## wp18236

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---

### Literature links and conceptual contributions
- Links to pre-1997 and post-1997 literature:
  - Third-generation currency crisis models: collateral constraints × currency mismatches can generate self-fulfilling currency runs (Krugman, 1999; Aghion, Bacchetta and Banerjee, 2000, 2001).
  - Debate on monetary response to balance-sheet disruptions: monetary stimulus vs. monetary tightening (Céspedes, Chang and Velasco (2004), Christiano, Gust and Roldos (2004), Gourinchas (2018)).
  - Global financial crisis motivated work showing currency mismatches and collateral constraints complicate monetary policy (Ottonello (2015); Farhi and Werning (2016)); monetary easing can depress domestic consumption via exchange-rate driven collateral tightening.
  - Other related mechanisms: bank profitability channel to ineffective policy at low rates (Brunnermeier and Koby (2016)); negative policy rates and bank profits (Eggertsson, Juelsrud and Wold (2017)).
  - Macroprudential and capital-control literatures on externalities from collateral constraints and policy tools (Jeanne and Korinek (2010); Bianchi (2011); Benigno et al. (2013, 2016); Korinek and Sandri (2016); Aoki, Benigno and Kiyotaki (2016)).

- Novel contributions of this paper:
  - Develops models in which monetary policy can determine whether collateral constraints bind, so that monetary policy may not be able to achieve a unique output target.
  - Generates an Effective Lower Bound (ELB) that places an upper bound on the level of output monetary policy can achieve.
  - Extends prior focus on currency mismatches by showing limits to output stimulation can also arise from carry-trade capital flows affecting domestic collateral constraints.
  - Emphasizes international/intermediary channels absent from closed-economy models in related literature.

### Key mechanism: ELB under carry-trade capital flows
- Core mechanism:
  - An interest rate cut reduces the expected excess return on domestic bonds.
  - This can trigger a capital outflow (carry-trade dynamics) if large enough.
  - Capital outflows tighten domestic collateral/leverage constraints.
  - Tighter constraints cause a domestic credit crunch, reducing aggregate demand and output.
  - Monetary easing can thus become contractionary, producing an ELB.

- Distinctions and mechanism details:
  - Foreign intermediaries in this model care about foreign-currency returns, creating carry-trade dynamics where policy rate cuts reduce expected excess returns and trigger outflows.
  - The ELB arises because monetary policy itself affects whether collateral constraints bind, not merely because of exogenous constraints on central bank actions.

### Model setup — agents, preferences, and markets
- Households:
  - Two types: borrowers (B) and savers (S). Variables denoted with B and S superscripts.
  - Objective:
    - E_0 ∑_{t=0}^∞ β^t ln C^i_t  (i = {B,S})
  - Consumption index:
    - C^i_t = (C^i_{H,t})^{1−α} (C^i_{F,t})^{α}, where α∈(0,1).
  - Borrowers' budget constraint:
    - P_t C^B_t + L_{t−1} I^L_{t−1} = Π^B_t + L_t
  - Savers' budget constraint:
    - P_t C^S_t + D_t = Π^S_t + D_{t−1} I^D_{t−1}
  - Euler equations:
    - 1 = β^B_t I^L_t E_t [ P_t C^B_t / (P_{t+1} C^B_{t+1}) ]
    - 1 = β^S_t I^D_t E_t [ P_t C^S_t / (P_{t+1} C^S_{t+1}) ]
  - Home-good spending:
    - P_{H,t} C^i_{H,t} = (1−α) P_t C^i_t
  - Foreign households:
    - 1 = β I^*_t E_t [ P^*_t C^*_t / (P^*_{t+1} C^*_{t+1}) ]
    - P^*_{H,t} C^*_{H,t} = α P^*_t C^*_t
  - Aggregates: C_t = C^B_t + C^S_t; Π_t = Π^B_t + Π^S_t

- Production and prices:
  - Continuum of monopolistically competitive firms producing differentiated Home varieties.
  - Consumption aggregator for Home goods:
    - C_{H,t} = ( ∫_0^1 C_{H,t}(j)^{ε−1/ε} d j )^{ε/(ε−1)} with ε>1.
  - Firms can set different domestic and foreign prices (LCP vs PCP).
  - Domestic and foreign prices of goods sold domestically and abroad are constant at ̄P_H and ̄P^*_H (normalized to 1) in periods 0 and 1; prices fully flexible from time 2 onward.

- Banking sector:
  - Bank balance sheet:
    - N_t + D_t = L_t + B_t + R_t
  - Networth evolution (equation (2)):
    - N_{t+1} = L_t I^L_t + B_t I^B_t + R_t I_t − D_t I^D_t
  - Leverage constraint (equation (3)):
    - L_t + λ B_t ≤ φ N_t, where φ>1 and λ∈(0,1)
    - Interpretation: government bonds have lower capital charge than loans.
  - Assumptions and rates:
    - Banks cannot issue new equity at time 0 and 1.
    - Competitive banks maximize period-by-period networth subject to leverage constraint.
    - No-arbitrage implies I^D_t = I_t.
    - First-order conditions and spreads:
      - I^L_t ≥ I_t
      - I^B_t = λ I^L_t + (1−λ) I_t
    - If leverage constraint binds: lending and bond rates increase above the policy rate; lending spread impairs monetary transmission and possibly reverses it (policy rate cut → lending rate increase).

- Foreign investors and intermediaries:
  - Foreign intermediaries purchase domestic bonds B^F_t financed by borrowing in foreign currency at rate I^*_t; balance sheet: B^F_t + e_t B^*_t = 0.
  - Expected foreign-currency return:
    - V^*_t = B^*_t E_t [ e_t e_{t+1} I^B_t − I^*_t ]
  - Agency friction / incentive compatibility constraint (equation (4)):
    - E_t [ e_t e_{t+1} I^B_t − I^*_t ] ≥ γ_t B^F_t I^*_t
  - Foreign demand for domestic government bonds (equation (5)):
    - B_Ft = 1/γ_t E_t [ e_t e_{t+1} I_Bt I*_t −1 ]
  - Interpretation of γ_t:
    - γ_t is an inverse measure of risk-bearing capacity (higher γ_t → smaller demand); γ_t stochastic captures global liquidity shocks.
    - If γ_t ∈ (0,1), the demand schedule generates carry-trade dynamics so domestic monetary easing triggers capital outflows.

- Public sector and market clearing:
  - Central bank sets policy rate I_t. Initially assume R_t ↓ 0; Section 2.3 later allows QE and FX intervention.
  - Government rolls over public debt B^G_t: B^G_t = B^G_{t−1} I^B_{t−1}.
  - Market clearing:
    - Y_{H,t} = C_{H,t} + C^*_{H,t}  (equation (6))
    - B^G_t = B_t + B^F_t  (equation (7))

### Model equilibrium — steady state and time-1 outcomes
- Steady-state characterization:
  - Assumptions: from time 2 onward the bank leverage constraint does not bind, prices are flexible, the model is in steady state and β=1 so that P2 Ci2 = Πi2 and P2 Ci2 = Mi2.
  - Market clearing in the Home economy: Π2 = (1−α)M2 + e2 αM*2.
  - Steady-state exchange rate:
    - e2 = M2 / M*2
    - Normalize steady-state money supplies to 1 in both countries, so that e2 = 1.

- Home output at time 1 (equation (8)):
  - YH,1 = (1−α)(ω2 IL1 + (1−ω2) I1) + α I*1
  - ω2 = ΠB2 / Π2 is the share of steady-state output appropriated by borrowers.

- Monetary easing when bank leverage constraint does not bind:
  - If constraint slack: IL1 = I1.
  - Effects of a policy rate cut:
    - Raises savers’ consumption (lower deposit rates).
    - Raises borrowers’ consumption (lower lending rates) → monetary easing is expansionary.
  - Ambiguous effect on capital flows; exchange rate depreciation required to restore foreign funds equilibrium; net effect depends on elasticity εeI.

- Elasticity of exchange rate to policy rate (unconstrained case) (equation (9)):
  - εeI = 1−γ1 BF1 − BF1 I1 + αγ1 I*1 / e1
  - Implications:
    - If current account deficit → elasticity > 1 → monetary easing generates capital inflows.
    - If current account surplus → elasticity < 1 → monetary easing triggers capital outflows.
    - Higher γ1 raises international borrowing costs and can lower εeI.

- Closed-form foreign bond holdings (equation (10)):
  - BF1 = BF1 / (1+γ1) α / I1
  - Interpretation:
    - Monetary easing (↓I1) reduces BF1 (increases capital outflows) when the country is a net debtor and γ1 > 0.

