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Simple New Keynesian Model:Foundations
Lawrence J. Christiano
Outline
• Preferences and Technology.
— First best, e¢cient allocations (benchmark for later analysis).
• Flexible Price Equilibrium
— E¢cient, as long as a monopoly distortion is eliminated.• Complete monetary neutrality.
• Equilibrium in sticky price version of the model (New Keynesianmodel).
— Private sector equilibrium conditions.— Ramsey Equilibrium.— Linearization (first order perturbation)— Taylor rule— Properties of Taylor rule equilibrium
Preferences and Technology• Household utility:
E
0
•
Ât=0
bt
log C
t
− exp (tt
)N
1+jt
1+ j
!, t
t
= ltt−1
+ #tt
• Goods production
— Final goods:
Y
t
=
[Z1
0
Y
#−1
#i,t
dj
] ##−1
.
— Intermediate goods:
Y
i,t
= exp (at
)N
i,t
, a
t
= ra
t−1
+ #a
t
• Labor:
N
t
=Z
1
0
N
it
di.
First Best Allocations
• Must solve two problems:
— For any given N
t
, how should employment be allocated acrosssectors?
— What should the level of total employment, N
t
, be?
E¢cient Sectoral Allocation of Employment
• Consider a given level of N
t
.
• Can show: e¢ciency dictates allocating employment equallyacross sectors:
N
it
= N
t
, i 2 [0, 1] .
• Consider, for example, the class of deviations from equal:
N
it
=
{2aN
t
i 2[0,
1
2
]
2 (1− a)N
t
i 2[
1
2
, 1
], 0 ≤ a ≤ 1.
This allocation is consistent with N
t
:
Z1
0
N
it
di =1
2
2aN
t
+1
2
2 (1− a)N
t
= N
t
Y
t
=
[Z1
0
Y
#−1
#i,t
di
] ##−1
=
"Z 1
2
0
Y
#−1
#i,t
di+Z
1
1
2
Y
#−1
#i,t
di
# ##−1
= e
a
t
"Z 1
2
0
N
#−1
#i,t
di+Z
1
1
2
N
#−1
#i,t
di
# ##−1
= e
a
t
"Z 1
2
0
(2aN
t
)#−1
#di+
Z1
1
2
(2 (1− a)N
t
)#−1
#di
# ##−1
= e
a
t
N
t
"Z 1
2
0
(2a)#−1
#di+
Z1
1
2
(2 (1− a))#−1
#di
# ##−1
= e
a
t
N
t
[1
2
(2a)#−1
# +1
2
(2 (1− a))#−1
#
] ##−1
= e
a
t
N
t
f (a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.88
0.9
0.92
0.94
0.96
0.98
1
D
f
Efficient Resource Allocation Means Equal Labor Across All Sectors
/ � 6
/ � 10
f�) � 12 �2)
/"1/ � 12 �2�1 " )
/"1/
//"1
Economy with Efficient N Allocation • Efficiency dictates
• So, with efficient production:
• Resource constraint:
• Preferences:
Yt � eatNt
Ct t Yt
E0!t�0
.
logCt " exp�A t Nt1�I
1 � I , A t � 5A t"1 � /tA, /tA~iid,
Nit � Nt all i
Efficient Determination of Labor • Lagrangian:
• First order conditions:
• or:
maxCt ,Nt
E0!t�0
.
* t
�logCt"exp�At Nt1�I
1�I
u�Ct,Nt,A t �5 t¡eatNt " Ct ¢
uc�Ct,Nt,At � 5t, un�Ct,Nt,At � 5teat � 0
un,t � uc,teat � 0
marginal cost of labor in consumption units�" dudNtdudCt
� dCtdNt
"un,tuc,t �
marginal product of labor¥eat
Efficient Determination of Labor, cont’d • Solving the fonc’s:
• Note: – Labor responds to preference shock, not to tech
shock
"un,tuc,t � eat
Ct exp�A t NtI � eat
eatNt exp�A t NtI � eat
v Nt � exp "A t1 � I
v Ct � exp at " A t1 � I
Response to a Jump in a
Leisure, 1-N
consumption Higher a
Indifference curve
Budget constraint
Key Features of First Best
• Employment does not respond to a
t
.
— This result is only somewhat sensitive to the absence of capital.— In RBC models response of N
t
to a
t
positive, but small.
• One reason for the small response is that income e§ectsmitigate substitution e§ects.
• First best consumption not a function of intertemporalconsiderations.
— Consumption is a function only of current realization of shocks.• Time series representations of shocks irrelevant.• For example, the impact of #a
t
and #tt
on E
t
a
t+j
and E
t
tt+j
, j > 0,
respectively have no e§ect on first best consumption.
— The parameter, b, does not enter first-best N
t
and C
t
.
Markets
• First, consider flexible prices.
— Equilibrium e¢cient as long as government subsidy neutralizesmonopoly power distortion.
— Monetary policy relevant to real allocations.
• Second, introduce price-setting frictions in goods market.
— Labor market has flexible wages.— We’ll see that equilibrium could be first best, or could be reallybad. It all depends on the quality of monetary policy.
