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) is prob of buying a good security, which rises with effort at rate b, + which can’t be monitored.

pe a be,b 0,pe 0,pe 0

Rg good return

Rb bad return

ER peRg 1 peRb

varR pe1 peRg Rb2

pe 0 e varR

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

c d y

C Rd

uc RuC

Period 1 budget constraint. Consume what’s left over from income after making deposits.

Period 2 budget constraint. Consume out of deposits in mutual funds, plus profits from bankers.

Usual Euler equation for consumption.

Mutual funds and banks’ constraints

peRgd 1 peRbd Rd

RgN d Rgdd 0

RbN d Rbdd 0

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 get on their investments from firms

What banks have to pay out to mutual funds who desposited money with them on behalf of households

The bankers’ problem

d,e,Rgd,Rbd

max peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

s.t. peRgd 1 peRbd Rd

RbN d Rbdd 0

Banker maximises its own profits,

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

L peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

peRgd 1 peRbd Rd vRbN d Rb

dd

Bankers’ profits Cost of effort

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

Rgd : pe pe 0

Rbd : 1 pe 1 pe v 0

e : peRg RbN d Rgd Rbdd e peRgd Rb

dd 0

First order conditions of bankers’ problem

FOC wrt Rgd ,Rbd ,v 0

V=0 means cash constraint not binding with observable effort

e bRg RbN d This equation, for optimal level of effort, comes from 3rd FOC.

Interpreting the optimal effort equation

e bRg RbN 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 peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

FOC wrt effort e

L peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

peRgd 1 peRbd Rd

vRbN d Rbdd

peRg RbN d Rgd Rbdd e

peRg RbN d Rgd Rbdd e 0

Lagrangian for mutual funds with unobserveable effort

L peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

peRgd 1 peRbd Rd

vRbN d Rbdd

peRg RbN d Rgd Rbdd 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

pe pe pe 0

1 pe 1 pe pe v 0

peRg RbN d Rgd Rbdd e

peRgd Rbdd

peRg RbN d Rgd Rbdd 1 0

v

vpe b

vbRgd Rbdd 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 peRgN d Rgdd 1 peRbN d Rbdd 0.5e2

peRgd 1 peRbd Rd

vRbN d Rbdd

peRg RbN d Rgd Rbdd e

peRg RbN d Rgd Rbdd e

peRgd Rbdd

peRg RbN d Rgd Rbdd 1 0

1. Use this effort constraint to get an eqn for e

2. And then substitute out for e in this FOC wrt e

vbRgd Rbdd 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’

RbN d Rbdd 0 This cash constraint does not bind so v, the multiplier

on that constraint, =0

With large net worth, bankers make non contingent payments to mutual funds

e bRg RbN d Substitute zero spread into effort constraint to get this, the same as the socially optimal effort when e is observed.

v 0,vpe b 0

vbRgd Rbdd 0 Rgd Rb

d 0

[imposing zero profit of mutual funds] R Rgd Rbd

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

vbRgd Rbdd 0

Rgd Rbd

vbd

First eq holds in eqm, so implies the second. Spread opened up.

peRg RbN d Rgd Rbdd e 0

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.)

It will be an exercise for you to show that effort falls. Full explanation will be given in exercise solutions.

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

RbN d Rbdd 0 Increasing N obviously makes it less

likely the cash constraint will bind, pushing v to 0.

v 0,vpe b 0

vbRgd Rbdd 0 Rgd Rb

d 0

[imposing zero profit of mutual funds] R Rgd Rbd

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

uc RuC

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 TFirm 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 pre-crisis years, banks’ net worth was very high, yet they put little effort into monitoring loans, particularly to sub-prime sector.