Sept 16 2015 Diamond Dybvig and Great Depression

Prévia do material em texto

Bank Runs and the Persistence 
of the Great Depression
Kenneth Rogoff
Harvard University, Fall 2015
(draws on Allen and Gale, Understanding Financial Crises,
2007, also Jeremy Stein notes)
Modern Theory of Banking from Diamond Dybvig
(1983) and Bryant (1980), Allen and Gale (1998)
(from Allen and Gale 2007)
• A maturity structure of bank assets in which less
liquid assets earn higher returns
• A theory of liquidity preference, involving uncer-
tainty about timing of consumption (or in some
versions, production)
• Banks are represented as an insurance mechanism
against liquidity shocks (in absence of complete
markets)
• Explanation of bank runs: DD: self fulilling. Bryant,
Allen Gale: Fundamentals
Still open questions
1. Are we sure we have the right theory of economic
role of bank deposits
2. Why do demand deposits need to be paired with
illiquid loans?
3. What happens when regulators allows this liq-
uid/illiquid mix (what about alternative of "narrow
banking")
We understand much more about question 3 than
questions 1 and 2
Some alternative approaches to questions 1
• Diamond Dybvig model demand deposits as an
insurance scheme uncertainty over consumption
timing
• Gordon Pennachi (1990) model bank deposits "in-
formation insensitive" — easy to value- claim use-
ful in transactions
• Diamond Rajan (2001) model demand deposits as
a discipline device on managers If they don;t have
cash, they face run
• Holmstrom Tirole (98) is about liability side of the
bank. Uncertainty is over short term investment
needs, otherwise like DD
Example (Jeremy Stein, 2008 
notes)
See also Allen and Gale, 
Understanding Financial Crises
4
The Diamond-Dybvig model:
3 time periods, long-run production technology.
–invest 1 in a tree at time 0.
–if cut down the tree at time 1, get 1 back.
–if wait til time 2, get R > 1 back.
Consumers have 1 unit of endowment, ex ante identical.
–at time 1, fraction t of consumers learn they are “early”:
get u(c1) if they consume at time 1, nothing if consume later.
–fraction (1 - t) are “late”, get u(c2) if they consume at
time 2.
–utility function u( ) has relative risk aversion > 1.
5
First-best benchmark: suppose types are contractible.
Optimal insurance sets u!(c1) = Ru!(c2); i.e., marginal
utility = marginal productivity.
Given relative risk aversion > 1, can show that optimal
insurance yields: c1 > 1; c2 < R.
–in limit with extreme risk aversion, would have c1 almost
equal to c2. 
Example: might have t = ½, and R = 2. Resource
constraint is then: c1/2+ c2/4 = 1.
–for right utility function, optimum might be something
like c1 = 1.2; c2 = 1.6.
6
What happens if everybody just invests directly in their
own tree?
–no insurance: early types must cut the tree at time 1, get
c1 = 1; late types don’t have to cut, get c2 = R.
Can we do better than this with unobservable types?
–should be able to, since optimal allocation is incentive-
compatible.
–in example, early and late consumers should reveal types
honestly: early types prefer 1.2 at time 1 to 1.6 at time 2; late
types prefer 1.6 at time 2.
7
Implementing the optimum insurance scheme: a bank with
demand deposits. 
–all consumers put their unit of endowment in the bank.
–bank invests in the trees, issues deposit contracts.
–contract promises (say) 1.2 to time-1 withdrawers.
–bank honors time-1 contract for as many people as it can 
until it runs out of assets (can think of time-1 customers having 
random places on line).
–time-2 withdrawers get what’s left over.
Note sequential service constraint: bank’s payoff to any
agent depends only on his place on line, not number of people
in line behind him–i.e., can’t take back some of his money if
more people come later.
8
Demand deposit contract can implement optimum risk-
sharing as a Nash equilibrium.
–early types withdraw at time 1; late types withdraw at
time 2.
–given this behavior, it is optimal for late types to wait.
 
But deposit contract also admits a “bank run” equilibrium.
–all types attempt to withdraw at time 1
–given this behavior, bank must cut down all the trees
–it’s optimal to run since this yields expected payoff of 1
at time 1; waiting until time 2 yields 0 for sure.
