as under price rigidity - october 2007
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THE NEW KEYNESIAN
MACROECONOMICS: AGGREGATE SUPPLY
Main feature:
The degree to which prices are determined in
advance and its consequence for the output-
inflation trade-off.
I - Households(Differentiated products and Money in the Utility
Function):
Max0
0
( ; ) ( ; ) ( ( ); )n
t to t t t t t
tt
ME u C h j dj
P
=
+
s.t.
+=+++++
n
tt
n
t
ttttttttttttttt
djjdjjhjw
BiBBifBMMTPCP
00
111**
1,1
*
1
)()()(
)1()1(
Bt = bond holdings at the end of date t
(denominated in the domestic currency)
Bt* = bond holdings at the end of date t
(denominated in the foreign currency)
Mt = money holdings at the end of date t
Pt = aggregate domestic price level
Ct = consumption index
ht(j) = supply of labor of type j by the
representative individual
wt(j) = Nominal wage rate of labor of type j
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it* = world interest rate
t(j) = Nominal profit of firm j (domestic) t = exchange rate in period t
Tt = government lump-sum transfers t = preference shock
ft,t+1 = forward exchange rate (the price paid in
period t in terms of domestic currency, of one unit
foreign currency to be delivered in period t+1)
Interest parity (can be obtained by obtaining the FOC
with respect to Bt and Bt*, without borrowing
constraints of any kind, and then dividing one FOC by
the other) :
t
tt
tt
fii
1,* )1(1++=+
There is a constant elasticity of substitution betweenany two goods in the economy. Ct is a composite of allthese goods.
1
0
11
*
1
)()(
+=
n
nttt djjcdjjcC
ct = goods produced at home
ct* = goods produced abroad (imports)
Corresponding price index (the minimum expenditurethat buys one unit of the consumption index):
+= 1
1
0
11*1 ))(()(
n
ntttt djjpdjjpP
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pt(j) = price of domestic good j (in domestic
currency)
pt*(j) = price of foreign good j (in foreigncurrency)
[The assumption is that the law of one price (PPP)
prevails,
1111 111 * 1 * 1 * 1
0 0
1( ) ( ) ( ) ( )
n n
t t t t t t t t t n n
t
P p j dj p j dj P p j dj p j dj
= + = = +
+= ==
+=
1
1
11*
0
1*
1
1
11*
0
1
)())(1
(
))(()(
nt
n
t
t
tt
ntt
n
tt
djjpdjjpP
djjpdjjpP
Taking a log approximation (where a hat (^) over a
variable indicates log deviation from steady state)
yields:
tt nP
= )1(
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A one percent movement in the exchange rate
will have an effect on domestic consumers
prices equal to the share of imports in
consumption.
Introduction of non-traded goods would allow
for deviations from PPP. Alternatively, a
fraction of firms set prices in the buyers
currency, or Local Currency Pricing (LCP).
Accordingly, let s represent the fraction of
foreign firms who set prices in domestic
currency and use p to indicate that a price is
fixed, we have:
+
+
+= =
++=
1
1
11*
0
1**
1
1
1
)1(
1*
0
)1(1*1
)()(
)}({)()(
nt
n
tttt
snntt
n snn
nttt
djjpdjjpP
djjpdjjpdjjpP
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It is useful to compare the case s=0, full
Producers Currency Pricing (PCP, which
amounts to PPP), with LCP. If PPP prevails
there is full pass through of exchange rate
movements to import prices.
tt nP
= )1( .
Whereas, if LCP prevails,
tt snnP
+= )))1((1( .
That is, the degree of Pass Through is
lessened. ]
Now lets go back to the PPP assumption.
