• Economic Agents Forecast Economic Magnitudes Making No
• Extreme Informational Requirements
• Policy Ineffectiveness Proposition
• Lucas Critique of Policy Evaluation
• Incomplete Information Leads to an Inability to Discern the
Difference Between Relative and Absolute Price Changes
• Lucas Supply Curve
• Provides an Explanation of Why Output Departs from
Potential GDP in a Market-Clearing Framework
Rational Expectations and Macroeconomic
Stabilization Policy: An Overview - McCallum
Labor Market: n d n 0 n1 (w t p t ) n 2 n t 1 v1t
n s n 3 n 4 (w t E t 1p t ) v 2t
n d n s ,n i 0i,0 n 2 1
Goods and Asset Markets:
IS : y t a 0 a1 i t E t 1 p e1 p t u1t ,a i 0i
LM : m t p t c0 c1y t c 2i t u 2t ,ci 0i
Monetary Policy Rule: mt u 0 u1mt 1 u 2 y t 1 et
Deriving the Lucas Supply Function: Eliminate w from
nd and ns
n 0 n 2 n t 1 v1t n t n t n 3 v 2t
pt E t 1p t
n 0 n 4 n1n 3 n1n 4 n 2n 4
nt p t E t 1p t n t 1
n1 n 4 n1 n 4 n1 n 4
n1n 4 n 4 v1t n1v 2t
n1 n 4
y t y0 y1n t , yi 0,i 1,2
yst 0 1 (p t E t 1p t ) 2 y t 1 v t
y 0 y1 (n1 n 4 ) y1 (n 0 n 4 n1n 3 ) y 0 n 2 n 4
y1 (n1 n 4 )
y1n1n 4 n 2n 4 y1n1n 4 n 4 v1t n1v 2t
1 0, 2 0, v t
n1 n 4 y1 (n1 n 4 ) n1 n 4
The Aggregate Demand Schedule: eliminate it with
m d ms m t
a 0 y t a1E t 1 p t 1 p t u1t c0 c1y t u 2t m t p t
yd 0 1 (m t p t ) 2 E t 1 p t 1 p t u t
c 2 a 0 c 0 a1 a1
0 , 1 0,
a 2 a1c1 a 2 a1c1
c 2 a1 c 2 u1t a1u 2t
2 , ut
a 2 a1c1 a 2 a1c1
The Rational Expectation of the Price Level: solve for pt with
y d ys y t
0 y t 1m t 2 E t 1 p t 1 p t u t
y t 0 1E t 1p t 2 y t 1 v t
0 0 1m t 2 E t 1 p t 1 p t 1E t 1p t u t 2 y t 1 v t
Agents use this last expression to form their expectations.
Now form the forecast error p t E t 1p t .
1 m t E t 1m t u t v t
p t E t 1p t
Insert this last expression into the supply schedule to get
1 m t E t 1m t u t v t
y 0 1
2 y t 1 v t
Expected Money: E t 1mt u 0 u1mt 1 u 2 yt 1
Unexpected Money: mt – Et-1mt = et
Policy Ineffectiveness Proposition:
1e t u t v t
y 0 1
2 y t 1 v t
Policy Invariance Proposition:
1 m t E t 1m t u t v t
y 0 1
2 y t 1 v t
yst 0 1 ( 1 1 ) 1 (1 (m t u 0 u1m t 1 u 2 y t 1 ) u t v t )
2 y t 1 v t
yst 0 u 0 11 ( 1 1 ) 1 11 ( 1 1 ) 1 m t u111 (1 1 ) 1 m t 1
[ 2 u 2 11 ( 1 1 ) 1 ]y t 1 v t 1 ( 1 1 ) 1 (u t v t )
Sometimes called the Lucas Critique
Reduced form parameters are functions of the parameters of the
policy rule obeyed by the central bank.
Few parameters are not functions of the ui: called “deep”
parameters if they are not.
Examples: parameters of technology and utility functions.
