Chapter 21. Stabilization policy
with rational expectations
ECON320
Prof Mike Kennedy
The role of expectations
• Activity today will depend importantly on what people expect to happen in
the future
• The convention is to assume that expectations are formed by looking at the
past – for example, our assumption that πe = π-1
• Such an assumption is reasonable when things are normal, but there are lots
of cases when it is not
– A change in the monetary policy regime
– Visible supply shocks that impinge on the structure of the economy
– A change in government with a new policy agenda
• The logical limit of forward looking expectations is the Rational Expectations
Hypothesis or REH
– Here it is assumed that expectations are based on all the information available today
– That includes information on the structure of the economy and policy changes
• The equation below is a formal way of saying that expectations for any
variable X today (time t) are based on all the available information, I, at
time t-1
Rational expectations and policy ineffectiveness
• Suppose that the central bank makes its interest rate decisions on
expected or forecast values of inflation and output, based on information
at time t-1
• The monetary policy rule is now
• Goods market equilibrium is still the same
• The SRAS curve is now
• We need to specify the stochastic properties of the exogenous variables,
which are assumed to be white noise
Policy ineffectiveness con’t
• Assume the central bank cannot observe current πt and yt
– Step 1: Express the endogenous variables yt and πt as functions of the
exogenous variables and expectations of y and π
Next substitute this expression into SRAS equation
– Step 2: Use the above two equations to calculate the rationally
expected values of yt and πt, remembering that E[st] = E[vt] = 0
The rationally expected values of y and π are their respective means
h( t,e t1   *) b(yt,e t1  y)  0
If we use this final condition we get the following for the forecasts of
y and π:
and

Policy ineffectiveness con’t
– Step 3: If we go back to the first two equations in Step 1, we see that
e
e
since h( t, t1   *) b(yt, t1  y)  0 then
 t   *  vt  st


 yt  y  vt
– Notice that the policy parameters π, h and b do not appear in the
equation for yt leading to the conclusion that systematic monetary
policy stabilisation is ineffective!
– Systematic demand management cannot influence real output and
employment when expectations are rational
– To see how this works, rewrite SRAS curve in terms of yt
yt  y  (1/  )( t  t,e

t 1
)  (1/  )st
– Since potential output and the shock are exogenous, the only way to
influence output is to create surprise inflation – create errors in the
forecasts of the private sector!
How robust is the Policy Ineffectiveness
Proposition (PIP)?
• A problem with PIP is that it assumes that the central bank cannot
act on the basis of new information as it becomes available
• While it may be true that the private sector cannot (due to fixed
contracts or a desire to have infrequent price changes) the central
can react and often does react to new information if it is important
• When the central bank can act after prices and wages have been set
then we get the familiar monetary policy rule
rt  r  h( t  *)  b(yt  y )
• While central bank may not know actual inflation, the above is a
convenient way of noting that the central bank can react to new
information
• With this policy rule, we will again proceed in three steps to solve
the model



Policy effectiveness under rational expectations
• Step 1: Solve the model for yt and πt in terms of exogenous
variables
– Inserting the new policy rule into the goods market equation
(2nd equation slide 3)
v  2 h( t   *)
yt  y  t
1 2 b
– From the aggregate supply equation (3rd equation slide 3) we have
 t   *   t,e t1   * (yt  y )  st
– This can be re-inserted into the 1st equation above to get
vt   2 hs t   2h( t,e t1   *)
yt  y 
1  2 (b  h)
– Next we substitute in the 1st equation on this slide into the 2nd to get
(1  2 b)(  t,e t1   *)  (1  2 b)s t  vt
 t  * 
1  2 (b  h)
Policy effectiveness under rational expectations con’t
• Step 2: Find πet, t-1 by taking the expected value of the final equation based on
information available at t-1 and remembering that E[st] = E[vt] = 0
e
(1

b)(

2
t, t1   *)
 t,e t1   * 

1  2 (b  h)
 t,e t1   *
• Agents expect the central bank to hit its target
• Step 3: Using this condition and the final two equations in Step 1 we get
vt  2 hs t
1  2 (b  h)
(1  2 b)s t  vt
 t   *
1  2 (b  h)
• The key point here is that the parameters of the policy function now appear as
determinants of yt and πt
• The reason for the effectiveness of policy is that the central bank can react to
vt and st after the private sector has been locked into contracts and prices
yt  y 
The Lucas Critique
• An econometric macro model with backward looking
expectations and which was estimated under a previous
policy regime cannot be used to predict economic behaviour
under a new policy regime
• The parameters of the model are likely to change
• An example is a lowering of the inflation target:
– If expectations are modelled as π-1 then the model will not be able to
predict
•
This critique applies to structural policies as well
– If the level of unemployment benefits or the degree of competition in
the economy changes then so will the natural unemployment rate
• The solution is to build macro modes based on micro
foundations


Optimal stabilisation under rational expectations
• Policy can influence real output and unemployment even when
expectations are rational as long as nominal wages and prices are fixed or
slow to adjust
• The question becomes: What are the optimal values of h and b in the
Taylor rule?
• We start by assuming that the bank wants to minimize the following
6 4 4 4 7SL 4 4 4 8
E[SL]  [(yt  y ) 2   ( t   *) 2 ]   y2   2 ,   0
• The parameter κ measures the social loss of inflation relative to output
• We can find the two variances from the final two equations in slide 8
2
2 2 2



