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Understanding Income Determination Under Keynesian Model

1-Introduction

Before we proceed to a detailed description of the Keynesian model of income determination, let us first understand some basic facts related to the classical perception regarding employment and income determination.

The classical perception regarding employment was, 'There is always full employment, except for frictional and voluntary unemployment, in the economy. If some unemployment exists in the short-run, the wage-price flexibility maintains full employment automatically in the long-run. In such a situation, there is no involuntary unemployment, and the economy  always remains in a state of equilibrium.' Based on the same perception, the classical economists believed that the full employment of resources not only produces goods and services but also generates income as equal to the market value of those goods and services. The generated income is further spent on the consumption of the goods and services produced. Hence, income equals expenditure and there is no general overproduction and no unemployment.

The classical perception or so-called classical theory of income and employment could not address the problem of economic depression that occurred in the 1930s and it was proved to be wrong. In 1936, invalidating the classical perception, Keynes developed a new theory of income and employment.

For the invalidation of classical perceptions, like full employment and the economy is always in the state of equilibrium, Keynes developed a new concept of  The Principle of Effective Demand and his own theory of income determination. His theory of income determination is also called the Keynesian model of income determination. The concept of the principle of effective demand and the Keynesian model of income determination earned much popularity at that time as it got success to overcome the problem of economic depression. Later on, other models of income determination were also developed by modern economists, yet the Keynesian model is equally important and also applicable in the present time.

2- Income Determination in Two-sector Economy Model

The two-sector economy model is that in which there are only household and business sectors actively involved in an economy. There is no government involvement and no links with the outside of the economy. Such an economy is not a real economy yet its study ease the understanding of starting point of the income determination of the economy. In this economy, the level of income is determined at the point where aggregate demand AD equals aggregate supply AS.
Symbolically,
`AD=AS`
`C+I=C+S`        `\because\AD=C+I`   &  `\because\AS=C+S`
`S=I`

Income determination in a two-sector economy is based on the following assumptions.
1- There are only two sectors.
2- AD is equal to consumption demand and investment demand.
3- No government interference,
4- No outside trade links,
5- All prices including factor cost remain constant.
6- Capital and technology is given,

2.1- AD-AS Approach
According to this approach, income is determined at the point at which AD equals AS as follows.
`AD=AS`
`C+I=C+S`
`\Or,\Y=C+I...........(i)`
Where, consumption function `C=a+bY` and `I` is constant as `\bar{I}`
Substituting `C=a+bY` for C and `\bar{I}` for `I` in equation (i) we get;
`Y=a+bY+\bar{I}`
`Or,Y-bY=a+\bar{I}`
`Or,Y(1-b)=a+\bar{I}`
`Or,Y=\frac{1}{(1-b)}\a+\bar{I}`

Diagrammatically, the income determination in a two-sector economy model is shown as follows.
In the above diagram, C is the consumption function and C + I is the AD curve. As shown in the diagram, the AD curve has intersected AS curve at point E1 where the aggregate output (income) supplied is equal to aggregate demand. Hence, the equilibrium identity AD = AS has been satisfied at point E1. Therefore, its corresponding income level OY1 is the determined equilibrium level of income.

2.2- S - I Approach
The saving-investment approach can be derived from the income equilibrium identity based on the AD-AS approach as follows.
`AD=AS`
`C+\bar{I}=C+S`
Since C is common to both sides of the equation and thus it gets canceled. Hence, under S - I approach, the equilibrium condition can be expressed as;
`bar{I}=S.......(i)`
Investment is assumed to be constant but saving is the function of income `S=f(Y)`. Therefore, we need to derive a saving function.

As we know that Saving `S=Y-C......(ii)`
By substituting `a+bY` for `C` in the equation (ii) we get'

`S=Y-(a+bY)`
`S=Y-a-bY`
`S=-a+Y-bY`
`S=-a+(1-b)Y......(iii)`
Given the saving function, the equilibrium condition under S - I approach can be expressed;
`\bar{I}=-a+(1-b)Y_d`            `\because\S=\bar{I}`

The determination of the equilibrium level of income under the S - I approach has been illustrated diagrammatically as follows.
In the above diagram `SS` is the saving function and `\bar{I}\bar{I}` is the investment function. The saving function or curve has intersected the investment curve at point E where there is an equality between saving and investment. This is the point of equilibrium and its corresponding income level OY is the determined level of income.

