Adding more periods to the Diamond Dybvig Model; fear of illiquidity not insolvency

David Andolfatto has a very good post on "Monetary Policy Implications of Blockchain Technology". In passing (it's not a central point of his post), David says:

"However, it's worth pointing out that the leading economic theory of bank sector fragility, the Diamond and Dybvig model, does not rely on the existence of opacity in the financial market. In that model, the portfolios of banks are perfectly transparent. A bank run may nevertheless be triggered by the expectation of a mass redemption event, which subsequently becomes a self-fulfilling prophecy. It is also interesting to note that (in the same model) bank-runs can be eliminated if banks adopt a credible policy of suspending redemptions once they run out of cash (this commits the bank not to firesale assets to meet short-term debt obligations)."

I think that David is right about suspending redemptions. But I think that David would be wrong if we made a small change to the Diamond Dybvig model. We simply add an extra time period, or periods. It's a friendly amendment to the model.

[My own views on Diamond Dybvig-type models have been influenced by Morgan Ricks' "The Money Problem", but I am not 100% sure whether Morgan would agree with what I say here, though I think he will.]

Agents are ex ante identical. Each agent has an endowment of apples. There is a costless storage technology for apples. There is also an investment technology (planting apples in the ground) which gives a strictly positive rate of return at maturity, but a negative rate of return if you cancel the investment before maturity. Each agent has a 10% probability of becoming impatient (getting the munchies) and wanting to eat all his apples this period. Those probabilities are independent across agents, and there is a large number of agents, so exactly 10% of agents will become impatient each period. Getting the munchies is private information.

The standard Diamond Dybvig model has 3 periods: an initial period where agents lend their apples to the bank; a second period where 10% of agents get the munchies and ask for their apples back; and a third period when the investment matures. Banks exist to provide insurance against risk of munchies by pooling assets; normal insurance won't work because the information is private.

Let's instead make it a 4 period model: an initial period where agents lend their apples to the bank; a second period where 10% of agents get the munchies and ask for their apples back; a third period where another 10% of agents get the munchies and ask for their apples back; and a fourth period when the investment matures.

Suppose the bank credibly commits that it will never cancel an investment before maturity, and stores 20% of apples in reserve. In the good equilibrium there is no sunspot and only agents who get the munchies ask for their apples back. Now suppose there is a sunspot and a run on the bank in the second period of the 4-period model. An agent who does not have the munchies in the second period will rationally join that run on the bank, falsely claiming that he does have the munchies. The reason is that he might get the munchies in the third period, and if the bank suspends redemptions he will be unable to satisfy his future cravings, so he wants to join the line before the bank runs out of stored apples, so he can store apples at home.

Alternatively, we could make an infinite period model, with the good equilibrium being a steady state with the bank's assets staying constant over time because flows in equal flows out. All that is required for runs, despite suspension of redemptions, is that investments take more than 2 periods to mature. [And in a continuous time version of the model, the bank would not be able to prevent runs by committing to stop redemptions whenever X% of apples have been redeemed.]

Even if people are 100% confident that the bank is solvent, there can still be bank runs if people cannot predict their own future needs for liquidity, and fear that the bank might become illiquid in future. Having a deposit in an illiquid bank is functionally not the same as having a deposit in a liquid bank, even if both are solvent.

Even if deposit insurance guarantees that your deposit is safe, you might see a bank run if there is expected to be a delay in paying out that insurance. Liquid assets are more desirable than illiquid assets that pay the same rate of return.

But the real problem with the Diamond Dybvig model is, ummmm, that it's a real model, not a monetary model. If there really were only one bank that created money, the problem of bank runs due to fear of illiquidity would not arise. Because any money spent by one agent immediately goes into the account of another agent at the same bank.

Comments

[Morgan Ricks]

Hi Nick, I think I agree with this on a first read.

I find nonmonetary models of banking surreal but others' mileage may vary.

[David Andolfatto]

Nick, let me take a look at that Morgan Ricks piece when I return (from conference).
Note: I have a monetary version of the DD model. Bank issue demandable liabilities, redeemable in cash (or gold).
The problem remains. Doesn't matter that you have one bank.

[Nick Rowe]

Hi Morgan: yes, I figured you probably would!

It's all in the "models as metaphors" question. We read "apples", but know I don't literally mean apples, but we don't know just how widely the metaphor can be stretched. Can it be stretched to "cash"? No, because apples get consumed, but cash doesn't get consumed, which is why the metaphor breaks down when we come to a single bank.