2h  s
 y2  E[(yt  y ) 2 ]  v
[1  2 (b  h)] 2
2 2
2 2



(1

b)
s
v
2
 2  E[( t   *) 2 ] 
[1  2 (b  h)] 2
Optimal stabilisation under
rational expectations con’t
• From the above we see that if the economy were hit by only demand shocks
then the variance of both output and inflation would be lower, the higher are
both b and h
• The economy is however hit by both types of shocks which will present tradeoffs due to supply shocks – they go in the opposite directions
• We can see this by taking the first order conditions of SL wrt h and b
 y2
E[SL]
 2
0

0
h
h
h
 y2
E[SL]
 2
0

0
b
b
b
• To lower the computational burden we will assume that b = 0; the central
bank focuses on just the inflation gap which changes the SL to:
 v2   22h 2 s2   ( 2 v2   s2 )
E[SL] 
(1  2h) 2
Optimal stabilisation under rational expectations con’t
•
Next we calculate the first order conditions to get
2h 22 s2 (1  2h) 2  2 2 (1  2 h)[ v2  h 2 22 s2   ( 2 v2   s2 )]
E[SL]
0
0
4
(1  2 h)
h
 h 2 s2 (1  2 h)  [ v2  h 2 22 s2   ( 2 v2   s2 )]  0




 h 2
 s2  h 2 22 s2  [ v2  h 2 22 s2   ( 2 v2   s2 )]
2

 

2
v
 h  (1   ) 2  
 2 
s

• The final expression says that the optimal value of h will depend
importantly on the relative variances of the demand and supply shocks
– When demand shocks are large, a strong interest rate response will serve to
close both the inflation and output gaps
– In the opposite case, the central bank should respond only moderately to an
inflation shock
– The coefficients α2 and γ as well as κ are also important
Note: The 3rd equation simplifies to the 4th because the terms in ovals cancel
Monetary policy in a liquidity trap
• Can forward-looking expectations be manipulated to get out of a liquidity
trap
• Noting that the demand shock (vt) represents future expected levels of
consumption and investment we can re-write the AD curve as:
e
vt  yt1,t
 y  at , E[at ]  0, E[at a j ]  0, t  j, E[at s j ]  0,
• The goods market equilibrium is still given by
e
yt  y  vt  2 (it   t1,t
r )
• It follows from the above that the rational expectation of next period’s
output gap is given by
e
e
e
e
e
e
e
yt1,t
 y  vt1,t
 2 (it1,t
  t2,t
r )  yt2,t
 y  2 (it1,t
  t2,t
r )
• Substituting the first and third equation into the second we get
e
e
e
e
yt  y  yt2,t
 y  2 (it1,t
  t2,t
r )  2 (it   t1,t
r )  at

Monetary policy in a liquidity trap, con’t
• The final equation on the previous slide may be written as
t2,t y
6 4 4 4 44y7
4 4 4 4 48
e
e
e
e
e
e
yt  y  yt3,t
 y   2 (it2,t
  t3,t
 r )   2 (it1,t
  t2,t
 r )   2 (it   t1,t
 r )  at
e
• Continuing to eliminate the output gap we get the following for the
current output gap

j1



t
t
2
t
t1,t
t
j,t
t
j1,t
y  y  a   i   e  r   (i e   e
 r )



• This expression shows that under rational expectations the current output
gap depends not only on current real interest rates but also on the future
path of monetary policy
• When the nominal interest rate is stuck at zero then the above becomes
yt  y  at   2 (r  

e
t1,t
)   2  (ite j,t   te j1,t  r )
j1
• Now the central bank can influence the output gap by promising both
lower interest rates but as well high inflation in the future


Announcement effects
• Under rational expectations, announcement effects can play an important
role in affecting the economy
• An announcement of a new policy can influence the economy even before
it is implemented
• To see how this works we return to our model of equity prices
e
e
Dt,te
Dt1,t
Dt2,t
Vt 


...
1 r (1 r) 2 (1 r) 3
• Importantly here define D as after-tax dividends and before-tax dividends
as d or (where ε is a stochastic ‘white noise’ variable)
d tn  d   tn
• If dividends are tax proportionally at a rate τ0, then
e
Dtn,t
 (1  0 )d




Announcement effects con’t
• As long as the tax rate, τ0, stays constant then
(1  0 )d
Vt 
r
• We now assume that at time t = t1 the government will lower the tax rate
to τ1
e
Dtn,t
 (1  0 )d for t n  t1 and t  t0
• If we insert this into the first equation on the previous slide we get
(1  0 )d (1  0 )d
(1  0 )d (1  1 )d
(1  1 )d
Vt 

...


... 
1 r
(1 r) 2
(1 r) t1 t (1 r) t1 1t (1 r) t1 2t
d 

 1 
1 
 for t0  t  t1
Vt   1
t1 t (1  0 )  
t1 t (1  1 )
r
(1
r)
(1
r)




 

• The above shows the price of equity today and how it will be affected by
policy changes in the future


Announcement effects con’t
• Between the time of the announcement and the actual tax
cut, the value of equity is given by the final equation on the
previous slide
• Once the tax cuts occurs, we get
Vt 
(1  1 )d
r
for t  t1
• Note that the market value of equity jumps immediately
when the policy is announced
Vt0 Vt0 1 
 0   1 (d /r)
(1 r) t1 t0
• Thereafter the price will rise and the speed will depend on
how long it takes to implement the policy
The effect of an announced dividend tax cut