### 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.

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