16

The IS–LM Model for a Two Sector Economy

INTRODUCTION

This chapter is a synthesis of the theory of income and output and the theory of money and interest. It analyses the linkages and the interactions between the goods and money markets to determine that level of income and the interest rate which bring about a simultaneous equilibrium in both the markets. The focus of this chapter is limited to a two sector economy which is later, in other chapters, extended to three sector and four sector economies.

The IS–LM model, which is the foundation of the short-run macroeconomics, was first introduced by J. R. Hicks in his famous article ‘Mr. Keynes and the Classics’ in 1937. While the IS curve represents the goods market equilibrium, investment (I) equals saving (S), the LM curve represents the money market equilibrium, demand for money (L) equals the supply of money (M).

Even today, after a period of more than 70 years since its inception, the IS–LM model continues to be popular because it is a useful tool for analysing the effects of monetary and fiscal policy on the level of output and the rate of interest.

We can analyse the IS–LM model in two ways. The earlier books on macroeconomics follow a four figure quadrants approach to derive the IS and LM curves while the recent books on the subject follow a different approach. It is difficult to say as to which of the two approaches is more correct. While most books on the subject deal with only one of the two approaches, here we attempt at dealing with both the approaches to have a complete picture (Refer to Appendix A).

THE IS–LM MODEL IN A TWO SECTOR ECONOMY

Many authors prefer the approach discussed in the chapter as compared to the recent approach to determine the simultaneous equilibrium in the goods and money markets.

The analysis is based on certain assumptions:

1. The price level is not a variable or is constant.
2. At that price level, the firms are willing to supply whatever output is demanded.
3. The short-run aggregate supply curve is perfectly elastic till the full employment level of output.

Thus, changes in aggregate demand alone can influence the output.

THE GOODS MARKET EQUILIBRIUM IN A TWO SECTOR ECONOMY: THE IS CURVE

In Chapter 4, we had seen that there are two approaches to determine the equilibrium level of income or in other words the goods market equilibrium:

(1) Aggregate demand–Aggregate supply approach where

 

Aggregate demand = Total value of output (or income)

or

Y = C + I

(2) Saving–Investment approach where

 

I = S

From the Aggregate demand–Aggregate supply approach, we have

 

Consumption function as C = C(Y)

Investment function as I = I(r)

Equilibrium condition as Y = C(Y) + I(r)

From the Saving–Investment approach, we have

 

Saving function as S = S(Y)

Investment function as I = I(r)

Equilibrium condition as S(Y) = I(r)

The two equilibrium conditions can be used to develop a graphical approach to the derivation of the IS curve as in Figure 16.1.

In Figure 16.1, there are four quadrants. Quadrant A depicts the investment function showing an inverse relationship between investment and the rate of interest.

Quadrant B gives the saving investment equality in the form of a 45 degree line drawn through the origin. At all points along this 45 degree line, saving equals investment.

Quadrant C depicts the saving function showing a direct relationship between saving and the income level.

Quadrant D shows the goods market equilibrium where the IS curve depicts the different combinations of the output levels and the interest rates at which planned investment equals saving (or planned spending is equal to income).

Figure 16.1 The Goods Market Equilibrium in a Two Sector Economy: The IS Curve

To understand the derivation of the IS curve in Quadrant D, we start in Quadrant A. Assume that the interest rate is r₁ indicating an investment at the level of I₁. Quadrant B shows that for saving investment equality to be maintained, saving must be equal to S₁ (where because of the 45 degree line I₁ = S₁). Quadrant C shows that saving will be S₁ only when the income level is Y₁. Bringing together r₁ of Quadrant A and Y₁ of Quadrant C gives one combination of income and the rate of interest at which the goods market is in equilibrium.

Now assume that the interest rate increases to r₂ indicating an investment at I₂. Quadrant B shows that saving must be equal to S₂. Quadrant C depicts that saving will be S₂ when income is Y₂. Once again bringing together r₂ of Quadrant A and Y₂ of Quadrant C yields another combination of income and the rate of interest at which the goods market is in equilibrium.

 

Similarly, the other combinations of income and the rate of interest at which the saving investment equality exists, indicating equilibrium in the goods market, can be arrived at. By joining such points we can trace out the IS curve. The IS curve is a graphic representation of the goods market equilibrium showing the different combinations of the output levels and the interest rates at which saving equals investment (or planned spending is equal to income). At each point on the IS curve, the goods market is in equilibrium while at all other points there is disequilibrium in the goods market. The IS curve is downward sloping showing that there is an inverse relationship between income and the rate of interest.