Method of Undetermined Coefficients: solution method
AD Schedule: y d 1 (m t p t ) 2 E t 1 p t 1 p t v t
AS Schedule: ys 1 (p t E t 1p t ) u t
Policy Rule: m t u1m t 1 e t
McCallum assumes that agents have current period information
when forming their expectations.
Conjecture the following solution.
y t 11m t 1 12 u t 13 v t 14e t
p t 21m t 1 22 u t 23 v t 24e t
Price equation implies that Et-1pt = 21mt-1. Further
Etpt+1 = 21Etmt = 21Et(u1mt-1 + et) = 21u1mt-1 + 21et
Put the price and output solutions into the AS schedule to get
11m t 1 12 u t 13 v t 14 e t 1 (p t E t 1p t ) u t
11m t 1 12 u t 13 v t 14 e t 1 (22 u t 23 v t 24 e t ) u t
Equating coefficients gives
11 0, 12 122 1, 13 123 , 14 124
Four more coefficients needed.
Use the aggregate demand schedule to get
11m t 1 12 u t 13 v t 14e t 1 (u1m t 1 e t )
(1 2 )(21m t 1 22 u t 23 v t 24e t )
2 (21u1m t 1 21e t ) v t
Equating coefficients gives
11 1u1 (1 2 ) 21 2 21u1 , 12 (1 2 ) 22
13 (1 2 ) 23 1, 14 1 (1 2 ) 24 2 21
Now solve the eight equations for the eight coefficients.
Optimal forecasting procedure used when agents cannot
directly observe a variable that they wish to forecast.
What they observe is noise plus the variable in which they are
Suppose that you are interested in getting an estimate of a
variable yt and you know it is related to the variables x1,…xn.
Suppose further that you know the first and second moments
(mean and variance) of the density functions associated with
each of these variables.
Further, you also know
Eyx1 Eyx 2 Eyx n
Ex1 Ex1x 2 Ex1x n
Ex n x1 Ex n x 2 Ex n
If we are restricted to the class of linear forecasting rules, then
we want to use
y a 0 aixi
If we want the minimum variance estimator,
E y y
then we must have
E y a 0 a i x i x i 0
E y y x i 0
The implication is that the forecast error must be orthogonal to
the information set (the xi). This means that we must choose
the parameters ai to minimize
E y aixi
giving the normal equations
2E y a i x i x i
Eyx E xx a
The xi need not be fixed in repeated sampling as in OLS. In a
two variable system with an intercept we get
Ey 1 Ex a 0
Eyx Ex Ex 2 a
Solve for the parameters a0 and a1 to get
E (y Ey)(x Ex) Cov(y, x)
a 0 Ey a1Ex,a1
E(x Ex) 2
Optimal Forecasting Equation: the projection of y on x
y Ey a1Ex a1x Ey a1 (x Ex)
Example: suppose an agent sees the composite data x = s + n
and wishes to forecast s.
E(s) E(n) E(sn) 0,E(s 2 ) s ,E(n 2 ) 2
E(x 2 ) s 2 ,E(xs) E[s(s n)] s
Using the projection formulas gives
a 0 0,a1 2
ˆ (s n)
As n gets noisier (its variance rises), an agent is more inclined
to associate variations in s+ n with n, thus having a smaller
impact on its estimate of s.
As s gets noisier (its variance rises), an agent is more inclined
to associate variations in s+ n with s, thus having a larger
impact on its estimate of s.
Some International Evidence on
Output-Inflation Tradeoffs – Lucas
Attempts to use incomplete information and signal extraction
to explain the existence of a Phillips curve in the aggregate
Market are indexed by z. Supply is given by
yt(z) = ynt + yct(z), ynt = + t
yct(z) = [Pt(z) – E(Pt | It(z))] + yct-1(z)
Pt is the general price level and 0 < < 1 due to adjustment
costs. Agents know normal supply and all lagged values of
Pt N(Pt , 2 )
Pt (z) Pt z,z N(0, 2 )
Agents must forecast Pt P
E( | z))
Agents observe Pt (z) Pt z
Using the optimal forecasting methods above gives
E( | z) 2 2 ( z) 2 2 Pt (z) Pt
E(Pt | I t (z)) Pt 2 2 (Pt (z) Pt )
E(Pt | I t (z)) E(Pt | Pt (z),Pt ) Pt (z) (1 )Pt
As z gets noisier (its variance rises), an agent is more inclined to
associate variations in Pt(z) with a relative price change, thus
having a smaller impact on the estimate of aggregate price.
