After studying this topic, you should be able to understand
Initial chapters have focused on income determination and the multiplier in a two sector economy, where there exist the households and the firms. This chapter focuses on extending the theory of income determination and the multiplier to a three sector model, the third sector being the government sector.
The action of the government relating to its expenditures, transfers and taxes is called the fiscal policy. Here, we focus on three fiscal policy models which are in increasing order of complexity, with the emphasis being on the government expenditure, taxation and the income level. We also discuss some of the fiscal policy multipliers, including the balanced budget multiplier.
Fiscal policy relates to the actions of the government regarding its expenditures, transfers and taxes.
Though the government is involved in a variety of activities, three of them are of greater relevance to us in the present context. Hence, we will focus on these activities of the government, which are discussed below:
Direct taxes are levied directly and include personal income and corporate income taxes.
Indirect taxes are levied indirectly and include sales tax and excise tax. They are paid as a part of the price of the goods.
We simplify our analysis by making a few assumptions, which are as follows:
Transfer payments are those government payments which do not involve any direct services by the recipient, for example welfare payments, unemployment insurance and others.
We here introduce the notion of an income leakage and an injection. In a two sector model, a part of the current income stream (which was not spent on consumption) ‘leaked’ out as saving whereas injections in the form of ‘investment’ were injected into the system. In a three sector model taxes, like saving, are income leakages whereas government expenditures, like investment, are injections.
This model is an extension of the two sector model with the following modifications:
Given the above modifications, we can now analyse the equilibrium in a three sector model.
(1) Aggregate Demand–Aggregate Supply Approach
Aggregate demand = Total value of output (or income)
or
where, | C = | Ca + bYd (consumption function) |
Yd = | Y − T = disposable income | |
T = | = tax (lump sum income tax) | |
I = | = investment (assumed to be autonomous) | |
G = | (assumed to be autonomous) |
Substituting for these values in the basic Eq. (1), we get
Y = Ca + bYd + +
Y = Ca + b(Y − ) + +
Or
Y − bY = Ca + + − b
Or
Y(1 − b) = Ca + + − b
Thus,
Equation (2) gives the equilibrium income in a three sector economy.
(2) Leakages Equal Injections Approach
In equilibrium, in a three sector model,
AD = AS
or
C + I + G = C + S + T
As C is common in both the sides, the equilibrium condition can be written as
where, | I = | = investment (assumed to be autonomous) |
G = | = government expenditure (assumed to be autonomous) | |
S = | Yd − C = saving function | |
C = | Ca + bYd (consumption function) | |
T = | = tax (lump sum income tax) |
Substituting for these values in the Eq. (3), we get
+ = S +
+ = (Yd − C) +
+ = [Y − − (Ca + bYd)] +
+ = [Y − − Ca − b(Y − )] +
Y − bY = Ca + + − b
Y (1 − b) = Ca + + − b
or
Y (1 − b) = Ca + + − b
Thus,
The above equation is the same as Eq. (2) above. Hence, both the approaches yield the same equilibrium level of income.
(1) Aggregate Demand–Aggregate Supply Approach
Equilibrium: The determination of the equilibrium income by the aggregate demand–aggregate supply approach in a three sector economy has been depicted in the Figure 7.1 (a).
where, | x-axis = | disposable income |
y-axis = | aggregate demand or aggregate planned expenditure | |
C = | aggregate consumption function, C = Ca + bYd = Ca + b(Y – ) |
|
Ca = | the intercept of the consumption function on the y axis showing consumption spending at zero income level. | |
b = | the MPC or the slope of the consumption function (it will remain constant since, in our analysis, the consumption function is a linear function.) | |
I = | = investment (assumed to be autonomous) | |
G = | = government expenditure (assumed to be autonomous) | |
AD1 = | aggregate demand function before a tax (which is obtained by adding the consumption function, the investment function and the government expenditures) | |
AS = | aggregate supply function (also called the guideline or the 45 degree line) | |
Point E = | the two sector equilibrium or the initial equilibrium where the aggregate demand curve AD (C + I) and aggregate supply curves intersect to determine the equilibrium income at Y*. |
Figure 7.1 Determination of Equilibrium Income or Output in a Three Sector Economy
A balanced budget exists when the entire government expenditure is financed by taxes or that G = T.