Households
• Problem:
max E
0
•
Ât=0
bt
log C
t
− exp (tt
)N
1+jt
1+ j
!, t
t
= ltt−1
+ #tt
s.t. P
t
C
t
+ B
t+1
≤ W
t
N
t
+ R
t−1
B
t
+ Profits net of taxest
• First order conditions ( ¯pt
≡ P
t
/P
t−1
):
1
C
t
= bE
t
1
C
t+1
R
t
¯pt+1
exp (tt
)C
t
N
jt
=W
t
P
t
.
Final Goods
• Final good firms:
— maximize profits:
P
t
Y
t
−Z
1
0
P
i,t
Y
i,t
dj,
subject to:
Y
t
=
[Z1
0
Y
#−1
#i,t
dj
] ##−1
.
— Foncs:
Y
i,t
= Y
t
(P
t
P
i,t
)#
!
"cross price restrictions"z }| {
P
t
=
(Z1
0
P
(1−#)i,t
di
) 1
1−#
Intermediate Good Producers• Demand curve for monopoly producer of Y
i,t
:
Y
i,t
= Y
t
(P
t
P
i,t
)#
.
• Production function:
Y
i,t
= exp (at
)N
i,t
, Da
t
= rDa
t−1
+ #a
t
• Competitive in labor market:— Wage rate, W
t
, taken as given.— (Real) marginal cost of production:
s
t
=(1− n)W
t
/P
t
e
a
t
,
where n is a government subsidy to firms.• Optimizing monopolist sets price as a markup, #/ (#− 1) , overmarginal cost:
P
i,t
=#
#− 1
P
t
s
t
.
Features of Flexible Price Equilibrium• Prices are all equal:
P
i,t
= P
t
, for all t,
so sectoral allocation of employment e¢cient and:
1 =#
#− 1
s
t
=#
#− 1
(1− n)W
t
/P
t
e
a
t
.
• Aggregate employment, N
t
, might not be e¢cient:
cost of labor e§ortz }| {−u
N,t
u
c,t
by household optimizationz}|{=
W
t
P
t
= e
a
t
1
##−1
(1− n)(∗)
• If n = 0 :
— Actual marginal cost < actual marginal product of labor.— Households get ‘wrong’ signal from the market about theiractual productivity, e
a
t
.
Profits, Government Budget Constraint andAggregate Resources (Walras’ Law)
• Government budget constraint and profits:
taxes = nW
t
Z
i
N
i,t
di
profits =Z
i
[Pi,t
Y
i,t
− (1− n)W
t
N
i,t
] di
• Household budget constraint at equality:
P
t
C
t
= W
t
N
t
+Z
i
[Pi,t
Y
i,t
− (1− n)W
t
N
i,t
] di+ nW
t
Z
i
N
i,t
di
because N
i,t
=N
t
, Y
i,t
=Y
tz}|{= W
t
N
t
+ P
t
Y
t
− (1− n)W
t
N
t
+ nW
t
N
t
= P
t
Y
t
! C
t
= Y
t
.
• Using e¢ciency of sectoral resource allocation: C
t
= e
a
t
N
t
. (∗∗)
Equilibrium• Sequences, {P
t
, P
i,t
, W
t
/P
t
, R
t
, C
t
, N
t
}•t=0
, such that
— Households’ problems are solved— Firms’ problems are solved— Markets clear.
• By equations (∗) , (∗∗)
C
t
= e
a
t
N
t
,
exp (tt
)C
t
N
jt
= e
a
t
1
##−1
(1− n).
• Then,
C
t
= e
a
t
N
t
N
t
=
"exp (−t
t
)#
#−1
(1− n)
# 1
1+j
.
Properties of Equilibrium• E¢cient if, and only if, monopoly power extinguished so thatreal wage correctly signals marginal product of work:
#
#− 1
(1− n) = 1,
so that equilibrium employment and consumption are first best:
N
t
= exp
(−
tt
1+ j
), C
t
= exp
(a
t
−t
t
1+ j
)
exp (at
) =W
t
P
t
= exp (tt
)C
t
N
jt
.
• Evidently, C
t
, N
t
, W
t
/P
t
are uniquely determined.— The objects, R
t
,
¯pt+1
, P
t
are not uniquely determined.— Presumably, monetary policy can determine these variables.But, note that monetary policy is ine§ective and uninterestingin the flexible price case, since consumption and employmentare completely determined without regard to monetary policy.
Real Interest Rate in Equilibrium• The real interest rate, R
t
/
¯pt+1
.— Absent uncertainty, R
t
/
¯pt+1
determined uniquely:
1
C
t
= b1
C
t+1
R
t
¯pt+1
.
— With uncertainty, household intertemporal condition simplyplaces a single linear restriction across all the period t+ 1
values for R
t
/
¯pt+1
that are possible given period t.
• The real interest rate, ˜
r
t
, on a risk free one-period bond thatpays in t+ 1 is uniquely determined:
1
C
t
= ˜
r
t
bE
t
1
C
t+1
.
• By no-arbitrage, the following weighted average of R
t
/
¯pt+1
across period t+ 1 states of nature is uniquely determined:
˜
r
t
=E
t
1
C
t+1
R
t
¯pt+1
E
t
1
C
t+1
.
Real Interest Rate, Continued
• The object, ˜
r
t
, plays no role in the computation of theequilibrium allocations.