THE MONEY MARKET EQUILIBRIUM IN A TWO SECTOR ECONOMY: THE LM CURVE

The money market is in equilibrium when the demand for money is equal to the supply of money. Thus, we have

 

m_s = m_d

As far as the supply of money is concerned, it is determined by the monetary authorities and hence assumed to be exogenous. Thus, we can express the supply of money as

Regarding the demand for money, according to the Keynesian theory the demand for money consists of the transactions demand (including the precautionary demand for money) which is a direct function of the income level and the speculative demand which is an indirect function of the interest rate. Thus, the total demand for money can be expressed as

 

m_d = m_t + m_sp

where	md = total demand for money
	mt = ky (transactions demand for money)
	msp = g(r) (speculative demand for money)

Thus, we can write

 

m_d = kY g(r)

From the above, we have

 

Supply of money as m_s = _s
Demand for money as m_d = kY + g(r)
Money market equilibrium condition as _s = kY + g(r)

The money market equilibrium condition can be used to develop a graphical approach to the derivation of the LM curve as in Figure 16.2.

In Figure 16.2, there are four quadrants. Quadrant A depicts the speculative demand for money showing an inverse relationship between speculative demand for money and the rate of interest.

Quadrant B shows the supply of money as a straight line meeting the two axes. At point A where it is meeting the y axis indicates that the entire supply of money is used for transactions demand while at point B where it is meeting the x axis indicates that the entire supply of money is used for purposes of speculative demand. All points along the line AB show the different ways in which the supply of money can be divided between the transactions demand and the speculative demand.

Quadrant C depicts the income level that will support the transactions demand determined in Quadrant B [given the transactions demand function as m_t = k(Y)].

Quadrant D shows the money market equilibrium where the LM curve shows the different combinations of the output levels and the interest rates at which the demand for money is equal to the supply of money.

To understand as to how the LM curve has been derived in Quadrant D we start in Quadrant A. Assume that the interest rate is r₁ indicating that the public wants to hold a speculative demand for money equal to m_sp1.

Quadrant B shows that when the speculative demand is m_sp1 the amount of money left as transactions balances will be m_t1. Quadrant C shows that transactions balances of m_t1 will be consistent with an income level of Y₁. Hence, combining together r₁ of Quadrant A and Y₁ of Quadrant C gives one combination of income and the rate of interest at which the money market is in equilibrium.

Figure 16.2 The Money Market Equilibrium in a Two Sector Economy: The LM Curve

Next assume that the interest rate increases to r₂ indicating that the speculative demand must be m_sp2. Quadrant B shows that the money available for transactions will be equal to m_t2. Quadrant C depicts that transactions balances of m_t2 will be consistent with an income level of Y₂. Once again bringing together r₂ of Quadrant A and Y₂ of Quadrant C yields another combination of income and the rate of interest at which the money market is in equilibrium.

 

In a similar manner, the other combinations of income and the rate of interest at which the demand for money equals the supply of money indicating equilibrium in the money market can be arrived at. By joining all such points we can trace out the LM curve. The LM curve is a graphic representation of the money market equilibrium showing the different combinations of the output levels and the interest rates at which the demand for money is equal to the supply of money. It is important to note that at each point on the LM curve, the money market is in equilibrium and at all other points there exists disequilibrium in the money market. The LM curve is upward sloping showing that there is a direct relationship between income and the rate of interest. This is because when the rate of interest increases, individuals demand less for speculative purposes and more for transactions which is consistent only with a higher level of income.

EQUILIBRIUM IN THE TWO MARKETS: THE GOODS MARKET AND MONEY MARKET

The IS curve represents all combinations of income and the rate of interest at which saving equals investment (or planned spending is equal to income) or, in other words, where the goods market is in equilibrium. The LM curve represents all combinations of income and the rate of interest at which the demand for money equals the supply of money or, in other words, where the money market is in equilibrium. But there is only one combination of income and the rate of interest at which both the goods and the money market are in equilibrium, as depicted in Figure 16.3. This combination is given by point E at which the IS and LM curves intersect to determine the equilibrium rate of interest at r* and the equilibrium level of income at Y*. At all other points, there exists disequilibrium in either the goods market or the money market or both the markets.

The IS Curve: An Algebraic Explanation

Aggregate Demand–Aggregate Supply Approach

The goods market is in equilibrium when

 

Aggregate demand = Total value of output (or income)

or

 

Y = C + I

But the linear forms of the consumption function is C = C_a + bY while investment function can be expressed as I = – hr.

