## 2. Decision Tree at t=1 Summarizes How a Bank Run Would Occur.

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### Purpose and setting
- Focus: relationship between capital controls (date-specific and maturity-specific reserve requirements) and the maturity structure of external private debt in a general equilibrium model with endogenous international lending.
- Policy instrument analyzed: date- and maturity-specific reserve requirement — a fraction of amount borrowed deposited in a non-interest bearing account at the central bank and returned when the bond matures.
- Key research questions:
  - What are the effects of date- and maturity-specific reserve requirements on the maturity structure of external debt?
  - Can such reserve requirements prevent a bank run?

### Model architecture (concise)
- Time horizon: three dates t=0, 1, and 2.
- Domestic economy:
  - Continuum of agents with unit-mass, each endowed with 0e at t=0, no endowments at t=1 and t=2.
  - Preference shock at t=1: with probability λ∈(0,1) agent is impatient (consume at t=1), with probability (1−λ) patient (consume at t=2).
  - Utility: U(c1,c2) = λ ln(c1) + (1−λ) ln(c2).
  - Investment technology: 1 unit invested at t=0 yields R>1 at t=2, but if liquidated at t=1 yields r with R>r.
  - Banks arise endogenously and choose k (long-term investment) and l (liquidation at t=1).
- International lenders:
  - Continuum unit-mass, homogeneous, utility U*(c1*,c2*) = ln(c1*) + β ln(c2*).
  - Lend in three instruments: d01 (t=0→t=1), d02 (t=0→t=2, long-term), d12 (t=1→t=2).
  - Perfect arbitrage across lenders implies ρ01 = ρ12 = ρ02 in equilibrium (equation (10)).

### Date- and maturity-specific reserve requirements
- Notation: θ01, θ12, θ02 denote fractions of corresponding debt placed as reserves at the central bank and returned at their maturities. Each θ∈(0,1).
- Budget constraints with reserve requirements (as given):
  - (2) 020201010 )1()1(ddekθθ−+−+≤
  - (3) 0101121201011 )1(drlddcθθρλ++−≤+,
  - (4) 02021212020212122 )()1(ddlkRddcθθρρλ++−≤++−
- Roles of reserve requirements:
  - Tax role: at t=0 a fraction θ01 and θ02 reduces investible funds (penalizes k); at t=1 θ12 acts like a tax on new issuance.
  - Liquidity role: at t=1 fraction θ01 of maturing short-term debt is returned to the bank, reducing need for early liquidation; at t=2 θ12 and θ02 return liquidity.

### Key propositions and equilibrium properties
- Proposition 1 (no reserve requirements):
  - Under perfect arbitrage, interior equilibrium has determinate and locally unique interest rates but indeterminate quantities of each bond.
  - With no reserve requirements, equations (10) and (11) coincide; lenders and borrowers indifferent over bond compositions so long as budget constraints satisfied.
  - Interest rates are pinned by exogenous parameters; long-term bond interest equals gross return R.
- Proposition 2 (with reserve requirements):
  - Under perfect arbitrage, there exist either one or two equilibria depending on the vector {θ01, θ12, θ02}.
  - Quantities of bonds are determinate for any set of equilibrium interest rates.
  - Reserve requirements can cause bifurcations in number of equilibria; specific combinations can yield uniqueness or multiplicity.
- Proposition 3 (equal reserve requirements):
  - If θ01 = θ12 = θ02 = θ with 0<θ<1, there is no unique equilibrium.
  - Two possible equilibria arise when Δ>0 (expression in proof), provided r≤1; if r>1 interior solution may not exist.
  - Equilibrium with lower ρ̂01 has lower domestic welfare than the other equilibrium.

### Emergence of bank runs — decision logic and illiquidity condition
- A bank run occurs at t=1 if:
  - (i) A “bad dream” is realized at t=1 (sunspot/shock).
  - (ii) Bank is illiquid: the bank cannot pay all deposit withdrawals even if it liquidates k.
- Illiquidity condition (without reserve requirements):
  - krqˆˆˆˆ01011 > +ρ (equation (12)). Stronger form: k rˆˆ1 > .
- Interior-solution illiquidity condition expressed in endowments and parameters (equation (13)):
  - 0*00*1**0*2*0 01 )(] )1([ ˆ eee eR eReeR rq−+ +++ > β βλ .
- Decision tree at t=1 (qualitative):
  - If bank liquid → no bank run.
  - If bank illiquid and bad dream → depositors run.
  - After run: international lenders re-optimize; outcomes include bank bailout or bank failure depending on re-lending and solvency.