• But, ˜
r
t
is important from an economic perspective since it playsa key role in the consumption decisions of households in themodel.
Real Interest Rate, Continued• To understand the economics of the model, it is useful toexamine response of ˜
r
t
to shocks:
˜
r
t
=1
b
1
E
t
C
t
C
t+1
! ˜
r
t
=1
b
1
E
t
exp
ha
t
− a
t+1
+ tt+1
−tt
1+j
i
=1
b
1
exp
hE
t
(at
− a
t+1
) + E
t
(t
t+1
−tt
1+j
)+V
i,
where c
t
≡ log C
t
and assuming #a
t+1
and #tt+1
are Normallydistributed (V is very small).
• Then,
r
∗t
≡ log
˜
r
t
= − log (b) + E
t
[a
t+1
− a
t
−(
tt+1
− tt
1+ j
)−V
]
Real Interest Rate, Continued
r
∗t
= − log (b) + E
t
[a
t+1
− a
t
−(
tt+1
− tt
1+ j
)−V
]
• The response of r
∗t
to #a
t
or #tt
depends how the forecast offuture a
t
and tt
responds.• Get opposite response of r
∗t
to #a
t
in trend stationary case:a
t
= ra
t−1
+ #a
t
.
• Suppose technology jumps 1 percent (i.e., #a
t
, a
t
rise by 0.01).— In di§erence stationary case (with r > 0), aggregate demand(i.e., C
t
), absent a change in r
∗t
, responds by more than 1
percent, for consumption smoothing reasons. A jump in r
∗t
ensures that aggregate demand only rises by the e¢cientamount, 1 percent.
— In the trend stationary case, aggregate demand rises by lessthan 1 percent, absent a change in r
∗t
. A fall in r
∗t
helps toboost aggregate demand so that it rises by the e¢cientamount, 1 percent.
Dynamic�Properties�of�the�Model
2 4 6 8 10 12 14 16 18 20 22 240
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Response to .01 Technology Shock in Period 1
(at - at-1) = 0.75(at-1 - at-2) + epsatlog
Ct
t
2 4 6 8 10 12 14 16 18 20 22 24
0.008
0.009
0.01
0.011
0.012
0.013
0.014
interest rate
log,
r t
t
log rt = -logE + 0.75at-1
2 4 6 8 10 12 14 16 18 20 22 24-5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
x 10-3
log
Ct
t
Wt = 0.5Wt-1 + epstaut
Response to .01 Preference Shock in Period 1
2 4 6 8 10 12 14 16 18 20 22 24
7.5
8
8.5
9
9.5
x 10-3
log,
r t
t
log rt = -logE - (0.5 - 1)Wt/(1 + 1)
interest rate
logrt � " log* � 0.75 at
Real Interest Rate, Continued
• Because of its attractive properties as a guide to aggregatedemand for goods (here, just C
t
), will later refer to r
∗t
as the‘natural rate of interest’.
• When we study equilibrium with price setting frictions,welfare-reducing discrepancies between equilibrium andfirst-best allocations will be traced to deviations between theactual and natural rate of interest.
Outline
• Preferences and Technology. (done)
— First best, e¢cient allocations (benchmark for later analysis).
• Flexible Price Equilibrium (done)
— E¢cient, as long as a monopoly distortion is eliminated.• Complete monetary neutrality.
• Equilibrium in sticky price version of the model (next).
— Private sector equilibrium conditions.— Ramsey Equilibrium.— Linearization (first order perturbation)— Taylor rule— Properties of Taylor rule equilibrium
The Model with Price Setting Frictions onIntermediate Good Producers
• Household and final good producers unchanged from before:
1
C
t
= bE
t
1
C
t+1
R
t
¯pt+1
exp (tt
)C
t
N
jt
=W
t
P
t
.
Y
t
=
[Z1
0
Y
#−1
#i,t
dj
] ##−1
, Y
i,t
= Y
t
(P
t
P
i,t
)#
! P
t
=
(Z1
0
P
(1−#)i,t
di
) 1
1−#
• Intermediate good firms’ problem changes.
Intermediate Good Producers• Demand curve for i
th monopolist:
Y
i,t
= Y
t
(P
t
P
i,t
)#
.
• Production function:
Y
i,t
= exp (at
)N
i,t
, a
t
= ra
t−1
+ #a
t
• Calvo Price-Setting Friction:
P
i,t
=
{˜
P
t
with probability 1− qP
i,t−1
with probability q .
• Real marginal cost:
s
t
=dCost
dworkerdoutputdworker
=
minimize monopoly distortion by setting = #−1
#z }| {(1− n) W
t
P
t
exp (at
)
Intermediate Good Firm • Present discounted value of firm profits:
• Each of the firms that can optimize price choose to optimize
Et!j�0
.
* jmarginal value of dividends to household�uc,t�j/Pt�j
¥Bt�j
period t�j profits sent to household
revenues
Pi,t�jYi,t�j "total cost
Pt�jst�jYi,t�j
1 " 2
P t
Et!j�0
.
* j
in selecting price, firm only cares aboutfuture states in which it can’t reoptimize
¥2j Bt�j¡P tYi,t�j " Pt�jst�jYi,t�j ¢.
