Download Section III. Business Cycles B. Rational Expectations Inflation

Section III. Business Cycles
B. Rational Expectations
Inflation expectations form a key part of the dynamics of the Neo-classical
synthesis model that was described in the previous chapter. However, there is a view by
some that the model of expectations formation is too ad hoc. In this section, we consider
a more theoretically well-founded model of expectation formation in the context of a very
simple version of the business cycle model.
We write down a linear function of the IS expenditure curve
qt  q  t  d   rt  r 
Where qt  ln(Yt ) , q  ln(Y P ) and αt is a demand shock affected by changes in consumer
or business confidence or fiscal policy. Consider if there is a long-run interest rate, r
which we might thing of as an interest rate along the balanced growth path. Assume in
the long-run, that spending cannot go above potential output.
Write monetary policy in the form
rt  r  t  b   tFED   tgt 
Where  t is a monetary policy shock and  tFED is the central bank’s measure of the
inflation rate. The parameter b is the inflation sensitivity of monetary policy.
We can combine these two equations to a version of the aggregate demand curve
qt  q  t  d  t  b   tFED   tgt   q  t  d  t  d  b   tFED   tgt 


The central bank has many statisticians measuring the economy and then makes decisions
about monetary policy on an intermittent basis. This means that when monetary policy is
being made, the central bank is using information from time t-1. We will then write the
Feds measure of inflation as a forecast made with information available at time t-1.
 tFED   tE   t Infot 1
must make its monetary p
A final equation will represent the supply curve of the economy.
1
qt  q   t  gtW 

Where is the logarithm of potential output. This equation says that firms will produce
potential output if inflation matches wage growth which would keep the real wage rate
constant. If inflation races ahead of nominal wage growth, real wage growth will decline,
firms will hire more workers and produce more than potential. If inflation is slower than
nominal wage growth, real wage growth increases and employment and production will
decline.
Wage growth is decided by contracts that are signed at time t-1. Therefore we can
say that wage growth is equal to expected inflation plus a cost push shock, νt.
gtW   tE  t
Therefore we can write the aggregate supply curve as
1
qt  q   t   tE  t    t   tE    qt  q   t

We can solve for the endogenous variables {qt, πt} as a function of the exogenous
variables {αt, νt, μt} and  tE . The question is how to write a model of  tE . In the previous
model we assumed that inflation expectations were just lagged inflation  tE   t 1 . If this
is the case, then monetary policy makers can choose the parameters of monetary policy to
stabilize the economy.
However, an alternative is to examine rational expectations. Rational
expectations is just another way of saying model consistent expectations. We think of the
exogenous variables as being shifts in the economy that are being driven by noneconomic events. These might be thought of as random shocks to the system. We know
from probability and statistics class that we can think of random variables as having a
distribution of possible outcomes and some probabilities associated with those outcomes.
Those random variables will then have a mean/expected value which is a weighted
average of the possible outcomes using the probabilities as weights. They will also have a
variance which is a weighted average of squared deviations of possible outcomes from
mean with the probabilities again used as the weights. Random variables also have higher
moments like skew or kurtosis etc. If we were rational, the best unbiased predictor for the
random variable will be the expected value. If xt is a random variable, we write the
expected value as E[xt]. We can write the variance of E[ (xt -E[xt])2].
When a variable follows a time series then we can decompose it into two parts:
the predictable component (which is know at time t-1) and the innovation which cannot
be predicted using information available at time t-1.
xt  predictablet Infot 1  innovationt
The expected value of the innovation term is E[innovationt]=0. This means that if we
have all information available to us at time t-1 the best forecast of xt is
E  xt Infot 1   predictablet Infot 1  E innovationt   predictablet
For example, if the random variable followed an AR(1) process, xt    xt 1   t where
 t is a white noise term then
predictablet 1   xt 1 innovationt   t
And we could write E  xt Infot 1     xt 1 . We could generalize this to the AR(p)
process xt  1  xt 1   2  xt 2  ...   p  xt  p   t . Then
xt Infot 1  1  xt 1  2  xt 2  ...   p  xt  p
Now consider the three exogenous variables that are impacting the economic
system we are modeling. {αt, νt, μt}. We can decompose each of them into their
predictable component and their unpredictable components
 t  at  Infot 1   t
E t   0  E  t Infot 1   at
t  mt  Infot 1    t
E  t   0  E  t Infot 1   mt
 t  nt  Infot 1   t
E t   0  E t Infot 1   nt
Think of the economic model is a numerical algorithm that accepts the exogenous
variables and the expectations of endogenous variables into outcomes for the endogeous
variables.
Expectations of
Endogenous
Variables
Model
Endogenous
Variables
Exogenous
Variables
Then the endogenous variable, given expectations, is a function of the exogenous
variables. If the exogenous variables are a random variable, the endogenous variables, as
functions of random variables, are themselves random variables. Thus, endogenous
variables have their own distribution, expected value and standard deviation. If we were
going to create the best forecast of the endogeonous variables we would use their
expected value. This expected value would be a function of the expectations that
economic agents had of them
E ( Endogenous Variablest )  f ( Expectations of Endogenous Variables)
The theory of rational expectations suggest that if we as outside observers were to
use the expected value of the endogenous as the forecast, then rational economic agents
such as workers and firms should do so as well. A model consistent set of expectations
should solve the equation
E( Endogenous Variablest )  f (E  Endogenous Variablest )
Given the dynamics of the exogenous variables, we apply the above equation to forecasts
using the information at time t+1
E  Endogenous Variablest Infot 1   f (E  Endogenous Variablest Infot 1 )
Lets apply the theory to examine the model when
This model writes the production decision of the firm as equal to
 tFED   tE  E  t Infot 1 
So our model is

