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Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Advanced MacroeconomicsAn Easy Guide
Campante, Sturzenegger & Velasco
Advanced Macroeconomics LSE PRESS 1/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
The classical model
Advanced Macroeconomics LSE PRESS 2/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
A money market equilibrium locus called the LM curve:
M
P= L
((−)i ,
(+)
Y
), (1)
and a goods market equilibrium called the IS curve:
Y = A
(−)r ,
(+)
Y︸︷︷︸<1
,(+)
Fiscal,(+)
RER
(2)
Advanced Macroeconomics LSE PRESS 3/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
or alternatively,
Y = A
((−)r ,
(+)
Fiscal,(+)
RER
). (3)
Finally a relationship between nominal and real interest rates:
r = i − πe . (4)
Advanced Macroeconomics LSE PRESS 4/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
The IS-LM model
Advanced Macroeconomics LSE PRESS 5/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
Classical version of the IS-LM model
Y = F (K ,N∗) . (5)
P =M
L(i , Y
) , (6)
which is an alternative way of writing the quantity equation ofmoney:
MV = PY . (7)
Advanced Macroeconomics LSE PRESS 6/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
The Keynesian version of the IS-LM Model
In the short run, the Keynesian assumption is that prices are fixedor rigid, and do not move to equate supply and demand:
P = P, (8)
Advanced Macroeconomics LSE PRESS 7/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
The IS-LM Model with an exogenous Interest Rate
Advanced Macroeconomics LSE PRESS 8/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
An interpretation: The Fed
M
P= L
(i ,Y
)(9)
Y = A(i − πe ,Fiscal, ...
). (10)
Advanced Macroeconomics LSE PRESS 9/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Keynesianism 101: IS-LM
AS-AD Model
Advanced Macroeconomics LSE PRESS 10/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
The model with perfect information
The representative producer of good i has production function
Qi = Li , (11)
so that his feasible consumption is
ci =PiQi
P. (12)
Let’s assume the specification
ui = ci −1
γLγi γ > 1. (13)
Advanced Macroeconomics LSE PRESS 11/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
Replacing (11) and (12) in (13) gives:
ui =PiLiP− 1
γLγi . (14)
The first order condition relative to L is:
Pi
P− Lγ−1i = 0, (15)
which can be written as a labor supply curve
Li =
(Pi
P
) 1γ−1
, (16)
Advanced Macroeconomics LSE PRESS 12/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
or, if expressed in logs (denoted in lower case letters), as
li =
(1
γ − 1
)(pi − p) . (17)
Demand depends on income, relative prices and agood-specific“taste shock” – in log format:
qi = y + zi − η (pi − p) η > 0, (18)
o introduce it consider a money demand function in log form:
y = m − p. (19)
Advanced Macroeconomics LSE PRESS 13/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
Equilibrium (1
γ − 1
)(pi − p) = y + zi − η (pi − p) , (20)
from which we obtain the individual price
pi =(γ − 1)
1 + ηγ − η(y + zi ) + p, (21)
Advanced Macroeconomics LSE PRESS 14/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
and from which we can obtain the average price. Averaging (21)we get:
p =(γ − 1)
(1 + ηγ − η)y + p (22)
which impliesy = 0.
You may find strange, but it is not: remember that output isdefined in logs. Replacing the solution for output in (19) we getthat
p = m, (23)
Advanced Macroeconomics LSE PRESS 15/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
Lucas’ supply curve
Denote relative prices as ri = (pi − p) , then the analog to (17) isnow
li =
(1
γ − 1
)E (ri |pi ) . (24)
E (ri |pi ) = α + βpi . (25)
With the assumption of normality, the solution to this problem is:
E (ri |pi ) =νr
νr + νp(pi − E (p)) , (26)
Advanced Macroeconomics LSE PRESS 16/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
Substituting in (24) yields:
li =
(1
γ − 1
)νr
νr + νp(pi − E (p)) . (27)
Aggregating over all the individual supply curves, and defining
b =1
γ − 1
vrvr + vp
(28)
we have thaty = b (p − E (p)) , (29)
Advanced Macroeconomics LSE PRESS 17/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
Solving the model
y = b (p − E (p)) = m − p, (30)
that can be used to solve for the aggregate price level and income:
p =m
1 + b+
b
1 + bE (p) , (31)
y =bm
1 + b− b
1 + bE (p) . (32)
Advanced Macroeconomics LSE PRESS 18/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Microfoundations of Incomplete Nominal Adjustment
we can take expectations of (31) to obtain:
E (p) =1
1 + bE (m) +
b
1 + bE (p) , (33)
which implies, in turn, that
E (p) = E (m) . (34)
Using this and the fact that m = E (m) + m − E (m) we have that
p = E (m) +1
1 + b(m − E (m)) , (35)
y =b
1 + b(m − E (m)) (36)
Advanced Macroeconomics LSE PRESS 19/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
Imperfect competition and nominal and real rigidities
Welfare effects of imperfect competition
Advanced Macroeconomics LSE PRESS 20/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The canonical New Keynesian modelTo describe the demand side we start from the Euler equation:
Ct = σ (rt − ρ)Ct , (37)
In a closed economy:Ct = Yt , (38)
and,Yt = σ (it − πt − ρ)Yt , (39)
Advanced Macroeconomics LSE PRESS 21/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Define the output gap as:
Xt ≡Yt
Yt, (40)
Xt
Xt=
Yt
Yt− g , (41)
xt = σ (it − πt − rn) , (42)
this equation is a dynamic New Keynesian IS equation (NKIS)
Advanced Macroeconomics LSE PRESS 22/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The price-setting technology allows firms to change prices whenthey receive a price change signal.
