2. REH AND ECONOMIC POLICY
What is REH or model-consistent expectations and its
implication for policy and macroeconomic modeling?
What is the so-called policy ineffectiveness proposition
(PIP) and the Lucas critique?
2.1. What is Rational Expectation Hypothesis?
Muth (1961) published an article on informational
assumption discussing using neoclassical synthesis.
Consider a neoclassical model with Adopted
Expectation Hypothesis
Consider Figure 2.1 to understand the model.
(3)
...
0
;
(2)
...
0
;
(1)
...
0
)
/
(
;
0
;
,
.
*
e
e
P
P
P
P
P
Y
Y
P
M
d
dAD
dG
dAD
P
M
G
AD
Y
3. 2.1. Rational Expectation Hypothesis: Meaning
Initial equilibrium is at E0 with output Y* and price P0.
There is an expectational equilibrium at P0=Pe.
Lets assume a monetary policy shock: increase in
money supply by the Central Bank
AD shifts to the right
Initial impact: equilibrium moves from EO to A;
Output increases and price rises;
At point A, because of wrong expectations dis-
equilibrium occurs; because P0Pe.
This discrepancy is slowly removed since agents
revise their expected prices upwards.
The economy moves to the new equilibrium at E1;
where Y=Y* and P=P1=Pe
1
5. 2.1. REH: Meaning…
Observation:
Under AEH, people follow wrong expectation adjustment
path;
Economic agents make systematic mistakes or errors
along the entire adjustment path; systematic errors in
their supply of labour decision;
They keep guessing wrongly; considered by Muth (1961)
as odd.
Systematic underestimation of the price level (Pe <P); the
error is negative and remained so along the adjustment
path;
Expectation errors caused by the money shock
disappear over time because of AEH through an upward
revision of expected prices; movement along AD1.
6. 2.1. REH: Meaning …
• Argument by Muth: “…expectations are informed
predictions of future events and are essentially the same as
the predictions of the relevant economic theory” (1961, p.
316.
• Agents hear at time t0 that money supply increased from M0
to M1. Using the relevant economic theory; agents calculate
that the correct price level for the new money supply is P1,
adjust their expectations so that so that:
They supply correct amount of labour;
Equilibrium moved directly from E0 to E1; Y =Y*; at P1.
It is similar to Perfect Foresight Hypothesis version of
policy ineffectiveness hypothesis.
• No uncertainty, higher money supply induces prices and
thus nominal wages upwards; real wages, output and
employment are unaffected.
1
1
P
P
e
7. Figure 2.2: Expectation error under AEH (given stochastic
shocks or with uncertainty such as climate or natural
disaster)
8. REH with stochastic shocks is clarified using the following
postulates.
Assume an isolated agricultural market with no inventory;
Where Ut captures all stochastic elements such as weather,
crop failure, animal disease, etc.
Eq (4), no stochastic element in the demand side of the
market; assume no change in income or tastes; demand
depends on actual price;
Eq (5), there is production lag; decision on the production
capacity is determined not by P but by Pe.
Assume rational agents do not waste scarce resources.
2.1. REH: Meaning …
(6)
...
(5);
....
0
;
...(4)
;
;
1
1
0
1
1
0
t
S
t
D
t
t
e
t
S
t
t
D
t
Q
Q
Q
b
U
P
b
b
Q
a
P
a
a
Q
9. 2.1. REH: Meaning …
When supply decision is made at time t-1,
information set available is given by
(a) agents do not forget past information (about P, Q,
etc.) and past realization of stochastic errors.
(b) agents know the parameters of the model
(c) agents know the stochastic process of shocks
although actual realization is not known at time t.
10. 2.1. REH: Meaning …
In mathematical form, REH can be expressed as:
Where Et-1 indicates expectation is conditional on
information set .
Eq (8) says the subjective expectation of the price
level in period t formed by agents in period t−1 or
coincides with the conditional objective expectation
of Pt, given the information set Ωt−1.
Model solution: Consider (4), (5) and (6), solve for
market equilibrium price and quantity.
t 1
)
8
...(
/ 1
1 t
t
t
t
e
t P
E
P
E
P
e
t
P
(10)
...