The First-Order Conditions
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Labor:
)());(( jwjhv tttth =
Consumption:
ttttc PCu =);(
= Lagrange multiplierSubstituting for t:
t
t
ttc
tth
Pjw
Cujh )(
);());(( =
(1) (labor supply)
The First-Order Inter-temporal Condition:
111
)1();(
);(
+++
+=tt
t
t
ttct
ttc
PE
Pi
CuE
Cu
)()1(
);(
);(
111 +++
+=t
t
tt
ttct
ttc
P
PEi
CuE
Cu
(2) (consumption-saving
choice)
The Fisher equation: )1();();(
11
t
ttct
ttc rCuE
Cu+
++
=>
1
)1(1
11
+
+=++
=+tt
t
t
t
t
tPE
Pi
ir
)()1(1
11
1+
+=++
=+t
t
tt
t
t
tP
PEi
ir
Demand for a variety:
t
t
t
t CP
jpjc
=
)()( (Dixit-Stiglitz demand for good j)
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Government budget:
t
tt
tP
MMT 10
+=
(Government income, seigniorage:t
tt
P
MM 1,
is rebated to the public in the form of a lump sum
transfer Tt).
II Producers
( ))()( jhfAjy ttt = (The production function)
At = random productivity shock
Variable cost of supplying:
=
t
tttt
A
jyfjwjhjw
)()()()(
1
Nominal Marginal cost:tt
tt
t
ttt
AAjyfjw
jyjhjwjx 1)()(
)())()(()(
'1 ==
Real marginal cost:ttt
t
t
t
t
tPAA
jyfjw
P
jxjs
1)()(
)()(
'1
== (3)
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Substituting (3) in (1) and assuming a symmetric
equilibrium (dropping the index j because of the
symmetry assumption):
tt
t
ttc
tth
tAA
yf
Cu
hs
1
);(
);( '1
=
=>tt
t
ttc
ttth
tttttAA
yf
Cu
AyfACys
1
);(
));/((),;,(
'11
=
World demand for the firm j product:
W H F H F
t t t t t Y Y Y C C = + = +
An index for all the goods produced around the world.
Producer j demand function:
=
t
tW
ttP
jpYjy
)()(
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III. The Labor Market
The market for each type of goods-specific skill of
labor service is characterized by workers as wage-
takers and producers as wage-makers, as in the
monopsony case.
Figure 1 describes equilibrium in one such market.
The downward-sloping, marginal-productivity curve,
is the demand for labor. Labor supply is implicitly
determined by the utility-maximizing condition for h.
The upward-sloping marginal factor cost curve is the
marginal cost change from the producer point of view.
It lies above the supply curve because, in order to elicit
more hours of work, the producer has to offer a higher
wage not only to that (marginal) hour but also to all the
(intra-marginal) existing hours. Equilibrium
employment occurs at a point where the marginal factor
costs is equal to the marginal productivity (point A).
Equilibrium wage is shown at point B, with the
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worker's real wage marked down below her marginal
product by a distance AB.1
Full employment obtains because workers are offered a
wage according to their supply schedule. This is why
our Phillips curve will be stated in terms of excess
capacity (product market version) rather than
unemployment (labor market version).
In fact, the model can also accommodate unemployment
by introducing a labor union, which has monopoly
power to bargain on behalf of the workers with the
monopsonistic firms over the equilibrium wage. In such
case, the equilibrium wage will lie somewhere between
the labor supply and the marginal productivity curves,
and unemployment can arise so that the labor market
version of the Phillips curve can be derived as well. To
simplify the analysis, we assume in this paper that the
workers are wage-takers.1 In the limiting case where the producers behave perfectly competitive in the labor market, the
real wage becomes equal to the marginal productivity of labor and the marginal cost of labor
curve is not sensitive to output changes. Thus, with a constant mark-up1
, the Phillips curve
becomes flat, i.e., no relation exists between inflation and excess capacity.
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Figure 1: The Labor Market
Equilibrium
h
W/P
Marginal Factor CostLabor Supply
Marginal Productivity
Mark
Downwage
Marginal
productA
B
Note: wages are perfectly flexible.
Price SettingA fraction of the firms set their prices flexibly at p1t,supplying y1t.
A fraction 1- of the firms set their prices one period inadvance (in period t-1) at p2t, supplying y2t.
