The Poole model is based on the standard IS–LM structure anduses output volatility as the main principle for evaluation. The model extendsthe IS-LM model by taking shocks into account. In this model, the CentralBank’s goal is to minimise the loss function:L = E(Y – Yf )²This function means that losses arise if the economy’soutput is above or below the target Yf. Monetary policies used to try and stabilisethe economy include the money-supply rule and the interest-rate rule. Themoney-supply rule goes under the assumption that the money stock is remainedconstant and the interest rate is adjusted to satisfy this. The interest-raterule involves the central bank setting the interest rate, and adjusting themoney supply as needed to achieve this target.
In general, output volatilitydiffers under the two rules reliant on particular characteristics of theeconomy. Both techniques are applied with the aim of minimising outputvolatility. The foundations of the Poole model expand on the basic closed-economyIS-LM model so that uncertainty can be accounted for:Y= ?? + ??i + u (1) M= ?? + ??Y + ??i + v (2)Where (1) is the IS curve and (2) is the LM curve. Y and Mare determined as the logarithms of output and money supply respectively. i is the interest rate and ??, ??, ??,??, and ?? are parameters where by, for example, ?? can be understood as theincome elasticity of money demand (Blanchard et al (2013)). u is a shock to the IS curve, possibly relating to investorconfidence, whereby the more positive terms relate to higher confidence,resulting in increased spending and higher equilibrium GDP, ceteris paribus. v is a shock to the LM curve, mainlyconcerns money demand. Bad economic times correlate to more positive values of v.
The shock terms u and v are usually 0, sothat E(u)=0 and E(v)=0. This doesn’t mean economists don’t expectshocks, however. The demand for money in bad economic times increases due tothe greater perception of downside risk involved with interest-bearing assets,resulting gin the fall in demand for corporate bonds. The graphs below show how the Poole model evaluates how boththe interest rates (i) and the moneysupply (Ms) are set, andhow the two components are adjusted in each situation to achieve theirrespective goals.
This was the principle objective mentioned in the originalPoole paper published in 1970.Figure 1.1 Figure 1.1 illustrates the money-supplyrule in action and it shows the LM curve with shock to money demand whilstfixing money supply (Ms). Setting themoney supply (Ms)creates a sloped LM curve as shown above. A shock to money demand changes thedemand for money (Md)at each interest rate and level of income from M’d to M”d. As show in Figure 1.
1, the LM curve is shifted to the left from LM’ to LM”which indicates a positive money demand shock. This shift of the LM curveincreases interest rates from i? to i? and consequently increases outputfrom Y? to Y?. In this case, theequation is as follows:Y = ????+??(M-??)+??u+??v/????+?? E(Y) = ????+??(M-??)/(????+??) To minimise expected losses, the Central Bank sets M so thatE(Y)=YfY = Yf+??u+??v/(????+??)Figure 1.2 Figure 1.2 illustrates theinterest-rate policy. It shows the LM curve with shock to money demand whilstfixing interest rates at iA.
Setting the rate of interest (i) creates a horizontal LM curve. Money supply adjusts between a andb to keep interest ratesconstant at iA. This policy option would be used in the face of increases inincome.
As a result of this increase in income, money supply would increasefrom M? to M’?. Leading to constant interest rates at iA and a new equilibrium fromA to D. The new LM curve, LM’, will be horizontal.
Inthe case of a fixed interest rate, the Central Bank will minimise the lossfunction, L, by setting interest rates, i,to ensure E(Y)=Yf. Since E(u)=0, then E(Y)=??+??i, Yf will equal ??+??i*. Therefore the optimal interest rate is defined as i*=Yf-??/??,and the solution for Y is Y=Yf+u. Figure 1.3 VV Figure 1.3 is showsa graph with private spending shocks only.
Private spending shocks will causethe IS curve to vary from IS? to IS?. With fixed money supply, output will varyfrom Y’? to Y’?. Similarly, with fixed interest rates, output varies from Y”?to Y”?. When concerning the volatility in the goods market, the money-supplyoption is more efficient as it has automatic stabilisers built in.
With amoney-supply policy where M is fixed, there will be a smaller variation in GDPthan with an interest-rate policy where iis fixed. Hence the difference between Y’? and Y’? due to a money-supply policyis smaller compared to the difference between Y”? and Y”? due to aninterest-rate policy. Fixing M will be more helpful in stabilising the economyin bad and good economic times than fixing i.This is because as liquidity demand increases, M and i will also increase, which results in the slowing down of theeconomy. A solitary source of shocks makes it easy for policy makersto set a rule for interest rates. They would fix i when money demand shocks occur, and fix M when spending shocksoccur.However, the idea that the Central Bank keeps the moneystock constant and lets the interest rate adjust when income changes is not agood portrayal of what central banks actually do in modern times.
Therefore,economists much prefer to derive the LM relation under the other idea that theCentral Bank fixes the interest rate, and adjusts the money supply as requiredto achieve that target. Figure 1.4 Figure 1.4 showsshocks in money demand and private spending occurring together.
With fixedmoney supply, output varies from Y’? to Y”?. Similarly, with fixed interestrates, output varies from Y”? to Y”?. In this case, a money-supply rule isfavoured as the changes in interest rates reduce the output volatility.However, say a financial market is more volatile than private spending, thenthis will make LM shift further than the IS curve, therefore an interest-raterule would be favoured. In the case of figure1.
4, the fixed money supply leads to a decrease in interest rates subjectto higher levels of output (Y). Intuitively, the money-supply rule lowersinterest rates in a recession, which subsequently reduces the consequence ofthe private spending shock. Conversely, a fixed M will increase i in a boom, again reducing theconsequence of the private spending shock. Instead of using a monetary policy, policymakers may chooseto use fiscal measures.
To conclude, the interest-rate rule should apply when thereis a horizontal displacement of LM>IS. If a horizontal displacement ofIS>LM occurs, then money supply should be fixed, and interest rates should beadjusted to reach the target.To derive loss from the interest-rate rule:Li = E(Y-Yf)² =E(Yf+u-Yf)² = E(u²)To derive the loss from the money rule: LM = E(Y-Yf)² = E(Yf + (??u–??v)/(????+??)–Yf)² = E((??u-??v)/(????+??))² = E(??²???-2???????? + ??²??²)/(????) The 2 monetary policy options can be compared by considering?=LM/Lr.
The Central Bank will use themoney-supply policy if ?<1 and the interest-rate policy if ?>1. The outcomes for a largeeconomy coincide with those deduced using the original Poole model. Whenrelating to welfare, this is also the case. Specifically, fixing interest ratesyields greater outcomes with respect to money shocks. Whereas fixing moneysupply is preferred when private spending shocks exist. In the case of a small openeconomy, the same outcomes exist from domestic shocks as they would in a largeeconomy. When foreign shocks are considered, an interest-rate rule isconsidered to improve welfare.
Welfare is improved by a greater amount relativeto a money supply rule when private spending shocks are considered. The case isthe opposite for foreign liquidity shocks. Inall situations, using the interest rate rule will stabilise domesticconsumption.