Lecture 3: Eggertson-Krugman’s ‘debt deleveraging’ as a two country model of the

global crisis

MSc Open Economy Macroeconomics,Birmingham Autumn 2015

Tony Yates

Overview and motivation

• Eggertson-Krugman, ‘Debt-deleveraging…’• Closed economy model of debtors and creditors• Interpreted as a global model of the financial

crisis.• Methodological reason for studying: it’s a step

towards a microfounded model.• This is the modern way, though not accepted as

such by all.

EK story

• Borrowers accumulate debt up to a limit.• More patient consumers lend.• Something causes the borrowing limit to drop. Cd

be failure of financial intermediation.• Drop in borrowing limit means borrowers have to

rebuild balance sheet, demand falls.• Real rate falls to clear the market for saving. • May potentially cause negative real rates,

deflation, and ‘debt-deflation’.

What is the model about?

• Model is about one economy.• But can be a parable of dynamics in the global

economy.• eg, the Eurozone, with core [Germany] lending

to the periphery [Greece, Ireland, Portugal…]• Or financially-induced ‘sudden stops’ in

emerging economies.• Or the entire global crisis

Source: Slides by Bean, Charles, linked to in the reading list.

Secular stagnation or debt hangover?

• Larry Summers and others claim that several factors mean the real interest rate will be very low for a long time.

• That means central bank rates will be low for a long time.

• That means little room to stimulate with cuts?• Which could mean more volatile business

cycles.

SS or debt hangover

• These factors are said to be:– Demographics, ie large number of savers; slower

technological progress; less demand for capital.• But the debt hangover story is an alternative.

Advanced eg by Rogoff.• Solutions are different, so important to

understand this alternative cause.

Policy significance

• Many policy implications.• Prudential policy to limit borrowing in the first place.• Role of monetary policy looseness in encouraging

borrowing/risk-taking.• Prudential regulation+bail-outs to prevent sudden

changes in ‘debt limits’.• Debt forgiveness, vs moral hazard.• Activist monetary and fiscal policy to counter falls in

aggregate demand.

Plan

• Solve a ‘real’ model, with no money or nominal price level, to see how the change in the debt limit drives down the real rate.

• Study a ‘nominal’ version, with additionally money, sticky prices and a price level, to show how the zero bound to interest rates may generate deflation.

• Look at what debt being denominated in nominal terms means.

Real model: falling debt limit drives down the short run real rate.

Constrained utility maximisation by patient and impatient consumers

maxE t t 0

i t logct, i s,b.

s b

Dti 1 rt 1Dt 1i 0.5y cti

Consumers try to maximise an infinitely long stream of utility, which comes from consuming, subject to the borrowing constraint.

Borrowers are more impatient, valuing the future less.

Resource constraint. Debt today=debt yesterday unpaid, times interest, less share in the economy’s income, plus consumption.

The borrowing limit:

Debt can’t be too high, and how high relates to the real interest rate. Higher real rates mean greater debt service cost relative to income.

Second line shows how we can derive the borrowing limit. It’s an exercise to figure out how this is done.

1 rtDti Dhigh 0

Dhigh 0.5 1 y

Consumption by savers and borrowers.

cb 0.5y r1 rDhigh

cb cs y

cs y cb

y 0.5y r1 r D

high

0.5y r1 r D

high

We meet borrowers after they have borrowed up to the debt limit. So consumption is income less the cost of debt.

Consumption of savers and borrowers has to be equal to total income.

And consumption by savers is equal therefore to total income, less consumption by borrowers.

Consumption Euler equation for savers

1c ts

1 rt E 1c t1s

Standard equation in modern macroeconomics, here for the case when consumers get utility from log (consumption).

The greater is the real rate, the lower consumption today relative tomorrow.

Higher real rate means the ‘price of consumption today relative to tomorrow’ has gone up, so consume less of it and save.

Higher rate of discounting=lower beta = lower saving.

NB borrowers just consume what’s left over from debt service, so ‘not on’ their Euler equation.

Deriving the Euler equation

• Dynamic, linear, constrained optimisation.• Not so easy, but very very useful technique.• Part of building microfounded model with

consumer optimisation (of course).• In a richer model with optimising firms we’d

also derive their price setting rules using the same techniques.

• Application of the Kuhn-Tucker theorem.

Dynamic optimisation with Lagrangians: the recipe

L t 0

s tlogcts tDts 1 rt 1Dt 1s 0.5yt cts

1) Form Lagrangian from period utility, and setting budget constraint=0.

2) Differentiate the infinite stream wrt choice variables c_t and D_t, and set=0 at the optimum.

3) Then eliminate the Larange multipliers [the lamdas] and solve for what you want to know. [In our case consumption, debt, real rates…]

Isolating 2 hypothetical periods in the infinite Lagrangian

L . . . . tlogcts tDts 1 rt 1Dt 1s 0.5yt c ts

t 1logc t 1s tDt 1

s 1 rtDts 0.5y t 1 ct 1s . . .

L t 0

s tlogcts tDts 1 rt 1Dt 1s 0.5yt cts

Imagine 2 hypothetical periods in the Lagrangian. Then differentiate these two wrt the choice variables.

Why? Because D_t [debt] shows up in both of these periods.

Our first order conditions for consumption and debt.

dLdc ts

0 1c ts

t t t 0

t 1cts

dLdDts

0 t t t 1 t 11 rt 0

t 1 1 rt t

1ct 1s 1 rt 1

cts

1ct 1s 1 rt 1

cts

The first FOC gives an expression for the Lagrange multiplier in terms of consumption, which must hold for all t.

We substitute this into the second FOC, twice.

This gives us our Euler equation for consumption.

You have just done dynamic optimisation.

