# Product and Money market Equilibrium: IS-LM Model

### 1- Introduction

The study of product and money markets together is a synthesis of the theory of income and output and the theory of money and interest. It analyzes the linkage and interaction between the product and money markets to determine the level of income and interest rate, which bring about a simultaneous equilibrium in both the markets. This concept of simultaneous equilibrium in both the markets was for the first time highlighted and developed as a model by J. R. Hicks in 1937. He integrated the product and money markets to show how both markets' equilibrium coincides at the same level of income and interest. His model is, then widely known, as the IS-LM model.
In the IS-LM model, the term IS refers to the equality between investment I and saving S and represents the product market equilibrium whereas the term LM refers to the equality between money demand L and money supply M, and it represents the money market equilibrium. The activities and variables of these markets are interrelated and, therefore, a change in the variables of one market affects the activities in the other market. In other words, changes in the variables in the product market affect the money market equilibrium whereas the changes in the variables of the money market influence the product market equilibrium.

#### 2- IS-LM Model in a Two-sector Economy

The IS-LM model in a two-sector model can be presented in an algebraic and diagrammatic form. The diagrammatical presentation of the IS-LM model shows the inter-variable linkage and working system of the product and money market. The IS-LM has been presented in the following three stages.
2.1- Derivation of IS curve,
2.2- Derivation of LM curve,
2.3- Presentation of IS-LM model of general equilibrium,

#### 2.1- Derivation of IS curve: Product Market Equilibrium

The IS curve is the curve or schedule which represents various combinations of the rate of interest i and the equilibrium level of income Y or output at which saving S equals investment I.

The IS curve's derivation can be explained by the AD-AS approach and the S-I approach. Let us first derive the IS curve under the AD-AS approach.
According to the AD-AS approach, the level of income Y is equal to consumption demand C and investment demand I and is expressed as follows.

Y=C+I.........(i)
The consumption function C=a+bY..........(ii)
Investment I is autonomous and the negative function of the rate of interest i and is expressed as follows.
I=\barI-hi..........(iii)
Now let us substitute a+bY for C and \barI-hi for I in equation (i) to get the equation for product market equilibrium as follows.
Y=C+I.........(i)
Y=a+bY+\barI-hi
Or,\Y-bY=a+\barI-hi
Or,\Y(1-b)=a+\barI-hi
Or,\Y=\frac{1}{(1-b)}\(a+\barI-hi)............(iv)
Where:
Y= Level of income,
a= Autonomous consumption,
b= Marginal propensity to consume,
\barI= Autonomous investment,
h= Sensitivity of investment to the rate of interest or ratio of a change in investment to the change in interest rate,
i= Rate of interest,
Diagrammatically, the derivation of the IS curve has been explained as follows.

In the diagram, in panel-A, it has been shown that with a certain amount of investment at the i2 rate of interest, the determined income level of income is 0Y2. The combination between the same rate of interest and the income level is represented by point B in panel-B. Now let us suppose that there is a fall in the rate of interest from i2 to i1. Consequently, there is an increase in investment which causes the AD curve to shift upward where the 0Y3 income level gets determined as shown in panel-A. The interest-income combination, 0i1 and 0Y3 represented by the point C as shown in panel-B. Similarly, with the rise in interest rate from i2 to i3, there is a fall in investment resulting in a fall in income level from 0Y2 to 0Y1. The interest-income combination 0i3 and 0Y1 is represented by point A as shown in panel-B. When all these interest-income combinations A, B, and C are joined together with a smooth line, we get a downward sloping curve called the IS curve, as shown in panel-B.

