The Poole model is used to analyse the effect on the basic

IS-LM model by including shock factors to both curves:

IS:

Y=?0+?1r+u

LM: M=?0+?1Y+?2r+v

Where

r is the interest rates and ?0, ?1, ?0, ?1, and ?2

are parameters; ?1< 0, ?1>0 and ?2<0 are assumed. u, and v, have been added as shock terms to
adjust both equations. These shock terms have following properties:
Their
means over long term will equal zero: E{u}=E{v}=0
This
does not however mean that economists expect no shocks, just that they will
average over time presumably based on the law of large numbers.
And
their variances and covariance are: E{u2}=?2u , E{v2}=?2v , E{uv}=?uv=??u?v
however,
?2u??2v so the expected shocks
are not expected to be equal, and ?=?uv/?u?v
(Poole,
1970, pp.203-205)
v
is the shock to the money demand, therefore the LM curve. Positive values are
the shocks in economic downturns, during which money is the preferred asset,
since other assets e.g. houses can rapidly decrease in value, like during the
2008 recession. Other assets such as bonds decrease in value during bad times
as the expected economic growth is low, so dividend returns, interest accounts
and other interest-bearing assets have greater risk resulting in higher money
demand.
The
shock to the IS curve, u is supposedly related to investor confidence, which
means that positive values of u would indicate higher levels of confidence, as
well as the opposite being true. Since confidence should be lower during
economic bad times, as people will be worried about losing their jobs etc., so ?<0 due to the negative
correlation between u and v.
Poole
himself named it the "instrument problem" (Poole, 1970, p.197) which is
essentially whether the money-supply or the interest-rate rule are better and
stabilising the economy.
The
Poole model itself: L=E(Y-Yf)2
This
being the loss function that the Central Bank has the objective of minimising.
Due to the function being in the format of a standard deviation/variance
(difference from the mean) it means that the losses will be equal regardless of
output being higher or lower than the target, Yf. (Poole, 1970,
p.205)
Central
banks can minimise that loss by setting a fixed interest rate, r=r*,
if the bank does this then only the IS curve will determine the level of
output. The bank will set the interest rate r* to get the expected
output of Yf, E{Y}= ?0+?1r+u but since E{u}=0 the optimal interest rate will be r*=(Yf- ?0)/ ?1 , and actual output is Y=Yf+u
by substituting r* into the IS curve equation.
The
alternative method is the fixed money supply, M=M*, by substituting
the IS curve into the LM curve we can solve to get: Y= ?0 ?2+ ?1 (M- ?0)+ ?2u- ?1v/ ?1 ?1+ ?2, thus the expected value of output is E{Y}=?0 ?2+ ?1 (M- ?0)/ ?1 ?1+ ?2, since the central bank will set M to target Yf, Yf=?0 ?2+ ?1 (M*- ?0)/ ?1 ?1+ ?2
hence,
M*= ?0-
(?0 ?2/?1) +(?1 ?1+ ?2)Yf/
?1 Substituting
M* into the equation for Y we get that the actual output equals Y=Yf+(?2u-
?1v)/
?1 ?1+ ?2
Next,
we derive the losses for both methods, substituting respective actual outputs
into the loss function: (Poole, 1970, pp.205-207) L=E(Y-Yf)2
For
the interest-rate rule we get: Lr=E(Yf+u-Yf)2= ?2u
And
for the money-supply rule: LM=E(Yf+(?2u-
?1v)/?1?1+?2-Yf)2
LM=E?22?2u+?21 ?2v-2 ?1?2??u?v/?1?1+ ?22
The
two policies are compared by working out the ratio, ?=LM/Lr
?=?22?2u+?21 ?2v-2 ?1?2??u?v/ ?2u?1?1+ ?22
?=?22+(?21 ?2v/?2u)-(2?1?2??v/?u)/?1?1+ ?22
Quite
obviously when ?>1 or LM>Lr the

bank will choose the interest rate rule as that has a lower cost. And when ?<1 or LM