[Nick Rowe]

David: thanks.

Suppose you have a single bank, that issues the only medium of exchange that people use, but its liabilities are redeemable at a fixed exchange rate in copper, which nobody uses as money. If that bank temporarily suspended convertibility into copper, people could continue to use their chequing accounts at that bank (or banknotes issued by that bank) as money.

But if the bank's liabilities are redeemable at a fixed exchange rate in gold, which people also use as money, we have a problem if the bank temporarily suspends convertibility. Because we now have two monies with a flexible exchange rate between them. Which brings us to your latest post...

[Jan Musschoot]

The real problem is that economists use Diamond-Dybvig, which overcomplicates things and misrepresents what banks actually do. Which in turn leads to confusing analogies like in this post.

Here is how it really works:
Suppose there is a world with 4 agents:
-A has $100 in $1 bills
-B is the bank without assets nor liabilities
-C has a house worth $100
-D has nothing

(1) A deposits the bills at B. B now has $100 assets, and creates a liability of $100, the deposit of A.
(2) D borrows $100 from B (for ease of the argument, we assume a 0% interest rate environment), to be paid back after 1 year. B now has created an additional asset: the $100 loan from D. In addition, B has created an extra liability: there is $100 in the account of D. So the bank has created money.
(3) D buys the house from C for $100, so the bank deletes the money in the account of D and writes $100 in the account of C.

Situation after these transactions:

-A has $100 in account at B
-B has $200 assets: $100 in bills + a loan of $100. B has $200 liabilities: $100 on the account of A and $100 on the account of C
-C has $100 in account at B
-D has a house worth $100 and a liability of $100 (the loan = debt to the bank)

The problem arises when A and C both want to withdraw the money from their accounts in physical $1 bills. B only has $100 worth of them, because it created money in accounts when it granted the loan.

The only way to prevent this problem without central bank intervention [see note at the end], is for B to match the maturity of its liabilities with that of its assets (the loan). So B needs to convince A, C or both of them to invest $100 from their current accounts into a 1 year term deposit (or a bond the bank issues). This guarantees that the bank can survive a run on its deposits.
D has one year to earn $100 from A and C. After the year, the bank deletes the loan from its assets and the $100 in the account of D from its liabilities.
The term deposit matures, so the proceeds are added back to the current accounts of A and C. B now has assets equal to $100 in bills, and liabilities of $100, so B is perfectly safe when the depositors suddenly want their money back.

Voilà, banking explained without fancy math or game theory.

[If there is a central bank, it can print $100 and buy the loan from the bank. When D pays back the loan, the extra $100 is removed from the economy after one year.]

[Nick Rowe]

Jan: That's just *accounting*! It doesn't *explain* anything!

(Your example has two banks, not one. There is the commercial bank that issues chequing account money; and there is the central bank that issues paper money (banknotes/bills).)

[Too Much Fed]

"Suppose you have a single bank, that issues the only medium of exchange that people use, but its liabilities are redeemable at a fixed exchange rate in copper, which nobody uses as money."

What is the medium of account here?

[Jan Musschoot]

Nick: "That's just accounting, it doesn't explain anything"
-> Yes it does, it shows why banks are vulnerable to runs and how they can protect themselves against runs. Please explain what DD explains more than my example.

"Your example has two banks"
-> No printing occurs in my example (except for my very last sentence between [], which describes another situation than the example). Of course the dollar bills once came from somewhere (created out of nothing by seignorage). But my example is completely unchanged when the physical money is not dollar bills but instead gold coins from ancient times (i.e. world without central bank), and there is no more gold to be found anywhere in the world.

The point is: commercial banks make money by making deposits. They cannot print the money, so they are always at risk of a run on physical money. The only sure protection they have is matching the duration of their assets and liabilities.

[Nick Rowe]

TMF: I was thinking the MoE is also MoA. But we could imagine copper is the MoA.

Jan: DD explains why banks exist (why people would rationally choose to use banks). It also explains why behaviour in a run is rational (and also why behaviour in a non-run is rational).

"But my example is completely unchanged when the physical money is not dollar bills but instead gold coins from ancient times (i.e. world without central bank), and there is no more gold to be found anywhere in the world."

Fair point.

[Majromax]

@Jan

Your explanation is not explanatory because it does not motivate steps 1 (the initial deposit) or step 4 (the hypothetical run). If loans are provided at no interest and deposits provide no interest, then why would anyone deposit money at the bank?

