Mausumi Das
Lecture Notes, DSE
August 5, 2014
In the last class we had speci…ed 4 di¤erent expectation formation rules.
Static Expectations:
xte =xt 1
Adaptive Expectations:
xte =xte 1+λ[xt 1 xte 1]; 0<λ<1 Perfect Foresight:
xte =xt Rational Expectations:
xte =E[xtjIt 1]
Recall that in the Classical system, when both workers and producers base their supply/demand decisions on the actual real wage Wt
Pt , then the labour market equilibrium is given by:
N¯ :g(N) =FN(N,K¯)
ThisN¯ - which we shall call the‘Natural Level of Employment’- is independent of the current price level (Pt).
On the other hand, when workers base their supply decisions on the expected real wage Wt
Pte , then the actual level of employement di¤ers from the natural rate(N¯ ) in the following way:
In other words,
Nt TN¯ according as Pte SPt. De…neY¯ as the‘natural level of output’such that
Y¯ =F(N¯ ,K¯).
When workers base their supply decisions on the expected real wage, then the actual output supplied di¤ers from the natural level(Y¯ ) in the following way:
Yts TY¯ according as Pte SPt.
This allows up to write the Aggregate Supply schedule in the following way:
Yts :Yt =Y¯ +f(Pt Pte); f(0) =0;f0 >0.
This representation of the aggregate supply schedule is called the Lucas Supply Function (after Robert Lucas, who postulated that (in the short run) workers’may not have perfect information about the price behaviour and hence their expectations may di¤er from the actual.)
Without any loss of generality, let us assume that the Lucas Supply Function (i.e., the AS schedule under inperfect information) is linear:
Yts :Yt =Y¯ +α[Pt Pte]; α>0. (I) The equilibrium price level will of course depend on aggregate
demand function, which we now turn to.
we know that the Aggregate Demand schedule is a decreasing function of the price level:
Ytd :Yt =h(Pt); h0 <0
We also know that aggregate demand increases corresponding to any increase in the the policy parameters G¯,M.¯ Thus
Ytd :Yt =h(Pt;G¯;M¯ ); h0 <0; ∂Y
∂G¯ >0;∂Y
∂M¯ >0 Without any loss of generality, let us again assume that the AD schedule is linear:
Ytd :Yt = µPt +γG¯ +µM;¯ γ,µ>0 (II) Question: In the AD schedule written above, why have we attributed same coe¢ cient (µ) to bothPt and M¯ ?
From the AS and the AD schedule (given by (I) and (II) repectively, we can solve for the equilibrium price level at time t as:
Pt : Y¯ +α[Pt Pte] = µPt+γG¯ +µM¯ ) Pt = 1
α+µ[γG¯ +µM¯ Y¯ +αPte] (III) Equation (III) gives us the precise relationship bewteen actual price level and expected price level in this economy at every point of time.
Let us now apply the di¤erent theories of expectation formation to equation (III) and see how the behaviour of the aggreagte economy changes (if at all) over time.
We know that the equilibrium price level at timet is determined by the following equation:
Pt = 1
α+µ[γG¯ +µM¯ Y¯ +αPte] (III) Under Static Expections:
Pte =Pt 1
Plugging this value ofPte in equation (III) we get a single di¤erence equation inPt, which will determine the movement of equilibrium price level over time:
Pt = α
α+µPt 1+ 1
α+µ[γG¯ +µM¯ Y¯ ]
Under Adaptive Expections:
Pte =Pte 1+λ Pt 1 Pte 1
Plugging this value ofPte in equation (III) we get a system of two di¤erence equations in two variables,Pt and Pte,which will simultaneously determine the movement of equilibrium price level as well as expected price level over time:
Pt = αλ
α+µPt 1+ α(1 λ)
α+µ Pte 1+ 1
α+µ[γG¯ +µM¯ Y¯ ](1) Pte = λPt 1+ (1 λ)Pte 1 (2)
Under Perfect Foresight:
Pte =Pt
Plugging this value ofPte in equation (III) we get a unique solution for the equilibrium price (Pt), which must be the ‘perfect foresight’
solution to the system:
1 α
α+µ Pt = 1
α+µ[γG¯ +µM¯ Y¯ ] ) Pt = 1
µ[γG¯ +µM¯ Y¯ ] =Pte
Under Rational Expectations:
Pte =E[PtjIt 1]
Notice that under perfect certainty, the information set,It 1, would include the information that the equilibrium price level in every period is determined by:
Pt = 1
α+µ[γG¯ +µM¯ Y¯ +αPte]
Hence when agents form their expectations, they will utilize this information. In other words:
E(Pt) = E 1
α+µ[γG¯ +µM¯ Y¯ +αE(Pt)]
= 1
α+µ[γG¯ +µM¯ Y¯ ] + α
α+µE(Pt) ) E(P ) = 1 [ G¯ + M¯ Y¯ ]
Notice that once we replace this value of E(Pt)in the equilibrium price determinantion equation (III), we get
Pt = 1
µ[γG¯ +µM¯ Y¯ ] =E(Pt)
Point to note: The rational expectation solution and the perfect forsight solution are identical, although the underlying mechanisms of arriving at the two solutions are di¤erent!
Let us now introduce some uncertainty in the system such that the price determinantion equation is given by:
Pt = 1
α+µ[γG¯ +µM¯ Y¯ +αPte] +et (IIIa) where et is a random variable with an expected value of e.¯
In this case the perfect foresight solution is given by:
Pt = 1
µ[γG¯ +µM¯ Y¯ ] +α+µ
µ et =Pte.
On the other hand the Rational Expectation solutions for expected price and actual price are given by:
E(Pt) = 1
µ[γG¯ +µM¯ Y¯ ] +α+µ µ e¯
1 α+µ
Notice that since the realized value of the random termet may not be equal to its expectated value e,¯
the solution for expected price level under perfect foresight and under rational expectation now di¤er.
In fact, under perfect foresight, the expected price level still coincides with its actual value.
However, under rational expectations, the expected price level now di¤ers from the actual price level due to the existence of a random surprise term (et e).¯
Food for Thought:
In this model (because the functional forms are assumed to be linear) we end up with ‘unique’perfect foresight/rational expectation solutions.
It is conceivable that these equations are not linear. Then multiple perfect foresight/rational expectation solutions may exist.
In such a scenario, would agents’expectaions be necessarily met even under the assumption of perfect foresight/rational expectation (with complete certainty and perfect information)?
The answer is "No" - unless all agents coordinate and everybody picks the same value among these multiple solutions! This in fact is one of the problematic areas of the perfect foresight/rational expectation hypothesis, which we shall come back to later in the course.
