The model expand on the basic closed-economy IS-LM model

The Poole model is based on the standard IS–LM structure and
uses output volatility as the main principle for evaluation. The model extends
the IS-LM model by taking shocks into account. In this model, the Central
Bank’s goal is to minimise the loss function:

L = E(Y – Yf )²

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This function means that losses arise if the economy’s
output is above or below the target Yf. Monetary policies used to try and stabilise
the economy include the money-supply rule and the interest-rate rule. The
money-supply rule goes under the assumption that the money stock is remained
constant and the interest rate is adjusted to satisfy this. The interest-rate
rule involves the central bank setting the interest rate, and adjusting the
money supply as needed to achieve this target. In general, output volatility
differs under the two rules reliant on particular characteristics of the
economy. Both techniques are applied with the aim of minimising output
volatility.

The foundations of the Poole model expand on the basic closed-economy
IS-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 M
are 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 the
income elasticity of money demand (Blanchard et al (2013)). u is a shock to the IS curve, possibly relating to investor
confidence, 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, mainly
concerns money demand. Bad economic times correlate to more positive values of v. The shock terms u and v are usually 0, so
that E(u)=0 and E(v)=0. This doesn’t mean economists don’t expect
shocks, however. The demand for money in bad economic times increases due to
the 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 both
the interest rates (i) and the money
supply (Ms) are set, and
how the two components are adjusted in each situation to achieve their
respective goals. This was the principle objective mentioned in the original
Poole paper published in 1970.

Figure 1.1

 

 

 

 

 

 

Figure 1.1 illustrates the money-supply
rule in action and it shows the LM curve with shock to money demand whilst
fixing money supply (Ms). Setting the
money supply (Ms)
creates a sloped LM curve as shown above. A shock to money demand changes the
demand 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 curve
increases interest rates from i? to i? and consequently increases output
from Y? to Y?.

 In this case, the
equation is as follows:

Y = ????+??(M-??)+??u+??v/????+??

E(Y) = ????+??(M-??)/(????+??)

To minimise expected losses, the Central Bank sets M so that
E(Y)=Yf

Y = Yf+??u+??v/(????+??)

Figure 1.2

 

                                                                                                                                                                               

 

 

 

 

 

Figure 1.2 illustrates the
interest-rate policy. It shows the LM curve with shock to money demand whilst
fixing interest rates at iA. Setting the rate of interest (i) creates a horizontal LM curve. Money supply adjusts between a and
b to keep interest rates
constant at iA. This policy option would be used in the face of increases in
income. As a result of this increase in income, money supply would increase
from M? to M’?. Leading to constant interest rates at iA and a new equilibrium from
A to D. The new LM curve, LM’, will be horizontal.   

In
the case of a fixed interest rate, the Central Bank will minimise the loss
function, 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 shows
a graph with private spending shocks only. Private spending shocks will cause
the IS curve to vary from IS? to IS?. With fixed money supply, output will vary
from Y’? to Y’?. Similarly, with fixed interest rates, output varies from Y”?
to Y”?. When concerning the volatility in the goods market, the money-supply
option is more efficient as it has automatic stabilisers built in. With a
money-supply policy where M is fixed, there will be a smaller variation in GDP
than with an interest-rate policy where i
is fixed. Hence the difference between Y’? and Y’? due to a money-supply policy
is smaller compared to the difference between Y”? and Y”? due to an
interest-rate policy. Fixing M will be more helpful in stabilising the economy
in 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 the
economy.

A solitary source of shocks makes it easy for policy makers
to set a rule for interest rates. They would fix i when money demand shocks occur, and fix M when spending shocks
occur.

However, the idea that the Central Bank keeps the money
stock constant and lets the interest rate adjust when income changes is not a
good portrayal of what central banks actually do in modern times. Therefore,
economists much prefer to derive the LM relation under the other idea that the
Central Bank fixes the interest rate, and adjusts the money supply as required
to achieve that target.

 

 

 

 

 

Figure 1.4

 

                                                                                                                               

    

 

 

 

 

Figure 1.4 shows
shocks in money demand and private spending occurring together. With fixed
money supply, output varies from Y’? to Y”?. Similarly, with fixed interest
rates, output varies from Y”? to Y”?. In this case, a money-supply rule is
favoured as the changes in interest rates reduce the output volatility.
However, say a financial market is more volatile than private spending, then
this will make LM shift further than the IS curve, therefore an interest-rate
rule would be favoured. In the case of figure
1.4, the fixed money supply leads to a decrease in interest rates subject
to higher levels of output (Y). Intuitively, the money-supply rule lowers
interest rates in a recession, which subsequently reduces the consequence of
the private spending shock. Conversely, a fixed M will increase i in a boom, again reducing the
consequence of the private spending shock.

 

Instead of using a monetary policy, policymakers may choose
to use fiscal measures.

 

To conclude, the interest-rate rule should apply when there
is a horizontal displacement of LM>IS. If a horizontal displacement of
IS>LM occurs, then money supply should be fixed, and interest rates should be
adjusted 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 the
money-supply policy if ?<1 and the interest-rate policy if ?>1.

 

The outcomes for a large
economy coincide with those deduced using the original Poole model. When
relating to welfare, this is also the case. Specifically, fixing interest rates
yields greater outcomes with respect to money shocks. Whereas fixing money
supply is preferred when private spending shocks exist.

In the case of a small open
economy, the same outcomes exist from domestic shocks as they would in a large
economy. When foreign shocks are considered, an interest-rate rule is
considered to improve welfare. Welfare is improved by a greater amount relative
to a money supply rule when private spending shocks are considered. The case is
the opposite for foreign liquidity shocks. In
all situations, using the interest rate rule will stabilise domestic
consumption.