The flexible price producer (type-1 firms) sets a
constant mark-up,
1
=
>1 ,
above the actualmarginal cost.:
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),;( 11
tttt
t
t ACysP
p= (4)
The producer who sets the price one period in advance
(type-2 firms), charging p2t . The objective function,
expected discounted profit, is:
( )
+
=
+
t
tt
W
t
tt
W
tt
t
ttttt
t
tA
PpYfwPYp
iEhwyp
iE
211
2
1
122
1
11
1
1
1.
The maximization problem:
+
t
tt
W
t
tt
W
tt
t
tp A
PpYfwPYp
iEMax
t
211
2
1
11
1
2
The FOC:
0)1(1
11
22'
12
1
1 =
+ +
t
tt
W
t
t
tt
W
tttWtt
t
tA
PpY
A
PpYfwPYp
iE
(5)
Substituting
=
=
t
t
tt
t
tt
t
ttc
ttth
ttttA
yf
PA
w
AA
yf
Cu
AyfACys
'1'1
11
);(
));/((),;,(
in (5), we get
[ ] 0),;,()1(1
1 11222
1
1 =
+
+
+
ttW
tttttt
W
tt
t
t PpYACysPYp
i
E
=> 01
),;,()1(1
1 11222
1
1 =
+
+
tttttttt
W
t
t
t PpACysPpYi
E
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=> 0),;,()1(1
12
211
2
1
1 =
+
+
tttt
t
t
tt
W
t
t
t ACysP
pPpY
iE (6)
A weighted average of the deviation of relative price
from the marked up marginal costs is set equal to zero.Where,
+
+
1121
)1(1
1tt
W
t
t
t PpYi
can be viewed as a weight at
a given state of nature.
Aggregate price index:
[ ]{ } ++= 1
1
1*1
2
1
1 )1()1( ttttt pnppnP
Potential Output
The potential (or the Natural level of ) output (Y tN) is
the output level under perfect price flexibility ( = 1).Using (4) and (6) with = 1 we get:{ }
1
11 *1 11
( , ; , )
(1 )
n ntt t t t
t t t
ps Y C A
np n p
=
+
If there are no capital flows (closed capital account),
then CtN = Yt
N. In this case the natural output is defined
by:
{ }
1
11 *1 11
( , ; , )
(1 )
n ntt t t t
t t t
ps Y Y A
np n p
=
+
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If there are no capital flows and no commodity trade,
then the economy is completely closed (A closed capital
account and closed current account), then n = 1 and CtN
= YtN
. The natural output is defined by:
),;,(1 ttn
t
n
t AYYs =
The natural output is independent of monetary
policy.
Note that the efficient output, ),;,(1**
tttt AYYs = is largerthan the natural output under monopolistic competition.
IV. The Aggregate Supply
The aggregate supply is a set of 6 equations:
[ ]{ } ++= 11
1*1
2
1
1 )1()1( ttttt pnppnP
),;( 11
tttt
t
t ACysP
p=
0),;,()1(1
12
211
2
1
1 =
+
+
tttt
t
t
tt
W
t
t
t ACysP
pPpY
iE
=
t
tW
ttP
pYy 11
=
t
tW
ttP
pYy 22
1
21
11
)1(
+=
ttt yyY
There are 6 endogenous variables that are determined in
the aggregate supply block of the model:
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THE Quantities-- ty1 , ty2 , tY
THE Nominal prices--- tp1 , tp2 , tP.
The Solution technique: log-linearization of the 6
aggregate-supply equations around the no shock steady
state.
IVa. The No Shock Steady State
Assume 1)1( * =+ r
Consider a deterministic steady-state, where 0=t and
Yttttt YCCppAA
====== ,,,,1 ** .
Log-linearization of equation (5),
tt
t
ttc
ttth
tttttAA
yf
Cu
AyfACys
1
);(
));/((),;,(
'11
=
, around the steady-
state point yields:
tc
h
t
c
h
ttt
AA
A
AyfCu
Ayf
A
AyfCu
Ayf
Cys
+
+
+=
]1
)/()0;(
)0);/((log[
]1
)/()0;(
)0);/((log[
_
__'1
__1
_
__'1
__1
1
(7)
Where: pw += ,A
y
v
fv
h
hh
w
'1
= ,A
y
f
fp '1
''1
= , andc
cc
u
Cu=1 .
