Download The Flexible Price Benchmark

1
NEW KEYNESIAN MACROECONOMICS
Why monopolistic competition?
To model price pre setting one must allow for
endogenous goods supply. This requires not
assuming endowments of goods. The determinants
of the cost of supplying goods are important
because the supply costs are a prime determinant
of the optimal pricing. Differentiated goods and
monopolistic competition among the suppliers of
these goods serve as a device to allow for
expansion/contraction of output whose price is pre
set in response to the realization of demand and
supply shocks. See Rothemberg (1982), Mankiw
(1985), Svensson (1986), and Blanchard and
Kiyotaki(1987) who assume monopolistic
competition rather than assuming a single good in
perfectly competitive supply.
A. Open Economy with Flexible Prices
I - Household (The Dixit and Stiglitz setup):



M   n
E o   t u (C t )  v t    ht ( j ) 2 dj 
t 0
 Pt  2 0


*
Pt C t  Pt Tt  M t  M t 1   t B t 1  f t 1,t (1  i t*1 ) B * t  Bt 1  (1  i t 1 ) Bt
Max
s.t.
n
n
0
0
  w t ( j )ht ( j )dj    t ( j )dj
2
n = number of domestically produced goods.
1 = number of domestically consumed goods.
The term    h ( j ) dj represents the disutility (convex)
n
2
0
2
t
function, of supplying labor of type j. We have
written it as if the representative household
simultaneously supplies all of the types of labor.
Bt = bond holdings at the beginning of date t
(denominated in the domestic currency)
Bt* = bond holdings at the beginning 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
t(j) = profit of firm j (domestic)
t = exchange rate in period t
Tt = government lump-sum transfers
Wt+1 = Mt + Bt+1* + Bt+1 = financial wealth
f
= forward exchange rate (the price paid in the
present of foreign currency in terms of domestic
currency to be delivered next period)
t ,t 1
3
There is a constant elasticity of substitution  between
any two goods (varieties). Ct is a composite of all these
goods.
 n
C t    ct ( j )
0

 1

1
dj   ct* ( j )
n
 1


dj 


 1
ct = goods produced at home
ct* = goods produced abroad (imports)
Elasticity of substitution:
 a ,b
 c (a) 

d  t
c
(
b
)

  t
d MRS a ,b 
 ct ( a ) 


c
(
b
)
 t

MRS a,b 
Pick any two goods “a” and “b” in

 1
 1
 n
  1
1
*

Ct   ct ( j ) dj   ct ( j )  dj 
0
n


MRS a ,b
 c (a) 

  t
 ct (b) 

then :
1

 a ,b 
1

1 
I.1 Prices
The corresponding price index (the minimum
expenditure that buys one unit of the consumption good
composite):
n
1
0
n
Min Z   pt ( j )ct ( j )dj    t pt* ( j )ct ( j )dj
{ct ( j )}

s. t.
 1
 1
 n
  1
1
Ct   ct ( j )  dj   ct* ( j )  dj   1
0
n


4
 Pt   0 pt ( j )

n
1
1
dj   ( t p ( j ))
*
t
n
1
dj 

1
1
pt (j) = price of domestic good j (in domestic
currency)
pt*(j) = price of foreign good j (in foreign currency)
I.2 The Inter and intra-temporal First-Order
Conditions
Substituting C from the budget constraint
into the Max problem yields:
t

E o   t [u{Tt 
t 0
1
{ M t  M t 1   t B * t 1  f t 1,t (1  i t*1 ) B * t  Bt 1
Pt
n
n
M  
 (1  i t 1 ) Bt   w t ( j )ht ( j )dj    t ( j )dj )}  v  t 1  
0
0
 Pt  2
Differentiating with respect to

n
0
ht ( j ) 2 dj ]
M t , Bt 1 , B * t 1 , ht ( j )
yields:
(1) Interest parity
1  it  (1  it* )
f t ,t 1
t
As long as there is no constraint on the size of debt (no
corner solution), this no-arbitrage condition must hold.
5
(2) First order conditions

P 
u ' (Ct )   (1  it ) Et  u' (Ct 1 ) t 
Pt 1 

(1)
The Euler Condition (saving rule)
M 
v'  t 
P
i
  t 1   t
u ' (C t )
1  it
(2)
Money Demand (as a function of
ht ( j )  u ' (Ct )
wt ( j )
Pt
C t and it ) .
(3)
Labor supply (of labor category j ) as a function of
C t and
wt ( j )
Pt
(3) Ttransversality conditions (Kuhn Tucker terminal
conditions) :
BT 1
0
T 
PT 1
M
lim  T Et u ' (CT 1 ) T  0
T 
PT 1
lim  T Et u ' (CT 1 ) T
lim  T Et u' (CT 1 )
T 
BT* 1
0
PT 1
Derivation of the Dixit-Stiglitz demand Functions
Maximize the consumption index subject to a given total
income, say, Z
 n
Max
C t    ct ( j )
*
0
ct ( j ),ct ( j )

