gr5215 fall 2021 lec 15 phillips curve
1
Nominal rigidities and the
Phillips curve
Miller GR5215
Wage and price dynamics
2
To get a useable model (say for policy) we
need to allow prices (and/or wages) to
move over time
Consider the basic static Keynesian model
with fixed wage. Now assume there is
lagged adjustment:
Wt = APt−1 A>0
Note we didn’t get this from anywhere. We
just assume the labor market works this way.
A “Phillips Curve”
3
Wt = APt−1 A>0
Labor demand: ′F (Lt )=Wt /Pt So:
′F (Lt )= A Pt−1 /Pt
′F (L
t
)= A
1+π
t
Which is a sort of Phillips curve: implies a
stable downward sloping relationship
between inflation and unemployment
Empirical Phillips curve (1950-
1969)
4
The natural rate hypothesis
5
Despite strong empirical evidence, Milton Friedman
and Ned Phelps (independently) argued that there
was no stable tradeoff: instead, whatever the rate
of inflation, unemployment would return to its
“natural rate”
The natural rate of unemployment” . . . is the level that would be
ground out by the Walrasian system of general equilibrium
equations, provided there is embedded in them the actual
structural characteristics of the labor and commodity markets,
including market imperfections, stochastic variability in demands
and supplies, the cost of gathering information about job
vacancies and labor availabilities, the costs of mobility, and so on.
Milton Friedman
Empirical Phillips curve (1970-
2009)
6
A natural rate “Phillips Curve”
7
Following Friedman and Phelps, economists tried
to “fix” the Phillips curve:
πt =πt
* +λ(lnY
t
− lnY
t
)+ ε
t
s
where Y-bar is the “natural rate of output” or “full-
employment output” – but what is the baseline
(core) rate of inflation, π*?
If πt
* =π
t−1 then Δπt = λ(lnYt − lnYt )+ εt
s
Which is sometimes called the “accelerationist
Phillips curve”
The expectations augmented
Phillips curve
8
But as Friedman noted, surely if inflation went up
1% every year, people would notice.
Led to assumption that “core inflation” was the
expected rate of inflation
πt =πt
e +λ(lnY
t
− lnY
t
)+ ε
t
s
If expectations are “rational” this turns out to have
strong (and incorrect) empirical implications, but
dependence on irrationality seems like a problem
too
A more modern model with
short-term fixed prices (I)
9
We have our NK IS curve:
lnY
t
≈ lnY
t+1 −
1
θ
r
t
And our LM curve:
Y
t
θ/ν =M
t
/P
t
i
t
1+ i
t
⎛
⎝⎜
⎞
⎠⎟
1/ν
But that’s hard to work with because of its
non-linearity and the role of expected inflation.
Instead we’ll use a “Taylor rule”:
rt = r(lnYt − lnYt ,πt )AKA, the MP (monetary policy) curve; policy
interest rate increasing in output and in inflation
Some notes on the Taylor rule
10
rt = r(lnYt − lnYt ,πt )
A rule like this was first proposed John Taylor as a
way of modelling Fed behavior; ”Taylor rules” can
be more general than this one
Idea is the Fed wants to keep Y close to its
“natural” level and wants to keep inflation low
So, when Y is high the economy is “overheating”
and you want to tighten monetary policy, i.e. raise
interest rates
And when inflation high also want to tighten to
reduce inflation
A more modern model with
short-term fixed prices (II)
11
Finally, we’ll have aggregate supply given by:
πt =πt
* +λ(lnY
t
− lnY
t
)+ ε
t
s
Note that this “modern Phillips curve” has not
been derived from any underlying fully-specified
model, yet
A more modern model with
short-term fixed prices (III)
12
So now we have a three equation model:
Can think of AD as arising from our IS/MP
relations, though in π/Y space rather than P/Y
space. An increase in inflation does not affect (IS)
but leads to an increase in r via (MP), reducing Y
πt =πt
* +λ(lnY
t
− lnY
t
)+ ε
t
s (AS)
lnY
t
≈ lnY
t+1 −
1
θ
r
t
(IS)
rt = r(lnYt − lnYt ,πt ) (MP)
An IS shock in an even
simpler version
13
Can simplify by linearizing MP and assuming core
inflation equals lagged inflation, but that’s still both
backward and forward looking, so further simplify:
πt =πt−1 +λ yt (AS)
y
t
= E
t
[ y
t+1]−
1
θ
r
t
+u
t
(IS)
rt = byt , b>0 (MP)
ut = ρut−1 +et (IS error process)
Where e is white-noise (uncorrelated, constant
variance)
The expectational difference
equation
14
Combine MP and IS:
y
t
=
θ
θ +b
E
t
[ y
t+1]+
θ
θ +b
u
t
yt =φEt[ yt+1]+φut Where:
φ =
θ
θ +b
Equations like this are called expectational
difference equations and need special solution
techniques
The law of iterated
expectations
15
The “law of iterated expectations” is a basic result
concerning conditional probabilities, applied to the
expectations operator. Simple form we will use is:
E” E”#$ 𝑥 = E”(x)
“I expect that tomorrow I will expect what I
expect today”
General version is for “strictly greater” information
sets: all the information available at t+j is strictly
greater than all the information at t
Solving the expectactional
difference equation
16
yt+ j =φEt+ j[ yt+ j+1]+φut+ j
Et[ yt+ j ]=φEt[ yt+ j+1]+φρ
ju
t
Take time-t expectations. For u:
yt =φEt[ yt+1]+φut ut = ρut−1 +et (IS error process)
This has to hold at every time t, so:
𝐸! 𝑢!”# = 𝜌𝑢! So: 𝐸! 𝑢!”$ = 𝜌
$𝑢!
Thus, using the law of iterated expectation:
Interpreting the result
17
y
t
=
θ
θ +b−θρ
u
t
πt =πt−1 +λ yt (AS)
And recall:
So:
π
t
=π
t−1 +λ
θ
θ +b−θρ
u
t
Recall that b parameterizes MP response to y and that
theta is inverse of the output response to r, (the CRRA
and 1/EIS) and rho measures shock persistence
So more persistent shock increases impact, bigger b
reduces impact, and increasing theta increases effect
(need to take the derivative to see that)
Solving the expectactional
difference equation (II)
18
yt =φut +φ(φEt[ yt+2]+φρut )
yt = (φut +φ
2ρu
t
)+φ2(φE
t
[ y
t+3]+φρ
2u
t
)
Now iterate forward in time:
yt = (φut +φ
2ρu
t
+φ3ρ2u
t
)+φ2(φE
t
[ y
t+3]+φρ
2u
t
)
Et[ yt+ j ]=φEt[ yt+ j+1]+φρ
ju
t yt =φEt[ yt+1]+φut
Use the second equation to sub out the
expectation in the first:
Solving the expectactional
difference equation (III)
19
yt = (φut +φ
2ρu
t
+φ3ρ2u
t
)+φ2(φE
t
[ y
t+3]+φρ
2u
t
)
So:
y
t
= (φ +φ2ρ +φ3ρ2 +…)u
t
+ lim
n→∞
φnE
t
[ y
t+n
]
If assume the limit goes to zero (which will be true
if y is not growing faster than rho), and recall:
y
t
=
φ
1−φρ
u
t
+ lim
n→∞
φnE
t
[ y
t+n
]
y
t
=
θ
θ +b−θρ
u
t
φ =
θ
θ +b
Then: This is a solution.