### Leverage constraint, ELB, and dynamics when constrained
- Outstanding loans at time 1 (equation (11)):
  - L1 = L1 + ω2 IL1 − ΠB1
  - To focus on capital flows, set ΠB1 = ω2 / I1 so borrowers’ demand does not respond to policy and L1 = L1 is constant.

- Leverage constraint slack iff foreigners purchase sufficiently high domestic bonds:
  - BF1 ≥ BF1, where BF1 ≡ BG1 − (φN1 − L1)/λ is the country’s capital shortfall.

- Expansionary Lower Bound (ELB) (equation (12)):
  - IELB1 = γ1 α BF1 / BF1 − 1
  - Comparative statics:
    - IELB1 increases with the country’s capital shortfall BF1.
    - IELB1 increases with global financial tightness γ1.
    - IELB1 decreases in foreign holdings of bonds at beginning of time 1, BF1.
    - If γ1 large enough, IELB1 > 1 and ELB can act as a stronger constraint than the Zero Lower Bound.

- Dynamics when I1 < IELB1 (credit crunch):
  - Capital inflows insufficient → lending rates and bond yields must rise above the policy rate; monetary transmission undermined.
  - Consistency condition (equilibrium of foreign bond holdings with leverage constraint) preserved exactly in source.
  - Left derivative of lending rate at IELB1 (equation (13)):
    - ∂IL1 / ∂I1 |− I1 = Ī1 = 1 − 1/(λ+ω2[ BF1/( αλ BF1 ) −1 ] + λ γ1 BF1)
  - Derivative < 1: when constrained, lending rate rises above policy rate; for sufficiently tight γ1 derivative can become negative so monetary easing raises IL1.
  - Bond yields increase above policy rate but monetary easing still reduces bond yields and triggers capital outflows even below ELB.

- Aggregate demand and contractionary monetary easing at ELB:
  - Easing raises lending rates when banks constrained → reduces borrowers’ consumption; savers’ consumption still stimulated.
  - Monetary easing is contractionary when increase in lending rates dominates; condition (equation (14)):
    - −ω2 ∂IL1 / ∂I1 |− I1 = IELB1 > 1−ω2
  - Central bank cannot raise output above level associated with ELB (equation (15)):
    - YELB H,1 = (1−α) 1 / IELB1 + α I*1
  - If target output < YELB H,1 → ELB not binding; if desired output > YELB H,1 → optimal policy sets I1 = IELB1.

- ELB and departures from Mundell’s trilemma:
  - With unconstrained banks, increase in γ1 absorbed via depreciation so output unaffected.
  - With ELB, increase in γ1 raises ELB and lowers YELB H,1, possibly forcing central bank to raise rates and accept lower output.
  - Foreign monetary tightening (↑I*1) reduces maximum attainable output but does not change IELB1 in this model (equation (12)).

### Time-0 expectations and monetary-space trade-offs
- At time 0 banks unconstrained, take limit γ0 ↓ 0, UIP holds, foreign investors supply any required capital.
- Binary distribution for γ1:
  - With probability ρ, γ1 high → ELB binding → central bank sets I1 = IELB1.
  - With probability 1−ρ, γ1 low (zero) → central bank unconstrained → sets I1 = Iopt1.
- How I0 affects IELB1:
  - IELB1 depends on ratio BF1 / BF1; dividing both terms by I0 and using BG1 = BG0 I0 and N1 = N0 I0 when leverage constraint does not bind at time 0.
  - Time-0 policy affects BF0 and L0:
    - BF0 = BF0 + α / I0 ( 1 / E0[I1] − 1 / E0[1/e1] )
    - L0 = L0 + ω2 / I0 E0[I1]
  - Holding expectations constant: tightening at time 0 reduces both capital inflows and domestic lending.
    - ↓BF0 tends to raise IELB1.
    - ↓L0 tends to lower IELB1.
    - Net effect ambiguous.
  - Full proportional expression (sign result preserved):
    - ∂IELB1 / ∂I0 ∝ [ BF0 + ρ α ̄γ1 BF1 / IELB1 E0[(1+ ̄γ1 BF1)/e1]^2 ] ∂L0/∂I0 − λ BF1 ∂BF0/∂I0 < 0
  - Conclusion: A monetary tightening at time 0 lowers IELB1 at time 1. Thus:
    - I0 negatively correlated with IELB1.
    - Calls for tighter monetary stance when financial conditions are supportive to create more monetary space later.

- Implications for effectiveness of time-0 monetary policy:
  - Time-0 policy less powerful in stimulating output:
    - YH,0 = (1−α) / I0 E0[I1] + α I*0 I*1
  - Reducing I0 raises expected IELB1 → tighter expected future stance E0[I1] → weakens impact on current output.
  - ELB constrains policy not only when binding but also weakens monetary transmission when banks unconstrained and global conditions supportive.

### Policies to escape the ELB (summary of model extension Section 2.3)
- Instruments introduced:
  - Government budget constraint with lump-sum taxes T_t and tax on foreign capital inflows χ_t:
    - B^G_t = B^G_{t−1} − T_t − χ_t B^F_t
  - Central bank balance sheet:
    - N^{CB}_t + R_t = B^{CB}_t + e_t X_t
  - Forward guidance via changes in steady-state money supply M_2.
  - Primary focus: effects at time 1 when ELB binds; preemptive use at time 0 also considered.

- Fiscal policy: taxes, capital inflow levies, Ricardian effects:
  - ELB arises because public debt crowds out private lending through bank leverage constraints.
  - Tax-based fiscal consolidation has two opposing effects:
    - Reduces public debt and relaxes bank leverage constraint in proportion to λ.
    - Raises loan demand via Ricardian equivalence: borrowers increase borrowing to smooth consumption despite higher taxes.
  - Condition determining whether tax-based consolidation tightens lending:
    - If T^B_t / T_t > λ then a tax-based fiscal consolidation tightens collateral constraints, raises the ELB, and lowers output.
  - Taxes on foreigners (χ_t):
    - Raising χ_t reduces foreign holdings of government bonds and forces banks to curtail private lending.
    - To lower the ELB it is optimal to subsidize capital inflows: χ_1 < 0 (implies higher public debt but overall lowers the ELB).

- Central bank balance-sheet operations:
  - Quantitative easing (QE):
    - Purchase government bonds B^{CB}_1 financed by an increase in R_1.
    - Central bank acts as a financial intermediary for government bonds, releasing liquidity to banks and lowering the ELB; stimulates output.
    - QE remains effective even when partially eroded by carry traders because lowering government bond yields can exacerbate capital outflows.
  - Unsterilized foreign exchange intervention:
    - Purchase foreign reserves X_1 against domestic reserves R_1 → depreciates exchange rate → increases expected return of government bonds to foreigners → stimulates capital inflows → can alleviate the ELB.
  - Sterilized foreign exchange intervention:
    - Sell foreign reserves and buy government bonds (combines unsterilized FX intervention with QE).
    - In equilibrium the QE-like effect prevails: sterilized intervention relaxes the ELB if the central bank reduces foreign reserves despite exchange rate appreciation.

- Forward guidance and limits when ELB endogenous:
  - Forward guidance increases M_2 and can raise time-1 spending for a given I_1.
  - However, because ELB is endogenous, higher M_2 also increases the ELB:
    - Higher M_2 → stronger depreciation of the exchange rate at time 2 than at time 1 → reduces foreign-currency return on domestic bonds → generates capital outflows → ELB rises.
  - Net effect: forward guidance increases the ELB and leaves the level of output at the ELB unchanged.

- Bank recapitalization and credit-easing measures:
  - Recapitalization of the banking sector is highly effective in overcoming the ELB.
  - Even if recapitalization is financed with lump-sum taxes on borrowers (captured by an increase in loan repayments at the beginning of period 1, L_1), the net effect can be positive because:
    - Lump-sum taxes increase loan demand one-to-one.
    - Recapitalization increases lending supply by a greater factor thanks to bank leverage, i.e. φ > 1.
  - Credit-easing policies (government lending subsidies or acting as financial intermediary) similarly relax lending constraints and stimulate aggregate demand.

- Preemptive interventions at time 0:
  - Fiscal consolidation at time 0 can lower a future ELB only if the tax burden on borrowers is smaller than the capital requirement on government bonds:
    - T^B_0 / T_0 < λ
  - Taxes on capital inflows have ambiguous preemptive effects (reduce public debt but reduce capital inflows).
  - If γ_0 > 0, foreign exchange intervention at time 0 can help by depreciating the exchange rate and attracting inflows, thereby lowering the ELB.

