1
The General Equilibrium Structure of the New
Keynesian Macro Model
The model combines 2 relationships:
• The relation between interest-rate targeting and intertemporal resource allocation;
• The relation between real activity and inflation.
I. Non-Linear Equilibrium Conditions
Employ the equilibrium condition, Ct  Yt  Gt .
(1)
(1  it )   1[

P 
1
Et  u ' (Yt 1  Gt 1 ,  t 1 ) t ]1
u ' (Yt  Gt ,  t ) 
Pt 1 
[Use has been made of the saving condition:

P 
u ' (Ct ,  t )   (1  it ) Et  u ' (Ct 1 ,  t 1 ) t  ].
Pt 1 

And the terminal condition:
(2) Lim  E u' (Y , 
T
T 
t
T
T

) BT / PT  0
it --- The one-period nominal interest rate controlled by the central bank.
2
Bt --- The government one-period debt.
In the flexible-price model Yt is an exogenous variable. It is implicitly solved from the
equation:
1  s(Yt , Yt ;  t , At ) .
Equations (1)-(2), together with the interest-rate rule linking the paths of it and Pt
determine the equilibrium paths of it and Pt .
In the rigid price Model equations (1)-(2) are still valid (the condition of optimal
spending by individual is the same). But instead of the exogenous process { Yt }, there is
the aggregate supply block within which Yt is endogenously determined.
The aggregate supply block:


1
1
(3)
Pt  p11t  (1   ) p12t 
(4)
p1t
 s( y1t , Yt  Gt ;  t , At )
Pt
3
(5)
 1 
p

(1   )Yt p 2t 1 Pt1  2t  s ( y 2t , Yt  Gt ;  t , At )   0
Et 1 
 Pt

 1  it 1 
(6)
p 
y1t  Yt  1t 
 Pt 
(7)
p 
y 2t  Yt  2t 
 Pt 


The system of equations (1) (3)-(7) jointly determine the evolution of
y1t , y 2 t , p1t , p 2t , Pt , and aggregate demand, Yt , given it , which is picked by the
monetary authority, and the evolution of the disturbances {  t , At }.
A monetary policy rule might be specified by a Taylor rule of the form
(8)
it   (
Pt
, Yt ;t )
Pt 1
t is an additional exogenous disturbance (error term).
Fiscal policy rule might be specified by an exogenous target path {Bt } for the value of
government liabilities.
A rational expectations equilibrium is the group of processes{ y1t , y 2 t , Yt
p1t , p 2t , Pt } that satisfy equations (1) –(7) given the exogenous disturbances.
4
Note that although the form of the equilibrium condition (1) is exactly the same as in the
flexible-price model its interpretation is somewhat different in the flexible price case: it
determined the equilibrium real rate of return, given the economy’s exogenous supply
of goods. In the rigid price case condition (1) can be viewed as the analog, in an intertemporal equilibrium model, of the IS curve. That is, it determines the level of real
aggregate demand associated with a given real interest rate. Since output is demand
determined, condition (1) determines the equilibrium level of output associated with a
given real interest rate. The real interest rate on the other hand, is determined by the
aggregate supply block and the interest rate policy rule.
A Log Linear Model
A log-linear approximation of (1) is



Y t  g t  Et (Y t 1  g t 1 )   ( i t  Et  t 1 )
.
gt = A composite exogenous disturbance that indicates the shift in the relation between
real income Yt and the marginal utility of real income, due to preference shocks and
variations in government purchases.
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Aggregate demand depends on expected future short real interest rates, and not simply
upon a current ex ante short real rate of return.
Log linearization of the aggregate supply block, equations (3) –(7), yields
 
 t  E t 1 t  
1 
    1  ˆ
(Yt  Yˆt N ) .

 1   
Log linearization of (8) yields
it  it   ( t   )  Y Y (Yˆt )
i t ,
--- exogenous intercept and constant policy coefficient, respectively.
,
Note that   1 for stability, as in the Taylor Rule.
We transform notation to the output gap, xt  (Yˆt  Yˆt N ) ,


I. xt  g t  Et ( xt 1 )   ( i t  Et  t 1  r t )

Aggregate Demand

xt  Et ( xt 1 )   ( i t  Et t 1  r t )
Aggregate Demand
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 
II.  t  Et 1 t   1  

    1 
( xt )

1




Aggregate Supply

i t  i t   ( t   )   x ( xt  x)
III.
Interest Rate Rule
it  it   ( t   )  Y Y ( xt  Yˆt N )
Where,

