Transcript Chapter 21

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 reasonable assumption
– 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
yt - y = vt - a2 (rt - r )
• 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, their respective means
The rationally expected values of y and π are
h(p t,e t-1 - p *) + b(yt,e t-1 - y ) = 0
If we use this final condition we get the following for the forecasts of y and π:
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(p t, t-1 - p *) + b(yt, t-1 - y ) = 0 then
p t = p * + gvt + 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/ g )(p t -p t,e
t -1
) + (1/ g )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(p t -p *) + 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 - a 2 h(p t - p *)
yt - y = t
1+ a 2 b
– From the aggregate supply equation (3rd equation slide 3) we have
p t - p * = p t,e t-1 - p *+g (yt - y ) + st
– This can be re-inserted into the 1st equation above to get
vt - a 2 hs t - a 2h(p t,e t-1 - p *)
yt - y =
1+ a 2 (b + gh)
– Next we substitute in the 1st equation on this slide into the 2nd to get
(1+ a 2b)(p t,e t-1 - p *) + (1+ a 2b)s t + gvt
p t - p* =
1+ a 2 (b + gh)
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
• 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 - a 2 hs t
1+ a 2 (b + gh)
(1+ a 2 b)s t + gvt
p t = p *+
1+ a 2 (b + gh)
• 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 past is not a good guide to the future as 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 what will happen if people have rational expectations
•
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 models based on micro
foundations and with rational expectations
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
• 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
s
+
a
2
2
v
2h s s
s y º E[(yt - y ) ] =
[1+ a 2 (b + gh)]2
2 2
2 2
g
s
+
(1+
a
b)
ss
2
2
v
2
s p º E[(p t - p *) ] =
[1+ a 2 (b + gh)]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
¶s y2
¶E[SL]
¶s p2
=0Þ
+k
=0
¶h
¶h
¶h
2
¶s y
¶E[SL]
¶s p2
=0Þ
+k
=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:
s v2 + a 22h 2s s2 + k (g 2s v2 + s s2 )
E[SL] =
(1+ ga 2h) 2
Optimal stabilisation under rational expectations con’t
•
Next we calculate the first order conditions to get
2ha 22s s2 (1+ ga 2h) 2 - 2ga 2 (1+ ga2 h)[s v2 + h 2a 22s s2 + k (g 2s v2 + s s2 )]
¶E[SL]
=0
=0Þ
4
(1+ ga2 h)
¶h
• 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 ¹ j, E[at s j ] = 0,
• The goods market equilibrium is still given by:
e
yt - y = vt - a2 (it - p 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
- a2 (it+1,t
- p t+2,t
- r ) = yt+2,t
- y - a2 (it=1,t
- p 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 - a2 (it+1,t
- p t+2,t
- r ) - a2 (it - p t+1,t
- r ) + at
Monetary policy in a liquidity trap, con’t
• The final equation on the previous slide may be written as
• Continuing to eliminate the output gap we get the following for the current
output gap
¥
é
ù
e
e
e
yt - y = at - a 2 êit - p t+1,t - r + å(it+ j,t - p t+ j+1,t - r )ú
êë
úû
j=1
• 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 + a 2 (r + p
¥
e
t+1,t
) - a 2 å (it+e j,t - p 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: e.g., lower real rates
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)
dt+n = d + et+n
• If dividends are tax proportionally at a rate τ0, then
e
Dt+n,t
= (1- t 0 )d
Announcement effects con’t
• As long as the tax rate, τ0, stays constant then
(1- t 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- t 0 )d
for t + n ³ t1 and t ³ t 0
• If we insert this into the first equation on the previous slide we get
æ d öìé
ü
é 1 ù
1 ù
Vt = ç ÷íê1(1- t 0 ) + ê
(1- t 1 )ý
t1 -t ú
t1 -t ú
r
(1+
r)
(1+
r)
û
ë
û
è øîë
þ
for t0 £ t £ t1
• 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- t 1 )d
r
for t ³ t1
• Note that the market value of equity jumps immediately
when the policy is announced
Vt0 -Vt0 -1 =
t 0 - t 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
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