Download Aggregate Supply Under Price Rigidity

1
THE NEW KEYNESIAN MACROECONOMICS:
AGGREGATE SUPPLY
Main feature:
The degree to which prices are determined in
advance and its consequence for the outputinflation trade-off.
I - Households
(Differentiated products and Money in the Utility
Function):

n


M
E o   t u(C t ; t )  ( t 1 ; t )    (ht ( j );  t )dj 
0
Pt
t 0


*
*
*
Pt Ct  Pt Tt  M t  M t 1   t B t  f t 1,t (1  it 1 ) B t 1  Bt  (1  it 1 ) Bt 1
Max
s.t.
n
n
0
0
  wt ( j )ht ( j )dj    t ( j )dj
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) = wage rate of labor of type j
it* = world interest rate
2
t(j) = 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) :
1  it  (1  it* )
f t ,t 1
t
There is a constant elasticity of substitution  between
any two goods in the economy. Ct is a composite of all
these goods.

 1
 1
 n
  1
1
Ct    ct ( j )  dj   ct* ( j )  dj 
0
n


ct = goods produced at home
ct* = goods produced abroad (imports)
Corresponding price index (the minimum expenditure
that buys one unit of the consumption index):
Pt    pt ( j )
 0
n
1
1
dj   ( t p ( j ))
n
*
t
1
dj 

1
1
pt(j) = price of domestic good j (in domestic
currency)
3
pt*(j) = price of foreign good j (in foreign currency)
[The assumption is that the law of one price (PPP)
prevails,
Pt    pt ( j )1 dj    t pt* ( j )1 dj 
 0

n
n
1
1
1
Taking a log approximation (where a hat (^) over a
variable indicates log deviation from steady state)
yields:

1
1
 n1
 1
  t Pt*   t 
pt ( j )1 dj   pt* ( j )1 dj 
n
 0 t


P t  (1  n)  t
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 buyer’s currency, or Local
4
Currency Pricing (LCP). Accordingly, let s
represent the fraction of foreign firms who set
prices in domestic currency and use
to indicate
p
that a price is fixed, we have:
Pt    pt ( j )
 0
n
1
dj  
n  (1 n ) s
n
*
t
p ( j)
1
1
dj  
n  (1 n ) s
 t p ( j)
*
t
1
dj 

1
1
1
n
1
1
  t Pt   t   pt* ( j )1 dj   pt* ( j )1 dj 
 0

n
*
It is useful to compare the case s=0, full Producer’s
Currency Pricing (PCP, which amounts to PPP),
with LCP. If PPP prevails there is full pass through
of exchange rate movements to import prices.


P t  (1  n)  t .
Whereas, if LCP prevails,


P t  (1  (n  (1  n) s))  t .
5
That is, the degree of Pass Through is lessened. ]
Now let’s go back to the PPP assumption.
The First-Order Conditions
Labor:
vh (ht ( j );  t )  t wt ( j )
Consumption:
u c (Ct ;  t )  t Pt
 = Lagrange multiplier
Substituting for t:
 h (ht ( j );  t ) wt ( j )

u c (Ct ;  t )
Pt
(1) (labor supply)
The First-Order Inter-temporal Condition:
uc (C t ; t )
Pt
  (1  i t )
E t uc (C t 1 ; t 1 )
E t Pt 1
choice)
(2)
(consumption-saving
6
uc (C t ; t )
  (1  rt ) =>
E t uc (C t 1 ; t 1 )
The Fisher equation:
1  rt 
1  it
Pt
 (1  i t )
1 t
E t Pt 1
Demand for a variety:

 p ( j) 
 C t
ct ( j )   t
 Pt 
(Dixit-Stiglitz demand for good j)
Government budget:
0  Tt 
M t  M t 1
Pt
(Government income, seigniorage:
M t  M t 1
,
Pt
is rebated to the public in the form of a lump sum
transfer Tt).
II –Producers
yt ( j )  At f ht ( j ) 
(The production function)
At = random productivity shock
Variable cost of supplying:
7
 y ( j) 

