B_S1 > B_S1. - Institutional choice at period 0: - Coalitions decide I ∈ {0,1} and γ_N ∈ [0,1] at start of period 0. I = 0: no crisis-time interventions; I = 1: regional coalitions bargain in period 1 over unlimited {δ_N, ξ_S, τ_S} taking B_S1 and γ_N as given. ### 4.5 Pareto-improving institutional design and simulations - Closed-form solutions not available due to second-best frontier; use illustrative simulations to demonstrate existence of Pareto improvements from pre-crisis perspective. - Simulation setup (parameters chosen to compare to section 3.3 results; note these parameter values are used for simulation illustrations described in the chapter): - φ = 0.05, φ_{ROW} = 0.01, α_T = 0.5, α_{NT} = 0.2, ν = 1, r = 1, e_T = 0.5, A_1 = 2, A_2 = 4. - Therefore Π_1 = 0.5, Π_2 = 1, H = 0.28, unconstrained interest rates r_1^* = 1.96 and r_2^* = 1.65. - Simulation strategy: - (i) For any B_S1 and B_S1 + B_N1, find unconstrained allocation, constrained allocation, and two post-intervention allocations indexed by {dU_i = 0} for i ∈ {N,S}, drawing on proof of Proposition 7. - (ii) Endogenize both B_S1 and B_S1 + B_N1 for a given H. - Simulation findings (qualitative and quantitative highlights): - Expectation of binding Southern borrowing constraint reduces B_S1 (Panel I figure 10). - Pre-crisis welfare changes V_i − V_i^* follow baseline predictions: when H just binds constraint, constraint benefits South and hurts North; for larger H it hurts South as well (Panel II figure 10). - Expectation of crisis-time intervention increases B_S1 above laissez-faire (Panel I figure 11); larger Southern bargaining power raises post-intervention B_S1. - Post-intervention B_S1 + B_N1 can increase above laissez-faire due to period-1 distortions dominating period-2 effects (Panel II figure 11). - Pre-crisis Pareto improvements: - For chosen parameters H > H̄ = 0.28, simulations show values of H and γ_N for which regional bargaining during crises is Pareto-improving from pre-crisis perspective. - V_{S,γ_N=0} always lies above laissez-faire V_S; V_{S,γ_N=1} lies above V_S for H ∈ (H̄ = 0.28, H_2 = 0.80]. - V_{N,γ_N=0} lies below laissez-faire V_N everywhere; V_{N,γ_N=1} dips below V_N for H ∈ (H̄ = 0.28, H_1 = 0.43) and then exceeds V_N for H ∈ (H_1 = 0.43, 1]. - Conclusion: even with production, financial openness to ROW, and only distortive interventions available, there exist parameter configurations (H, γ_N) such that crisis-time regional bargaining yields Pareto improvements from the pre-crisis perspective. *Source: IMF Working Paper — "4. A More Elaborate Financial Union" (wp18205)* ### 5. Conclusion ### 5. Conclusion ### Financial-integration problems in heterogeneous unions - The financial integration of heterogenous regions of countries generates novel economic problems. - Over and above the risk-sharing concerns that lay at the heart of the literature on optimal currency areas, institutions must facilitate the efficient flow of capital from the core to the periphery, and mitigate financial crises that arise when flows are misdirected and/or excessive. - Waves of financial integration mediated by asymmetric debt flows are