The Poole model was first hypothesized by William Poole in 1970.
The Poole model builds on the stochastic IS-LM model to incorporate unexpected shocks to investment and the money market. The model works under the assumption that policymakers primary objective is to minimise output volatility (Poole, 1970). To achieve this primary goal of minimised volatility, policymakers can adopt a monetary approach. Under this approach they can either set the money supply and control interest rates or set interest rates and control money supply. Policymakers have a trade-off between these two options as they can change the money supply and interest rates, however cannot do these independently of one another.
Firstly, take the scenario where an economy experiences an unexpected increase in money demand (MD) and investment is fixed (IS curve stationary). Under a fixed money supply rule the change in MD (M’D to MD”, Figure 1) would subsequently cause an increase in real interest rates as Money supply is set as constant causing a shift in the LM curve and a change in output (LM’ to LM”, Y0 to Y- Figure 1) . Figure 1.When policymakers decide to set interest rates and there is a shock to MD, money supply can subsequently be changed to maintain the set interest rate and counteract the shock.
As shown in Figure 2, the shift in MS means the interest rate remains constant, thus the LM curve is perfectly elastic and Y does not fluctuate when adverse shocks to the money demand occur.Figure 2.In the case of an economy which is exclusively affected by shocks to MD, policymakers should always choose to set interest rates and control the money supply. This eliminates any output changes (Y?Y0) compared to fluctuating interest rates and maintained money supply (Figure 3, Y can fluctuate from Y0 to Y+/ Y-). Figure 3.In the scenario of a shock only to investment policymakers favour a money supply rule.
Interest rates can efficiently fall or rise to counteract the adverse shift in the IS curve however there will still be some variation in Y (Figure 4, Y’+/ Y’-). However, the variation is smaller in comparison to the fixed IR rule which sees output vary from Y”+/ Y”-.Figure 4.
Furthermore, when there is a shock to both MD and Investment the choice for policymakers is not straightforward. As an interest rate rule counteracts a shock to MD more efficiency and a money supply rule counteracts a shock to the Investment more efficiently, policymakers must quantify which policy will reduce the effect of the shocks to output the most.To explore which route central banks should take we must first express the IS/LM curves mathematically: IS: Y= ?0 + ?1i + uLM: M= ?0 + ?1Y + ?2i + v Where u and v are stochastic random variables (the respective shocks to the IS/LM curve) and it is assumed ?1<0 and ?1>0 and ?2<0.Since u and v are shocks we assume their means to be zero and have constant variances:E(u) = 0 and E(v) = 0E(u2) = ?2 and E(v2) = ?2We must now consider the role of the central bank. The primary objective of the central bank is to minimise volatility thus its loss function is the variation of output from equilibrium levels of output. L = E(Y-Yf)2 If the central bank follows an interest rate rule, output is affected only by shocks to investment as money supply can be altered to counteract any unexpected changes in MD. Y= ?0 + ?1i + uE(Y) = ?0 + ?1i (since E(u) = 0)The central bank will set i to ensure: E(Y) = YfYf = ?0 + ?1i*This can be rearranged to provide the optimal IR:i* = (Yf - ?0) / ?1Solution for Y:Y = Yf + uIf the central bank choses to the fix money supply they must take into account both u and v:IS: Y= ?0 + ?1i + u LM: M= ?0 + ?1Y + ?2i + vFirstly we must rearrange to solve for Y in terms of M:Y=1?1?1+ ?2(?0?2+ ?1(M- ?0)+ ?2u- ?1v) Since E(u) = 0 and E(v) = 0:Y=1?1+ ?2(?0+ ?1(M- ?0)To minimise the loss function, M will be set so that E(Y) = Yf:Y=Yf= 1?1?1+ ?2(?0?2+ ?1(M*- ?0)Final output:Y=Yf+ 1?1?1+ ?2(?2u- ?1v) Now that the two respective output functions have been derived they can be substituted into the central banks loss function.