### Can reserve requirements prevent bank runs?
- With reserve requirements, the illiquidity condition becomes (equation (14)):
  - k r c qˆˆˆ)( 1010101 > + − θ ρ .
- Investment with reserve requirements at t=0 (equation (15)):
  - 01010202 *00 ˆ )()1(ˆ qeek θθθ −+−+ = .
  - For θ01>0 and θ02>0: k̂ < e0* + e0 (investment lower than without reserve requirements).
- Detailed illiquidity condition with reserve requirements (equation (16)) — preserved exactly as in source:
  - ))1( ˆ (1 )}1(} 1 ˆ {),,( ˆ 010201 02 *00 01 0101 02120101 θθρ θ θ θρ θθθ −+− −+ − − −> rr reerq .
- Result 1 (no-reserve case sensitivity):
  - Since ˆq01∈(0, e0*), if ˆq01 → 0 the equilibrium is less vulnerable to illiquidity; if ˆq01 → e0* it is more vulnerable.
- Result 2 (reserve requirement design):
  - If {θ01, θ12, θ02} guarantee:
    - (i) Uniqueness via equation (C.8) in Appendix.
    - (ii) ))1( ˆ 1 )}1(} 1 ˆ {),,( ˆ 010201 02 *00 01 0101 02120101 θθρ θ θ θρ θθθ −+− −+ − − −≤ rr reerq .
  - Then there exists an equilibrium invulnerable to a bank run but with lower investment.
  - Reserve requirements must be date- and maturity-specific; equal θ across maturities does not guarantee uniqueness.
- Additional inequality linking θ01 and θ02 (equation (17)):
  - For 0 12 > ρ̂, require 02 1212 01 )1( )1( θ θθ θ +− − ≥ R R .
  - Implies 1 0212 − ≤ + R R θθ (equation (18)). This highlights that reserve requirements on short-term borrowing should be higher relative to long-term, echoing Chilean practice.

### Market incompleteness and policy interpretation
- Indeterminacy without reserve requirements reflects market incompleteness: lenders cannot offer type-contingent contracts to match depositors’ private types; banks pool risks and are indifferent between borrowing structures.
- Reserve requirements that make specific bonds relatively costlier can correct for this incompleteness by inducing determinacy in maturity composition.
- Only specific combinations of date- and maturity-specific reserve requirements resolve the market failure without creating multiplicity (Result 2).

### International lenders and post-run bailouts
- After a bank run, international lenders re-optimize and decide net new lending x̂12 at t=1.
- Bailout condition (equation (21)):
  - r l x c + ≤ 121 ˆ .
- Solvency condition at t=2 for bank (equation (22)):
  - ) ˆ ( ˆ 02021212 lkRxx − ≤ + ρρ .
- Interior-solution expressions for re-optimization (equations (23)–(25)) specify l̂, x̂02, x̂12 as functions of parameters and t=0 allocations.
- Key insight: international lenders may provide new short-term lending after a bank run to protect and retrieve value of their long-term claims, but bailout occurs only if the bank is illiquid (not insolvent) and lenders anticipate repayment at t=2.

### Welfare and trade-offs
- Reserve requirements:
  - Reduce vulnerability to bank runs when appropriately designed (date- and maturity-specific).
  - Act as taxes ex-ante (reducing k and welfare) while providing liquidity at specific dates ex-post.
  - Can reduce scope of indeterminacy in debt composition and thus correct market failure.
- Trade-off: preventive liquidity role versus ex-post tightening of liquidity due to higher cost of borrowing; effectiveness depends on the full policy mix and exact θ-vector.

### Conclusion (concise)
- Date- and maturity-specific reserve requirements can shift external debt toward longer maturities and reduce vulnerability to bank runs by (i) providing liquidity at bond maturities and (ii) inducing determinacy in bond quantities.
- Effectiveness requires reserve requirements that (a) guarantee liquidity and (b) yield a locally unique equilibrium (specific condition given by equation (C.8) in Appendix).
- International lenders may still bail out an illiquid bank after a run to preserve the value of their long-term claims; bailout feasible when bank is illiquid not insolvent.

