6.5. Productivity growth, the Phillips curve, and the natural rate. (Braun, 1984, and Ball and Moffitt, 2001.) Let gt be growth of output per worker in period t, πt inflation, and π W t wage inflation. Suppose that initially g is constant and equal to gL and that unemployment is at the level that causes inflation to be constant. g then rises permanently to gH > gL . Describe the path of ut that would keep price inflation constant for each of the following assumptions about the behavior of price and wage inflation. Assume φ > 0 in all cases.

(a) (The price-price Phillips curve.) πt = πt −1 − φ(ut − u), πw t = πt + gt.

(b) (The wage-wage Phillips curve.) πw t = πw t −1 − φ(ut − u), πt = πw t − gt.

(c) (The pure wage-price Phillips curve.) πw t = πt −1 − φ(ut − u), πt = πw t − gt.

(d) (The wage-price Phillips curve with an adjustment for normal productivity growth.) πw t = πt −1 + gˆt − φ(ut − u), ˆgt = ρgˆt −1 + (1 − ρ)gt, πt = πw t − gt.

Assume that 0 < ρ < 1 and that initially ˆg = gL .