Let us give a numerical example of income determination.
Suppose, given the consumption function is as, `C=200+0.8Y` and investment function as  `\bar{I}=100`  find the equilibrium level of national income.

We know that the equilibrium of income can also be expressed as follows.
`Y=C+I.......(i)`
Substituting `C` and `\bar{I}` by their respective values, we get'

`Y=200+0.8Y+100`
`Or,\Y-0.8Y=300`
`Or,\Y(1-0.8)=300`
`Or,\Y=300/0.2`
`Or,\Y=1500`

3- Income Determination in a Three-Sector Economy  (A Simple Model)

A three-sector economy, which is also known as a closed economy, consists of household, business as well as government sectors. With the inclusion of the government sector, economic activities are affected in several ways with the implementation of fiscal policy and monetary policy in an economy but for simplicity, an effort to include only the government expenditure has been made in this simple model. Similar to the two-sector model, the income level, in this model is also determined by AD-AS and S-I approaches.

3.1- AD-AS Approach

Under this approach equilibrium income level is determined at the point where aggregate demand equals aggregate supply. Hence, the equilibrium condition is expressed as follows.

`AD=AS`
`AD` consists of the demand for consumption expenditure `C,` Investment expenditure `I,` and government expenditure `G,` and thus `AD` is expressed as follows.

`\AD=C+I+G........(i)`
The total amount of national income `Y` of an economy is equal to aggregate expenditure including the government expenditure. So the equilibrium identity in a three-sector economy can also be written as;

`\Y=C+I+G.......(ii)`
This analysis of income determination is based on the following assumptions.
a- There are no trade links with the rest of the world.
b- No transfer payments,
c- The government expenditure is met by printing currency.
d- No tax imposition, or zero tax,
e- Expenditure is influenced by external factors but not by govt revenue.

Being the consunption function as `C=a+bY,` now substitute `a+bY` for `C` in equation `(ii),` and we get;

`\Y=a+bY+I+G`
`Or,\Y-bY=a+I+G`
`Or,\Y(1-b)=a+I+G`
`Or,\Y=\frac{1}{(1-b)}\a+I+G...........(iii)`

The income determination in the three-sector economy has also been described diagrammatically as follows.
In the above diagram, the `AD` curve `(C+I),` without government expenditure, intersects the `AS` curve at point `E` where national income is determined equal to `OY1`. This part of the analysis is similar to as explained in the two-sector model. Let us now include the government sector in the two-sector model and assume that the government makes its expenditure equal to `FE1` by printing the currency but not by the tax revenue. With the inclusion of `G` under this assumption, the `AD` curve shifts upward to `C+I+G` and it intersects the `AS` curve at point `E1`. As a result, the initial equilibrium point `E` also shifts to `E1` and it determines the equilibrium level of income equal to `OY1` as shown in the above diagram. This is how national income is determined in a simple three-sector economy model.

4- Income Determination in a Three-Sector Economy; An Extended Model

As already mentioned, the inclusion of the government affects economic activities in several ways through its financial involvement and policies. However, in this somewhat extended model, we confine to the effect of government expenditure including transfer payment and taxation. So, the new variables tax `T` government expenditure `G` and transfer payments `G_T` has been included in this model.

In this model, we assume a simple system of government taxation, government expenditure as well as a transfer payment which includes the following fiscal operations.

a- The government levies only direct taxes on households.
b- It spends its tax revenue on goods and factor services.
c- It makes transfer payments in the form of pensions and subsidies.

Based on the above assumptions, the extended model of the three-sector economy can be analyzed under the following four different sub-models. 

4.1- Model-I: Income Determination with Government Expenditure and Tax

Income determination in the three-sector model with government expenditure and taxation is an extended form of the three-sector simple model. This model includes two variables such as government expenditure `G` and taxation `T`. This model also based on some assumptions as follows.

1- The provision of transfer payment is nill.
2- There is a provision for only lump-sum tax.
3- Government expenditure is determined by exogenous factors.
4- Government adopts a balanced budget policy.

4.1.1- AD-AS Approach
Under this approach, income is determined at the point where `AD` equals `AS`. The variable of `AD` and `AS` of this model can be expressed as follows.