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The expression for the real marginal cost, evaluated at
the natural level of output, is:
t
c
h
t
c
h
N
t
N
t
N
t
AA
A
AyfCu
Ayf
A
AyfCu
Ayf
CYs
+
+
+=
]1
)/()0;(
)0);/((log[
]1
)/()0;(
)0);/((log[
_
__'1
__1
_
__'1
__1
1
(7)
Subtracting (7) from (7):)()(
1 N
tt
N
tt
N
tt CCYyss += (7)
Log-linearizing (4), ),;( 11
tttt
t
t ACysP
p= , around the steady-
state yields:
ttt sPp
1 +=
Subtracting the (log-linearized version of the ) equation
evaluated at the natural level of output, substitutingN
t
N
t Pp =1 , and using (7) yields:
)()( 111N
tt
N
tttt CCYyPp ++= (8)
We go through a similar procedure for equation (6)
N
tt
tttt
t
t
tt ACysP
pE
=
=
0),;,( 22
1 (in this case the relevant part
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of the equation is the term inside the square brackets)
and get:
)]()([ 1
212
N
tt
N
ttttt
CCYyPEp ++=
(9)
Log-linearizing the price index yields:
))(1(])1([*
21 ttttt pnppnP +++= (10)
Assume now that in steady-state there is zero inflation
; then:
)log()log()log( 1111 tttt pppp ==
)log()log()log( tttt PPPP ==)log()log()log( 2222 tttt pppp ==
)log()log()log(****
tttttttt pppp ==+
The rate of inflation rate is given by:
1
11
1 log
=
= tt
t
t
t
tt
t PPP
P
P
PP
=> tttttt PEPE 11 = (the surprise rate of
inflation)
The real exchange rate is defined as:
t
tt
tP
Pe
*=
IV c. Deriving the Aggregate Supply Relationship
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Log-linearizing the (Dixit-Stiglitz) demand for the good
produced by firm j,
=
t
tW
ttP
jpYjy
)()( :
][ tjtw
tjt PpYy =
, with
FtN
t
w
t YnYnY)1( +=
With symmetry between firms of type 1 (flexible price
firms) and between firms of type 2 (sticky price firms),we have:
][ 11 ttw
tt PpYy =
][ 22 ttw
tt PpYy =
Substituting for),(
21 tt
yy
in (8) and (9):
)(1
)(1
1
1
N
tt
N
t
w
ttt CCYYPp ++
+=
(8)
++
+=
)(
1)(
1
1
112
N
tt
N
t
w
ttttt CCYYEPEp
(9)
=> tttpEp
112
= (11)
From (10):
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[ ]))(1(])1([))(1(])1([
*
211
*
2111
ttttt
tttttttttt
pnppnE
pnppnEPEP
+++
+++==
Using (11)ttt pEp 112 =
andtttt pEpE 1121 =
yields:
)]())[(1(][*
1
*
211 tttttttttt pEpnppnE +++= (12)
From the definition of the real exchange rate we get:
tttt PPe * += => tttt PeP * +=+
Substituting in (12) yields:
])[1(])[1(][ 11211 ttttttttttt PEPneEenppnE ++=
=> ])[1(][)( 1211 tttttttt eEenppnEn += (13)
From (10), tp2 is given by:
])1([)1(
1
12 tttt enpnPnn
p
=
Substituting tp2 in (13); we have:
+
+
= )(1
1
1
1][
1111 ttttttttt eEee
n
nPpE
+
+
= )(1
1][
1111 ttttttttt eEee
n
nPpE
Using (8) )()( 111N
tt
N
tttt CCYyPp ++= to substitute for tt Pp 1
in this expression, we obtain the open-economy
Aggregate Supply (Phillips) Curve:
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+
++
+=
tttN
tt
N
t
w
tttt eEen
nCCYYE
1
11)(
1)(
111
1
1
But because that the world output is divided between
the domestic and foreign world as:
f
t
h
t
w
t YnYnY)1( +=
we have
=> ))(1()( Ntf
t
N
t
h
t
N
t
w
t YYnYYnYY +=
and
+
++
+
++
=
ttN
tt
N
t
f
t
N
t
h
tttt Een
nCCYY
nYY
nE
1
11)(
1)(
1
)1()(
11
1
1
[If LCP prevails,
tt snndomesticP
++= )))1((1( .