;
...(9)
;
1
1
0
0
1
0
1
0
a
U
P
b
b
a
P
U
P
b
b
P
a
a
Q
t
e
t
t
t
e
t
t
t
11. 2.1. REH: Meaning …
Take expectations of both sides of Equation 10
based on the information set to get:
(a) take out expectations operator since the
parameters are in the information set and known.
(b) expectation of a constant is a constant itself
(c) as there is no better prediction of
Et-1Ut=0 than Et-1Ut-1=0;
(11)
...
1
)
(
1
1
1
1
1
1
0
0
1
1
1
0
0
1
1
t
t
e
t
t
t
t
t
e
t
t
t
t
U
E
a
P
E
a
b
a
b
a
P
E
a
U
P
b
b
a
E
P
E
12. 2.1. REH: Meaning …
Thus, we have
Equation (12) states that the objective expectation
(a) equals the subjective expectation (b).
But REH in Equation (8) states that the objective
expectation, Et−1Pt, and the subjective expectation,
coincide or
Substituting and Solving for
...(12)
1
1
1
0
0
)
(
1
b
e
t
a
t
t P
a
b
a
b
a
P
E
e
t
P
e
t
t
t P
P
E
1
e
t
P
...(14)
...(13)
;
)
(
1
1
0
0
1
1
1
1
0
0
b
a
b
a
P
E
P
P
a
b
a
b
a
P
t
t
e
t
e
t
e
t
13. 2.1. REH: Meaning …
If we substitute (14) into (10); the actual market
clearing price Pt in its reduced form as:
where is the equilibrium price that could be
obtained if there is no stochastic shock.
Equation (15) states that actual price Pt fluctuates
randomly around the equilibrium price .
P
...(15)
1
1
1
1
1
1
1
0
0
1
1
0
0
1
0
0
1
t
t
t
t
U
a
P
U
a
b
a
b
a
U
b
a
b
a
b
b
a
a
P
1
1
0
0
(
b
a
b
a
P
P
14. 2.1. REH: …
Expectational Error equals = Pt-Et-1Pt = -(1/a1) Ut.
Implications: Economic agents are rational; they
do not make systematic errors in forecasting
actual prices;
Many economists accept the REH as the standard
assumption to make in macro-models involving
uncertainty.
The equilibrium described by models involving the
REH is inherently stochastic.
For that reason, REH solutions for models can be
referred to as stochastic steady-state solutions.
15. 2.2 REH in Macroeconomics
New Classical economists like Lucas,Sargent,
Wallace, and Barro introduced the REH into
macroeconomics.
Consider a simple IS-LM-AD and As model with REH.
All variables are in logarithms.
AS and AD are expectation based aggregate supply
and aggregate demand curves.
(18)
...
Rule
Supply
Money
;
...(17)
(AD)
;
)
(
)
(
16
...
(AS),
;
1
2
1
1
0
1
1
2
1
0
1
1
0
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
e
y
m
m
v
p
p
E
p
m
y
u
p
E
p
y
16. 2.2 REH in Macroeconomics
The random terms are distributed as:
Error terms are independent from themselves in
time;
Error terms are independent from each other
);
,
0
(
~
),
,
0
(
~
),
,
0
(
~ 2
2
2
v
t
v
t
u
t N
e
N
v
N
u
s
t
e
e
E
v
v
E
u
u
E s
t
s
t
s
t
for
0
]
[
;
0
]
[
;
0
]
[
0
]
[
;
0
]
[
;
0
]
[
t
t
t
t
t
t e
v
E
e
u
E
v
u
E
17. 2.2 REH in Macroeconomics
Equation 16 states that if HHs under estimate the
price level (pt> Et−1pt), they supply more labour than
otherwise and output expands.
In Equation 16,
The expected inflation rate, Et−1(pt+1−pt), represents a
Tobin effect.
Approximations used:
Expected inflation affects AD curve.
Why and how? Answer: Tobin Effect.
level
employment
Potential
ln *
*
0
Y
y
18. 2.2 REH in Macroeconomics…
What is Tobin Effect?