### The ELB and currency mismatches (introductory findings)
- Alternative setup:
  - All households are borrowers; banks issue foreign-currency debt D^*_t and hold domestic loans L_t and reserves R_t:
    - N_t + e_t D^*_t = L_t + R_t
  - Bank networth evolution:
    - N_{t+1} = L_t I^L_t + R_t I_t − e_{t+1} D^*_t I^*_t
  - Leverage constraint:
    - L_t ≤ φ N_t with φ ≥ 1 and λ = 0.
  - No carry-trade frictions; UIP holds.

- Output at time 1 (equation (17)):
  - Y_{H,1} = (1−α) I^L_1 + e_1 α I^*_1

- Exchange rate depreciation from monetary easing erodes bank networth:
  - N_1 = L_1 − e_1 D^*_1  where L_1 ≡ L_0 I_0 and D^*_1 ≡ D^*_0 I^*_0.

- Lending rate when constraint binds (equation (18)):
  - I^L_1 = α (φ−1) L_1 − e_1 (φ D^*_1 − α I^*_1)
  - If φ D^*_1 > α / I^*_1, depreciation-induced networth losses dominate and the lending rate increases with monetary easing.

- Condition for contractionary easing and ELB formula:
  - Monetary easing becomes contractionary once collateral effects dominate export-demand gains if:
    - φ D^*_1 > α (1−α) I^*_1   (condition (19))
  - The ELB under currency mismatches (equation (20)):
    - I_{ELB,1} = φ D^*_1 I^*_1 / ((φ−1) L_1)
  - Implications:
    - ELB depends on extent of currency mismatches (foreign-currency debt relative to domestic loans).
    - ELB can occur at positive interest rates and be stronger than ZLB.
    - Foreign policy rate I^*_1 affects ELB: ↑I^*_1 depreciates domestic currency, raises ELB, and reduces maximum output achievable by monetary policy.

- Time-0 policy and inter-temporal trade-off:
  - D^*_1 = D^*_0 I^*_0 + δ α E_0[I^*_1]   (equation (21))
  - L_1 = L_0 I_0 + δ α E_0[I_1]         (equation (22))
  - δ ≡ 1/β_0 − 1 captures impatience of domestic households relative to foreigners.
  - Monetary tightening at time 0 (holding E_0[I_1] constant) increases L_1 but does not increase D^*_1 → reduces time-1 ELB.
  - If δ > 0, reduction in ELB lowers E_0[I_1], increasing loan demand at time 0 and amplifying the ELB reduction.
  - Trade-off: greater easing at time 0 reduces future monetary space by raising the ELB; normative case for tighter domestic stance when global conditions are favorable.

- International policy interactions and moral hazard:
  - Foreign tightening raises ELB in EMs and can push them into recession.
  - If major central banks commit to avoid sharp hikes to protect EMs, EMs have incentives to accumulate more foreign-currency debt.
  - Ways to limit accumulation:
    - Commit foreign policy to be looser when ELB binds but tighter otherwise to stabilize E_0[I^*_1].
    - EM macroprudential regulation to constrain foreign-currency borrowing.
  - Any foreign commitment to avoid tight policy when ELB binds is partially ineffective because it induces endogenous additional foreign-currency borrowing.

### Policies to escape the ELB when arising from currency mismatches (Section 3.3)
- Forward guidance and balance-sheet operations:
  - Forward guidance is ineffective at escaping ELB from currency mismatches because promise of looser future monetary stance leads to immediate depreciation tightening the bank leverage constraint.
  - An increase in M2 through forward guidance raises the ELB while leaving Y_ELB_H,1 unchanged.
  - Balance-sheet operations ineffective here since exchange rate pinned by UIP.

- Recapitalization, capital controls, and subsidies:
  - Recapitalization relaxes collateral constraints and the ELB.
  - Capital controls can be effective because they sever link between exchange rate and domestic monetary conditions.
  - Government subsidy χ1 on foreign currency debt can stimulate capital inflows and support exchange rate, placing a wedge in UIP:
    - e1 = e2 (1−χ1) I∗1 / I1
    - This leads to exchange rate appreciation, relaxes ELB, and allows greater monetary stimulus.

- Macro-prudential capital controls:
  - Taxing capital inflows at time 0 can reduce foreign currency debt carried into period 1, lowering I_ELB1 and allowing higher output.
  - Can be used preemptively.

### Key findings and policy recommendations (paper summary)
- Definition and implications:
  - The ELB is an interest rate below which monetary easing becomes contractionary; it places an upper bound on the level of output achievable through monetary policy.
  - A tightening in global liquidity or monetary conditions may raise the ELB and push emerging markets into recession while central banks are forced to raise policy rates.

- Multiple origins of ELB:
  - Carry-trade capital flows: monetary easing triggers capital outflows, forcing domestic banks to absorb bonds liquidated by foreign investors, pushing banks against leverage constraints and reducing domestic credit.
  - Currency mismatches: banking sector borrows abroad in foreign currency and lends domestically in local currency; monetary easing reduces bank net worth and moves banks closer to leverage constraints, potentially making further easing contractionary.

- Dependence on past policy:
  - ELB depends on past monetary stance: tighter ex-ante monetary conditions tend to lower the ELB and create more monetary space to offset shocks.
  - Normative implication: keep a somewhat tighter monetary stance when global conditions are supportive to preserve future policy space.

- Policy toolkit implications:
  - Active use of the central bank’s balance sheet (quantitative easing and foreign exchange intervention) is warranted to overcome the ELB and restore monetary transmission.
  - Capital controls and macro-prudential policies provide a new rationale: they can relax tensions between domestic collateral constraints and capital flows.
  - Fiscal policy can help to overcome the ELB.
  - Forward guidance is ineffective because the ELB increases with the expectation of looser future monetary conditions.
  - Preemptive measures (recapitalization, capital-flow measures, tighter ex-ante stance) should be considered to lower the ELB and preserve monetary space.

### Directions for future research
- Provide additional empirical evidence about the existence of the ELB; this would require showing that monetary easing is contractionary when domestic or international financial conditions are tight.
- Use quantitative DSGE models to analyze the circumstances under which the ELB may arise, its level, and the extent to which monetary authorities should keep the economy below potential when the ELB is expected to bind in the near future.

*Source: wp18236*

### 3. We summarize key findings and avenues for future research in the concluding section.

### 3. We summarize key findings and avenues for future research in the concluding section.

### Literature links and conceptual contributions
- Links to pre-1997 and post-1997 literature:
  - Third-generation currency crisis models: collateral constraints × currency mismatches can generate self-fulfilling currency runs (Krugman, 1999; Aghion, Bacchetta and Banerjee, 2000, 2001).
  - Debate on monetary response to balance-sheet disruptions: monetary stimulus vs. monetary tightening (Céspedes, Chang and Velasco (2004), Christiano, Gust and Roldos (2004), Gourinchas (2018)).
  - Global financial crisis motivated work showing currency mismatches and collateral constraints complicate monetary policy (Ottonello (2015); Farhi and Werning (2016)); monetary easing can depress domestic consumption via exchange-rate driven collateral tightening.
  - Other related mechanisms: bank profitability channel to ineffective policy at low rates (Brunnermeier and Koby (2016)); negative policy rates and bank profits (Eggertsson, Juelsrud and Wold (2017)).
  - Macroprudential and capital-control literatures on externalities from collateral constraints and policy tools (Jeanne and Korinek (2010); Bianchi (2011); Benigno et al. (2013, 2016); Korinek and Sandri (2016); Aoki, Benigno and Kiyotaki (2016)).

- Novel contributions of this paper:
  - Develops models in which monetary policy can determine whether collateral constraints bind, so that monetary policy may not be able to achieve a unique output target.
  - Generates an Effective Lower Bound (ELB) that places an upper bound on the level of output monetary policy can achieve.
  - Extends prior focus on currency mismatches by showing limits to output stimulation can also arise from carry-trade capital flows affecting domestic collateral constraints.
  - Emphasizes international/intermediary channels absent from closed-economy models in related literature.

### Key mechanism: ELB under carry-trade capital flows
- Core mechanism (summary):
  - An interest rate cut reduces the expected excess return on domestic bonds.
  - This triggers a capital outflow (carry-trade dynamics) if large enough.
  - Capital outflows tighten domestic collateral/leverage constraints.
  - Tighter constraints cause a domestic credit crunch, reducing aggregate demand and output.
  - Monetary easing can thus become contractionary, producing an ELB.

- Distinction from other models:
  - Unlike Gabaix and Maggiori (2015), foreign intermediaries in this model care about foreign-currency returns, creating carry-trade dynamics where policy rate cuts reduce expected excess returns and trigger outflows.
  - The ELB here arises because monetary policy itself affects whether collateral constraints bind, not merely because of exogenous constraints on central bank actions.