If demand deposits are vulnerable to bank runs then they are less liquid than cash, so A would not make the deposit without some incentive. If that incentive is overcome (say there's a better-than-cash electronic payments system), then you have the opposite problem of arguing why a run is plausible – everyone should prefer demand deposits to cash.

Your point is correct that runs are impossible if the term structure of assets is the same as or longer than the term structure of liabilities, but banks don't work that way. Despite borrowing short and lending long, they seem to have only sporadic problems.

[Nick Edmonds]

You can get something similar in a monetary version of DD with one bank and non-redeemability, but it arises as a run on the value of the bank's issued money rather than the bank's liquidity. The bank can maintain the value of its money, provided not too many people want to offload it at once. The bank controls the value by calling loans which creates demand for its money. If too many people want to get rid of their money earlier than expected, the bank can no longer create sufficient demand to be able to control the value.

I touched on this in this post: http://monetaryreflections.blogspot.co.uk/2014/11/diamond-dybvig-and-monetary-circuit.html

[Nick Rowe]

Nick E: Good point. I think it's right. But those falls in value are self-limiting, because people will expect the value of money to rise again when the bank eventually redeems some of it, which increases the current demand for money.

It's a bit like using indirect convertibility to target the price of the CPI basket, rather than using direct convertibility on demand (redeemability) to target the price of (say) gold.

[Too Much Fed]

I am going to be picky here because I believe it matters. The term "money" should not be used if there is going to be a difference between MOA and MOE.

OK. There is a single bank. *Some* of its liabilities are both MOA and MOE.

Copper is not MOE.

Now fix the exchange rate between copper and the bank liabilities that are MOA and keep the fixed exchange rate there.

I am going to say there is dual MOA now (both the some liabilities of the single bank and copper).

I believe there is a general idea involved here.

Select a MOA. Fix the MOA to something else. The something else also becomes MOA. There is a dual MOA.

[Jan Musschoot]

@Majromax

*) Why would my agent A make a deposit? For convenience, so he doesn’t have to store his bills under the matrass. Reality proves me right on this point: most Belgian current accounts offer 0% interest rates. Despite this, billions of euros are stored in current accounts at the moment. Undoubtedly the same is true in Canada. To elaborate on the existence of deposits: neglect my agent A and look at my steps (2) and (3). It is the bank that creates a deposit when granting a loan. The client cannot choose it, he can only withdraw the account money in cash form (or wire it to another bank in a world where more than one bank exists).

*) There are loans yielding 0% at the moment (for example mortgages for very good clients). The bank tries to make money on fees. I didn’t include this in the example to keep the math simple.

*) Why can there be a run in my example? As I explained, the bank never has enough cash when it has made loans. If there are doubts that the bank will not be able to pay your money back, it’s rational to panic first and withdraw. But usually it’s just convenient to store money in the bank.

*) It’s true that the term structures of banks’ assets and liabilities don’t match in practice. Banks kind of assume that not everybody will withdraw their money. But that is just wishful thinking that explains why they are always vulnerable to runs, just ask Northern Rock or Dexia.

Note that under Basel III rules, they just need to proof to the regulator that they have enough liquid assets to survive some bogus stress scenario that describes 30 days liquidity outflows (just google 'liquidity coverage ratio'). I should know, because I have made these kinds of reports for years ;-)

[Nick Edmonds]

I don't think it's self-limiting in the way I set it up, because of the way I've incorporated DD. Essentially we have an arrangement where in the normal course, there is, say, $100 in circulation and $100 of loans to be repaid. But if too many loans have to be called early, you might end up with, say, $60 in circulation and only $50 of loans outstanding. It then becomes clear that someone will be left holding the last $10, at which point they are worthless bits of paper, because the bank has exhausted any ability to manipulate the value. Of course, the money may retain some value if people still accept it hoping to be able to pass it on to someone else, but we're then running on steam.

[Jan Musschoot]

@Nick: when the loan in my accounting model comes with an interest rate of 6%, and the bank offers 2% on the bank deposits, it explains banking as well as DD does. After a year, the bank earns $6, and the depositors earn $4 on the combined $200 in their accounts. So the depositors win (they earn 2% instead of 0% when money is kept at home), the bank makes $2 profit ($6 interest income - $4 interest on accounts). But a run is still rational if there is doubt that the bank will not be able to pay the depositors back. Banks never have enough cash to cover their deposits. The loan is illiquid. A buyer would offer cents on the dollar for the loan, even if the credit worthiness of D is not questioned. So the bank can be solvent (it will get $106 from D after a year, exceeding the liabilities), but it doesn't have the money available as long as the loan has not matured. The run (and corresponding fire sale of assets) makes the bank both illiquid and insolvent.

