# Multiplier in 2, 3, 4 Sector

To simplify the analysis, it has been classified into a two-sector model, a three-sector model and a four-sector model.

First two sectors are related to a closed economy in which there is no foreign trade and the last sector is concerned with the open economy.

**TWO SECTOR MODEL**

A two-sector model of income determination of an economy consists only of domestic and business sectors.

**Assumptions**

**The income determination in a closed economy is based on the following Assumptions**

- It is a two-sector economy where only consumption and investment expenditures take place. Thus the total output of the economy is the sum of consumption and investment expenditure.
- Investment relates to net investment after deducting depreciation.
- It is a closed economy in which there are no exports or imports.
- There are no corporate firms in the economy so that there are no corporate undistributed profits.
- There are no business taxes, no income taxes and no social security taxes so that disposable personal income equals NNP.
- There are no transfer payments.
- There is no government.
- There is autonomous investment.
- The economy is at less than full employment level of output.
- The price level remains constant up to the level of full employment.
- The money wage rate is constant.
- There is stable consumption function.
- The rate of interest is fixed.
- The analysis relates to the short period.

**Explanation**

Given these assumptions, the equilibrium level of national income can be determined by the equality of aggregate demand and aggregate supply or by the equality of saving and investment.

Aggregate demand is the summation of consumption expenditure on newly produced consumer goods by households and on their services (C), and investment expenditure on newly produced capital goods and inventories by businessmen (I).

**It is shown by the following identities:**

Y = C+I ….(1)

Personal Income: Y_{d} = C+S….(2)

But Y= Y_{d}

C + I = C + S

Or I = S

Where Y = national income, Y_{d} = disposable income, C = consumption, S = saving, and I = investment.

In the above identities, C + l relate to consumption and investment expenditures which represent aggregate demand of an economy. C is the consumption function which indicates the relation between income and consumption expenditure.

The consumption function is shown by the slope of the C curve in Fig. 1 which is MPC (marginal propensity to consume). I is investment demand which is autonomous. When investment demands (I) is added to consumption function (C), the aggregate demand function becomes C+I.

C+S identity is related to the aggregate supply of an economy. That is why, consumer goods and services are produced from total consumption expenditure and aggregate savings are invested in the production of capital goods.

In an economy, the equilibrium level of national income is determined by the equality of aggregate demand and aggregate supply (C+I=C+S) or by the equality of saving and investment (S=I).

We explain these two approaches one by one with the help of Figure 1 (A) and (B).

**Equality of Aggregate Demand and Aggregate Supply**

The equilibrium level of national income is determined at a point where the aggregate demand function (curve) intersects the aggregate supply function. The aggregate demand function is represented by C+I in the figure. It is drawn by adding to the consumption function C the investment demand I.

The 45° line represents the aggregate supply function, Y = C+S. The aggregate demand function C+I intersects the aggregate supply function Y= C+S at point E in Panel (A) of Figure 1 and the equilibrium level of income OY is determined.

Suppose there is disequilibrium in aggregate supply and aggregate demand of the economy. Disequilibrium can be in either case, aggregate supply exceeding aggregate demand or aggregate demand exceeding aggregate supply. How will the equilibrium level of income be restored in the two situations?

First, take the case when aggregate supply exceeds aggregate demand. This is shown by OY_{2}level of income in Panel (A) of the figure. Here aggregate output or supply is Y_{2}E_{2} and aggregate demand is Y_{2}k. The disposable income is OY_{2} (=Y_{2}E_{2}). At this income level OY_{2}, consumers will spend Y_{2}d on consumption goods and save dE_{2}.

But businessmen intend to make investment equal to dk in order to buy investment goods. Thus the aggregate demand for consumption goods and investment goods is Y_{2}d + dk = Y_{2}k. But aggregate supply (or output) Y_{2}E_{2} is greater than aggregate demand Y_{2}k by kE_{2} (=Y_{2}E_{2} – Y_{2}k).

Therefore, the surplus output of goods worth kE_{2 }accumulated by businessmen in the form of unintended inventories. In order to avoid further inventory accumulation, they will reduce production. As a result of the reduction in output, income and employment will fall and the equilibrium level of income will be restored at OY where the aggregate supply equals aggregate demand at point E.

The second situation of disequilibrium when aggregate demand exceeds aggregate supply is shown by the income level of OY_{1} in Panel (A) of the figure. Here the aggregate demand is Y_{1}E_{1} and the aggregate output is Y_{1}a. The disposable income is OY_{1} (=Y_{1}a).

At this income level, consumers spend Y_{1}b on consumption goods and save ba. But businessmen intend to invest bE, to buy investment goods. Thus the aggregate demand is Y_{1}b + bE_{1}= Y_{1}E_{1} which is greater than the aggregate supply of goods Y_{1}a by aE_{1}.