especially subject to sudden stops when financial frictions bind. - Lessons from the emerging markets literature may become relevant—and indeed exacerbated via an endogenous interest rate—even if the member countries of the union are all advanced. - A key element is that the institutions to efficiently manage capital flows may exist within but not between such countries, because cross-border flows are new and political sovereignty constraints limit the creation of supranational entities until crises occur. ### Model design and interventions analyzed - The authors design a model with the above environment in mind. - They identify two interventions—governmental loans and/or tax-subsidy packages with debt relief—which could be used to tackle union-wide financial imperfections, going beyond the standard toolkit of monetary and fiscal policies. - The analysis focuses on generating Pareto improvements so as to respect political sovereignty constraints. - The study analyzes changes in pre-crisis welfare to address the concern that crisis-time bargaining may generate pre-crisis overborrowing, finding that such concerns are dominated by the size of crisis-time Pareto gains when the degree of heterogeneity of the financial union is sufficiently high. ### Baseline model versus more elaborate financial union - The baseline model: - Captures the fundamental mechanisms while being simple enough to generate all final allocations, with and without intervention, in closed form. - Allows the authors to prove all main results and trace through all relevant economic channels. - The more elaborate financial union: - Preserves the result that Pareto improvements are possible. - Shows final allocations move from first-best to second-best. - Demonstrates that the two interventions have quite different effects on tradable and nontradable output. - Shows that the current account balance of the union as a whole becomes excessively large when the governmental loan is overused relative to tax-subsidies and debt relief. - Pre-crisis welfare results do not yield closed-form solutions in the more elaborate financial union, but numerical simulations are interpretable via comparison with the baseline-model proofs. ### Main policy lesson - For sufficiently heterogeneous financial unions, the main lesson holds for both the baseline model and the more elaborate financial union, and from both crisis-time and pre-crisis perspectives: - Inter-regional lending and debt relief institutions need to be built. *wp18205 - 5. Conclusion* ### References ### wp18205 - References ### References - Bayoumi, Tamim, and Barry Eichengreen. 1993. “Shocking Aspects of European Monetary Unification,” in Adjustment and Growth in the European Monetary Union, Francisco Torres and Francesco Giavazzi (eds.), Cambridge: Cambridge University Press. - Bayoumi, Tamim, and Paul R. Masson. 1995. “Fiscal Flows in the United States and Canada: Lessons for Monetary Union in Europe,” European Economic Review, Vol. 39 (2), 253-274. - Beetsma, Roel M. W. J., and Henrik Jensen. 2005. “Monetary and Fiscal Policy Interactions in a Micro-Founded Model of a Monetary Union,” Journal of International Economics, Vol. 67 (2), 320-352. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young. 2013. “Financial Crises and Macro-prudential Policies,” Journal of International Economics, Vol. 89, 453-470. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young. 2016. “Optimal Capital Controls and Real Exchange Rate Policies: A Pecuniary Externality Perspective,” Journal of Monetary Economics, Vol. 84, 147-165. - Benigno, Pierpaulo, and Federica Romei. 2014. “Debt Deleveraging and the Exchange Rate,” Journal of International Economics, Vol. 93, 1-16. - Bianchi, Javier. 2010. “Credit Externalities: Macroeconomic Effects and Policy Implications,” American Economic Review: Papers and Proceedings, Vol. 100 (2), 398-402. - Bianchi, Javier. 2011. “Overborrowing and Systematic Externalities in the Business Cycle,” American Economic Review, Vol. 101 (7), 3400-3426. - Blanchard, Olivier, Christopher J. Erceg, and Jesper Lindé. 2015. “Jump Starting the Euro Area Recovery: Would a Rise in Core Fiscal Spending Help the Periphery?,” NBER Working Paper No. 21426. - Blanchard, Olivier, and Francesco Giavazzi. 2002. “Current Account Deficits in the Euro Area: The End of the Feldstein-Horioka Puzzle?,” Brookings Papers on Economic Activity, Vol. 33 (2), 147-210. - Bolton, Patrick, and Olivier Jeanne. 2011. “Sovereign Default Risk and Bank Fragility in Financially Integrated Economies,” IMF Economic Review, Vol. 59 (2), 162-194. - Broner, Fernando, Aitor Erce, Alberto Martin, and Jaume Ventura. 2014. “Sovereign Debt Markets in Turbulent Times: Creditor Discrimination and Crowding-Out Effects,” Journal of Monetary Economics, Vol. 61, 114-142. - Caballero, Ricardo J., and Arvind Krishnamurthy. 2001. “International and Domestic Collateral Constraints in a Model of Emerging Market Crises,” Journal of Monetary Economics, Vol. 48 (3), 513-548. - Calvo, Guillermo A. 1998. “Capital Flows and Capital Market Crises: The Simple Economics of Sudden Stops,” Journal of Applied Economics, Vol. 1 (1), 35-54. - Calvo, Guillermo A., Alejandro Izquierdo, and Luis-Fernando Mejia. 2004. “On the Empirics of Sudden Stops: The Relevance of Balance-Sheet Effects,” NBER Working Paper No. 10520. - Chen, Ruo, Gian Maria Milesi-Ferretti, and Thierry Tressel. 2012. “External Imbalances in the Euro Area,” IMF Working Paper No. 12/236. - Cremers, Emily T., and Partha Sen. 2009. “Transfers, the Terms of Trade, and Capital Accumulation,” The Canadian Journal of Economics, Vol. 42 (4), 1599-1616. - de Grauwe, Paul. 1993. “The Political Economy of Monetary Union in Europe,” The World Economy, Vol. 16 (6), 653-661. - de Grauwe, Paul. 1996. “Monetary Union and Convergence Economics,” European Economic Review, Vol. 40 (3-5), 1091-1101. - de Grauwe, Paul. 2012. “The Governance of a Fragile Eurozone,” Australian Economic Review, Vol. 45 (3), 255-268. - Djajić, Slobodan, Sajal Lahiri, and Pascalis Raimondos-Møller. 1998. “The Transfer Problem and the Intertemporal Terms of Trade,” The Canadian Journal of Economics, Vol. 31 (2), 427-436. - Dogra, Keshav. 2014. Optimal Debt Restructuring and Lending Policy in a Monetary Union,” Mimeo, Columbia University. - Dornbusch, Rudiger, Ilan Goldfajn, and Rodrigo O. Valdés. 