Point E1 = | the three sector equilibrium where the aggregate demand curve AD1 (C + I + G) and aggregate supply curves intersect to determine the equilibrium income at Y1. It is important to note that at this stage, the entire government expenditure is deficit financed and that the taxes are zero. Thus, increase in government spending will lead to an increase in income where ΔY > ΔG or ΔY = mΔG. Hence, the multiplier m > 1. | |
C′ = | the consumption function after the tax. As a tax is a withdrawal, it reduces the disposable income (Yd) thus leading to a shift of the consumption function from C to C′ and a shift of the aggregate demand curve from AD1 to AD2. | |
AD2 = | aggregate demand function after a tax | |
Point E2 = | the three sector equilibrium where the aggregate demand curve AD2 (C′ + I + G) and aggregate supply curves intersect to determine the equilibrium income at Y2. The entire government expenditure is financed by taxes or that G = T. In other words, it is a balanced budget. |
(2) Leakages Equals Injections Approach
Equilibrium: The determination of the equilibrium income by the leakages and injections approach in a three sector economy has been depicted in the Figure 7.1(b).
where, x-axis = | disposable income |
y-axis = | planned saving, planned investment, taxes and government expenditure |
S = | planned saving |
I = | I = investment function |
S + = | planned saving plus taxes |
I + = | planned investment plus government expenditure |
E = | the two sector equilibrium or the initial equilibrium where the saving curve, S and the investment curve, intersect to determine the equilibrium income at Y*. |
Point E1 = | the three sector equilibrium where the saving curve, S and the investment plus government expenditure curve, + intersect to determine the equilibrium income at Y1. Here, taxes are zero. |
S′ = | saving after the imposition of the tax |
Point E2 = | the three sector equilibrium where the planned saving and taxes curve, S′ + and investment plus government expenditure curve, + intersect to determine the equilibrium income at Y2. The entire government expenditure is financed by taxes or = . In other words, it is a balanced budget. |
We find that the two approaches to the determination of the equilibrium income, the aggregate demand—aggregate supply approach and the leakages equals injections approach, both yield the same result.
It is important to observe that though government expenditure has an expansionary effect, taxes have a contractionary effect on the income level. However, the contractionary effect of taxes is less than the expansionary effect of government expenditure even though ΔG = ΔT. This is because though an increase in government spending is entirely an addition to the aggregate demand, an increase in T is not entirely a decrease in the aggregate demand. Some part of the increase in T involves a reduction in the savings whereas the rest is absorbed by a reduction in consumption and, hence, in aggregate demand.
Although before the imposition of the tax the equilibrium level of income was Y1, after the tax it reduces to Y2 and not to Y* (even though G = T). Hence the imposition of a tax, given the level of government expenditure, causes a reduction in the equilibrium level of income from Y1 to Y2, though it is still larger than the initial income level Y*.
An important implication for fiscal policy is that an economy can achieve full employment output by an expansion in its budget, financing every rupee of additional expenditure with a rupee of additional taxes.
The fundamental equations in an economy are given as:
C = | 150 + 0.75(Y – T) | |
= | 300 T | |
T = | 40 + 0.2Y | |
= | 150 |
Find the equilibrium level of income.
Solution
The equilibrium condition in a three sector economy is given as Y = C + I + G.
Thus, | Y = | 150 + 0.75[Y – (40 + 0.2Y)] + 300 + 150 |
Y = | 150 + 0.75(Y – 40 – 0.2Y) + 300 + 150 | |
Y = | 600 + 0.75(–40 + 0.8 Y) | |
Y = | 600 – 30 + 0.6 Y | |
Y – | 0.6 Y = 570 | |
0.4Y = | 570 | |
Y = | ||
Y = | 1425 |
The equilibrium level of income is 1600.
