Money and Modern Bank Runs

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Money and Modern Bank Runs

• David Skeie
• Federal Reserve Bank of New York
• Workshop on Banking, Financial Stability
• and the Business Cycle
• August 27, 2004
• Sveriges Riksbank
• The views expressed in this presentation are
  those of the author and do not necessarily
  reflect the views of the Federal Reserve Bank of
  New York or the Federal Reserve System.

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Traditional vs. Modern BankingDoes Money
Effect Bank Run Theory?

• Real World
• Its a Wonderful Life
• Currency withdrawn from banking system
• Wheelbarrow Lending
• Wire transfers
• Same day electronic payment systems

• Theory
• Diamond-Dybvig
• Demand deposits paid in goods
• Clearinghouse model
• Deposits paid with inside money

Traditional Banking Introduce Modern Bankin
g
Depletion from banks ? Bank Runs
Interbank market lending ? No Bank Runs
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Outline of Results

• Results show how modern banking avoids
  traditional bank runs.
• Bank Runs Model Two innovations
• I. Clearinghouse
• ? No panic Redeposit Runs
• Due to interbank lending
• II. Demand deposits paid in money
• ? No Purchase Runs
• Monetary prices for goods adjust
• Money as unit of account

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Bank Run Literature

• Models of interbank lending with goods
• Bhattacharya and Gale (1987), Bhattacharya and
  Fulghieri (1994), Allen and Gale (2000a).
• Models of interbank lending with money
• Allen and Gale (1998, 2000b), Chang and Velasco
  (2000), Gale and Vives (2002), Freixas et al.
  (2000, 2003), Freixas and Holthausen (2001),
  Rochet and Vives (2002).
• Bank runs because money withdrawn from banks and
  economy.
• Diamond and Rajan (2003b)
• Money in general equilibrium.
• Traditional currency withdrawal from banking
  system ? Bank runs.
• Heterogeneous asset shocks ? Bank runs.

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Real Model Setup

• Three periods t 0,1,2.
• Consumers have total endowment of 1 good at t0.
  
• Privately observed types at t1
• ? early u(C1).
• 1-? late u(C2).
• Storage returns 1 over a period.
• Investment at t0 returns
• R gt 1 at t2
• or r lt 1 at t1 if liquidated.

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Real Model Results

• First Best maxC1,C2,a ?u(C1) (1-?)u(C2)
• s.t. feasibility constraints
• a 1 is amount of goods invested.
• FOCs C1, C2 and a.
• ?C1 1-a
• (1-?)C2 aR
• Spot Market
• t0 Consumers choose a to invest and store 1-a.
• t1 Early and late consumers trade.
• ? Outcome a, C11, C2R generally not first
  best.
• Lemma 1 Enforced storage of 1-a ? First
  best C1, C2.
• Bank with Real Contracts (Diamond-Dybvig)
• Real demand deposits (C1,C2) Multiple
  equilibria.

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• Bank with Money
• and Nominal Contracts
• Add to Diamond-Dybvig (real banking) model
• 1. Entrepreneur Store invest goods to sell.
• Bank ensures entrepreneur stores 1-a.
• 2. Money and nominal contracts.
• t0 Central Bank sets P01.
• Money Introduced at t0

Central Bank
1 good
1 good
1
1
Entrepreneur
Consumers
1
1
(D1, D2)
K1 and K2
Bank
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II. Single Bank Money Deposits

• Consumers
• t1 ?p ? withdraw D1 and purchase goods.
• t2 1-?p D2
  
• Bank run ?p gt ? (Purchase Run).
• Prices Market clearing condition
• Entrepreneur sells Qt goods at time t.
• Assumptions
• Bank pays on t1 to t2 deposits competitive D1,2
  
• CRRA 1 ? D1,2 R
• Bank provides insurance, not lottery ticket.
• Sequential-service constraint or pro-rata.

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Results No Purchase Runs

• Step 1. Banks choice of contracts.
• Step 2. Entrepreneurs maximization problem and
  choice of Q1 and Q2.
• Step 3. Late consumers decision whether to run
  the bank.

