Rational Expectations

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Rational Expectations

• Economic Agents Forecast Economic Magnitudes
  Making No Systematic Errors
• 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

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Rational Expectations and Macroeconomic
Stabilization Policy An Overview - McCallum JMCB
1980
Labor Market
Goods and Asset Markets
Monetary Policy Rule
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Deriving the Lucas Supply Function Eliminate w
from nd and ns
Production Function
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The Aggregate Demand Schedule eliminate it with
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The Rational Expectation of the Price Level
solve for pt with
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Agents use this last expression to form their
expectations. Now form the forecast error
Insert this last expression into the supply
schedule to get
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Expected Money
Unexpected Money mt Et-1mt et
Policy Ineffectiveness Proposition
Policy Invariance Proposition
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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.
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Method of Undetermined Coefficients solution
method
AD Schedule AS Schedule Policy Rule
McCallum assumes that agents have current period
information when forming their expectations. Conje
cture the following solution.
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Price equation implies that Et-1pt ?21mt-1.
Further Etpt1 ?21Etmt ?21Et(u1mt-1 et)
?21u1mt-1 ?21et
Put the price and output solutions into the AS
schedule to get
Equating coefficients gives
Four more coefficients needed.
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Use the aggregate demand schedule to get
Equating coefficients gives
Now solve the eight equations for the eight
coefficients.
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Signal Extraction
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 interested. 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
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If we are restricted to the class of linear
forecasting rules, then we want to use
If we want the minimum variance estimator,
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then we must have
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
giving the normal equations
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The xi need not be fixed in repeated sampling as
in OLS. In a two variable system with an
intercept we get
Solve for the parameters a0 and a1 to get
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Optimal Forecasting Equation the projection of y
on x
Example suppose an agent sees the composite data
x s n and wishes to forecast s. Statistical
assumptions
Using the projection formulas gives
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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.
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Some International Evidence on Output-Inflation
Tradeoffs Lucas AER 1973
Attempts to use incomplete information and signal
extraction to explain the existence of a Phillips
curve in the aggregate economy. 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)
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Pt is the general price level and 0 lt ? lt 1 due
to adjustment costs. Agents know normal supply
and all lagged values of economic
data. Statistical assumptions
Agents must forecast
Agents observe
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Using the optimal forecasting methods above gives
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.
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Output Supply
Aggregate Output Supply
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 coefficients.
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?xt has a mean of ?. Implication ? has an output
effect of zero. Unanticipated changes in xt have
output effects.
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Rational Expectations and the Role of Monetary
Policy JME 1976

• Objective
• Provides an analysis of the impact of monetary
  policy in an incomplete information
  market-clearing model with rational expectations.
• Provides an explanation of the Phillips curve.

Assumptions Money enters the economy as a
transfer. Economy-wide variables are known to
agents with a one period lag.
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Localized Supply
Localized Demand
Wealth Effects higher wealth increases leisure,
reducing labor supply. Higher wealth increases
goods demands.
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Market-Clearing Conditions
Assumptions for Stochastic Processes
Price Solution
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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 ? gt ?.
  Unanticipated money is not neutral.

Output Solution
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• Implications of the Output Solution
• Mt-1 has no effect upon output. Only
  unanticipated money affects output and, if H gt 0,
  has a positive relationship to real output.
• Aggregate and relative shifts have the same
  effects since agents cant tell the difference.
  Output has no persistence.

Phillips Curve Slope
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.
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• Monetary Policy
• Let denote the full information value of an
  economic magnitude.
• Assume that agents have current period
  information about random variables.

Mt-1 and mt now each have unit coefficient.
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mt no longer affects output. Aggregate and
relative shocks now have different
effects. Policy Criterion minimize
Barro shows that minimizing this requires that
the variance of unanticipated money should be
zero. The best policy is a predictable
one. Feedback Rule
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vt-1 last periods real shift in aggregate
demand. If the Fed has the same information set
as the public,
Output equation is unaffected. Public knows the
feedback rule to ?vt-1 is in the price
equation. Since vt affects next periods money
supply, there is an additional effect of ? on Pt.
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Superior Information suppose that the Fed knows
vt but the public does not.
Policy Rule
Variance of Aggregate Demand
Now ? can be chosen to minimize the variance of
aggregate demand.
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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
Policy Rule
Substitute the policy rule in the equation above
it to get
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Solve for the unconditional mean of y to get
Difference equation for y gives
If the goal of policy is to minimize the variance
of y then
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• Friedmans 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.
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Policy with Rational Expectations

