004: Macroeconomic Theory

Transcription

1 004: Macroeconomic Theory Lecture 5 Mausumi Das Lecture Notes, DSE August 5, 2014 Das (Lecture Notes, DSE) Macro August 5, / 16

2 Various Theories of Expectation Formation: In the last class we had specified 4 different expectation formation rules. Static Expectations: Adaptive Expectations: x e t = x t 1 x e t = x e t 1 + λ [x t 1 x e t 1] ; 0 < λ < 1 Perfect Foresight: Rational Expectations: x e t = x t x e t = E [x t I t 1 ] Das (Lecture Notes, DSE) Macro August 5, / 16

3 Various Expectation Formation Rules: An Application (Classical Labour Market) Recall that in the Classical system, when both workers and producers ( ) Wt base their supply/demand decisions on the actual real wage, then the labour market equilibrium is given by: N : g(n) = F N (N, K ) P t This N - which we shall call the Natural Level of Employment - is independent of the current price level (P t ). Das (Lecture Notes, DSE) Macro August 5, / 16

4 On the other hand, ( when) workers base their supply decisions on the Wt expected real wage, then the actual level of employement P e t differs from the natural rate ( N ) in the following way: Das (Lecture Notes, DSE) Macro August 5, / 16

5 In other words, N t N according as P e t P t. Define Ȳ as the natural level of output such that Ȳ = F ( N, K ). When workers base their supply decisions on the expected real wage, then the actual output supplied differs from the natural level (Ȳ ) in the following way: Y s t Ȳ according as P e t P t. This allows up to write the Aggregate Supply schedule in the following way: Y s t : Y t = Ȳ + f (P t P e t ); f (0) = 0; f > 0. Das (Lecture Notes, DSE) Macro August 5, / 16

6 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 differ 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: Y s t : Y t = Ȳ + α [P t P e t ] ; α > 0. (I) The equilibrium price level will of course depend on aggregate demand function, which we now turn to. Das (Lecture Notes, DSE) Macro August 5, / 16

7 we know that the Aggregate Demand schedule is a decreasing function of the price level: Y d t : Y t = h(p t ); h < 0 We also know that aggregate demand increases corresponding to any increase in the the policy parameters Ḡ, M. Thus Y d t : Y t = h(p t ; Ḡ; M); h < 0; Y Ḡ > 0; Y M > 0 Without any loss of generality, let us again assume that the AD schedule is linear: Y d t : Y t = µp t + γḡ + µ M; γ, µ > 0 (II) Question: In the AD schedule written above, why have we attributed same coeffi cient (µ) to both P t and M? Das (Lecture Notes, DSE) Macro August 5, / 16

8 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: P t : Ȳ + α [P t Pt e ] = µp t + γḡ + µ M P t = 1 α + µ [γḡ + µ M Ȳ + αpt e ] (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 different theories of expectation formation to equation (III) and see how the behaviour of the aggreagte economy changes (if at all) over time. Das (Lecture Notes, DSE) Macro August 5, / 16

9 We know that the equilibrium price level at time t is determined by the following equation: P t = 1 α + µ [γḡ + µ M Ȳ + αp e t ] (III) Under Static Expections: Pt e = P t 1 Plugging this value of Pt e in equation (III) we get a single difference equation in P t, which will determine the movement of equilibrium price level over time: P t = α α + µ P t α + µ [γḡ + µ M Ȳ ] Das (Lecture Notes, DSE) Macro August 5, / 16

10 Under Adaptive Expections: Pt e = Pt 1 e + λ [ P t 1 Pt 1 e ] Plugging this value of Pt e in equation (III) we get a system of two difference equations in two variables, P t and Pt e, which will simultaneously determine the movement of equilibrium price level as well as expected price level over time: P t = αλ α + µ P α(1 λ) t 1 + α + µ Pe t α + µ [γḡ + µ M Ȳ ](1) P e t = λp t 1 + (1 λ)p e t 1 (2) Das (Lecture Notes, DSE) Macro August 5, / 16

11 Under Perfect Foresight: Pt e = P t Plugging this value of Pt e in equation (III) we get a unique solution for the equilibrium price (P t ), which must be the perfect foresight solution to the system: ( 1 α ) 1 P t = α + µ α + µ [γḡ + µ M Ȳ ] P t = 1 µ [γḡ + µ M Ȳ ] = P e t Das (Lecture Notes, DSE) Macro August 5, / 16

12 Under Rational Expectations: P e t = E [ P t I t 1 ] Notice that under perfect certainty, the information set, I t 1, would include the information that the equilibrium price level in every period is determined by: P t = 1 α + µ [γḡ + µ M Ȳ + αp e t ] Hence when agents form their expectations, they will utilize this information. In other words: [ ] 1 E (P t ) = E α + µ [γḡ + µ M Ȳ + αe (P t )] = 1 α + µ [γḡ + µ M Ȳ ] + α α + µ E (P t ) E (P t ) = 1 µ [γḡ + µ M Ȳ ] Das (Lecture Notes, DSE) Macro August 5, / 16

13 Notice that once we replace this value of E (P t ) in the equilibrium price determinantion equation (III), we get P t = 1 µ [γḡ + µ M Ȳ ] = E (P t ) 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 different! Das (Lecture Notes, DSE) Macro August 5, / 16

14 Difference between Perfect Foresight and Rational Expectation Solutions Under Uncertainty: Let us now introduce some uncertainty in the system such that the price determinantion equation is given by: P t = 1 α + µ [γḡ + µ M Ȳ + αpt e ] + ɛ t (IIIa) where ɛ t is a random variable with an expected value of ɛ. In this case the perfect foresight solution is given by: P t = 1 µ [γḡ + µ M Ȳ ] + α + µ µ ɛ t = P e t. On the other hand the Rational Expectation solutions for expected price and actual price are given by: E (P t ) = 1 µ [γḡ + µ M Ȳ ] + α + µ µ ɛ P t = 1 µ [γḡ + µ M Ȳ ] + α + µ µ ɛ t Das (Lecture Notes, DSE) Macro August 5, / 16

15 Perfect Foresight and Rational Expectation Solutions Under Uncertainty Notice that since the realized value of the random term ɛ t may not be equal to its expectated value ɛ, the solution for expected price level under perfect foresight and under rational expectation now differ. In fact, under perfect foresight, the expected price level still coincides with its actual value. However, under rational expectations, the expected price level now differs from the actual price level due to the existence of a random surprise term (ɛ t ɛ). Das (Lecture Notes, DSE) Macro August 5, / 16

16 Existence of Multiple Perfect Foresight/Rational Expectation Solutions: 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. Das (Lecture Notes, DSE) Macro August 5, / 16