### Model setup — agents, preferences, and markets (preserve expressions as in source)
- Households:
  - Two types: borrowers (B) and savers (S). Variables denoted with B and S superscripts.
  - Identical preferences, heterogeneous income; borrowers borrow at time 0 and 1, savers save.
  - Objective:
    - E_0 ∑_{t=0}^∞ β^t ln C^i_t  (i = {B,S})
  - Consumption index:
    - C^i_t = (C^i_{H,t})^{1−α} (C^i_{F,t})^{α}, where α∈(0,1).
  - Borrowers' budget constraint:
    - P_t C^B_t + L_{t−1} I^L_{t−1} = Π^B_t + L_t
  - Savers' budget constraint:
    - P_t C^S_t + D_t = Π^S_t + D_{t−1} I^D_{t−1}
  - Euler equations (households smooth consumption):
    - 1 = β^B_t I^L_t E_t [ P_t C^B_t / (P_{t+1} C^B_{t+1}) ]
    - 1 = β^S_t I^D_t E_t [ P_t C^S_t / (P_{t+1} C^S_{t+1}) ]
  - Home-good spending:
    - P_{H,t} C^i_{H,t} = (1−α) P_t C^i_t
  - Foreign households:
    - 1 = β I^*_t E_t [ P^*_t C^*_t / (P^*_{t+1} C^*_{t+1}) ]
    - P^*_{H,t} C^*_{H,t} = α P^*_t C^*_t
  - Aggregates: C_t = C^B_t + C^S_t; Π_t = Π^B_t + Π^S_t

- Production and prices:
  - Continuum of monopolistically competitive firms producing differentiated Home varieties.
  - Consumption aggregator for Home goods (as in footnote):
    - C_{H,t} = ( ∫_0^1 C_{H,t}(j)^{ε−1/ε} d j )^{ε/(ε−1)} with ε>1.
  - Firms can set different domestic and foreign prices (Local Currency Pricing vs Producer Currency Pricing noted).
  - Domestic and foreign prices of goods sold domestically and abroad are constant at ̄P_H and ̄P^*_H (normalized to 1) in periods 0 and 1; prices fully flexible from time 2 onward.

- Banking sector:
  - Bank balance sheet:
    - N_t + D_t = L_t + B_t + R_t
  - Networth evolution (equation (2)):
    - N_{t+1} = L_t I^L_t + B_t I^B_t + R_t I_t − D_t I^D_t
  - Leverage constraint (equation (3)):
    - L_t + λ B_t ≤ φ N_t, where φ>1 and λ∈(0,1)
    - Interpretation: government bonds have lower capital charge than loans.
  - Assumptions:
    - Banks cannot issue new equity at time 0 and 1.
    - Competitive banks maximize period-by-period networth subject to leverage constraint.
    - No-arbitrage between deposits and reserves implies I^D_t = I_t (deposit rate equals policy rate).
    - First-order conditions and spreads:
      - I^L_t ≥ I_t
      - I^B_t = λ I^L_t + (1−λ) I_t
    - If leverage constraint binds: lending and bond rates increase above the policy rate; lending spread impairs monetary transmission and possibly reverses it (policy rate cut → lending rate increase).

- Foreign investors and intermediaries:
  - Foreign intermediaries purchase domestic bonds B^F_t financed by borrowing in foreign currency at rate I^*_t; their balance sheet: B^F_t + e_t B^*_t = 0.
  - Their expected foreign-currency return:
    - V^*_t = B^*_t E_t [ e_t e_{t+1} I^B_t − I^*_t ]
  - Agency friction / incentive compatibility constraint (equation (4)):
    - E_t [ e_t e_{t+1} I^B_t − I^*_t ] ≥ γ_t B^F_t I^*_t
  - Foreign demand for domestic government bonds (equation (5), preserved exactly as in source):
    - B_Ft = 1/γ_t E_t [ e_t e_{t+1} I_Bt I*_t −1 ]
  - Interpretation of γ_t:
    - γ_t is an inverse measure of risk-bearing capacity (higher γ_t → smaller demand); γ_t stochastic captures global liquidity shocks.
    - If γ_t ∈ (0,1), the demand schedule generates carry-trade dynamics so domestic monetary easing triggers capital outflows.

- Public sector and market clearing:
  - Central bank sets policy rate I_t. Initially assume R_t ↓ 0 (no central-bank balance-sheet operations); Section 2.3 later relaxes this to allow QE and FX intervention.
  - Government rolls over public debt B^G_t: B^G_t = B^G_{t−1} I^B_{t−1}; later extensions include taxes and capital-flow taxes.
  - Market clearing:
    - Y_{H,t} = C_{H,t} + C^*_{H,t}  (equation (6))
    - B^G_t = B_t + B^F_t  (equation (7))

### Implications for monetary policy and policy tools
- Core policy insight:
  - Monetary policy can face an ELB because policy-rate cuts can generate carry-trade outflows that tighten collateral constraints and induce a credit crunch; therefore conventional monetary easing can be contractionary.
- Role of other policy instruments (previewed):
  - Section 2.3 expands analysis to incorporate fiscal and unconventional monetary policy tools to understand how they can help overcome restrictions imposed by the ELB (quantitative easing, FX intervention, taxes on capital flows are all considered in the broader model extension).
- Comparative notes:
  - Unlike the closed-economy channels that make policy contractionary via bank profitability, this model stresses international capital-flow/intermediary margins as a source of an ELB in open economies.

*Source: wp18236 - 3. We summarize key findings and avenues for future research in the concluding section.*

### 2.2    Model equilibrium

### 2.2    Model equilibrium

### Steady-state characterization
- Assumptions: from time 2 onward the bank leverage constraint does not bind, prices are flexible, the model is in steady state and β=1 so that P2 Ci2 = Πi2 and P2 Ci2 = Mi2.
- Market clearing in the Home economy: Π2 = (1−α)M2 + e2 αM*2.
- Steady-state exchange rate:
  - e2 = M2 / M*2
  - Without loss of generality normalize steady-state money supplies to 1 in both countries, so that e2 = 1.

### Model equilibrium at time 1 — output and spending
- Home output at time 1 (consumption of Home goods by domestic and foreign households):
  - YH,1 = (1−α)(ω2 IL1 + (1−ω2) I1) + α I*1  (equation (8))
  - ω2 = ΠB2 / Π2 is the share of steady-state output appropriated by borrowers.
- Interpretation:
  - First term: domestic consumption where lending and deposit rates control borrowers’ and savers’ consumption.
  - Second term: foreign demand, unaffected by domestic policy because export prices are sticky in foreign currency.

### Monetary easing when bank leverage constraint does not bind
- If constraint slack: IL1 = I1.
- Effects of a policy rate cut:
  - Raises savers’ consumption (lower deposit rates).
  - Raises borrowers’ consumption (lower lending rates) → monetary easing is expansionary.
- Ambiguous effect on capital flows:
  - Monetary easing raises import consumption (↑ demand for foreign funds).
  - Monetary easing reduces bond yields (IB1 = I1) → curbs supply of foreign capital.
  - Exchange rate must depreciate to restore foreign funds equilibrium.
  - Net effect depends on elasticity of exchange rate with respect to policy rate, εeI.

### Elasticity of exchange rate to policy rate (unconstrained case)
- Formula:
  - εeI = 1−γ1 BF1 − BF1 I1 + αγ1 I*1 / e1  (equation (9))
  - where BF1 = BF0 I0 are government foreign liabilities at beginning of time 1.
- Implications:
  - Effect of monetary policy on capital flows depends on sign of current account (net repayment of foreign debt).
  - If current account deficit → elasticity > 1 → monetary easing generates capital inflows.
  - If current account surplus → elasticity < 1 → monetary easing triggers capital outflows.
  - Global financial conditions (γ1) affect current account: higher γ1 raises international borrowing costs and induces deleveraging if BF1 > 0, lowering εeI.

### Closed-form foreign bond holdings and capital flows
- Equilibrium foreign bond holdings:
  - BF1 = BF1 / (1+γ1) α / I1  (equation (10))
- Interpretation:
  - Monetary easing (↓I1) reduces BF1 (increases capital outflows) when the country is a net debtor and γ1 > 0.
  - If banks unconstrained, domestic banks absorb bond sales by increasing leverage funded by more domestic deposits, so transmission is not impaired.