In the first model we had included only two activities of the government, namely, government expenditure and taxes. Here in this second model, we bring transfer payments also into the picture.
As already mentioned, transfer payments are those government payments which do not involve any quid pro quo or in other words do not involve any direct services by the recipient; for example, welfare payments. Transfer payments are, in fact, just the reverse of taxes or in other words they are negative taxes. Taxes reduce the spending capacity whereas transfer payments increase the spending capacity of the households leading to, ultimately, an increase in the equilibrium level of income.
We have our basic Eq. (1) Y = C + I + G
where, | C = | Ca + bYd = consumption function |
Yd = | Y – T + R = disposable income | |
T = | = tax (lump sum income tax) | |
I = | = investment (assumed to be autonomous) | |
G = | = government expenditure (assumed to be autonomous) | |
R = | = transfer payments (assumed to be autonomous) |
Substituting for these values in the basic equation, we get
Y = Ca + b (Y −( + ) + +
or
Y − bY = Ca + + − b + b
or
Y (1 − b) = Ca + + − b + b
Thus,
Similar to the first model, here also a change in any of the values within brackets will lead to a change in income which will be equal to the change in that particular value times the multiplier.
It is important to observe that both the transfer payments and the government expenditure have an expansionary effect on the income level. However, the expansionary effect of an increase in the transfer payments will be less than the effect of an increase in the government expenditure even though ΔG = ΔR (as long as the marginal propensity to consume is less than 1). Thus,
This is because while the entire increase in government spending is an addition to the aggregate demand, only a part of the increase in R will be an addition to the aggregate demand (through an increase in the consumption spending). Some part of the increase in R is directed towards savings. Hence, the increase in income
It is important to note that though a change in government expenditure affects aggregate demand directly, a change in transfer payments affects aggregate demand indirectly through a change in disposable income.
In recent years, a large part of the tax receipts of the government consists of personal and corporate income taxes. Other taxes like sales tax, excise duty and service taxes also vary indirectly with the income level though less as compared to personal and corporate income taxes. Thus, we introduce tax as a linear function of income in this model.
T = + t Y +
where, | T = | the total tax |
T = | = autonomous tax receipts (tax receipts at, theoretically, zero income level) | |
t = | proportional income tax rate (the fraction of any income that will be taxed) | |
= | Transfers | |
Y = | C + I + G | |
where, | C = | Ca + bYd = consumption function |
Yd = | Y – T + = disposable income | |
T = | + tY + = net tax (where the tax is a function of the income level) | |
= | transfer payments | |
I = | = investment (assumed to be autonomous) | |
G = | = government expenditure (assumed to be autonomous) |
Substituting for these values in the Eq. (1), we get
Y = Ca + b [Y − ( + tY + )] + +
or
Y − bY = Ca + + − b − btY + b
or
Y − bY + btY = Ca + + − b + b
Y (1 − b + bt) = Ca + + − b + b
Thus,
In the second model where the tax receipts are independent of the income level, the multiplier is larger than the multiplier in the third model where the tax receipts are dependent on the income level.
Given an increase in government expenditure, the expansion in income will be smaller with a proportional income tax than for a lump sum income tax.
Numerical Illustration 2
In the Numerical Illustration 1 suppose the government transfer payments are at Rs. 60 crores. Find the equilibrium level of income.
