We may turn to a detailed discussion of the multiplier effects of change in tax. As we already know, the effects of two types of taxes on equilibrium national income, viz., lumpsum tax and proportional tax.

Both types of taxes reduce Y, but in different ways. We will now see that both have a multiplier effect in the sense that a certain change in the tax rate or tax total will have a multiplier effect on Y. For analytical simplicity, we consider the multiplier effect of a lumpsum tax.

It may be recalled that taxes are a leakage from the circular flow of income. So an increase in lumpsum tax will lead to_{ }a multiple drop in Y through the tax multiplier. The converse is also true. A tax cut will lead to a multiple increase in Y through the tax multiplier.

**The reduced form equation of equilibrium national income for a three-sector economy with lumpsum taxes is written as: **

Y = 1/1 – b (a – bT + I + G) … (1)

Where T is lumpsum income tax. We know that an increase in tax will reduce consumption demand (as also saving) by reducing disposable income. The fall in consumption is the MPC times the tax, i.e., ΔC = bΔT.

Now suppose national income rises by AY and the government imposes an additional lumpsum tax on this extra income.

**As a result the income equation (1) now becomes: **

Y + ΔY = 1/1 – b [a – b (T + ΔT) + I + G]

= 1/1 – b [a – bT – bΔT + l + G] … (2)

**Now if we subtract equation (1) from (2) we get: **

ΔY = 1/1 – b (-bΔT) … (3)

**Finally by dividing both sides of equation (3) by ΔT, we get: **

ΔY/ΔT = -b/1 – b … (4)

Which is indeed the tax multiplier. It may be noted that the imposition of an additional lump-sum tax (AT) will reduce national income not by the same amount but by a multiple of it. Suppose, for example, that b = 0.80 and AT = Rs 20 crores.

So the tax multiplier is ΔY/ΔT = -b/1 – b = -0.80/1 – 0.80 = -0.80/0.20 = -4

The tax multiplier is negative because taxes act as a leakage from the circular flow of income Thus if ΔT = Rs. 20 crores, ΔY will be – Rs 80 crores.

However, a cut in taxes will increase national income by the same amount since the multiplier will now be positive

ΔY/ΔT = b/1 – b = 4

and the multiplier effect will be favourable.

**The Balanced Budget Multiplier****: **

An important theorem of macroeconomics is the balanced budget multiplier theorem, or the unit budget multiplier theorem. The multiplier is derived by assuming that the budget of the government is in balance, i.e., G = T. So an increase in government spending (ΔG) is financed by an equal increase in tax (ΔT). Here we consider the multiplier effect of an equal increase in G and T, assuming that the government imposes only lumpsum tax (T).

**We know that if the government imposes a lumpsum tax, the equation of equilibrium national income is: **

Y = 1/1 – b (a – bT + I + C) … (1)

where T is lumpsum tax which remains constant at all levels of income.

Now suppose the government increases its expenditure by (ΔG) and finances it by imposing lumpsum taxes of ΔT. So the budget of the government is in balance, i.e., ΔG = ΔT.

In this case two things will happen at the same time. National income will rise due to an increase in G. It will fall due to an increase in T. However, the net increase in Y will be positive since the government expenditure multiplier is greater than the tax multiplier.

So equation (1) now becomes

This is the balanced budget (or unit budget) multiplier. This means that if ΔG = ΔT= Rs 20 crores y will also increase by Rs 20 crore. In other words if the budget of the government remains balanced, i.e., if an additional G. is financed by an additional T of the same amount, the net increase in national income will be exactly equal to the amount, by which G and T increased in the first instance.

We can use an alternative method to prove that the balance budget multiplier (BBM) is always equal to 1. We know that the government expenditure multiplier (m_{G}) = 1/1 – b, where b is the MPC. We also know that lumpsum tax multiplier (m_{r}) = -b/1 – b.

**Now if we combine the two multipliers, we get the BBM: **

m_{G} + m_{T} = 1/ 1 – b + -b/1 – b = 1 – b/1 – b = 1

Thus once again we prove that the BBM is always equal to one. The reason is that the government expenditure multiplier is always greater than the lumpsum tax multiplier by 1. For example, when b = 0.80, m_{G} = 5 and m_{T} = 4 and when b = 0.50. m_{G} = 2 and m_{T} = 1.

Now suppose ΔG = ΔT= Rs. 20 crores and b = 0.75. In this case F will increase by Rs. 80 crores, since m_{G} = 4. It will also fall by Rs. 60 crores since m_{T} = 3. So the net increase in income is ΔY = Rs. 80 crores – Rs. 60 crores = Rs. 20 crores. This is exactly equal to the amount by which G and T increased initially.

**Example: **

**Suppose an economy is described by the following equations: **

C = 50 + 0.75Y

I = 60

G = 50

T = 50.

All the terms have their usual meaning and all figures are in crores of rupees.

**Now answer the following questions: **

(i) What is the equilibrium level of Y?

(ii) Suppose government spending increases by Rs. 30 crores. What will be effect of this on equilibrium Y?

(iii) Find out the tax multiplier as also the BBM.

(iv) Suppose T = T + tY = 50 + 0.20Y

Find out the equilibrium value of Y.

**Solution: **

(i) In equilibrium

Y = C + I + G

= a + bY_{d} + I + G

= a + b(Y – T) + I + G

**Now it we substitute the numerical values of b, T, 1 and G in the previous equation we get: **

Y = 50 + 0.75 (Y – 50) + 60 + 50

= 50 + 0.75Y – 40 + 60 + 50

or Y – 0.75Y = 120

or 0.25Y = 120

or Y = 120 × 4 = 480 crores.

This is indeed the equilibrium value of Y.