Money demand(and thus the LM curve)depends on
nominal interest rate whilst investment
demand(and thus the IS curve) depends on the real
interest rate- “Tobin effect”.
Higher expected inflation rate implies lower real
interest rate that induces an increase in investment
and AD.
19. 2.2REH in Macroeconomics…
Special cases of the model
Friedman: A constant money supply growth rule
In Equation (18)
A Keynesian like Tobin would suggest a
countercyclical policy rule
If output in the previous period is low relative to its
potential, then it is advisable to raise money supply
at time t to stimulate the economy.
20. 2.2 REH in Macroeconomics…
Recall: Actual market price Pt is stochastic in the
case of REH as indicated in Equation (15).
The best prediction or the rational expectation of Pt
is the equilibrium price; P-bar.
Question1: What is the RE solution of the IS-LM-AD
and AS model or Equations 16-18?
Equate AD and AS to solve for the price level:
1
1
t
t U
a
P
P
21. 2.2 REH in Macroeconomics…
Solving for pt becomes
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
u
v
p
p
E
p
E
m
p
p
v
p
p
E
p
m
u
p
E
p
)
(
)
(
1
1
2
1
1
1
0
0
1
1
1
1
2
1
1
0
1
1
1
0
)
19
....(
)
(
)
(
)
( 1
1
2
1
0
1
1
0 t
t
t
t
t
t
t
t
t
t v
p
p
E
p
m
u
p
E
p
AD
AS
)
20
....(
]
[
2
1
1
1
2
1
1
1
0
0
t
t
t
t
t
t
t
t
t
u
v
p
p
E
p
E
m
p
22. 2.2 REH in Macroeconomics…
Take expectations based on the information set
dated t-1
We need the price error, pt –Et-1pt,in the AS curve
Rules:
Parameters are constants, known by the agents;
thus their expectation is a constant itself and can be
taken out of the expectations’ operator.
The conditional expectation of a conditional
expectation is simply the conditional itself.
)
21
...(
]
[
1
1
1
1
2
1
1
1
0
0
1
1
t
t
t
t
t
t
t
t
t
t
t
u
v
p
p
E
p
E
m
E
p
E
23. 2.2 REH in Macroeconomics…
By assumption there is no autocorrelation of
shocks:
Imposing all the above rules, we find:
Get the expression for the price error term or price
surprise (subtract (22) from (20):
)
22
...(
]
[
1
1
1
1
2
1
1
1
1
0
0
1
t
t
t
t
t
t
t
t
t
p
p
E
p
E
m
E
p
E
*
(16)
...
]
[
1
]
[
:
(16)
in
Insert(23)
)
23
]...(
[
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
u
u
v
m
E
m
Y
u
v
m
E
m
p
E
p
24. 2.2 REH in Macroeconomics…
Equation (23) says, the price is higher than
rationally expected if
The money supply is higher than what was rationally
expected
AD shock was higher than what was rationally
expected
The AS shock was lower than what was rationally
expected
Only unanticipated shocks to AD and AS, and
unanticipated changes in money supply can cause
agents to be surprised in period t or bring real effect.
25. 2.2 REH in Macroeconomics
Lets use Money Supply Rule (Equation 18).
Economic agents rationally forecast the money
supply in period t based on the information set at t-
1 as:
Hence, the “money surprise” = the difference
between Equation (18) and the above equation is
given as:
Mt–Et-1Mt=et …. (19)
(19) Shows the difference is simply stochastic.
26. 2.2 REH in Macroeconomics
Substitute the “money surprise –(24)” into modified AS
equation (16*) to get the REH solution for output:
In short it is given as
Surprising results . Why?
Output does not depend on any of the policy variables
(the coefficients)!
Hence, policy makers cannot influence output in this
model! Monetary policy is completely ineffective.
This is the strong policy ineffectiveness proposition
[PIP].
i
)
25
...(
]
[
1
1
1
1
1
1
1
0 t
t
t
t
t u
u
v
e
y
)
26
...(
1
1
1
1
1
1
0
t
t
t
t
u
v
e
y
27. 2.2 REH in Macroeconomics
The Lucas critique:
The macro-econometric models used in the 1960 and
1970s are not good for policy simulation
Their coefficients are not invariant with respect to the
policy stance.