So at the same time, the accounting model offers the rationale for a central bank as a lender of last resort. (And while we're at it, the rationale for banking regulation.)

[Too Much Fed]

Jan said: " The loan is illiquid. A buyer would offer cents on the dollar for the loan, even if the credit worthiness of D is not questioned. So the bank can be solvent (it will get $106 from D after a year, exceeding the liabilities), but it doesn't have the money available as long as the loan has not matured. The run (and corresponding fire sale of assets) makes the bank both illiquid and insolvent.

So at the same time, the accounting model offers the rationale for a central bank as a lender of last resort. (And while we're at it, the rationale for banking regulation.)"

Assume an ECB type central bank.

What happens if the ECB acts as a lender of last resort to the bank experiencing a bank run (redemption for currency) here?

[Nick Rowe]

Nick E: currency has been running on steam for the last few centuries. People will hold it at 0% nominal interest, and negative real interest. It's a profitable business (depending on competition from other issuers). In the long run, it's usually an asset, not a liability, to the bank that issues it. You only need assets for short run drops in demand.

[Ralph Musgrave]

[Ralph: I am trying to train TypePad to stop putting you in Spam, so deleted the content of this comment, since it's a duplicate. Sorry about that. Nick]

[RalphMus]

Given that central banks / governments can issue infinite amounts of money, I'm baffled as to what the point of letting private banks issue money / liquidity is: particularly since (as correctly pointed out by Diamond) private banks can't create or issue money without risking runs, instability, credit crunches, etc.

Moreover, private banks act in a PRO-CYCLICAL manner: that is, they exacerbate booms and recessions - which central banks and governments have to counter with publicly created money. Witness the HUGE increase in the stock of base money over the last few years resulting from interest rate cuts and QE.

[Jan Musschoot]

@Too Much Fed:

The central bank (CB) should buy the loan from the commercial bank for a price below its (nominal value + accrued interest).

For ease of argument, suppose there is a run after half a year. The $200 deposits have earned 1/2 of $4 (=$2), which the bank needs to pay at the end of the year.

The accrued interest on the 6% loan after half a year is $3.

So the CB could buy the loan from the commercial bank for $102, which is smaller that the $100 (nominal) + $3 (accrued interest). The CB prints $102. The bank now has $202 in physical cash assets, and it can pay back the $200 of its depositors.

The debtor D still has half a year to earn $106 from the other agents in the economy.

At the end of the year, D pays $106 to the CB.

Net results:

*)the commercial bank has earned $0 (it needs to pay the $2 it had left to the depositors), so it is punished for having run into trouble and needing saving by the CB

*)the CB has made $4

*)the depositors have earned only $2 instead of $4 because they panicked and withdrew their money after half a year

*)nothing has changed for the debtor: he fulfilled his obligations to the owner of the loan

(In practice the commercial bank has a capital buffer, so the central bank can even pay less for the loan. So the CB punishes the shareholders for the imprudent behavior of their bank)

[Too Much Fed]

Jan said: "The central bank (CB) should buy the loan from the commercial bank for a price below its (nominal value + accrued interest)."

What if the ECB-like central bank does not buy the loan, but uses the loan as collateral?

[Jan Musschoot]

@Too Much Fed
"What if the ECB-like central bank does not buy the loan, but uses the loan as collateral?"

In that scenario, the central bank lends the commercial bank $100 in printed money and gets loan as collateral. The commercial bank uses the cash to pays its depositors $200 cash (remember it still had $100 from the deposit of agent A). At end of the year, debtor D pays $106 in cash to commercial bank. The commercial bank returns the $100 cash to the central bank.

The terms of the emergency credit from the central bank to the commercial bank will determine how the profit from the interest income of the loan to D is distributed between central bank and commercial bank.

For example: suppose the emergency credit from central bank to commercial bank is granted after half a year, when there's a run. Say the central bank demands 6% interest for its $100. At the end of the year, the commercial bank receives $106 from debtor D. The commercial bank needs to pay back $100 principal to central bank, plus $3 interest (6%/year on $100 for half a year). So the central bank earns $3, the depositors have earned $2 (2% on $200 for half a year) and the commercial bank has earned $1.

[Too Much Fed]

Jan, I think that sounds good. I am going to look at it a different way.