### Role of loan demand in bank leverage dynamics
- Outstanding loans at time 1:
  - L1 = L1 + ω2 IL1 − ΠB1  (equation (11))
  - where L1 = L0 I L0 is beginning stock of loans.
- Ambiguous effect of easing on loan demand:
  - Lowers lending rates → increases borrower consumption (↑ loan demand).
  - Raises borrower income (ΠB1) via higher output and export revenues → may reduce loan demand.
- To focus on capital flows, set ΠB1 = ω2 / I1 so borrowers’ demand does not respond to policy and L1 = L1 is constant.

### Leverage constraint, capital shortfall and minimum foreign capital
- Leverage constraint slack iff foreigners purchase sufficiently high domestic bonds:
  - BF1 ≥ BF1 (where BF1 ≡ BG1 − (φN1 − L1)/λ is the country’s capital shortfall).
- Interpretation:
  - BF1 is the minimum foreign capital required to satisfy domestic credit demand L1 + BG1 at prevailing policy rate.
  - Foreign capital directly funds government bonds; domestic deposits must be intermediated by banks.

### Expansionary Lower Bound (ELB)
- Policy rate level at which leverage constraint becomes binding:
  - IELB1 = γ1 α BF1 / BF1 − 1  (equation (12))
- Comparative statics:
  - IELB1 increases with the country’s capital shortfall BF1.
  - IELB1 increases with global financial tightness γ1.
  - IELB1 decreases in foreign holdings of bonds at beginning of time 1, BF1.
  - If γ1 large enough, IELB1 > 1 and ELB can act as a stronger constraint than the Zero Lower Bound.

### Dynamics when policy rate falls below IELB1 (credit crunch)
- If I1 < IELB1:
  - Capital inflows insufficient → lending rates and bond yields must rise above the policy rate; monetary transmission undermined.
  - Consistency condition (equilibrium of foreign bond holdings with leverage constraint):
    - 1/(1+γ1) α / IB1 [ BF1 + α IL1 − I1 / IB1 ( λ(1−ω2)/I1 − ω2 (1−λ)/IL1 ) ] = BF1 + ω2 λ (1/IL1 − 1/I1)
    - with IB1 = λ IL1 + (1−λ)I1.
  - Left derivative of lending rate at IELB1:
    - ∂IL1 / ∂I1 |− I1 = Ī1 = 1 − 1/(λ+ω2[ BF1/( αλ BF1 ) −1 ] + λ γ1 BF1)  (equation (13))
  - derivative < 1: when constrained, lending rate rises above policy rate; for sufficiently tight γ1 derivative can become negative so that monetary easing raises IL1.
  - Bond yields increase above policy rate but monetary easing still reduces bond yields and triggers capital outflows even below ELB.

### Aggregate demand and contractionary monetary easing at ELB
- Easing raises lending rates when banks constrained → reduces borrowers’ consumption; savings consumption still stimulated (deposit rates fall).
- Monetary easing is contractionary when increase in lending rates dominates; condition:
  - −ω2 ∂IL1 / ∂I1 |− I1 = IELB1 > 1−ω2  (equation (14))
- Non-monotonic relation between policy rate and output around IELB1:
  - Central bank cannot raise output above level associated with ELB:
    - YELB H,1 = (1−α) 1 / IELB1 + α I*1  (equation (15))
  - If target output < YELB H,1 → ELB not binding; if desired output > YELB H,1 → optimal policy sets I1 = IELB1.

### ELB and departures from Mundell’s trilemma
- With unconstrained banks, increase in γ1 absorbed via depreciation so output unaffected.
- With ELB, increase in γ1 raises ELB and lowers YELB H,1, possibly forcing central bank to raise rates and accept lower output.
- Foreign monetary tightening (↑I*1) reduces maximum attainable output but does not change IELB1 in this model (equation (12)).

### Model equilibrium at time 0 — expectations and policy trade-offs
- Assumptions: At time 0 banks unconstrained, take limit γ0 ↓ 0, UIP holds, foreign investors supply any required capital.
- Binary distribution for γ1:
  - With probability ρ, γ1 high → ELB binding → central bank sets I1 = IELB1.
  - With probability 1−ρ, γ1 low (zero) → central bank unconstrained → sets I1 = Iopt1.
- Borrowers have no income at time 0: ΠB0 = 0 → they accumulate debt.

### How I0 affects IELB1 (monetary space at time 1)
- IELB1 depends on ratio BF1 / BF1; dividing both terms by I0:
  - BF1 / BF1 = BF0 / ( BG0 − (φN0 − L0)/λ )
  - used BG1 = BG0 I0 and N1 = N0 I0 when leverage constraint does not bind at time 0.
- Time-0 policy affects BF0 and L0:
  - BF0 = BF0 + α / I0 ( 1 / E0[I1] − 1 / E0[1/e1] )
  - L0 = L0 + ω2 / I0 E0[I1]
- Holding E0[I1] and E0[1/e1] constant: tightening at time 0 reduces both capital inflows and domestic lending.
  - ↓BF0 tends to raise IELB1 (more appreciated exchange rate → less foreign demand for gov bonds).
  - ↓L0 tends to lower IELB1 (banks can absorb more gov bonds given leverage constraint).
  - Net effect ambiguous.
- Full expression (proportionality):
  - ∂IELB1 / ∂I0 ∝ [ BF0 + ρ α ̄γ1 BF1 / IELB1 E0[(1+ ̄γ1 BF1)/e1]^2 ] ∂L0/∂I0 − λ BF1 ∂BF0/∂I0 < 0
- Conclusion: A monetary tightening at time 0 lowers IELB1 at time 1 (Appendix A result). Thus:
  - I0 negatively correlated with IELB1.
  - Calls for tighter monetary stance when financial conditions are supportive to create more monetary space later (keeping economy below potential).

### Implications for effectiveness of time-0 monetary policy
- Time-0 policy less powerful in stimulating output:
  - YH,0 = (1−α) / I0 E0[I1] + α I*0 I*1
  - Reducing I0 raises expected IELB1 → tighter expected future stance E0[I1] → weakens impact on current output.
- ELB thus constrains policy not only when binding but also weakens monetary transmission when banks unconstrained and global conditions supportive.

*Source: wp18236 - 2.2    Model equilibrium (PDF chapter/section)*

### 2.3    Policies to escape the ELB

### 2.3    Policies to escape the ELB

### Model extensions and policy instruments analyzed
- Government budget constraint with lump-sum taxes on domestic households and a tax on foreign capital inflows:
  - B^G_t = B^G_{t−1} − T_t − χ_t B^F_t
  - T_t are lump-sum taxes on domestic households.
  - χ_t is a tax on foreign capital inflows.
- Central bank balance sheet:
  - N^{CB}_t + R_t = B^{CB}_t + e_t X_t
  - N^{CB}_t is networth; R_t are domestic reserves; B^{CB}_t are holdings of government bonds; X_t are foreign reserves.
- Forward guidance enters via changes in the steady-state level of money supply M_2.
- Primary focus: effects at time 1 when the ELB binds; also consider preemptive use at time 0.
- Formal derivations are in Appendix A (referenced in source).

### Fiscal policy: taxes, capital inflow levies, and Ricardian effects
- Key mechanism: the ELB arises because public debt crowds out private lending through bank leverage constraints.
- Tax-based fiscal consolidation has two opposing effects:
  - Reduces public debt and relaxes bank leverage constraint in proportion to the capital requirement λ.
  - Raises loan demand via Ricardian equivalence: borrowers increase borrowing to smooth consumption despite higher taxes.
- Condition determining whether tax-based consolidation tightens lending:
  - If T^B_t / T_t > λ then a tax-based fiscal consolidation tightens collateral constraints, raises the ELB, and lowers output.
- Taxes on foreigners (χ_t):
  - Raising χ_t reduces foreign holdings of government bonds and forces banks to curtail private lending.
  - To lower the ELB it is optimal to subsidize capital inflows: χ_1 < 0 (implies higher public debt but overall lowers the ELB).

### Central bank balance-sheet operations (quantitative easing and FX intervention)
- Quantitative easing (QE):
  - Purchase government bonds B^{CB}_1 financed by an increase in central bank reserves R_1.
  - Central bank acts as a financial intermediary for government bonds, releasing liquidity to banks and lowering the ELB; stimulates output.
  - QE remains effective even when partially eroded by carry traders because lowering government bond yields can exacerbate capital outflows.
- Unsterilized foreign exchange intervention:
  - Purchase foreign reserves X_1 against domestic reserves R_1 → depreciates the exchange rate → increases expected return of government bonds to foreigners → stimulates capital inflows → can alleviate the ELB.
- Sterilized foreign exchange intervention:
  - Sell foreign reserves and buy government bonds (combines unsterilized FX intervention with QE).
  - In equilibrium the QE-like effect prevails: sterilized intervention relaxes the ELB if the central bank reduces foreign reserves despite the exchange rate appreciation.