Thus,

 

Y = C_a + bY + – hr

Hence,

Y – bY = C_a + – hr

or

Figure 16.3 Equilibrium in the Two Markets: The Goods Market and Money Market

Equation (1) represents the IS curve.

Saving–Investment Approach

In equilibrium,

 

I = S
But the saving function is S = – C_a + (1 – b)Y while investment function is taken as I = – hr.

Thus,

 

– hr = –C_a + (1 – b)Y

or

This is the same equation as Eq. (1) above.

Hence, both the approaches yield the same equation for the IS curve.

Numerical Illustration 1

Suppose the consumption and investment functions are as follows:

 

C = 100 + 0.75 Y

I = 250 – 5r

Find the equation of the IS curve.

Solution

The equation of the IS curve is

 

Y = C + I
Y = 100 + 0.75Y + 250 – 5r
Y – 0.75Y = 350 – 5r
0.25Y = 350 – 5r
Y = 1400 – 20r

Figure16.4 shows the IS curve Y = 1400 – 20r.

The LM Curve: An Algebraic Explanation

The money market is in equilibrium when

 

m_d = m_s

But
	md = mt + msp
where	md = total demand for money
	mt = kY (transactions demand for money)
	msp = g(r) (speculative demand for money)

However for the sake of convenience, we assume that the speculative demand for money is a linear function (rather than a curve). Hence, we have m_sp= – g(r)

From the above, we have

 

Supply of money as m_s = _s
Demand for money as m_d = kY + _sp g(r)

The money market equilibrium condition can be written as

Thus,

Equation (2) represents the LM curve.

Numerical Illustration 2

Suppose that the supply of money is Rs. 400. The transactions and speculative demand for money functions are as follows:

 

m_t = 0.25Y

m_sp = 100 – 4r

Find the equation of the LM curve.

Solution

 

m_d = m_t + m_sp

m_d = 0.25Y + 100 – 4r

In equilibrium, m_d = m_s

Figure 16.4 The IS Curve Equation Y = 1400–20r

Thus,

0.25Y + 100 – 4r = 400

0.25Y = 300 + 4r

Y = 1200 + 16r

Figure16. 5 shows the LM curve Y = 1200 + 16r.

Equilibrium in the Two Markets: The Goods Market and Money Market: An Algebraic Explanation

A simultaneous equilibrium in both the goods and money markets can be determined by solving the equations for the IS and LM curves.

Equation of the IS curve:

Equation of the LM curve:

These are a set of simultaneous equations which can be solved to determine the equilibrium values of Y and r.

Numerical Illustration 3

Suppose the consumption and investment functions are as follows:

 

C = 100 + 0.75Y

I = 250 – 5r

Also assume that the supply of money is Rs. 280. The demand for money function is as follows:

 

m_d = 0.25Y – 2r

Figure 16.5 The LM Curve Equation Y = 1200 + 16r

1. Find the equation of the IS curve.
2. Find the equation of the LM curve.
3. Find the simultaneous equilibrium for the IS curve and LM curves.

Solution

(1) IS equation:

 

Y = C + I

Y = 100 + 0.75 Y + 250 – 5r

Y – 0.75Y = 350 – 5r

0.25Y = 350 – 5r

Y = 1400 – 20r

(2) LM equation:

 

m_d = 0.25Y – 2r

m_s = 280

In equilibrium,

 

m_d = m s

Thus,

 

0.25Y – 2r = 280

0.25Y = 280 + 2r

Y = 1120 + 8r

(3) Simultaneous equilibrium for the IS curve and LM curves:

 

IS = LM

1400 – 20r = 1120 + 8r

28r = 280

r = 10%

Y = 1400 – 20 × 10

Y = 1200

Simultaneous equilibrium for the IS curve and LM curves exists when Y = 1200 and r = 10%.

Figure 16.6 shows the simultaneous equilibrium for the IS curve and LM curves when Y = 1200 and r = 10%.

• The IS curve is downward sloping because there exists an inverse relationship between income and the rate of interest.
• The LM curve is upward sloping because there is a direct relationship between income and the rate of interest.
• It is only at the intersection of the IS and LM that both the goods and the money market are in equilibrium.

Figure 16.6 Simultaneous Equilibrium for IS and LM Curves When Y = 1200 and r = 10%

DISEQUILIBRIUM TO EQUILIBRIUM: THE PROCESS OF ADJUSTMENT

In Figure 16.3, we have already observed that at points, other than point E*, there will exist disequilibrium in either the goods market or the money market or in both the markets.

As far as the IS curve is concerned, any combination of income and interest that lies on the IS curve represents a goods market equilibrium. However, at all other points there exists goods market disequilibrium.