To meet this excess demand worth aE_{1}, businessmen will have to reduce inventories by this amount. In order to stop further reduction in their inventories, businessmen will increase production. As a result of the increase in production, output, income and employment will increase in the economy and the equilibrium level of income OY will be restored again at point E.

**Equality of Saving and Investment**

The equilibrium level of income can also be shown by the equality of the saving and investment functions. Since the equilibrium level of income is determined when aggregate supply (C+S) equals aggregate demand (C + I) in the economy, intended (or planned) saving also equals intended (or planned) investment. This can be shown algebraically

C + S = C + l

S = I

The equilibrium level of income in terms of the equality of saving and investment is shown in Panel (B) of Figure 1, where I is the autonomous investment function and S is the saving function. The saving and investment functions intersect at point E which determines the equilibrium level of income OY.

If there is disequilibrium in the sense of inequality between saving and investment, forces will operate in the economy and the equilibrium position will be restored. Suppose the income level is OY_{2} which is above the equilibrium income level OY.

At this income level OY_{2}, saving exceeds investment by gE_{2}. It means that people are consuming and spending less. Thus aggregate demand is less than aggregate supply. This will lead to the accumulation of unintended inventories with businessmen. To avoid further accumulation of inventories, businessmen will reduce production. Consequently, output, income and employment will be reduced till the equilibrium level of income OY is reached at point E where S=I.

On the contrary, if the income level is less than the equilibrium level, investment exceeds saving. This is shown by OY_{1} level of income when investment Y_{1}E_{1} is greater than saving. The excess of intended investment over intended saving means that aggregate demand is greater than aggregate supply by eE_{1}.

Since aggregate output (or supply) is less than aggregate demand, businessmen will decrease inventories held by them. To stop further reduction in their inventories, they will increase production. Consequently, output, income and employment will increase in the economy and the equilibrium level of income OK will be again reached at point E.

The determination of equilibrium level of income simultaneously by the equality of aggregate demand and aggregate supply and of saving and investment is explained in Table I below.

**Table 1**

**THREE-SECTOR MODEL**

A three-sector model of income determination consists of a two-sector model and the government sector. The government increases aggregate demand by spending on goods and services, and by collecting taxes.

**Government Expenditure**

**First, we take government expenditure. To explain it, given all the above assumptions except the government sector in the two-sector model, income determination is as follows**

By adding government expenditure (G) to equation (1) of the two-sector model, Y – C + l, we have

Y= C+I+G

Similarly, by adding government expenditure (G) to the saving and investment equation, when we have

Y = C + I + G

Y = C + S [S= Y-C]

I+G=S

Both are illustrated in Figure 2(A) and (B). In Panel (A), C+I+G is the new aggregate demand curve which intersects the aggregate supply curve 45° line at point E_{1} where OY_{1} is the equilibrium level of income. This income level is more than the income level OY without government expenditure.

Similarly, according to the concept of saving and investment, the new investment curve I+G intersect the saving curve 5 at point in Panel (B). Consequently, the income level OY_{1} is determined which is more than the income level OY without government expenditure.

It should be noted that by adding government expenditure to consumption and investment expenditure (C + I), the national income increases by YY_{1} which is more than the government expenditure, ∆Y>G in Panel (A) of the figure. This is due to the multiplier effect which depends upon the value of MPC or MPS where MPC or MPS < 1.

**Taxation**

Now we explain the effects of taxes on the level of national income. When the government imposes a tax, the amount of tax is reduced from the national income and what remains is the disposable income. Thus

Y-T=Y_{d}

Where Y-national income, T=tax, and Y_{d }= disposable income. Now disposable income will be less than national income by the amount of tax, Y_{d}<Y. With the fall in disposable income, people will reduce expenditure on consumption. This will lead to reduction in national income, which will depend on the amount or rate of tax and the value of MPC.

Given all the above mentioned assumptions in which government expenditure is constant, the effects of taxes on national income are illustrated in the following figures.

First, the effect of a lump-sum tax on income is shown in Fig. 3. The equilibrium level of income without a tax is at point E where the aggregate demand curve (C+I+G) intersects the aggregate supply curve 45° line and the income level OY is determined. By imposing a lump-sum tax, the consumption function is reduced by the amount of tax.

As a result, the aggregate demand curve C+I+G shifts downwards to C_{1 }+ I +G and intersects the aggregate supply curve 45° line at point E_{1}. This result in the reduction of income level from OY to OY_{1 }Thus with the imposition of a lump-sum tax, the national income is reduced by YY_{1}.

