Lecture Note #6. Page1 Advanced Macroeconomics
Expectations
We are going to endogenize expectations in this chapter, which will make the macroeconomic model complete. We will deal with it in two ways: the first is a traditional way of adaptive expectation. The second is Rational Expectations. The second one requires some ground work that we are going to spend some time now.
Of course, we would like to have ‘rational’, as opposed to ‘less than rational’ expectations. The, what constitutes rationality of expectations?
I. Basics
1. Notations
1) Actual values
: actual value of x at time t
: actual value of x at time t + 1, which is revealed at time t + 1
2) Expected values
: expectations formed at time t as to the value of x at time t + 1, or expectations formed at time t as to , : expectations formed at time t as to .
3) General Concept of Conditional Expectations
Concept of Conditional Expectations
:
forming expectations as to with information available up to time t.
:
forming a i – period - ahead expectations about x by using information at time t.
2. Various assumptions about Expectations Formation.
1) Static Expectations:
2) Adaptive Expectations:
The expected future value is a weighted average of the current and past values of the variable, such as:
, where is called an ‘expectation coefficient’ with the value ranging between 0 and 1.
Proof:
Error Learning Model
, where 0<
The revision of expectation between t-1 and t is proportional to the revealed forecast error or the difference between the actual value and the previous expectations.
Solving for , we get
…. (1)
Let’s work on the second term of above equation (1):
By lagging (1) for one period, we get
…….(2)
By plugging (2) into (1), we get
= ….. (3)
Now, let’s work on the third term of equation (3):
Again, by lagging (1) for two periods, we get
…..(4)
Thus, plug (4) into (3) to get:
The weights are geometrically declining ones.
Eg) when
3) Rational Expectations
Mathematically, rational expectations are the expectations that are consistent with the solution of the given set of equations and with all the relevant variables and data. Let’s deal with the Rational Expectations separately.
3. Rational Expectations:
1)A New Operational Tool: Concept of Conditional Expectations
:
forming expectations as to with all availableinformation available up to time t.
:
forming a i – period - ahead expectations about x by using all available information at time t.
All available information includes all the relevant economic theories, values of the determinant variables (data set), and other institutional information. No information is to be wasted.
- What can you tell me about the information set used in the static expectations and the adaptive expectations model in the framework of the general concept of conditional expectations?
2) Operational Rules for Conditional Expectations:
3) Meanings of Rationality in Expectations
Note that “rationality” in expectations formation does not mean “superpower” or “perfect foresight”.
The Rationality means that the information set used for expectations formation includes all available information at the time of expectations formation. Although all the relevant information is used, still the expectations will be only a statistical statement.
The expected value will inevitably differ from the realized value by the amount of “unexpected random value”.
The unexpected random value will have the average value of zero; In other words, t+1 has the expected value of zero at the time of t.
Expected value + Expectations errors = Actual Value
Expectations errors are certainly there in most cases,
but their expected/average value is zero:
If , then, that certain value should be added to the expectation of x at the time of expectation formation.
4) Mathematical and Econometric Implications (Requirements) for Rationality in expectation formation.
(1) Orthogonality
Conditional Expectation of the Forecast Errors should be zero.
; Rational economic agents make only genuine errors which are not foreseeable; if she/he can predict any errors, she/he should correct them now, rather than wait until the next period and suffer; there are no predicting forecast errors.
proof:
We can rewrite the above into
, where
Taking expectations of both sides of the equation (1);
The left side is:
Therefore: , the left hand side.
*Econometrically, this is equivalent to the following test:
Step 1. Run the original regression equation:
X = a + b Z +
And get the estimated value for the constant term or a hat, and the coefficient b or b hat.
In fact, the regression package gives you
X hat = a hat + b hat * Z with R squares, and its statistics separately.
Step 2. Get the fitted value of X or X hat (= a hat + b hat * Z)
Step 3. Get the residual terms or error terms, as the difference between X and X hat.
X – X hat
Step 4. Now run a regression of on Z again:
a + b Z.
If the Rational Expectations are correct here, what value would you get for the estimated coefficient for b?
(2) Consistency (over time)
If you form an expectation a few periods or multiple periods ahead of time, you know that your expected value will change over time as you get new information. However, you would not know what the future information set would be like, and thus you would not know what your future revision(s) of the expected value would be.
For instance, a two-period-ahead Forecast is not the same as one period ahead forecast in most cases as you get more information and thus revises your expectations.
There is no way of predicting forecast improvement. What matters is what information to be used, not when to form expectations.
proof:
Suppose
Forecast improvement to be made
between time periods t and t+1……(1)
Take the conditional expectations (on all the available information set) of both sides of the above equation:
The left hand side of equation (1):
The right hand-side of equation (1) = E[Forecast Improvement / It].
This means that E [ForecastImprovement / It] = 0; We don’t know what the future forecast improvement will be.
II. Macroeconomics Model with Adaptive Expectation Model:
- AD: Y = C(Y) + I(r) + G
= L(Y, i)
- AS:
- Adaptive Expectation Model: ; []
What will happen to Y when goes up?– What will happen to the real national income when inflation rates accelerate?
To answer this question, we will examine the impact on real interest rate and investment under the adaptive expectation model:
*Some ground work: We should know that there are two different concepts of real interest rates:
i)ex-ante real interest rate; or
This is Fisher equation. It is used for the determination of the Nominal Interest Rate in the money market: This is what the financial investor would like to get for the compensation for his loaned money.
* Dichotomy between real and monetary sectors:
= MPK
Thus r is constant in the short-run other things being equal.
ii) ex-post real interest rate;.