wt ( j )ht ( j )  wt ( j ) f 1  t
 At 
Nominal Marginal cost:
Real marginal cost:
st ( j ) 
xt ( j ) 
wt ( j )ht ( j )
'  y ( j)  1

 wt ( j ) f 1  t
yt ( j )
 At  At
xt ( j )
'  y ( j)  1

 wt ( j ) f 1  t
Pt
A
 t  At Pt
(3)
Substituting (3) in (1) and assuming a symmetric
equilibrium (dropping the index j because of the
symmetry assumption):
st 
=>
 h (ht ; t ) 1'  y t
f 
uc (C t ; t )
 At
 1

 At
 h ( f 1 ( y t / At );  t ) 1'  y t
s t ( y t , C t ; t , At ) 
f 
uc (C t ; t )
 At
 1

 At
World demand for the firm j product:
YtW  Yt H  Yt F  C t  C tF
An index for all the goods produced around the world.
Producer j demand function:
 p ( j) 

y t ( j )  Yt  t
P
 t 
W

8
III. The Labor Market
The market for each type of goods-specific skill of
labor service is characterized by workers as wagetakers 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.
implicitly
determined
by
Supply of labor, Sh, is
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
9
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 (inra-marginal)
existing hours. Equilibrium employment occurs at a
point where the marginal factor costs is equal to the
marginal productivity. Equilibrium wage is shown at
point B, with the worker'’ 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
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-up
no relation exists between inflation and excess capacity.

, the Phillips curve becomes flat, i.e.,
 1
10
monopsonistic firms over the equilibrium wage. In such
case, the equilibrium wage will lie somewhere between
Sh and M Ph, 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.
W/P
Figure 1: The Labor Market
Equilibrium
Marginal Factor Cost
Marginal
product Mark
Down
wage
Labor Supply
Marginal Productivity
h
Note: wages are perfectly flexible.
Price Setting
A fraction  of the firms set their prices flexibly at p1t,
supplying y1t.
11
A fraction 1- of the firms set their prices one period in
advance (in period t-1) apt p2t, supplying y2t.
The flexible price producer (type-1 firms) sets a
constant mark-up,
   >1 ,
 1
above the actual marginal cost.:
p1t
 s( y1t Ct ;  t , At )
Pt
(4)
The producer who sets the price one period in advance
(type-2 firms), charging p2t , the objective function,
expected discounted profit, is:
 1   1 W 
 1 

 Y W p  P 
 p 2t y 2t  wt ht   Et 1 
  p 2t Yt Pt  wt f 1  t 2t t
Et 1 
At

 1  it 1 

 1  it 1  
 
  .

 
The maximization problem:
W
 
 1   1 W 
1  Yt p 2 t Pt



Max Et 1 
  p 2t Yt Pt  wt f 
p2 t
At

 1  it 1  
 
 

 
The FOC:

 1 
(1   ) p 2t YtW Pt  wt f
Et 1 

 1  it 1 
Substituting
1 '
 YtW p 2t Pt


At

 YtW p 2t 1 Pt


At



  0
 

(5)
12
s( yt , Ct ;  t , At ) 
 h ( f 1 ( yt / At );  t ) 1'  yt
f 
u c (Ct ;  t )
 At
 1
w
' y

 t f 1  t
 At At Pt
 At



in (5), mwe get
 1 

 (1   ) p 2t YtW Pt  s ( y 2t , Ct ;  t , At )YtW p 2t 1 Pt1   0
Et 1 
 1  it 1 



=>
 1 


(1   )YtW  p 2t Pt  s ( y 2t , Ct ;  t , At )
Et 1 
p 2t 1 Pt1    0
 1


 1  it 1 
=>
 1 
p

(1   )YtW p 2t 1 Pt1  2 t  s( y 2 t , C t ;  t , At )   0
E t 1 
 Pt

 1  i t 1 
(6)
A weighted average of the deviation of relative price
from the marked up marginal costs is set equal to zero.
Where,    1 (1   )Y p  P  can be viewed as a weight at
t
 1  i t 1 
W
t
 1
2t
1
t
a given state of nature.
Aggregate price index:
P  np
1
1t
t
 (1   ) p
1
2t
 (1  n)
t
p

