Economics 302 Spring 2011 University of Wisconsin Madison Menzie D Chinn Social Sciences 7418 Notes on the Phillips Curve Take the combination of the price and wage setting equations P P e 1 F u z Linearize the F function F u z 1 u z Substitute the second equation into the first which yields equation 8 1 and put in time t subscripts Pt Pt e 1 1 u t z t Divide both sides by Pt 1 Pt Pe t 1 1 u t z t Pt 1 Pt 1 8A 1 Pt e Pt 1 Pt e Pt 1 Pt Pt 1 Pt Pt 1 1 t and 1 te Since P Pt 1 Pt 1 Pt 1 t 1 Then 1 t 1 te 1 1 u t z t e Divide both sides by 1 t 1 to obtain 1 t 1 u t z t 1 te 1 1 e t Approximate 1 e 1 1 t t Then t 1 t te 1 u t z t Rearranging t te z t u t 8 3 If expected inflation is always zero one obtains the original Phillips curve t z t u t 8 4 This seems to apply to the 1948 69 period shown below However it breaks down post 1969 1948 69 1970 onward How can one explain the post 1969 data If instead of zero expected inflation expectations are formed adaptively backward looking one could write expected inflation as te t 1 8 5 Which yields a Phillips curve of the form t t 1 z t u t When 1 one obtains the accelerationist hypothesis t t 1 z t u t implies t t 1 z t u t 8 6 It would be useful to express the Phillips curve as a function of the gap between unemployment and the natural rate of unemployment One can solve for the natural rate of unemployment by setting the change in inflation in equation 8 6 equal to zero 0 z t u t zt un also u n z t 8 8 Substitute 8 8 into 8 3 t te z t u t Bring expected inflation to the left hand side t u n u t t te u t u n e t 8 9 Using the accelerationist hypothesis t t 1 u t u n 8 10 E302 phillips s11 21 3 2011
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