`AD=C+I+G` and `AS=C+S+T`
`C+I+G=Y=C+S+T`
Thus, the equilibrium identity is `Y=C+I+G.........(iv)`

The consumption Function in this model is as follows.
`C=a+b(Y-T)..........(v)`             `\becauseY_d=(Y-T)`
where;
`Y` = National Income,
`Y_d` = Disposable income,
`T` = Lump-sum tax,

Substituting equition `(v)` for `C` in equation `(iv)` , we get,
`Y=a+b(Y-T)+I+G`
`Y=a+bY-bT+I+G`
`Y-bY=a-bT+I+G`
`Y(1-b)=a-bT+I+G`
`Y=\frac{1}{(1-b)}(a-bT+I+G)`

The income determination in this model has been illustrated in the following diagram.
In the above diagram, let us suppose that the government initially makes its expenditure by creating (Printing) currency as has been mentioned in the three-sector simple model. Consequently, the `AD` curve shifts to `AD1`, and the equilibrium point also shifts from `E` to `E1`. This point of equilibrium determines income equal to `OY1` as shown in the diagram. Now, suppose that the government levies lump-sum tax. It is a leakage from the income stream which reduces disposable income and as a result, private consumption expenditure falls as shown by the downward shifted blue dotted curve `C2` in the diagram. It causes the `AD2` curve to shift downward to `AD3` where it intersects the `AS ` curve at point `E2`. This is the new point of equilibrium after the imposition of the lump-sum tax, where the income level equal to `OY2` is determined. If the government adopts a balanced budget policy `(G=T)`, for which the total amount of tax revenue will be equal to the government expenditure. In this situation, the equilibrium point `E1` and income level `OY1` will be restored again.

4.1.2- S-I Approach with G & T
According to the `S-I` approach, national income with government expenditure and lump-sum tax is determined at the point where `S+T=I+G........(vi)`

To determine income in this approach, we need to derive a saving function. The saving function for the three-sector model is given as;

`S=Y-C`
`S=(Y-T)-C......(vii)`            `\becauseY_d=(Y-T)`

Substituting `a+bY_d` for `C` in equation `vii` we, get;
`S=(Y-T)-(a+bY_d)...........(viii)`
By substituting equations iii for `S` in equation `vi` we arrive at the equilibrium level of national income.

The equilibrium level of national income with `G` and `T` can be illustrated diagrammatically as shown in the following diagram.
In the above diagram,  `S` and `I` curve intersect each other at point `E` where the equilibrium income level is equal to `OY`, as similar to a two-sector model.  with the inclusion of lump-sump tax, the saving curve shifts to `S+T` and there is an addition to government expenditure `G` equal to `T` which causes the `I` curve shift to `I+G` resulting in income determination equal to `OY1` at point `E1`.  In case the government makes its expenditure by creating currency but not by levying the lump-sump tax, the autonomous investment with government expenditure `I+G` shifts upward and intersects the existing saving curve at point `E2` where the equilibrium level of national income `OY2` is determined.

4.2- Model-II: Income Determination with Transfer Payments

Transfer payments to the people by the government in different forms boost up the spending capacity of households and it has a positive impact on the equilibrium level of national income. Transfer payments may be financed from tax revenue or they may be autonomous. The analysis of income determination will be similar to the first model of income determination if it is assumed that transfer payments are financed from tax revenue. Therefore this analysis confines to the analysis of autonomous transfer payments.

In a three-sector model, the income equilibrium identity is as follows.
`Y=C+I+G........(ix)`
With the inclusion of transfer payments, the consumption function changes to `C=a+b(Y-T+G_T).......(xi)`

Substituting `a+b(Y-T+G_T)` for `C` in equation `(ix)` we get;
`Y=a+b(Y-T+G_T)+I+G`
`Y=a+bY-bT+bG_T+I+G`
`Y-bY=a-bT+bG_T+I+G`
`Y(1-b)=a-bT+bG_T+I+G`
`Y=\frac{1}{(1-b)}\(a-bT+G_T+I+G)...........(x)`

In the equation `(x)` the term `bG_T` represents an increase in consumption expenditure as a result of the implementation of transfer payments and with the increment in transfer payments, the equilibrium level of national income goes up, else it falls down.

