gr5215 fall 2021 lec 14 nominal rigidities
1
Effects of nominal rigidities:
the New Keynesian IS curve
Miller GR5215
Some direct evidence on price
stickiness
Direct evidence on price
stickiness (II)
From Nakamura and Steinsson (2013)
Are sale prices different?
From Nakamura and Steinsson (2013)
Why don’t businesses change
prices?
Heterogeneity in price adjustment
A household optimizing model
with fixed prices – introduction
7
We are going to build a small dynamic, sort-
of GE model with money and optimizing
consumers
We assume firms simply supply whatever is
demanded at a fixed price level, like IS/LM –
that’s why it’s not really GE
No K, to simplify, but households can choose
between holding bonds and holding money,
and they choose their labor supply (though
we won’t do much with the latter, for now)
A household optimizing model
with fixed prices – setup
8
Mt is the stock of money held by the
household; money in the utility function is
sometimes called a Sidrauski model
Y = F(L) ′F >0 ′′F <0 U = β t[u(C t )+Γ(M t /P t )−V(L t )] t=0 ∞ ∑ 0<β <1 u has usual assumptions; V is a disutility function, because we assume households don’t like to work: it has V>0,V’’>0, V’’’>0
A household optimizing model
with fixed prices – comments
9
Economists struggled for decades to put
money into GE models
Putting money directly into the utility
function is the easiest way to do it: implicitly
assume households use money for
transactions and get some utility out of that
directly
M/P appears in the function, as when prices
go up need proportionately more money for
transactions
A household optimizing model with
fixed prices – setup (II)
10
2 assets now, bonds and money, with
nominal interest rate on bonds it. So budget
constraint and (assets = money + bonds):
u(C
t
)= Ct
1−θ
1−θ
, θ >0
Γ(M
t
/P
t
)= (Mt /Pt )
1−ν
1−ν
, ν >0
At+1 =Mt +(At +WtLt −PtCt −Mt )(1+ it )
Household takes P,W,i as given, chooses C,
M, though M is set in aggregate by CB
Assume both stage functions in preferences
are CRRA, so
Consumption Euler equation
11
Define:
MU
t
= β
(1+ i
t
)P
t
P
t+1
(MU
t+1)
This is the exact definition of the real interest rate –
the usual Fisher equation version is an approximation:
Follow the usual perturbation argument:
1+ r
t
≡
(1+ i)P
t
P
t+1
log(1+ rt )≈ rt
log(1+ r
t
)= log(1+ i
t
)+ log(P
t
)− log(P
t+1)
≈ i−π
t
so: it ≈πt + rt
The New-Keynesian IS curve
12
Take logs:
With that definition and putting in CRRA utility:
Ct
−θ = (1+ r
t
)βC
t+1
−θ
Use log approximation, suppress constant (regard
as local approximation) and note that, in
aggregate, consumption must equal output (recall
no capital stock!):
lnY
t
≈ lnY
t+1 −
1
θ
r
t
This is called the “New-Keynesian IS curve”
lnC
t
= lnC
t+1 −
1
θ
ln[(1+ r
t
)β ]
Effects of fixed prices: Money
FOC
13
Now FOC wrt M. We are trading off money
holdings for consumption. Increase Mt/Pt and
decrease Ct so as to leave At+1 unchanged – want
an intratemporal FOC.
Recall the budget constraint:
At+1 =Mt +(At +WtLt −PtCt −Mt )(1+ it )
So with dAt+1=0:
𝑑 “#
$#
𝑖& = −(1 + 𝑖&)𝑑𝐶&
dC
t
= −
i
t
1+ i
t
dmDefine dm=d(M/P), thus
(d𝑀&)𝑖& = −(1 + 𝑖&)𝑑(𝑃&𝐶&)
Effects of fixed prices: Money
FOC (II)
14
U = β t[u(C
t
)+Γ(M
t
/P
t
)−V(L
t
)]
t=0
∞
∑ 0<β <1 And recall our objective fn.: ′Γ (M t /P t )= ′u (C t ) it 1+ i t So: dC t = − i t 1+ i t dm is a necessary condition for optimization Effects of fixed prices: FOCs (IV) 15 ′Γ (M t /P t )= ′u (C t ) it 1+ i t Incorporate functional forms: So this is like an LM curve: an upward-sloping relationship between i and Y. And we have NK-IS M t /P t =Y t θ/ν 1+ it i t ⎛ ⎝⎜ ⎞ ⎠⎟ 1/ν Y t θ/ν =M t /P t i t 1+ i t ⎛ ⎝⎜ ⎞ ⎠⎟ 1/ν lnY t ≈ lnY t+1 − 1 θ r t u(C t )= Ct 1−θ 1−θ , θ >0
Γ(M
t
/P
t
)= (Mt /Pt )
1−ν
1−ν
, ν >0
Effects of fixed prices
16
Now assume P is fixed
Consider a temporary increase in M: looks just like
regular IS, though it’s basically a Pigou effect (that
is, a wealth effect of increased money)
Similarly, the IS curve shifts with a temporary
increase in government purchases (increases
current Y without altering future)
Y
t
θ/ν =M
t
/P
t
i
t
1+ i
t
⎛
⎝⎜
⎞
⎠⎟
1/ν
lnY
t
≈ lnY
t+1 −
1
θ
i
t