Determination of Equilibrium Income in an Economy with Three Sectors – Explained!

Determination of equilibrium level of income in an economy that has only two sectors, namely, the households’ and the producers’ sectors. Such economies do not survive in real world for long.

They need a regulatory authority for their smooth functioning sooner or later. Invariably, a government sector becomes inevitable, more so, when the state is a welfare state. Apart from acting as a producer, the government of a welfare state acts as a watchdog to all the activities of production and consumption. Taxation – direct and indirect – provides a source of income for the government and public expenditure and transfer payments, as channels of its spending.

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(A) When Taxation is Autonomous:

The aggregate demand (AD) for an economy with three sectors, namely, the households’ sector, the producers’ sector and the government sector, can be expressed as

AD = C + I + G

Here, G = government spending on goods and services.

I = I_a, the autonomous investment.

(In this section, we will proceed with the assumption that investment is fixed) The Aggregate Supply (AS), in like manner, can be expressed as

AS = C + S + T

Here, T = tax revenue net of transfer payments. Taxation assumed fixed at T_a and the transfer payments assumed fixed at R, the tax revenue net of transfer payments, T = T_a– R.

AS = Y, as in 6.1, would again lead to a 45° line while AD = C + I_a + G would be a vertical displacement of C = C_a + bY^d, first by I_a and then by G. Here, Y^d is the disposable income given as

Y^d = National Income – Tax Revenue net of Transfer payments.

= Y-[T_a-R]

= Y-T_a+R

Equilibrium level of income would refer to the condition AD = AS, as in section 6.1. The reader can easily verify that the equilibrium condition would stand modified to

I_a + G = S + T

where, I_a + G = injections

and, S + T = withdrawals.

Graphically, equilibrium is shown in Fig. 6.6. The upper panel shows determination of income through the equality of AD and AS while the lower panel, that of it through the equality of injections and withdrawals.

Algebraically, combining AS = Y and AD = C + I_a + G, we have

Y = C + I_a + G

= C_a+ bY^d + I_a + G

= C_a + b[Y – (T_a – R)] + I_{a +} G

= C_a+ bY – bT_a + bR + I_a + G

(1 – b)Y = C_a– bT_a + bR + I_a + G

Y = 1 / (1 – b) [C_a – bT_a + bR + I_a + G]

Expression in equation 6.20 gives the equilibrium level of income in a three-sector economy.

This, as evident, has been obtained from the equality of AD and AS. It refers to the upper panel of Fig. 6.6. The same result can be obtained through the equality of injections and withdrawals as in the lower panel.

I_a + G = S + T

= – C_a + (1 – b) Y^d + T

= -C_a + (1 – b) [Y – T_a + R] + T_a – R

(Y^d = Y – T = Y – T_a + R and T = T_a – R)

= – C_a + (1 – b) Y – (1 – b)T_a + (1 – b)R] + T_a – R

= – C_a + (1 – b) Y – T_a + bT_a + R – bR + T_a – R

(1 – b)Y = C_a – bT_a + bR + I_a + G

Y = 1 / (1 – b) [C_a – bT_a + bR + I_a + G]

Note that the expressions in equations 6.20 and 6.21 for the equilibrium level of income are the same.

Following the expression in equation 6.12 for investment multiplier, we can define three more multipliers, namely, the government expenditure multiplier (dY/dG), the taxation multiplier (dY/dT_a) and the transfer payments multiplier (dY/dR) as follows:

K_G = dY / dG = 1 / (1 – b)

K_Ta = dY / dT_a = -b / (1 – b)

K_R = dY / dR = b / (1 – b)

An illustration would be in order here to explain the concepts.

Illustration 6.3:

For a hypothetical economy with three sectors, the following specifications are provided:

C = 20 + 0.75Y^d

I_a = 20

G = 25

T_a = 25

Determine:

(i) The equilibrium level of income through the equality of the AS and AD as well as through that of the injections and withdrawals.

