Answer to Question #219909 in Macroeconomics for Collins

Question

Consider the expectations adjusted Phillips’s curve and assume that expected inflation
is given by πet = πt-1. Suppose that unemployment is initially equal to the natural rate
and that π=10%. The central bank decides that inflation is too high and that, starting in
year t, it will maintain the unemployment rate 1% point above the natural rate until the
inflation rate has decreased to 2%.
(a) What is the sacrifice ratio in this economy [Hint: the sacrifice ratio is the percentage of
a year’s excess unemployment needed to reduce inflation by 1%. For a Philips curve
given as  t −  t −1 = −α (ut − un ), the sacrifice ratio is 1/α]?
(b) Compute the rate of inflation for year t, t+1, t+2, t+3, …, t+8.
(c) For how many years must the central bank keep the unemployment rate above
the natural rate of unemployment? Is the implied sacrifice ratio consistent
with your answer to (a)?

Answer 1

Phillips curve is a concept which describes an inverse relationship between inflation and unemployment . This concept has been mutated in the form and the new concept depicts an analytical relation between the change in inflation rate and the difference between natural and actual level of unemployment . Following equation contains this relation:

π_t − π^e_t = −α (ut − un ) 

Phillips Curve Equation : 

π_t − π^e_t = −α (ut − un ) 

where , π_t^e = π_{t −1} 

a.) Sacrifice Ratio refers to the change in unemployment which is to tolerated to reduce the inflation by 1 percentage point . We find the sacrifice ratio by taking the difference inflation we want and the unemployment rate which is to be maintained .

Given :

, π_t = 10 %

(ut − un) to be maintained as 1% 

& 

inflation rate is to be reduced to 2%

so the sacrifice ratio given by :

$\frac{1}{α}= \frac{change\space in\space inflation\space required }{unemployment \space above\space natural\space level } =\frac{ 10\% − 2\%}{1\%} = 8$

b.) 

inflation rate in t+1

π_t+1 − π^e_t+1 =

π_t+1 − π_t = −α (ut − un ) 

$π_{t+1} = -\frac{ 1}{8}\times0.01+0.1(simce,\pi_t=10\%\space \\and \space (ut-un)=1\%)$

π_t+1 = 9.8%

Similarly 

π_t+2 - π_t+1 = −α (ut − un ) 

$\pi_{t+2}=-\frac{1}{8}\times0.01+0.098\\=9.6\%$