Financial Market part 2: Diamond-Dybvig model Flashcards
(17 cards)
DD model considers the possibility of a bank run - deposits want to be able to withdraw savings at any time they want.
Why is this problematic
Maturity mismatch - banks borrow short term (from depositors putting in money) yet lend long term (e.g loans to firms)
I.e short term liabilities and long term assets
E.g if we all decided to withdraw saving tomorrow, the system will collapse since they wonโt have since lent out to firms.
and when may this happen IRL when people wanna withdraw savings at same time? during crisis! like northern rock and lehman!
DD model - 3 time periods (0,1,2)
Each agent is endowed with 1 unit of the economyโs single good in period 0.
If good is invested in period 0, yields return R>1 units if it is held until p2, but only 1 unit if the investment is liquidated in p1 early (so no return, just get investment back)
Now consider individuals:
In period 0, all individuals are the same.
In period 1 they learn their type
Fraction ๐ of individuals learn they only like period 1 consumption (C1) (impatient)
While others (1-๐) discover that they value period 2 consumption (C2) as well. (patient)
Typeโs cannot be observed by other individuals (an information friction):
How to express type a and type b individualsโ utility?
Type a:
Ua = lncโa
Type b:
Ub = pln(cโb+cโb)
where p is between 0 and 1, measures how type B individuals values consumption less than type A people
1st result under autarky (i.e no trade or risk sharing)
What do individuals do in period 0?
b) what about onwards?
All invest their endowment in period 0, as in period 0 consumption isnโt valued by anyone (recall both utility functions did not include period 0)
b) Type Aโs who only value consumption in P1, liquidate investments early to consume in period 1 (consume 1 unit)
Type B hold on to enjoy consumption in period 2 only (consume R where R>1)
Given type 1 payoff=1, and type 2=R
What is the expected utility under Autarky?
Uautarky
(hint: once found, this can be simplified)
Uautarky = ๐ln1 + (1 โ ๐)๐ln๐
simplifies to = (1 โ ๐)๐ln๐
(since ln1= 0! so just basically only 2nd term stays)
Simple: What will planner choose for cโa and cโb?
Since Aโs get no utility from P2 consumption, set cโa = 0
Since Bโs are patient so rather to get R>1, so set cโb = 0
That was simple: but what about cโan and cโb?
What is the fraction of investment projects liquidated early,
B) vs ones held till maturity
Fraction of investment projects liquidated early (๐๐โ๐)
and the ones that hold till maturity (1-๐๐โ๐) to get return R, which is shared out amongst type Bโs
So to do this; social planner faces optimisation problem
it has to uphold cโa=cโb=0 as we mentioned.
What is the resource constraint i.e Cโb
Cโb = (1 โ ๐๐โ๐)๐ / 1 โ ๐
Numerator is how many hold till maturity to get return R.
and then since split amongst type Bโs, shown by dividing by denominator which is the amount of type Bโs!
What is expected utility? (pg 15 i did workings)
First
E[U] = ๐lncโa + (1 โ ๐)๐lncโb
(only include them as cโa=cโb=0 as we mentioned)
then sub in our cโb resource constraint into it, and take logs to get final expected utility!
Why is it useful to know expected utility in this log form?
Since taking FOC, with respect to cโa and cโb gives us optimal levels of them!
Derive it!
So derive FOC and find them (hint: they have same denominator, just rmb the different top!) Final answers
Overall findings for optimal cโa and cโb (sociallly optimal outcome)
c*โa = 1/ ๐ + (1 โ ๐)๐ > 1
c*โb = pR/ ๐ +(1 โ ๐)๐ < R
Compared to autarky, where cโa=1 and cโb=R
now we have risk sharing, we find type B (patient) people lose out, now consume < R!
Impatient people type A are now better off than autarky, consume >1 rather than =1!
So when risk can be shared, patient people lose out more. Still happy to be patient as benefit, but lose out compared to autarky with no trade/risk sharing
Now introduce a bank!
Bank takes deposits, makes investments and accepts withdrawals whenever depositors wish, so long as the money is available.
What is the equilibrium
The nash equiliibrium from the previous slide! Type Aโs get their cโa, type Bโs get their cโb
Type B wait, type A do not
What is the 2nd less desireable nash equilibrium - how can this occur
(hint: donโt wanna be last man standing!)
If type Bโs believe all agents (not just Aโs) will withdraw in period 1. Asymmetric info means cannot tell the type of withdrawers, so panic!
Since if all withdraw, bank has to liquidate all investment projects eaely, so nothing for type B in period 2! (they donโt wanna be left with nothing!)
i.e everybody acts like type A - a bank run
Financial contagion
One bank getting into difficulty can cause worry for other banks, whether warranted or not - contagion
Reasons for financial contagion (4 types of contagion)
Counterparty contagion: banks hold claims on one another, so can trigger domino effect
Confidence contagion
Fire sale contagion: forced liquidations can bring down market prices and valuations for all banks
Macroeconomic contagion: one bank can cause negative macro effects to economy, causing scare
How should policymakers respond to possibilty of bank runs and financial contagion (seminar 3 q3d) (2)
b) what can we note about the motive of both
- CB could act as lender of last resort to support banks in difficulty (usually in secret so prevents panic i.e contagion!)
eval: may not be sufficient as it just extends loan, not a gift.
- UK gov have deposit guarantee system (if bank fails, guaranteed ยฃ85k PP) (this was strenghtened following Northern Bank bank run, was originally a 100% guarantee for only the first ยฃ2000 of savings, then get back 90% of next ยฃ33k
b) both motives/aim is to manage expectations and provide reassurance to prevent self-fulfilling prophecys i.e bank run!
For reference, how much is deposite guarantee system for US?
they get more generous, 250k per person! compared to 85k.