Interest rate rule:Li=E(Y-Yf)2 Y=Yf + uLi = E(Yf+u-Yf)2Li = Eu2Li = ?2uMoney supply rule:LM = E(Y-Yf)2 Y=Yf+ 1?1?1+ ?2(?2u- ?1v))LM = E(Yf+ 1?1?1+ ?2(?2u- ?1v)- Yf)2LM = E( 1?1?1+ ?2(?2u- ?1v)2By expanding the brackets the following loss function is derived:LM=E( 1(?1?1+ ?2)2(?12?v2-2?1?2??u?v+ ?22?u2) Since the two loss functions for both the money supply and interest rate rule have been derived the two policies can be compared through the ratio of the respective losses: Li = ?u2 LM=E( 1(?1?1+ ?2)2(?12?v2-2?1?2??u?v+ ?22?u2) ?=LM/Li=1(?1?1+ ?2)2(?12?v2/?u2-2?1?2?v/?u)+ ?22)If ?>1 then the central bank will choose the interest rate rule (as LM>Li), if ?<1 money supply rule (LM
Vice versa, if v is greater than u, output variation is greater under a money supply rule (Figure 6: Y”+/- to Y > Y’+/- to Y), central banks should choose an interest rate rule (?>1).Figure 5. Figure 6.
In the decades following the initial publication of the Poole model (1970-1990) the instability of MD rose substantially due to mass deregulation and innovation within the financial industry. This led central banks to move from a money supply rule to an interest rate rule to counteract this new volatility in the money market as it outweighed volatility within Investment (Hoffmann and Kempa, 2009).From expressing the Poole model mathematically it shows the importance of ?1 (income elasticity of money demand). The relationship can be derived from the ratio of the loss-functions:?=1(?1?1+ ?2)2(?12?v2/?u2-2?1?2?v/?u)+ ?22)?=(?12?v2/?u2-2?1?2?v/?u+ ?22)(?1?1+ ?2)2Simplified as a2 + b2 + 2ab = (a + b)2?=(?1?v/?u+ ?2)2(?1?1+ ?2)2Setting ?=1?=(?1?v/?u+ ?2)2(?1?1+ ?2)2=1?=(?1?v/?u+ ?2)(?1?1+ ?2)=1(?v/?u)/?1=1(?v/?u)=?1This shows that ?>1 if (?v/?u)>?1and ?<1 if (?v/?u)1. This demonstrates that even if the shock to MD (v) is larger than the shock to Investment (u) then ? may still be smaller than 1 (LM
Usually this is carried under the auspices of Keynesian policies, increasing/ decreasing government expenditure to correct any shocks to Investment (Figure 7: Moving the IS curve +/- back to original level thus maintaining equilibrium output).Figure 7The common perception advocating for these Keynesian policies is that the impact is felt immediately in the economy in comparison to monetary adjustments which often have substantial time lags (Gruen, Romalis and Chandra, 1999). By eliminating these time lags the output shocks will be corrected in a shorter time compared to using a monetary approach.
This factor has brought Keynesian policies to the forefront of economic policy. For example, at the start of the 2008 financial crisis, the US had experienced a sudden negative shock to investment and a liquidity freeze. This shock was caused by firms not wanting to invest, citing an dampened economic outlook (Cukierman, Web and Neyapti, 1992). To combat this, the Obama administration authorised an unprecedented fiscal stimulus of $787bn, this aimed to stimulate the economy and investment to move output back to the equilibrium level (Y- to Y, Figure 7). After US GDP falling from $14,719bn (2008) to $14,419bn (2009) it recovered to $14,964bn (2010) (Bea.gov, 2018) thus implying a quick recovery had occurred.To conclude, to reduce macroeconomic volatility policy by monetary policy, policymakers must first identify the cause of shock.
If the shock is purely caused by a change in MD, central banks should use an interest rate rule, as keeping interest rates constant and allowing money supply to adjust will ensure output remains constant. If there is a shock only to Investment, policymakers should use a money supply to allow interest rates to effectively fall/rise to counteract the shift in the IS curve. Finally, if there is a shock to both, policymakers must identify which shock has taken a bigger effect on output. This can also be deduced mathematically by expressing the central banks loss function in terms of output variation, whichever policy produces the lowest value should be used. Fiscal policies can also be used to reduce volatility by counteracting any adverse shocks to investment for such policies are often found to reduce time-lags that are particularly associated with monetary policy.