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Step 1. Bank

• Bank
• Chooses Contracts
• D1 C1
• D2 C2
• K1 ?D1 ( 1-a)
• K2 (1-?)D2 ( aR)
• a a.

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Step 2. Entrepreneur

• t1
• Chooses Q1 goods to sell.
• May liquidate investments or store good till t2.
• Repays K1 if possible.
• Excess revenues (P1Q1-K1) deposited for return
  D1,2.
• t2
• Chooses Q2 goods to sell.
• Repays K2 if possible.
• Consumes excess goods p2.
• Optimization
• maxQ1,Q2 p2 ?p
• s.t. feasibility constraints.

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Step 2. Entrepreneur (Contd)

• Result Entrepreneur never defaults.
• t1 Revenues P1Q1 ?pD1 ?D1 K1
• t2 Revenues P2Q2 (P1Q1-K1)D1,2
• (1-?)D2 K2
• Result Entrepreneur provides optimal goods at
  t1 and t2.
• FOCs ? Liquidation only if P1 gt .
• Key Bank enforces optimal storage at t0.

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Step 3. Late Consumers

• Suppose there is a Purchase Run ?pgt?
• ? Requires ? P1 lt P2
• ? No liquidation Q1 ?D1 1-a
• ?
• ? P1 gt P2 ? Contradiction
• Result There is never a Purchase Run
• ?p ?.

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• I. Clearinghouse
• Transactions at t1

Original Bank Account Balances
Early consumers
Late Consumers
withdraw
?D1
K1
loan
?D1P1goods

redeposit
?D1
goods
deposit
Entrepreneur
Entrepreneur
Second Bank Account Balances
Off-equilibrium Bank Run Late consumers
withdraw at t1
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• I. Clearinghouse
• Transactions at t2

Original Bank Account Balances
Late Consumers
withdraw
K2
goods
loan repayment

goods
Late Consumers
Entrepreneur
withdraw

withdraw
Second Bank Account Balances
Off-equilibrium Bank Run Late consumers
withdraw at t1
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Assumptions

• Second Bank makes Take-It-Or-Leave-It loan offer
  to Original.
• Bank has special relationship ability to collect
  on its loans.
• If Original Bank defaults
• Calls in K2 loan.
• Demand deposits senior to interbank loans.
• Prices are analogous to those above.

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Results No Redeposit Runs

• Step 1. Original Banks budget constraint and
  choice of contracts.
• Second Banks interbank lending decision.
• Step 2. Entrepreneur Same choice as in
  Purchase Runs.
• Step 3. Late consumers decision whether to run
  the bank.

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Step 1. Banks

• Original Bank chooses same contracts as above.
• Second Bank lends fully at return .
• Original Bank Budget Constraint holds for t1
  and t2.
• All funds net owed by Original at t1 are
  available to borrow from Second Bank.
• Original then owes one-period return to
  Second Bank instead of to late consumers who run.
• Lemma 2 Original Bank does not default even if
  all consumers withdraw early.
• Key Second Bank (interbank market) lends
  efficiently.

D2
D1
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Step 3. Late Consumers

• Suppose there is a Redeposit Run
• ? Requires either
• Original Bank is expected to default. ?
  Contradiction
• ?
• ? Contradiction
• Result There is never a Redeposit Run

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Unique First Best Outcome

• Consumption is uniquely determined
• Proposition 1 Unique equilibrium is the first
  best outcome with no bank runs.

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New Bank Run Mechanism

• Breakdown in interbank lending ? Runs and
  Contagion
• Original Bank liquidation ? Deflation P1 ?
• ? Sub-optimal risk sharing for all consumers.
• ? All banks lose liquidity.
• ? Monetary prices cause contagion.
• Lender of Last Resort

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Conclusion

• Model of Banking with Money
• Deposits paid in money? No purchase runs
• Clearinghouse payment system
• Interbank lending ? No redeposit runs
• Lack of bank runs in U.S. post-Great Depression
• Flexible value of money
• Modern electronic clearinghouse