MakØk3, Fall 2010 (blok 2)
“Business cycles and monetary stabilization policies”
Henrik Jensen
Department of Economics
University of Copenhagen
Lecture 3, November 30: The Basic New Keynesian Model (Galí, Chapter 3)
c 2010 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personal
use or distributed free.
Introduction
The classical ‡ex-price model(s) covered so far cannot account for realistic short-run dynamics following monetary shocks
–In case there are any real e¤ects, the magnitude appears modest
–No liquidity e¤ect present (long run = short run)
Some “sand in the wheels”are necessary to explain the e¤ects of money on output seen in data
Obvious candidate is incomplete nominal adjustment. In ‡ex-price models, money neutrality holds
both in the short and long run
–If money neutrality fails in the short run, the impact of monetary policy will clearly be stronger
Classical model is therefore amended with nominal rigidities (and explicit determination of prices by
…rms)
Result is Dynamic Stochastic General Equilibrium (DSGE) model of the New-Keynesian type
1
A basic New-Keynesian model
Goods market
–Demand side: Households consume a basket of goods (based on utility maximization)
–Supply side: Firms produce di¤erent consumption goods (maximize pro…ts under monopolistic
competition)
Labor market
–Demand side: Firms hire labor (maximize pro…ts under monopolistic competition)
–Supply side: Households supply labor (based on utility maximization)
Financial markets
–Households optimally invest in a one-period risk-less bond
–Households also hold money. We again just assume it (and generally do not make much use of it
here, as focus will be on interest-rate policies)
2
Wages are perfectly ‡exible (this assumption is relaxed in Chapter 6)
All goods prices are subject to in‡exibility, i.e., stickiness:
–Every period, each …rm faces a state-independent probability
–So, 0
of “being stuck”with its price
1 is natural “index”for stickiness in the economy
The probability is independent of when the …rm last changed its price; i.e., time-dependent price
stickiness (as opposed to state-dependent price stickiness)
Stylistic representation of “staggering.”The independence of history facilitates aggregation:
– is the fraction of …rms not adjusting prices in a period
–1
is the fraction of …rms adjusting is a period
–1 + +
2
+
3
+ :::: = 1= (1
) is average duration of a price contract
When “allowed”to change the price, expectations about future prices become of importance
3
A view at prices changes: Belgium, 2001-2005
Source: Cornille and Dossche (2006)
4
A view at prices changes: France and Italy, 1996-2001
Source: Dhyne et al. (2005)
5
A view at prices changes: (Monthly) frequencies of price adjustments; Euro area vs. US
Source: Dhyne et al. (2005) (with US data based on Bils and Klenow, 2004)
6
A view at prices changes: France, hazard rates in clothing industry
Source: Baudry et al. (2004)
7
A view at prices changes: France: Hazard Rates in service industry
Source: Baudry et al. (2004)
8
Summary of …ndings from data:
Aggregate prices are sticky (as also indicated by VAR analyses)
Price setting in di¤erent sectors di¤ers;
–some sectors exhibit long periods of …xed prices (time dependent price setting)
–some sectors exhibit continuous changing prices
Price setting is asynchronized across sectors
In some sectors, the probability of price adjustment is (nearly) independent of the life-span of the
price
Relation to theoretical modelling of price rigidities:
The staggered price setting scheme captures many of these features, and their aggregate implications,
in a simple and handy way
Modelling due to Calvo (1983); now just called “Calvo pricing”
9
Household behavior
The essentials of household behavior are like in the basic classical model
Household maximize expected utility
E0
1
X
t
U (Ct; Nt) ;
0<
< 1:
t=0
Main di¤erence: Ct is not one good, but a basket of the di¤erent goods produced by the di¤erent
producers (indexed by i 2 [0; 1])
"
Z 1
" 1
" 1
Ct (i) " di
;
">1
Ct
0
Budget constraint therefore becomes:
Z 1
Pt (i) Ct (i) di + QtBt
Bt
1
+ W t Nt
Tt
0
(and a No-Ponzi Game constraint ruling out explosive debt)
Household chooses optimal paths of Ct, Nt and Bt (taking prices Pt, Wt and Qt as given), but must
now choose the composition of Ct as well
10
Finding the optimal composition of Ct:
Assume that total nominal expenditures on consumption are Zt
Z 1
Pt (i) Ct (i) di = Zt
0
Optimal demand for Ct (i) ; is found by
Z 1
" 1
max
Ct (i) " di
Ct (i)
Z
"
" 1
s.t.