Intermediate Good Firm Problem • Substitute out the demand curve:
• Differentiate with respect to :
• or
Et!j�0
.
�*2 jBt�j¡P tYi,t�j " Pt�jst�jYi,t�j ¢
� Et!j�0
.
�*2 jBt�jYt�jPt�j/ ¡P t1"/ " Pt�jst�jP t"/ ¢.
P t
Et!j�0
.
�*2 jBt�jYt�jPt�j/ ¡�1 " / �P t "/ � /Pt�jst�jP t"/"1 ¢ � 0,
Et!j�0
.
�*2 jBt�jYt�jPt�j/�1 P tPt�j
" // " 1 st�j � 0.
Intermediate Good Firm Problem • Objective:
• or
Et!j�0
.
�*2 juU�Ct�j Pt�j
Yt�jPt�j/�1 P tPt�j
" // " 1 st�j � 0.
Et!j�0
.
�*2 j�Xt,j "/ p tXt,j " /
/ " 1 st�j � 0,
p t � P tPt, Xt,j �
1=� t�j=� t�j"1���=� t�1
, j u 1
1, j � 0., Xt,j � Xt�1,j"1
1=� t�1
, j � 0
v Et!j�0
.
�*2 jPt�j/ P tPt�j
" // " 1 st�j � 0.
Intermediate Good Firm Problem • Want in:
• Solution:
• But, still need expressions for
Et!j�0. �*2 j�Xt,j "/ p tXt,j " /
/"1 st�j � 0
p t
Kt, Ft.
p t �Et!j�0
. �*2 j�Xt,j "/ /
/"1 st�jEt!j�0
. �*2 j�Xt,j 1"/ � Kt
Ft
Kt � Et!j�0
.
�*2 j�Xt,j "/ // " 1 st�j
� // " 1 st � *2Et!
j�1
.
�*2 j"1 1=� t�1
Xt�1,j"1"/ // " 1 st�j
� // " 1 st � *2Et
1=� t�1
"/!j�0
.
�*2 jXt�1,j"/ /
/ " 1 st�1�j
� // " 1 st � *2
�Et by LIME
EtEt�11
=� t�1
"/!j�0
.
�*2 jXt�1,j"/ /
/ " 1 st�1�j
� // " 1 st � *2Et
1=� t�1
"/
exactly Kt�1!
Et�1!j�0
.
�*2 jXt�1,j"/ /
/ " 1 st�1�j
� // " 1 st � *2Et
1=� t�1
"/Kt�1
• From previous slide:
• Substituting out for marginal cost:
Kt � // " 1 st � *2Et
1=� t�1
"/Kt�1.
// " 1 st �
// " 1 �1 " 7
dCost/dlabor
Wt/PtdOutput/dlabor
exp�at
� // " 1 �1 " 7
� WtPtby household optimization
exp�A t NtICt
exp�at .
In Sum • solution:
• Where:
p t �Et!j�0
. �*2 j�Xt,j "/ /
/"1 st�jEt!j�0
. �*2 j�Xt,j 1"/ � Kt
Ft,
Kt � �1 " 7t // " 1
exp�At NtICt
exp�at � *2Et
1=� t�1
"/Kt�1.
Ft q Et!j�0
.
�*2 j�Xt,j 1"/ � 1 � *2Et
1=� t�1
1"/Ft�1
To Characterize Equilibrium • Have equations characterizing optimization by
firms and households. • Still need:
– Expression for all the prices. Prices, , will all be different because of the price setting frictions.
– Relationship between aggregate employment and aggregate output not simple because of price distortions:
• This part of the analysis is the reason why it made Calvo famous – it’s not easy.
Pi,t, 0 t i t 1
Yt p eatNt, in general
Going for Prices • Aggregate price relationship
• In principle, to solve the model need all the prices, – Fortunately, that won’t be necessary.
Pt � ;0
1Pi,t�1"/ di
11"/
� ;firms that reoptimize price
Pi,t�1"/ di � ;
firms that don’t reoptimize pricePi,t�1"/ di
11"/
all reoptimizers choose same price¥� �1 " 2 P t
�1"/ � ;firms that don’t reoptimize price
Pi,t�1"/ di
11"/
Pt, Pi,t, 0 t i t 1
Key insight
;firms that don’t reoptimize price in t Pi,t�1"/ di
add over prices, weighted by # of firms posting that price¥�
;‘number’ of firms that had price, P�F , in t"1 and were not able to reoptimize in t
f t"1,t�F P�F �1"/ dF
Applying the Insight
• By Calvo randomization assumption
• Substituting:
f t"1,t�F � 2 �
total ‘number’ of firms with price P�F in t"1
f t"1�F , for all F
;firms that don’t reoptimize price
Pi,t�1"/ di � ; f t"1,t�F P�F �1"/ dF
� 2 ; f t"1�F P�F �1"/ dF
� 2Pt"1�1"/
Something hard got very simple!
Applying the Insight
• By Calvo randomization assumption
• Substituting:
f t"1,t�F � 2 �
total ‘number’ of firms with price P�F in t"1
f t"1�F , for all F
;firms that don’t reoptimize price
Pi,t�1"/ di � ; f t"1,t�F P�F �1"/ dF
� 2 ; f t"1�F P�F �1"/ dF
� 2Pt"1�1"/
Something hard got very simple!