### Forward guidance and limits when the ELB is endogenous
- Forward guidance increases M_2 and can raise time-1 spending for a given policy rate I_1.
- However, because the ELB is endogenous, higher M_2 also increases the ELB:
  - Higher M_2 → stronger depreciation of the exchange rate at time 2 than at time 1 → reduces foreign-currency return on domestic bonds → generates capital outflows → ELB rises.
- Net effect in the model: forward guidance increases the ELB and leaves the level of output at the ELB unchanged.

### Bank recapitalization and credit-easing measures
- Recapitalization of the banking sector is highly effective in overcoming the ELB (also noted in Kollmann, Roeger and in’t Veld (2012); Sandri and Valencia (2013)).
- Even if recapitalization is financed with lump-sum taxes on borrowers (captured by an increase in loan repayments at the beginning of period 1, L_1), the net effect can be positive because:
  - Lump-sum taxes increase loan demand one-to-one.
  - But recapitalization increases lending supply by a greater factor thanks to bank leverage, i.e. φ > 1.
- Credit-easing policies (government lending subsidies or acting as financial intermediary) would similarly relax lending constraints and stimulate aggregate demand (referenced literature: Gertler and Karadi (2011); Gertler, Kiyotaki and Queralto (2012); Negro et al. (2011)).

### Preemptive interventions at time 0
- Fiscal consolidation at time 0 can lower a future ELB only if the tax burden on borrowers is smaller than the capital charge on government bonds:
  - T^B_0 / T_0 < λ
- Taxes on capital inflows have ambiguous preemptive effects:
  - Reduce public debt → lower ELB.
  - Reduce capital inflows → raise ELB.
  - Net effect depends on model parameters and the probability the ELB may bind in the future.
- If γ_0 > 0 (as in Appendix A), foreign exchange intervention at time 0 can also help by depreciating the exchange rate and attracting inflows, thereby lowering the ELB.

---

### 3    The ELB and currency mismatches (introductory findings)

### Mechanism: currency mismatches generate an ELB via balance-sheet effects
- Setup differences from prior model:
  - All households are borrowers; banks issue foreign-currency debt D^*_t and hold domestic loans L_t and reserves R_t:
    - N_t + e_t D^*_t = L_t + R_t
  - Bank networth evolution:
    - N_{t+1} = L_t I^L_t + R_t I_t − e_{t+1} D^*_t I^*_t
  - Leverage constraint:
    - L_t ≤ φ N_t with φ ≥ 1 and λ = 0 in this setup.
  - No carry-trade frictions; UIP holds.
- Exchange rate and nominal spending link: e_2 = M_2 / M^*_2.
- Output at time 1:
  - Y_{H,1} = (1−α) I^L_1 + e_1 α I^*_1  (equation (17))
  - Domestic and foreign spending on home goods are captured by the two terms.
- Exchange rate depreciation from monetary easing erodes bank networth:
  - N_1 = L_1 − e_1 D^*_1  where L_1 ≡ L_0 I_0 and D^*_1 ≡ D^*_0 I^*_0.
- If the leverage constraint binds, lending rate can rise above the policy rate to clear the credit market:
  - I^L_1 = α (φ−1) L_1 − e_1 (φ D^*_1 − α I^*_1)   (equation (18))
  - When φ D^*_1 > α / I^*_1, depreciation-induced networth losses dominate and the lending rate increases with monetary easing.

### Condition for monetary easing to become contractionary and the ELB formula
- Monetary easing becomes contractionary once collateral effects dominate export-demand gains if:
  - φ D^*_1 > α (1−α) I^*_1   (condition (19))
- The ELB under currency mismatches:
  - I_{ELB,1} = φ D^*_1 I^*_1 / ((φ−1) L_1)   (equation (20))
  - This prevents the central bank from increasing output above Y^{ELB}_{H,1} = 1 / I_{ELB,1}.
- Implications:
  - The ELB depends on the extent of currency mismatches (foreign-currency debt relative to domestic loans).
  - If mismatches are severe, the ELB can occur at positive interest rates and be a stronger constraint than the ZLB.
  - The ELB is affected by global monetary conditions: an increase in the foreign policy rate I^*_1 depreciates the domestic currency, raises the ELB, and reduces the maximum output monetary policy can achieve.

### Time-0 (preemptive) monetary policy and the inter-temporal trade-off
- When the leverage constraint does not bind at time 0, banks’ beginning-of-time-1 balances depend on time-0 policy and expectations:
  - D^*_1 = D^*_0 I^*_0 + δ α E_0[I^*_1]   (equation (21))
  - L_1 = L_0 I_0 + δ α E_0[I_1]         (equation (22))
  - δ ≡ 1/β_0 − 1 captures impatience of domestic households relative to foreigners.
- Monetary tightening at time 0 (holding E_0[I_1] constant) increases L_1 (loan repayments at time 1) but does not increase D^*_1 → reduces the time-1 ELB (equation (20)).
- If δ > 0, the first-round reduction in the ELB lowers E_0[I_1], which increases loan demand at time 0, raising L_1 further and amplifying the ELB reduction.
- Key trade-off:
  - Greater easing at time 0 reduces future monetary space by raising the ELB (negative correlation between time-0 policy rate and time-1 ELB).
  - This calls for tighter domestic monetary stance when global conditions are favorable to preserve space for future foreign shocks.
- Transmission is weakened even when the ELB does not bind because time-0 cuts raise expected future tightening E_0[I_1], offsetting current stimulus.
- Time-0 output expression (illustrating offsetting effects):
  - Y_{H,0} = (1−α) / (β_0 I_0 E_0[I_1]) + e_0 α I^*_0 / E_0[I^*_1]
  - Exchange rate e_0 = I^*_0 E_0[I^*_1] / (I_0 E_0[I_1])

### International policy interactions and moral hazard
- A foreign tightening (e.g., Fed hikes) raises the ELB in EMs and can push them into recession (illustrated in Figure 3).
- If major central banks commit to avoid sharp hikes to protect EMs, EMs have incentives to accumulate more foreign-currency debt:
  - D^*_1 depends inversely on E_0[I^*_1] when δ > 0 (equation (21)).
- Two ways to limit accumulation of foreign-currency debt:
  - Commit foreign policy to be looser when the ELB binds but tighter when it does not, stabilizing E_0[I^*_1].
  - EM macroprudential regulation to constrain foreign-currency borrowing.
- Any foreign commitment to avoid tight policy when the ELB binds would be partially ineffective because it induces endogenous additional foreign-currency borrowing.

*Source: wp18236 - 2.3    Policies to escape the ELB*

### 3.3    Policies to escape the ELB

### 3.3    Policies to escape the ELB

### Forward guidance and balance-sheet operations
- Forward guidance is ineffective at escaping the ELB when it arises from currency mismatches because the promise of a looser future monetary stance leads to an immediate depreciation of the exchange rate that tightens the bank leverage constraint.
- Formally, an increase in M2 through forward guidance raises the ELB, while leaving the upper bound on the level of output attainable through monetary policy, Y_ELB_H,1, unchanged.
- Balance-sheet operations by the central bank are also ineffective in this model since the exchange rate is pinned down by the UIP condition and not by quantity conditions.

### Recapitalization, capital controls, and subsidies
- Recapitalization of the banking sector can relax collateral constraints and thereby relax the ELB.
- Capital controls can be effective because they can sever the link between the exchange rate and domestic monetary conditions.
- A government subsidy χ1 on foreign currency debt can stimulate capital inflows and support the domestic exchange rate. This places a wedge in the UIP condition, e1 = e2 (1−χ1) I∗1 / I1, that leads to an appreciation of the exchange rate, relaxes the ELB, and allows for greater monetary stimulus.

### Macro-prudential capital controls
- Taxing capital inflows at time 0 can reduce the amount of foreign currency debt carried into period 1, lowering the time-1 ELB, I_ELB1, and allowing for a higher level of output.
- Such macro-prudential capital controls can be put in place in anticipation of the ELB becoming binding.

### Key findings on the ELB (summary of paper’s conclusions)
- The ELB is an interest rate below which monetary easing becomes contractionary; it places an upper bound on the level of output achievable through monetary policy.
- A tightening in global liquidity or monetary conditions may raise the ELB and push emerging markets into a recession while central banks are forced to raise policy rates.
- The ELB can arise under multiple mechanisms:
  - Carry-trade capital flows: monetary easing triggers capital outflows, forcing domestic banks to absorb bonds liquidated by foreign investors, pushing banks against leverage constraints and reducing domestic credit.
  - Currency mismatches: banking sector borrows abroad in foreign currency and lends domestically in local currency; monetary easing reduces bank net worth and moves banks closer to leverage constraints, potentially making further easing contractionary.
- The ELB depends on past monetary stance: tighter ex-ante monetary conditions tend to lower the ELB and create more monetary space to offset shocks. This implies a normative case for keeping a somewhat tighter monetary stance when global conditions are supportive.