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The Pass Through from exchange rate fluctuations to
domestic inflation is lessened, and the effect of the real
exchange rate on surprise inflation is:
+ ttt eEe
n
snn
1
1))1((11
.
s = The fraction of foreign producers which
preset prices in a domestic buyers currency.]
IV.1 Perfect Capital Mobility
If capital is perfectly mobile, then the domestic agent
has a costless access to the international financial
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market. As a consequence, household can smooth
consumption similarly in the rigid price and flexible
price cases.
=> Ntt CC =
The Aggregate Supply curve becomes:
+
+
++
= tttN
t
f
t
N
t
h
tttt eEen
nYY
nYY
nE
1
11)(
1
)1()(
1111
IV. 2 Closed Capital Account
If the domestic economy does not participate in the
international financial market, then there is no
possibility of consumption smoothing, and we have that:
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NtYt
NtCttYttCt YPCPYPCP ;
==
=
+=
1
1
1
0
1
1
1
0
11*1
)(
))(()(
djjpP
djjpdjjpP
ttY
n
nttttC
In this case, the Aggregate Supply Curve becomes:
+
+
+++
=
tttN
t
f
t
N
t
h
tttt eEen
nYY
nYY
nE
1
11)(
1
)1()(
111
1
1
IV.4 Closed Economy
If both the capital and trade accounts are closed, then
the economy is an autarky, completely isolated of the
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rest of the world. In this case, all the goods in the
domestic consumption index are produced domestically,
which means that n = 1.
The Aggregate Supply Curve becomes:
)(11
1
1
N
t
h
tttt YYE
++
=
IV.4 Slopes (Sacrifice Ratios)
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The slope of the Aggregate Supply Curve under each
scenario is:
(i))1)(1(
1
+
=n
(perfect capital mobility)
(ii))1)(1(
)( 1
2
+
+=
n(closed capital account)
(iii))1)(1(
)(1
3
+
+=
(closed economy)
It can be seen that 321
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V. Revised Aggregate Supply: Accountingto the Assumption that Producers Are
Wage-makers.
II Producers
( ))()( jhfAjy ttt = (The production function)
At = random productivity shock
Variable cost of supplying:
=
t
tttt
A
jyfjwjhjw
)()()()(
1
Monopsonistic-Wage-adjusted Nominal Marginal cost:
)(
))()(()(
jy
jhjwjx
t
tt
t
=
However, we assume the workers are wage takers and
the producers are wage makers, so )( jwt is a function of)( jht :
This relation is characterized by equation (1):
);(
));(()(
ttc
tth
ttCu
jhPjw
=
As a result,
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tt
t
t
t
c
hh
t
c
h
t
t
t
t
t
t
t
t
t
t
tt
t
AA
jyf
A
jyf
u
vP
u
vP
jhjdy
jdh
jh
jw
jdy
jdhjw
jy
jhjwjx
1)()(
)()(
)(
)((
)((
)(
)()(
)(
))()(()(
'11
+=
+=
=
And the wage-adjusted real marginal cost is:
tt
t
t
t
c
hh
c
h
t
t
tAA
jyf
A
jyf
u
v
u
v
P
jxjs
1)()()()(
'11
+== (3)
Substituting (3) in (1) and assuming a symmetric
equilibrium (dropping the index j because of thesymmetry assumption):
=>
tt
t
t
t
ttc
ttthh
ttc
ttth
tttttAA
yf
A
yf
Cu
Ayfv
Cu
AyfvACys
1
);(
));/((
);(
));/((),;,(
'1111
+=
World demand for the firm j product:
W H F H F
t t t t t Y Y Y C C = + = +
An index for all the goods produced around the world.
