Determination of Equilibrium Income in an Economy with Four Sectors – Explained

When the economy opens up to foreign trade, it is said to have the fourth sector also to deal with. The fourth sector is the Rest of the World (ROW) sector. Exports to ROW and imports from it disturb the equilibrium of an economy as mentioned in the previous section.

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(A) When Taxation and Imports are Both Autonomous:

Value of exports (X) constitutes another source of income and value of imports (M), as another channel of expenditure. Aggregate Demand (AD) and Aggregate Supply (AS) can for an economy with four sectors be expressed as

AD = C + I + G+ X

And, AS = C + S + T + M

An alternative expression for AD may be given as

AD = C + I + G + (X – M)

And that for AS may be given as

AS = C + S + T

To simplify matters, let us assume again that I = I_a, T = T_a and M = M_a. We know that G, R and X are already independent of income, Y. With these specifications, AD curve can be obtained by the vertical displacement of the consumption curve first by I_a, then by G and finally by (X – M_a).

We know that the Aggregate Supply (AS) curve representing the identity AS = Y is always a straight line inclined at 45° to the x-axis, the equilibrium level of income can be determined at the equality of the AD and AS. Thus,

Equations 6.39 and 6.40 both yield the same value of the equilibrium level of income.

Graphical representation of the equilibrium of a four-sector model is left to the reader. What the reader needs to do is to give a further upward shift by (X – M_a) to the AD curve of Fig. 6.6 (upper panel) and to the injections curve (lower panel). Rest of the two panels remains unchanged.

Besides the multipliers given in equations 6.21, 6.22, 6.23 & 6.24; we can define yet another two as the export multiplier and the import multiplier for this model by differentiating equation 6.40.

K_x = 1 / (1 – b)

K_Ma = -1 / (1 – b)

Illustration 6.5:

Consider an economy with following specifications:

C = 80 + 0.80Y^d

Y^d = Y – T_a + R

T_a = 10

R = 20

I_a = 50

G = 50

X = 20

M_a = 28

And calculate:

(i) Equilibrium level of income

(ii) The relevant multipliers

(iii) Effect on income of each of the following changes:

1. Increase in investment by 10

2. Decrease in autonomous taxation by 5

3. Increase in transfer payments by 5

4. Increase in exports by 10

5. Decrease in autonomous imports by 8

6. Decrease in public expenditure by 10

The solution is left to the reader as an exercise. Answers are provided below for reader’s verification.

[Ans.: (i) 900; (ii) K_Ia = K_G = K_x = 5.00; K_Ta = – 4.00; K_R = 4.00; K_Ma = – 5.00

(iii) 1. + 50; 2. + 20; 3. + 20; 4. + 50.00; 5. + 40.00; 6. – 50.00]

(B) When Tax Is Progressive/Digressive and Import a Function of Income:

We have so far discussed the four-sector model with taxation and import both wholly autonomous in character. Let us now take it up in its more general form with taxation progressive/digressive and import, as a function of income.

We have seen in the earlier section that the tax function under progressive/digressive taxation transforms to one given in Equation 6.25. The import function on the same lines transforms to

M = M_a + m Y

Where, M_a is autonomous component and ‘mY’ the induced component. It can be seen that m is the marginal propensity to import, dM / dY

Aggregate demand (AD) now transforms to

AD = C + I_a + G + (X – M)

= C_a + bY^d + I_a + G + (X – M)

= C_a + b [Y – (T_a + t Y – R)] + I_a + G + (X – M_a – mY)

= C_a + b [Y – T_a – t Y + R] + I_a + G + X – M_a – mY

= C_a + b Y – b T_a – btY + bR + I_a + G + X – M_a – mY

Aggregate supply (AS) remaining unchanged,

AS =Y

Illustration 6.6:

For a hypothetical economy with following specifications,

C = 2000 + 0.80 Y^d

I_a = 500

G = 400

R = 100

T = 100 + 0.25 Y

X = 200

M = 50 + 0.20 Y

Determine:

1. Equilibrium level of Income, tax revenue, disposable income, consumption, savings and imports.

2. Relevant multipliers

3. Effect on income of each of the following

(a) Increase in government spending by 20

(b) Increase in autonomous taxation by 20

(c) Decrease in autonomous investment by 50

(d) Decrease in transfer payments by 50.

(e) Increase in tax rate by 5%.

4. Budget Surplus at equilibrium

5. Net Exports (NX) at equilibrium

(Note that net exports being negative, the balance is unfavourable).