 1

1
dj   ct* ( j )
n
 1


dj 


 1
6
subject to: 
n
0
1
pt ( j )ct ( j )dj    t pt* ( j )ct* ( j )dj  Z
n
pick any two goods, say good a and good b:
First Order Conditions:

 1
... 11   1 ct (a)




1
 t p t ( a )

... 11   1 ct (b)

 1

1
=>

1

1
pt (b)ct (b)  pt (a)ct (a)


1
=>
 t pt (b)
pt (b)ct (b)  pt (a)ct (a) ct (b)

 1

Integrating both sides for all b’s:
1

0
 1
 1
 1

1
*

pt ( j )ct ( j )dj    t p ( j )ct ( j )dj  pt (a)ct (a)   ct ( j ) dj   ct ( j )  dj 
n
0
n


1
1
Z  pt (a)ct (a) Ct

=>
=>
1
*
t

 1

pt (b) ct (b)  pt (a) ct (a)
pt (b)ct (b)  pt (a) ct (a) pt (b)1
Integrating both sides for all b’s:
1

0
1
n
1
pt ( j )ct ( j )dj    t pt* ( j )ct ( j )dj  pt (a) ct (a)  pt ( j )1 dj   ( t pt* ( j ))1 dj 
 0

n
n
7
=>
Z  pt (a) ct (a) Pt

1
1
1
=>
pt (a) ct (a) Pt
=>
 p (a) 
 C t
ct (a )   t
P
 t 
 p t ( a )c t ( a ) C t

 1


I.3 The government budget
0  Tt 
M t  M t 1
Pt
(Seigniorage revenue is rebated to the public in the
form of a lump sum transfer Tt)
II – Producers
Monopolistic competitive firm:
Max  t ( j )  pt ( j ) yt ( j )  wt ( j )ht ( j )
ht ( j )
That is:
Max  t ( j )  ( f (ht ( j ))
ht ( j )
1
1

1
* 
Pt (Yt  Yt )  wt ( j )ht ( j )
subject to:
yt ( j )  f ht ( j )
The production function

 p ( j) 
 [Yt  Yt* ]
yt ( j )   t
 Pt 
producer of good j.
Dixit-Stiglitz demand, facing
8

 1
 n
  1
Yt   yt ( j )  dj  .
0


The number of domestically produced goods is n.
Returning to the profit maximization problem:
Max  t ( j )  ( f (ht ( j ))
1
ht ( j )
1

1
* 
Pt (Yt  Yt )  wt ( j )ht ( j )
A First Order Condition:

wt ( j )
 1
( f ht ( j ) )  Pt (Yt  Yt* )  

f ' ht ( j ) 
1
pt ( j )  
1
wt ( j )
f ' ht ( j ) 
Demand for hours worked of type j
wt ( j )
 mc( j )
f ' ht ( j ) 


 1
the monopolist mark up
,
III – Targeting Monetary Aggregates (but not the
interest rate)
IIIa. a closed economy
Bt*  0 ,
, C 0
(Aggregate demand has no foreign demand component)
Bt  0
*
t
9

Financial wealth = W
t 1
 Mt
Equilibrium Conditions:
Ct  Yt
M t  M ts
(Goods Demand = Goods supply)
(Money Demand = Money supply)
The idea is to approximate the behavioral and
equilibrium conditions around a fixed point so as to be
able to solve for the equilibrium price level.
We study equilibrium around a no-shock, constant –
money-growth steady state, characterized by:
The Purely Deterministic Steady State
Mt
 mt  m
Pt
P
t  t  
Pt 1
it  i
Mt
=>

M t 1
M
M  P
1
mt  t  t 1 t t 1  mt 1  t
Pt
Pt 1 Pt
t
t 
(4)


In the steady-state:
m m
In the steady-state:
Ct  Ct 1  ...  C
From (1):
From (2):
1 i 


=>
 u (Yt ) it 
Mt
 (v) 1 

Pt
  1  it 
=>
i
 

 
10
M ts
 LYt , it 
Pt
=>
The “LM curve”
The steady-state LM Curve:
   

m  L Y ,
 

, the steady state real money balances
III. The dynamic equilibrium in terms of proportional
Deviations from the Steady State)
Hat overstrikes denote proportional deviations from the
steady state for all variables (except for interest rates i
and r):
X X
X 
Xˆ t  t
 log  t 
X
 X 
Let
X 
Yt  log  t 
 X 
and take a linear approximation around
Xt  X
=>
X
Yt  log 
X
1