N
N
r t   [( gt  Y t )  Et ( gt 1  Y t 1 )]
_1
represents deviations of the natural real rate of interest from the value consistent with a
zero inflation steady state (as in Wicksell).
The monetary picks the interest rate that through I. and II. Determine inflation and output
gap. The interest that the monetary picks up is consistent with the monetary rule in III.
and inflation and output gaps that are derived from this rate.
Rational-expectations’ solution of I. II. and III , for the endogenous variables

xt , i t ,  t
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represents the basic New Keynesian macro equilibrium model, in which one can
simulate the effects of different monetary policy rules.
Solution
The new-Keynesian DSGE model is described by the following equations:
AS curve
 t  Et  t 1  xt  u tS
AS curve
 t  Et  t 1  xt
IS curve
xt  Et xt 1   (it  Et  t 1 )  u tD
(current conditions, not forward)
Policy rule
it   st t   st x  Ct
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Solution
The new-Keynesian DSGE model is described by the following equations:
AS curve
 t  Et  t 1  xt  u tS
IS curve
xt  Et xt 1   1 (it  Et  t 1 )  u tD
Policy rule
it   st t   st xt
(current conditions, not forward)
To write the new-Keynesian model in a compact form, substitute the policy rule into the
IS curve, rearrange the terms in the AS curve and the IS curve, to get
Fs y t  HE t y t 1  u t
yt 
ut 
 
t
 

Fs
H
xt
utS
utD

 1 st 1 1 s
 0
 1 1
1



let
ut 
     a  v
aS
a
D
Et vt 1  0
utS
utD
t
t
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Note that the exogenous vector (denoted by 'u') is decomposed to predictable vector,'a',
and to vector of unpredictable innovations, 'v'. The predictable vector can be written as
intercepts and serial-correlation random variables.
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1
y t  Fs [ HE t y t 1  at  vt ]
1
1
1
 Fs {HFs [ HE t y t  2  ( Ha t 1  Hv t 1 )]  Fs [at  vt ] 
1
1
1
1
 ( Fs H ) 2 Et y t  2  ( Fs ) 2 Et at  2  Fs Et at 1  ( Fs )( at  vt )
1
1
1
1
1
 ( Fs H ) 3 ( Et y t  2  ( Fs ) 3 Et at 3  ( Fs H ) 2 Et at  2  ( Fs H )1 Et at 1  ( Fs )( at  vt ) 
T
 lim [( Fs H ) T ( Et y t T )   ( Fs H ) i Et at i ]  ( Fst )( at  vt )
1
T 

1
1
i 1
y t   ( Fs H ) i Et at i ]  ( Fs )( at  vt )
1
1
i 1
An Alternative Setup
AS curve
 
 t  Et 1 t  
1 
IS curve
xt  Et xt 1   (it  Et  t 1 )  u tD

Policy rule
Rewriting:
_
_
    1  ˆ
(Yt  Yˆt N )

1




_

i t  i   ( t   )  Y Y ( xt  Yt N )
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(1) AS curve
 t  Et 1 t  xt
(2) IS curve
xt  Et xt 1   ( i t  Et  t 1 )  u tD


i t  a t  bxt  C t
(3) Policy rule
Substitute (3) into (2) yields:
xt  Et ( xt 1 )   (a t  bxt  Ct  Et  t 1 )  u tD
(4) (1  b ) xt   (a t )  Ct  Et  t 1  u tD  0
because 0  Et ( xt 1 )
(1) imply that
xt 
1

[ t  Et 1 t ]
substitute in to (4) to get
(1  b )
1

[ t  Et 1 t ] +  (a t )  Ct  Et t 1  utD  0
 t  AEt 1 t + BE t  t 1  d t
Et 1 t  AEt 1 t + BEt 1 t 1  Et 1 d t
Et 1 t  ( B /(1  A))Et 1 t 1  (1 /(1  A))Et 1 d t
 t  { A( B /(1  A))  B}Et 1 t 1  (1  A /(1  A)) Et 1d t ]
this is the fundamental stochastic first-order difference equation.
By forward iterations we get:
12

B
2 A
B s
) Et 1 t t 
Et 1  (
) d ts
1 A
1 A
s 0 1  A
 t  lim (
T 
we have to show that
B
lim (
) Et 1 t t =0
T  1  A
therefore
t 

