ECO 3320
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Lecture # 8
Rational Expectations alone do not provide a perfect solution: we need some additional institutional
arrangement to have the optimum. That institution is an exogenous factor in addition to the economic
model. In a sense this is a beginning of political economy.
Time Inconsistency
The issue of time inconsistency illustrates gam-theoretic complication in the Rational
Expectation setting.
Background - Optimal Inflation in Theory and Practice
According to the PIT, a full announced and this fully anticipated disinflation policy
should not have any negative impact on Y or real income.
Y  Y f   (   e ) t ,
Pf
   e ;    e ; still    e  0 .
[note:   1 ]
Refer to
Supplement
#3-1
If there are some welfare gains from lower rate of inflation – we have covered this
“Optimal Rate of Inflation”; there will be a Pareto Welfare Improvement in disinflation
policy. Click Here for the Supplement of the Optimal Monetary Policy

“Pareto Improvement” is said to occur when nothing gets worse or better.
However, in reality, disinflation policy may not work due to time inconsistency of the
policy itself.
1.) What is Time Inconsistency in gerneral?
“Time inconsistency” cause changes in optimality from one time to another; in the end
the result is sub-optimality or the second best, not the first best. The reality in life has
abundance of examples:
E.g. -
U.S. Immigration Policy towards Illegal Aliens;
U.S. government Policy towards Frequently Flooded Areas
Parents’ Push for a child’s Summer Job;
Exams in universities;
Disinflation(Lower rate of inflation) Policy.
If anything is time inconsistent, then it cannot be trusted, believed or acted upon from the
beginning.
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Lecture # 8
‘I am here illustrating that Disinflation Policy, particularly aiming at zero
inflation, is time inconsistent in the setting of Rational Expectations.’
2.) Model of Time Inconsistency of Disinflation Policy
1._Asumptions:
i.) Government is playing a game to maximize its utility subject to constraints.
Max U st. constaint.
ii.) People have rational expectations.
iii.) Government announces its commitment to the zero inflation policy.
iv) There will be games played between the government and people.
2._Government/Monetary Authority Utility Function:
i.) Suppose Utility Function is given for government as follows:
U  U (Y ,  )  Y    ;
2
Y when   0
 in unemployment
 popularity
losing face by not
achieving that target
of   0
ii.) Constraints are given as follows:
Y  Y f   (   e ) t
: Lucas AS curve , where
is what the government chooses and e is people’s expected rate of inflation.
For simplicity let’s assume   1 , t  0 .
Y  Yf   e
   e Yf Y
iii.) Utility Maximization
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Lecture # 8
So government’s optimization is to choose  that maximizes its utility given the
constraints. Thus here  is called a ‘choice’ variable.
Max U  Y    st. Y  Yf   (   ) .
e
2
{ }
We can combine the utility function and the constraint:
Max U  Yf   (   )   2
{ }
e
As long as ded= 0, the first order condition is:
  2 * 0
Solving for we get
 *


2
Note: The optimal rate of inflation for government is not “zero” as is
announced, but it is some positive number.
3._Time Inconsistency Issue: Intuition First
 * 0 (Disinflation Policy to “zero” rate of inflation) may be optimal for the
whole society ex-ante, However, when people are convinced, from the
perspective of utility maximization, it is ex-post no longer optimal. In other
words, the government’s announced disinflation policy is time-consistent.
“Would there be gains in any way to government?”
(    e  Y  above Yf → and thus more popularity)
That is only what government wishes. Would this be possible in the long-run?
The answer is negative. People have rational expectations, they figure out
what is optimal for government.

 0 ).
Their expectations = optimal rate for government. (  e   
2

 0 , then
ie. If ,  e   
2
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Lecture # 8
• National Income: Y  Y f   (   e )  Y f , which is no better than Y f or
anything.
• Popularity: U  Y    < Y f , which means no gain in government
popularity at all.