AD : qt  q   t  d  t  d  b  E  t Infot 1    tgt

AS :  t  E  t Infot 1     qt  q   t
First examine the model under the assumption of  t  0 so we could focus on demand
shocks. Using the AS supply curve, if expectations are model consistent, then the
expected value of the left-hand side of the equation were equal to the expected value of
the right hand side of the equation
AS : E  t Infot 1   E  E  t Infot 1     qt  q  Infot 1 


Expectations and forecasts are linear functions. By this we mean if the random variable yt
were a linear function of another random variable, yt = a∙xt + b, then E[yt] = a∙E[xt]+ b.
Therefore we can write the right hand side as
AS : E  t Infot 1   E  E  t Infot 1  Infot 1     E  qt  q  Infot 1 


To solve this problem, we need to apply the Law of Iterated Expectations. The Law states
that the expectation is its own expected value. By this we mean that a forecast is made
with the information available at time t-1. If we have that same information, we should be
able to know what that forecast is. Therefore we could write
L.I .E.: E  E  t Infot 1  Infot 1   E  t Infot 1 


Combine this with the AS curve to get
AS : E  t Infot 1   E  t Infot 1     E  qt  q  Infot 1  






E  qt  q  Infot 1   0  E  qt  Infot 1   q
Since output is different from potential output only if inflation is different than workers
expectations, then workers should expect that output is equal to potential output. This
might be why we call model consistent expectations, rational expectations. It would not
be rational to forecast a forecast error; if you did, why not simply improve your forecast.
Since the output gap is dependent on the worker’s making forecast errors, it is not
rational to forecast an output gap.
We can use the AD curve to forecast inflation. The forecast of output should be
equal to the forecast of the right hand side of the equation.
AD : E  qt  q  Infot 1   E   t  d  t  d  b  E  t Infot 1  Infot 1   d  b   tgt 


0  E  t  Infot 1   d  E  t Infot 1   d  b  E  E  t Infot 1  Infot 1   d  b   tgt






0  at  d  mt  d  b  E  t Infot 1   d  b   tgt 
a  d  mt
E  t Infot 1    tgt  t
d b
Now that we have solved for model consistent expectations, we can now solve for
the outcome of the model. The aggregate demand curve becomes
 a  d  mt 
AD :  qt  q   t  d  t  d  b   t

 d b 
 t  at   d   t  mt   tA  d   t
Then we can plug this into the aggregate supply curve to get the level of inflation
 a  d  mt 
A
 t   tgt   t
   t  d   t 
d

b


We find a number of results.
1. Output is white noise. All the fluctuations are driven by the innovations in
demand shocks rather than by the systematic component. This is a clear
implication of the rational expectations model. If people use all relevant
predictable information to form their expectations, the only reason that inflation
can differ from expected inflation is unpredictable innovations. When firm’s
produce output different from potential output, only when inflation is different
from the inflation that people expected when labor contracts are signed, then
output can only differ from potential due to unpredictable shocks.
2. Systematic monetary policy cannot impact the stability of output. Assuming no
covariance between money and demand shocks, we can write the variance of the
output gap as  2  d 2   2 . Note that regardless of the degree of inflation
sensitivity of monetary policy, b. Since systematic policy cannot take the
innovations into account when real interest rates are set, it cannot moderate the
impact of these innovations. Moreover, since workers and firms can predict the
systematic component of monetary policy, it will automatically be incorporated
into inflation expectations and not impact output.
3. Monetary policy shocks can destabilize output. The only avenue along which
monetary policy can work to reduce output volatility is to eliminate monetary
policy shocks  2  0 .
4. Systematic monetary policy can impact inflation. The degree to which the
expected component of demand, at  d  mt , pushes up prices depends on the
counter-veiling monetary policy response.
Examine a couple of variants of the model. First, lets assume that the central bank can set
the real interest rate in light of the current inflation rate (i.e.  tFED   t ). Then our model
becomes
AD :  qt  q    t  d  t  d  b   t   tgt 
AS :  t  E  t Infot 1     qt  q 
First solve for model consistent expectations. From the aggregate supply equation, we say
that the expected value of the left hand side of the aggregate supply equation is equal to
the expected value of the right hand side.
AS : E  t Infot 1   E  t Infot 1     E  qt  q  Infot 1  
E  qt  q  Infot 1   0  E  qt  Infot 1   q
Similarly for the demand equation
AD : E  qt  q  Infot 1   E t  d  t  d  b   t  Infot 1  
a  d  mt
 at  d  mt  d  b  E  t Infot 1   E  t Infot 1   t
d b
Notice that this is the same as the previous model. The central bank can respond to
inflation rather than expected inflation. Therefore, this difference may result in a
difference in the actual actions of the central bank However, when we try to predict the
central bank’s behavior we can only base our prediction on our own expectations of
inflation, therefore there will be no predictable difference in the behavior of the central
bank.
Insert the model consistent expectations to solve for actual output. First insert expected
inflation into the supply equation.
a  d  mt
AS :  t  t
   qt  q 
d b
Now insert the aggregate supply equation into the aggregate demand equation
 a  d  mt