The probability of receiving such a signal s periods from now is:
αe−αs , α > 0. (43)
Advanced Macroeconomics LSE PRESS 23/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The total number of firms that set their price at time s < t, a share
e−α(t−s) (44)
will not have received the signal at time t. Therefore,
αe−α(t−s) (45)
is the share of firms that set their prices at time s and have not yetreceived a price-change signal at time t > s
Advanced Macroeconomics LSE PRESS 24/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
pt is the arithmetic average of all prices dt weighed by the share offirms with the same vt
pt = α
∫ t
−∞vse−α(t−s)ds. (46)
Optimal price vt is:
vt = pt + α
∫ ∞t
[(vs − ps) + ηxs ] e−(α+ρ)(s−t)ds, (47)
Advanced Macroeconomics LSE PRESS 25/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
We can use Leibniz’s rule to differentiate the expressions for pt andvt with respect to time, obtaining:
pt = πt = α (vt − pt) , (48)
andvt − pt = −αηxt + ρ (vt − pt) . (49)
Combining the two we have
vt − pt = −αηxt +ρ
απt . (50)
Advanced Macroeconomics LSE PRESS 26/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Differentiating the expression for the inflation rate πt , again withrespect to time, yields
πt = α (vt − pt) . (51)
Finally, combining the last two expressions we arrive at
πt = ρπt − κxt , (52)
This is the New Keynesian Phillip Curve (NKPC)
Advanced Macroeconomics LSE PRESS 27/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Solving this equation forward we obtain
πt =
∫ ∞t
κxs .e−ρ(s−t)ds (53)
Inflation is the present discounted value of all future expectedoutput gaps.
Advanced Macroeconomics LSE PRESS 28/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The steady state is
π = i − rn ( from xt = 0) (54)
ρπ = κx ( from πt = 0) (55)
Advanced Macroeconomics LSE PRESS 29/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
In matrix form the dynamic system is[πtxt
]= Ω
[πtxt
]+
[0
σ (i − rn)
](56)
where
Ω =
[ρ −κ−σ 0
](57)
The system is saddle path stable.But both x and π are jump variables. This means we have acontinuum of perfect-foresighted convergent equilibrium.
Advanced Macroeconomics LSE PRESS 30/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Indeterminacy in the NK Model
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Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
A Taylor rule in the canonical New Keynesian model
it = rnt + φππt + φxxt , (58)
Using the Taylor rule in the NKIS equation yields
xt = σ [(rnt − rn) + (φπ − 1)πt + φxxt ] , (59)
Advanced Macroeconomics LSE PRESS 32/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The resulting dynamic system can be written as[πxt
]= Ω
[πtxt
]+
[0
σ (rnt − rn)
](60)
where
Ω =
[ρ −κ
σ (φπ − 1) σφx
](61)
φπ > 1 is sufficient to ensure both eigenvalues are positive. Thesteady state is now unique.
Advanced Macroeconomics LSE PRESS 33/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Active interest rule in the NK Mode
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Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
A reduction in the natural Rate
Advanced Macroeconomics LSE PRESS 35/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Back to solving the model
In discrete time the NKPC becomes
πt = βEtπt+1 + κxt , (62)
To derive the IS curve, again start from the Euler equation, whichin logs can be written as
yt = Etyt+1 − σ log (1 + rt) + σ log(1 + ρ), (63)
the equation above becomes
yt = Etyt+1 − σ (rt − ρ) (64)
Advanced Macroeconomics LSE PRESS 36/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
Finally, subtracting y from both sides yields
xt = Etxt+1 − σ (it − Etπt+1 − ρ) , (65)
If the natural rate of output is not constant, so that
yt+1 = yt + ∆ (66)
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Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
we then have
xt = Etxt+1 + ∆− σ (it − Etπt+1 − ρ) (67)
xt = Etxt+1 − σ (it − Etπt+1 − rn) , (68)
where
rn = ρ+∆
σ(69)
this is the discrete time version of the NKIS.
Advanced Macroeconomics LSE PRESS 38/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
To close the model, we can again appeal to an interest rule of theform
it = in + φπEtπt+1 + φxxt (70)
Substituting the interest rate rule into the NKIS equation (in thesimple case of constant y ) yields
xt = Etxt+1 − σ [(φπ − 1)Etπt+1 + φxxt + (in − rn)] (71)
Advanced Macroeconomics LSE PRESS 39/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
To analyze formally the dynamic properties of this system, rewritethe NKPC as
Etπt+1 = β−1πt − β−1κxt (72)
Next, use this in the NKIS above to yield
Etxt+1 =
σ (φπ − 1)β−1πt + xt + σ[− (φπ − 1)β−1κ+ φx
]xt + σ (in − rn)
(73)
Advanced Macroeconomics LSE PRESS 40/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
The last two equations together constitute the canonical NewKeynesian model in discrete time. In matrix form the dynamicsystem is [
Etπt+1
Etxt+1
]= Ω
[πtxt
]+
[0
σ (in − rn)
](74)
where
Ω =
[β−1 −β−1κ
σβ−1 (φπ − 1) 1 + σ[− (φπ − 1)β−1κ+ φx
] ] (75)
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Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
NowDet(Ω) = β−1 (1 + σφx) = λ1λ2 > 1 (76)
and
Tr(Ω) = β−1 + 1 + σ[φx − (φπ − 1)β−1κ
]= λ1 + λ2, (77)
Advanced Macroeconomics LSE PRESS 42/43
Chapter 15: (New) Keynesian Theories of Fluctuations – A Primer
New Keynesian DSGE models
For both λ1 and λ2 to be larger than one, a necessary andsufficient condition is that
Det(Ω) + 1 > Tr(Ω) (78)
which, using the expressions for the determinant and the trace, isequivalent to
(φπ − 1) x + (1− β)φx > 0 (79)
This condition clearly obtains if φπ > 1.
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