0
The relevant …rst-order condition is
Z 1
" 1
Ct (i) " di
Z
1
Pt (i) Ct (i) di = Zt
0
1
" 1
Ct (i)
1
"
=
Pt (i) ;
=
Pt (i) Ct (i)
0
1
Ct (i)
" 1
"
di
1
" 1
Ct (i)
" 1
"
0
where
is the Lagrange multiplier associated with the constraint
11
Integrating over all goods:
Z
1
Ct (i)
" 1
"
1
" 1
di
0
Z
1
Ct (i)
" 1
"
Z
di =
0
1
Pt (i) Ct (i) di
0
Ct =
Zt
Let Pt be the price index associated with Ct, then PtCt = Zt and
=
Inserted into …rst-order condition:
Z 1
Ct (i)
" 1
"
1
Pt
1
" 1
Ct (i)
di
1
"
0
1
"
Ct Ct (i)
1
"
Pt (i)
;
Pt
Pt (i)
=
Pt
=
The relative demand for good i is therefore
Pt (i)
Pt
Ct (i) =
"
Ct
(1)
Note that " is the elasticity of substitution between goods. (Hence, the limiting case of perfect
competition applies when " ! 1.)
12
The price index
The precise form of the price index Pt follows from
Z 1
Pt (i) Ct (i) di
PtCt =
0
and the expression for relative demand
I.e., we get
PtCt =
Z
1
0
Pt =
Z
"
Pt (i)
Pt (i)
Pt
1
Pt (i)1
0
13
"
Ct di
1
1 "
di
Remaining household optimality conditions
Precisely as in the simple classical model:
Un;t Wt
=
Uc;t
Pt
Qt = Et
With U (Ct; Nt)
Ct1
1
Uc;t+1 Pt
Uc;t Pt+1
1+'
Nt
1+'
, ; ' > 0, we have the same log-linear expressions:
–Labor supply schedule:
ct + 'nt = wt
(2)
pt
–Consumption-Euler equation:
1
ct = Et fct+1g
with it
log Qt and
log
14
(it
Et f
t+1 g
)
(3)
Firms’behavior and aggregate in‡ation dynamics
Generally, optimal price-setting of a …rm that can change its price in period t:
1
X
k
max
Et Qt;t+k Pt Yt+kjt
t+k Yt+kjt
Pt
k=0
s.t. Yt+kjt =
Pt
Pt+k
"
(8)
Ct+k
I.e., the maximal expected discounted stream of pro…ts of choosing Pt taking into account the
probabilities that it is …xed in the future
t+k
is total costs of sales at price Pt in period t + k (will depend on labor costs and productivity)
Qt;t+k is the “stochastic discount factor”(or present-value operator), equal to the one facing consumers
between periods t and t + k:
Pt
k Ct+k
Qt;t+k
Ct
Pt+k
Note
Ct+1
Pt
Ct
Pt+1
Ct+2
Pt
Ct+1
= 2
=
Ct
Pt+2
Ct
= Qt;t+1Qt+1;t+2:::::Qt+k 2;t+k 1Qt+k
Qt;t+1 =
Qt;t+2
Qt;t+k
15
Pt
Pt+1
1;t+k
Ct+2
Ct+1
Pt+1
Pt+2
= Qt;t+1Qt+1;t+2
First-order condition for Pt :
1
X
k
Et Qt;t+k Yt+kjt + Pt
k=0
where
0
t+k
t+kjt
1
X
k
@Yt+kjt
t+kjt
@Pt
@Yt+kjt
@Pt
= 0;
Yt+kjt is nominal marginal costs
Et Qt;t+k Yt+kjt 1 +
k=0
1
X
k
Pt @Yt+kjt
Yt+kjt @Pt
@Yt+kjt
t+kjt
Yt+kjt @Pt
= 0;
"
t+kjt
Pt
= 0:
1
Et Qt;t+k Yt+kjt 1
"+
k=0
We then get the central pricing equation:
1
X
k
Et Qt;t+k Yt+kjt Pt
k=0
with M
"= ("
M
t+kjt
= 0;
1) > 1 being the mark-up; a measure of the monopoly distortion
16
(9)
This is rewritten as
1
X
k
Pt
Pt 1
Et Qt;t+k Yt+kjt
k=0
MM Ct+kjt
where M Ct+kjt
t+kjt =Pt+k is real marginal costs and
between t 1 and t + k
t 1;t+k
= 0;
Pt+k =Pt
t 1;t+k
Optimality condition is log-linearized around a zero-in‡ation steady state:
Pt
= 1;
Yt+kjt = Y; M C = M 1; Qt;t+k =
t 1;t+k = 1;
Pt 1
One gets
1
X
(
)k Et pt
pt
mc
c t+kjt
1
k=0
where mc
c t+kjt
mct+kjt
pt+k + pt
1
(10)
1
is gross in‡ation
k
=0
mc = mct+kjt + log M = mct+kjt + log " " 1 = mct+kjt +
