The main difference between the cross model and the IS-LM model is that the nominal interest rate is exogenous in the cross model but endogenous in the IS-LM model. In this chapter we will explain how the nominal interest rate is determined in the IS-LM.

P remains exogenous and constant in the IS-LM model. Therefore, inflation and expected inflation is zero. This in turn implies that the nominal interest rate is equal to the real interest rate: R = r. This will allow us to talk about "the interest rate" without specifying whether we mean the nominal or real interest rates.

Aggregate demand

The investment function in the IS-LM model

Investment was an exogenous variable in the cross model due to the fact that the interest rate was exogenous. Now that the interest rate is endogenous, investment will be endogenous. As for the classical model, investment depends negatively on the real interest rate but since R = r in the IS-LM model, we can make investment a function of R: I = I(R).

The consumption function in the IS-LM model

The consumption function will be the same as in the cross model, consumption will depend positively on Y. In the classical model, consumption depends negatively on the real interest rate. You may allow consumption to depend negatively on interest rates in the IS-LM as well. You must then write C = C(Y, R). In the literature, both variants are found but since the results will be largely the same, we choose to let C depend on Y only, C = C(Y). We will also, for the same reason, model imports as a function of Y only even though it may depend on R as well.

Aggregate demand

Aggregate demand depends on Y and R in the IS-LM model

Since investments depend on R and consumption and imports depend on Y, the aggregate demand will depend on both Y and R. In the cross model, we used the notation YD(Y) for aggregate demand. In the IS-LM model, we must instead use the notation YD(Y, R). We have

It does not make much of a difference if we allow C and Im to depend on R as well, YD will depend positively on Y and negatively on R in any case.

It should also be clear that we can no longer determine GDP the way we did it in the cross model. We cannot successfully solve the equation YD(Y, R) = Y as we have only one equation but two unknowns (Y and R). We need one more equation if we want to solve for both Y and R. This equation will come from the money market.

The money market

Demand for money

The demand for money depends negatively on R and positively on the Y in the IS-LM model

As for any kind of goods, there is a demand for money and a supply of money. Remember that the demand for an arbitrary good is the amount an individual wishes to buy (and pay for with money) under different conditions. The demand for an arbitrary good is always related to money. But the demand for money cannot relate to money itself - how much money we want to "buy" with money becomes a rather useless definition.

Instead, we define the demand for money as the amount out of your wealth that you wish to hold as money. We use the symbol MD to the demand for money. In the IS-LM model, there is only one alternative to money and that is bonds.

If your total wealth is 1.000 euro and you wish to keep 100 euro in cash or in an account connected to a debit or credit card and the rest in government bonds then your demand for money is precisely 100 euro. It is the amount that you want to have easily accessible for immediate payments. Note that having a low demand for money does not mean that you do not want money. Instead, it means that you prefer to hold most of your wealth in other types of assets

Demand for money and the interest rate

Money has one important advantage and one important disadvantage compared to bonds:

Advantage: Money is more liquid than bonds. If most of your wealth is invested in bonds, you must first sell some of the bonds whenever you want to make a payment.

Disadvantage: You receive interest payments on bonds but not on money.

At 0% interest, there would be no reason to hold bonds and the demand for money would be maximized. The higher the interest rate, the more you lose by holding money instead of bonds. Therefore, we would expect the demand for money to fall when R increases and this is the assumption in the IS-LM model.

Demand for money and GDP

The demand for money also depends on the GDP as GDP is closely related to national income. If you choose to hold a fixed proportion of your wealth as money, you will want to hold more money when Y increases (you will want to hold more bonds as well). In the IS-LM model we assume that the demand for money is positive function of GDP.

As the demand for money depends on Y and R in the IS-LM model, we write MD(Y, R) for the demand for money. Remember that it depends positively on Y and negatively on R.

Supply of money

The supply of money is an exogenous variable in the IS-LM model

The money supply is completely under the control of the central bank in all models in this book. Money supply is therefore an exogenous variable not affected by either interest rates or GDP. We denote the money supply by MS.

Equilibrium in the money market

In the IS-LM-model, we have equilibrium in the money market when MD(Y, R) = Ms

This is our "missing equation" as discussed in section xx. It is now possible to determine all endogenous variables in the IS-LM model:

We now have two equations and two unknown (Y and R) and in most cases we can find a unique solution to the system of equations. Exactly how this is done is best illustrated by the IS-LM diagram which is presented in section xxx.

Money market diagram

Let us begin by studying the money market when the GDP is given. When Yis given, MD will only depend (negatively) on R and we can draw a diagram with supply and demand for money as functions of R.

Fig. 12.1

In the diagram above, R* is the interest rate in which the demand for money is exactly equal to the supply of money (again for a given Y). The IS-LM model, R will always tend to R* until they are equal and we have an equilibrium in the money market.

The justification for why R will tend to R* is not entirely straightforward:

• Say that R < R*.

• In this case, MD > Ms, that is, people want to hold more money than what is available.

• People increase the amount of money they hold by selling bonds so there is an excess supply of bonds.

• This excess supply of bonds will drive down the price of bonds.

• When the price of bonds falls, interest rates increase. We discussed this negative relationship between the price of bonds and the interest rate in section 7.2.3.

• The interest rate will increase until R = R*. Only then will the demand for money have decreased enough such there is no longer an excess demand for money. Then there is no excess supply of bonds either. The money market is in equilibrium.

• The case of R > R* can be analyzed in the same way.

The money market diagram can be used to determine the equilibrium rate of interest if we know GDP.

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