Recap on objective

• We are going to consider a fall in the debt limit, and derive what happens to the real rate.

• We’ll show that it falls, the greater the fall in the debt limit.

• And potentially can go negative.• Which is going to have implications for central

bank nominal rates.

Steady state real rate.

1c 1 r 1c

r 1

Compute the steady state real rate.

Do this by dropping the time subscripts in the Euler equation….

And then solving for r.

Debt limit falls from d_high to d_low

c lb 0.5y r

1 r Dlow

0.5y 1

1 1

Dlow

0.5y 1 Dlow

Work out borrowers consumption at the new, lower debt limit.

C_l_b means ‘borrowers long run consumption’.

Substitute in our expression for the steady state real interest rate.

And there’s the answer.

The lower is the new lower debt limit, the higher is consumption, obviously, since debt service is lower.

But in the deleveraging period, things are different…. As we will see.

Short run consumption of borrowers in the deleveraging period

csb 0.5y Dhigh Ds

Ds Dlow1 rs

csb 0.5y Dhigh D low

1 rs

Consumption of borrowers not optimal, and constrained by the change in debt limit.

The greater the gap between the discounted value of the new low debt limit, and the old high debt limit, the lower is consumption of borrowers.

Short and long run consumption of savers

c ls 0.5y r

1 rDlow 0.5y 1 Dlow

LR savers’ consumption takes the same form as before. Long run consumption of borrowers settles at a level equal to their share of the endowment income….

…plus the proceeds from lending [including the interest].

y css csb

css y csb y 0.5y Dhigh Dlow1 rs

0.5y Dhigh Dlow1 rs

Short run savers’ consumption we can get from the national income identity, ie total income less borrowers short run consumption.

Deriving the short run real rate that prevails during deleveraging

cls 1 rs css

0.5y 1 Dlow 1 rs 0.5y Dlow1 rs

Dhigh

0.5y Dlow

0.5y Dhigh 1 rs

Savers are on their Euler equation, since they are unconstrained.

So we derive the real rate by substituting in our expressions for s and l consumption.

…When we do that, we find that the higher the gap between the low and high debt limit, the lower is the real rate.

This is one of the punchlines of the model, and how it seeks to explain what we see today.

Conditions for a negative real rate in the deleveraging period

rs 0 0.5y Dlow

0.5y Dhigh 1

0.5y Dlow 0.5y Dhigh

0.5y 0.5y Dhigh Dlow

0.5y1 Dhigh Dlow

We can show that it’s quite easy to get negative real rates in the short run.

You are asked to say why these conditions mean that it is ‘easy’ in an exercise.

But it should be sort of shocking. Negative real rates means paying someone to borrow from you!

Samuelson: borrow money to bulldoze mountains for a highway.

• Since nominal and real rates are linked via arbitrage… • Possibility of negative real rates raises issue of

whether central banks can accommodate this fall in demand.

• And if central banks can’t reduce nominal rates below zero, whether that might mean there is deflation.

• And if debt is nominal, how this can aggravate the problem.

Nominal model with money, [sort of] sticky prices, nominal interest rates and a zero bound

Euler equation with nominal rates

1c ts

1 it E 1c t1s

p tp t1 ; it 0

Euler equation for savers, or patient consumers, with inflation and nominal interest rates.

Consumers’ real rate will be the nominal rate i_t ‘deflated’ by the expected rate of inflation.

If inflation is expected to be higher, then the existing nominal rate translates to a lower expected real rate.

The Fisher equation

1 rs 1 ispsp 1

1 is 1 rs

psp

1 is 1 rsp

ps

Important equation in macro and finance, named after Irving Fisher.

Says, crudely ‘the nominal interest rate should equal the real rate plus expected inflation’

Here we suppose there is a long run p, p_*, fixed by monetary policy.

And there is a short run p, p_s.

Implications of constant prices for trajectory of nominal rates i_t

1 is 0.5yD low

0.5yDhigh

is 0 1 1 rsp

p s p s

p 1 rs

Suppose p_s=p_*

Then this equation holds, and we know that the RHS can be <1.

That implies i_t<0, which is no allowed because of the zero lower bound.

If instead we hold i=0, then this implies with r<0 that prices fall.

Debt denominated in nominal terms

Dhigh Bhighp s

Bhighp s D low

1 rs

1 rs 0.5yD low

0.5y Bhighps

Suppose the debt borrowers start out with is denominated in nominal terms.

The real burden of deleveraging/ paying down the debt now higher the lower is the price level, as this raises the RHS term here.

This old equation for the real rate shows that a fall in the price level will cause the real rate to fall.

Hence the problem with the zero bound in a closed economy.

Recap

• Minsky Moment that lowers the debt limit drives down the real rate

• Borrowers forced to lower consumption and pay down debt.

• Savers demand to save unchanged, so real rate has to fall to clear the market.

• Quite easy even to get negative real rates.

Recap 2

• EK offer this as an explanation of low real rates in a closed economy.

• But we can just as well use it to explain pattern between two economies where there have been substantial capital flows, and a large net foreign asset position built up.

• Eg, the Eurozone crisis, involving lenders [Germany] and borrowers [periphery].

Recap 3

• We saw that if we want to maintain constant prices, negative real rates can imply negative nominal rates.

• By implication, nominal rates bounded at zero mean falling prices.

• If debt priced in nominal terms, this will aggravate the fall in the real rate.

Policy lessons

• Policies to avoid sudden drops in the debt limit.– Prudential policy to stop excessive borrowing in the first

place– Bail out policy to prevent collapse in intermediation.

• Policies to substitute for negative nominal rates.– Fiscal policy– Unconventional monetary policy

• Wisdom or otherwise of fixed exchange rate systems between countries with large borrowing and lending positions.