### Policy implications and recommendations
- Active use of the central bank’s balance sheet (for example, quantitative easing and foreign exchange intervention) is warranted to overcome the ELB and restore monetary transmission.
- Capital controls and macro-prudential policies provide a new rationale: they can relax tensions between domestic collateral constraints and capital flows.
- Fiscal policy can help to overcome the ELB.
- Forward guidance is ineffective because the ELB increases with the expectation of looser future monetary conditions.
- Policymakers should consider preemptive measures (recapitalization, capital-flow measures, tighter ex-ante stance) to lower the ELB and preserve monetary space.

### Directions for future research
- Provide additional empirical evidence about the existence of the ELB; this would require showing that monetary easing is contractionary when domestic or international financial conditions are tight, though such episodes may be limited.
- Use quantitative DSGE models to analyze the circumstances under which the ELB may arise, its level, and the extent to which monetary authorities should keep the economy below potential when the ELB is expected to bind in the near future.

*Source: wp18236 - 3.3    Policies to escape the ELB*

### References

### wp18236 - References

### References (bibliography)
- Comprehensive list of cited works including (selected entries):
  - Acharya, Viral; Cecchetti, Stephen; De Gregorio, José; Kalemli-Ozcan, Sebnem; Lane, Philip R.; Panizza, Ugo.2015. “Corporate Debt in Emerging Economies: A Threat to Financial Stability?” Committee on International Economic Policy and Reform.
  - Aghion, Philippe; Bacchetta, Philippe; Banerjee, Abhijit.2000. “A Simple Model of Monetary Policy and Currency Crises.” European Economic Review, 44(4-6): 728 – 738.
  - Benigno, Gianluca; Chen, Huigang; Otrok, Christopher; Rebucci, Alessandro; Young, Eric R.2013. “Financial crises and macro-prudential policies.” Journal of International Economics, 89(2): 453–470.
  - Bianchi, Javier.2011. “Overborrowing and Systemic Externalities in the Business Cycle.” American Economic Review, 101(7): 3400–3426.
  - Brunnermeier, Markus K., and Yann Koby.2016. “The “Reversal Interest Rate”:  An Effective Lower Bound on Monetary Policy.” Manuscript.
  - Calvo, Guillermo A, and Carmen M Reinhart.2002. “Fear of floating.” The Quarterly Journal of Economics, 117(2): 379–408.
  - Eggertsson, Gauti B., and Michael Woodford.2003. “The Zero Bound on Interest Rates and Optimal Monetary Policy.” Brookings Papers on Economic Activity, 2003(1): 139–211.
  - IMF.2012. “The Liberalization and Management of Capital Flows - An Institutional View.” IMF Policy Paper.
  - Rey, Hélène.2015. “Dilemma not Trilemma: The Global Financial Cycle and Monetary Policy Independence.” NBER Working Paper 21162.
  - Wu, Jing Cynthia, and Fan Dora Xia.2016. “Measuring the macroeconomic impact of monetary policy at the zero lower bound.” Journal of Money, Credit and Banking, 48(2-3): 253–291.
- (Complete alphabetical reference list present in source.)

### Appendix A — Generalization of the model with carry traders: model structure and key steady-state and comparative-static expressions
- Household and bank accounting and definitions:
  - Borrowers and Savers budget constraints:
    - P_t C^B_t − Π^B_t = L_t − L_t
    - P_t C^S_t − Π^S_t = −D_t + D_t
    - where L_t = L_{t−1} I^L_{t−1}, D_t = D_{t−1} I_{t−1}
  - Net incomes:
    - Π^B_t = ω^B_t P_{H,t} C^B_{H,t} + ω^S_t P_{H,t} C^S_{H,t} + ω^F_t e_t P^*_ {H,t} C^*_{H,t} + ω^G_t (P_{H,t} G_{H,t} − T_t) + ω^N_t (N_t − N_t)
    - Π^S_t = (1−ω^B_t) P_{H,t} C^B_{H,t} + (1−ω^S_t) P_{H,t} C^S_{H,t} + (1−ω^F_t) e_t P^*_{H,t} C^*_{H,t} + (1−ω^G_t) (P_{H,t} G_{H,t} − T_t) + (1−ω^N_t) (N_t − N_t)
  - Euler equations:
    - 1 = β I^L_t E_t [ P_t C^B_t / (P_{t+1} C^B_{t+1}) ]
    - 1 = β I_t E_t [ P_t C^S_t / (P_{t+1} C^S_{t+1}) ]
  - Demand for Home goods: P^i_{H,t} C^i_{H,t} = (1−α) P^i_t C^i_t
  - Banking sector balance sheet:
    - N_t + D_t = L_t + B_t + R_t → N_t = L_t + B_t + R_t − D_t
    - B_t = B_{t−1} I^B_{t−1}, R_t = R_{t−1} I_{t−1}, I^B_t = λ I^L_t + (1−λ) I_t
    - Leverage constraint: L_t + λ B_t ≤ φ N_t
  - Foreign demand for domestic government bonds:
    - B^F_t = 1/γ_t E_t [ e_t / e_{t+1} (1−χ_t) I^B_t I^*_{t−1} − 1 ]
  - Central bank and public sector consolidated balances:
    - N^{CB}_t + R_t = B^{CB}_t + e_t X_t → N^{CB}_t = B^{CB}_t + e_{t+1} X_t − R_t
    - Consolidated budget: P_{H,t} G_{H,t} − T_t = B^G_t − B^G_{t−1} + χ_t B^F_t + N^{CB}_{t−1} − N^{CB}_t
  - Market clearing:
    - Y_{H,t} = C_{H,t} + C^*_{H,t} + G_{H,t}; B^G_t = B_t + B^F_t + B^{CB}_t
  - Aggregate budget constraint:
    - α P_t C_t = α e_t P^*_t C^*_t + (1+χ_t) B^F_t − B^F_{t−1} + e_t (X_{t−1} − X_t)
- Steady state with γ_t = χ_t = 0 yields:
  - e_t = (α β P_t C_t + (1−β) B^F_t) / (α β P^*_t C^*_t + (1−β) X_t)
  - with B^F_t = β B^F_2 and X_t = β X_2; at time 2:
    - e_2 = (α M_2 + (1−β) B^F_2) / (α M^*_2 + (1−β) X_2)
    - Π_2 = M_2 + (1−β)(L_2 − D_2)
    - P_2 C^B_2 = ω_2 M_2 − (1−β)[(1−ω_2) L_2 + ω_2 D_2]
    - P_2 C^S_2 = (1−ω_2) M_2 + (1−β)[(1−ω_2) L_2 + ω_2 D_2]
    - with P_2 C_2 = M_2, P^*_2 C^*_2 = M^*_2, and ω_B2 = ω_S2 = ω_F2 = ω_G2 = ω_N2 = ω_2
- Time-1 and time-0 demand/supply and collateral constraints are given explicitly — see source for full equations:
  - Time-1 foreign funds:
    - (1+χ_1) B^F_1 = B^F_1 + α ( P_2 C^B_2 β I^L_1 + P_2 C^S_2 β I_1 ) + e_1 ( X_1 − X_0 − α P^*_2 C^*_2 β I^*_1 )
    - B^F_1 = 1/γ_1 E_1 [ e_1 / e_2 (1−χ_1) I^B_1 I^*_1 − 1 ]
  - Collateral constraint at time 1:
    - (1+χ_1) λ B^F_1 ≥ L_1 + λ ( B^G_1 − N^{CB}_1 + P_{H,1} G_{H,1} − T_1 − R_1 + e_1 X_1 ) − φ N_1
- ELB expressions and comparative statics (selected exact formulas from source):
  - Baseline parameter specialization: set β = 1, ω_B1 = ω_F1 = 0, ω_S1 = (1/(1−α)) ω_2/(1−ω_2) yields P_2C^B_2 = ω_2 M_2 and P_2C^S_2 = (1−ω_2) M_2 and e_1, B^F_1, L_1 expressions as in source.
  - ELB implicit equation (symbolic form reproduced from source; see source for full displayed expression):
    - ELB equation (time-1) given by long algebraic identity involving (1−χ_2 1) B^F_0, (1+χ_1) M_2 I_ELB_1, αγ_1, X_0 I^*_1 M^*_2, γ_1 X_1 M_2/M^*_2 I^*_1 I_ELB_1, ω_G1, T_1 − P_{H,1} G_{H,1}, R_1, B^F_1, and other terms — set equal to 0.
  - Comparative static results using implicit function theorem (exact derivatives presented in source):
    - ∂I_ELB_1 / ∂T_1 = ( (1 / (B^F_1 I_ELB_1))^2 B^F_0 α γ_1 ω_G1 − λ ) / λ  (as presented in source)
    - ∂I_ELB_1 / ∂χ_1 = (1 / I_ELB_1)^3 1 / (γ_1 B^F_1)  (as presented in source)
    - ∂I_ELB_1 / ∂X_Ster_1 = (1 / I_ELB_1)^4 (1+γ_1 B^F_1) / (B^F_1)  (as presented in source)
    - ∂I_ELB_1 / ∂X_Unster_1 = − (1 / I_ELB_1)^3 (1+γ_1 B^F_1) α γ_1 B^F_1  (as presented in source)
    - ∂I_ELB_1 / ∂R_1 = − (1 / I_ELB_1)^4 αγ_1 B^F_0 (B^F_0 − B^F_1)^2  (as presented in source)
    - ∂I_ELB_1 / ∂M_2 = M_2 / I_ELB_1
  - Special case: with all unconventional tools zero at time 1 and M_2 = M^*_2 = 1:
    - I_ELB_1 = α ̄ γ_1 / ( λ B^F_0 λ B^G_0 + L_0 − φ N_0 − 1 )
- Additional comparative-static and generalized-income-profile results (exact functional forms and sign conditions preserved in source):
  - Expressions for ∂I_ELB_1 / ∂I_0, ∂I_ELB_1 / ∂T_0, ∂I_ELB_1 / ∂χ_0, ∂I_ELB_1 / ∂X_0 contained in source with conditions such as sufficient condition ω_2 > λ for denominator positivity.
  - Generalized income profile: expressions for e_1, B^F_1, L_1, bank leverage constraint λ B^F_1 − L_1 − λ B^G_1 + φ N_1 ≥ 0, and implications for ∂(λ B^F_1 − L_1)/∂I_1 and ∂I^L_1/∂I_1 evaluated at I_1 = I_ELB_1.