Solution
The equilibrium condition in the three sector economy is given as Y = Ca + b (Y – + ) + +
Thus, | Y = | 150 + 0.75[Y – (40 + 0.2Y) + 60] + 300 + 150 |
Y = | 150 + 0.75(Y – 40 – 0.2Y + 60) + 300 + 150 | |
Y = | 600 + 0.75(20 + 0.8Y) | |
Y = | 600 + 15 + 0.6Y | |
Y – | 0.6Y = 615 | |
0.40 = | 615 | |
Y = | ||
Y = | 1537.50 |
The equilibrium level of income is 1537.50
Through its fiscal policy, the government is in a position to influence the economic activities in an economy. To what extent its fiscal operations have an impact on the equilibrium level of income depends on the fiscal multipliers. Here, we analyse three multipliers.
Government Expenditure Multiplier: In the First Model, we had arrived at
where, | C = | Ca + bYd (consumption function) |
Yd = | Y – T = disposable income | |
T = | = tax (lump sum income tax) | |
I = | = investment (assumed to be autonomous) | |
G = | (assumed to be autonomous) |
Substituting for these values in the basic Eq. (1), we get the equilibrium level of income as
Assume that there is an increase in government expenditure by ∆G. Hence,
Subtracting Eq. (2) from Eq. (3), we get
or
where, | ∆G = change in government expenditure |
b = marginal propensity to consume | |
ΔY = change in income | |
GM = government expenditure multiplier |
An increase in government expenditure by ΔG leads to an increase in aggregate demand and, hence, in the equilibrium level of income by ΔY. The government expenditure multiplier has the same value as the investment multiplier in the two sector model, as discussed in Chapter 5, of . As the value of b is always between 0 and 1, the multiplier will always have a value greater than 1. Hence, a change in government expenditure by ΔG will lead to a change in the equilibrium level of income by ΔY where ΔY > ΔG.
(1) Tax Multiplier (Lump Sum Tax)
We have Eq. (2) as
Assume that there is a change in tax by Δ. Hence, we get
Subtracting Eq. (2) from Eq. (4), we get
or
where, | Δ = change in tax |
b = marginal propensity to consume | |
ΔY = change in income | |
GT = government tax multiplier |
As the tax multiplier is negative, an increase in tax leads to a decrease in the equilibrium level of income.
(2) The Balanced Budget Multiplier
The budget is in balance when the government expenditures plus transfer payments equal the gross tax receipts, or in other words, G = T. It follows that when there is an increase in the government expenditure, it will be financed by an increase in taxation. Thus, ΔG = ΔT.
Balanced budget multiplier is the increase in the output as a consequence of equal increases in the government expenditure and taxes.
It is important to note that an increase in the government expenditures, which is balanced by an increase in taxes of an equal amount, will not leave the income level unchanged. In fact, there will be an increase in income by the same amount as the increase in the government expenditures and the tax. This implies that it is incorrect to assert that government expenditures and taxes of an equivalent amount off set one another and that there is no increase in the income level if the budget is balanced. In fact, the increase in the income is exactly equal to the amount by which there is an increase in the government expenditures and tax. Hence the value of the ‘balanced budget’ multiplier, which is the increase in the output as a consequence of equal increases in the government expenditure and taxes, is equal to one. This is what is known as the ‘balanced budget’ or ‘unit multiplier’ theorem.
We have Eq. (2) as
Assume that there is a change in government expenditure by ΔG and in tax by ΔT, and ΔG = ΔT; hence, we get
Subtracting Eq. (2) from Eq. (5), we get
But
ΔG = ΔT
Thus, we can write
or
ΔY(1 − b) = Δ(−b + 1)
or
ΔY(1 − b) = Δ(1 −b)
or
where, | ΔG = change in government expenditure |
ΔT = change in tax | |
b = marginal propensity to consume | |
ΔY = change in income |
Alternatively, the balanced budget multiplier can also be obtained by summing up the government expenditure multiplier and the tax multiplier to get
The budget is in balance when the government expenditures plus transfer payments equal the gross tax receipts, or in other words, G = T.
Whatever the value of b, the sum of the government expenditure multiplier and the tax multiplier will always be equal to unity.
Regardless of the value of b (the marginal propensity to consume), the government expenditure multiplier will always be one greater than the tax multiplier.