Producer j demand function:
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=
t
tW
ttP
jpYjy
)()(
III. The Labor Market
The market for each type of goods-specific skill of
labor service is characterized by workers as wage-
takers and producers as wage-makers, as in the
monopsony case.
Figure 1 describes equilibrium in one such market.
The downward-sloping, marginal-productivity curve,
is the demand for labor. Labor supply is implicitly
determined by the utility-maximizing condition for h.
The upward-sloping marginal factor cost curve is the
marginal cost change from the producer point of view.
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It lies above the supply curve because, in order to elicit
more hours of work, the producer has to offer a higher
wage not only to that (marginal) hour but also to all the(intra-marginal) existing hours. Equilibrium
employment occurs at a point where the marginal factor
costs is equal to the marginal productivity (point A).
Equilibrium wage is shown at point B, with the
worker's real wage marked down below her marginalproduct by a distance AB.2
Full employment obtains because workers are offered a
wage according to their supply schedule. This is why
our Phillips curve will be stated in terms of excess
capacity (product market version) rather than
unemployment (labor market version).
In fact, the model can also accommodate unemployment
by introducing a labor union, which has monopoly
power to bargain on behalf of the workers with the2 In the limiting case where the producers behave perfectly competitive in the labor market, the
real wage becomes equal to the marginal productivity of labor and the marginal cost of labor
curve is not sensitive to output changes. Thus, with a constant mark-up1
, the Phillips curve
becomes flat, i.e., no relation exists between inflation and excess capacity.
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monopsonistic firms over the equilibrium wage. In such
case, the equilibrium wage will lie somewhere between
the labor supply and the marginal productivity curves,and unemployment can arise so that the labor market
version of the Phillips curve can be derived as well. To
simplify the analysis, we assume in this paper that the
workers are wage-takers.
Figure 1: The Labor Market
Equilibrium
h
W/P
Marginal Factor CostLabor Supply
Marginal Productivity
MarkDown
wage
Marginal
product
A
B
Note: wages are perfectly flexible.
Price Setting
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A fraction of the firms set their prices flexibly at p1t,supplying y1t.
A fraction 1- of the firms set their prices one period inadvance (in period t-1) at p2t, supplying y2t.
The flexible price producer (type-1 firms) sets a
constant mark-up,
1= >1 ,above the actualmarginal cost.:
),;( 11 ttttt
t ACysPp = (4)
The producer who sets the price one period in advance
(type-2 firms), charging p2t . The objective function,
expected discounted profit, is:
( )
+=
+
t
tt
W
t
tt
W
ttt
tttttt
t A
PpYfwPYp
iEhwyp
iE
211
21
1221
1 1
1
1
1
.
The maximization problem:
+
t
tt
W
t
tt
W
tt
t
tp A
PpYfwPYp
iEMax
t
211
2
1
11
1
2
When we take the first order condition, we need to bare
in mind that );());((
)(ttc
tth
ttCu
jhPjw
= ,
And ).()(21
t
tt
w
t
tA
PpYfjh
= =
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=> 0),;,()1(1
12
211
2
1
1 =
+
+
tttt
t
t
tt
W
t
t
t ACysP
pPpY
iE (6)
A weighted average of the deviation of relative price
from the marked up marginal costs is set equal to zero.Where,
+
+
1121
)1(1
1tt
W
t
t
t PpYi
can be viewed as a weight at
a given state of nature.
