MACROECONOMICS: INTEREST RATE RULES, BARRO-GORDON MODEL

INTEREST RATE RULES

An interest rate rule gives a direct relationship between the primary tool of a Central Bank (the interest rate) and inflation and output.

It is essentially the maths behind the three-equation model developed previously.

We shall therefore be using the following equations:

IS: yt – ye = -a(rt – rs)

IAPC: πt = πt-1 + α(yt – ye)

MR: πt – πT = -(1/αβ)(yt – ye)

INTEREST RATE RULES

Substitute the IAPC into the MR; πt-1 + α(yt – ye) – πT = -(1/αβ)(yt – ye)

(yt – ye) = -a (rt – rs) (The IS equation), so subbing in:

πt-1 -aα(rt – rs) – πT = (a/αβ)(rt – rs)

πt-1 - πT = (aα + a/αβ)(rt – rs) Thus, the above equation can be arranged to

arrive at:

INTEREST RATE RULES Here:

The bottom equation holds if all parameters are 1. The important point here is that ALL the

parameters are included in the consideration of interest rate changes.

A problem, however, is that only INFLATION is considered, when output is also in the loss function.

Ttst

Ttst

rr

a

rr

1

1

5.0

1

1

THE DOUBLE LAG

In order to include output considerations, we must realise that there are lags in the economy.

Interest rates take 1 period to affect output and output takes a further period to affect inflation:

We can thus use this knowledge and update the time indicators on the 3-equation model we used previously.

πt-1

πt

πt+1

yt

yt-1 rt-1

THE TAYLOR RULE

IS: yt – ye = -a(rt-1 – rs)

IAPC: πt = πt-1 + α(yt-1 – ye)

MR: πt+1 – πT = -(1/αβ)(yt – ye) Follow the same steps as last time, but at the

end, sub in inflation to get the output expression:

THE TAYLOR RULE DERIVATION

etT

tst

stT

ett

etT

t

etT

ett

yy

a

rr

rrayy

yy

yyyy

111

111

1

1

1

1

1)(

THE BARRO-GORDON MODEL The Barro-Gordon Model (BGM) is concerned

with what happens when the targeted level of output ‘y’ is GREATER than ye.

Next slide derives it diagrammatically and it is derived mathematically following that.

Assume, however, that there is a Lucas ‘surprise’ supply function and a loss function of the form:

L = (y – kye)2 + (πt – πT)2

Where ‘k’ is a positive constant greater than one.

IMPORTANT: RESULT HOLDS FOR IAPC/EAPC AS WELL, BUT IT TAKES TIME.

THE BARRO-GORDON MODEL The Diagram to the right shows

the effects of setting an output target greater than the natural rate of output.

Initially we are at A with inflation πT and output ye.

However, the Government may cheat and decide to try and boost output to kye (i.e, move to point B).

However, because they are constrained to the SRAS curve, they will choose the optimal point along it (C) as it is tangential to their preference circles.

This leads to a higher inflation rate π1 which shifts the SRAS curve up next period.

The economy will then move to point D. This continues until point E, a point on the LRAS curve.

Hence, output ends up at ye and inflation is higher than targeted.

πB – πT is the INFLATION BIAS.

π

Ykyeye

πTA B

C

D

E

LRAS

SRAS (πT)

SRAS (π1)

SRAS (πB)

π1

π2

πB

BARRO GORDON MATHS:

2

2

22

22

22

1

122

222122.

022

:Substitute

:toSubject

:FunctionLoss

ky

yky

ykyV

ykyV

yy

ykyV

Te

Te

Te

Te

e

T

BARRO-GORDON MATHS That equation describes the reaction function

(the grey line in the previous diagram). In order to find what rate will be selected, we must TAKE EXPECTATIONS.

The size of the inflation bias, therefore, is (k-1)θYe

We can put this value back into the reaction function equation to see what point the Gov. Will choose.

If we do this, we get the SAME level of inflation.

yk

ykE

TBe

eTe

1

1

12

2

LASH IT TO THE MAST! The problem with the Government being in charge of

policy is therefore evident, should we consider game theory.

We can think of this as a finite game, with the end known (elections). Thus, the game will ‘unravel’ such that the Government will ‘cheat’ straight away.

Hence the Central Bank – no elections, therefore it is an infinite game and there is no incentive to cheat.

The Central Bank itself must build credibility over time; one possible way to do this is to tie itself to a reputable central bank (Europe did this with the Bündersbank).

Another way of establishing credibility is for ‘Walsh Contracts’. That is, a set of incentives provided by the Government for the Central Bank to reach the targeted inflation level (New Zealand).

There may be little incentive to cheat anyway in developed capital markets – although there are lags between the stimulus and economic activity, there are NO lags between the stimulus and capital flight that will follow, should investors decide the stimulus signals a weakness in the currency.