As Pt gets noisier (its variance rises), an agent is more inclined
to associate variations in Pt(z) with a change in the general
price level, thus having a larger impact on the estimate of
aggregate price. 20
Output Supply: y t (z) ynt (1 ) Pt (z) Pt yc,t 1 (z)
Aggregate Output Supply:
y t ynt (1 ) Pt Pt y t 1 y n,t 1
Slope of the aggregate supply curve varies with the fraction
of total variance due to relative price changes. The smaller
is this variance, the more vertical is the supply schedule.
Aggregate Demand : yt + Pt = xt
Model is solved using the method of undetermined
(1 ) 1 (1 )
Pt xt x t 1
1 (1 ) 1 (1 ) 1 (1 )
y t 1 (1 )y nt
(1 ) (1 )
yt x t y t 1 (1 )y nt
1 (1 ) 1 (1 )
xt has a mean of .
Implication: has an output effect of zero.
Unanticipated changes in xt have output effects.
Rational Expectations and the Role of
Monetary Policy – JME 1976
•Provides an analysis of the impact of monetary policy in
an incomplete information market-clearing model with
•Provides an explanation of the Phillips curve.
Money enters the economy as a transfer.
Economy-wide variables are known to agents with a one
period lag. 23
ys (z) k s (z) s Pt (z) EPt 1 | I t (z)
s M t M t 1 | I t (z) EPt 1 | I t (z) u s s (z)
s 0, s 0, s s
yd (z) k d (z) d Pt (z) EPt 1 | I t (z)
d M t M t 1 | I t (z) EPt 1 | I t (z) u d d (z)
d 0, d 0, d d
Wealth Effects: higher wealth increases leisure,
reducing labor supply. Higher wealth increases goods
k t (z) k d (z) k s (z),k t (z) d (z) s (z),u t u d u s
t t t t t t
Assumptions for Stochastic Processes:
u t u t 1 v t , v t iid(0, 2 )
M t M t 1 M t g m t ,m t iid(0, m )
Pt M t 1 1 2 ( / )(1 1 2 ) m t 1v t 1u t 1
s d , s d , 1 2 2 A 2 , A 2 m 2
Total Aggregate Disturbance: mt +-1vt
Implications of the Price Solution:
•Mt-1 has unit coefficient. Anticipated money is neutral.
•mt coefficient is less than unity if > . Unanticipated
money is not neutral.
Yt (H / )(1 1 2 )m t 1 s (H / )(1 2 ) v d d (z)
1 d (H / )(1 2 ) vs s (z) (d / )u d1 (s / )u s1
t t t t
H s d d s
Implications of the Output Solution:
•Mt-1 has no effect upon output. Only unanticipated money
affects output and, if H > 0, has a positive relationship to
•Aggregate and relative shifts have the same effects since
agents can’t tell the difference. Output has no persistence.
1 1 2 2 2 ,
Phillips Curve Slope: m v
2 1 1 2
Slope is the fraction of total excess demand variance
attributable to relative disturbance.
As the variance of unanticipated money rises, mt has a
smaller effect upon output. Agents are more likely to
attribute price changes to money. 27
-Let * denote the full information value of an economic
- Assume that agents have current period information
about random variables.
Pt* (z) M t 1 m t 1 (u t 1 v t ) 1 t (z)
Mt-1 and mt now each have unit coefficient.
y* (z) 1 (s (H / ))v d ( d (H / ))v s
t t t
( s / ) d (z) ( d / )s (z) (d / )u d (z) (s / )u s (z)
t t t t
mt no longer affects output.