#### 2.2- Derivation of LM curve: Money market Equilibrium

The LM curve shows all the combinations of the rate of interest and levels of income at which demand for and supply of money are equal to each other. In other words, the LM schedule shows the various combinations of interest rates and levels of income where demand for money L and supply of money M are equal such that the money market is in equilibrium.
As mentioned above, the money market is in equilibrium where demand for money is equal to the supply of money. Demand for money is also denoted by Md whereas the supply of money is also denoted by Ms and the equality between Md and Ms is expressed as follows.
Md=Ms........(v)
As far as the supply of money concerns, it is supplied by the monetary authorities and assumed to be constant for a time period. Therefore the supply of money is expressed as;
Ms=\barMs.......(vi)
Regarding the demand for money, based on the Keynesian theory of demand for money, it consists of transaction demand for money (including precautionary demand for money) Mt which is a direct function of income level Y and speculative demand for money Msp which is an indirect or negative function of the rate of interest. Thus, the total demand for money can be expressed as follows.
Md=Mt+Msp........(vii)
Where;
Md= Total demand for money,
Mt= Transaction demand including precautionary demand for money,
Msp= Speculative demand for money,
Mt is a positive function of income level and it is expressed as;
Mt=kY.........(viii)
Where k= ratio of change in Mt to Y or k=\frac{△Mt}{△Y} and assumed to be constant to some extend.

Msp is the negative function of interest and expressed as;
\barL-li.........(x)
Where L= autonomous demand for Msp and l= ratio of change in Msp to i or l=\frac{△Msp}{△i}.
Thus, the total demand for money is expressed as follows.
Md=kY+\barL-li.........(x\i)
And the money market equilibrium is expressed as follows.
Ms=Md
Or,\Ms=kY+\barL-li..........(x\ii)

Diagrammatically, the derivation of the LM curve has been illustrated as follows.

In the above diagram, panel-A shows the equality between Md and Ms at different rates of interest. The vertical line is the Ms curve which is income inelastic or is fixed by monetary authorities. On the other hand, demand for money is determined by the income level. With an increase in the income level, demand for money Md also increases for the transaction, precautionary and speculative motives. So the liquidity preference or Md corresponds to the level of income. In panel-A, the point A shows Md=Ms and interest rate i1 is determined at income level Y1. The equilibrium rate of interest i1 and income level maintains a combination D as shown in panel-B. Likewise, an increase in income level from Y1 to Y2, causes money demand Md to increase and shift, the curve upward which equals Ms at point B and rate of interest i2 gets determined. The new rate of interest i1 and income level Y1 maintains another point of combination denoted by E in panel-B. The similar process happens when the demand for money further increases as shown by the point of intersection at C in panel A and its effect in panel B. When all these points of combination between the rates of interest and income levels in panel-B are joined together with a line, we derive the LM curve.
Thus, the LM curve represents the various combinations between rates of interest and levels of income at which demand for money Md equals the supply of money Ms.

#### 2.3- IS-LM Model: A General Equilibrium or Simultaneous Equilibrium of IS and LM

Having derived the IS and LM curves, we can integrate them to find the general equilibrium or simultaneous equilibrium of product and money markets at the same interest rate and income level. The simultaneous equilibrium of both the markets has been illustrated with a help of the following diagram.
In the above diagram, the IS curve represents the equilibrium levels of income at different interest rates with a condition that I=S in the product market. Similarly, the LM curve represents the equilibrium levels of income at different rates of interest with a condition that Md=Ms in the money market. These two curves intersect each other and maintain equality between the product and money market equilibriums as shown in the diagram at point E. It is the point of simultaneous equilibrium of product and money markets at the 0i interest rate and 0Y income level where investment equals saving in the product market and demand for money equals the supply of money in the money market.

#### 3- Causes of a shift in IS curve and its effect on equilibrium income

There are different causes of a shift in the IS curve which have been mentioned as follows.

#### 3.1- A change in consumption expenditure

A change in consumption expenditure causes the IS curve to shift either to the right or to the left resulting in a change in equilibrium income level. If there is an increase in consumption expenditure for whatever reason, it causes the AD curve to shift upward resulting in an increase in income level. Consequently, the IS curve shifts to the right. On the contrary, with a fall in consumption expenditure, the IS shifts to the left.

#### 3.2- A change in investment

A change in autonomous investment also causes the IS curve to shift to the right or to the left. When there is an increase in autonomous investment with positive business expectations, the IS curve shift to the right, and it shifts to the left on its contrary situation.