Once you attempt to use the macro-econometric model
for setting policy; its parameters will change; so not
possible to have constant parameters. This is PIP.
Should we take PIP seriously?
Are macroeconomists useless?
To disprove a supposedly general proposition all that
is needed is one counter-example.
28. 2.2 REH in Macroeconomics
The Keynesian economist Stanley Fischer provided
this counter-example
Key idea: If there are nominal (non-indexed with
price changes or inflation)wage contracts which are
renewed less frequently than new information
becomes available, the government has an
informational advantage over the public
Result: Stabilization is possible and is
desirable[raises welfare]. PIP is invalid
29. 2.2 REH in Macroeconomics
To show the informational advantage of policy
makers is crucial, study a case with one-period and
the general case with two-period contracts.
Model 1: Single period nominal wage contracts
Consider Fischer’s (1977) model:
All variables are in logarithms. (27) says the growth
of AD is a function of only real money supply: pure
(SR) monetarist and government consumption is
absent.
It is a special case of Equation (17).
)
27
...(
t
t
t
t v
p
m
y
30. 2.2 REH in Macroeconomics
AD shock is assumed to display autocorrelation:
Supply side: Consider workers signing a one period
wage contract; after that labour demand determines
actual level of employment and thus output.
Assume that workers aim for nominal wage that ensures
full employment in the contract period.
The nominal wage is set in period t-1 to hold for period t
such that full employment of labour is expected in period t.
Workers are assumed to know the demand and supply
schedules of labor and estimate the market clearing real
wage, .
)
,
0
(
~
1
);
28
,...(
2
1
N
v
v
t
t
t
t
31. 2.2 REH in Macroeconomics
The contract is specified before the price in period t (Pt)
is known.
Workers use to determine market clearing real wage.
Full employment occurs at a point where
If actual price exceeds the expected price ,
employment is higher than full employment; because
actual real wage rate is higher than full
employment real wage rate .
The opposite happens when
Reason: Real wage rate is too high and the economy
settles in lower level of employment.
)
( 0
t
t p
w
)
( e
t
t p
w
0
t
P e
t
P
e
t
t P
P
e
t
P
e
t
t P
P
32. 2.2 REH in Macroeconomics
Workers use the expected wage rate to determine
the market clearing real wage.
Why? The contract is signed before the price in
period t is known.
Let be the nominal wage (in logarithm)
specified at the end of period t-1 to hold in period t.
The equilibrium real wage rate is normalized to unity
(so that its logarithm is zero).
Then is set as:
where (t-1) is date of contract settlement and log of
the normalized real wage rate equated to 1 is 0,, .
)
1
(
t
wt
)
29
....(
1 1 t
t
t p
E
t
w
)
1
(
t
wt
.
0
33. 2.2 REH in Macroeconomics…
Note once again: Labour demand determines the
quantity of labour traded and thus the supply of output
depends on the actual real wage in period t.
But inserting (29) into the above equation gives;
(30)
The shock in output supply is auto-correlated
)]
1
(
[ t
t
t
t u
t
w
p
y
34. 2.2 REH in Macroeconomics…
Supply equation (30) is a special case of the Lucas-
type AS curve:
Assume the policy rule of the monetary policy maker
has the form:
…(31)
In principle the policy maker can react to all past
shocks in AD and AS.
In practice it is only needed to react to shocks
lagged once and lagged twice, so that
35. 2.2 REH in Macroeconomics…
Rational expectations solution for the price error:
This can be obtained by equating AD and Lucas-
type AS
First, solve for pt :
Taking the conditional expectations on both sides:
Subtracting Et-1Pt from Pt to get:
…(32)
t
t
t
t
t
t
t
t u
p
E
p
v
p
m
y
1
36. 2.2 REH in Macroeconomics…
a) mt- Et-1mt =0 as the MSR only contains variables
that are in the information set of the agent at time t-
1 (no stochastic term in the MSR); even if we have
we have a stochastic error term et.