Applying the Insight
• By Calvo randomization assumption
• Substituting:
f t"1,t�F � 2 �
total ‘number’ of firms with price P�F in t"1
f t"1�F , for all F
;firms that don’t reoptimize price
Pi,t�1"/ di � ; f t"1,t�F P�F �1"/ dF
� 2 ; f t"1�F P�F �1"/ dF
� 2Pt"1�1"/
Something hard got very simple!
Expression for in terms of aggregate inflation
• Conclude that this relationship holds between prices: – Only two variables here!
• Divide by :
• Rearrange:
Pt � �1 " 2 P t�1"/ � 2Pt"1
�1"/ 11"/ .
1 � �1 " 2 p t�1"/ � 2 1
=� t
�1"/ 11"/
Pt
p t �1 " 2=� t
�/"1
1 " 2
11"/
p t
Relation Between Aggregate Output and Aggregate Inputs
• Technically, there is no ‘aggregate production function’ in this model – If you know how many people are working, N, and
the state of technology, a, you don’t have enough information to know what Y is.
– Price frictions imply that resources will not be efficiently allocated among different inputs.
• Implies Y low for given a and N. How low?
• Tak Yun (JME) gave a simple answer.
Tak Yun Algebra
• Where:
Yt' � ;0
1Yi,tdi � ;
0
1AtNi,tdi
labor market clearing¥� AtNt
demand curve¥� Yt ;
0
1 Pi,tPt
"/di
� YtPt/ ;
0
1�Pi,t "/di
� YtPt/�Pt
' "/
Pt' q ;
0
1Pi,t
"/di"1/
� ¡�1 " 2 P t"/ � 2�Pt"1' "/¢
"1/
Calvo insight
Relationship Between Agg Inputs and Agg Output
• Rewriting previous equation:
• ‘efficiency distortion’:
Yt �Pt'
Pt
/Yt'
� pt'eatNt ,
pt' :t 1� 1 Pi,t � Pj,t , all i, j
Example of Efficiency Distortion
Pj,t �P1 0 t j t )
P2 ) t j t 1. pt' � Pt
'
Pt
/�
) � �1 " ) P2
P1"/ "1
/
) � �1 " ) P2
P11"/ 1
1"/
/
) � 0.5,/ � 10
-1 -0.5 0 0.5 1
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
logP1/P2
Collecting Equilibrium Conditions • Price setting:
• Intermediate good firm optimality and restriction across prices:
Kt � �1 " 7 // " 1
exp�At NtICt
At� *2Et=� t�1
/ Kt�1 (1)
Ft � 1 � *2Et=� t�1/"1Ft�1 (2)
�p t by firmoptimality¥KtFt
�
�p t by restriction across prices
1 " 2=� t�/"1
1 " 2
11"/
(3)
Equilibrium Conditions • Law of motion of (Tak Yun) distortion:
• Household Intertemporal Condition:
• Aggregate inputs and output:
• 6 equations, 8 unknowns:
• System under determined!
pt' � �1 " 2 1 " 2=� t�/"1
1 " 2
//"1
� 2=� t/
pt"1'
"1
(4)
1Ct
� *Et1Ct�1
Rt=� t�1
(5)
Ct � pt'eatNt (6)
7, Ct, pt',Nt,=� t,Kt,Ft,Rt
Tak Yun Distortion: a Closer Look• Law of motion, Tak Yun distortion:
p
∗t
= p
∗ (¯p
t
, p
∗t−1
)≡
2
4(1− q)
1− q ¯p
(#−1)t
1− q
! ##−1
+ q¯p#
t
p
∗t−1
3
5−1
— Distortion, p
∗t
, increasing function of lagged distortion, p
∗t−1
.— Current shocks a§ect current distortion via ¯p
t
only.
• Derivatives:
p
∗1
(¯p
t
, p
∗t−1
)= − (p∗
t
)2 #q ¯p#−2
t
2
4 ¯pt
p
∗t−1
−
1− q ¯p
(#−1)t
1− q
! 1
#−1
3
5
p
∗2
(¯p
t
, p
∗t−1
)=
p
∗t
p
∗t−1
!2
q ¯p#t
.
Linear Expansion of Tak Yun Distortion inUndistorted Steady State
• Linearizing about ¯pt
= ¯p, p
∗t−1
= p
∗:
dp
∗t
= p
∗1
( ¯p, p
∗) d
¯pt
+ p
∗2
( ¯p, p
∗) dp
∗t−1
,
where dx
t
≡ x
t
− x, for x
t
= p
∗t
, p
∗t−1
,
¯pt
.
• In an undistorted steady state (i.e., ¯pt
= p
∗t
= p
∗t−1
= 1) :
p
∗1
(1, 1) = 0, p
∗1
(1, 1) = q.
so that
dp
∗t
= 0× d
¯pt
+ qdp
∗t−1
! p
∗t
= 1− q + qp
∗t−1
• Often, people that linearize NK model ignore p
∗t
.
— Reflects that they linearize the model around aprice-undistorted steady state.