#### 3.3- A change in government expenditure

If the economy is operating under a three-sector or a four-sector model, a change in the government expenditure makes the IS curve shift either to the right or to the left. If the government adopts a deficit budgeting (expansionary) policy there is a rise in government expenditure and with that the IS curve shifts to the right. When the government adopts a surplus budgeting (contractionary) policy, there is a fall in government expenditure and it causes the IS curve to shift to the left.

#### 3.4- A change in tax policy

A change in tax policy also brings about a shift in the IS curve eighter to the right or to the left. When the government imposes a higher tax rate, it brings about a fall in disposable income which causes a fall in consumption expenditure. Consequently, the IS curve shifts to the left. On the other hand, with a tax release, there is an increase in disposable income and also increases in consumption expenditure resulting in a shift in the IS curve to the right.

The shift in the IS curve has been explained with a help of the following figure.

In the above figure-7, E2 is the initial equilibrium where 0Y2 income level and 0i2 rate of interest has been determined. Now, let us suppose that there is an increase in autonomous investment, and with that, the AD curve shifts upward resulting in an increase in the income level. Consequently, the IS curve shifts to the right from IS2 to IS3 where the new equilibrium point E3 is determined as shown in the diagram. The equilibrium point represents competitively a higher level of income and interest rate. On the contrary, given the initial equilibrium point, if there is a fall in autonomous investment, there is a fall in AD resulting in a fall income level. Consequently, the IS curve shifts to the left from IS2 to IS1 and intersects the LM curve at point E1 where the income level and rate of interest both are underdetermined. This is how a shift in the IS curve affects the equilibrium level of income and interest rate.

#### 4- Causes of a shift in LM curve and its effect on equilibrium income

The main causes of the shift in the LM curve have been mentioned as follows.

#### 4.1- A change in money supply

Given the demand for money, a change in the money supply causes the LM curve to shift either to the right or to the left. When there is an increase in money supply or the government adopts an expansionary monetary policy, the income level increases with a decrease in the rate of interest. Consequently, the LM curve shifts to the right. On the contrary, when there is a fall in the money supply or the government adopts a contractionary monetary policy, the income level falls with a rise in the rate of interest. Hence, the LM curve shifts to the left.

#### 4.2- A change in money demand

Given the quantity of money supply, a change in demand for money also causes the LM curve to shift either to the right or to the left. if there is an increase in demand for money for the transaction, precautionary and speculative motives, the LM curve shifts to the left with an increase in interest rate. On the other hand, with a fall in demand for money, the LM curve shifts to the right with a fall in the rate of interest.

The shift in the LM curve has been explained with a help of the following diagram.
In the above diagram, let us suppose that E2 is the initial equilibrium point. Again suppose that there is an increase in the quantity of money supply and as a result the LM curve shifts to the right from LM2 to LM3 where the new point of equilibrium E3 is attained resulting in a rise in the level of income and fall in the interest rate as shown in the diagram. On the contrary, if it is assumed that there is a fall in the quantity of money supply, the LM curve shifts to the left from LM2 to LM1 where the equilibrium point establishes at E1 resulting in a fall in the income level and rise in the interest rate. This is how the shift in the LM curve affects the equilibrium income level and rate of interest.

#### 5- A simultaneous shift in IS and LM curves

A simultaneous shift in the IS and LM curves has been explained with the help of the following diagram.
In the above diagram, the initial equilibrium point is E1 at which initial income level 0Y1 and interest rate 0i1 have been determined. When there is a shift in the IS and LM curves to the right due to an increase in autonomous investment and a rise in money supply, the IS and LM curves intersect each other at point E3. Consequently, the income level goes up from 0Y1 to 0Y3 with no change in interest rate (ambiguous) as shown in the diagram.  If only the IS curve shifts to the right, the new equilibrium will attain at point E2 where the income level will rise up to 0Y2 leading to the rise in interest rate up to 0i2. On the other hand, if only the LM curve shifts to the right with the given IS curve, the income level will be higher than before but the interest rate falls to a minimum level. Thus, the simultaneous shift in both the curves brings about a multivariate result in the income level and rate of interest, depending upon the direction and magnitude of the shift in the curves.