## _wp15218 ## Source details **Canonical URL:** [_wp15218](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15218.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15218.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15218.pdf.json) --- ### 2.1 Setup — model environment, agents, and financial constraint - Small open economy with three time periods t ∈ {0,1,2}. - Agents: unit mass of borrowers B and unit mass of savers S; foreigners F trade bonds at exogenous net interest rate of zero. - Consumption: - Traded goods c^i_T in all periods; non-traded goods c^i_N consumed only in period 1. - Period utility: U^i = u(c^i_T,0) + u(c^i_1) + u(c^i_T,2), u(c) = ln c. - Period-1 consumption index: c^i_1 = (c^i_T,1)^α (c^i_N,1)^{1−α} with expenditure share α and 1−α. - Budget constraints: - Period 0: c^i_T,0 + b^i_1 = b^i_0 + y^i_T,0 (b<0 denotes borrowing). - Period 1: c^i_T,1 + p c^i_N,1 + b^i_2 = y^i_T,1 + p y^i_N,1 + b^i_1 (p = relative price of non-traded to traded goods). - Period 2: c^i_T,2 = y^i_T,2 + b^i_2. - Financial constraint and amplification: - Courts recover at most fraction φ of period-1 income if borrower defaults. - Borrowing limit in period 2: −b^B_2 ≤ φ ( y^B_T,1 + p y^B_N,1 ). - φ is a pledgeability parameter; the relative price p appears in the constraint and generates financial amplification and pecuniary externalities. - Amplification: φ = 0 → no amplification; higher φ → greater amplification. - Regularity: Assumption 1: φ < ˆφ (ˆφ defined in Appendix A.1). ### 2.2 Decentralized equilibrium — mechanism, constraints, and planner comparison - Equilibrium definition: each agent maximizes utility subject to budget and financial constraint; nontraded goods market clears: ∑i (CiN,1 − yiN,1) = 0. - Period-1 liquid net worth: mi = yiT,1 + bi1; aggregate state (MB, MS). - Agent problem (period 1) yields Euler uiT,1 = uiT,2 + λi, with λS ≡ 0 (savers never constrained). - Traded/non-traded consumption and p: - p = uiN,1 / uiT,1 = (1 − α)/α · (ciT,1 / ciN,1). - Aggregated: p = (1 − α)/α · (CBT,1 + CST,1) / YN,1 (9). - p is strictly increasing in aggregate tradable spending CBT,1 + CST,1. - Unconstrained region ((MB, MS) ∈ Munc): - uiT,1 = uiT,2 for all agents. - ciT,1 = α/2 · (mi + p yiN,1 + yiT,2) (10). - p(MB, MS) = (1 − α)/(1 + α) · (MB + MS + YBT,2 + YST,2) / YN,1. - ∂p/∂MB = ∂p/∂MS = (1 − α)/[(1 + α) YN,1] > 0. - Marginal propensity to consume out of wealth (MPCi) = 1/2. - Constrained region ((MB, MS) ∈ Mcon) when borrowers constrained: - Borrower traded consumption: cBT,1 = α [ mB + p yBN,1 + φ(yBT,1 + p yBN,1) ] (11). - p(MB, MS) = (1 − α)/D · (MB + φ YBT,1 + MS + YST,2)/2 (12) with D = YN,1 − (1 − α)[YBN,1(1 + φ) + YSN,1/2] > 0 under Assumption 1. - Comparative statics: ∂p/∂MB = (1 − α)/D > ∂p/∂MS = (1 − α)/(2D) > 0. - MPCB = 1, MPCS = 1/2. - Mechanism: financial amplification—higher MB raises demand for non-traded goods, increases p, relaxes collateral, further raising demand. - Welfare derivatives with respect to aggregate net worth Mi: - ∂Vj/∂Mi = ujT,1 · (∂p/∂Mi) (yjN,1 − cjN,1) ︸︷︷︸ Rj i + λj · (∂p/∂Mi) φ yjN,1 ︸︷︷︸ Φj i (13). - Rj i: redistribution via price changes; Φj i: welfare gains from relaxation of collateral constraints. - Assumption 2: 1 + RB B − RB S > 0 (rules out immiserizing transfers). - Planner (period 0) internalizes how MB, MS affect equilibrium: - Planner picks Bi1 to maximize weighted welfare subject to period-0 feasibility. - Planner’s intertemporal condition introduces social benefit terms γi ∂Vi/∂Mi + γj ∂Vj/∂Mi; rewritten form highlights redistributive and constraint-relaxation channels (17). - Proposition 2: - (i) Constrained-efficient allocation satisfies equations (18) and (19) that introduce wedges in marginal rates of substitution proportional to λB and Φ terms. - (ii) If λB = 0, planner's conditions coincide with decentralized equilibrium. - (iii) If λB > 0, planner acts more precautionary in period 0 (uiT,0 > uiT,1); wedge larger for borrowers. - Implementation via taxes/subsidies and transfers: - Instruments: τi on bond purchases and lump-sum Ti in period 0; modified Euler: (1 − τi) uiT,0 = uiT,1. - Corollary 3: - (i) Any constrained efficient equilibrium implemented by taxes (τB, τS) with τB > τS plus transfers satisfying TB + TS = τB bB1 + τS bS1. - (ii) τB / τS = MPCB / MPCS. - (iii) Policymaker can achieve a Pareto improvement by solving planner problem and implementing taxes/transfers. - Mapping to capital controls τCC and macroprudential τMP: - 1 − τS = (1 − τCC) - 1 − τB = (1 − τCC)(1 − τMP) - Corollary 4: set τCC = τS and choose τMP so that 1 − τMP = (1 − τB) / (1 − τS). ### 2.5 Uncertainty — complete vs incomplete markets, underinsurance, and policy implications - Environment: stochastic shock ω ∈ Ω realized at beginning of period 1 that affects y_i^T,1(ω) or φ(ω); lowest realization makes borrower constraint binding. - Two market environments: - Complete markets in period 0 (Arrow securities available). - Incomplete markets (only uncontingent bonds). - 2.5.1 Complete markets: - Private Euler condition across states: u_i^T,0 = u_i^T,1(ω) (22) where constraint may bind between periods 1 and 2. - Planner problem parallels deterministic case but with expectations. - Proposition 5 (Underinsurance): - u_i^T,1(ω) / u_i^T,0 = 1 − λ_B(ω) u_B^T,0 Φ_B^i(ω) / (1 + R_B^B(ω) − R_B^S(ω)) (24). - When λ_B = 0 planner does not deviate from private smoothing; when λ_B > 0 planner insures more against