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Diamond Dybvig Model (1983). Captures elements of what a bank does. Shows that there is a basic problem of bank runs. The model consists of two parties. Depositors Banks The model has three time periods: yesterday, today and tomorrow. Depositors.

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Diamond dybvig model 1983 l.jpg
Diamond Dybvig Model (1983)

  • Captures elements of what a bank does.

  • Shows that there is a basic problem of bank runs.

  • The model consists of two parties.

    • Depositors

    • Banks

  • The model has three time periods: yesterday, today and tomorrow.

Depositors l.jpg

  • Depositors placed money (say £1000) in a bank (yesterday) before learning when they need the money.

  • Depositors either need their money today (impatient) or tomorrow (patient). There is a 50% chance of being either type.

  • The ones that need their money tomorrow can always take the money today and hold onto it.

  • The ones that need money today get relatively very little utility for the money tomorrow.

Banks l.jpg

  • Banks have both a short term and a long term investment opportunity for the money.

    • The short term investment (reserves) is locking the money in the vault. This investment returns the exact amount invested.

    • The long term investment returns an amount R tomorrow. It is illiquid and returns only L<1 today.

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Deposit Contract

  • The depositors invested £1000 yesterday have a contract with the bank.

  • The depositors can withdraw their money today and receive £1000 or wait until tomorrow and receive R*£1000.

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Bank’s decision

  • How can the bank meet this contract?

    • The bank can divide into two parts.

      • Take half and keep it as reserves.

      • Take the other half and put it in the long term investment.

  • Say there are 10 depositors: 5 patient and 5 impatient. The bank puts £5000 in the vault and invests £5000.

  • Demands today are 5*1000, and 5*R*1000. The bank has 5000 and R*5000 tomorrow.

  • Thus, a bank makes zero profit.

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Multiple equilibria

  • This leads to multiple (Nash) equilibria.

  • It is inherent in banking.

  • Here is an example with 2 patient depositors (and 2 impatient depositors).

  • This forms a 2x2 game between the patient depositors.

  • R=1.5 and L=.5

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Game between patient depositors

Depositor 1








Depositor 2






R=1.5, L=.5

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Our experiment

credit crunch

Normal conditions

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What is not captured in the model

  • Uncertainty in depositor’s preferences.

    • Too many actually need the money today.

  • Riskiness in technology.

    • Perhaps there really isn’t enough to meet demand tomorrow.

    • Implication: some bank runs will be rational.

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Early Solutions to Bank Runs

  • Put money in the windows

  • Slow up payments.

Solutions l.jpg

  • Make sure R is not risky.

  • Pay early withdrawers less than 1 or pay late withdrawers less than R (and keep more reserves)

    • Problems: not best contract.

  • Suspend payments/ Partial Suspension.

    • Problem when number needing money today is uncertain.

  • Creditor Coordination.

    • Long Term Capital Management ran into trouble in 1998.

    • The NY FED organized a bailout with creditors.

  • Lender of last resorts.

    • Central bank will stop in and loan the bank money to replace deposits.

    • This should work with depositors in the case of a problem with liqudity

    • In 1975,

      • April 14th, Credit Suisse announced lost some money in one of its branches. It didn’t mention details.

      • April 25th, The Swiss Central Bank announced it was willing to lend money.

      • This had the opposite result cauing share price to tumble 20%.

  • Deposit Insurance.

    • This works well. Risk-Sharing between banks.

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Better Contract

  • Why should the bank pay the depositors withdrawing early only 1?

  • The bank can pay them more.

  • This would “insure” a depositor against needing the money early.

  • For R=1.5, what would the full insurance contract look like. In other words, the payment is the same in either period.

  • The amount would solve (2000-X)*R=X

  • This amounts to a gamble of having either 1000 or 1500 or 1200 for sure. Risk-averse enough people would prefer 1200.

  • Note the best contract (and perhaps fairest) will pay depositors withdrawing today somewhere 1000 and 1200.

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Hidden assumption

  • Depositors withdraw sequentially: a bank cannot count the number of people wanting to withdraw today and then decide how much to pay them.

  • Otherwise, they can just pay them 5000/N where N is the number withdrawing early (for the 10 depositor case).

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Insurance Problem: Moral hazard

  • Todd buys theft insurance for his laptop.

  • Because he buys the insurance, he is more likely to leave the laptop in his car.

  • Ideally, he would like to commit to not leaving the computer in his car.

  • Sometimes, we can contract on it.

  • Other times, we can’t.

  • Do we have a moral hazard problem with deposit insurance?

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Answer: Yes.

  • Marc is the manager of a Springfield S&L.

  • Marc pays higher interest than a bigger and safer bank claiming his small size helps him cut costs.

  • Springfield has deposit insurance (100%).

  • Todd puts money in Springfield.

  • Springfield lends money to a dodgy lecturer at Springfield State University at a higher rate.

  • When there is no default, everyone wins.

  • When there is a default, Todd still gets paid.

  • Without insurance, Todd wouldn’t invest if he sees Springfield’s risky behavior.

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Model of Moral Hazard.

  • The bank can choose any investment x, where 3>x=>1.

  • Any investment costs £.95 and is either successful and pays of x or unsuccessful and pays £0.

  • The probability of the investment being successful is


  • Choosing x=1 is safe, choosing x close to 3 is unsafe.

  • Todd is close to risk neutral and wants to earn at least as much as £1 (in expectation) which the other banks are offering as a risk free investment. He wants R where R*P(x)=1.