Intuition: actions of late types are strategic complements. 
When one runs, it raises appeal to others of running, since
more trees are cut and remaining value falls.
Bank run not only ruins risk-sharing, but is worse than
autarky solution of direct ownership: all trees are cut down.
9
Preventing runs: suspension of convertibility.
Suppose bank’s policy is to suspend convertibility if too
many people (e.g., more than fraction t of population) try to
withdraw at time 1.
–bank commits to never cutting down more trees than in
optimum case.
–bad equilibrium is gone: dominant strategy for late type
is now to wait, irrespective of what other late types do.
But this only works well if t is non-stochastic and known
in advance.
Proposition 1: deposit contracts with sequential service
cannot achieve optimal risk sharing when t is stochastic.
10
Proof: for any t, optimal risk sharing requires: i) all early
types get same c1(t); and ii) u!(c1(t)) = Ru!(c2(t)).
–bank doesn’t know t when early types start to show up at
time 1; only way to treat them all equally is for c1 to be a
constant, independent of t.
–but then resource constraint implies c2(t) decreasing in t:
bank has to cut more trees at time 1 when t is high.
–having c2 depend on t while c1 doesn’t contradicts
optimal risk sharing.
Some intuition: suspension of convertibility can still help
avert runs, but now run risk of freezing out some early types.
11
The role of deposit insurance:
Proposition 2: demand deposits plus gov’t deposit
insurance can achieve optimum as unique Nash equilibrium if
insurance is financed with right kind of tax.
Basic idea: gov’t can achieve something that bank can’t
with sequential service constraint–can tax people after they’ve
withdrawn at time 1.
–so gov’t can base tax on realized time-1 withdrawals
(i.e., on t) in such a way as to pick desired c1(t).
–tax proceeds plowed back into bank so it doesn’t have to
cut down too many trees.
–removes incentive to run without having to use
suspension of convertibility.
Example: if number of time-1 withdrawers exceeds tmax,
gov’t sets high tax on them so as to create c1 = 1, c2 = R.
–even if he conjectures that all other late types will run, a
given late type is now better off being patient. 
More general case
World has 3 periods: T = 0, 1, 2.
Technology requires −1 units in period 0.
Then, if interrupted in period 1, yields 1;
if not interrupted, yields R > 1. CHOICE made in
period 1
Utility may be of 2 types:
U(c1, c2; θ) =
⎧
⎪⎪⎨
⎪⎪⎩
u(c1) if j is type 1 in θ
ρu(c1 + c2) if j is type 2 in state θ
ρ < 1, but ρR > 1
(so equilibrium involves redistribution).
u0 > 0, u00 < 0
Assume −cu00(c)/u0(c) > 1 example c1−θ
1−θ
EXACTLY t% of all individuals are type 1.
U(c1, c2) = u(c1) if individual turns out to be of type
1 when state is revealed in period 1
ρu(c1 + c2) f individual turns out to be of type 2
when state is revealed in period 1
where ρR > 1
Individual has endowment of 1 in period 0
Absent banks, assume individual must hold assets
directly
Easy to show that period zero price of period 1 con-
sumption is 1
Period 0 price of period two consumption is 1/R
There is no trade at these prices, everyone is the
same
If types were publicly observable in period 1 when
type is revealed, then it would be possible to have
optimal risk sharing
optimal contract has consumption of "type 1" con-
sumers at zero in period 2, and also sets period1
consumption of "type 2 consumers to zero
Suppose that banks do not know t , percent of early
withdrawals.
Assume now that type is private information.
Then no risk sharing is possible. No public informa-
tion on which to condition contracts.
Suppose, however, that types were observable. Then
optimal contract satisfies
c2∗1 = c
1∗
2 = 0 (where * denotes "optimal").
Those who can, delay consumption
u·
³
c1∗1
´
= ρRu0
³
c2∗2
´
.
Marginal utility in line with productivity
tc1∗1 +
h
(1− t)c2∗2 /R
i
= 1.