"In that scenario, the central bank lends the commercial bank $100 in printed money and gets loan as collateral."

The depositor had a demand deposit. I consider demand deposits to be a bond. The depositor wants to sell the bond back to the commercial bank for currency. The commercial bank does not have enough currency to exchange for the bond and is solvent. Instead of selling assets to raise currency in the market or to the central bank, the commercial bank goes to the central bank as a lender of last resort.

Basically, the commercial bank buys the bond/demand deposit back from the depositor for currency and then sells the bond/demand deposit for currency to the central bank with the loan acting as collateral against the bond/demand deposit.

This allows the demand for currency to be met and maintaining the fixed exchange rate between currency and demand deposits, which just happens to be 1 to 1.

Sound good?

Also, are the conditions for lender of last resort a valid demand deposit and the commercial bank being solvent?

[Jan Musschoot]

@Too Much Fed

I think we agree :)

"Also, are the conditions for lender of last resort a valid demand deposit and the commercial bank being solvent?"
I would say yes. The central bank insures bank deposits by acting as the lender of last resort to the commercial bank (the deposit insurance itself reduces the risk for a bank run).
The condition of commercial bank solvency is exactly like Bagehot recommended (see https://en.wikipedia.org/wiki/Lombard_Street:_A_Description_of_the_Money_Market#Lender_of_last_resort): the central bank should lend freely at a high rate of interest (so as to punish the bank for its lousy risk management) on good banking securities (i.e. good collateral, the loan in my example).

[Too Much Fed]

Jan, sounds like we agree. Good! A couple of other things.

1) The price inflation target has nothing to do with the lender of last resort function of the central bank?

2) What if the commercial bank does not have lousy risk management? What if the preferences of the private entities has changed so they want to hold more currency and fewer demand deposits even though the commercial bank is solvent and entities believe that?

[Jan Musschoot]

@Too Much Fed

Good points, I would need to think them over, I don't have a well-founded opinion at the moment :)

[Too Much Fed]

Jan, about #1.

It seems to me that if the price inflation target has nothing to do with the lender of last resort function of the central bank, then the system works one way.

If the price inflation target affects the lender of last resort function of the central bank, then the system works a different way.

[Frances Coppola (@Frances_Coppola)]

Nick, this is not a comment aimed at you specifically - it is aimed at David too.

I disagree that bank runs can be eliminated by expectation that banks will suspend redemptions once it has run out of free cash. Expectation that a bank will suspend redemptions at some point during a run could actually trigger a run even in a three-period model, since those with the munchies will stampede for the exit. If you add in your fourth period, or make this an infinite-period model (which in effect assumes that investors expect the bank will never re-open its doors), then the run is likely to be far worse, since there is no reason for anyone not to join it.

Lender-of-last-resort liquidity does not stop runs if investors believe the bank is insolvent. And as you rightly point out, deposit insurance only prevents runs if it is both credible AND liquid. I can't remember how many times I've tried to explain to people that deposit insurance that doesn't pay out for 7 days is no damn use!

You have ignored the velocity effects of bank runs. If there were only one bank, the bank would not fail, but there would be hypervelocity as everyone desperately tried to convert bank money to some other kind of asset. Suspending redemptions does not solve this problem. Greece last summer is a interesting example (though you have to think of Greek bank money as different from ECB-supplied Euros). There was a run on the entire Greek banking system, coupled with severely restricted physical currency withdrawals and capital controls preventing money leaving the country. We can think of this as approaching a one-bank model. Greek bank money became all but worthless as the population spent it as fast as they could because of the fear of redenomination. So in a one-bank model, a bank run would be hyperinflationary. Does this make sense?

I like the Diamond & Dybvig model. It has some useful insights. I wrote about it a while ago: http://www.coppolacomment.com/2015/04/rediscovering-old-economic-models.html

[notsneaky]

Maybe someone mentioned it above (sorry for not reading all the comments) but it seems like the problem re-appears in the >2 period model because of the either/or nature of the "trigger" (completely suspend redemption). The either/or redemption works in the two period model because there's only two periods. But it seems like there'd be some kind of policy function in a multiperiod model which could also accomplish the task. Basically the bank would have a pre-announced path for what % they're willing to redeem in any one period.

Of course I'm just speculating (and maybe it won't work in an infinite period model) but this phrasing sort of suggests that it might be possible

"The reason is that he might get the munchies in the third period, and if the bank suspends redemptions he will be unable to satisfy his future cravings"

(or maybe some kind of combination of threshold and interest rate?)