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PHILIPS - RATIONAL EXPECTATIONS AND PHILLIPS CURVE HAKAN...

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RATIONAL EXPECTATIONS AND PHILLIPS CURVE HAKAN YILMAZKUDAY 1. RATIONAL EXPECTATIONS In this section, I will give a short and brief definition of rational expectations. Then some of the solution methods for rational expectations will be introduced. “The theory of rational expectations was first proposed by John F. Muth of Indiana University in the early sixties. He used the term to
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RATIONAL EXPECTATIONS AND PHILLIPS CURVE 2 describe the many economic situations in which the outcome depends partly upon what people expect to happen.” 1 Errors in expectations are costly to those who make them. So agents try to avoid these errors by using all the information they have. This can be thought as avoiding systematic errors. This behaviour is known as Rational Expectations. “Of course, expectations will actually differ across individuals. But as Muth noted in his original paper [47, p.321], these differences will be unimportant in the aggregate unless they are significantly correlated with other cross-sectional differences among agents.” 2 Rational expectations are forward-looking. That is to say, rational people take the new information into account in order to modify their expectations for the future behaviour of a variable. After this simplest introduction, I will try to show the solution method for a rational expectation. Suppose that we have: y t = a 0 + a 1 E t y t+1 + u t (1.1) 1 Thomas J. Sargent 2 McCallum (Nov., 1980).
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RATIONAL EXPECTATIONS AND PHILLIPS CURVE 3 where u t is the white noise. According to this simple model, we are trying to find a value for the variable, y t , by taking the expectation of the future value of that variable into account. In the equation (1.1) , we can say that the determinant of y t is u t and a constant. So the solution of y t must be at the form of: y t = φ 0 + φ 1 u t (1.2) If we assume (1.2) is true, than we can write an equation for E t y t+1 : E t y t+1 = E t ( φ 0 + φ 1 u t+1 ) = φ 0 since E t u t+1 = 0. Then we can rewrite (1.1) as follows: φ 0 + φ 1 u t = a 0 + a 1 φ 0 + u t (1.3) For this equation to hold, φ 0 must be equal to a 0 + a 1 φ 0 ; and φ 1 u t must be equal to u t . So we find the values for φ 0 and φ 1 as a 0 /(1-a 1 ) and 1 respectively. Consequently, we find the solution of y t as follows:
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RATIONAL EXPECTATIONS AND PHILLIPS CURVE 4 0 t t 1 a y = +u 1-a (1.4) So we found a solution for a variable which depends on the future value of itself. This solution does not involve any expectation anymore. What if we add a lagged endogenous variable to (1.1) ? Then the equation becomes: y t = a 0 + a 1 E t y t+1 + a 2 y t-1 + u t (1.5) In the equation (1.5) y t-1 is another determinant of y t . So our solution model needs some modification. Now, we will try to find a solution in the form of: y t = φ 0 + φ 1 y t-1 + φ 2 u t (1.6) Then; E t y t+1 = φ 0 + φ 1 y t E t y t+1 = φ 0 + φ 1 ( φ 0 + φ 1 y t-1 + φ 2 u t ) (1.7) Putting (1.6) and (1.7) into (1.5) yields
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