binding states by carrying greater net worth into constrained states. - Policy implication: decentralized underinsurance → role for risk-sensitive capital controls and macroprudential policies; uncontingent dollar debt magnifies externalities and calls for higher regulation; contingent instruments (equity) create smaller externalities. - 2.5.2 Incomplete markets: - Private Euler: u_i^T,0 = E[u_i^T,1(ω)]. - Planner intertemporal condition (state-aggregate): - γ_i u_i^T,0 = γ_i E[u_i^T,1] + E[(γ_i u_i^T,1 − γ_j u_j^T,1) R_i^i] + γ_B E[λ_B Φ_B^i] - = γ_i E[u_i^T,1] + E(γ_i u_i^T,1 − γ_j u_j^T,1) E[R_i^i] + Cov(γ_i u_i^T,1 − γ_j u_j^T,1, R_i^i) + γ_B E[λ_B Φ_B^i] (25). - Covariance term is generally positive when borrowers constrained: planner's wedge is larger under incomplete markets because shoring up net worth has greatest redistributive effect when borrowers are most constrained. - Proposition 6 (Excessive Leverage, Incomplete Markets): - E[u_i^T,1(ω)] / u_i^T,0 = 1 − E[λ_B(ω) Φ_B^i(ω)] + Cov(γ_B u_B^T,1(ω) − γ_S u_S^T,1(ω), R_B^i(ω)) / γ_B u_B^T,0 · (1 + E[R_B^B(ω) − R_B^S(ω)]). - Policy takeaway: risk-sensitive regulation targeting liabilities that amplify externalities in bad states; prefer contingent instruments that provide insurance. ### 2.6 Numerical illustration — calibration, baseline results, and comparative statics - Calibration and baseline: - Endowment schedule: Y_i_T,0 = 1, Y_i_T,1 = α, Y_i_N,1 = 1 − α, Y_T,2 = 1. - α = 0.3. - Exit condition: B_i_3 = B_i_0 so B_i_0 = B_i_1 = B_i_2 = B_i_3 if constraints non-binding. - Target initial NFA: average NFA of East Asian crisis countries over five years prior to crisis = −40 percent of GDP. - Baseline assumption: borrowers carry all this debt, B_B_0 = −0.8; savers B_S_0 = 0. - Two states in period 1: sudden stop (low) and good times (high). - π = 5percent (probability of sudden stop). - Baseline φ(L) = 0.65 calibrated to match current account surplus of 10 percent and exchange rate depreciation of 25 percent during East Asian crisis. - φ(H) = ∞ (constraint loose in good times). - Baseline numerical results and optimal policy: - Optimal taxes: τ_B = 4 percent, τ_S = 2 percent. - These translate into capital control and macroprudential taxes of 2 percent each. - Quantitative effects of implementing policies: - Reduces current account reversal during crisis by about 2 percent. - Keeps the real exchange rate about 4 percent more appreciated. - Emphasis: intervention smooths excess volatility but does not fully avoid macro adjustment. - Comparative statics: wealth heterogeneity (Figure 5 summary) - Increasing |B_S_0 − B_B_0| with other parameters constant: - Example: increase borrowers’ initial debt from 40 to 50 percent of GDP and savers’ initial savings from 0 to 10 percent of GDP. - Results: - Greater wealth heterogeneity → borrowers more constrained → deeper crises (larger current account reversals and real depreciations). - Increasing wealth heterogeneity by 10 percent of GDP (B_B_0 = −1 and B_S_0 = 0.2) more than doubles current account reversal and real depreciation, and increases optimal taxes by more than threefold. - Footnote: if that heterogeneity increase accompanied by φ(L) = 0.9 to match 1997 crisis, optimal τs would increase only to 3.1 percent. - Comparative statics: borrowing limit φ(L) (Figure 6 summary) - For φ(L) ≥ 0.8 borrowing constraint loose → first-best allocation → no policy required. - For lower φ(L) borrowers deleverage → current account surplus and real depreciation emerge; magnitudes increase as φ(L) declines. - Optimal taxes rise as φ(L) declines. - Asset-price externalities (model variant): - Remove non-traded goods; agents hold capital k_i_1 and produce F_i(k_i_2); borrowers linear F_B(k_B_2) = A; savers with decreasing returns beyond first unit. - Borrowing constraint: −b_B_2 ≤ φ q k_B_2. - Asset price condition (26): q = F_i′(k_i_2) [φ + (1 − φ) u_i_T,1 / u_i_T,2]^(−1). - Key results: - Fisherian separation for savers implies ∂q/∂M_S = 0. - If borrowers constrained, ∂q/∂M_B > 0 → raising M_B raises asset prices and relaxes constraints. - Assumption 3: 1 + R_B_B > 0. - Planner implications: - No reason to distort savers’ intertemporal decisions: u_S_T,0 = u_S_T,1. - Planner imposes macroprudential restrictions on borrowers only: - τ_MP = τ_B = λ_B u_B_T,0 · Φ_B_B / (1 + R_B_B) > 0. - Planner does not impose capital controls on savers: τ_CC = τ_S = 0. - Corollary 8: macroprudential regulation suffices to address fire-sale asset-price externalities; capital controls are unnecessary in this channel. ### Policy conclusions and key quantitative takeaways - Exchange-rate-driven pecuniary externalities (emerging markets, significant foreign-currency debt): - Both capital controls and macroprudential regulation are generally necessary. - Capital controls raise net worth of both borrowers and savers; macroprudential regulation raises borrowers’ net worth further. - Instruments should be risk-sensitive and vary with crisis risk (higher with larger debt burdens, greater exposure to negative shocks, higher wealth inequality). - Asset-price-driven pecuniary externalities (advanced economies, domestic-currency debt): - Macroprudential regulation targeted at borrowers suffices; capital controls unnecessary because savers’ net worth does not affect asset prices. - Baseline quantitative takeaway: - Baseline optimal τ_B = 4 percent and τ_S = 2 percent → capital control and macroprudential taxes of 2 percent each produce measurable but partial mitigation (≈2 percentage point reduction in current account reversal; ≈4 percentage point smaller real depreciation). *Source: _wp15218 (selected excerpts from sections 2.1–2.6).