1
*1 1
t
Potential Output
The potential (or the Natural level of ) output (YtN) is
the output level under perfect price flexibility ( = 1).
Using (4) and (6) with  = 1 we get:
np
1
1t
pt
 (1  n) t p

1
*1 1
t
 s(Yt n , Ctn ;  t , At )
13
If there are no capital flows (closed capital account),
then CtN = YtN. In this case the natural output is defined
by
np
pt
1
t
 (1  n) t p

1
*1 1
t
 s(Yt n , Yt n ; t , At )
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  s (Yt n , Yt n ;  t , At )
 The natural output is independent of monetary
policy.
Note that the efficient output, 1  s(Y , Y ; , A ) is larger
*
t
*
t
t
than the natural output.
IV. The Aggregate Supply
The aggregate supply is a set of 6 equations:

1
1t
Pt  n p  (1   ) p
p1t
 s( y1t Ct ;  t , At )
Pt
1
2t
 (1  n)
t
p

1
*1 1
t
 1 
p

(1   )YtW p 2t 1 Pt1  2 t  s( y 2 t , C t ;  t , At )   0
E t 1 
 Pt

 1  i t 1 
p 
y1t  Yt  1t 
 Pt 
W

t
14
p 
y 2t  Yt  2t 
 Pt 

W


Yt  y1t  (1   ) y 2t 


 1

 1


 1
There are 6 endogenous variables that are determined in
the aggregate supply block of the model:
THE Quantities-- y , y , Y
THE Nominal prices--- p
1t
2t
t
1t
, p 2t , Pt .
The Solution technique: log-linearization of the 6
aggregate-supply equations around the no shock steady
state.
IVa. The No Shock Steady State
 (1  r * )  1
Assume
Consider a deterministic steady-state, where
A  A  1,    , p  p , C  C , Y  Y .
t
t
*
t
*
t
Log-linearization
s t ( y t , C t ; t , At ) 
state point yields:
and
t
of
 h ( f ( y t / At );  t ) 1'  y t
f 
uc (C t ; t )
 At
1
t  0

 1

,
A
 t
equation
(5),
around the steady-
15
sˆ t  yˆ t   1Cˆ t
_
 log[
_
 h ( f 1 ( y/ A);0)
uc (C ;0)


_
 log[
_
 h ( f 1 ( y/ A);0)
uc (C ;0)

A
_ _
'
1
f 1  ( y/ A)  _ ]

A 
t
_ _
'
1
f 1  ( y/ A)  _ ]

A
(7)

At
'
Where:   
w
  p , w 
vhh f 1 y
vh A
,
p 
f
1 ''
1 '
f
y
A
, and 
1

u cc C
uc
.
The expression for the real marginal cost, evaluated at
the natural level of output, is:
sˆ tN  Yˆt N   1Cˆ tN
_
 log[
_
 h ( f 1 ( y/ A);0)
uc (C ;0)


_
 log[
_
 h ( f 1 ( y/ A);0)

uc (C ;0)
A
_ _
'
1
f 1  ( y/ A)  _ ]

A 
t
_ _
'
1
f 1  ( y/ A)  _ ]

A
(7’)

At
Subtracting (7’) from (7):
sˆt  sˆtN   ( yˆ t  Yˆt N )   1 (Cˆ t  Cˆ tN )
Log-linearizing (4),
state yields:
pˆ 1t  Pˆt  sˆt
(7’’)
p1t
 s( y1t Ct ;  t , At )
Pt
, around the steady-
16
Subtracting the (log-linearized version of the ) equation
evaluated at the natural level of output, substituting
p  P , and using (7’’) yields:
N
1t
N
t
pˆ 1t  Pˆt   ( yˆ1t  Yˆt N )   1 (Cˆ t  Cˆ tN )
(8)
We go through a similar procedure for equation (6)
 p