(ii) Investment multiplier, Govt expenditure multiplier and tax multiplier.

(iii) Disposable income, consumption expenditure and the level of savings at equilibrium.

(iv) Effect on the national income of each of the following changes:

1. Investment increases to 30.

2. Govt expenditure increases to 40.

3. Tax decreases to 20.

(v) (dY/dT_a) + (dY/dG). Can you interpret the result?

Solution:

(i) AD = AS leads to equation 6.20. We have

Y = 1 / (1 – b) [C_a – bT_a + bR + I_a + G]

Substituting data from the question, b = 0.75, C_a = 20, G = T_a = 25, I_a = 20, R = 0; we have

Y = 1 / (1 – 0.75) [20 – 0.75 x 25 + 0.75 x 0.00 + 20 + 25]

Y = 4.00 [20 – 18.75 + 0.00 + 45] = 185.

Equality of injections (I_a + G) to withdrawals (S + T) can be verified to yield the same result.

(ii) From equation 6.12, the investment multiplier, K_Ia (using subscript I_a to distinguish it from other multipliers and to have identical notation for it), we have

K_Ia = 1 / (1 – b)

= 1 / (1 – 0.75) = 4.00

From equation 6.22, government expenditure multiplier, K_G can be calculated as

K_G = 1 / (1 – b)

= 4.00 (as above)

Tax multiplier, K_Ta from equation 6.23, is

K_Ta = -b / (1 – b)

= – 0.75 / (1 – 0.75)

= – 3.00

Likewise transfer payments multiplier,

K_R = b / (1 – b)

= 0.75 / (1 – 0.75) =3.00

(iii) Disposable income, Y^d, at the equilibrium,

Y^d = Y – T_a+ R

= 185 – 25 + 0.00 = 160

Consumption expenditure at the equilibrium level of income,

C = C_a + bY^d

= 20 + 0.75 x 160

= 20 + 120 = 140

Savings at the equilibrium,

S = – C_a + (1 – b) Y^d

= – 20 + (1 – 0.75) × 160 = 20

Even directly, savings,

S = Y^d – C = 160 – 140 = 20

(iv) ∆I = 30 – 20 = 10, K_Ia = 4.00

∆Y= (K_Ia) ∆I_a = 4.00 x 10 = 40.

Thus when investment increases from 20 to 30, increase in income would be 40.

∆G = 40 – 25 = 15, K_G = 4.00

∆Y = (K_G) ∆G

= 4.00 × 15 = 60.00

That is, an increase in government spending by 15 increases national income by 60.

∆T_a = 20 – 25 = – 5.00, K_Ta = – 3.00

∆Y = (K_Ta ) × ∆T_a

= (- 3.00) × (- 5.00) = 15.00

That is a decrease in tax from 25 to 20, increases income by 15.00.

(v) Note that the sum of the government expenditure multiplier and the tax multiplier,

K_G + G_Ta = 4.00 + (-3.00) = 1.00

Even directly, [1 / (1 – b)] + [-b / (1 – b)] = (1 – b) / (l – b) = 1.00

The result that the sum of the government expenditure multiplier and the tax multiplier is unitary, is at times referred to as the unit budget multiplier theorem or even as the balanced budget multiplier theorem. The significance of this theorem is that the national income increases only by the amount of the increase in tax financed government spending under a balanced budget. To explain, let

∆T = 10

∆G = 10

Then, increase in income due an increase in taxes by 10 is

∆Y = (K_Ta) × ∆T

= (- 3.00) × (10)

= – 30.00

Increase in income due to an increase in government expenditure by 10 is given as

∆Y = (K_G) ∆G

= 4.00 × 10 = 40.

Total increase in income due to an increase in taxes by 10 under the balanced budget, thus, is given as (-30) + (40) = 10. That explains the significance of the theorem.