* ### 2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ### _wp15218 - 2.1 Setup ### Model environment and agents - Small open economy with three time periods t∈{0,1,2}. - Two domestic agent types: a unit mass of borrowers B and a unit mass of savers S. - Foreigners F trade bonds at an exogenous net interest rate of zero (world interest rate = zero). - Domestic agents consume traded goods c^i_T and non-traded goods c^i_N; non-traded goods are consumed only in period 1. - Period utility specification: - U^i = u(c^i_T,0) + u(c^i_1) + u(c^i_T,2) - u(c) = lnc - consumption index in period 1: c^i_1 = (c^i_T,1)^α (c^i_N,1)^{1−α} with relative expenditure shares α and 1−α. - Initial conditions: - Agents enter period 0 with bond stocks b^i_0 and traded endowments y^i_T,0. - Period-0 budget constraint: c^i_T,0 + b^i_1 = b^i_0 + y^i_T,0, where b<0 corresponds to borrowing. - Period-1 budget constraint: c^i_T,1 + p c^i_N,1 + b^i_2 = y^i_T,1 + p y^i_N,1 + b^i_1, where p is the relative price of non-traded to traded goods (measure of real exchange rate). - Period-2 consumption: c^i_T,2 = y^i_T,2 + b^i_2. - Distribution of endowments and initial debt is such that in periods 0 and 1 borrowers optimally borrow (b^B_t < 0) and savers optimally save (b^S_t > 0). ### Financial constraint and amplification - Commitment/default friction: borrowers may divert income after receiving loans in period 1; courts can recover up to a fraction φ of their period-1 income. - Borrowing limit for borrowers in period 2: - −b^B_2 ≤ φ ( y^B_T,1 + p y^B_N,1 ) - Interpretation: φ is a pledgeability parameter; the relative price p appears in the constraint and generates financial amplification and pecuniary externalities. - Amplification properties: - If φ = 0, no amplification (borrowing limit is constant). - Higher φ implies greater amplification effects. - Regularity condition imposed: - Assumption 1: φ < ˆφ, where the upper limit ˆφ is characterized in Appendix A.1. ### Key mechanisms and economic interpretations (from surrounding sections in the source) - Capital controls vs macroprudential regulation: - Capital controls discriminate by residency of parties in financial transactions and segment domestic and international financial markets; international lenders and domestic agents face different effective interest rates. - Macroprudential policies restrict borrowing by domestic agents regardless of lender origin and segment borrowers from all lenders; borrowers and lenders face different effective interest rates. - Sudden-stop feedback loop when collateral constraints bind: - Depreciation reduces the value of collateral → tightening constraints → capital outflows → further exchange rate depreciation. - Pecuniary externality: private agents do not internalize their collective contribution to exchange rate declines and sudden-stop dynamics. - Policy implications highlighted in the source: - Combining capital controls and macroprudential regulation can be optimal in emerging economies vulnerable to sudden stops. - Macroprudential regulation reduces overborrowing; capital controls create an interest rate differential that induces domestic savers to save more, increasing aggregate wealth and resilience to sudden stops. - When borrowers delever, repayments to foreign lenders cause purchasing power to flow out (exchange rate depreciation), while repayments to domestic lenders keep purchasing power domestically, supporting demand and mitigating exchange rate pressure. - Optimal adjustment: higher risk of sudden stops (e.g., larger aggregate debt burden, greater exposure to negative shocks, higher wealth inequality) calls for higher settings of both instruments. As an economy develops, the role of capital controls diminishes relative to macroprudential regulation. ### Empirical illustration and numerical magnitudes (from the source) - East Asian crisis of 1997 example: - Current account reversal: more than 10 percentage points of GDP within one year. - Real exchange rate correction: about 25 percent. - Numerical illustration calibrated to these numbers with a 5 percent crisis probability finds optimal ex ante macroprudential taxes and capital controls of about 2 percent each, implying a combined tax burden on borrowers of 4 percent. *Source: _wp15218 - 2.1 Setup (excerpt provided).* ### 2.2 Decentralized Equilibrium ### _wp15218 - 2.2 Decentralized Equilibrium ### Definition and setup - Equilibrium: allocations and prices in which each agent i ∈ {B,S} maximizes utility (1) subject to budget constraints (2), (3), (4) and financial constraint (5), and markets for nontraded goods clear: ∑i (CiN,1 − yiN,1) = 0 (6). - Notation convention: individual variables lower-case, sector-wide/aggregate variables upper-case (e.g. CiN,1 is total nontraded consumption of sector i). - Period 1 liquid net worth: mi = yiT,1 + bi1. Aggregate state in period 1: (MB, MS) with mi = Mi in equilibrium. - Individual period 1 problem (backward induction): Vi(mi; MB, MS) = max_{bi2, ciT,1, ciN,1, ciT,2} u(ciT,1, ciN,1) + u(yiT,2 + bi2) subject to - bi2 + φ(yiT,1 + p yiN,1) ≥ 0 - mi + p(yiN,1 − ciN,1) ≥ ciT,1 + bi2 - Euler equation (from optimization): uiT,1 = uiT,2 + λi (8). Domestic