Aggregate price index:
[ ]{ } ++= 1
1
1*1
2
1
1 )1()1( ttttt pnppnP
Potential Output
The potential (or the Natural level of ) output (Y tN) is
the output level under perfect price flexibility ( = 1).Using (4) and (6) with = 1 we get:{ }
1
11 *1 11
( , ; , )
(1 )
n ntt t t t
t t t
ps Y C A
np n p
=
+
If there are no capital flows (closed capital account),
then CtN = Yt
N. In this case the natural output is defined
by:
{ }
1
11 *1 11
( , ; , )
(1 )
n ntt t t t
t t t
ps Y Y A
np n p
=
+
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If there are no capital flows and no commodity trade,
then the economy is completely closed (A closed capital
account and closed current account), then n = 1 and CtN
= YtN
. The natural output is defined by:
),;,(1 ttn
t
n
t AYYs =
The natural output is independent of monetary
policy.
Note that the efficient output, ),;,(1**
tttt AYYs = is largerthan the natural output under monopolistic competition.
IV. The Aggregate Supply
The aggregate supply is a set of 6 equations:
[ ]{ } ++= 11
1*1
2
1
1 )1()1( ttttt pnppnP
),;( 11
tttt
t
t ACysP
p=
0),;,()1(1
12
211
2
1
1 =
+
+
tttt
t
t
tt
W
t
t
t ACysP
pPpY
iE
=
t
tW
ttP
pYy 11
=
t
tW
ttP
pYy 22
1
21
11
)1(
+=
ttt yyY
There are 6 endogenous variables that are determined in
the aggregate supply block of the model:
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THE Quantities-- ty1 , ty2 , tY
THE Nominal prices--- tp1 , tp2 , tP.
The Solution technique: log-linearization of the 6
aggregate-supply equations around the no shock steady
state.
IVa. The No Shock Steady State
Assume 1)1( * =+ r
Consider a deterministic steady-state, where 0=t and
Yttttt YCCppAA
====== ,,,,1 ** .
Notice that the assumption 0=t will not cause trouble if
we actually let redefine the system in terms of )exp( t .
Log-linearization of equation (5),
tt
t
t
t
ttc
ttthh
ttc
ttth
tttttAA
yfA
yfCu
Ayfv
Cu
AyfvACys 1
);(
));/((
);(
));/((),;,(''1111
+=
around the steady-state point yields:
ttt
t
tt
ttt
As
A
A
ACys
s
ACys
Cys
+
+
+=
)],0;,(
)],0;,(
'
__
__
1
(7)
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Where:
s
yy
AA
jyf
A
jyf
u
v
Af
u
Afv
tt
t
t
t
c
hh
c
hh
+
=
1)()(
1
1 '11
'1
'1
, and
c
cc
u
Cu=1 .
The expression for the real marginal cost, evaluated at
the natural level of output, is:
ttt
t
tt
N
t
N
t
N
t
As
A
A
ACys
s
ACys
CYs
+
+
+=
)],0;,('
)],0;,('
__
__
1'
(7)
Subtracting (7) from (7):)()('' 1 Ntt
N
tt
N
tt CCYyss += (7)
Log-linearizing (4), ),;(' 11
tttt
t
t ACysP
p= , around the steady-
state yields:
ttt sPp
1 +=
Subtracting the (log-linearized version of the ) equation
evaluated at the natural level of output, substitutingN
t
N
t Pp =1 , and using (7) yields:
)()( 111N
tt
N
tttt CCYyPp ++= (8)
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We go through a similar procedure for equation (6)
N
tt
tttt
t
t
tt ACysP
pE
=
=
0),;,(' 22
1 (in this case the relevant part
of the equation is the term inside the square brackets)
and get:
)]()([ 1212N
tt
N
ttttt CCYyPEp ++=
(9)
Log-linearizing the price index yields:
))(1(])1([ *21 ttttt pnppnP +++= (10)
Assume now that in steady-state there is zero inflation
; then:
)log()log()log( 1111 tttt pppp ==
)log()log()log( tttt PPPP ==)log()log()log( 2222 tttt pppp ==
)log()log()log( **** tttttttt pppp ==+
The rate of inflation rate is given by:
1
11
1 log
=
= tt
t
t
t
tt
t PPP
P
P
PP
=> tttttt PEPE
11 = (the surprise rate ofinflation)
The real exchange rate is defined as:
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t
tt
tP
Pe
*=
37