Hence,
1
X
k=0
(
k
) pt =
1
X
k=0
pt = (1
(
)k Et mc
c t+kjt + pt+k
)
1
X
k=0
(
)k Et mc
c t+kjt + pt+k
Optimal price is a function of expected current and future real marginal costs and aggregate prices
17
Marginal costs depend on production function and factor payments
Each …rm has the following production function (i.e., identical technology):
Yt (i) = AtNt (i)1
0
;
<1
(5)
Total costs
T Ct = WtNt (i)
Real marginal costs
M Ct (i) =
Wt
) Nt (i)
PtAt (1
0
@=
Wt
=
Pt M P Nt PtAt (1
) (Yt (i) =At)
Wt
=
1
1
Pt (1
) At1 Yt (i)
=0
Special case of constant returns to scale,
M Ct+kjt = M Ct+k
pt = (1
Wt
)
1
X
(
k=0
=
Wt+k
Pt+k At+k
)k Et fmc
c t+k + pt+k g
This is the unique stationary solution to the …rst-order rational expectations di¤erence equation:
pt =
Et fpt+1g + (1
18
) (mc
c t + pt )
1
1
A
Aggregate price dynamics
Remember, is fraction of price setters that keep past period’s price;
that set new prices
By de…nition of the price index:
h
i 1
1 "
1 " 1 "
Pt = (Pt 1) + (1
) (Pt )
1 "
t
=)
is fraction of price setters
Pt
)
Pt 1
= + (1
1 "
Log-linearized around the zero-in‡ation steady state:
(1
")
t
= (1
") (1
pt 1 )
) (pt
=)
t
= (1
pt 1 )
) (pt
Rewrite the di¤erence equation for optimal price setting:
pt
pt
1
=
pt
ptg + (1
Et fpt+1
pt
1
=
) (mc
c t + pt )
ptg + (1
Et fpt+1
Use aggregate dynamics to obtain:
(1
t
= Et f
)
1
t+1 g
t
= (1
)
1
Et f
(1
+ mc
c t;
t+1 g
) mc
ct +
+ (1
) (1
pt
)
;
+
pt
t
) mc
ct +
lim
This is the basic in‡ation-adjustment equation in New-Keynesian theory
19
1
!0
t
= 1:
(7)
General case of decreasing returns to scale, 0 <
<1
We de…ne aggregate real marginal costs (in logs):
mct
wt
pt
= wt
pt
mpnt
1
(at
1
yt )
log (1
)
We have for a price-changing …rm:
mct+kjt = wt+k
pt+k
= mct+k +
1
1
at+k
1
yt+kjt
yt+kjt
log (1
)
yt+k
By the demand function in logs (here using equilibrium condition yt+k = ct+k ):
"
(pt pt+k )
mct+kjt = mct+k
1
(14)
Optimal price setting:
pt = (1
)
1
X
k=0
Associated in‡ation dynamics
t
= Et f
c t;
t+1 g + mc
(
)k Et mc
c t+k
(1
) (1
"
(pt
1
)
1
1
20
pt+k ) + pt+k
+ "
;
lim
!0
= 1:
(16)
Equilibrium outcomes
Goods market clearing on each goods market
Yt (i) = Ct (i) ;
all 0
i
1
In the aggregate:
Yt = Ct;
where
Yt
Z
1
Yt (i)
" 1
"
"
" 1
di
:
0
Asset market equilibrium (as in classical economy):
1
yt = Et fyt+1g
21
(it
Et f
t+1 g
)
(12)
Aggregate employment:
Nt =
Z
1
Nt (i) di =
0
Nt =
Z
1
0
Yt
At
1
Yt
At
1
1
Z
1
Yt (i)
Yt
0
1
Z
1
nt =
(1
) nt = yt
1
(yt
0
at) + log
Z
1
0
at + dt;
dt
1
(1
1
1
di
1
di
Pt (i)
Pt
In logs:
1
Yt (i)
At
1
"
di
Pt (i)
Pt
Z
) log
0
1
"
;
1
Pt (i)
Pt
1
"
:
The variable dt is a natural measure of price dispersion ( proportional to the variance of prices)
When we consider log deviations from a symmetric zero-in‡ation steady state, dt can be ignored (it
is of second order— see Appendix 3.2 and 3.3)
Hence, as a log-linear approximation:
yt = at + (1
as in the classical case
22
) nt ;
(13)
Relationship between real marginal cost and output