  X
 Xt
X
X
t
 X
=>
X  X X
log  t   t
X
 X 
Xt X
For interest rates i and r, the deviation notation is :
Xˆ  X  X .
t
t
The LM Curve
C
= equilibrium consumption elasticity of money
demand
11
i
= equilibrium interest semi-elasticity of money
demand
ˆ t  C Yˆt  i iˆt
m
(5)
From (4):
ˆt  m
ˆ t 1  ˆ t  ˆ t
m
(6)
We use the fact from consumer theory that in
equilibrium the marginal rate of inter-temporal
substitution ( u' (C ) / E [u' (C )] ) has to be equal to the
price-ratio (1  r ) . Using equation (1):
t
t
t 1
t
=>
1  rt  (1  i t )
Pt
E t Pt 1
This is known as the Fisher equation.
By log-linearizing it, we obtain:
rˆt  iˆt  E t ˆ t 1
(7)
Substituting (5) into (7) to eliminate iˆ , substituting into
(6), and then pushed forward by one period, yields:
t
Et mˆ t 1  mˆ t  Et ˆ t 1  rˆt 

where:
Since 
i
1  i
i
 0 ,   1.
C ˆ
Yt
i
12
Solving for
m̂t ,
in the forward-looking manner



( L1   ) Et mˆ t  Et  ˆ t 1  rˆt  Y Yˆt 
i 

1
 (L) 1
1

  [(L) 1  (L) 2  (L) 3  ...]
1
1  L 1  (L)
L
L is an operator which does not affect the timing of the
expectation operator. Hence: LE mˆ  E mˆ
t
t 1
t
t


1 
mˆ t      Et  ˆ t 1 s  rˆt  s  C Yˆt  s 
 s 0    
i

1
s

We use mˆ  Mˆ  Pˆ , and get the unique equilibrium value
for the
Equilibrium Price Level:
t
s
t
t


1  1 
Pˆt     Et  ˆ t 1 s  rˆt  s  C Yˆt  s   Mˆ ts
 s 0    
i

s
III.b The open economy
Equilibrium:
Ct  Ct*  Yt  Yt*
worldwide goods market equilibrium
M t  M tS
(The domestic money is effectively non-
tradable)
13
We study the equilibrium around a steady-state that is
characterized by
Mt
 mt  m
Pt
P
t  t  
Pt 1
it  i
Mt

M t 1
M
M  P
1
mt  t  t 1 t t 1  mt 1  t
Pt
Pt 1 Pt
t
t 
(8)
mt  m ,  t     t    
Recall that,
1
n
1
1
Pt    pt ( j )1 dj    t pt* ( j )1 dj  .
 0

n
*
Assume Pt*  1 .
Pt 1
Because pt ( j)  Pt   =>  t   .
pt 1 ( j ) Pt 1
 t 1
III.b.1 Flexible Exchange Rate
pt ( j )
P

 t  t
pt 1 ( j ) Pt 1  t 1
(9)
Log-linearizing equations (8) and (9) around the steadystate:
ˆt  ˆt  ˆt 1
ˆt  m
ˆ t 1  ˆ t  ˆ t
m
(10)
(11)
14
LM curve, again from (2):
 u (Yt  Yt*  Ct* ) it 
Mt
 (v) 1 

Pt

1  it 

Proportional deviations from the steady-state:
(5’)
ˆ t  C (Yˆt  Yˆt*  Cˆ t* )  i iˆt
m
Log-linearizing the Fisher equation:
rˆt  iˆt  E t ˆ t 1
(12)
Substituting (12) into (5’) to eliminate iˆ :
t
ˆ t  C (Yˆt  Yˆt*  Cˆ t* )  i rˆt  iˆ t 1
m
Substituting into (11):
mˆ t 1  mˆ t  ˆ t 1 
C ˆ C ˆ * C ˆ *
1
Yt 
Yt 
Ct  rˆt  mˆ t
i
i
i
i
Solving for
m̂t
mˆ t  
(13)
as we did before:

 *  * 
 1  
  Et  ˆ t 1 s  rˆt  s  C Yˆt  s  C Yˆt  s  C Cˆ t  s 

 s 0    
i
i
i

1
s

(14)
By using (14) and (11) we can solve for the exchange
rate:
ˆt  ˆ t

ˆt  ˆ t   M t 
where:

 *  * 
 1  
  Et  ˆ t 1 s  rˆt  s  C Yˆt  s  C Yˆt  s  C Cˆ t  s 

 s 0    
i
i
i

1

s
ˆt  ˆt  ˆt 1
15
III.b.2 Fixed Exchange Rate
ˆt  0





1
0      E t  ˆ t 1 s  rˆt  s  C Yˆt  s  C Yˆt* s  C Cˆ t* s  (15)
 s 0    
i
i
i

1

s
the ̂ path is endogenously determined so as to satisfy
equation (15).
t