savers S are never constrained so λS ≡ 0. ### Traded/non-traded consumption and real exchange rate - Optimality condition between traded and non-traded consumption yields exchange rate: - p = uiN,1 / uiT,1 = (1 − α)/α · (ciT,1 / ciN,1) - Aggregating and using nontraded goods market-clearing (6) gives: - p = (1 − α)/α · (CBT,1 + CST,1) / YN,1 (9) - Conclusion: the real exchange rate p is a strictly increasing function of aggregate tradable spending (CBT,1 + CST,1). ### Unconstrained Period 1 equilibria ( (MB, MS) ∈ Munc ) - When borrowers unconstrained, uiT,1 = uiT,2 for all i ∈ {B,S}. - Period 1 traded consumption: - ciT,1 = α/2 · (mi + p yiN,1 + yiT,2) (10) - Agent spends half of her income in period 1 and a fraction α on traded goods. - Exchange rate as function of (MB, MS): - p(MB, MS) = (1 − α)/(1 + α) · (MB + MS + YBT,2 + YST,2) / YN,1 - Comparative statics: - ∂p/∂MB = ∂p/∂MS = (1 − α)/[(1 + α) YN,1] > 0 (10) - Marginal propensity to consume out of wealth (MPCi) in unconstrained region: - MPCi = ∂[ciT,1 + p ciN,1]/∂mi = 1/2 ### Constrained Period 1 equilibria ( (MB, MS) ∈ Mcon ) - If borrowers constrained, they spend entire liquid net worth in period 1. Borrower traded consumption: - cBT,1 = α [ mB + p yBN,1 + φ(yBT,1 + p yBN,1) ] (11) - Savers’ traded consumption still given by (10). - Exchange rate linear function: - p(MB, MS) = (1 − α)/D · (MB + φ YBT,1 + MS + YST,2)/2 (12) - where D = YN,1 − (1 − α)[YBN,1(1 + φ) + YSN,1/2] > 0 under Assumption 1, φ < ˆφ. - Comparative statics in constrained region: - ∂p/∂MB = (1 − α)/D > ∂p/∂MS = (1 − α)/(2D) > 0 - MPCB = 1, MPCS = 1/2 - Mechanism: financial amplification—additional net worth for borrowers raises demand for non-traded goods, increases their price, relaxes collateral constraints, further raising prices and loosening constraints. ### Lemma 1 (summary) - (i) The real exchange rate p is an increasing function of period 1 spending on traded goods, which is increasing in MB and MS. - (ii) If unconstrained, ∂p/∂MB = ∂p/∂MS > 0. - (iii) If constrained, ∂p/∂MB > ∂p/∂MS > 0. ### Comparative statics of welfare with respect to aggregate net worth Mi - Derivative of value function of agent j with respect to aggregate net worth Mi: - ∂Vj/∂Mi = ujT,1 · (∂p/∂Mi) (yjN,1 − cjN,1) ︸︷︷︸ Rj i + λj · (∂p/∂Mi) φ yjN,1 ︸︷︷︸ Φj i (13) - Interpretation: - Rj i: redistribution between agents from price changes. Since nontraded goods only traded domestically, Rii + Rji = 0. If agent j is net seller (yjN,1 > cjN,1), higher price benefits j. - Φj i: relaxation of collateral constraints; higher p relaxes constraints and yields welfare benefit proportional to λj. - Assumption 2 imposed for well-behaved model: - 1 + RB B − RB S > 0 - Rules out “immiserizing” transfers (transferring one dollar from savers to borrowers should not reduce borrowers’ wealth via price changes). ### Period 0 equilibrium and planner question - Period 0 optimization: max_{bi1} u(bi0 + yiT,0 − bi1) + Vi(yiT,1 + bi1; MB, MS) leading to Euler equation uiT,0 = uiT,1 (perfect smoothing of traded consumption). - Question: Can a planner improve on decentralized allocation? ### Constrained planning problem (Section 2.3) - Planner maximizes weighted sum of domestic agents’ welfare with weights γi, subject to period 0 resource constraint: - max_{CiT,0, Bi1} ∑_{i∈{B,S}} γi [ u(CiT,0) + Vi(mi; MB, MS) ] - s.t. ∑_{i∈{B,S}} (CiT,0 + Bi1 − Bi0 − YiT,0) ≤ 0 (14) - Planner chooses Bi1 in period 0; subsequent allocations in periods 1 and 2 left to decentralized agents. Planner internalizes how (MB, MS) affect equilibrium and welfare. - Envelope condition: ∂Vi/∂mi = uiT,1. - Planner’s optimality conditions: - Weighted intratemporal equality: γi uiT,0 = γj ujT,0 (15) - Intertemporal condition (Euler with social effects): - γi uiT,0 = γi uiT,1 ︸︷︷︸ private benefit + γi ∂Vi/∂Mi + γj ∂Vj/∂Mi ︸︷︷︸ social benefit (16) - Rewriting (using (13), market clearing, λS = 0): - γi uiT,0 = γi uiT,1 + (∂p/∂Mi)[ (γi uiT,1 − γj ujT,1)(yiN,1 − ciN,1) ︸︷︷︸ redistributive + γB λB φ yBN,1 ︸︷︷︸ relax constraint ] (17) ### Proposition 2 (characterization of constrained efficient allocation) - (i) Any constrained efficient allocation satisfies: - uB T,1 / uB T,0 = 1 − λB / uB T,0 · ΦB B / (1 + RB B − RB S) (18) - uS T,1 / uS T,0 = 1 − λB / uB T,0 · ΦB S / (1 + RB B − RB S) (19) - (ii) If financial constraint loose (λB = 0), planner’s optimality conditions coincide with decentralized agents. - (iii) If financial constraint binding (λB > 0), planner introduces a wedge in the marginal rate of substitution and acts more precautionary in period 0 (uiT,0 > uiT,1). The wedge is larger for borrowers than for savers. - Intuition: - When unconstrained, pecuniary externalities (redistributions) cancel and allocation is Pareto efficient. - When borrowers constrained, planner can relax constraint by increasing net worth of borrowers and savers because both consume nontraded goods; higher net worth raises p, mitigating depreciations. Planner favors larger wedge for borrowers due to higher MPC. ### Implementation via taxes/subsidies and transfers (Section 2.4) - Policy instruments: tax/subsidy τi on bond purchases and lump-sum transfer Ti in period 0. Period 0 budget constraint becomes: - ciT,0 + (1 − τi) bi1 + Ti = bi0 + yiT,0 - Interpretation: if bi1 > 0 (saver), τi > 0 is subsidy to saving; if bi1 < 0 (borrower), τi > 0 is tax on borrowing. Positive τi induces agent i to carry more liquid net worth into