Aggregate and relative shocks now have different
Policy Criterion: minimize
2 H 2 (22 2 )
E y t (z) y (z) | I t (z)
() 2 (22 2 )
Barro shows that minimizing this requires that the variance
of unanticipated money should be zero.
The best policy is a predictable one.
M t m t v t 1 ,m t iid(0, m )
vt-1: last period’s real shift in aggregate demand.
If the Fed has the same information set as the public,
Pt M t 1 1 2 ( / )(1 1 2 ) 2 m t 1 (v t t (z))
1u t 1 v t 1
Output equation is unaffected.
Public knows the feedback rule to vt-1 is in the price equation.
Since vt affects next period’s money supply, there is an
additional effect of on Pt.
Superior Information: suppose that the Fed knows vt but
the public does not.
Policy Rule: M t m t v t ,m t iid(0, m )
Pt M t 1 1 2 ( / )(1 1 2 ) m t 1 ((1 )v t t (z))
1u t 1
Variance of Aggregate Demand: 2 22 (1 )2
A m v
Now can be chosen to minimize the variance of
Rational Expectations and the Theory
of Economic Policy – Sargent and
Wallace JME 1976
First consider how to carry out policy without regard for how
expectations are formed. Let a “goal” variable be given by
y t y t 1 m t u t ,u t iid(0, u )
Policy Rule: mt g 0 g1y t 1
Substitute the policy rule in the equation above it to get 32
y t ( g 0 ) ( g1 )y t 1 u t
Solve for the unconditional mean of y to get
1 ( g1 )
Difference equation for y gives
1 ( g1 ) 2
If the goal of policy is to minimize the variance
of y then
(y * ) (y * )
g0 ,g1 m t y t 1
Friedman’s k% rule is inferior to this policy because
Friedman advocated g1 = 0.
Why should policymakers look at everything?
•There is extensive simultaneity so that a shock in any
sector impinges upon all sectors.
•Shocks have distributed lag effects on economic variables
so that their effects are to some degree predictable.
•This distributed lag structure is stable.
Advocates of k% rules may not believe that last point.
Policy with Rational Expectations:
y t 0 1 (m t E t 1m t ) 2 y t 1 u t
m t g 0 g1y t 1 t , E t 1m t g 0 g1y t 1
y t 0 1g0 1mt (2 1g1 )y t 1 u t
Under rational expectations,
y t 0 2 y t 1 u t 1 t
Implication: the parameters of the policy rule do not
affect output. 35
Phillips Curve Example:
Pt Pt 1 0 1U t t 1 Pt* Pt 1 t
With correct, fully anticipated inflation expectations
0 0 1U t t
UN U 36
AD Curve: Pt amt bx t cU t
Exogenous Variables: xt = xt-1 + ut
Case I: “Irrational” Expectations t 1 Pt* Pt 1
U t (c 1 ) 1 0 a(m t m t 1 ) b(x t x t 1 ) cU t 1 t
Money has real effects if the public is fooled by the
Case II: Rational Expectations t 1 Pt* E t 1Pt
Policy Rule: mt Gt 1 t E t 1mt Gt 1
U t (c 1 ) 1 at b(x t E t 1x t ) t (0 / 1 )
Policy parameters do not appear in the unemployment
Econometric Policy Evaluation: A
Critique – Lucas CRCS 1976
•Lucas argues that econometric practice is ill-suited to the
evaluation of the effects of alternative policies.
The Theory of Economic Policy:
-the economy is described by yt+1 = f(yt, xt, t)
- y is a vector of state variables
- x is a vector of exogenous variables
- f is unknown and is a vector of iid shocks
Task of the Empiricist: estimate f().
In practice, f(y, x, ) = F(y, x, , )
Policy: a sequence of present and future values of some or all
of x is assumed, translating into predictions about (y, x, ).
Two Aspects: theory has a secondary role in specifying F()
and the variance of the forecasts decline with the variance of
implying that the accuracy of forecasts of y declines as well.
Long-run forecasts thus appear to be reliable.
Econometricians (forecasters) ignore this theory.