0.9 0.95 1 1.05
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
πt
Tack
Yun
dis
torti
on, p
* t
Current Period Tack Yun Distortion as a Function of Current InflationGraph conditioned on two alternative values for p*t−1 and θ = 0.75, ε = 6.00
lagged Tack Yun distortion = 1 (i.e., no distortion)lagged Tack Yun distortion = .9
Ignoring Tak Yun Distortion, a Mistake?
1950 1960 1970 1980 1990 2000 2010
0.98
0.99
1
1.01
1.02
1.03
Tack Yun Distortion and Quarterly US CPI Inflation (Gross)
Tack Yun distortioninflation
Underdetermined System • Not surprising: we added a variable, the
nominal rate of interest.
• Also, we’re counting subsidy as among the unknowns.
• Have two extra policy variables.
• One way to pin them down: compute optimal policy.
Ramsey-Optimal Policy • 6 equations in 8 unknowns…..
– Many configurations of the 8 unknowns that satisfy the 6 equations.
– Look for the best configurations (Ramsey optimal) • Value of tax subsidy and of R represent optimal policy
• Finding the Ramsey optimal setting of the 6 variables involves solving a simple Lagrangian optimization problem.
Ramsey Problem
max7,pt' ,Ct,Nt,Rt,=� t,Ft,Kt
E0!t�0
.
* t£ logCt " exp�At Nt1�I
1 � I
� 51t 1Ct
" Et*Ct�1
Rt=� t�1
� 52t 1pt'
" �1 " 2 1 " 2�=� t /"11 " 2
//"1
� 2=� t/
pt"1'
� 53t¡1 � Et=� t�1/"1*2Ft�1 " Ft ¢
� 54t �1 " 7 // " 1
Ct exp�A t NtI
eat � Et*2=� t�1/ Kt�1 " Kt
� 55t Ft1 " 2=� t/"11 " 2
11"/" Kt
� 56t¡Ct " pt'eatNt ¢¤
Solving the Ramsey Problem (surprisingly easy in this case)
• First, substitute out consumption everywhere
max7,pt' ,Nt,Rt,=� t,Ft,Kt
E0!t�0
.
*t£ logNt � logpt' " exp�A t Nt1�I
1 � I
� 51t 1pt'Nt
" Eteat*
pt�1' eat�1Nt�1
Rt=� t�1
� 52t 1pt'
" �1 " 2 1 " 2�=� t /"11 " 2
//"1
� 2=� t/
pt"1'
� 53t¡1 � Et=� t�1/"1*2Ft�1 " Ft ¢
� 54t �1 " 7 // " 1 exp�A t Nt
1�Ipt' � Et*2=� t�1/ Kt�1 " Kt
� 55t Ft1 " 2=� t/"11 " 2
11"/" Kt ¤
defines R
defines F
defines tax
defines K
Solving the Ramsey Problem, cnt’d • Simplified problem:
• First order conditions with respect to
• Substituting the solution for inflation into law of motion for price distortion:
max=� t,pt' ,Nt
E0!t�0
.
*t£ logNt � logpt' " exp�At Nt1�I
1 � I
� 52t 1pt'
" �1 " 2 1 " 2�=� t /"11 " 2
//"1
� 2=� t/
pt"1'¤
pt', =� t, Nt
pt' � *52,t�12=� t�1/ � 52t , =� t �
�pt"1' /"1
1 " 2 � 2�pt"1' /"1
1/"1
, Nt � exp " AtI � 1
pt' � �1 " 2 � 2�pt"1' �/"1 1
�/"1 .
Solution to Ramsey Problem
pt' � �1 " 2 � 2�pt"1' �/"1 1
�/"1
=� t �pt"1'pt'
Nt � exp " At1 � I
1 " 7 � / " 1/
Ct � pt'eatNt .
Eventually, price distortions eliminated, regardless of shocks
When price distortions gone, so is inflation.
Efficient (‘first best’) allocations in real economy
Consumption corresponds to efficient allocations in real economy, eventually when price distortions gone
Eventually, Optimal (Ramsey) Equilibrium and Efficient Allocations in Real Economy Coincide
2 4 6 8 10 12 14 16 18 200.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
p-st
ar
Convergence of price distortion
2 � 0.75, / � 10
pt' � �1 " 2 � 2�pt"1' �/"1 1
�/"1
• The Ramsey allocations are eventually the best allocations in the economy without price frictions (i.e., ‘first best allocations’)
• Refer to the Ramsey allocations as the ‘natural allocations’…. – Natural consumption, natural rate of interest, etc.
