Macroeconomics and financial
frictions, lectures 2 and 3
Part 2: lazy banker model
Lecture to Bristol MSc Macro, Spring
2014
Tony Yates
Overview and motivation!
• We will cover Christiano and Ikeda’s simplified
lazy banker model.
• By way of motivation, ex colleagues of mine
who worked in retail banks for a while
witnessed a lot of golf playing and lunching!
• Still to do after this: BGG, and problems with
financial frictions models and new directions.
Quick recap on the thieving banker
• Laziness a form of theft, of course, so quite
similar.
• When net worth fell, the benefits of keeping
the bank as a going concern fell, relative to
running off with the stealable resources.
• So depositors had to offer a cut in deposit
rates to entice them into not stealing.
Lazy bankers story
• Consumers deposit funds with mutual funds.
• Mutual funds lend to bankers, diversifying themselves
across bankers.
• This implies neither consumers nor mutual funds in a
position to monitor bankers’ effort.
• Bankers then buy securities in one firm each, choosing
how much effort to spend monitoring the firms.
• Effort only affects probability of getting a good firm, so
returns don’t reveal effort.
• Empirical observation that bank profits differ suggests
banks not diversified.
Lazy bankers’ story, ctd..
• If funding cost is independent of whether returns
are good or bad, the bank captures all the returns
to extra marginal effort monitoring.
• This contract requires net worth to be high
enough. If it isn’t, can’t pay back funds if project
turns out to be bad.
• And depositors then require compensation by
way of a higher return if project is good.
• And banks then don’t get all the returns to effort,
so expend less of it.
Remarks about model construction
• Learning points in model construction;
• How formulation of problem of agents differs
depending on whether they act in concert or
are in conflict.
• I’ll explain more about what I mean later.
Moral hazard model of banking –
monitoring bankers’ effort
s N d
Bankers buy securities with their funds, [net
worth N and deposits d as before.
p
ea be, b 0, p 
e0, p 
e0
Rg good return
Rb bad return
P(e) is prob of buying a good
security, which rises with effort
at rate b, + which can’t be
monitored.
E
Rp
e
Rg 
1 p
e

Rb
var
Rp
e

1 p
e


Rg Rb 2
p
e0  e 
 var
R
It will be an exercise to
compute the variance.
Answer is not in the paper.
Analytical strategy in the lazy banker
model
• Derive equilibrium when effort is observeable
and show that it is optimal
• Derive equilibrium when effort is
unobservable and the no default constraint is
binding, and show that effort is lower than
socially optimal and spreads therefore higher.
• Study effect of 2 government policies, one
that works [equity injection] and one that
doesn’t ]tax financed deposits in the banks].
Household: same as thieving banker
model
Period 1 budget constraint. Consume
what’s left over from income after
making deposits.
c d y
Period 2 budget constraint. Consume
out of deposits in mutual funds, plus
profits from bankers.
C Rd 
u
cRu 
C


Usual Euler equation for
consumption.
Mutual funds and banks’ constraints
p
e
Rdg 
1 p
e

Rdb Rd
Rg 
N dRdg d 0
Rb 
N dRdb d 0
What banks get on
their investments from
firms
Zero profit condition for mutual funds.
LHS=expected value of returns paid to mutual
funds by banks.
RHS=returns on deposits paid to households.
Two cash constraints for bankers, one for
good and bad times. CI show that either
they don’t bind, or if they do it’s only in bad
times, so without loss of generality we
consider only the bad-times constraint.
What banks have to pay out to
mutual funds who desposited
money with them on behalf of
households
The bankers’ problem
max p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2
d,e,R dg ,R db
Banker maximises its own profits,
s.t. p
e
Rdg 
1 p
e

Rdb Rd
R 
N
b
dRdb d
0
less utility cost of effort,
subject to its own bad-times cash
constraint
and the mutual funds’ zero profit
condition.
Lagrangian for the banker when effort
is observeable
Cost of
effort
Bankers’ profits
L p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2

p
e
Rdg 
1 p
e

Rdb Rdv
Rb 
N dRdb d
Mutual funds’ zero profit
condition
Bankers’ bad-times
cash constraint
Lamda=value of marginal consumption from funds remitted by
banker to the household
From FOCs for banker, to an equation
for effort, when it’s observable
First order conditions of bankers’ problem
Rdg : p
ep
e0
Rdb : 
1 p
e

1 p
e
v 0
e : p 
e


Rg Rb 

N d
Rdg Rdb 
de p 
e

Rdg Rdb 
d 0
FOC wrt
Rdg , Rdb
  , v 0
e b
Rg Rb 

N d
V=0 means cash constraint not
binding with observable effort
This equation, for optimal level
of effort, comes from 3rd FOC.
Interpreting the optimal effort
equation
e b
Rg Rb 

N d
Effort higher when:
- Net worth of banks is higher – more skin in the game.
- The return to effort [b] is higher.
- The difference between good and bad returns is greater [what’s the
point of scrutinising your investments if they are all the same?]
Now: unobservable effort
• Banker doesn’t worry about profit constraint
on mutual funds, and cash constraint in bad
states of world.
• Instead, mutual fund sets contract [not
conditional on effort] in light of these
constraints.
• Now banker simply chooses effort e and takes
d,Rdg, Rdb as given.
Lagrangian for banker with
unobserveable effort
L p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2
FOC wrt effort e
p 
e