*Source: _wp08108 - 2. Decision Tree at t=1 Summarizes How a Bank Run Would Occur.*

### References..............................................................................................................

### References...................................................................................................................................25

### References
- References...................................................................................................................................25

### Figures
- 1. Structure of the Model ..............................................................................................................9

*Source: _wp08108 - References..........................................................................................................*

### 2. Decision Tree at t=1 Summarizes How a Bank Run Would Occur. ......................................13

### 2. Decision Tree at t=1 Summarizes How a Bank Run Would Occur.

### Purpose and setting
- Focus: relationship between capital controls (date-specific and maturity-specific reserve requirements) and the maturity structure of external private debt in a general equilibrium model with endogenous international lending.
- Policy instrument analyzed: date- and maturity-specific reserve requirement — a fraction of amount borrowed deposited in a non-interest bearing account at the central bank and returned when the bond matures.
- Key research questions:
  - What are the effects of date- and maturity-specific reserve requirements on the maturity structure of external debt?
  - Can such reserve requirements prevent a bank run?

### Model architecture (concise)
- Time: three dates t=0, 1, and 2.
- Domestic economy:
  - Continuum of agents with unit-mass, each endowed with 0e at t=0, no endowments at t=1 and t=2.
  - Preference shock at t=1: with probability λ∈(0,1) agent is impatient (consume at t=1), with probability (1−λ) patient (consume at t=2).
  - Utility: U(c1,c2) = λ ln(c1) + (1−λ) ln(c2).
  - Investment technology: 1 unit invested at t=0 yields R>1 at t=2, but if liquidated at t=1 yields r with R>r.
  - Banks arise endogenously and choose k (long-term investment) and l (liquidation at t=1).
- International lenders:
  - Continuum unit-mass, homogeneous, utility U*(c1*,c2*) = ln(c1*) + β ln(c2*).
  - Lend in three instruments: d01 (t=0→t=1), d02 (t=0→t=2, long-term), d12 (t=1→t=2).
  - Perfect arbitrage across lenders implies ρ01 = ρ12 = ρ02 in equilibrium (equation (10)).

### Date- and maturity-specific reserve requirements
- Notation: θ01, θ12, θ02 denote fractions of corresponding debt placed as reserves at the central bank and returned at their maturities. Each θ∈(0,1).
- Budget constraints with reserve requirements (as given):
  - (2) 020201010 )1()1(ddekθθ−+−+≤
  - (3) 0101121201011 )1(drlddcθθρλ++−≤+,
  - (4) 02021212020212122 )()1(ddlkRddcθθρρλ++−≤++−
- Roles of reserve requirements:
  - Tax role: at t=0 a fraction θ01 and θ02 reduces investible funds (penalizes k); at t=1 θ12 acts like a tax on new issuance.
  - Liquidity role: at t=1 fraction θ01 of maturing short-term debt is returned to the bank, reducing need for early liquidation; at t=2 θ12 and θ02 return liquidity.

### Key propositions and equilibrium properties
- Proposition 1 (no reserve requirements):
  - Under perfect arbitrage, interior equilibrium has determinate and locally unique interest rates but indeterminate quantities of each bond.
  - With no reserve requirements, equations (10) and (11) coincide; lenders and borrowers indifferent over bond compositions so long as budget constraints satisfied.
  - Interest rates are pinned by exogenous parameters; long-term bond interest equals gross return R.
- Proposition 2 (with reserve requirements):
  - Under perfect arbitrage, there exist either one or two equilibria depending on the vector {θ01, θ12, θ02}.
  - Quantities of bonds are determinate for any set of equilibrium interest rates.
  - Reserve requirements can cause bifurcations in number of equilibria; specific combinations can yield uniqueness or multiplicity.
- Proposition 3 (equal reserve requirements):
  - If θ01 = θ12 = θ02 = θ with 0<θ<1, there is no unique equilibrium.
  - Two possible equilibria arise when Δ>0 (expression in proof), provided r≤1; if r>1 interior solution may not exist.
  - Equilibrium with lower ρ̂01 has lower domestic welfare than the other equilibrium.