1995. “Currency Crises and Collapses,” Brookings Papers on Economic Activity, 219-293. - Eggertsson, Gauti B., and Michael Woodford. 2003. “The Zero Bound on Interest Rates and Optimal Monetary Policy,” Brookings Papers on Economic Activity, Vol. 34, Issue 1, 139-235. - Farhi, Emmanuel, and Iván Werning. 2012. “Dealing with the Trilemma: Optimal Capital Controls with Fixed Exchange Rates,” NBER Working Paper No. 18199. - Farhi, Emmanuel, and Iván Werning. 2013a. “Fiscal Unions,” NBER Working Paper No. 18280. - Farhi, Emmanuel, and Iván Werning. 2013b. “Fiscal Multipliers: Liquidity Traps and Currency Unions,” NBER Working Paper No. 18381. - Fornaro, Luca. 2015. “International Debt Deleveraging,” CEPR Discussion Paper No. DP10469. - Galí, Jordi, and Tommaso Monacelli. 2008. “Optimal Monetary and Fiscal Policy in a Currency Union,” Journal of International Economics, Vol. 76 (1), 116-132. - Giavazzi, Francesco, and Luigi Spaventa. 2010. “Why the Current Account May Matter in a Monetary Union: Lessons from the Financial Crisis in the Euro Area,” CEPR Discussion Paper No. 8008. - Goyal, Rishi, Petya Koeva Brooks, Mahmood Pradhan, Thierry Tressel, Giovanni Dell’Ariccia, Ross Leckow, and Ceyla Pazarbasioglu. 2013. “A Banking Union for the Euro Area,” IMF Staff Discussion Note No. 13/01. - Gourinchas, Pierre-Olivier, Thomas Philippon, and Dimitri Vayanos. 2016. “The Analytics of the Greek Crisis,” NBER Working Paper No. 22370. - Guerrieri, Veronica, and Guido Lorenzoni. 2011. “Credit Crises, Precautionary Savings, and the Liquidity Trap,” NBER Working Paper No. 17583. - Hale, Galina. 2013. “Balance of Payments in the European Periphery,” FRBSF Economic Letter, 2013-01. - Iacoviello, Matteo. 2005. “House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle,” American Economic Review, Vol. 95 (3), 739-764. - Jeanne, Olivier, and Anton Korinek. 2010. “Excessive Volatility in Capital Flows: A Pigouvian Taxation Approach,” American Economic Review, Vol. 100 (2), 403-07. - Jeanne, Olivier, and Anton Korinek. 2013. “Macroprudential Regulation Versus Mopping Up After the Crash,” NBER Working Paper No. 18675. - Kenen, Peter. 1969. “The Theory of Optimum Currency Areas: An Eclectic View,” in Monetary Problems of the International Economy, Robert A. Mundell and Alexander K. Swoboda (eds.), Chicago: University of Chicago Press. - Keynes, John M. 1929. “The German Transfer Problem,” The Economic Journal, Vol. 39, 1-7. - Korinek, Anton. 2011. “The New Economics of Prudential Capital Controls,” IMF Economic Review, Vol. 59 (3), 523-561. - Korinek, Anton, and Enrique G. Mendoza. 2014. “From Sudden Stops to Fisherian Deflation: Quantitative Theory and Policy,” Annual Review of Economics, Vol. 6, 299-332. - Lane, Philip R.. 2012. “The European Sovereign Debt Crisis,” Journal of Economic Perspectives, Vol. 26 (3), 49-68. - Martin, Philippe, and Thomas Philippon. 2014. “Inspecting the Mechanism: Leverage and the Great Recession in the Eurozone,” NBER Working Paper No. 20572. - Midrigan, Virgiliu, and Thomas Philippon. 2011. “Household Leverage and the Recession,” NBER Working Paper No. 18675. - McKinnon, Ronald. 