b) vt- Et-1vt = t as agents know the stochastic process
for vt;
c) ut- Et-1ut = t as agents know the stochastic process
for ut
Note that autocorrelation of shocks imply that agents
can use information on the shocks in period t-1 to
forecast the shocks in period t:
37. 2.2 REH in Macroeconomics…
Using this information we have:
Hence the rational expectation solution for price
error is:
…(33)
Substituting the price surprise (33) into the Lucas-
type AS equation, we get:
…(34)
t v t 1 t
t u t 1 t
v v
u u
38. 2.2 REH in Macroeconomics…
Conclusion
For model 1 we still have Policy Infectiveness
Proposition (PIP). The policy parameters do not
influence aggregate output at all despite the fact that
there are nominal contracts.
Anticipated monetary policy is unable to cause
deviations of output from its natural path. Why?
The reason is that the policy maker is as much in the
dark as the private agents are and thus has no
informational advantage.
39. 2.2 REH in Macroeconomics…
Model 2:Two-period over lapping nominal wage
contracts
AD curve and MSR the same as before:
….(35)
Nominal wage contracts
Run for two periods
Each period, half of the work force is up for renewal of
their contract
Wage is set such that market clearing of labour market
is expected
40. 2.2 REH in Macroeconomics…
In period t half of the work force receive wt(t-1) and
the other half receives wt(t-2):
Half of the work force is on wages based on “stale or
decayed information”(i.e. dated t-2)
Firms are perfectly competitive[law of one price for
output of two periods].
AS is given by:
…(36)
41. 2.2 REH in Macroeconomics…
(a) supply by firms which renewed their workers’
contract in the previous period plus
(b) supply by firms which renewed their workers’
contract two period ago
Substituting wages
into AS, we have:
…(37)
This is AS when there are overlapping nominal wage
contracts.
The AS curve contains two surprise terms differing in
their information set.
42. 2.2 REH in Macroeconomics…
What is the rational expectations solution?????
This can be obtained by repeated substitution:
First solve for pt using AD and AS:
Taking expectations conditional upon t-2 on both
sides to get:
Notice that:
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
p
p
p
p
p
E
p
E
u
v
m
u
p
E
p
p
E
p
v
p
m
y
2
2
/
1
2
/
1
2
/
1
2
/
1
]
[
2
/
1
2
/
1
2
1
2
1
43. 2.2 REH in Macroeconomics…
It indicates what agents expect in period t-1 about the
price level in t (based on information in t-2).
It is difficult for agents to know ahead how they are
going to change their mind; law of Iterated
Expectations. Only new information enables you
change your expectation; but not available currently.
Hence, we have
Then we get:
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
t
u
E
v
E
m
E
p
u
E
v
E
m
E
p
E
p
E
p
E
E
u
E
v
E
m
E
p
E
2
2
2
2
-
t
2
2
2
2
2
2
2
2
2
2
2
E
:
have
we
So
]
[
2
/
1
)
(
2
/
1
.
2
.
2
1
2
1
44. 2.2 REH in Macroeconomics…
Next take expectation conditional upon t-1 on pt
Notice and since
is known in period t-1 and hence it is constant.
We have
Et-1pt=1/2[Et-1mt+Et-1vt-Et-1ut+1/2(Et-1pt+Et-2pt)]
By substituting the value of Et-2pt into the Et-1pt
equation, we get
45. 2.2 REH in Macroeconomics…
By substituting the expressions for Et-2 pt and Et-1 pt
into pt, we get the REH solution for price level:
Upon substitution the expression for the price level into
AD, we get the intermediate REH output level:
Notice that Et-1mt =mt and can be simplified as:
The money surprise is given by:
46. 2.2 REH in Macroeconomics…
Taking the expectation we have:
Notice that and Et-2 vt-1=
With little algebra, we have:
47. 2.2 REH in Macroeconomics…
Note that:
Thus we have:
Substituting this equation into the intermediate
REH output we obtain,
t 1 v t 2 t 1
t 1 u t 2 t 1
v v
u u
t t 2 t 11 t 1 21 t 1
m E m
48. 2.2 REH in Macroeconomics…
This provides the counter example to PIP.
The expression contains policy parameters.
Output can be affected by monetary policy even
under REH.
