APPENDIX B

The IS–LM Framework for a Four Sector Model

INTRODUCTION

In Chapters 16 and 17, we have focused on a closed economy. We are now in a position to relax this assumption and focus on an open economy in which there are four sectors: households, firms, government and the foreign sector.

THE IS–LM MODEL FOR A FOUR SECTOR ECONOMY

In the construction of a four sector model, two new variables which pertain to the foreign sector are included, exports and imports. The analysis is based on certain assumptions, some of which are the same as in the earlier models:

1. The price level is constant.
2. At that constant 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.
4. Government expenditure is autonomous. Hence, G = .
5. Taxes are a linear function of income. Thus, T = + tY, where T is the autonomous tax (which is independent of the income level) while t is the income tax rate.
6. Exports involve an outflow of domestic goods and services and, hence, an inflow of income. It is assumed that exports are constant, or X = .
7. Imports involve an inflow of foreign goods and services and, hence, an outflow of income. It is assumed that imports are a function of income, or M = + mY.

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

As already discussed, there are two approaches to determine the equilibrium level of income. In a four sector economy, they are:

1. Aggregate Demand—Aggregate Supply Approach

    

   	Aggregate demand = Total value of output (or income)
   or	Y = C + I + G + X – M

2. Injections equal Leakages Approach

    

   I + G + X = S + T + M

We have consumption function as	C = C(Y)
Investment function as	I = I(r)
Government expenditure	G =
Saving function as	S = S(Y)
Tax as	T = + tY
Exports as	X = (where exports are constant)

Imports function as M = + m Y (where M_a represents autonomous imports and m is the marginal propensity to import)

The equilibrium condition can be written as

 

Y = C(Y) + I(r) + + + M(Y)

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

In Figure B.1, in Quadrant A exports have been added to the I + G curve to get the I + G + X. As exports are independent of the interest rate, I + G + X curve lies to the right of I + G curve of the three sector economy.

Quadrant B gives the injections—leakages equality in the form of a 45 degree line drawn through the origin. At all points along this 45 degree line, injections equal leakages or I + G = X = S + T + M.

Figure B.1 The Goods Market Equilibrium in a Four Sector Economy: The IS Curve

 

Quadrant C shows the saving plus tax plus imports function. To introduce imports in the model, imports are added vertically to the S + T curve to get the S + T + M curve.

The investment plus government expenditure plus exports (I + G + X) curve of Quadrant A and the saving plus tax curve plus imports, (S + T + M) curve of Quadrant C yields the curve IS_o for the open economy in Quadrant D.

As compared to the IS curve of the closed economy, the slope of the IS curve is greater than the IS_o curve of the closed economy.

THE MONEY–MARKET EQUILIBRIUM IN A THREE SECTOR ECONOMY: THE LM CURVE

In a four sector economy, the analysis of the money market will remain the same as in the two and three sectors economy since the introduction of the foreign sector does not influence either the demand for money or the supply of money.

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

The IS Curve: An Algebraic Explanation

The goods market is in equilibrium when

Aggregate demand = Total value of output (or income)

 

or	Y = C + I + G + X – M
But
	C = Ca + bYd,
	I = – hr,
	G = ,
	Yd = Y – T,
	T = + tY
	X =
	M = + mY
Thus,	Y = Ca + bYd + – hr + + – ( + mY)
	Y = Ca + b(Y – T) + – hr + + – ( + mY)
	Y = Ca + b[Y – ( + tY)] + – hr + + – ( + mY)
	Y = Ca + bY – b– btY + – hr + + – – mY
	Y – bY + btY + mY = Ca – b + – hr + + – Ma
	Y(1 – b + bt + m) = Ca – b + – hr + + –

 

Equation (1) represents the IS curve in a four sector economy.

The LM Curve: An Algebraic Explanation

The money market equilibrium is similar to that of a two and three sector economy. Thus,

 

m_d = m_s

But
	md = mt + msp
where,	mt = kY
	msp = g(r)

 

We assume that the speculative demand for money is a linear function

We have

 

Supply of money as                      

Demand for money as                   

The money—market equilibrium condition can be written as

 

Thus,

 

Equation (2) represents the LM curve.

Equilibrium in the Goods and the Money Market in Three Sector Economy

We have already observed in Chapter 17 that there is only one combination of income and the rate of interest at which there exists a simultaneous equilibrium in the goods and the money markets in a four sector economy. This has been depicted in Figure B.2. This combination exists at point E_o at which the IS_o and LM curves intersect to determine the equilibrium rate of interest at r_o and the equilibrium level of income at Y_o. At all other points, there exists disequilibrium in either the goods market or the money market or both the markets. This disequilibrium will remain unchanged until a shift in the IS or LM curve disturbs the equilibrium.

 

 

Figure B.2 Equilibrium in the Goods and the Money Market in a Four Sector Economy

 

 

Figure B.3 Shifts in the IS Curve in a Four Sector Economy

SHIFTS IN THE IS CURVE IN A FOUR SECTOR ECONOMY

A shift in the IS curve can occur due to a change in both internal and external factors. Here, we are focusing on only the external factors or the foreign sector, as we have already analysed the internal factors in much detail.

In Figure B.3, the initial equilibrium exists at point E_o at which the IS_o and LM curves intersect to determine the equilibrium rate of interest at r_o and the equilibrium level of income at Y_o.

Assume now that there occurs a rightward shift in the IS curve from IS_o to IS_r This could occur due to:

1. An increase in exports which could arise due to
   1. a change in the tastes in the other countries in favour of exports.
   2. an increase in the incomes in the other countries.
   3. a decrease in the duties on imports in the other countries.
2. A decrease in imports which could arise due to
   1. a change in the tastes in the domestic economy in favour of the domestic goods.
   2. an increase in the duties on imports from the other countries.

Given that there is no change in the relative price levels or any other changes, the new equilibrium will occur at point E₁ at which the IS₁ and LM curves intersect. The equilibrium rate of interest will be r₁ and the equilibrium level of income will be Y₁. Hence, there will occur an increase in both the equilibrium income and the rate of interest. Similarly, it can be shown that a leftward shift in the IS curve will lead to a decrease in both the equilibrium income and the rate of interest.