The ex-post real interest rate is the nominal interest rate minus the actual inflation rate.(Note: is an expected value, and is an actual value.)
This is the relevant determinant for the Investment in the goods market; This is what the physical investor would regard as his actual cost of capital.
What will happen if people form expectations in the adaptive manner?
In the Adaptive Expectations Model, when ↑, adjust upwards in a gradual fashion:
-This means that when does rise, does take time to be adjusted, and that the increase in i or nominal interest ratewill be smaller than the increase in.
when ↑ by a certain amount,
a smallerin ↑ in the short-run; and thus ‘Adaptive Expectations’
a smaller increase in ↑. ‘Fisher Equation’
Then effective ‘actual’ real interest or ‘ex-post’ real interest rate does fall:
↓r’ = i - = (i[↑] - [↑])
This will increase investment. This leads to an increase in AD and adds to the existing inflationary pressures. Potentially, this destabilizes the whole economic system.
Illustration of this case:
i) Recall the Convergence case.
Suppose that government expenditures or G rises.
When the expectations are more or less “perfect foresight”, there is no gap between the expected inflation and the actual rate of inflation. The system shows convergence.
First, an increase in G shifts the IS curve to the right as indicated by (1).
In the long-run Y = Yf , inflationary pressure P↑ ↑
LM shifts to the left (2).
ii) Non-Convergence Case
However, if people form adaptive expectations, the IS curve shifts to the right once again (3).
Y > Yf : inflation gap
Recall that inflation gap:
Mild inflation > expectations
i↑ ↑
→r↓→I↑ →IS ↑
- A critical Assessment of Monetary Policy with Rational Expectations:
Complete Macroeconomic Model
1. Introduction
There is a one-page illustrated summary of the Rational Expectations Theory with Monetary Policy.
2. Rational Expectation Model:
Let’s endogenize the expectations variable in a way consistent with rational expectations theory.
Expectations are formed in such a way that, and the expectations error of the price levelis a random term, iid, with
,
where et is an aggregate shock.
This is Lucas’ Supply Curve. For simplicity, we may assume that Then
,
,
where Ut is an aggregate demand shock.
In fact, this AD equation is a simplified version of
Implicitly, money demanded is assumed to be perfectly interest inelastic, and we also ignore the fiscal policy.
M – P may be interpreted as the log value of (M/P or real money supply).
* Here and are aggregate supply and demand shocks, iid, with (0, e.) and (0, M) respectively.
3. Solution for Y:
Step 1: Solve for reduced form equation of as follows by combining the AD and AS curves:
….. (1)
*This is incomplete as we have in the right side of the equality. We have to replace this with exogenous variables.
Step 2: Invoke the Rational Expectations Hypothesis by taking expectations, conditional on , of bothsides of the above equation (1):
*Note that the expected value of and are equal to zero, and thus they drop out.
In addition, for a notational simplicity, we use and for the conditional expectations for and .
…..(2)
In (2), we have Pe on both sides of equation. By transposing all Peto the left side of the equality sign, and solve for :
(2’)
The expected price level is an increasing function of the expected money supply and a decreasing function of the full employment income.
Step 3: Plug(2’) from above Step 2 into the place of in equation (1) from Step 1.
…….. (1’)
Now we have the reduced form solution for and , we can get
… (3)
-The difference between the actual and the expected price level is nothing but the forecast error.
* In order for the expectations mechanism to be rational, what conditions should the forecast error meet?
(Forecast errors):
**Is an alternative expectations model, say , rational?
No.
Pf
Step 4: Solve for using :
Going back to the original AS curve at the beginning:
*
Thus,
The equilibrium national income consists of the full employment income and some deviation from it, which is given by a function of the (last period’s) forecast error of money supply and the random aggregate supply and demand shocks.
4. Policy Implications:
1) Policy Invariance Theorem:
“From the last equation, we know that if the money supply is fully anticipated, , and thus the money supply has no bearing with the equilibrium level of output, ”
When the monetary authorities are adopting a mechanical rule in the determination of money supply, as long as the rule is known, deterministic Monetary Policy, which is based on the ‘policy-known’ feedback rule, cannot influence
For example, the monetary authorities may set the money supply to counter the cyclical movement in the unemployment rate as follows:
,
where UEt-1 is the current period’s unemployment rate; a >0 and b >0.
In this case, as long as the rule is known and the last period’s unemployment rate is known by now without any information delay (lag), the general public can form the rational expectations of the money supply which is the same as the actual money supply.
Thus, the forecast errors of money supply are nil, and thus the national income will be
We note that there is no money supply in this equilibrium national income (solution), and that the national income is determined only by the full employment income and the random shocks;
An anticipated change in money supply cannot influence the national income even in the very short-run.
The national income fluctuates around the full employment income in a random fashion over time.
2) Information Advantage:
What if the monetary authorities have ‘information advantage’?
Imagine a very unlikely case where the monetary authorities know what e and U will be in the next period, and thus set the money supply in advance in accordance with their formula, for instance,.
What will be the one-period-ahead forecast of money supply, forecast error and the actual national income from the Lucas supply curve?
And thus
The result is the complete stabilization of the national income. However,how likely is this in reality?
3) Random Monetary Policy
What if the monetary authorities adopt a monetary policy (set with the money supply target), which defeats anyone’s expectations?
Mt = t where
is a random variable with the expected value equal to zero and the variance of
In other words,
Then, the forecast and the national income will be
Here, an unanticipated change in money supply can influence the national income.
However, compared with the case of an anticipated money supply, the variance of Yt with this random monetary policy is larger now by 1/2 .
This means that the national income is more fluctuating now than before, and thus it is welfare reducing.
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