As the tax multiplier is negative, an increase in tax leads to a decrease in the equilibrium level of income.
Numerical Problem 1
In a two sector economy, the basic equations are as follows: the consumption function is C = 100 + 0.80 Yd and investment is - I = 150 crores. The equilibrium level of income is Rs. 1250 crores. Suppose the government sector is added to this two sector model, which then becomes a three sector economy. The government expenditure is at Rs. 50 crores.
Numerical Problem 2
In an economy, the full employment output occurs at Rs. 1000 crores. The marginal propensity to consume is 0.80 and the equilibrium level of output is currently at Rs. 800 crores. Suppose the government aspires to achieve the full employment output, find the change in
Numerical Problem 3
Suppose the consumption function is C = 50 + 0.80Yd and investment is = 100 crores. The government expenditure is at Rs. 90 crores whereas the tax function is a proportional income tax function where T = 0.10 Y.
Numerical Problem 4
In an economy, the marginal propensity to consume is 0.80. The tax is a lump sum tax or, in other words, not related to the income level. Find the change in the equilibrium output for the following:
Numerical Problem 5
In an economy, C = 50 + 0.80 Yd, = 100 crores, government expenditure is at Rs. 50 crores whereas T = 20 crores.
Solution 1
Thus, | Y = | 100 + 0.80 Y + 150 + 50 |
Y – 0.80Y = | 100 + 150 + 50 | |
0.20Y = | 300 | |
Y = | 300/0.20 | |
Y = | 1500 |
The equilibrium level of income in the three sector economy is Rs. 1500 crores, which is an increase by Rs. 250 crores over the two sector economy.
Investment Multiplier,
(where b is the marginal propensity to consume)
Thus, the magnitude of the multiplier effect (of a change in the autonomous investment) is the same as that of a change in government expenditure.
Thus, | C = | 100 + 0.80(Y – 50) |
C = | 100 – 40 + 0.80Y | |
C = | 60 + 0.80Y | |
But | Y = | C + I + G |
Y = | 60 + 0.80Y + 150 + 50 | |
Y – 0.80Y = | 60 + 150 + 50 | |
0.20Y = | 260 | |
Y = | 26/0.20 | |
Y = | 1300 |
The new equilibrium level of income in the three sector economy, when there exists a balanced budget, is Rs. 1300 crores.
Solution 2
where, | ∆G = change in government expenditure |
b = marginal propensity to consume | |
∆Y = change in income | |
GM = government expenditure multiplier |
For example,
b = 0.80
ΔY = 1000 − 800 = 200
Thus,
ΔG = 200(0.20) = 40
Thus, the level of government expenditures required to achieve the full employment output is Rs. 40 crores.
where | ∆T = change in tax |
b = marginal propensity to consume | |
∆Y = change in income | |
GT = government tax multiplier |
As the tax multiplier is negative, an increase in tax leads to a decrease in the equilibrium level of income.
For example,
b = 0.80
ΔY = 1000 − 800 = 200
Thus,
−0.80ΔT = 200(0.20) = −50
The net lump sum tax is Rs. –50 crores. There should be a decrease in lump sum tax by Rs. 50 crores.
But
ΔG = ΔT
Thus, we can write
or
ΔY(1 − b) = Δ (−b + 1)
or
ΔY(1 − b) = Δ G (1 − b)
or
Thus,
ΔY = ΔG = Rs. 200 crores.
The required increase in the level of government expenditures and the net lump sum tax is Rs. 200 crores.
Solution 3
Here | C = 50 + 0.80Yd |
C = 50 + 0.80(Y − T) | |
C = 50 + 0.80(Y − 0.10Y) | |
C = 50 + 0.80 (0.9Y) | |
C = 50 + 0.72Y | |
Thus, | Y = 50 + 0.72Y + 100 + 90 |
Y − 0.72Y = 50 +100 + 90 | |
0.28Y = 240 | |
Y = 240/0.28 | |
Y = 85.71 |
The equilibrium level of income is Rs. 857.14 crores.
