As Under Price Rigidity - October 2007

8/2/2019 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

4


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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)

7


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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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9/37

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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10/37

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.

10


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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.:

11


12/37

),;( 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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13/37

=> 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

=

+

13


14/37

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


),;( 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 .

15


16/37

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

16


17/37

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

17


18/37

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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19/37

[ ]))(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( .

20


21/37

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

21


22/37

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:

22


23/37

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

23


24/37

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)

24


25/37

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


26/37

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,

26


27/37

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:

27


28/37

=

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.

28


29/37

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.

29


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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

30


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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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32/37

=> 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


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


),;( 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:

33


34/37

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)

34


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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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36/37

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

Documents

As Under Price Rigidity - October 2007