The factors to both curves: IS: Y=?0+?1r+u LM: M=?0+?1Y+?2r+v

 

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

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                                                          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