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Assume that unemployment, u, is related to inflation, π, according to the following Phillips curve: u = u − φ (π−πe), where u is the natural rate of unemployment and πe is the expected rate of inflation. Assume rational expectations and that the central bank’s preferences are given by the loss function L(u,π) =λu+π2, where λ denotes the weight that the central bank assigns unemployment.


a. Suppose that φ = 1. Show what rate of inflation a central bank with λ = .04 will choose under discretion. What will the unemployment rate be?


b. Assume that a less conservative executive board of the central bank is appointed with the weight λ = .08 on unemployment. What is the new inflation rate? How is unemployment affected?


c. What will inflation and unemployment be if the executive board of the central bank does not care at all about unemployment, i.e. if λ = 0?


d. What can we learn from the calculations above?


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Solution:-

Given that

As given in the question

u=\bar{u}-\phi (\pi - \pi^e) ............(1)

where

\bar{u} = Natural rate of unemployment

\pi^e = Expected rate of inflation.

u = Actual unemployement Rate

\pi= Actual inflation rate

Equation (1) can be written as

\pi = \pi^e -\phi (u - \bar{u}) ..........(2)

when \pi =1 , then Actual inflation rate depends directly on past year inflation rate and in that case\pi =\pi^e ,

Now '\lambda' which is the weight assign to unemployment depends on actual inflation and '1-\lambda' depends on expected inflation.

So, eqn (2)

becomes

T = AT + (1 - ) – ou - u

(1 - \lambda) \pi =(1-\lambda )\pi^e-\phi (u-\bar{u})

(1 - \lambda) (\pi-\pi^e)=-\phi (u-\bar{u})

\pi-\pi^e=\frac{-\phi}{1-\lambda} (u-\bar{u}) .........(3)

a)

When\phi=1,\ \lambda =0.4

As we know when \phi=1 , then\pi = \pi^e

so from eqn (1)

we get

\pi - \pi^e=-\frac{\phi}{1-\lambda}(u-\bar{u})

\frac{\phi}{1-\lambda}=u-\bar{u}

\frac{1}{1-0.04}=u-\bar{u}

u-\bar{u}=0.96

u=0.96+\bar{u}

unemployement rate will be 0.96 more than natural rate of unemployement

b)

When \lambda =0.08

then \frac{\phi}{1-\lambda}=u-\bar{u}

\frac{1}{0.02}=u-\bar{u}

u=50+\bar{u}

unemployment rate will be 50 more than natural rate of unemployment.

Inflation will still depend on part year inflation as\phi = 1

c)

When 0

Then from eqn (3)

\pi - \pi^e =-\phi (u-\bar{u}) .......(4)

From eqn (4) it follow that change in inflation rate depends on deviation of unemployment rate from natural rate and its magnitude is determined by the values of \phi .

d)

The above calculation show that, in case of high inflation, \phi is positive and the relation between unemployment rate and inflation rate is likely to change, as a result of which the manner in which expectation are formed get changed.

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