We want a dynamic system in output and in‡ation; hence, marginal cost is replaced appropriately
by output
We have
mct = wt
1
pt
(at
1
yt )
log (1
)
Using the labor supply schedule
1
mct = ct + 'nt
mct =
=
Under ‡exible prices,
yt +
'
1
(1
1
(yt
at )
)+'+
1
= 0, mc
ct = 0
yt
(at
yt )
1
log (1
(at
yt )
1
1+'
at log (1
1
)
log (1
)
)
(17)
mct = mc
=
(=
ln M)
Hence, denoting ytn the ‡exible-price output, or, the natural rate of output,
(1
)+'+ n 1+'
mc =
=
yt
at log (1
1
1
23
)
The natural rate of output is given as
ytn =
=
Under perfect competition,
1+'
at
(1
)+'+
n
n
ya at + #y
[
log (1
)] (1
(1
)+'+
)
(19)
= 0, this is output as in the classical model
We get the marginal cost in deviation from steady state:
We denote yet
yt
mc
c t = mct
mc =
(1
)+'+
1
(yt
ytn)
(20)
ytn the output gap
The “New-Keynesian”Phillips Curve (“NKPC”):
t
= Et f
t+1 g
(1
+ yet;
)+'+
1
24
(21)
Rephrase the Euler equation in terms of the output gap:
yt = Et fyt+1g
1
(it
1
(it
yet = Et fe
yt+1g
1
(it
yet = Et fe
yt+1g
where
rtn
Et f
t+1 g
)
Et f
t+1 g
rtn)
Et f
t+1 g
)
n
ytn + Et fyt+1
g
(22)
n
+ Et f yt+1
g
is the natural rate of interest
–A dynamic “IS”curve (“DIS”)
–Let rbtn
rtn
New-Keynesian Phillips curve and dynamic IS curve determines output gap and in‡ation conditional
on monetary policy (it).
So, in contrast with the classical model, monetary policy matters for output determination!
25
Interest rate policy in the New Keynesian model
The case of a simple Taylor-type interest rate rule:
it =
where
t
is a policy shock, vt =
+
v vt 1
t
+
+ "vt
et
yy
+
t;
;
y
0;
(25)
vt ) ;
(26)
The policy rule, NKPC and DIS give the dynamics
yet
= AT
t
AT
1
+
+
Et fe
yt+1g
Et f t+1g
;
+ BT (b
rt
1
BT
;
y
1
+
+
y
A unique non-explosive equilibrium requires the eigenvalues of AT to be stable (within the unit circle)
Uniqueness depends on policy-rule parameters:
0<(
1) +
y
(1
)
is necessary and su¢ cient condition (the “Taylor principle”)
–Note
> 1 is su¢ cient condition (an “active”Taylor rule)
26
(27)
27
28
Concluding remarks
The New-Keynesian model is (compared to the classical model) a simple but powerful tool for analyzing monetary policy
Given monetary policy works more in accordance with evidence, how can the model be used for policy
evaluation?
To come:
–Welfare e¤ects of di¤erent policies
–Optimal monetary policy and credibility issues
29
Next time(s)
Monday, December 6: Exercises:
Derive the dynamics of the New-Keynesian model on matrix form (or, state-space form), (26)
Derive the condition for determinacy, (27) in Galí, Chapter 3. Hint: A 2 2 matrix has two stable
roots, when the coe¢ cients in the characteristic polynomial 2 + a1 + a0 satisfy ja0j < 1 and
ja1j < 1 + a0
Derive the explicit solutions for the output gap, in‡ation, and the nominal interest rate and output
in New-Keynesian model of Galí, Chapter 3 (page 51) under an interest-rate rule (and in the absence
of technology shocks).
–Interpret the solutions economically
Tuesday, December 7:
Lecture: Monetary policy design and welfare in the simple NK Model (Galí, Chapter 4)
30