following period. - Modified private Euler: (1 − τi) uiT,0 = uiT,1 - Corollary 3 (implementation results): - (i) Any constrained efficient equilibrium can be implemented by taxes (τB, τS) with τB > τS together with lump-sum transfers satisfying government budget: TB + TS = τB bB1 + τS bS1. - (ii) Relative size of taxes pinned down by MPCs: - τB / τS = MPCB / MPCS - (iii) Policymaker can achieve Pareto improvement on any decentralized equilibrium with binding constraints (where savers obtain US,DE) by solving the constrained planner problem maximizing u(CBT,0) + VB(MB; MB, MS) subject to feasibility and saver utility constraint, then implementing via taxes and transfers as in (i). - Explicit mapping of tax instruments into capital controls τCC and macroprudential regulation τMP: - Capital controls impose wedge between domestic agents and foreigners, increasing both τB and τS. - Macroprudential measures increase the rate at which domestic agents borrow but do not affect savers’ lending rate—i.e., increase τB but not τS. - Mapping: - 1 − τS = (1 − τCC) - 1 − τB = (1 − τCC)(1 − τMP) - Corollary 4: The regulated equilibrium in Corollary 3 can be implemented by setting τCC = τS and choosing τMP so that 1 − τMP = (1 − τB) / (1 − τS). *Source: _wp15218 - 2.2 Decentralized Equilibrium (PDF chapter content).* ### 2.5 Uncertainty ### 2.5 Uncertainty ### Overview - The section extends the baseline perfect-foresight framework to explicitly account for uncertainty via a stochastic shock ω∈Ω realized at the beginning of period 1 that affects either period 1 traded income y_i^T,1(ω) or the tightness of the financial constraint φ(ω). - The lowest realization of the two shocks is assumed sufficiently low to make the financial constraint on borrowers binding. - Analysis proceeds in two environments: - Complete markets in period 0 (Arrow securities available). - Incomplete period 0 financial markets (only uncontingent bonds). ### 2.5.1 Complete Markets - Setup: - Private agents contract state-contingent payoffs b_i^1(ω) sold by foreigners at price E[b_i^1(ω)] in period 0. - Reduced-form utility V_i(·) used; m_i(ω) = y_i^T,1(ω) + b_i^1(ω). - Private agents solve: max_{b_i^1(ω)} u(b_i^0 + y_i^T,0 − E[b_i^1(ω)]) + E[V_i(m_i(ω); M_B(ω), M_S(ω), φ(ω))] (equation 21) - Private Euler condition (consumption smoothing between periods 0 and 1 and across states): u_i^T,0 = u_i^T,1(ω) (equation 22) - Where the financial constraint is binding, smoothing between periods 1 and 2 is inhibited. - Planner problem under uncertainty: - Constrained social planner maximizes weighted sum of domestic welfare: max_{C_i^T,0, B_i^1(ω)} Σ_{i∈{B,S}} γ_i { u(C_i^T,0) + E[V_i(m_i(ω); M_B(ω), M_S(ω), φ(ω))] } subject to resource and accounting constraints (equation 23). - Planner’s intra- and inter-temporal optimality conditions yield allocations mirroring Proposition 2. - Proposition 5 (Underinsurance) - Any constrained efficient allocation in the domestic economy satisfies: u_i^T,1(ω) / u_i^T,0 = 1 − λ_B(ω) u_B^T,0 Φ_B^i(ω) / (1 + R_B^B(ω) − R_B^S(ω)) for i∈{B,S} (equation 24) - Interpretation: - When borrower constraint is loose (λ_B = 0), planner does not deviate from private smoothing condition (22). - When borrower constraint is binding (λ_B > 0), planner introduces a wedge so that u_i^T,0 > u_i^T,1(ω); wedge larger for borrowers. - Planner insures more against states with binding constraints by carrying greater net worth into constrained states of period 1 to push up the exchange rate and relax the constraint — trading off smoothing between periods 0–1 for better smoothing between periods 1–2. - Policy implication highlighted: - Decentralized equilibrium underinsurance implies a role for risk-sensitive capital controls and macroprudential policies: - Borrowers take excessively risky liabilities (e.g., dollar debt instead of local currency debt and equity). - Savers hold insufficient insurance (e.g., insufficient dollar reserves vs. risky local currency assets). - Uncontingent dollar debt imposes larger externalities and calls for higher regulation. - Contingent instruments that provide insurance (e.g., equity) create much smaller externalities. ### 2.5.2 Incomplete Markets - Setup: - Domestic agents can only borrow or save in uncontingent bonds; y_i^T,1(ω) and φ(ω) are stochastic. - Choice variable is b_i^1 (uncontingent) and m_i(ω) = y_i^T,1(ω) + b_i^1. - Private Euler condition (smoothing expected marginal utility): u_i^T,0 = E[u_i^T,1(ω)] - Planner inter-temporal optimality condition (state-aggregate form): γ_i u_i^T,0 = γ_i E[u_i^T,1] + E[(γ_i u_i^T,1 − γ_j u_j^T,1) R_i^i] + γ_B E[λ_B Φ_B^i] = γ_i E[u_i^T,1] + E(γ_i u_i^T,1 − γ_j u_j^T,1) E[R_i^i] + Cov(γ_i u_i^T,1 − γ_j u_j^T,1, R_i^i) + γ_B E[λ_B Φ_B^i] (equation 25) - Role of terms when saving one additional unit of net worth in period 0: - Reduces expected marginal utility of traded consumption. - Changes the exchange rate and expected redistribution between agents. - Leads to expected relaxation in the collateral constraint. - Redistributive effect decomposed into expected redistribution plus a covariance term. - Sign of covariance: - R_B^i = ∂p/∂M_i · (y_B^N,1 − c_B^N,1) with ∂p/∂M_B > 0 constant. - Cov(γ_B u_B^T,1 − γ_S u_S^T,1, R_B^B) = ∂p/∂M_i · Cov(γ_B u_B^T,1 − γ_S u_S^T,1, y_B^N,1 − c_B^N,1) - When borrowers are constrained, both γ_B u_B^T,1 − γ_S u_S^T,1 and y_B^N,1 − c_B^N,1 are above average; thus the covariance term is generally