- ignore data prior to 1947 despite its availability
- frequent refitting
- adjust intercepts to improve accuracy 40
Consider yt+1 = F(yt, xt, , t). Theory of economic policy
requires that (F, ) not vary with xt.
-Theory implies that this cannot be true.
- Economic agents set their optimal decision rules based
upon their views about the paths of future economic
- If (F, ) is stable, this implies that agents views about
shocks are invariant to changes in the true behavior of
- This is not true and may explain why parameters change
- Also x obeys some sort of stochastic process that is
ignored in the traditional way of doing policy analysis.
Example: Permanent Income
cpt = kypt, ct = cpt + ut, yt = ypt + vt
Muth’s definition of permanent income:
y pt (1 ) i E(y t i | I t )
Actual Income Process: yt = a + wt + vt
- vt is transitory income
- wt are independent increments with zero mean and
Using the minimum variance estimator for yt+i and
inserting this into the consumption function gives
c t k(1 )y t k(1 ) j y t j u t
Now suppose that income rises permanently by an
The theory of economic policy states that to evaluate the
effects of these increments, add x to y and insert the new
values into the consumption equation above. The
prediction derived in this way is incorrect and the error
does not go to zero as time passes to infinity.
Theory predicts the change in
consumption to be: kx
Empirical equation above predicts
the change at time t to be:
c t kx (1 ) (1 )
y t 1 F(y t , x t , , t )
y t 1 F(y t , x t , (), t )
x t G(y t , , t )
A policy is a change in the parameter .
Rules Rather Than Discretion: The
Inconsistency of Optimal Plans –
Kydland and Prescott JPE 1977
•Establishes the proposition that even with a well-defined
social welfare function and if policymakers know the
timing and effects of policies, choosing the best policy will
not maximize social welfare because rational agents will
react to policy.
•Cannot use control theory to manage an economy
•Application of the Lucas Critique
Optimal Control Theory: max S(x , v )
x t 1 x t f (x t , v t )
State dynamics depend upon current values of the
instruments and past policy choices as represented by
the current state xt.
-Not true – future policy actions affect the current state.
-Does not depend upon complete accuracy
-Argument only requires some reaction by economic
agents in a forward-looking manner.
Let = (1, 2, …, T) be a sequence of policies over time.
Let x = (x1, x2, …, xT) be a sequence of agents’ decisions.
Using the social welfare function
S(x1, x2, …, xT, 1, 2, …, T),
it is reasonable to assume that
xt = xt(x1, x2, …, xT, 1, 2, …, T).
Consistent Policy: one which maximizes social welfare,
taking as given x1,…, x,t-1.
-future choices are made in the same way.
Two Period Example: maximize
S(x1, x2, 1, 2) subject to
x1 = x1(1, 2) and x2 = x2(x1, 1, 2)
Consistent plan would give
S x 2 S
2 : 0
x 2 2 2
But x1 depends upon 2 so full optimality requires
S x 2 S x1 S S x 2
2 : x x x 0
x 2 2 2 2 x 2 1
The view has been expressed that, even if there is a
natural unemployment rate, it is possible to choose an
optimal inflation-unemployment combination.
u t x e x t u*
xt = Inflation, e = Expectation, u = Unemployment
Typically it is assumed that expected inflation depends
upon lagged inflation. But if expectations are rational, a
fully optimal policy sets inflation at zero but a consistent
policy will not generate this choice. Suppose S(xt, ut) is a
social welfare function.
xt Consistent Plan
ut – u*
Optimality Condition: 1
yt: State Variables, t = Policy Variables, = Random
Shocks, xt = Decision Variables for Economic Agents
State Dynamics: yt+1 = F(yt, t, xt, t)
Feedback Policy Rule for Future Periods: s = f(ys), s > t
Decision Rules of Rational Agents: xs = df(ys; f)
Important Point: changes in the policy rule f change the
functional form of df.
Current Period Decision Rules of Agents: xt = dc(yt, t; f )
-current period decisions depend upon future policy
If social welfare is given by
s q(x s , ys , s ),0 1
t will depend upon yt and f.
