William Poole’s model extends the Investment Savings – Liquidity Money model to include uncertainty. Monetary policy is a macroeconomic policy which involves management of the interest rate and the money supply in order to achieve government macroeconomic objectives such as stable economic growth.

Monetary authorities can directly fix the interest rate in the economy under the interest rate rule. Money supply adjusts to the level of money demand in the financial markets, therefore, an increase in money demand will be matched with an increase in money supply. The financial sector is of no concern to monetary authorities resulting in a horizontal and flat LM curve. However, “poor performance in the financial market is usually matched with bad macroeconomic performance in the overall economy.” (Pickering, 2017)

Equilibrium in the goods market determines the IS curve. The most volatile component of the goods market is investment; therefore, a fixed interest rate increases the stability of investment which aids the objective of reducing macroeconomic volatility. However, an interest rate set too low results in a liquidity trap this is where the interest rate becomes useless at stimulating growth in the economy.

Figure 1

i

i

i

i

l”

IS’

IS

l

1

Y”

l

Y

Y

l

l’

Figure 1 illustrates the investment shock under a fixed interest rate rule. An increase in income caused by a boom leads to an increase in investment (l – l’), causing the IS curve to increase from (IS – IS’). A fixed interest rate decreases investment volatility as investment is responsive to a change in interest rates.

The IS curve is given by 2 where (Y) denotes output, (r) denotes the interest rate and (U) denotes the investment shock. The (IS) equation is rearranged to set (r) to the natural level of output (Yf) such that 2 where E(U) = 0. When (r) is substituted into the original (IS) equation the parameters (a0 and a1) cancel out. This results in the actual level of output (Y) being equal to the expected rate of output (Yf) plus the stochastic variable (U) Such that 2.

The loss function 2 measures output which is above or below actual output (Yf). Substituting the actual level of output 2 into the loss function we obtain

2. This is the expected loss under an interest rate rule.

Monetary authorities can directly set the money supply in the economy using the money supply rule. Volatility in money demand is due to liquidity preferences. During a recession, liquidity preference will be high due to consumers and investors experiencing low confidence in institutions which hold intangible assets. Low confidence results in consumers and investors liquidising assets which leads to increasing levels of money demand. The LM curve represents equilibrium in the financial market, therefore, an increase in money demand causes an increase in the LM curve which results in a higher interest rate. However, income levels decline during a recession due to decreasing household expenditure which leads to a decrease in money demand. Liquidity preferences and the level of income work in the opposite direction during a boom compared to a recession. This shows that output volatility is dependent on money demand and income levels under a money supply rule.

Figure 2

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i

i

i’

Ms

Md”

Md

Md & Ms

i’

LM”

LM

i

1

Y

Y0

Figure 2 illustrates a positive shock to money demand (Md – Md”) with a fixed money supply (Ms). The LM curve increases from (LM – LM”) resulting in the readjustment of the interest rate from (i – i’).

The LM equation is given by 2 where (M) denotes money demand, (Y) denotes output, (r) denotes the interest rate and (V) denotes money demand.

Money demand depends upon the stochastic variable (V) and the income elasticity of money demand (B1). The variable (V) and parameter (B1) have an inverse relationship. A high enough level of (B1) may offset the extent to which (V) decreases or increases.

The expected level of output (Yf) is where (r) becomes the subject of the IS equation and substituted into the (LM) equation where E(U) and E(V) are both equal to zero. (Yf) is made the subject of the (LM) equation such that 2. The fixed money supply that will produce the expected level of output (Yf) is where M* is made the subject such that 2. The actual level of output will combine (Yf) and the stochastic shocks such that 2. Substituting actual output 1 into the loss function 2 we obtain

2 This is the expected loss under a money supply rule.

When money demand is the only stochastic variable in the economy the optimal monetary instrument is the interest rate rule. Shocks to money demand have an inverse relationship with interest rates. Interest rates worsen a recession or further stimulate economic growth in a boom causing overheating. A fixed interest rate will eliminate the counterproductive effect of a fluctuating interest rate.

When private spending is the only stochastic shock variable the optimal monetary instrument is the money supply rule. The interest rate works as a stabiliser to investment shocks because of the positive relationship between the interest rate and economic growth.

Figure 3

i

Private spending shocks

i

Money demand shocks

LM(Ms=Md)

LM’

LM

i

i

LM(Ms=Md)

LM

IS

IS

IS

1

Y

Y-‘

Y-

Y+’

Y+

Y-

Y

Y+

Y

The money demand diagram illustrates the interest rate rule as the optimal solution to minimising output volatility (Y) as greater output volatility exists under the money supply rule ((Y-) – Y+). The private spending diagram illustrates that the money supply rule will result in the optimal solution in terms of minimising output volatility ((Y-) – y+) compared to the interest rate rule ((Y-‘) – y+’).

The optimal choice of Monetary instrument with stochastic shocks both to money demand and private spending is dependent upon the horizontal displacement between the (IS) and the (LM) curves. Under the interest rate rule, the horizontal displacement between the (IS) curves is important. Under the money supply rule, the horizontal displacement between the equilibrium points of the (IS) and (LM) curves are important. The policy instrument with the smallest horizontal displacement produces the least output volatility which results in the most desirable instrument choice.

Figure 4

Money and real shocks (1)

Money and real shocks (2)

i

IS+

LM+

IS+

LM+

i

IS-

LM-+

LM(Ms=Md)

LM(Ms=Md)

LM-+

IS-

1

Y

Y-‘

Y-

Y-‘

Y-

Y+

Y+’

Y+

Y+’

Y

The money and real shocks (1) diagram illustrate greater horizontal displacement between the IS curves showing that private spending shock (U) is greater than the money demand shock (V/B1). Output volatility under the money supply rule ((Y-) – Y+) is smaller compared to the interest rule ((Y-‘) – Y+). This shows the money supply rule is the optimal solution in minimising output volatility.

The money and real shocks (2) diagram illustrate greater horizontal displacement between the LM curves showing the money demand shock (V/B1) is greater than the private spending shock (U). Output volatility under the interest rate rule ((Y-) – Y+) is smaller compared to the money supply rule ((Y-‘) – Y+). This shows that the interest rate rule is the optimal solution in minimising output volatility.

The loss function ratio compares the loss function of the interest rate rule and the money supply rule such that

.2

The money supply instrument is the optimal choice when (Lm/Lr) 1. The central bank is indifferent between the two instruments when (Lm/Lr) = 1. Under the condition where the money demand shock and the IS shock are perfectly negatively correlated and the ratio of the standard deviations > B1 the fixed interest rate will be the optimal solution. When the standard deviations