1. All combinations of income and interest that lie above and towards the right of the IS curve indicate a situation where Y > C + I or saving is greater than planned investment. Hence, the level of income will fall.
2. All combinations of income and interest that lie below and towards the left of the IS curve indicate a situation where Y < C + I or saving is less than planned investment. Hence, the level of income will rise. It has been divided into four spaces.

   Space	Goods Market	Money Market
   Space 1	S > I, Y> C + I	md < ms
   Space 2	S > I, Y > C + I	md > ms
   Space 3	S > I, Y < C + I	md > ms
   Space 4	S < I, Y < C + I	md < ms

Regarding the LM curve, any combination of income and interest that lies on the LM curve represents money market equilibrium. However, at all other points there exists money market disequilibrium.

1. All combinations of income and interest that lie below and towards the right of the LM curve indicate a situation where the demand for money is greater than the supply of money or there is an excess demand for money. Hence, the rate of interest will rise.
2. All combinations of income and interest that lie above and towards the left of the LM curve indicate a situation where the demand for money is less than the supply of money or there is an excess supply of money. Hence, the rate of interest will fall.

As far as the goods market is concerned,

1. When S > I or Y > C+ I, there will occur a decrease in income.
2. When S < I or Y< C+ I, there will occur an increase in income.

As far as the money market is concerned,

1. When m_d < m_s, there will occur a decrease in the rate of interest.
2. When m_d > m_s, there will occur an increase in the rate of interest.

Suppose that in Figure 16.3 the economy is at point P, a position of a disequilibrium located in the space 3. Here, there exists

1. an excess demand for goods or S < I, or Y < C+ I. Thus, there will occur an increase in the income.
2. an excess demand for money or m_d > m_s. Thus, there will occur an increase in the rate of interest.

The economy will gradually move to a point like Q. But since point Q is on the IS curve, the goods market will be in equilibrium. However, the demand for money is greater than the supply of money since point Q is located to the right of the LM curve. Thus, there will occur an increase in the interest rate and a movement to a point like R. Thus, adjustments will occur and ultimately the forces that are pushing the income level and the interest rate will move the economy to point E which is the only point at which the goods and the money markets are both in equilibrium.

Figure 16.7 A Shift in the IS Curve

The economy could be anywhere in the system in any of the different spaces. However, it is important to remember that the forces which are at work will go on till the final equilibrium is attained at point E. Once point E is reached, the income level and the rate of interest will remain unchanged until there occurs a shift in the IS or LM curve or in both of them which can disturb the equilibrium and start off a new round of adjustments.

• If there exists disequilibrium in the economy, then the forces will be at work till the final equilibrium is attained at the intersection of the IS–LM curves.

A SHIFT IN THE IS–LM CURVES

A shift in the IS or LM curve or in both of them will disturb the equilibrium; Hence, it is imperative to examine these shifts.

A Shift in the IS Curve

The shifts in the IS curve can occur due to a shift in the investment function or the saving function or due to a change in any of the factors which are responsible for a shift in these functions.

In Figure. 16.7, the IS₁ curve has been derived diagrammatically from the investment curve I₁ in Quadrant A, the saving–investment equality in Quadrant B and the saving function in Quadrant C. The intersection of IS₁ and the LM curves at point E₁ determines the equilibrium income as Y₁ and the equilibrium interest rate at r₁.

Assume there is rightward shift in the investment function from I₁ to I₂ in Quadrant A of the Figure 16.7. This implies an increase in investment at all interest rates. Given the interest rate at r₂, to match the increase in the investment, the equilibrium level of saving in Quadrant B will also increase from S₁ to S₂ leading to an increase in the income level from Y₁ to Y₂ in the Quadrant C. Thus, combining the income Y₂ with the interest rate, r₂ we can trace out the new IS curve IS. The intersection of IS₂ and the LM curves at point E₂ determines the new equilibrium income as Y₂ and the equilibrium interest rate at r₂.

For a leftward shift in the investment function, the results that follow will be just the opposite.

A shift in the consumption function also leads to a shift in the IS curve. An upward shift in the consumption function (or in other words a downward shift in the saving function) implies a decrease in the saving at any income level. To maintain the saving investment equality, the income level would have to increase leading to eventually a rightward shift in the IS curve. For a downward shift in the consumption function, the results that follow will just be the opposite.

A Shift in the LM Curve

The shift in the LM curve can occur due to a shift in the transactions demand, speculative demand or the money supply function.

