Theory of Income II - Winter 2025
Guido Lorenzoni
Lecture notes 3 - Household Borrowing
1
Household borrowing in the NK model
• Based on Eggertsson-Krugman
• Preferences of consumer i
X
βi ln Cit
Qt Bit+1 + Pt Cit = Wt Nt + Bit + Tt
• We remove disutility of labor: agents just have a unitary fixed endowment
of labor 1 and supply it inelastically
• We introduce two elements:
– heterogeneity, some consumers borrow some lend
– a borrowing constraint
• Simplest reason for borrowing and lending: different preferences
• Two groups of agents with discount factors
βs > βb
• Borrowing constraint
Bit ≥ −Φ
1
1
≥ (1 + it ) βi
Cit
Cit+1
with inequality only if Bit = −φ
• Mass α of impatient (b) agents and 1 − α of patient (s) agents
• Firms use the linear technology
Yt = Nt
• We are interested in what happens when there is a shock to the borrowing
limit
Φ
1
1.1
Flexible prices
• We begin with the neoclassical benchmark
• Full employment equilibrium must have clearing in labor market
Nt = 1
• Natural output is
Y ∗ = N∗ = 1
1.1.1
Steady state
• Steady state:
– impatient agents have constant consumption
C b = Y ∗ − Φ + QΦ
– patient agents have constant consumption
C s = Y ∗ + B − QB
• Bond price
1
1+r
(r real interest rate, nominal price of goods constant and normalized to 1)
Q=
• Bond market clearing
α (−Φ) + (1 − α) B = 0
or
B=
α
Φ
1−α
• Patient agents are unconstrained, so their Euler equation must hold
1
1
= (1 + r) β s s
s
C
C
which requires
1+r =
1
βs
• Impatient agents are always constrained, easy to check that
1
1
> (1 + r) β b b
Cb
C
just because
βb < βs
2
• Equilibrium allocation
r
Φ
1+r
α
r
Cs = Y ∗ +
Φ
1−α1+r
Cb = Y ∗ −
• Is the goods market in equilibrium?
• Yes, by Walras Law since we checked bonds market equilibrium we automatically get goods market equilibrium:
αC b + (1 − α) C s = Y ∗
1.1.2
Financial shock
• Suppose there is a shock and Φ goes from some positive level to zero
• Then new steady state is especially simple
Cb = Cs = Y ∗
and the same interest rate
1+r =
1
1
= s
q
β
• But how do we reach the new steady state?
• Transitional dynamics
• They only last one period (guess and verify)
• First period after the shock, call it time t, has
– For impatient agents
Ctb = Y ∗ − Φ
– So for patient agents we need
Cts = Y ∗ +
α
Φ
1−α
• (Again this can come from bond market clearing or goods market clearing)
• Check Euler equations
1
1
= (1 + rt ) β s ∗
α
Y ∗ + 1−α
Φ
Y
1
Y∗−Φ
> (1 + rt ) β b
3
1
Y∗
• So we need a temporary drop in the interest rate
1 + rt =
1
Y∗
1
< s
α
s
∗
β Y + 1−α Φ
β
• Check Euler equation of borrowers
α
Y ∗ + 1−α
Φ
Y∗−Φ
>
βb
βs
b
but this always holds because ββs < 1
• (Transition from low Φ to high Φ can last many periods, but transition
from high Φ to low Φ always lasts only one period)
• Result: a contraction in the borrowing capacity of households leads to a
temporary drop in the interest rate
• What happens when we add nominal rigidities?
2
With nominal rigidities
• An especially simple assumption
• Downward inflexible nominal wages
Wt ≥ Wt−1
• If
Nt < 1
wages remain at
Wt−1
• (We could add gradual adjustment, or richer forms of stickiness, the basic
message is similar)
• Competitive firms produce identical goods, so no markup and prices are
always
Pt = W t
• What happens after shock?
• Suppose
1
Y∗
<1
α
β s Y ∗ + 1−α
φ
• Now we need negative real interest rates in the flexible price economy
4
(1)
• Suppose we have zero inflation in all periods t + 1, t + 2, ...
• (This is an implicit assumption on monetary policy behavior in future
periods, we’ll go back to this)
• Guess and verify that in all periods t + 1, t + 2, ... we reach the new flexible
price steady state with
Yt+j = Y ∗
and
Ci = Y ∗
for all agents and nominal interest rate
1+i=
1
>1
βs
• What happens at period t?