(In fact, U  as  
rather than 0 ).
2
2
If government does not attempt disinflation policy,
Y  Yf
U  Y  Y f itself
This situation is better than the disinflation policy results; disinflation
does not bring any gains in Y, but loss to government popularity
( U  Y  something ).
4._Illustrative Solutions for different cases:
1) Let’s illustrate the above point:
First, in any utility maximization or optimization problem, we have to get (set
of) indifference curses and constraint curve;
i.) Indifference Curve:
U  Y   2 - Slope of IC is given by ‘Marginal Rate of Substitution’.
We wish to get the slope of U  Y   2 in case we draw s graph which
shows the relationship between Y and  for a given fixed value of U .

1
Implicit Function rule says
 
 0.
Y
 2

Y
U  Y   2
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Lecture # 8
Curvature is given by derivative of the slope.
  


 Y    1  0

(2 ) 2
as

  , the slope is decreasing
U  Y   2
U  Y   2
A small slope
A large slope
Y
U  Y   2
U1
Y
U  Y   2
As   , U  . Therefore, moving up, the IC denote or corresponds to a lower
utility level.

U1 > U 2 > U 3
U  Y   2
U  Y   2
U3
U2
U1
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Lecture # 8
ii.) Constraint Curve:
Y  Yf     e  t , for our graph, we must write/ solve for  .
  Yf   e  1  Y
Intercept

Slope
In general  e > 0
U  Y   2
Y
U  Y   2
Yf -
U  Y   2
Yf
-Yf+e
iii.) Utility maximization by government:
Let’s combine the above (1.) and (2.) to get the optimal point which
maximizes government utility subject to the given constraint.
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“a” is better than “b”
or any other point on
the constraint curve.
c
U  Y   2
b
a
Y
U  Y   2
5._ Let’s apply the above to the three possible cases:
There are 3 possible cases:
i.) Government does disinflation in honest way and people are rational.
  0
and  e    0
Results
Y1  Y f
U g  Y  Yf
Given by ‘a’
as  e  0

  Y f   e  Y
  Y f  Y
Yf-0
 Y f +0
Y
a
Lecture # 8
ECO 3320
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Lecture # 8
Comment: In Practice this is unlikely as government has an incentive to break its
promise   0 .
ii.) Government announces   0 , people believe it (people are
naïve, and in that sense, are not rational). And in reality, government
generates ‘surprise’ positive inflation.
 e  0 but  e  0
as  e  0
  Y f   e  Y

  Y f  Y
Yf
b
Y
Yf
What is the Utility level and the National Income?
(1) + (2) shows that where both curves have the same slops, they are tangent to
each other.
1
2
 1 ,   
1
1
,  
2
2
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Lecture # 8
Y  Y f   e  
( e  0)

  Y f  Y
1
2
U1
b
Slope(s) = 1
U  Y   2
Y
U  Y   2
Yf
U  Y   2
The national income corresponding to b is equal to:
Y2  Y f 
1
 Yf
2
The level of utility corresponding to b is equal to
U 2  Y   2
1
 1 
 

2
 2 
1
1
 Yf 

2 4
1
 Yf 
4
2
 Yf 
This is also unlikely as people are rational: ‘you cannot fool everybody all the time.’
iii.) (Most likely case) People are rational so that they know all these
1
optimization and thus    e 
.
2
This raises the constraint curve up by the now non-zero expected inflation rate.
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Lecture #
  Y f   e  Y
  Y f  Y 

U3
1
2
1
2
Slope(s) = 1
U  Y   2
U1
U2
c
b
a  Yf
Y
U  Y   2
Now U1 (and c) is unattainable as the constraint curve shifts up.
The best is to get c or U 2 : there will be a tangency point at
1
as the
2
slopes of the two curves will be equal to each other (= 1).
Y  Y f  (   e ) 
t
1
1
 Yf 

2 4
 Yf
U 2  Y f   2
 1 
 Yf  

 2 
1
 Yf 
4
2
Now the core question is: How to ensure that we get to the Optimal, which is Point a?
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Lecture #
It is obvious that we cannot obtain this optimal within the economic model. We will end
up at point c.
Thus we need an external or exogenous factor that forces us to stick to point a. It is an
institutional arrangement, not an economic factor.
It is a Rule that forces us to stick to Point a, and prevents us from being carried by
economic incentives to Point c. This Rule should be there to eliminate any chance for us
to get there, and to remove Discretion for the economic policy maker (to follow the
economic incentive).