In Figure 16.8, LM₁ has been derived diagrammatically from the transactions demand and speculative demand for money functions. The intersection of IS and the LM₁ curves at point E₁ determines the equilibrium income as Y₁ and the equilibrium interest rate at r₁.

Assume there is an increase in the money supply. Hence, there is a rightward shift in the money supply curve from m_s1 to m_s2 in Quadrant B of Figure 16.8. Given the rate of interest at r₂, this will lead to an increase in the transactions demand for money from m_t1 to m_t2. Since a larger transactions demand for money is consistent only at a higher income level, this will involve an increase in the income level from Y₁ to Y₂. Hence, combining the change in income Y₂ with the change in the interest rate, r₂ we can trace out the new LM curve, LM₂. The intersection of IS and the LM₂ curves at point E₂ determines the new equilibrium income as Y₂ and the equilibrium interest rate at r₂.

Figure 16.8 A Shift in the LM Curve

For a leftward shift in the money supply function, the results that follow will be just the opposite.

A shift in the speculative demand function also leads to a shift in the LM curve. An upward shift in the speculative demand function implies a decrease in the transactions demand for money by an equal amount. This would necessitate a decrease in the income level. Hence for an upward shift in the speculative demand function, the LM curve will shift to the left. Similarly, a shift in the transactions demand for money function will lead to a shift in the LM curve.

A Simultaneous Shift in Both IS and LM Curves

Till now we have analysed the shifts in the IS and LM curves separately. Now we analyse the two shifts simultaneously.

In Figure 16.9, suppose the initial IS and LM curves are given as IS₁ and LM₁ They intersect at point E₁ to determine the equilibrium income at Y₁ and the rate of interest at r₁.

Next suppose there is:

Figure 16.9 A Simultaneous Shift in Both IS and LM Curves

1. an increase in investment (or a rightward shift in the investment function) which shifts the IS curve from IS₁ to IS₂.
2. an increase in the money supply (or a rightward shift in the money supply function) which shifts the LM curve from LM₁ to LM₂.

The curves IS₂ and LM₂ intersect at point E₂ to determine the equilibrium income at Y₂ while the rate of interest remains unchanged at r₁. The reason why there is no change in the interest rate is that the extent of the shifts in the IS and LM curves are equal. However, in reality the extent of the shifts may not be equal in which case there will not only be a change in the income level but also a change in the interest rate.

• The shifts in the IS curve occur due to a shift in the investment function or the saving function.
• The shift in the LM curve occur due to a shift in the transactions demand, speculative demand or the money supply function.

INTRODUCTION

1. The chapter analyses the linkages and the interactions between the goods and money markets to determine that level of income and the interest rate which bring about a simultaneous equilibrium in both the markets.
2. The IS–LM model, which is the foundation of the short-run macroeconomics, was first introduced by J. R. Hicks.
3. The IS curve represents the goods market equilibrium.
4. The LM curve represents the money market equilibrium.
5. We can analyse the IS–LM model in two ways.

THE IS–LM MODEL FOR A TWO SECTOR ECONOMY

The analysis is based on certain assumptions: price level is constant; at that price level, the firms are willing to supply whatever output is demanded; short-run aggregate supply curve is perfectly elastic till the full employment level of output.

THE GOODS MARKET EQUILIBRIUM: THE IS CURVE

1. There are two approaches to determine the goods market equilibrium.
2. According to the aggregate demand–aggregate supply approach, the goods market equilibrium exists where Y = C(Y) + I(r).
3. According to the savings–investment approach, the goods market equilibrium exists where S(Y) = I(r).
4. The two equilibrium conditions can be used to develop a graphical approach to the derivation of the IS curve.
5. The IS curve is a graphic representation of the goods market equilibrium showing the different combinations of the output levels and the interest rates at which saving equal investment (or planned spending is equal to income).
6. The IS curve is downward sloping showing that there is an inverse relationship between income and the rate of interest

THE MONEY MARKET EQUILIBRIUM: THE LM CURVE

1. The money market is in equilibrium when the demand for money is equal to the supply of money or m_s= m_d.
2. The supply of money is assumed to be exogenous or as m =
3. The total demand for money can be expressed as m_d = m_t + m_sp.
4. The money market equilibrium condition can be used to develop a graphical approach to the derivation of the LM curve.
5. The LM curve is a graphic representation of the money market equilibrium showing the different combinations of the output levels and the interest rates at which the demand for money is equal to the supply of money.
6. The LM curve is upward sloping showing that there is a direct relationship between income and the rate of interest. This is because when the rate of interest increases, individuals demand becomes less for speculative purposes and more for transactions which is consistent only with a higher level of income.