positive. - Proposition 6 (Excessive Leverage, Incomplete Markets) - Any constrained efficient allocation in the domestic economy with uncertainty and bond markets only satisfies: E[u_i^T,1(ω)] / u_i^T,0 = 1 − E[λ_B(ω) Φ_B^i(ω)] + Cov(γ_B u_B^T,1(ω) − γ_S u_S^T,1(ω), R_B^i(ω)) / γ_B u_B^T,0 · (1 + E[R_B^B(ω) − R_B^S(ω)]) - Proof sketch: - Combine inter-temporal conditions (25) for the two agents to obtain: E[γ_B u_B^T,1 − γ_S u_S^T,1] = − E[γ_B λ_B (Φ_B^B − Φ_B^S)] + Cov(γ_B u_B^T,1 − γ_S u_S^T,1, R_B^B − R_B^S) / (1 + E[R_B^B − R_B^S]) - Use R_B^S = R_B^B / 2 and Φ_B^S = Φ_B^B / 2 to derive planner’s optimal wedges. - Interpretation: - Because the covariance term is positive, the planner’s wedge is greater under incomplete markets than suggested by E[λ_B Φ_B^B] alone. - Intuition: shoring up net worth has the greatest redistributive effects when borrowers are most constrained and cannot insure, increasing the planner’s incentive to shore up net worth of both agents. ### Key Findings and Policy Recommendations - Under uncertainty, decentralized equilibria display underinsurance relative to constrained-efficient allocations; the planner introduces precautionary wedges when borrower constraints bind. - Financial instruments and contractual form matter for the pecuniary externality: - Uncontingent dollar debt that requires repayment in bad states magnifies externalities and warrants higher regulation. - Contingent instruments that provide insurance (e.g., equity) reduce externalities and are less problematic. - Policy implication: capital controls and macroprudential policies should be risk-sensitive, targeting liabilities and instruments that create larger externalities in bad states. *Source: _wp15218 - 2.5 Uncertainty* ### 2.6 Numerical illustration ### _wp15218 - 2.6 Numerical illustration ### Calibration and baseline parametrization - Endowment schedule (steady state if constraints non-binding): - Y_i_T,0 = 1, Y_i_T,1 = α, Y_i_N,1 = 1−α, Y_T,2 = 1 - α = 0.3 (weight of traded goods in period 1 consumption) - Exit condition: agents exit period 2 with the same amount of debt B_i_3 = B_i_0, implying B_i_0 = B_i_1 = B_i_2 = B_i_3 if constraints non-binding. - Initial net foreign asset (NFA) calibration: - Target: average NFA position of East Asian crisis countries over the five years prior to the crisis = -40 percent of GDP. - Baseline conservative assumption: borrowers carry all this debt, B_B_0 = −0.8; savers have no asset holdings, B_S_0 = 0. - Probabilities and financial states in period 1: - Two states: sudden stop (low) and good times (high). - π = 5percent (probability of sudden stop). - Baseline φ(L) = 0.65 to match a current account surplus of 10 percent and exchange rate depreciation of 25 percent during the East Asian crisis. - φ(H) = ∞ (borrowing constraint loose in good times). ### Baseline numerical results and optimal policy - Under the baseline parametrization the model predicts: - Optimal taxes on borrowers τ_B = 4 percent. - Optimal taxes on savers τ_S = 2 percent. - These translate into capital control and macroprudential taxes of 2 percent each. - Quantitative effects of implementing these policies: - Reduces the current account reversal during the crisis by about 2 percent. - Keeps the real exchange rate about 4 percent more appreciated. - Purpose emphasized: prudential policy intervention aims to smooth excess volatility by internalizing associated externalities rather than to completely avoid macroeconomic adjustment. ### Comparative statics: Wealth heterogeneity (Figure 5 summary) - Experiment: increase wealth heterogeneity |B_S_0 − B_B_0| while holding other parameters constant. - Example change: increase borrowers’ initial debt from 40 to 50 percent of GDP and savers’ initial savings from 0 to 10 percent of GDP. - Key behaviors illustrated: - As wealth heterogeneity increases, borrowers become more constrained → deeper crises: greater current account reversals and larger real exchange rate depreciations. - Planner vs decentralized allocations: - Dashed (red): decentralized equilibrium. - Solid (blue): planner’s optimal allocation. - Dotted (black): first-best equilibrium (no financial constraints). - Magnitudes reported: - Increasing wealth heterogeneity by just 10 percent of GDP (i.e. setting B_B_0 = −1 and B_S_0 = 0.2) more than doubles the current account reversal and real depreciation, and increases optimal taxes by more than threefold. - Note (footnote): If that increase in wealth heterogeneity is accompanied by a simultaneous increase in borrowing capacity in the low state to φ(L) = 0.9 to match the 1997 crisis current account and real depreciation, the optimal capital controls and macroprudential taxes would increase only to 3.1 percent. ### Comparative statics: Borrowing limit φ(L) (Figure 6 summary) - Behavior as φ(L) varies: - For φ(L) ≥ 0.8 the borrowing constraint is loose and the economy replicates the first-best allocation → no policy intervention required. - For lower values of φ(L), borrowers are forced to deleverage → current account surplus and real depreciation appear; magnitude increases as φ(L) declines (tighter borrowing limit). - Effect on optimal taxes: optimal taxes rise as φ(L) declines and constraints tighten. ### Asset price externalities: model variant and implications - Model modification: - Drop non-traded goods; agents obtain endowment k_i_1 of capital goods traded domestically in period 1 and