Linearizing around E¢cient Steady State• In steady state (assuming ¯p = 1, 1− n = #−1
# )
p
∗ = 1, K = F =1
1− bq, s =
#− 1
#, Da = t = 0, N = 1
• Linearizing the Tack Yun distortion, (4), about steady state:ˆ
p
∗t
= q ˆ
p
∗t−1
!ˆ
p
∗t
= 0, t large
• Denote the output gap in ratio form by X
t
:
X
t
≡C
t
A
t
exp
(− t
t
1+j
) = p
∗t
N
t
exp
(t
t
1+ j
),
so that (using x
t
≡ ˆ
X
t
):
x
t
= ˆ
N
t
+ dtt
1+j
NK IS Curve• The intertemporal Euler equation, (5), after substituting for C
t
in terms of X
t
:1
X
t
A
t
exp
(− t
t
1+j
) = bE
t
1
X
t+1
A
t+1
exp
(− t
t+1
1+j
) R
t
¯pt+1
1
X
t
= E
t
1
X
t+1
R
∗t+1
R
t
¯pt+1
,
where
R
∗t+1
≡1
bexp
(a
t+1
− a
t
−t
t+1
− tt
1+ j
)
then, use
dz
t
u
t
= ˆ
z
t
+ ˆ
u
t
,
\(u
t
z
t
)= ˆ
u
t
− ˆ
z
t
to obtain:ˆ
X
t
= E
t
[ˆ
X
t+1
−(
ˆ
R
t
− b̄pt+1
− ˆ
R
∗t+1
)]
NK IS Curve• Let:
R
t
≡ exp (rt
)
! ˆ
R
t
=d exp (r
t
)
exp (r)=
exp (r) dr
t
exp (r)= dr
t
≡ r
t
− r.
• Also,
R
∗t+1
= exp
(log R
∗t+1
)
! ˆ
R
∗t+1
=d exp
(log R
∗t+1
)
exp (log R
∗)= d log R
∗t+1
= log R
∗t+1
−
=r in e¢cient steady state, with ¯p=1z }| {log R
∗
• So, (letting r
∗t
≡ E
t
log R
∗t+1
)
E
t
(ˆ
R
t
− ˆ
R
∗t+1
)= dr
t
− E
t
d log R
∗t+1
= r
t
− r
∗t
.
NK IS Curve• Substituting
E
t
(ˆ
R
t
− ˆ
R
∗t+1
)= r
t
− r
∗t
into
ˆ
X
t
= E
t
[ˆ
X
t+1
−(
ˆ
R
t
− b̄pt+1
− ˆ
R
∗t+1
)], x
t
≡ ˆ
X
t
,
and usingb̄p
t+1
= pt+1
, when ¯p = 1,
we obtain NK IS curve:
x
t
= E
t
x
t+1
− E
t
[rt
− pt+1
− r
∗t
]
• Also,
r
∗t
= − log (b) + E
t
[a
t+1
− a
t
−t
t+1
− tt
1+ j
].
Linearized Marginal Cost
• Marginal cost (using da
t
= a
t
, dtt
= tt
because a = t = 0):
s
t
= (1− n)¯
w
t
A
t
,
¯
w
t
≡W
t
P
t
= exp (tt
)N
jt
C
t
! b̄w
t
= tt
+ a
t
+ (1+ j) ˆ
N
t
• Then,
ˆ
s
t
= b̄wt
− a
t
= (j+ 1)h
tt
j+1
+ ˆ
N
t
i= (j+ 1) x
t
Linearized Phillips Curve• Log-linearize equilibrium conditions, (1)-(3), around steadystate:
ˆ
K
t
= (1− bq) ˆ
s
t
+ bq(# b̄p
t+1
+ ˆ
K
t+1
)(1)
ˆ
F
t
= bq (#− 1) b̄pt+1
+ bq ˆ
F
t+1
(2)ˆ
K
t
= ˆ
F
t
+ q1−qb̄p
t
(3)
• Substitute (3) into (1)
ˆ
F
t
+q
1− qˆp
t
= (1− bq) ˆ
s
t
+ bq
(# b̄p
t+1
+ ˆ
F
t+1
+q
1− qˆp
t+1
)
• Simplify the latter using (2), to obtain the NK Phillips curve:
pt
= (1−q)(1−bq)q ˆ
s
t
+ bpt+1
The Equilibrium Conditions
x
t
= x
t+1
− [rt
− pt+1
− r
∗t
]
pt
= (1−q)(1−bq)q ˆ
s
t
+ bpt+1
ˆ
s
t
= (j+ 1) x
t
r
∗t
= − log (b) + E
t
ha
t+1
− a
t
− tt+1
−tt
1+j
i
Taylor rule (in deviation from steady state):
r
t
= ar
t−1
+ (1− a) [fppt
+ fx
x
t
]
Shocks:Da
t
= rDa
t−1
+ #a
t
, tt
= ltt−1
+ #tt
.
Solving�the�Model
st � atAt
�> 00 5
at"1At"1
�/t/tA
st � Pst"1 � .t* 0 0 01@ 1 0 00 0 0 00 0 0 0
= t�1xt�1rt�1rt�1'
�
"1 4 0 00 "1 " 1@ 1
@
�1 " ) C= �1 " ) Cx "1 00 0 0 1
= txtr trr t'
�
0 0 0 00 0 0 00 0 ) 00 0 0 0
= t"1xt"1rt"1rt"1'
�
0 0 00 0 00 0 00 0 0
st�1 �
0 00 00 0
"@E> " 1@�I �1 " 5
st
Et¡)0zt�1 � )1zt � )2zt"1 � *0st�1 � *1st ¢ � 0
Solving�the�Model
• Solution:
• As�before:
Et¡)0zt�1 � )1zt � )2zt"1 � *0st�1 � *1st ¢ � 0
st " Pst"1 " .t � 0.
zt � Azt"1 � Bst
)0A2 � )1A � )2I � 0,
F � �*0 � )0B P � ¡*1 � �)0A � )1 B¢ � 0
Cx � 0, C= � 1.5, * � 0.99, I � 1, > � 0.2, 2 � 0.75, ) � 0, - � 0.2, 5 � 0.5.