1963. “Optimum Currency Areas,” The American Economic Review, Vol. 53, 717-724. - Mendoza, Enrique G. 2002. “Credit, Prices and Crashes: Business Cycles with a Sudden Stop,” in Preventing Currency Crises in Emerging Markets, S. Edwards and J. Frankel (eds.), Chicago: University of Chicago Press. - Mendoza, Enrique G. 2006. “Lessons from the Debt-Deflation Theory of Sudden Stops,” American Economic Review, Vol. 96 (2), 411-416. - Mendoza, Enrique G. and Katherine A. Smith. 2006. “Quantitative Implications of a Debt-Deflation Theory of Sudden Stops and Asset Prices,” Journal of International Economics, Vol. 70, 82-114. - Mendoza, Enrique G., Linda L. Tesar, and Jing Zhang. 2014. “Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies,” NBER Working Paper No. 20200. - Merler, Silvia, and Jean Pisani-Ferry. 2012. “Sudden Stops in the Euro Area,” Bruegel Policy Contribution No. 2012/06. - Mundell, Robert A. 1961. “A Theory of Optimum Currency Areas,” The American Economic Review, Vol. 51 (4), 657-665. - Mundell, Robert A. 1973. “Uncommon Arguments for Common Currencies,” in The Economics of Common Currencies, H. G. Johnson and A. K. Swoboda (eds.), London: Allen and Unwin Limited. - Obstfeld, Maurice. 2013. “Finance at Center Stage: Some Lessons of the Euro Crisis,” CEPR Discussion Papers 9415, C.E.P.R. Discussion Papers. - Ohlin, Bertil. 1929. “The Reparation Problem: A Discussion,” The Economic Journal, Vol. 39, 172-173. - Reis, Ricardo. 2013. “The Portuguese Slump and Crash and the Euro Crisis,” Brookings Papers on Economic Activity, Vol. 46 (1), 143-210. - Shambaugh, Jay C. 2012. “The Euro’s Three Crises,” Brookings Papers on Economic Activity, Spring 2012, 157-231. - Waysand, Claire, Kevin Ross, and John C. De Guzman. 2010. “European Financial Linkages: A New Look at Imbalances,” IMF Working Paper No. 10/295. ### Proofs (lemmas, propositions, and derived expressions) - Lemma 1 - c_{i,j}^{Tt} = 2 y^{Tt} λ_{i,j} / ∑_{i∈{N,S}} ∫_{0}^{1} λ_{i,j} dj ⇒ c_{i,j}^{T2} / c_{i,j}^{T1} = y^{T2} / y^{T1} for all i,j. - Setting λ_{i,j} = λ_{i} for all i,j yields the result. - Proposition 1 - Laissez-faire equilibrium is Pareto efficient iff b_{S}^{1} ∈ [0, \bar b_{S}^{1}]. - For b_{S}^{1} > \bar b_{S}^{1}, market and shadow interest rates: - r_{2} = y^{T2} (1/2 + φ(1+ν)/(1+φν))^{1/2} b_{S}^{1} + 1/2 y^{T1} - R_{S}^{2} = y^{T2} (1−φ)/(1+φν) (1/2 + φ(1+ν)/(1+φν)) − b_{S}^{1} (1/2 + 1/2 φ(1+ν)/(1+φν)) + y^{T1} (1/2 + 3/2 φ(1+ν)/(1+φν)) - Welfare derived by substituting equations (25) and (26) and market-clearing into expression (1). - Proposition 2 - Existence of Pareto improvements starting from b_{S}^{1} > \bar b_{S}^{1}. As intervention continues dr_{2} > 0. - Derivatives: - dc_{S}^{T2} = − (1−φ)/(1 + φν) dz_{N} < 0 - dc_{S}^{T1} = (2ρ/(x z (γ_{N})^{-1} r_{2})) (1−φ)/(1 + φν) dz_{N} > 0 ⇒ dR_{S}^{2} < 0 - dp_{S}^{2} = − 1 / y_{N}^{T2} (1−φ)ν/(1 + φν) dz_{N} < 0 - db_{S}^{2} = − φ(1 + ν)/(1 + φν) dz_{N} < 0 - When R_{S}^{2} = r_{2} = c_{i}^{T2}/c_{i}^{T1} = y^{T2}/y^{T1} for all i, union restored to first-best; frontier equation (29) derived. - Consumption at ends of Pareto frontier given by