  • Without insurance, the bank maximizes

    • P(X)*(X-R) where R=1/P(x)

  • With insurance, Todd only needs R=1. So the bank maximizes

    • P(X)*(X-R) where R=1

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Savings and Loans scandal

  • In the 1980s about 1000 S&L’s went bankrupt.

  • They originally lent money out at fixed rates of 6% and paid deposits 3%.

  • With inflation, they lost money.

  • Took gambles to catch up, went to Vegas.

  • They were able to take high risk due to the deposit insurance.

  • This cost US taxpayers $120 billion.

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Solution to Moral Hazard

  • One solution is for insurance to not be 100% (co-pay as in the UK).

  • However, this requires the depositors to be savvy and this still keeps the multiple equilibrium problem.

  • In the US, in 2006 Bush signed a law allowing the FDIC to charge premiums based upon risk.

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Lender of Last Resorts: commitment

  • Gambling Jim has a rich uncle.

  • Jim’s uncle loves him very much.

  • Jim blows his money in a poker game. His rich uncle bails him out.

  • His uncle says that is the last time.

  • Jim gambles again and loses. His rich uncle can’t bear to see Jim’s legs broken.

  • The problem is that Jim knows his uncle will always be there for him.

  • The uncle can either find some way to commit not to help Jim afterwards, or sacrifice Jim to stop his other nephews from gambling.

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Northern Rock and the Subprime crisis.

  • Jim Cramer on subprime.

  • Bill Poole says that it is risky lenders that got what they deserved.

  • Jim Cramer more or less says everyone is in trouble.

  • Bernanke is thinking about whether to cut rates.

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Subprime mortgages

  • Miriam, a divorced mother, was offered a mortgage on a 2-28 deal: 2 years of a teaser rate of a mortgage and the rest floating.

  • Finally, the dream of owning a home is a reality.

  • Miriam did not have to verify income or assets. She got a piggyback loan to cover the down payment.

  • She was told that she can refinance after two years and with the prices the way they are going get some money out as well.

  • She took some extra credit cards and agreed.

  • The bank took her mortgage packaged it up with others. The rating agency (paid by the bank) rates the package high. It is sold to a hedge fund.

  • The bank now only collects the money.

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Sub-prime continued

  • Unfortunately, rates went up and she couldn’t make the payments.

  • Housing prices didn’t go up and she has no equity.

  • The local bank doesn’t want to work out a deal so forecloses (actually collects extra fees doing so).

  • These packages drop faster than one would have thought.

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Hedge fund

  • Hedge funds are highly leveraged. The price of these securities drop more than they should given the state of the economy, interest rate, etc.

  • People loaning the hedge funds want to take their money out. Forcing the hedge funds to sell more. This further suppresses the price.

  • Hedge funds start to go broke.

  • Banks also have these mortgages on their books. It isn’t clear who owns what.

  • Banks don’t want to lend to each other for two reasons:

    • Afraid of the financial state of other banks.

    • Want to keep extra reserves in case they can’t borrow.

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Northern Rock

  • Had mortgages not necessarily subprime.

  • To provide funding for these mortgages, it had deposits and borrowed from other banks.

  • The other banks were in essence another depositor.

  • When the credit dried up, the other banks needed the money (became impatient).

  • Northern Rock couldn’t continue to borrow.

  • They had to borrow from the lender of last resorts.

Analogy l.jpg

  • There are a limited number of homes around a lake.

  • The owners of the homes only sometimes go there for a few days. They only know if they can go last minute.

  • When they don’t go, they rent them out. Say that they stay there only 1/5 the time and there is 3 non-owners for every owner during a summer.

  • The owners are indifferent to staying at their home or someone else’s home.

  • Thus, the owners are not too worried about renting out their home since they can always rent someone else’s home.

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Lake Analogy

  • Suddenly, a rumour spreads that it will be hard to rent.

  • The owners want to stay very much when they are free and take their homes off the market.

  • Since all the owners do so, there is no rental market and it is self-fulfilling.

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Northern Rock Bank Run

  • Depositors now started to run.

  • Was it rational for shareholders to run as well?

  • Not enough deposit insurance.

  • It also wasn’t clear how much lending was to Northern Rock.

  • It isn’t clear how good their loans are.

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Should Mervyn King act?


  • Was there a risk of contagion.

  • Was it a solvency problem or a liquidity problem?

  • Would it cost tax-payers?

  • Does this set a bad example?

    Actions: Cut Rate, Lend, Organize a bailout (LTCM)


  • Rewrite the deposit insurance?

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Other applications

  • Farepak.

    • People saved during the year and got coupons at end worth their savings.

    • Company used next year’s money to pay for this year’s coupons.

  • Defined-Benefit Pension schemes/Social Security.

    • Young pay for old.

    • Usually mandatory to stop runs.

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What we learned

  • Theoretical Model of Bank Runs.

  • That these may actually happen (experiment).

  • Possible solutions to the problem.

  • The Moral Hazard problem.

  • A bit about the current crisis.

Homework l.jpg

  • Take the DD model with L=.5 and R=2. Let us say that deposits are insured up to fraction f. For what values of f is there only one equilibrium and what values are there two equilibria? (Early withdrawers are guaranteed to get 1*f and late get 2*f.)

  • How would you modify our classroom experiment to test different deposit insurance schemes? Under what parameters do you think we will get a bank run.