EQUILIBRIUM IN THE TWO MARKETS: THE GOODS MARKET AND MONEY MARKET

1. The IS curve represents all combinations of income and the rate of interest at which the goods market is in equilibrium.
2. The LM curve represents all combinations of income and the rate of interest at which the money market is in equilibrium.
3. There is only one combination of income and the rate of interest at which both the goods and the money market are in equilibrium, point E at which the IS and LM curves intersect.

THE IS CURVE: AN ALGEBRAIC EXPLANATION

Both the aggregate demand–aggregate supply approach and the saving–investment approach yield

 

as the equation of the IS curve.

THE LM CURVE: AN ALGEBRAIC EXPLANATION

 

is the equation of the LM curve.

EQUILIBRIUM IN THE TWO MARKETS: THE GOODS MARKET AND MONEY MARKET: AN ALGEBRAIC EXPLANATION

A simultaneous equilibrium in both the goods and money markets can be determined by solving the equations for the IS and LM curves and thus the equilibrium values of Y and I can be determined.

A SHIFT IN THE IS–LM CURVES

1. A shift in the IS or LM curve or in both of them will disturb the equilibrium; hence, it is imperative to examine these shifts.
2. The shifts in the IS curve can occur due to a shift in the investment function or the saving (or consumption) function.
3. A rightward shift in the investment function leads to a rightward shift in the IS curve. For a leftward shift in the investment function, the results that follow will be just the opposite.
4. The shift in the LM curve can occur due to a shift in the transactions demand, speculative demand or the money supply function.
5. A rightward shift in the money supply curve leads to a leftward shift in the LM curve. For a leftward shift in the money supply function, the results that follow will be just the opposite.
6. A simultaneous shift in both IS and LM curves bring about a change in the income level and also a change in the interest rate.

TRUE OR FALSE QUESTIONS

1. The IS–LM model is the foundation of long run macroeconomics.
2. The IS curve is upward sloping showing that there is a direct relationship between income and the rate of interest.
3. The LM curve is upward sloping showing that there is a direct relationship between income and the rate of interest.
4. There is only one combination of income and the rate of interest at which both the goods and the money market are in equilibrium.
5. The shift in the LM curve can occur due to a shift in the investment function or the saving function.

VERY SHORT-ANSWER QUESTIONS

1. Which are the two approaches to determine the goods market equilibrium?
2. Why does the LM curve slope upward? Explain.
3. At which point does a simultaneous equilibrium occur in both the goods and money markets?
4. Give the equations for the IS and LM curves.
5. How is simultaneous equilibrium in the two markets determined? Give an algebraic explanation.

SHORT-ANSWER QUESTIONS

1. Write a short note on the IS curve.
2. Write a short note on the LM curve
3. ‘There is only one combination of income and the rate of interest at which both the goods and the money market are in equilibrium.’ Explain.
4. Show the algebraic derivation of
   1. IS Curve
   2. LM Curve
5. Examine the effects of a simultaneous shift in both the IS and LM curves.

LONG-ANSWER QUESTIONS

1. ‘The two equilibrium conditions aggregate demand–aggregate supply approach and S–I approach can be used to develop a graphical approach to the derivation of the IS curve.’ Explain.
2. ‘The money market equilibrium condition can be used to develop a graphical approach to the derivation of the LM curve’. Explain.
3. How is simultaneous equilibrium in the goods and money market achieved? Explain with the help of diagram/diagrams.
4. Examine the effects of a shift in the IS curve.
5. Examine the effects of a shift in the LM curve.

SOLVED NUMERICAL PROBLEMS

Numerical Problem 1

Suppose the consumption and investment functions are as follows:

 

C = 50 + 0.75Y

I = 80 crore – 2r

Find

1. The equation of the IS curve and plot it.
2. The equation of the IS curve when investment increases by Rs. 20 crore.
3. By how much does the IS curve shift?

Numerical Problem 2

Suppose that the supply of money is Rs. 240 crore. The demand for money is m_d = 0.20 Y – 5r. Find

1. The equation of the LM curve and plot it.
2. The equation of the LM curve when the supply of money increases by Rs. 60 crore to Rs. 300 crore.
3. By how much does the LM curve shift

Numerical Problem 3

Suppose the consumption, investment, demand for money and supply of money functions are as follows:

 

C = 0.75 Y

I = 107.5 crore – 0.25r

m_d = 0.25Y – 2.5r

m_s = 80 crore

Find

1. The equilibrium income and the rate of interest.
2. The equilibrium income and the rate of interest when autonomous investment increases to Rs. 135 crore.

