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

maxE t  
it log
ct 
, i s, b.
t0

s 
b
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.
Dt 
i
1 rt1 
Dt1 
i0. 5y c t 
i
Resource constraint. Debt
today=debt yesterday unpaid,
times interest, less share in the
economy’s income, plus
consumption.
The borrowing limit:

1 r t 
Dt 
iDhigh 0
Dhigh 0. 5

y
1 
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.
Consumption by savers and borrowers.
r
high
c b 0. 5y  1
D
r
c b c s y
c s y c b
y 0. 5y  r Dhigh
1 r
0. 5y  r Dhigh
1 r
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
1
c st

1 rt 
E c1s
t
1
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  
s t 
logc st t 
Dst 
1 r t1 
Dst1 0. 5y t c st 

t0
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  
s t 
logc st t 
Dst 
1 r t1 
Dst1 0. 5y t c st 

t0
L . . . . t 
log
c st t 
Dst 
1 r t1 
Dst1 0. 5y t c st 

t1 
log
c st1 t 
Dst1 
1 r t 
Dst 0. 5y t1 c st1 

...
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.
dL  0  1 t  t 0
t
dc st
c st
 t 1s
ct
dL  0   t  t1 
1 r t 0
t
t
1
dDst
 t1 
1 r t  t
 1
1 r t  1s
s 
c t1
ct
 1
1 r t  1s
s 
c t1
ct
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.
1 
1
1

r

c
c
1 
r

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 bl
Work out borrowers consumption at
the new, lower debt limit.
0. 5y  r Dlow
1 r
1
0. 5y 

1
1
0. 5y 
1 
Dlow
Dlow
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
c bs
0. 5y 
D
high
Consumption of borrowers not
optimal, and constrained by the
change in debt limit.
Ds 
low
D
Ds 
1 r s
low
c bs 0. 5y 
Dhigh  D1r 
s
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
r
c sl 0. 5y 1
Dlow 0. 5y 
1 
Dlow
r
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 c ss c bs
low
c ss y c bs y 0. 5y 
Dhigh  D 
1 r s
low
D
high
0. 5y 
D


1 r s
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
Savers are on their Euler equation, since they
are unconstrained.
c sl 
1 rs 
c ss
So we derive the real rate by substituting in
our expressions for s and l consumption.
low
0. 5y 
1 
Dlow 
1 r s 
0. 5y  D
Dhigh 
1 r s
0. 5y Dlow
1 r s
high
0. 5y D 
…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
0. 5y Dlow
rs  0 
1
high
0. 5y D 
 0. 5y Dlow  
0. 5y Dhigh 
 0. 5y 0. 5y  Dhigh Dlow
 0. 5y
1  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
1
c st

1 it 
Ec1s
t
1
pt
;it
p t1 
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
p
p
1 
1 r s  
1 is  s 
1 r s  s
p
p
1 is
p
 1 is 
1 r s 
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 
0.5y
D low
Suppose p_s=p_*

0.5y
D high 
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.
p
is 0  1 
1 r s 
ps

ps
p
1 r s
If instead we hold i=0, then
this implies with r<0 that
prices fall.
Debt denominated in nominal terms
Suppose the debt borrowers start out with
is denominated in nominal terms.
high
Dhigh  Bp s
B high
ps
The real burden of deleveraging/ paying
down the debt now higher the lower is the
price level, as this raises the RHS term
here.
D low
 1r
s
1 r s 
0.5y
D low
high
0.5y
B p s
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.