Economics 302 Menzie D. Chinn Spring 2011 Social Sciences 7418 University of Wisconsin-Madison Notes on the Phillips Curve Take the combination of the price and wage setting equations: ),()1( zuFPPeμ+= Linearize the F(.) function: zuzuF +−=α1),( Substitute the second equation into the first (which yields equation 8.1), and put in time t subscripts: )1)(1(ttettzuPP +−+=αμ Divide both sides by 1−tP: )1)(1(11tttetttzuPPPP+−+=−−αμ (8A.1) Since tttttttPPPPPPπ+≡−+≡−−−−1)(1111 and etttetttetPPPPPPπ+≡−+≡−−−−1)(1111 Then: )1)(1)(1()1(ttettzu +−++=+αμππ Divide both sides by )1)(1(μπ++etto obtain: )1()1)(1()1(ttettzu +−=+++αμππ Approximate μππμππ−−+≈+++ettett1)1)(1()1( . Then: ttettzu +−=−−+αμππ11 Rearranging: ttettuzαμππ−++= )( (8.3) If expected inflation is always zero, one obtains the original Phillips curve: tttuzαμπ−+= )( (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: 1−=tetθππ (8.5) Which yields a Phillips curve of the form: ttttuzαμθππ−++=−)(1 When θ = 1, one obtains the accelerationist hypothesis: ttttuzαμππ−++=−)(1 implies ttttuzαμππ−+=−−)(1 (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. ttuzαμ−+= )(0 αμ)(tnzu+= also )(tnzu +=μα (8.8) Substitute (8.8) into (8.3): ttettuzαμππ−++= )( Bring expected inflation to the left hand side. tnettuuααππ−=− )(ntettuu −−=−αππ (8.9) Using the accelerationist hypothesis: )(1 ntttuu −−=−−αππ (8.10) E302_phillips_s11
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