• Lucas Critique
• Under rational expectations,

Implication the parameters of the policy rule do
not affect output.
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Phillips Curve Example
With correct, fully anticipated inflation
expectations
?
U
UN
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AD Curve
Exogenous Variables xt ?xt-1 ut
Case I Irrational Expectations
Money has real effects if the public is fooled by
the monetary authorities.
Case II Rational Expectations
Policy Rule
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Policy parameters do not appear in the
unemployment equation.
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Econometric Policy Evaluation A Critique Lucas
CRCS 1976

• Objective
• 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 yt1 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

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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

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• Consider yt1 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 variables.
• If (F, ?) is stable, this implies that agents
  views about shocks are invariant to changes in
  the true behavior of these shocks.
• This is not true and may explain why parameters
  change over time.
• Also x obeys some sort of stochastic process
  that is ignored in the traditional way of doing
  policy analysis.

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Example Permanent Income cpt kypt, ct cpt
ut, yt ypt vt Muths definition of permanent
income

• Actual Income Process yt a wt vt
• vt is transitory income
• wt are independent increments with zero mean and
  constant variance.

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Using the minimum variance estimator for yti and
inserting this into the consumption function gives
Now suppose that income rises permanently by an
increment xt. 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.
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Theory predicts the change in consumption to be
Empirical equation above predicts the change at
time t to be
Policy Considerations
A policy is a change in the parameter ?.
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Rules Rather Than Discretion The Inconsistency
of Optimal Plans Kydland and Prescott JPE 1977

• Objective
• 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

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Optimal Control Theory max

• 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.

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Consistent Policy 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.
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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
But x1 depends upon ?2 so full optimality
requires
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Inflation-Unemployment Example The view has been
expressed that, even if there is a natural
unemployment rate, it is possible to choose an
optimal inflation-unemployment combination.
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.
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xt
Consistent Plan
ut u
Optimal Solution
Optimality Condition
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Infinite Horizon yt State Variables, ?t
Policy Variables, ? Random Shocks, xt
Decision Variables for Economic Agents
State Dynamics yt1 F(yt, ?t, xt, ?t) Feedback
Policy Rule for Future Periods ?s ?f(ys), s gt
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
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If social welfare is given by

• ?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.

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Long-Term Contracts, Rational Expectations, and
the Optimal Money Supply Rule

• Uses Long-Term Overlapping Contracts to Explain
  Why Monetary Policy Can Have a Stabilization Role
  Even in a World with Rational Expectations

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One Period Contracts
Wage Setting
Output Supply
?
Aggregate Demand
Shocks
Solve for Price
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Impose Rational Expectations
Policy Rule
?
Forecast Error
Implication Monetary policy has no effect upon
output.
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Two-Period Nonindexed Labor Contracts
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
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Output Supply
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.
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Indexed Contracts
This contracting scheme implies that
Indexing Formula implied by this is
This is not the sort of contracting observed in
the economy.
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Consider a more likely form of contracting with
an inflation adjustment.
If the first period wage is set to minimize the
variance of the real wage in the first period,
then
Monetary policy can minimize the variance of
output in this case at the cost of destabilizing
real wages.
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Staggered Wage Setting in the Macro Model

• 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
  employment.

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Model Setup Contract Wage Determination
x log of the contract wage rate, y excess
demand Hat Conditional Expectation Wages Set
for Two Periods
Money Demand
W Aggregate Wage Rate, m Money Stock, v
Velocity Shock
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Policy Rule mt gwt Aggregate Demand yt
-?wt vt, ? 1 g Aggregate Wage Rate wt
.5(xt xt-1) These assumptions generate
Assuming a stable solution for x yields
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Aggregate Wage Equation
Wage inertia is determined by ? which in turn is
determined by ?, ?, and 0 lt d lt 1.
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• 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 potential
  GDP.
• 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).