2 A
B s
Et 1  (
) d ts
1 A
s 0 1  A
The Optimal Problem posed
In the log-linear version of the "New Keyenesian model", inflation
and (log) output are determined by aggregate supply equation
 t  Et t 1  xt  ut
and an aggregate demand equation
xt  Et xt 1   (it  Et  t 1  rt n )
The "cost-push shock" ut , the "natural rate of interest " rt n
y tN ( xt  y t ) ,
are exogenous disturbances.
, and "natural rate of output"
rt n   t   1 ( Et y tn1  y tn )
If we treat the nominal interest rate as being directly under controls
of the central bank, these equations suffice to indicate the paths of
inflation and output that can be achieved through alternative
interest rate policies.
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If one wishes to treat the central bank instrument as some measure
of the money supply, then the interest rate is determined by the
market, and we can add another equilibrium equation
mt  pt   y yt  iit   tm
Combining this with the identity
 t  pt  pt 1
there is a 4-equation system with 4 endogenous variables
yt , pt ,  t , it .
In fact, to allow for the zero bound possibility the we should write
mt  pt   y yt  i it   tm
it  0
Together with the complementary slackness requirement that at
least one of the two inequalities must hold with equality at any
point of time.
Let us suppose that goal of policy is to minimize a discounted loss
function of the form
Et0 t t  t t0 [ t2   ( xt  x * ) 2 ]

t0
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x*
is the target level for output gap(positive, if there are
monopolistic competition distortions).
The problem is to choose  t , it , it ( the central bank policy tool is the
interest rate) to minimize the loss function under commitment. The
maximization is subject to the aggregate supply.
The solution for the optimal state-contingent paths of inflation and
the output gap depends only on the evolution of the exogenous
disturbance process
{u t } but
not on the evolution of the
disturbances { ytn ,  t ,  tm } , to the extent that the latter have no
consequence for {u t } . ytn affects the path of output but not output
gap,
t
affects the nominal interest rate but not inflation and output
gap, and
 tm
affects the path of the money supply.
Lagrangian:

1
Lt0  E t0 t t  t t0  [ t2   ( x t  x * ) 2   t [ t  E t  t 1  x t ]
t0
2

1
 E t0 t t  t t0  [ t2   ( x t  x * ) 2   t [ t   t 1  x t ] 
t0
2

Because Et0 t Et  t 1  Et0 Et t  t 1  Et0 t  t 1
Differentiation the lagrangian yields the first order conditions
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 t  t  t 1  0
 ( xt  x * )   t  0
For each
t  t0
, where for
t  t0
we substitute the value
t 1  0 into
 t  t  t 1  0 .
Using the first order conditions to substitute for
t
and
xt
respectively we obtain a stochastic difference equation for the
evolution of the multipliers
Et [ t 1  (1   
2
)t  t 1 ]  x *  u t

The characteristic equation
 2  (1   
has roots
2
)  1  0

0  1  1   2
writing the difference equation with lag-lead operators
Et [  (1  1 L)(1   2 L)t 1  x *  u t
The solution is of the form
( (1  1 L)t   1221 Et [(1   112 L1 (x*  ut )]
Or alternatively
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t  1t 1  1  j 0  j [x *  Et u t  j ]

This is an equation that can be solved each period for
t
given the
previous values of the multiplier and current expectations
regarding the "cost push"terms. Starting from initial conditions and
given the law of motion for {u t } that allowed conditional
expectations to be computed, it is possible to solve this equation
for complete state contingent evolution of the multiplier.
Substituition into
 t  t  t 1  0 and  ( xt  x * )   t  0
allows us to
solve for the implied state contingent evolution of the inflation.
If we assume the the unconditional (ex ante) expected value of
each of the cost-push terms is zero, then the deterministic part of
the solution to {t } and { t } is given by

x * (1  1t t 1 )


 t  (1  1 ) x * 1t t )

t  
0
1
0
1
Thus if the target output gap is zero (no monopolistic distortions),
the anticipated inflation aimed by the central bank is equal to zero.
Discretionary policy maker
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Discretionary policy maker in period t expects his decision to in
the loss function. All other terms arealready given by the time of
the decision, or expected to be determined by factors that will not
be changed by the current period decision.
Minimizing the current loss yields
t 

 
2
[x *  te  u t ]
A Markov-subgame perfect rational expectations equilibrium is
then a pair of functions  (st ),  e (st ) such that:
(i)  ( st ) is a solution to
 ( st ) 

 
2
[x *  e ( st )  u t ]
(ii)  (st )  E ( (st 1 ) / st ) given the law of motion of
{st }
The solution is:
 t   ( st ) 


2

j 


j 0
j
(
 
2
) j [x *  Et u t  j ]
The deterministic comp[onenet of the solution is a constant
positive inflation rate if x* is positive—the inflation bias story.
18