### Emergence of bank runs — decision logic and illiquidity condition
- A bank run occurs at t=1 if:
  - (i) A “bad dream” is realized at t=1 (sunspot/shock).
  - (ii) Bank is illiquid: the bank cannot pay all deposit withdrawals even if it liquidates k.
- Illiquidity condition (without reserve requirements):
  - krqˆˆˆˆ01011 > +ρ (equation (12)). Stronger form: k rˆˆ1 > .
- Interior-solution illiquidity condition expressed in endowments and parameters (equation (13)):
  - 0*00*1**0*2*0 01 )(] )1([ ˆ eee eR eReeR rq−+ +++ > β βλ .
- Decision tree at t=1 (qualitative):
  - If bank liquid → no bank run.
  - If bank illiquid and bad dream → depositors run.
  - After run: international lenders re-optimize; outcomes include bank bailout or bank failure depending on re-lending and solvency.

### Can reserve requirements prevent bank runs?
- With reserve requirements, the illiquidity condition becomes (equation (14)):
  - k r c qˆˆˆ)( 1010101 > + − θ ρ .
- Investment with reserve requirements at t=0 (equation (15)):
  - 01010202 *00 ˆ )()1(ˆ qeek θθθ −+−+ = .
  - For θ01>0 and θ02>0: k̂ < e0* + e0 (investment lower than without reserve requirements).
- Detailed illiquidity condition with reserve requirements (equation (16)) — preserved exactly as in source:
  - ))1( ˆ (1 )}1(} 1 ˆ {),,( ˆ 010201 02 *00 01 0101 02120101 θθρ θ θ θρ θθθ −+− −+ − − −> rr reerq .
- Result 1 (no-reserve case sensitivity):
  - Since ˆq01∈(0, e0*), if ˆq01 → 0 the equilibrium is less vulnerable to illiquidity; if ˆq01 → e0* it is more vulnerable.
- Result 2 (reserve requirement design):
  - If {θ01, θ12, θ02} guarantee:
    - (i) Uniqueness via equation (C.8) in Appendix.
    - (ii) ))1( ˆ 1 )}1(} 1 ˆ {),,( ˆ 010201 02 *00 01 0101 02120101 θθρ θ θ θρ θθθ −+− −+ − − −≤ rr reerq .
  - Then there exists an equilibrium invulnerable to a bank run but with lower investment.
  - Reserve requirements must be date- and maturity-specific; equal θ across maturities does not guarantee uniqueness.
- Additional inequality linking θ01 and θ02 (equation (17)):
  - For 0 12 > ρ̂, require 02 1212 01 )1( )1( θ θθ θ +− − ≥ R R .
  - Implies 1 0212 − ≤ + R R θθ (equation (18)). This highlights that reserve requirements on short-term borrowing should be higher relative to long-term, echoing Chilean practice.

### Market incompleteness and policy interpretation
- Indeterminacy without reserve requirements reflects market incompleteness: lenders cannot offer type-contingent contracts to match depositors’ private types; banks pool risks and are indifferent between borrowing structures.
- Reserve requirements that make specific bonds relatively costlier can correct for this incompleteness by inducing determinacy in maturity composition.
- Only specific combinations of date- and maturity-specific reserve requirements resolve the market failure without creating multiplicity (Result 2).

### International lenders and post-run bailouts
- After a bank run, international lenders re-optimize and decide net new lending x̂12 at t=1.
- Bailout condition (equation (21)):
  - r l x c + ≤ 121 ˆ .
- Solvency condition at t=2 for bank (equation (22)):
  - ) ˆ ( ˆ 02021212 lkRxx − ≤ + ρρ .
- Interior-solution expressions for re-optimization (equations (23)–(25)) specify l̂, x̂02, x̂12 as functions of parameters and t=0 allocations (preserved exactly in form).
- Key insight: international lenders may provide new short-term lending after a bank run to protect and retrieve value of their long-term claims, but bailout occurs only if the bank is illiquid (not insolvent) and lenders anticipate repayment at t=2.

### Welfare and trade-offs
- Reserve requirements:
  - Reduce vulnerability to bank runs when appropriately designed (date- and maturity-specific).
  - Act as taxes ex-ante (reducing k and welfare) while providing liquidity at specific dates ex-post.
  - Can reduce scope of indeterminacy in debt composition and thus correct market failure.
- Trade-off: preventive liquidity role versus ex-post tightening of liquidity due to higher cost of borrowing; effectiveness depends on the full policy mix and exact θ-vector.