• Central bank sets it = 0 and expected inflation is 0
• Savers satisfy
1
1
= βs ∗
Cts
Y
• Borrowers
Ctb = Yt − Φ
• Let’s look at the goods market equilibrium condition
Yt = α (Yt − Φ) + (1 − α)
or
Yt =
Y∗
βs
Y∗
α
Φ<Y∗
−
βs
1−α
• The inequality follows from the negative natural rate condition (1)
• There is unemployment caused by deleveraging and the inability of the
central bank to counter its effects by lowering rates
3
Aggregate demand externalities
• Suppose agents anticipate at t − 1 that there will be a financial shock to
Φ at date t
• Suppose impatient agents are sufficiently impatient that they still borrow
Φ at t − 1 (see condition (2) below)
5
• Suppose there is zero inflation in all periods and the equilibrium at t − 1
has
Yt−1 = Y ∗
• We will check below that the central bank can choose a positive interest
rate at t − 1 to get that equilibrium
• Equilibrium consumption levels are
b
Ct−1
= Y ∗ − (1 − Qt−1 ) Φ
α
s
(1 − Qt−1 ) Φ
Ct−1
=Y∗+
1−α
• We derive the interest rate from
1
α
Y ∗ + 1−α
1 − 1+i1t−1
= β s (1 + it−1 )
Φ
1
1
2
= (β s ) (1 + it−1 ) ∗
Cts
Y
(using the s Euler equation at t − 1 and at t)
• Notice that since it−1 is close to 1 the real rate required for this equation
2
is close to 1/ (β s ) so we have it−1 > 0 and the zero lower bound is not
binding
• To do a simple experiment we choose β b such that Euler equation of b
agents holds exactly
1
Y ∗ − 1 − 1+i1t−1 Φ
= (1 + it−1 ) β b
1
Yt − Φ
(2)
where Yt was derived above
• Suppose we introduce a regulation to lower Φ a bit to Φ0 = Φ + dΦ with
dΦ < 0
• Keep zero inflation throughout, suppose interest rate is kept at natural
level at t − 1
• What are the welfare effects?
• Consumption at t − 1 and t is
b
Ct−1
= Y ∗ − Φ + Qt−1 Φ0
Ctb = Yt − Φ0
and
α
(Φ − Qt−1 Φ0 )
1−α
Y∗
Cts = s
β
s
Ct−1
=Y∗+
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• Consumption is still Y ∗ for all consumers at t + 1, t + 2, ...
• Output at t is
Yt =
Y∗
α
−
Φ0
βs
1−α
so the effect is to increase output at t
dYt = −
α
dΦ > 0
1−α
• This is an aggregate demand externality: by entering period t with less
debt, b agents increase aggregate spending and thus aggregate activity
• It is an externality because individual borrowers do not internalize the
general equilibrium effect of their spending on aggregate income
• Real rate at t − 1 from
1
1
= β s (1 + rt−1 ) β s ∗
α
Y ∗ + 1−α
(φ − qt−1 φ0 )
Y
• It can be shown that dφ < 0 causes dqt−1 > 0 and d (qt−1 φ) < 0 so we
have
b
dCt−1
<0
s
dCt−1
>0
and also
dCtb > 0
dCts = 0
• So patient agents are better off, what about impatient agents?
• Notice that
b
dCt−1
= qt−1 dφ + φdqt−1
and
dCtb = dYt − dφ
• The marginal effect on borrowers’ welfare is
1
1
1
1
b
dCt−1
+ β b b dCtb = b (qt−1 dφ + φdqt−1 ) + β b b (dYt − dφ)
b
Ct−1
Ct
Ct−1
Ct
• Now the terms with dφ cancel due to condition (2) as
1
1
b 1
b 1
q
dφ
−
β
dφ
=
−
(1
+
r
)
β
qt−1 dφ = 0
t−1
t−1
b
b
Ct−1
Ctb
Ct−1
Ctb
7
• So we are left with
1
1
φdqt−1 + β b b dYt > 0
b
Ct−1
Ct
• This effect is positive due to the combination of a pecuniary externality
(lower borrowing reduces interest rates, transfering resources to borrowers)
and of an aggregate demand externality
• So the regulatory intervention to reduce debt at t − 1 leads to a Pareto
improvement
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