The tax function is a proportional income tax function where T = 0.10 Y.
Thus, T = 0.10(857.14) = 85.71 crores
Hence, the revenue from taxes at the equilibrium level of income is Rs. 85.71 crores whereas the government expenditure is at Rs. 90 crores. Therefore, there is a budget deficit of Rs. 4.29 crores.
Y = 50 + 0.72Y + 120 + 90
Y − 0.72Y = 50 + 120 + 90
Y = 260/0.28
Y = 928.57
The equilibrium level of income is Rs. 928.57 crores.
T = 0.10(928.57) = 92.857 crores
Hence, the revenue from taxes at the equilibrium level of income is Rs. 92.857 crores whereas the government expenditure is at Rs. 90 crores. Therefore, there is a budget surplus of Rs. 2.857 crores. Due to the higher income level, there are larger tax revenues leading to a budget surplus.
Solution 4
For example, b = 0.80 and ∆G = 10.
Thus,
Thus for a 10 crore increase in the level of government expenditures, the equilibrium output increases by 50 crores.
For example,
b = 0.80
∆T = 15
Thus,
Thus, for a 15 crore increase in taxes, the equilibrium output decreases by 60 crores.
Thus, a 10 crore increase in transfers, increases the equilibrium output by 40 crores.
Solution 5
Here | C = 50 + 0.80Yd |
C = 50 + 0.80(Y − T) | |
C = 50 + 0.80(Y − 20) | |
C = 50 + 0.80Y − 16 | |
C = 50 + 0.80Y − 16 | |
Thus, | Y = 50 + 0.80Y − 16 + 100 + 50 |
Y − 0.8Y = 50 − 16 + 100 + 50 | |
0.2Y = 184 | |
Y = 184/0.2 | |
Y = 920 |
The equilibrium level of income is Rs. 920 crores.
Equilibrium level of consumption:
C = 50 + 0.80Yd | |
C = 50 + 0.80(Y − T) | |
C = 50 + 0.80(Y − 20) | |
C = 50 + 0.80 Y − 16 | |
C = 50 + 0.80Y − 16 | |
C = 50 + 0.80 (920) − 16 | |
C = 770 |
The equilibrium level of consumption is 770 crores.
Equilibrium level of saving: S = Y – C – T
S = 920 – 770 – 20 = 130
The equilibrium level of saving is 130 crores.
Leakages = S + T = 130 + 20 = 150
This depicts the injections leakages equality at the equilibrium level.
(b) The equilibrium level of income decreases by 346.15 when there is an increase in autonomous tax by 150.
(c) The equilibrium level of income decreases by 230.77 when there is an increase in transfers by 150.
(b) There should be a decrease in lump sum tax by Rs. 33.33 crores.
(c) The required increase in the level of government expenditures and the net lump sum tax is Rs. 100 crores.
(b) The revenue from taxes at the equilibrium level of income is Rs. 147.692 crores whereas the government expenditure is at Rs. 180 crores. Therefore, there is a budget deficit of Rs. 32.31 crores.
(c) The equilibrium level of income is Rs. 1938.46 crores.
(d) The revenue from taxes at the equilibrium level of income is Rs. 193.846 crores whereas the government expenditure is at Rs. 180 crores. Therefore, there is a budget surplus of Rs. 13.84 crores.
(b) The equilibrium level of income decreases by 56.25 when there is an increase in autonomous tax by 150.
(c) The equilibrium level of income decreases by 37.5 when there is an increase in transfers by 150.
(b) The equilibrium level of consumption is 640 crores.
The equilibrium level of saving is 130 crores.
(c) Injections = I + G = 80 + 40 = 120
Leakages = S + T = 110 + 10 = 120
This depicts the injections leakages equality at the equilibrium level.