The determination of national income with transfer payments has been illustrated with the help of the following diagram as;
In the above diagram, let us suppose that `E1` is the initial equilibrium point with the inclusion of the government sector, where it makes its expenditure by creating currency. Now, there is an inclusion of autonomous transfer payments that causes an increase in the aggregate demand of households. As a result, the `AD1` curve shifts upward to `AD2` and intersects the `AS` curve at point `E2` where the equilibrium level of national `OY2` is determined.

4-3 Model-III: Income Determination with Tax as a Function of Income

In the previous analysis, the tax was assumed as a lump-sum tax. In fact, the government levies direct and indirect taxes. Therefore, this analysis assumes tax as a function of income rather than a lump-sum tax. Hence, we assume both autonomous tax as `T` as well as proportional income tax as `tY`, as both are effective.

The tax function in this model is expressed as follows.
`T=\barT+tY \.......(i)`
Where
`T=` Total tax,
`\barT=` Autonomous tax,
`tY=` Proportional tax rate,

Given the tax function, the consumption function can be expressed as follows.

`C=a+bY`
`C=a+b(Y-T)`
`C=a+b{Y-(\barT+tY)}`
`C=a+bY-b\barT-tY`
`C=a-b\barT+bY-btY`
`C=a-b\barT+bY(1-t)`
`C=a-b\barT+b(1-t)Y...........(ii)`

Income determination with tax as a function of income has been illustrated with the help of the following diagram.
As shown in the diagram, with the imposition of the proportional tax, the amount of tax goes on increasing with the increase in income level. As a result, consumption expenditure tends to fall and with that, the `C` curve rotates downward to the right. This causes a fall in aggregate demand as well. Consequently, the `AD1` curve also rotates downward to `AD2` and intersects the `AS` curve at point `E1` where the new equilibrium level of national income equal to `OY1` is determined. As shown in the diagram, this new equilibrium income level is less than that of the initial income level because of the imposition of proportion tax.

4.4 Model-IV: Income Determination with `T`, `G` and `G_T` 

When tax function, transfer payments an and government expenditure are simultaneously included in the economy, the consumption function changes as shown in the following function.

`C=a+bY`
`C=a+b(Y-T)`
`C=a+b{Y-(\barT-tY)}`
`C=a+b{Y-\barT-tY+G_T..........(iii)`

By substituting the consumption function in the equation `Y=C+I+G............(vi)` in place of `C` we get the income determination equation of this model.

`Y=C+I+G`
`Y=a+b(Y-\barT-tY+G_T+I+G`
`Y=a+bY-b\barT-btY+bG_T+I+G`
`Y-bY+btY=a-b\barT+bG_T+I+G`
`Y(1-b+bt)=a-b\barT+bG_T+I+G`
`Y{1-b(1-t)}=a-b\barT+bG_T+I+G`
`Y=\frac{1}{1-b(1-t)}\(a-b\barT+bG_T+I+G).............(v)`

5- Income Determination in a Four-sector Economy (A Simple Model)

A four-sector economy is an open economy that consists of household, business, government, and foreign sectors. The equilibrium level of national income in a four-sector economy is also determined at the point where aggregate demand `AD` equals aggregate supply `AS` or `AD=AS`.

Symbolically it is expressed as follows.
`Y=C+I+G+(X-M)........(i)`

In a four-sector economy, the foreign sector consists of import and export, and these variables are assumed to be autonomous. Export of goods generates income for a domestic country whereas import passes income from home country to foreign countries. Hence, the national income of a country is also influenced by the net export `X_n`. When the net export is positive, there will be an addition to the aggregate demand and with that income level will also go up whereas a negative net export will cause a fall in aggregate demand as well as the income level of the country. With all these transactions with foreign countries, the national income of a home country is determined under a four-sector economy

Mathematically, the income determination in a four-sector economy is treated as follows.

`Y=C+I+G+(X-M)........(i)`
Substituting `a+b(Y-T)` for `C` in equation `(i)`, we get;

`Or,\Y=a+b(Y-T)+I+G+X_n`         `\becauseC=a+b(Y-T)` and `\becauseX_n=(X-M)`
`Or,\Y=a+bY-bT+I+G+X_n`
`Or,\Y-bY=a-bT+I+G+X_n`
`Or,\Y(1-b)=a-bT+I+G+X_n`
`Or,\Y=\frac{1}{(1-b)}\(a-bT+I+G+X_n)........(ii)`

Income determination in a four-sector economy has been illustrated with a help of the following diagram as well. 
In the above diagram, the aggregate demand curve, without the inclusion of foreign sector, has intersected the aggregate supply curve at `E` where the determined income level is equal to `OY`. When a foreign sector with positive net export is included, aggregate demand increases, and with that `AD` curve shifts upward. It intersects the `AS` curve at point `E1` where income level `OY1` is determined.