### Appendix A — Generalized discount factor β and its implications
- When ω_B1 = ω_F1 = 0, ω_S1 = (1/(1−α)) P_2 C^B_2 / P_2 C^S_2 and alternative tools zero, with M_2 = M^*_2 = 1:
  - e_1 = I^*_1 α (1+γ_1 B^F_1) [ α I^B_1 + (1−β) B^F_1 ]^{-1} (exact form in source)
  - B^F_1 and loan/deposit supply/demand detailed; when constraint does not bind:
    - B^F_1 − B^F_1 + αγ_1 / (β I_1) B^F_1 + (1−β)/β B^F_1 (1+γ_1 B^F_1) = 0
    - L_1 = L_1
  - Derivative when constraint does not bind:
    - ∂B^F_1 / ∂I_1 = αγ_1 B^F_1 / I^2_1 β + γ_1 α I_1 + (1+2 γ_1 B^F_1) (1−β)  (exact formula in source)
  - When constraint binds, lending rate and derivatives satisfy exact algebraic relations in source; ∂I^L_1/∂I_1 at binding given by specified rational expression.

### Appendix B — Generalization of the model with currency mismatches: model structure and ELB results
- Representative agent and banking sector:
  - Budget constraint: P_t C_t − Π_t = L_t − L_t, with L_t = L_{t−1} I^L_{t−1}
  - Net income: Π_t = P_{H,t} C_{H,t} + e_t P^*_{H,t} C^*_{H,t} − T_t + N_t − N_t
  - Euler equation: 1 = β_t I_t E_t [ P_t C_t / (P_{t+1} C_{t+1}) ]
  - Demand for Home goods: P_{H,t} C_{H,t} = (1−α) P_t C_t
  - Banking sector balance sheet with foreign currency deposits: N_t + e_t D^*_t = L_t + R_t → N_t = L_t + R_t − e_t D^*_t
  - Modified UIP condition with collateral shadow cost μ_{t+1}:
    - E_t [ ( e_t I_t − e_{t+1} I^*_t (1−χ_t) ) ( I_{t+1} + φ μ_{t+1} ) ] = 0
  - Leverage constraint: L_t ≤ φ N_t
  - Central bank: N^{CB}_t + R_t = e_t X_t → N^{CB}_t = e_t X_t − R_t
  - Public sector consolidated budget: T_t = N^{CB}_t − N^{CB}_t + χ_t e_t D^*_t
  - Market clearing: Y_{H,t} = C_{H,t} + C^*_{H,t}
  - Aggregate budget: α P_t C_t = e_t [ α P^*_t C^*_t + (1−χ_t) D^*_t − D^*_t + X_t − X_t ]
- Steady state with χ_t = 0:
  - e_t = (α β^1 P_t C_t) / (α β^*_1 P^*_t C^*_t + (1−β)(X_t − D^*_t))
  - At time 2: e_2 = (α M_2) / (α M^*_2 + (1−β)(X_2 − D^*_2)); Π_2 = M_2 + (1−β) L_2
- Time-1 and time-0 loan demand, collateral constraints, and e_t expressions given exactly in source:
  - Time-1 loan demand:
    - L_1 = 1/(1−χ_1) α M_2 / β 1/I^L_1 − e_1/(1−χ_1) ( α M^*_2 / β^*_1 I^*_1 − χ_1 D^*_1 ) + L_1 + (N_1 − N_1) + χ_1/(1−χ_1) ( R_1 − R_1 )
  - Collateral constraint: L_1 ≤ φ ( L_1 + R_1 − e_1 D^*_1 )
  - e_1 = (1−χ_1) I^*_1 / I_1 α M_2 / α M^*_2 + (1−β)(X_2 − D^*_2)
- ELB closed-form (exact expression reproduced in source):
  - I_ELB_1 = ( M_2 / M^*_2 α M^*_2 1−χ_1 − α M^*_2 + [ φ − χ_1 (φ−1) ] D^*_1 ̄I^*_1 (φ−1)(L_1 + N_1 − N_1) − χ_1/(1−χ_1) R_1 + φ(1−χ_1)+χ_1/(1−χ_1) R_1 ) (as presented in source)
  - Partial derivatives (exact formulas in source):
    - ∂I_ELB_1 / ∂χ_1 = − (φ−1) D^*_1 ̄I^*_1 − α (φ−1) L_1
    - ∂I_ELB_1 / ∂M_2 = I_ELB_1 M_2
    - ∂I_ELB_1 / ∂N_1 = − I_ELB_1 L_1
- Special case with unconventional tools zero and M_2 = M^*_2 = 1:
  - I_ELB_1 = φ D^*_1 ̄I^*_1 / ( (φ−1) L_1 )
  - Time-0 relationships for L_1 and D^*_1, and comparative-static proportionality:
    - ∂I_ELB_1 / ∂z ∝ ( φ/(φ−1) ∂D^*_1/∂z ̄I^*_1 − ∂L_1/∂z I_ELB_1 )
  - Sign results and proportionalities for ∂I_ELB_1 / ∂I_0, ∂I_ELB_1 / ∂χ_0, ∂I_ELB_1 / ∂ ̄I^*_1, ∂I_ELB_1 / ∂E_0[I^*_1] provided in source.
- Generalized betas and ELB expression with β, β^*_1:
  - e_1, L_1, D^*_1, and lending-rate expressions in presence of generalized betas provided; ELB:
    - I_ELB_1 = α / β_1 [ α ( β^*_1 − β_1 ) + [ (φ−1)(1−β) + φ β_1 ] β^*_1 D^*_1 I^*_1 ]^{-1} × (φ−1) L_1 [ α (1−β+β^*_1) − (1−β) β^*_1 D^*_1 I^*_1 ] (exact expression in source)
  - Effect on output around ELB:
    - ∂Y_{H,1} / ∂I_1 |_{−I_1 = I_ELB_1} = − (1 / I_ELB_1)^2 [ 1−α + α^2 β^*_1 / (1−β) β_1 / [ α(1−β+β^*_1) − (1−β) β^*_1 D^*_1 I^*_1 ] ] × ∂I^L_1 / ∂I_1 |_{−I_1 = I_ELB_1} − (1 / I_ELB_1)^2 α^2 β^*_1 / [ α(1−β+β^*_1) − (1−β) β^*_1 D^*_1 I^*_1 ]
    - Condition for output contraction in constrained region given by inequality in source involving (D^*_1 I^*_1)^2, α, φ, β^*_1, β_1, and other parameters.

*Source: wp18236 - References (PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18236.pdf_