produce output F_i(k_i_2) in period 2. - Assume borrowers have linear production: F_B(k_B_2) = A k_B_2. - Savers: F_S′(0) = A but F_S′′(k_S_2) < 0 (decreasing returns after first unit). - Borrowing constraint linked to asset value q: −b_B_2 ≤ φ q k_B_2. - Asset price condition (equation (26)): - q = u_i_T,2 F_i′(k_i_2) / ((1−φ) u_i_T,1 + φ u_i_T,2) = F_i′(k_i_2) [φ + (1−φ) u_i_T,1 / u_i_T,2]^(−1) - Key mechanics and lemmas: - Fisherian separation for unconstrained savers implies λ_S = 0 and u_S_T,1 = u_S_T,2, hence q = F_S′(k_S_2). - Asset prices are independent of savers’ liquid net worth: - ∂q/∂M_S = 0. - If borrowers unconstrained, q = A (unconstrained level) and is independent of borrowers’ net worth. - If borrowers constrained, ∂q/∂M_B > 0: increasing borrowers’ net worth raises asset prices (relaxes borrowing constraints). - Redistributive effect on borrowers when constrained: - R_B_B = (∂q/∂M_B) (k_B_1 − k_B_2). - Impact of higher M_B on borrowing constraint: - Φ_B_B = (∂q/∂M_B) φ k_B_2 > 0. - Assumption 3: 1 + R_B_B > 0 (ensures extra liquidity to borrowers does not immiserize them). - Planner’s implications: - Planner recognizes ∂q/∂M_S = 0 → no rationale to distort savers’ intertemporal decisions (planner’s Euler for savers identical to laissez-faire: u_S_T,0 = u_S_T,1). - Planner intervenes on borrowers when constraint binding; planner’s Euler for borrowers: - u_B_T,0 = u_B_T,1 + λ_B Φ_B_B / (1 + R_B_B) - Planner shores up liquid net worth of borrowers to reduce fire sales and asset price declines without distorting savers. - Corollary 8 (Asset Price Externalities): - Planner imposes macroprudential restrictions on borrowers: - τ_MP = τ_B = λ_B u_B_T,0 · Φ_B_B / (1 + R_B_B) > 0 - Planner does not impose capital controls on savers: - τ_CC = τ_S = 0 - Interpretation: macroprudential regulation is sufficient to address pecuniary externalities linked to fire sales in asset prices; capital controls are superfluous in this channel. ### Summary of policy conclusions (numerical-illustration insights) - For exchange-rate-driven pecuniary externalities (emerging markets with significant foreign currency debt): - Both capital controls and macroprudential regulation are necessary; capital controls raise net worth of both borrowers and savers, macroprudential regulation raises borrowers’ net worth further. - Optimal capital control and macroprudential taxes should be varied over time with crisis risk (affected by level of debt, wealth inequality, incidence of adverse shocks). - For asset-price-driven pecuniary externalities (advanced economies with domestic-currency debt): - Macroprudential regulation targeted at borrowers suffices; capital controls are unnecessary because savers’ net worth does not affect asset prices. - Quantitative takeaway from baseline calibration: - Baseline optimal τ_B = 4 percent and τ_S = 2 percent → capital control and macroprudential taxes of 2 percent each yield measurable but partial mitigation (≈2 percentage point reduction in current account reversal; ≈4 percentage point smaller real depreciation). *Source: _wp15218 - 2.6 Numerical illustration* ### References ### _wp15218 - References ### References - Alfaro, Laura, Anusha Chari and Fabio Kanczuk, 2014, “The Real Effects of Capital Controls: Liquidity Constraints and Firm Investment,” manuscript, Harvard. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young, 2010, “Revisiting Overborrowing and Its Policy Implications”, in Central Bank of Chile: Analysis and Economic Policies Series, Vol. 14. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young, 2012, “Optimal Policy for Macro-Financial Stability,” manuscript, LSE. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young, 2013a, “Capital Controls or Real Exchange Rate Policy? A Pecuniary Externality Perspective,” manuscript, LSE. - Benigno, Gianluca, Huigang Chen, Christopher Otrok, Alessandro Rebucci, and Eric R. Young, 2013b, “Central Bank of Chile“Financial Crises and Macro-Prudential Policies”, Journal of International Economics 89(2), pp. 453-470. - Bianchi, Javier, 2011, “Overborrowing and Systemic Externalities in the Business Cycle,” American Economic Review 101(7), pp. 3400-3426. - Bianchi, Javier, and Enrique Mendoza, 2010, “Overborrowing, Financial Crises and ’Macro-prudential’ Taxes,” NBER Working Paper w16091. - Blanchard, Olivier J., Jonathan D. Ostry, Atish R. Ghosh, and Marcos Chamon, 2014, “Capital Flow Management,” manuscript. - Caballero, Ricardo J., and Arvind Krishnamurthy, 2003, “Excessive Dollar Borrowing: Financial Development and Underinsurance,” Journal of Finance 58(2), pp. 867-894. - Claessens, Stijn, Swatish R. Ghosh, Roxana Mihet, 2013, “Macro-Prudential Policies to Mitigate Financial System Vulnerabilities,” Journal of International Money and Finance 39, pp. 153–185. - Farhi, Emmanuel and Iván Werning, 2012, “Dealing with the Trilemma: Optimal Capital Controls with Fixed Exchange Rates,” NBER Working Paper w18199. - Farhi, Emmanuel and Iván Werning, 2013, “A Theory of Macroprudential Policies in the Presence of Nominal Rigidities,” NBER Working Paper w19313. - Farhi, Emmanuel and Iván Werning, 2014, “Dilemma not Trilemma? Capital Controls and Exchange Rates with Volatile Capital Flows,” manuscript, Harvard. - Federico, Pablo M., Carlos A. Végh and Guillermo Vuletin, 2013, “Effects and Role of Macroprudential Policy: Evidence from Reserve Requirements Based on a Narrative