Numerical Problem 4

Suppose autonomous consumption is Rs. 60 crore, investment is Rs. 120 crore, marginal propensity to consume is 0.75 and the value of h, the behavioral coefficient which measures the sensitivity of investment to the rate of interest, is 4. Find

1. The equation of the IS curve and plot it.
2. The equation of the IS curve when h increases to 8.
3. What is the effect on the slope of the IS curve when the value of g increases

Numerical Problem 5

Suppose that the value of k is 0.25. Find the direction and the amount of shift in the LM curve when

1. the increase in the money supply is Rs. 10 crore.
2. the decrease in the money supply is Rs. 25 crore.

UNSOLVED NUMERICAL PROBLEMS (WITH ANSWERS)

1. Suppose the saving and investment functions are as follows:

    

   S = –50 + 0.5 Y

   I = 120 crore – 5r

   Find the equation of the IS curve and plot it.

2. Suppose the consumption and investment functions are as follows:

    

   C = 20 + 0.5Y

   I = 120 crore – 5r

   Find

   1. The equation of the IS curve and plot it.
   2. The equation of the IS curve when the investment function changes to I = 120 crore – 10r
3. Suppose the consumption and investment functions are as follows:

    

   C = 20 + 0.75Y

   I = 400 – 2500r

   Find

   1. The equation of the IS curve by the aggregate demand–aggregate supply approach.
   2. The equation of the IS curve by the saving–investment approach.
4. In a two sector model, assume that the consumption and investment functions are as follows:

    

   C = 100 + 0.75Y

   I = 1500 – 10r

   Find

   1. The equation of the IS curve
   2. The equation of the IS curve when the investment function changes to I = 1500 – 5r while the consumption function remains the same.
5. Suppose in a two sector model consumption and investment functions are as follows:

    

   C = 600 + 0.80Y

   I = 1160 – 20r

   The demand for money and the supply of money are

    

   L = 0.20Y – 50r

   M = 1200

   1. Find the equation of the IS curve
   2. Find the equation of the LM curve
   3. Find the simultaneous equilibrium for the IS and LM curves.

TRUE OR FALSE QUESTIONS

1. False. The IS–LM model is the foundation of the short run macroeconomics and was first introduced by J. R. Hicks.
2. False. The IS curve is downward sloping showing that there is an inverse relationship between income and the rate of interest.
3. True. The LM curve is upward sloping showing that there is a direct relationship between income and the rate of interest. This is because when the rate of interest increases, individuals demand less for speculative purposes and more for transactions which is consistent only with a higher level of income.
4. True. There is only one combination of income and the rate of interest at which both the goods and the money market are in equilibrium and that is the point at which the IS and LM curves intersect. At all other points, there exists disequilibrium.
5. False. The shift in the LM curve can occur due to a shift in the transactions demand, speculative demand or the money supply function

SOLVED NUMERICAL PROBLEMS

Solution 1

1. Equation of the IS curve:

    

   Y = C + I

   Y = 50 + 0.75Y + 80 – 2r

   Y – 0.75Y = 130 – 2r

   0.25Y = 130 – 2r

   Y = 520 – 8r

2. Equation of the IS curve when investment increases by Rs. 20 crore:

    

   Y = C + I

   Y = 50 + 0.75 Y + 100 – 2r

   Y – 0.75 Y = 150 – 2r

   0.25Y = 150 – 2r

   Y = 600 – 8r

3. When investment increases by Rs. 20 crore, the IS curve shifts horizontally by Rs. 80 crore, which is equal to the increase in investment times the multiplier, Rs. 20 crore × 4.

Figure 16.10 gives the diagrammatic picture of the IS curves where equation Y = 520 – 8r is represented by IS₁ and equation Y = 600 – 8r is represented by IS₂.

Solution 2

1. Equation of the LM curve

    

   In equilibrium, m_d = m_s

   Thus,

    

   0.20Y – 5r = 240

   0.20Y = 240 + 5r

   Y = 1200 + 25r

2. Equation of the LM curve when the supply of money increases by Rs. 60 crore to Rs. 300 crore:

    

   In equilibrium, m_d = m_s

   Thus,

   0.20Y – 5r = 300

   0.20Y = 300 + 5r

   Y = 1500 + 25r

3. When the supply of money increases by Rs. 60 crore, the LM curve shifts horizontally by Rs. 300 crore, which is equal to

Figure 16.11 gives the diagrammatic picture of the LM curves where equation Y = 1200 + 25r is represented by LM₁ and equation Y = 1500 + 25r is represented by LM₂.