equations (30)-(31) (logs shown in source). - Bounds for z_{N,γ_{N}=0} and z_{N,γ_{N}=1} provided explicitly in source. - Proposition 3 - Similar existence of Pareto improvements when using tax-subsidy τ_{S}; derivatives: - dc_{S}^{T2} = − c_{S}^{T2} φ(1 + ν)/(1 + φ(ν+τ_{S}+ντ_{S})) dτ_{S} < 0 - dc_{S}^{T1} = 2ρ (γ_{N})^{-1} r_{2} c_{S}^{T2} φ(1 + ν)/(1 + φ(ν+τ_{S}+ντ_{S})) dτ_{S} > 0 ⇒ dR_{S}^{2} < 0 - dp_{S}^{2} = 1 / y_{N}^{T2} c_{S}^{T2} (1 + ν)/(1 + φ(ν+τ_{S}+ντ_{S})) dτ_{S} > 0 - db_{S}^{2} = c_{S}^{T2} φ(1 + ν)/(1 + φ(ν+τ_{S}+ντ_{S})) dτ_{S} > 0 - Frontier equation (32) and set [τ_{S,γ_{N}=0}, τ_{S,γ_{N}=1}] characterized explicitly in source. - Lemma 2 - A tax on the non-collateralizable nontradable good, θ_{S} < 0, can generate Pareto improvements starting from b_{S}^{1} > \bar b_{S}^{1}. - Amended Pareto frontier: - x_{S} = b_{S}^{1} − 2 y^{T1} φ (1 + ν + τ_{S} + ντ_{S} − θ (1−θ)^{-1} (γ + γ τ_{S})) / (1 + φ (ν+τ_{S}+ντ_{S} − θ (1−θ)^{-1} (γ + γ τ_{S}))) - Lemma 3 - Follows by substituting dx_{S} > 0 and dz_{N} = dτ_{S} = 0 into subsection 2.3 equations. - Lemma 4 - For H > \bar H, Southern debt level is intersection as in panel I of figure 6. - Comparisons and derivatives: c_{S}^{*T1}(b_{S}^{1}) = c_{S,C}^{T1}(b_{S}^{1}) = c_{S,γ_{N}=1}^{T1}(b_{S}^{1}) = c_{S,γ_{N}=0}^{T1}(b_{S}^{1}) - dc_{S}^{*T1}/db_{S}^{1} = −1/2 while dc_{S,C}^{T1}/db_{S}^{1} = − (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) < −1/2 - Further derivatives and inequalities presented in source, establishing ordering of b thresholds. - First-order condition Q = 0 and implication dc_{S,γ_{N}}^{T1}/dγ_{N} < 0 and db_{S,γ_{N}}^{1}/dγ_{N} < 0. - Lemma 5 - First-order conditions of first-best problem as in lemma 1; set λ_{i,j} = λ_{i} for all i,j. - Proposition 4 - Laissez-faire equilibrium Pareto efficient iff H ∈ [0, \bar H]. - When H ∈ (\bar H,1], b_{S,C}^{1} > b_{S}^{1}, c_{S,C}^{T2} = c_{S}^{T2} and R_{S}^{2} > y^{T2}/y^{T1} > r_{2}; other consumption relations stated. - Unconstrained and constrained consumption levels: - c_{S}^{*T1} = y^{T1} (1 − 1/3 H) - c_{S}^{*T2} = y^{T2}/y^{T1} c_{S}^{*T1} - c_{S}^{*T0} = e^{T}/2 y^{T1} c_{S}^{*T1} - c_{S,C}^{T1} = y^{T1} (2 + 4 φ + 6 φ ν)/(2 + 3 φ + 5 φ ν) − (1 + φ + 2 φ ν)/(2 + 3 φ + 5 φ ν) H - c_{S,C}^{T2} = c_{S}^{T2}, c_{S,C}^{T0} = e^{T}/2 y^{T1} c_{S,C}^{T1} - Welfare derived by substitution into expression (33). - Lemma 6 - Follows by applying section 2 analysis to model in section 3 and using relations c_{i}^{T0} = e^{T}/2 y^{T1} c_{i}^{T1}, relation (44), and utility expression (33). - Proposition 5 - Post-intervention consumption levels for γ_{N} = 1 and γ_{N} = 0: - c_{S,γ_{N}=1}^{T1} = (1/2) y^{T1} √( (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) (1−φ)/(1 + φ ν) )^{2} + 4 (2 − (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) H ) (1−φ)/(1 + φ ν) − (1/2) (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) (1−φ)/(1 + φ ν) y^{T1} - c_{S,γ_{N}=0}^{T1} = (1/2) (4 + (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) (1 + φ + 2 φ ν)/(1 + φ ν)) y^{T1} − (1/2) y^{T1} √{4 (1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) (1 + φ + 2 φ ν)/(1 + φ ν) (2 + H) + ((1 + φ + 2 φ ν)/(1 + 2 φ + 3 φ ν) (1 + φ + 2 φ ν)/(1 + φ ν))^{2}} - Welfare expressions V_{S*}, V_{S,C}, V_{S,γ_{N}=1}, V_{S,γ_{N}=0}, V_{N*}, V_{N,C}, V_{N,γ_{N}=1}, V_{N,γ_{N}=0} provided explicitly; κ = e^{T} / (2 y^{T2} Π_{t={0,1,2}} y_{N}^{Tt}). - Comparisons at H = \bar H and H = H_{max} = 2 (1 + 2 φ + 3 φ ν)/(1 + φ + 2 φ ν) discussed. - Lemma 7 - First-order conditions of first-best problem with ROW and production: - c_{i,j}^{T1} = (− (B_{N}^{1} + B_{S}^{1}) + 2 φ_{ROW} r Π_{2}) λ_{i,j} / ∑_{i∈{N,S}} ∫_{0}^{1} λ_{i,j} dj - c_{i,j}^{T2} = 2 (1 − φ_{ROW}) Π_{2}/(1 + α_{NT} ν) λ_{i,j} / ∑_{i∈{N,S}} ∫_{0}^{1} λ_{i,j} dj ⇒ c_{i,j}^{T2}/c_{i,j}^{T1} = r_{2}^{*} for all i,j - k_{i,j}^{T1} = A^{2} (α_{T} r)^{1/(1−α_{T})} and k_{i,j}^{NT1} = α_{NT} ν c_{i,j}^{T2}/r for all i,j. Assuming r_{2}^{*} > r, condition (67) holds with equality. - Proposition 6 - Laissez-faire equilibrium Pareto efficient iff B_{S}^{1} ∈ [0, \bar B_{S}^{1}]. - For B_{S}^{1} > \bar B_{S}^{1}, Southern tradable consumption levels given by equation (77); Northern tradable consumption: - c_{N,C}^{T1} = 1/(2 + α_{NT} ν) (− B_{N}^{1} + (φ_{ROW} r + (1−φ_{ROW}) r_{2}) Π_{2}) - c_{N,C}^{T2} = r_{2} c_{N,C}^{T1} - Market and shadow interest rates derived; welfare from substitution into expression (47). - Proposition 7 - Four-step proof establishing Pareto-optimal intervention combinations and existence of second-best frontier using reaction functions ξ_{S,SB}(τ_{S}) and τ_{S,SB}(ξ_{S}). - Conditions for positive Pareto gains for governmental loan and tax-subsidy: - R_{S}^{2}/r_{2} > (1+α_{NT} ν)(1−φ_{ROW}−φ) / [1 + (α_{NT} + φ(1−α_{NT}))(ν+τ_{S}+ντ_{S})] (1 + ξ_{S} α_{T}/(1−ξ_{S}−α_{T})) (1+α_{NT}(ν+τ_{S}+ντ_{S}))(1−φ_{ROW}−φ) / [1 + (α_{NT} + φ(1−α_{NT}))(ν+τ_{S}+ντ_{S})] − (φ_{ROW} r_{2}/r + (1−φ_{ROW})) ξ_{S} α_{T}/(1−ξ_{S}−α_{T}) - R_{S}^{2}/r_{2} > φ(1−α_{NT})(ν+τ_{S}+ντ_{S}) + α_{NT} (1 + ν) τ_{S} / (φ(1−α_{NT}) (ν+τ_{S}+ντ_{S})) - Interventions along Pareto boundaries generate dc_{S}^{T2} < 0, dc_{S}^{T1} > 0, and if r_{2} > r then dr_{2} > 0; reaction functions and compactness arguments lead to intersection and existence of second-best solutions. - Argument extends to interior of Pareto set and cases where r_{2} = r initially. ### Figures (captions and parameters) - Figures included in the source: Figures 1 through 11 with panel labels and axis annotations (e.g., "Laissez-faire equilibrium", "Impact of intervention", "How intervention begins", "How intervention ends", "Endogenizing the debt level b_{S}^{1}", "Institutional design", "Laissez-faire equilibrium with ROW and production", "Simulation: impact of constraint on pre-crisis welfare", "Simulation: institutional design"). - Selected figure captions and panel labels are provided verbatim in the source (e.g., "Figure 1. Laissez-faire equilibrium", panel labels I–IV; "Figure 5. How intervention ends", panels I–V; etc.). - Simulation parameters (used in figures): - φ = 0.05, φ_{ROW} = 0.01, α_{T} = 0.5, α_{NT} = 0.2, ν = 1, r = 1, e^{T} = 0.5, A_{1} = 2, A_{2} = 4. *Source: wp18205 - References (PDF chapter/section).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18205.pdf_