Resource constraint
Since ρR > 1 and risk question >1
c11 > 1 c
2
2 < R
c1∗−θ1 = ρRc
2∗−θ
2
µ
c2
c
¶θ
= ρR c2c1 = (ρR)
1
θ
Banks can provide this insurance [as long as we rule
out period-1 trading].
Assume each agent in period 1 who withdraws re-
ceives a fixed amount r1.
Withdrawal orders are served sequentially.
Assume bank is a "mutual," liquidated pro rata in
period 2.
V1 is payoff to agent in period 2 (depends on place
in line).
V2 is payoff in period 1.
V1(f1, r·1) =
⎧
⎪⎪⎨
⎪⎪⎩
r1 if f1 ≤ 1r
0 if f1 > 1δ
(since when f = r, fr = 1).
V2(f, r1) = max {R(1− r, f)/(1− f), 0}
Two equilibria
When r·1 = c1∗1 Good equilibrium
"Bad" equilibrium
Bank run
Multiplicity of equilibria
Suspension of Deposits
V1(f1, r1) =
⎧
⎪⎪⎨
⎪⎪⎩
r1 if f1 ≤ f
0 if f1 > cf
V2(f, r1) = max
(
(1−fr·1)R
1−f ,
(1− bfr1)R
1− bf
Assume 1− cfr1 > 0
But what if banks do not know t?
Then a suspension of withdrawals is not enough to
get efficient outcome
Proof: Optimal risk sharing involves all early con-
sumers get same consumption. But then c(1) must
be constant independent of t. But then c(2) will de-
pend on t, while c(1) does not, contradicts optimal
risk sharing
Suspections of convertibility: risks that some early
consumers get nothing
Diamond Dybvig propose deposit insurance, although
government is subtly given superior information to
make system self contained
Still a big debate
DD show that demand deposits combinsed with right
kind of government deposit insurance can achieve
optimal risk sharing
Why: Government can take actions based on su-
perior information because they can tax people after
they have withdrawn at time t
tax proceeds go to bank so it does not have to liqui-
date too many illiquid assets
removes incentives to run
Most robust point in DD is mix of liquid and illiquid
assets leads to financial fragility
Does not just have to be about banks but can be any
similar system (see Gorton 2009 on run on wholesale
banking system in 2007)
Fundamental role of deposits is less clear. Jacklin
(1987) shows that demand deposits unravel if there
is a securities market so that late consumers can buy
illiquid assets at time 1. Then it becomes impossible
to subsidize early consumers (since late consumers
will take out money and reinvest in illiquid assets —
unless price is one, in which case demand deposits
do not provide insurance.
More likely that Gordon Pennachi or Diamond Rajan
— or something else - -explains demand deposits
Further readings on Bank Runs
• *Allen, Franklin, and Douglas Gale, Understanding 
Financial Crises, Oxford University Press, 2007, ch 3.
• Geanokoplous, John, Leverage Cycles, forthcoming in 
NBER Macroeconomics Annual 2009, D. Acemoglu, K. 
Rogoff, and M. Woodford (eds). April 10, 2009. 
• Bolton, Patrick, Tony Santos and Jose Scheinckman, 
“Inside and Outside Liqudity”, mimeo, Princeton 
University, March 30, 2009, Diamond, Douglas and 
Raghuram Rajan, “Liquidity Shortages and Banking 
Crises,” NBER working paper 10071, October 2003.
• Bolton, Patrick, and Mathias Dewatriapoint, Contract 
Theory, Cambridge: MIT Press. 2005, pp. 397-418
• Gorton, Gary, “Slapped in the Face by the Invisible 
Hand: Banking and the Panic of 2007, mimeo, Yale 
University, April 3 2009
Kenneth Rogoff, Harvard University
YEAR Private Hours Total Hours Farm Hours
1932 72.4 73.5 98.6
1933 70.8 72.7 98.8
1934 68.7 71.8 89.1
1935 71.4 74.8 93.1
1936 75.8 80.7 90.9
1937 79.5 83.1 98.8
1938 71.7 76.4 92.4
1939 74.4 78.8 93.2
From Kehoe and Prescott’s Depressions volume (www.greatdepressionsbook.com)
US GREAT DEPRESSION: TOTAL HOURS WORKED (Per capita) Relative to 1929