(B) When Taxation Is Progressive/Digressive:

In our analysis of equilibrium of a three-sector economy, we assumed that taxation is purely autonomous in character. The assumption was intended to simplify the model at its introduction stage. Now that the purpose is served, we relax the assumption and take taxation in its most common form, the one in which it is a function of income as shown in Equation 6.25.

The equation describes it as a function of income comprising of two components, one, purely autonomous (T_a) and the other, purely progressive (tY) in character. A tax is termed as progressive when its rate increases with increasing income.

The incomes for the purpose are divided into several income-slabs with the lowest slab subjected to the lowest rate of taxation and the highest, to the highest rate. When incomes, high or low, are subjected to the same rate, taxation is termed as proportional and when incomes up to a certain slab are subjected to progressive taxation and thereafter to proportional taxation, the system of taxation is termed as a digressive one. Income tax in India, for example, is a digressive tax. Returning to the tax function,

T = T_a + tY

Where, T_a is autonomous component and ‘t’, the rate of tax and ‘Y’, the income. Aggregate demand (AD) under purely progressive component, tY transforms as

AD = C + I_a + G

= C_a + bY^d + I_a + G

= C_a + b[Y – (T – R)] + I_a + G

= C_a + b [Y – T + R] + I_a + G

= C_a + b [Y – tY + R] + I_a + G

= C + bY – btY + bR + I_a + G

Aggregate supply (AS) remaining unchanged, is given as

AS =Y

For equilibrium, AD = AS = Y.

Hence,

Y = C_a+ bY – btY + bR + I_a + G

(1 – b + b t)Y = C_a + b R + I_a + G

Y = 1 / (1 – b + bt) [C_a + bR + I_a + G]

Equation 6.26 gives the equilibrium level of income, with the following multipliers

dY / dG = 1 / (1 – b + bt)

dY / dI_a = 1 / (1 – b + bt)

dY / dR = b / (1 – b + bt)

Apart from the above, with which the reader is already familiar, we have yet another multiplier, the tax rate multiplier, given as

dY / dt = [-b / (1 – b + bt)²] [C_a + bR + I_a + G]

= [-b / (1 – b + bt)] × [1 / (1 – b + bt)] [C_a + bR + I_a +G]

= -bY / (1 – b + bt)

Y = {1 / (1 – b + bt)} {C_a + bR + I_a +G}, and t = (t + dt)}

Aggregate demand (AD) under the tax function given by Equation (6.25), transforms to

AD = C + I_a + G

= C_a + bY^d + I_a + G

= C_a + b [Y – (T_a + tY – R)] + I_a + G

= C_a + b [Y – T_a – tY + R] + I_a + G

=C_a + bY – bT_a – btY + bR + I_a + G

Aggregate supply (AS) remaining unchanged,

AS = Y

For equilibrium, AD = AS. Hence,

Y = C_a + bY – bT_a – btY + bR + I_a + G

= (1 – b +bt)Y = C_a – bT_a + bR + I_a + G

Y = 1 / (1 – b + bt) [C_a – bT_a + bR + I_a + G]

Equation 6.31 gives the equilibrium level of income, with the following multipliers

dY / dG = 1 / (1 – b + bt)

dY / dI_a = 1 / (1 – b + bt)

dY / dR = b / (1 – b + bt)

dY / dT_a = -b / (1 – b + bt)

From Equation 6.32 and 6.35, we have

dY / dT_a + dY / dG = (1 – b) / (1 – b + bt) < 1

In part (A) of this section, we showed that the sum of the government expenditure multiplier (dY/ dG) and the tax multiplier (dY / dT_a) is 1 under autonomous taxation,

dY / dT_a + dY / dG = 1

We also identified this result as unit multiplier theorem.

Under digressive taxation, as Evident from equation 6.36, the unit multiplier theorem collapses.

Let us have an illustration to demonstrate the three-sector model with digressive taxation.

Illustration 6.4:

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

Determine:

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

2. Relevant multipliers

3. Effect on income of each of the following

(а) 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