0 2 4 6
0.01
0.02
0.03
inflation
0 2 4 6
0.05
0.1
0.15
output gap
0 2 4 6
0.05
0.1
0.15
0.2nominal rate
natural nominal rateactual nominal rate
0 2 4 6
0.05
0.1
0.15
0.2natural real rate
0 2 4 60
0.5
1
log, technology
0 2 4 61
1.05
1.1
1.15
1.2
output
natural outputactual output
natural employmentactual employment
Dynamic Response to a Technology Shock
0 5 100
0.05
0.1
0.15
0.2employment
0 2 4 6
0.02
0.04
0.06
0.08
0.1inflation
0 2 4 6
0.050.1
0.150.2
0.25
output gap
0 2 4 6
0.05
0.1
0.15
0.2
0.25nominal rate
natural real rateactual nominal rate
0 2 4 6
0.05
0.1
0.15
0.2
0.25natural real rate
0 2 4 60
0.5
1preference shock
0 2 4 6-0.5
-0.4
-0.3
-0.2
-0.1
output
natural outputactual output
natural employmentactual employment
Dynamic Response to a Preference Shock
0 2 4 6-0.5
-0.4
-0.3
-0.2
-0.1
employment
Why Is Output Ine¢ciently High or LowSometimes?
• Brief answer: the Taylor rule sets the wrong interest rate(should be the natural rate).
• Households equate costs with the private benefit of working,W
t
:−u
N,t
u
c,t
= exp (tt
)C
t
N
jt
=W
t
P
t
.
— So, the only possible reason e¢ciency may not occur is ifW
t
/P
t
does not correspond to the actual marginal product oflabor.
• The relationship between W
t
/P
t
and labor productivity may beunderstood by studying the markup of price over marginal cost.
Price over Marginal Cost (Markup)
•i
th firm sets P
i,t
as a markup, µi,t
, over marginal cost
P
i,t
=
markupz}|{µ
i,t
×
marginal costz }| {(1− n)W
t
e
a
t
,
where we have been setting 1− n = #−1
# .
• In the flexible price version of the model the markup is trivial,µ
i,t
= #−1
# so
P
i,t
= P
t
=W
t
e
a
t
,
and W
t
/P
t
corresponds to the marginal product of labor.
Price over Marginal Cost (Markup)
• In the sticky price version of the model, the markup is morecomplicated.
— firms currently setting prices, do so to get the markup to be#/ (#− 1) on average in the current and future periods (seeearlier discussion)
— for firms not able to set prices in the current period, themarkup is
µi,t
=P
i,t−1
(1−n)Wt
e
a
t
.
• for these firms the markup moves inversely with a shock tomarginal cost.
• Need to look at some aggregate of all markups.
Price over Marginal Cost (Markup)• Weighted average markup, µ
t
:
µt
≡(Z
1
0
µ(1−#)i,t
di
) 1
1−#
=P
t
(1−n)Wt
e
a
t
=1
1−ne
a
t
e
tt
C
t
N
jt
=1
(1− n) e
tt
p
∗t
N
1+jt
=1
(1− n) p
∗t
e
− tt
1+j
N
t
!1+j
.
• Consider the output gap:
X
t
=C
t
e
a
t
−tt
/(1+j)=
p
∗t
e
a
t
N
t
e
a
t
−tt
/(1+j)= p
∗t
N
t
e
−tt
/(1+j)
• Use this to substitute into the markup
µt
=(p∗
t
)j
(1− n)X
−(1+j)t
.
µt
moves inversely with the output gap.
Price over Marginal Cost (Markup)• The preceding implies:
−u
N,t
u
c,t
source of e¢ciencyz}|{=
W
t
P
t
=e
a
t
(1− n) µt
= e
a
t (p∗t
)−jX
1+jt
.
• Suppose p
∗t
= 1.• When the output gap is high, X > 0, then the markup is lowand the real wage exceeds e
a
.
— There is too much employment in this case because the privatebenefit to the workers of working exceeds the actual benefits(the di§erence comes out of lump sum profits).
• Why are firms willing to produce with a low markup? Becauseon average it is high (µ = #/ (#− 1)) and so cutting it (not toomuch!) allows them to still get some profits out of the workers.
• Interpretation: model implies markups are countercyclical.— Nekarda and Ramey (‘The Cyclical Behavior of the Price-CostMarkup’, UCSD 2013) argue that the evidence does notsupport this.
Wrap Up• Suppose monetary policy puts the interest rate below thenatural rate, driving up the output gap.
• The low interest rate gives people an incentive to spend more.• Marginal cost rises and since prices sticky, markups
µt
=P
t
MC
t
,
go down (MC
t
= (1− n)W
t
/e
a
t).• Low markup means high wage when there is a monetary shock,encouraging more work.
• Profits are reduced, but presumably remain positive:
profits = P
i,t
Y
i,t
− (1− n)W
t
N
i,t
= P
i,t
Y
i,t
− e
a
t
MC
t
×N
i,t
= [Pi,t
−MC
t
]Yi,t
=hµ
i,t
− 1
iMC
t
× Y
i,t