Solution 3

1. Equilibrium income and the rate of interest:

    

   Equation of the IS curve:

   Y = C + I

   Y = 0.75Y + 107.5 – 0.25r

   Y – 0.75Y = 107.5 – 0.25r

   0.25Y = 107.5 – 0.25r

   Y = 430 – r

   Equation of the LM curve:

   In equilibrium, m_d = m_s

   Thus,

    

   0.25Y – 2.5r = 80

   0.25Y = 80 + 2.5r

   Y = 320 + 10r

   Simultaneous equilibrium for the IS and LM curves:

    

   IS = LM

   430 – r = 320 + 10r

   

   Figure 16.10 IS Curves of Equation Y = 520–8r and Y = 600–8r

    

   11r = 110

   r = 10%

   Y = 430 – 10

   Y = 420

   Simultaneous equilibrium for the IS curve and LM curves exists when Y = 420 and r = 10%.

2. The equilibrium income and the rate of interest when autonomous investment increases to Rs. 135 crore: Equation of the new IS curve:

    

   Y = C + I

   Y = 0.75Y + 135 – 0.25r

   

   Figure 16.11 LM Curves of Equation Y = 1200 + 25r and Y = 1500 + 25r

    

   Y – 0.75Y = 135 – 0.25r

   0.25Y = 135 – 0.25r

   Y = 540 – r

   Equation of the LM curve: Y = 320 + 10r

   Simultaneous equilibrium for the IS curve and LM curve:

    

   IS = LM

   540 – r = 320 + 10r

   11r = 220

   r = 20%

   Y = 540 + 20

   Y = 520

   Simultaneous equilibrium for the IS curve and LM curves exists when Y = 520 and r = 20%.

Solution 4

1. Equation of the IS curve:

   The consumption and investment functions are as follows:

    

   C = 60 + 0.75Y

   I = 120 – 4r

   We have

    

   Y = C + I

   Y = 60 + 0.75Y + 120 – 4r

   Y – 0.75Y = 180 – 4r

   0.25Y = 180 – 4r

   Y = 720 – 16r

   This is represented by the IS₁ curve in the Figure 16.12.

   

   Figure 16.12 IS Curves of Equation Y = 720–16r and Y = 720–32r

2. Equation of the IS curve when h increases to 8.

   The consumption and the new investment functions are as follows:

    

   C = 60 + 0.75Y

   I = 120 – 8r

   We have

    

   Y = C + I

   Y = 60 + 0.75Y + 120 – 8r

   Y – 0.75Y = 180 – 8r

   0.25Y = 180 – 8r

   Y = 720 – 32r

   This is represented by IS₂ in Figure 16.12.

3. The effect on the slope of the IS curve when the value of h increases: The slope of the IS₂ curve is less than that of IS₁.

Solution 5

1. The direction and the amount of shift in the LM curve when the increase in the money supply is Rs. 10 crore:

   Amount of shift in the LM curve: × 20 = 80

   Since there is an increase in the money supply, the LM curve will shift to the right.

2. The direction and the amount of shift in the LM curve when the decrease in the money supply is Rs. 25 crore:

   Amount of shift in the LM curve: × 25 = 100

   Since there is a decrease in the money supply, the LM curve will shift to the left.

Space	Goods Market	Money Market
Space 1	S > I, Y> C + I	md < ms
Space 2	S > I, Y > C + I	md > ms
Space 3	S > I, Y < C + I	md > ms
Space 4	S < I, Y < C + I	md < ms

UNSOLVED NUMERICAL PROBLEMS

1. Equation of the IS curve: Y = 340 – 10r

   Different values of r can be taken with the corresponding values of Y and then the IS curve can be plotted. When r = 5, Y = 290; r = 7, Y = 270; and r = 10, Y = 240.

2. (a) Equation of the IS curve: Y = 280 – 10r

   (b) Equation of the IS curve when investment changes:
   Y = 280 – 20r

3. (a) Equation of the IS curve by the aggregate demand–aggregate supply approach: Y = 1680 – 10000r

   (b) Equation of the IS curve by the saving–investment approach:
   Y = 1680 – 10000r

4. (a) Equation of the IS curve: Y = 6400 – 40r

   (b) Equation of the IS curve when the investment function changes to I = 1500 – 5r while the consumption function remains the same: Y = 6400 – 20r

5. (a) IS equation: Y = 8800 – 100r

   (b) LM Equation: Y = 6000 + 250r

   (c) Simultaneous equilibrium for the IS and LM curves:
   r = 8%, Y = 8000