Now we take a proportional tax which is imposed on income as a constant percentage. With the increase in the rate of tax, consumption and national income will decrease and vice versa. The effect of such a tax on income level is shown in Figure 4.

The aggregate demand curve C+I+G before the imposition of tax intersects the aggregate supply curve 45° line at point E and the income level OY is determined. After imposing the tax, the C+I+G curve shifts downward to C_{1}+I+G due to a fall in consumption, and it intersects the 45° line at point E_{1} consequently, the equilibrium level of national income is reduced by YY_{1}.

**Effect on Saving and Investment**

**The effect of a tax on saving and investment also determines the equilibrium of national income as follows**

Y = C+I+G

And Y=C+S+T

Y= C+I+G=C+S+T

Or K=I+G=S+T

It is clear from the above equation that when planned investment (I) plus government expenditure on goods and services (G) equal planned saving (S) plus tax (T), the equilibrium of national income is established. I+G are inflows or injections in the national income and S+T are outflows or leakages. If they are equal to each other, the national income is in equilibrium.

This is shown in Fig. 5. Here, E is the equilibrium point before imposing the tax where S and I+G curve intersects and the income level OY is determined. With the imposition of a tax, the S curve shifts upward to the left as S + T and the new equilibrium is established at point E_{1}with I+G and the national income falls from OK to OY_{1}.

**FOUR SECTOR MODEL**

We shall now show how national income is determined in an open economy. For this, we relax the assumptions that there are no exports or imports and government expenditures. This means that we shall have to add imports and exports and government expenditures and taxation in our analysis.

It may be noted that government expenditures are like investment because they raise the demand for goods. They are injections in the national income. On the other hand, taxes are leakages in the national income like savings because they tend to reduce the demand for consumer goods.

The impact of exports and imports is similar to that of the government expenditure. Exports are injections because they increase the demand for goods in the same economy. Imports, on the other hand, are leakages in the national income because they represent the supply of goods to the given economy.

**Assumptions**

**The analysis of the determination of income in an open economy is based on the following assumptions**

- The domestic economy’s international trade is small relative to total world trade.
- There is less than full employment in the economy.
- The general price level is constant up to the full employment level.
- Exchange rates are fixed.
- There are no tariffs, trade and exchange restrictions.
- Gross exports are determined by external factors.
- Exports (A), investment (I) and government expenditure (G) are autonomous.
- Consumption (C), imports (M), savings (S) and taxes (I) are each a fixed proportion of national income (Y) and their relationships with national income are linear.

**Determination of Equilibrium Level of Income**

Given these assumptions, an open economy is in equilibrium when its national expenditure (E) is equal to its national income (Y).

**This can be shown in the following equation for the equilibrium level of income**

Y= E=C+I+G+(X-M)

But Y = C+S+T

C+S+T= C+I+G+(X-M)

In the above analysis, C+S+T is gross national income (GNI) and C+I+G+(X-M) is gross national expenditure (GNE). Thus the equilibrium level of income in an economy is determined when aggregate supply, GNI=GNE, aggregate demand, or, C+S+T=C+I+G+(X-M).

This is shown in Figure 6 where C is the consumption function. On this curve, T autonomous investment is superimposed to form the C+I function, and autonomous government expenditure G is superimposed on C+I to form the C+I+G function. When net exports of X-M are superimposed on C+I+G, we get the aggregate demand function C+I+G+(X-M). The 45° line is the aggregate supply function which represents C+S+T.

It should be noted that so long as C+I+G+(X-M)>C+I+G, exports exceed imports and there is net addition to aggregate demand. At point D in Panel (A) of the figure, X-M=O. Beyond point D,C+I+G>C+I+G+(X- M) and imports exceed exports, and this gap continues to grow as income increases. This leads to net reduction in aggregate demand so that the aggregate demand function C+I+G+(X-M) lies below the domestic demand function C+I+G.

The equilibrium level of income in an open economy, OY is determined at point E where the aggregate demand function C+I+G+(X-M) intersects the aggregate supply function C+S+T.

This analysis shows that in the absence of foreign trade, the equilibrium level of income would have been at a higher level, as determined by the equality of C+I+G=C+S+T at point F whereas with foreign trade it is at a lower point E.

There is also an alternative method for determining the equilibrium level of income in an open economy in terms of saving and investment equality.

Accordingly,

C+S+T=C+I+G+(X-M)

Or S+T=I+G+(X-M)

Or S+T+M=I+G+X

Where S+T+M refers to total income and I+G+X to total expenditure. When S+T+M is equal to I+G+X, the equilibrium level of income is determined. This is shown in Panel (B) of Fig. 6 where the S+T+M curve intersects the I+G+X curve at point E and the equilibrium level of income OY is determined.

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