E t 1  t  2 t  s( y 2 t , C t ;  t , At )   0

  Pt
(in this case the relevant part
N

t  t
of the equation is the term inside the square brackets)
and get:
pˆ 2t  Et 1[ Pˆt   ( yˆ 2t  Yˆt N )   1 (Cˆ t  Cˆ tN )]
(9)
Log-linearizing the price index yields:
Pˆt  n[pˆ 1t  (1   ) pˆ 2t ]  (1  n)(ˆt  pˆ t* )
(10)
Assume now that in steady-state there is zero inflation
; then:
pˆ 1t  log( p1t )  log( p1t )  log( p1t )
Pˆ  log( P )  log( P )  log( P )
t
t
t
t
pˆ 2t  log( p2t )  log( p2t )  log( p2t )
ˆt  pˆ t*  log(  t pt* )  log(  t pt* )  log(  t pt* )
The rate of inflation rate is given by:
17
t 
 P 
Pt  Pt 1
 log  t   Pˆt  Pˆt 1
Pt 1
 Pt 1 
=>
 t  Et 1 t  Pˆt  Et 1 Pˆt
(the surprise rate of inflation)
The real exchange rate is defined as:
et 
 t Pt*
Pt
IV c. Deriving the Aggregate Supply Relationship
Log-linearizing the (Dixit-Stiglitz) demand for the good
produced by firm j,
 p ( j) 

y t ( j )  Yt  t
 Pt 
W

:
yˆ jt  Yˆt w   [ pˆ jt  Pˆt ]
, with
Yˆt w  nYˆt N  (1  n)Yˆt F
With symmetry between firms of type 1 (flexible price
firms) and between firms of type 2 (sticky price firms),
we have:
yˆ 1t  Yˆt w   [ pˆ 1t  Pˆt ]
yˆ 2t  Yˆt w   [ pˆ 2t  Pˆt ]
Substituting for
( yˆ 1t , yˆ 2 t )
in (8) and (9):
18
pˆ 1t  Pˆt 

 1 ˆ
(Yˆt w  Yˆt N ) 
(Ct  Cˆ tN )
1  
1  
(8’)
 

 1 ˆ
w
N
ˆ
ˆ
ˆp2t  Et 1 Pˆt  Et 1 
(Yt  Yt ) 
(Ct  Cˆ tN )
1  
1  

=>
(9’)
(11)
pˆ 2t  Et 1 pˆ 1t
From (10):
Pˆt  Et 1 Pˆt   t  Et 1 t  n[pˆ 1t  (1   ) pˆ 2t ]  (1  n)(ˆt  pˆ t* ) 

 Et 1 n[pˆ 1t  (1   ) pˆ 2t ]  (1  n)(ˆt  pˆ t* )
Using (11)
pˆ 2t  Et 1 pˆ 1t
and

Et 1 pˆ 2t  Et 1 pˆ 1t
yields:
 t  Et 1 t  n [ pˆ 1t  pˆ 2t ]  (1  n)[(ˆt  pˆ t* )  Et 1 (ˆt  pˆ t* )]
(12)
From the definition of the real exchange rate we get:
eˆt  ˆt  Pˆt*  Pˆt
=>
ˆt  Pˆt*  eˆt  Pˆt
Substituting in (12) yields:
 t  Et 1 t  n [ pˆ 1t  pˆ 2t ]  (1  n)[eˆt  Et 1eˆt ]  (1  n)[ Pˆt  Et 1 Pˆt ]
=>
n( t  Et 1 t )  n [ pˆ 1t  pˆ 2t ]  (1  n)[eˆt  Et 1eˆt ]
From (10),
pˆ 2t
is given by:
(13)
19
pˆ 2t 
1
[nPˆt  npˆ 1t  (1  n)eˆt ]
n(1   )
Substituting
 t  E t 1 t 
pˆ 2t
in (13); we have:


