Question

Suppose that the Phillips curve is given by: ?? = ?? ? + ? − ??? where ?? is the inflation rate, ?? ? is the expected inflation, ?? is the rate of unemployment and, a and b are two positive parameters. Before the COVID-19 crisis, no wage contract was indexed to the inflation. After the crisis a fraction ? > 0 of wage contracts are indexed to inflation.

a. Derive the new equation for the Phillips curve after the crisis.

b. Derive an algebraic expression for the natural rate of unemployment after the crisis. It is believed that the Phillips curve for Canada will be: ?? − ??−1 = 3.2% − 0.4?? 3

c. Plot this curve on a graph with ?? on x-axis and ?? − ??−1 on y-axis.

d. Assume that ? = 0.5, find the numerical values of parameters a and b as implied by the above estimated Phillips curve.

e. What will be the natural rate of unemployment?

Answer 1

Given : ?? = ??? + ? − ???

a) In case of wage indexation,  ? proportion of nominal wages contracts depend on actual inflation and the (1- ? ) proportion, the non-indexed contracts assume inflation to be same as previous period.

Therefore, ??? =  ? (??) + (1- ? )(??-1)

Therefore, substituting the value, we get ,

?? = ??? + ? − ??? =  ? (??) + (1- ? )(??-1) + ? − ???

?? - ? (??) = (1- ? )(??-1) + ? − ???

(1-?)?? = (1- ? )(??-1) + ? − ???

or ?? = (??-1) + ?/(1-?) − [?/(1-?)]??

The new phillips curve will be steeper as 1/(1 − λ) > 1 and that this factor is increasing in λ, implying that the higher the proportion of indexed contracts the higher the effect of unemployment (represented by (?/(1-?) on inflation.

b) Natural rate of unemployment is when ?? = ??? therefore, ? − ??n= 0,

or ?n = a/b, since, ?n does not depend on indexation, therefore there will be no change in the natural rate of unemployment.

c) ?? − ??−1 = 3.2% − 0.4??

d) ?? = (??-1) + ?/(1-?) − [?/(1-?)]?? is estimated to be ?? − ??−1 = 3.2% − 0.4??

This implies, ?/(1-?) = 3.2, if ?= 0.5, this implies ?/0.5 = 3.2, or a = 1.6

similarly, b/(1-?) = 0.4 , or b/0.5 = 0.4, which implies, b=0.20

e) ?n = a/b = 1.6/0.20 = 8%