# The Keynesian Model Of Income Determination In A Four Sector Economy

Determination of Equilibrium income or output in a Four Sector

The inclusion of the foreign sector in the analysis influences the level of aggregate demand through the export and import of goods and services. Hence it is necessary to understand the factors that influence the exports and imports.

The volume of exports in any economy depends on the following factors:

1. The prices of the exports in any domestic economy relative to the price in the other countries.
2. The income level in the other economies.
3. Tastes, Preferences, customs and traditions in the other economies.
4. The tariff and trade policies between the domestic economy and the other economies.
5. The domestic economy’s level of imports.

Few illustrations could explain us the Keynesian model of income determination in a four sector economy.

Illustration 23

The fundamental equations in an economy are given as:

Consumption Function           C         =          200 + 0.8Yd

Investment Function               I           =          300

Tax                                          T          =          120

Government Expenditure       G         =          200

Exports                                    X         =          100

Imports                                    M         =          0.05Y

Find the following.

1. The equilibrium level of income
1. The net exports

The consumption function is
C         =          200 + 0.8Y

C         =          200 + 0.8 (Y – T)

C         =          200 + 0.8 (Y – 120)

The equilibrium condition is given as

Y         =          C + I + G + X – M
Thus,
Y         =          200 + 0.8 (Y – 120) + 300 + 200 (100 – 0.05Y)

Y         =          200 + 0.8 Y – 96 + 600 – 0.05Y

Y – 0.8Y+ 0.05Y        =          704

0.25Y              =          704

Y         =          704 / 0.25

The equilibrium level of income is 2,816.

In equilibrium in a four sector model, leakages equal injections or

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

The consumption function is C           =          200 + 0.8Y

C         =          200 + 0.8 (2,816 – 120)

C         =          200 + 0.8 (2,696)

C         =          200 + 2,156.8

C         =          2,356.8

The saving function is           S          =          Yd – C

S          =          (Y – 120) – 2,356.8

S          =          2,816 – 2476.8

S          =          339.2

Thus,
I + G + X        =          S + T + M

300 + 200 + 100         =          339.2 + 120 + 0.05Y

600      =          459.2 + 0.05 Y

600      =          459.2 + 0.05 (2,816)

600      =          459.2 + 140.8

600      =          600

Imports                                    M         =          0.05Y = 0.05 (2,816)

=          140.8

Net Exports:                X – M              =          100 – 140.8

X - M              =          - 40.8

There is a deficit in the balance of trade.

Illustration 24

For Credentials of the numerical illustration 23, find the following:

1. The increase in the income if both government expenditure and tax increased by an amount of 20 each.
2. The net exports, if exports increased by an amount of 60.
3. The increase in the government expenditure if the economy were to achieve the full employment income level of 3200.

Solution

1. If both government expenditure and tax increased by an amount of 20 each, G = 220 and Tax = 140

The equilibrium condition is given as Y = C + I + G + X – M

Thus,
Y         =          200 + 0.8 (Y - 140) + 300 + 220 + (100 – 0.05Y)

Y         =          200 + 0.8Y – 112 + 620 – 0.05Y

Y – 0.8 Y + 0.05Y      =          708

0.15Y  =          708

Y         =          708 / 0.25

Y         =          2,832

The equilibrium level of income is 2,832. Hence, there is an increase in the income by 16.

1. If the exports increased by an amount of 60, X = 160

The equilibrium condition is given as Y = C + I + G + X – M

Thus,
Y         =          200 + 0.8 (Y – 120) + 300 + 200 + (160 – 0.05Y)

Y         =          700 – 96 + 160 + 0.8Y – 0.05Y

Y         =          764 + 0.75Y

Y – 0.75Y       =          764

0.25Y  =          764

Y         =          764 / 0.25

The equilibrium level of income is 3,056.

Imports M = 0.05 Y = 0.05 (3,056) = 152.8

Net Exports X – M = 160 – 152.8 = 7.2

X – M             =          7.2

There is a surplus in the balance of trade.

1. We have GM = Δ Y     =               1
Δ G                 1 – b + m

Where,
Δ G      =          Change in government expenditure

b          =          Marginal propensity to consume

Δ Y      =          Change in income

GM       =          Government expenditure multiplier

m         =          Marginal propensity to import

In the present example,

b          =          0.80

Δ Y      =          3,200 – 2,816

Δ Y      =          384
Thus,
384      =                      1
Δ G                  1 – 0.80 + 0.05

Δ G      =          384 (0.25)

Δ G      =          96

The level of government expenditures required to achieve the full employment output is 96.

Illustration 25

The equations in an economy are given as:
C = 260 + 0.8 Yd,
Investment function Ī = 320
Tax = 300
Government Expenditure G = 300
Exports X = 300 – 0.05Y

You are required to ascertain the following:

1. Find the equilibrium level of income
2. Find the net exports at equilibrium level of income
3. Find the equilibrium level of income and the net exports when there is an increase in investment from 320 to 340
4. Find the equilibrium level of income and the net exports when the net export function becomes 280 – 0.05Y

Solution

(a) The consumption function is

C = 260 + 0.8Yd

C = 260 + 0.8 (Y – T)

C = 260 + 0.8 (Y – 300)

The equilibrium condition is give as Y = C + I + G + X – M

Thus,               Y         =          260 + 0.8 (Y – 300) + 320 + 300 + (300 – 0.05Y)

Y         =          260 + 0.8Y – 240 + 920 – 0.05Y

Y – 0.8 + 0.05Y          =          940

0.25Y  =          940

Y         =          940 / 0.25

The equilibrium level of income is 3,760.

(b) Imports M = 0
Net Exports X – M     =          300 – 0.05(3,760) – 0

X – M                          =          300 – 188        =          112

There is a surplus in the balance of trade.

(c)        Y         =          260 + 0.8 (Y – 300) + 340 + 300 + (300 – 0.05Y)

Y         =          260 + 0.8Y – 240 + 340 + 300 + 300 – 0.05Y

Y – 0.8Y + 0.05Y       =          960

0.25 Y             =          960

Y         =          960 / 0.25

The equilibrium level of income (Y) is 3,840 which is an increase by 80

Imports M = O,

Net Exports X – M
= 300 – 0.05 (3,840) – 0         =          108

(d) There is a surplus in the balance of trade.

Y         =          260 + 0.8(Y – 300) + 320 + (280 – 0.05Y)

Y         =          260 + 0.8Y – 240 + 900 – 0.05Y

Y – 0.8Y + 0.05Y       =          920

0.25Y              =          920

Y         =          920 / 0.25

Thus the equilibrium level of income is 3,680 which is a decrease by 80.

Imports M       =          0

Net Exports X – M     =          280 – 0.05(3,680) – 0

X – M =          96

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