Rg Rb 

N d
Rdg Rdb 
de 0
L  p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2

p
e
Rdg 
1 p
e

Rdb Rd
v
Rb 
N dRdb d

p e

Rg Rb 

N d
Rdg Rdb 
de
Lagrangian for mutual funds with
unobserveable effort
L  p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2

p
e
Rdg 
1 p
e

Rdb Rd
v
Rb 
N dRdb d

p e

Rg Rb 

N d
Rdg Rdb 
de
Lagrangian for mutual funds is the same as the old one for the bankers, [with
observeable effort] …
…Except with the additional constraint that the mutual funds have to recognise
that the bankers behave according to their own FOC.
Deriving and manpiulating FOCs for
mutual funds
p
ep
ep 
e0

1 p
e

1 p
e
p 
ev 0
p e

Rg Rb 

N d
Rdg Rdb 
de
p 
e

Rdg Rdb 
d

p e

Rg Rb 

N d
Rdg Rdb 
d10
   v
vp
e b

 vbRdg Rdb d 0
First 2 FOCs combine to give this eqn
for mu.
Then substitute eqn for mu back into
first FOC to get this eqn in p.
Third equation got by…..
Manipulating the FOCs of the mutual
fund
L  p
e

Rg 
N dRdg d
1 p
e
Rb 
N dRdb d
0. 5e 2

p
e
Rdg 
1 p
e

Rdb Rd
v
Rb 
N dRdb d

p e

Rg Rb 

N d
Rdg Rdb 
de
1. Use this effort constraint to
get an eqn for e
p e

Rg Rb 

N d
Rdg Rdb 
de
p 
e

Rdg Rdb 
d

p e

Rg Rb 

N d
Rdg Rdb 
d10
2. And then substitute out for e in this
FOC wrt e

v
b
Rdg Rdb 
d 0
3. To get this equation!
Remarks: analogous model structures
that crop up
• Notice how we encounter repeatedly situations where
one agent takes another’s FOC as its own constraint.
• Another example : optimal monetary policy in the
New Keynesian model that Engin Kara taught you.
• Policymaker takes economy’s laws of motion [basically
FOC for consumers and producers] as its own
constraints.
• Distinct from finding ‘planner’s problem’, who simply
takes preferences and resource constraints, and finds
optimal allocation.
‘Normal times, lots of N’
Rb 
N dRdb d 0
This cash constraint does not bind so v, the multiplier
on that constraint, =0
v  0, vp
eb  0
 
 v
b
Rdg Rdb 
d 0  Rdg Rdb 0
[imposing zero profit of mutual funds]  R Rdg Rdb
With large net worth, bankers make non contingent payments to mutual funds
e b
Rg Rb 

N d
Substitute zero spread into effort constraint
to get this, the same as the socially optimal
effort when e is observed.
Normal times, lots of N, in words
• Cash constraint doesn’t bind; mutual funds don’t
have to worry about it affecting bankers’
behaviour.
• Socially optimal level of effort obtains, and there
is no gap between the rates paid on mutual fund
deposits (made on behalf of households) in good
and bad times.
• They can offer contract that doesn’t differentiate
between good and bad times without fear that
bankers will sit back on their sofas.
‘Abnormal times’ ie not enough N

 vbRdg Rdb d 0 

Rdg Rdb 
First eq holds in eqm, so implies the
second. Spread opened up.


 vbd
p 
e


Rg Rb 

N d
Rdg Rdb 
de 0
It will be an exercise for you to
show that effort falls. Full
explanation will be given in
exercise solutions.
Recall effort chosen by
bankers is given by this eqn. If
we can show R_d spread is
positive when cash constraint
binds, this will imply effort
falls (with lower
intermediation, lower returns,
lower welfare. (This is exactly
what happens.)
Equity injections by governments
• Recall that in the last model, of thieving
bankers, equity injections helped.
• They reduced deposits, and reduced the
incentive to run off with the money, and this
eased financing frictions and spreads.
• What about in the lazy banker model?
• Turns out they have NO effect at all in bad
times, and are counter productive in good
times.
Effect of equity injections by govt in
good times
• Amounts to a tax hike in period 1, followed by
reduction in period 2.
• Satisfies part of demand of household to save.
• So it reduces deposits. That reduces incentive
for banker to make effort. So that increases
probability of a bad return.
• And this raises spreads.
Effect of equity injections by govt in
bad times
• As well as the bad effect in good times, there’s
a good effect. Which cancels.
• In bad time, net worth too low to allow non
state contingent returns. Because not sure
enough money to cover if returns are bad.
• Lower deposits that follow govt equity
injection mean less to pay back, so reduces
necessary state contingency.
Effect of net worth transfer to banks
by government
Rb 
N dRdb d 0
Increasing N obviously makes it less
likely the cash constraint will bind,
pushing v to 0.
v  0, vp
eb  0
 
 vbRdg Rdb d 0  Rdg Rdb 0
[imposing zero profit of mutual funds]  R Rdg Rdb
We already saw before that if cash constraint binds we can deduce
that there is no spread, and that this brings about socially optimal
level of effort.
Effect of lump-sum, tax-financed
deposits by government


c d y, d d T

C Rd RT
u 
cRu 
C
Period 1 and 2 budget
constraints the same, except
modified form for d
Household Euler equation still
holds
Households problem unaffected; d falls to accommodate T
Firm problems similarly unaffected. Doesn’t matter from their
point of view whether deposits made by households directly or by
government on behalf of them.
Does the model ‘get’ the crisis?
• Well spreads rise during a downturn, which
affects net worth of financial intermediaries
negatively.
• But did ‘effort’ in monitoring returns really fall
as a result?
• Anecdotally, we could say that during the precrisis years, banks’ net worth was very high,
yet they put little effort into monitoring loans,
particularly to sub-prime sector.