AD :  qt  q    t  d  t  d  b   t
   qt  q   
 d b

 qt  q    t  at   d   t  mt   d  b   qt  q  
 qt  q   ta  d   t  d  b   qt  q  
1  d  b     qt  q   ta  d   t 
 qt  q  
ta  d   t
1  d  b  
We can also solve for inflation
AS :  t   tgt 
at  d  mt


 ta  d   t 
d b
1

d

b




We can make two points.
1. Output is white noise. It is an integral point of the rational expectations model that
if output differs from potential due to expectations errors and expectations errors
can only be caused by new unpredictable information, then output must be
unpredictable.
2. The central bank’s ability to stabilize output can only come from an information
advantage over private agents. We see that here the volatility of the output gap is
 2  d 2   2
2
given by   qt  q  
which is a negative function of the inflation
2
1  d  b  
sensitivity of the monetary policy rule, b. When there is a demand shock, that will
translate into inflation, the central bank will raise interest rates to counter-act it.
Now lets examine the model with supply shocks. Continue to use the assumption that the
central bank can respond to current inflation. Simplify by assuming that there are no
demand shocks.
AD :  qt  q   d  b   t   tgt 
AS :  t  E  t Infot 1     qt  q   t
Again, solve for model consistent expectations, using the aggregate supply curve so that
the expected value of the right hand side is equivalent to the expected value of the left
hand side.
AS : E  t Infot 1   E  E  t Infot 1     qt  q   t Infot 1 




 E  E  t Infot 1  Infot 1     E  qt  q  Infot 1   E  t Infot 1 


 E  t Infot 1     E  qt  q  Infot 1   nt 
n
E  qt Infot 1    q   t

If we expect a rise in the cost push variable this will raise real wages and induce firms to
cut back on production. Therefore, the expected level of the cost push shock at time t, nt,
has a negative effect on expected supply next period. We can also then calculate expected
inflation using the aggregate demand curve.
E  qt  q Infot 1   d  b  E  t Infot 1  
E  qt  q Infot 1 
nt
E  t Infot 1    tgt  
  tgt 
d  b
d  b 
The cost push shock of higher wages will push through into higher inflation.
Now that we have solved for model consistent inflation we can now solve the model.
 qt  q   d  b   t   tgt   d  b  E  t   tgt
Infot 1     qt  q    t

 n

 d  b   t    qt  q   t 
 d  b 

n
 qt  q    t  d  b   qt  q   d  b  t 

n
 n  t 
 
 d  b    t
  1  d  b     t  d  b     t 



  
 
d  b 
n
  qt  q    t 
 1  d  b   t
1  d  b     qt  q   
nt
Plug the results into the AD curve
 qt  q   d  b   t   tgt   
 t   tgt 
nt

nt
t

d  b  1  d  b   

 d  b
1  d  b  
 t 
We can say a few things about this result.
1. Cost push shocks directly impact that supply curve and therefore impact the level
of production regardless of whether they are expected or not. We see that in the
equilibrium both output and inflation are a function of nt in a staglationary way.
That is an increase in labor market inefficiency increases inflation and reduces the
level of output.
2. Because expected cost push shocks directly impact output, monetary policy has
1
no effect on the impact of cost push shocks on output. The coefficient on nt,  ,

is not a function of the monetary policy parameter, b. We see that the only
impact of monetary policy on output come through the ability of the central bank
to change inflation in a way unexpected by workers when wage contracts are
signed. Any way that monetary policy respond to information that the workers
also have, will not affect  t  E  t Infot 1  and will not impact output.
3. The central bank can affect the impact of the cost push shock t but must make a
trade-off. An inflation sensitive monetary policy rule will lead to a bigger impact
on output and a smaller impact on inflation following a cost-push shock.
There will be, by definition, no difference between the predict of inflation and the
prediction