-The best current-period policy is functionally related to
the future policy rule.
-The authors argue that policymakers will likely settle for
consistent but suboptimal policies.
Long-Term Contracts, Rational
Expectations, and the Optimal Money
• Uses Long-Term Overlapping Contracts to
Explain Why Monetary Policy Can Have a
Stabilization Role Even in a World with
One Period Contracts
Wage Setting: Wt t 1 Pt
Output Supply: Yts Pt Wt u t
Yts Pt t 1 Pt u t
Aggregate Demand: Yt M t Pt v t
u t 1u t 1 t , 1 1
v t 2 v t 1 t , 2 1
Solve for Price: 2Pt M t t 1 Pt u t v t 54
Impose Rational Expectations
t 1 t t 1 M t t 1 u t v t
Policy Rule: M t a i u t i bi v t i t 1 Mt Mt
i 1 i 1
Forecast Error: Pt t 1 Pt t t
Implication: Monetary policy has no effect upon
Two-Period Nonindexed Labor Contracts
t i Wt t i Pt ,i 1,2
Wage rate, Wt, is specified by contracts drawn up in
periods t-1 and t-2
In each period, half of the firms are in the first period of
a contract drawn up at t-1 and half are in the second
year of a contract drawn up in t-2
Output Supply: Y Pt t 1Wt u t
Yt Pt t 1 Pt u t
Yt t t t 1 a1 21 t 1 b1 2 1 u t 2
Parameters from the monetary policy rule now affect output
because the Fed knows realizations of shocks that cannot be
known by the public and so the Fed can affect second-period
real wage and thus output.
Parameters of the policy rule can be chosen to minimize the
variance of output.
t i Wt P ,i 1,2
t 1 t
This contracting scheme implies that
Yt t t 1u t 1
Indexing Formula implied by this is
Wt 2 M 1 2 Pt 1 2 1 Yt 1 1Wt 1
This is not the sort of contracting observed in the
Consider a more likely form of contracting with an
t i Wt t i Wt i1 Pt 1 Pt i
If the first period wage is set to minimize the variance of
the real wage in the first period, then
Yt 6 4L 2L2 2M t 1 L2 u t 3 1 1 L 1L2
v t 3 3 2 L 2 2 L2
Monetary policy can minimize the variance of output in
this case at the cost of destabilizing real wages.
Staggered Wage Setting in the Macro
• Uses a Keynesian model of staggered wage
setting to show that staggered wage setting can
produce business cycle-like properties of the
economy even with rational expectations
• The model attempts to disentangle the effects
of expectations and contracts on output and
Contract Wage Determination:
x t bx t 1 dx t 1 by t dy t 1 t
ˆ ˆ ˆ
x = log of the contract wage rate, y = excess demand
Hat = Conditional Expectation
Wages Set for Two Periods
Money Demand: mt yt w t vt
W = Aggregate Wage Rate, m = Money Stock,
v = Velocity Shock
Policy Rule: mt = gwt
Aggregate Demand: yt = -wt + vt, = 1 – g
Aggregate Wage Rate: wt = .5(xt + xt-1)
These assumptions generate
bx t 1 cx t dx t 1 0,c (1 .5) /(1 .5)
ˆ ˆ ˆ
Assuming a stable solution for x yields
x t x t 1 t
c c 4d(1 d)
Aggregate Wage Equation:
w t w t 1 .5 t t 1
Wage inertia is determined by which in turn is
determined by , , and 0 < d < 1.
•The higher is , the less accommodative is economic policy,
the lower is . Wage inertia declines as policy becomes less
accommodating to wages but output fluctuates more around
•As d declines, the more “backward-looking” are wages and
the larger is . As d falls, the more wage inertia there will be.
•As d rises (wage-setting is more forward-looking), aggregate
demand has a bigger impact upon wages. Aggregate demand
can change by less to reduce inflation.
•Aggregate demand shocks have a hump-shaped impact upon
the output gap.
•The peak occurs at one year and it requires a contract length
of about the same length (relatively short).