### Conclusion (concise)
- Date- and maturity-specific reserve requirements can shift external debt toward longer maturities and reduce vulnerability to bank runs by (i) providing liquidity at bond maturities and (ii) inducing determinacy in bond quantities.
- Effectiveness requires reserve requirements that (a) guarantee liquidity and (b) yield a locally unique equilibrium (specific condition given by equation (C.8) in Appendix).
- International lenders may still bail out an illiquid bank after a run to preserve the value of their long-term claims; bailout feasible when bank is illiquid not insolvent.

*Source: _wp08108 - 2. Decision Tree at t=1 Summarizes How a Bank Run Would Occur.*

### REFERENCES

### REFERENCES

### Books and Monographs
- Azariadis, C., 1993, Intertemporal Macroeconomics (Cambridge, Massachusetts: Blackwell Publishers).
- Fisher, I., 1930, The Theory of Interest, (New York: The Macmillan Company).
- Keynes, J. M., 1935, The General Theory of Employment, Interest and Money (New York: Harcourt Brace).
- Sargent, T., 1987, Dynamic Macroeconomic Theory (Cambridge, Massachusetts: Harvard University Press).

### Working Papers and NBER Papers
- Auernheimer, L., and R. Garcia-Saltos, 2000, “International Debt and the Price of Domestic Assets,” IMF Working Paper 00/177 (Washington: International Monetary Fund).
- Chang, R., and A. Velasco, 1998, “Financial Crises in Emerging Markets: a Canonical Model,” NBER Working Paper No. 6606 (Cambridge, Massachusetts: National Bureau for Economic Research).
- Diamond, D., and R. Rajan, 2000, “Banks, Short-term Debt and Financial Crises: Theory, Policy Implications and Applications,” NBER Working Paper No. 7764. (Cambridge, Massachusetts: National Bureau of Economic Research).

### Journal Articles and Chapters
- Aizenman, J., and S. Turnovsky, 2002, “Reserve Requirements on Sovereign Debt in the Presence of Moral hazard on Debtors or Creditors,” The Economic Journal, Vol.112, pp. 107–32.
- Dadush, U., D. Dasgupta, and D. Ratha, 2000, “The Role of Short-term Debt in Recent Crises,” Finance and Development, Vol. 37, No. 4.
- De Gregorio, J., S. Edwards, and R.O.Valdes, 2000, “Controls on Capital Inflows: Do They Work?,” Journal of Development Economics, Vol. 63, pp. 59–83.
- Diamond, D., 2007, “Banks and Liquidity Creation: a Simple Exposition of the Diamond-Dybvig Model,” Economic Quarterly, Vol. 93, pp. 189–200.
- Diamond, D., and P. Dybvig, 1983, “Bank Runs, Deposit Insurance, and Liquidity,” The Journal of Political Economy, Vol. 91-3, pp. 401–19.
- Forbes, K., 2007, “One Cost of the Chilean Capital Controls: Increased Financial Constraint for Smaller Traded Firms,” Journal of International Economics, Vol. 71, pp. 294–323.
- Goldstein, I. and A. Pauzner, 2005, “Demand-Deposit Contracts and the Probability of Bank Runs,” The Journal of Finance, Vol. 9, No. 3.
- Kaminsky, G., and C. Reinhart, 1999, “The Twin Crises: The Causes of Banking and Balance-of-Payments Problems,” American Economic Review, Vol. 89, No. 3, pp. 473–500.
- Reinhart, C., and V. Reinhart, 1999, “On the Use of Reserve Requirements in Dealing with Capital Flow Problems,” International Journal of Finance and Economics, Vol. 4, pp. 27–54.
- Reinhart, C., and T. Smith, 2002, “Temporary Controls on Capital Inflows,” Journal of International Economics, Vol. 57, pp. 327–51.
- Rogoff, K., 2002, “Rethinking Capital Controls: When Should We Keep an Open Mind?,” Finance and Development, Vol. 39, No. 4.

*Source: _wp08108 - REFERENCES*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08108.pdf_