5.1- The four-sector Economy Model Further Extended

In the previous analysis of income determination under a four-sector economy model, Import and export were assumed to be autonomous or independent of income level but in fact, export is exogenously influenced and still assumed to be independent of income level whereas import is treated as a function of both autonomous import as well as income level. The import function in this model is expressed as follows.

`M=\barM+mY..........(iii)`
Where;
`M` = Total amount of import,
`\barM`= Autonomous import,
`m` = Rate of change in import due to change in income level,
`Y` =Income level,

By applying the import function, we can express the four-sector model of income determination as follows.

`Y=C+I+G+(X-M)`
`Or,\Y=a+bY_d+I+G+{X-(\barM+mY)}`                     `\because\a+bY_d`  &  `\becauseM=\barM+mY`
`Or,\Y=a+b(Y-T)+I+G+X-\bar\M-mY`                `\becauseY_d=Y-T`
`Or,\Y=a+bY-bT+I+G+X-\barM-mY`
`Or,\Y-bY+mY=a-bT+I+G+X-\barM`
`Or,\Y(1-b+m)=a-bT+I+G+X-\barM`
`Or,\Y=\frac{1}{(1-b+m)}\(a-bT+I+G+X-\barM)........(iv)`
The above equation `(iv)` is the extended form of income determination model of a four-sector economy.

The income determination in an extended form of the four-sector economy has been illustrated with the help of the following diagram.
In general, with the inclusion of the foreign sector, import increases with an increase in income level and it causes a fall in aggregate demand function. It is because import constitutes leakages of income from the income stream and ultimately it causes a fall in income level as well.
As it has been shown in the above diagram, initially the economy was in an equilibrium position at point `E1` where the equilibrium income level was `OY1`. When the foreign sector is included with the import function, the aggregate demand curve tends to decline and intersects the aggregate supply curve at point `E`. The causes the equilibrium point to shift downward from `E1` to `E`. Consequently, the level of income determined is equal to `OY` which is less than the previous level of income as shown in the diagram. This analysis of income determination shows that the import function underdetermines the level of national income.

5.2- S-I Approach Or Augmented S-I approach of income determination in a four-sector economy

In this approach, leakages, and injections are applied to explain the determination of income. In a four-sector model, saving, tax, and import `S,T&M` are the leakages whereas investment, government expenditure, and export `I,G&X` are the injections. The economy is in equilibrium at the point where the sum of `S+T+M` is equal to the sum of `I+G+X`. Hence, the sum of leakages and injections can be expressed symbolically as follows.

`S+T+M=I+G+X.........(v)`
Rearranging the equation (v), we get;
`S+(T-G)=I+(G+X).............(vi)`
In equation (vi), `T-G` represents government saving whereas `X-M` represents net export. The aggregate saving that includes private as well as government saving `S+(T-G)` is called augmented saving and it increases with an increase in income level. Therefore, the augmented saving curve slopes upward with the increases in income level. On the other hand, the import is influenced by the income level of domestic inhabitants, their decisions to buy foreign goods, import policy etc. So the import generally rises with an increase in income level but the export is affected by external factors and remains somewhat stable in comparison to the import. In such a situation the net export decreases with an increase in domestic income level. So, the sum of investment and net export `I+(X-M)`, which is also called augmented investment, falls. Consequently, the augmented investment curve slopes downward with an increase in income level. Hence, there is an interaction between augmented saving and investment curves and the equality between these curves determines the income level under this approach. 

Income determination in a four-sector economy under the `S-I` approach has been explained diagrammatically as follows.
In the above diagram, the augmented saving curve is sloping upward and the augmented investment curve is sloping downward as described above. These curves have intersected each other at point E where `S+(T-G)` is equal to `I+(X-M)`. The point `E` is the point of equilibrium and its corresponding level of income equal to `OY` is the determined level of income. This how the level of income in a four-sector economy under the `S-I` approach is determined.




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