1 n  
1

[ pˆ 1t  Pˆt ] 
eˆ t 
(eˆ t  E t 1eˆ t ) 
1
n 1 
1

Using (8) pˆ  Pˆ   ( yˆ  Yˆ )   (Cˆ  Cˆ ) to substitute for pˆ  Pˆ
in this expression, we obtain the open-economy
Aggregate Supply (Phillips) Curve:
N
1t
 t  Et 1 t 
t
1t
t
1
t
N
t
1t
t
1
 1 n  1

  
ˆ w  Yˆ N )  

(
Y
(Cˆ t  Cˆ tN ) 
eˆt  Et 1eˆt 
t
t

1   1  
1  
n 1 


But because that the world output is divided between
the domestic and foreign world as:
Yˆt w  nYˆt h  (1  n)Yˆt f
we have
20
=>
Yˆt w  Yˆt N  n(Yˆt h  Yˆt N )  (1  n)(Yˆt f  Yˆt N )
and
 t  Et 1 t 
 1 n  1

  n ˆ h ˆ N (1  n) ˆ f ˆ N
 1 ˆ

(
Y

Y
)

(
Y

Y
)

(Ct  Cˆ tN ) 
eˆt  Et 1eˆt 
t
t
t
t

1   1  
1  
1  
n 1 


[If LCP prevails,


P t  domestic  (1  (n  (1  n) s))  t .
The Pass Through from exchange rate fluctuations to
domestic inflation is lessened, and the effect of the real
exchange rate on surprise inflation is:

1  (n  (1  n) s)  1

eˆt  Et 1eˆt 
n
1 
.
21
s = The fraction of foreign producers which
preset prices in a domestic buyer’s currency.]
IV.1 Perfect Capital Mobility
If capital is perfectly mobile, then the domestic agent
has a costless access to the international financial
market. As a consequence, household can smooth
consumption similarly in the rigid price and flexible
price cases.
22
=>
Cˆ t  Cˆ tN
The Aggregate Supply curve becomes:
 t  E t 1 t 

  n ˆ h ˆ N (1  n) ˆ f ˆ N  1  n  1

(Yt  Yt ) 
(Yt  Yt ) 
eˆt  E t 1 eˆt 

1   1  
1  
n 1 


IV. 2 Closed Capital Account
23
If the domestic economy does not participate in the
international financial market, then there is no
possibility of consumption smoothing, and we have that:
N
N
ˆ
ˆ
ˆ
ˆ
Ct  Yt ; Ct  Yt
In this case, the Aggregate Supply Curve becomes:
 t  E t 1 t 

  n   1 ˆ h ˆ N (1  n) ˆ f ˆ N  1  n  1

(Yt  Yt ) 
(Yt  Yt ) 
eˆt  E t 1 eˆt 

1    1  
1  
n 1 


24
IV.4 Closed Economy
If both the capital and trade accounts are closed, then
the economy is an autarky, completely isolated of the
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:
 
 t  E t 1 t  
1 
    1  ˆ h ˆ N
(Yt  Yt )

 1   
25
IV.4 Slopes (Sacrifice Ratios)
The slope of the Aggregate Supply Curve under each
scenario is:
(i)  1 
n
(1   )(1   )
(perfect capital mobility)
 (n   1 )
(ii)  2  (1   )(1   )
(closed capital account)
 (   1 )
(iii)  3  (1   )(1   )
(closed economy)
It can be seen that 
1
 2  3
 Successive opening of the economy will flatten the
Aggregate Supply Curve.
26
(Note however that the fraction of flex- and fixedprice firms is assumed to be given exogenously.
Intuitively, liberalizing trade account transactions
may increase the number of flex-price firms. If so,
opening may increase the slope.)