Approach,” manuscript, Johns Hopkins University. - Forbes, Kristin, Marcel Fratzscher and Roland Straub, 2013, “Capital Controls and Macro-prudential Measures: What Are They Good For?” DIW Discussion Paper 1343. - Galati, Gabriele, and Richhild Moessner, 2013, “Macroprudential policy – a literature review, ” Journal of Economic Surveys 27(5), pp. 846-878. - IMF, 2012, The Liberalization and Management of Capital Flows: An Institutional View, Washington, DC. - Jeanne, Olivier, 2014, “Macroprudential Policies in a Global Perspective,” NBER Working Paper w19967. - Jeanne, Olivier, and Anton Korinek, 2010a, “Excessive Volatility in Capital Flows: A Pigouvian Taxation Approach,” American Economic Review P&P, pp. 403-407. - Jeanne, Olivier, and Anton Korinek, 2010b, “Managing Credit Booms and Busts: A Pigouvian Taxation Approach,” NBER Working Paper w16377. - Jeanne, Olivier, and Anton Korinek, 2013b, “Macroprudential Regulation versus Mopping Up After the Crash,” NBER Working Paper w18675. - Keynes, John M., 1929, “The German Transfer Problem,” Economic Journal 39(1929), pp. 1-7. - Korinek, Anton, 2007, “Excessive Dollar Borrowing in Emerging Markets,” manuscript, University of Maryland. - Korinek, Anton, 2010, “Regulating Capital Flows to Emerging Markets: An Externality View,” manuscript, University of Maryland. - Korinek, Anton, 2011a, “The New Economics of Prudential Capital Controls,” IMF Economic Review 59(3), pp. 523-561. - Korinek, Anton, 2011b, “Systemic Risk-Taking: Amplification Effects, Externalities, and Regulatory Responses,” ECB Working Paper 1345. - Korinek, Anton and Alp Simsek, 2014, “Liquidity Trap and Excessive Leverage,” NBER Working Paper w19970. - Lorenzoni, Guido, 2008, “Inefficient Credit Booms,” Review of Economic Studies 75(3), pp. 809-833. - Magud, Nicolas E., Carmen M. Reinhart, and Kenneth S. Rogoff, 2011, “Capital Controls: Myth and Reality - A Portfolio Balance Approach,” NBER Working Paper w16805. - Mendoza, Enrique G., 2002, “Credit, Prices and Crashes: Business Cycles with a Sudden Stop,” in S. Edwards and J. Frankel (editors), Preventing Currency Crises in Emerging Markets, pp. 335-392, NBER, University of Chicago Press. - Ohlin, Bertil, 1929, “The German Transfer Problem: A Discussion,” Economic Journal 39(1929), pp. 172-182. - Ostry, Jonathan D., Atish R. Ghosh, Karl Habermeier, Marcos Chamon, Mahvash S. Qureshi and Dennis B.S. Reinhardt, 2010, “Capital Inflows: The Role of Controls,” IMF Staff Position Note 10/04. - Ostry, Jonathan D., Atish R. Ghosh, Karl Habermeier, Luc Laeven, Marcos Chamon, Mahvash S. Qureshi and Annamaria Kokenyne (2011), “Managing Capital Inflows: What Tools to Use?” IMF Staff Discussion Note 11/06. - Schmitt-Grohe, Stephanie and Martin Uribe, 2012, “Prudential Policy for Peggers,” NBER Working Paper w18031. ### Mathematical Appendix - A.1 Model with Non-Traded Goods - The technical condition that characterizes the upper limit ˆφ on the pledgeability parameter in the model with non-traded goods is Y_N,1 −(1−α)[Y_B^N,1(1 + ˆφ) + Y_S^N,1/2] = 0 - Given this definition, the assumption φ < ˆφ implies that the denominator D in expression (12) is strictly positive so as to avoid degenerate equilibria. - The unconstrained region, i.e. the set (M_B,M_S) ∈ M_unc, is determined by the fact that the borrowing level that ensures perfect consumption smoothing is no greater than the constrained limit: 1/2 (M_B + p Y_B^N,1 − Y_B^T,2) ≥ −φ (Y_B^T,1 + p Y_B^N,1) - By substituting out the definition of the price level in the unconstrained region: p = (1−α)/(1 + α) (M_B + M_S + Y_T,2) where Y_T,2 = Y_B^T,2 + Y_S^T,2, we can derive the following inequality that pins down the set (M_B,M_S) ∈ M_unc: M_B (1/2 + Y_B^N,1 (1−α)/(1 + α) (1/2 + φ)) + M_S Y_B^N,1 (1−α)/(1 + α) (1/2 + φ) ≥ Y_B^T,2/2 − φ Y_B^T,1 − Y_T,2 Y_B^N,1 (1−α)/(1 + α) (1/2 + φ) - A.2 Model with Capital Goods - To understand how M_B affects the price of capital goods when borrowers are constrained, note that savers set their marginal product of capital equal to the price q. Using the market clearing condition K_S^2 = K − K_B^2, where K is the total stock of capital, we infer that ∂q/∂M_B = −F_S′′ ∂K_B^2/∂M_B - Since F_S′′ < 0, we see that ∂q/∂M_B > 0 if and only if ∂K_B^2/∂M_B > 0. This latter derivative can be analyzed by considering that the optimality condition (26) implies F_S′ = F_B′ φ + (1−φ) u_B^T,1 / u_B^T,2 - where the consumption levels of constrained borrowers are given by: C_B^T,1 = M_B + q (K_B^1 − K_B^2) + φ (Y_B^T,1 + q K_B^2) C_B^T,2 = F_B + Y_B^T,2 − φ (Y_B^T,1 + q K_B^2) - By using the implicit function theorem, we can show that ∂K_B^2/∂M_B > 0 if the following (sufficient but not necessary) conditions are satisfied: - ∂C_B^T,1/∂K_B^2 = −F_S′′ (K_B^1 − K_B^2 + φ K_B^2) − F_S′ (1−φ) < 0 - ∂C_B^T,2/∂K_B^2 = A − φ (−F_S′′ K_B^2 + F_S′) > 0 - The first condition implies that an increase in K_B^2 should come at the cost of a reduction in C_B^T,1. This requires placing an upper bound on the collateral parameter φ and ensuring that the second derivative of the savers’ production function is not too high in order to limit the responsiveness of prices to the demand for capital. The second condition requires that a marginal increase in K_B^2 should lead to greater net worth at time 2 and thus higher C_B^T,2. This condition also places an upper bound on φ. *Source: _wp15218 - References (PDF).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15218.pdf_