Since the government plays a very important role in a modern economy, we include the government in the analysis of income determination. The government is the third basic unit of an economy. Its basic objective is to create more jobs, promote faster economic growth and thus ensure maximum welfare.

This is why it spends lots of money on construction of roads, bridges, schools, colleges, hospitals, etc. The expenditure of the government does not depend on national income or its rate of change. This is why in the Keynesian three-sector model of income determination, government expenditure is taken as autonomous and thus independent of income.

To start with we assume that government spends money in the economy on various types of goods and services but does not impose any tax. We will relax the assumption later and extend the model further by taking into account the effect of the imposition of taxes on equilibrium national income. Here we consider several models.

**Model 1: Income Determination with Government Expenditure but Without Taxes: **

We denote government expenditure by the symbol G.

**So in a three-sector economy, the equation for the equilibrium level of national income is written as: **

Y = C + I + G

where Y is national income and output and C + I + G represents aggregate demand (which now includes G, along with C and I).

Now the equilibrium condition of national income can be derived.

**We first express the equilibrium level of national income as follows: **

Y = C + I + G . . . (1)

**Since the consumption function is C = a + bY where a is autonomous consumption and b is MPC, equation (1) can be rewritten as: **

Y = a + bY + I + G

or Y – bY = a + I + G

or Y(1 – b) = a + l + G

Y = 1/1 – b (a + l + G)

= 1/1 – b (A)

where A = a + I + G = total autonomous spending (which includes autonomous consumption, private investment and government spending on goods and services).

Thus we see that the equilibrium value of national income can be calculated by multiplying autonomous expenditure A by the autonomous expenditure multiplier (1/1 – b)

Table 6A.1 gives an illustration of the simple Keynesian model of income determination (figures in Rs crores).

Here we assume that MPC is ⅘. We also assume that I = I̅ = 90 and G = G̅ = 10 at all levels of income. Thus when Y is 100, C + I + G is 180. So there is unintended inventory shortfall of 80. This will cause output to rise in order to meet the excess demand pressure.

When output increases to 500, there is no unintended shortfall of inventories. Current output is just sufficient to meet current demand. Actual (realised) investment is equal to desired investment (which also includes desired inventory investment) and undesired inventory investment is zero.

If producers overreact to the excess demand pressure and produce an output of 600, there will be the problem of excess supply (surplus). So there is undesired accumulation of inventory of 20. This will cause output to fall to the equilibrium level (500).

Thus only when Y=C + I + G, national income attains its equilibrium value. At this level of income S + T =I + G=100.So both the conditions of national income equilibrium are satisfied in the Keynesian model.

In Fig. 6A.1 we graphically illustrate how the equilibrium value of national income is determined in the Keynes’s three-sector model.

**We proceed in five steps: **

First we draw the 45° income line (which acts as the guideline). Then we draw the consumption line C. Next we draw the investment line I. Then we superimpose the I line on C line to arrive at the combined C + I line. Finally we add the fixed amount of government spending (G) to C + I to arrive at the aggregate demand schedule (C + I + G).

The aggregate demand curve intersects the income line or the aggregate supply curve at point E. If we drop a perpendicular from E it touches the horizontal axis at point Y_{E}, which is indeed the equilibrium value of national income.

The reason is that national income received (OY_{E}) is equal to the desired expenditure (EY_{E}). And the three components of aggregate demand (expenditure) are shown separately. We know that G is autonomous, i.e., independent of income. So the C + I + G line is parallel to the C + I line.

Since 0Y_{E} is the equilibrium level of income, no other level of income can be treated as an equilibrium level of income. The reason is easy to find out. If actual income is less than 0Y_{E}, say 0Yd, aggregate expenditure C + l + G will exceed aggregate supply.

This means that the sales plans of the business firms are fulfilled but production plans are not. As inventories are exhausted, business firms will be eager to produce more to meet the extra demand pressure.

As output increases, national income will rise and ultimately reach its equilibrium level (0Y_{E}). The converse is also true. If actual income (such as 0Y_{E}) exceeds its equilibrium level, aggregate demand will fall short of aggregate supply.

Due to such deficiency of demand, business firms will have unsold stocks of final goods. This unintended increase in inventories will force the firms to reduce production of goods. As production falls, national income will fall and come back to the equilibrium level (0Y_{E}).

**Two related points may be noted in this context: **

**1 Balance between Saving and Autonomous Spending: **

In the three-sector model aggregate saving (= Y- C) is equal to not only autonomous investment (I) but also government spending (G), i.e. at the equilibrium level of income S = I + G, since taxes are zero. In Fig. 6A.1 it is measured by the vertical distance EE’.

**2. Multiplier Effect: **

Keynes pointed out that any increase in autonomous spending generates a multiplier effect. Thus like autonomous investment, government spending has also a multiplier effect.

**Application of the Three-Sector Model: **

We may now apply Keynes’s three-sector model to study inflationary and deflationary gaps.

**Inflationary and Deflationary Gaps: **

In the context of the three-sector model of income determination, we may refer to two important concepts, viz., inflationary and deflationary gaps. These two concepts may now be explained one by one.

**Inflationary Gap: **

It was Keynes who first introduced the concept of ‘inflationary gap’ while discussing the theory of demand-pull inflation. According to Keynes, true inflation occurs only when the economy reaches the state of full employment. In his view, ‘full employment’ is the economy’s inflation threshold.

And the end of full employment is the beginning of inflation; demand-pull inflation starts only when the full-employment boom comes to an end (i.e., when actual output or GNP = the potential or maximum capacity output).

According to Keynes, for demand-pull inflation to occur there has to exist an inflationary gap in the economy. It is defined as the excess of desired expenditure over full-employment output (GNP) at the current price level.

Keynes introduced the concept to show the factors which cause demand-pull inflation. A simple example will make the concept clear.

**A Numerical Example: **

Suppose in 20.13, the value of total output in a country is Rs 1,600 crores at the current price level when all resources are fully employed. In other words, this is the full-employment level of output or income. Now the government imposes a tax on Rs 200 crores.

As a result disposable income falls to Rs 1 400 crores. If the net total money income of the people is Rs 1,400 crores, the income is equal to the value of output at the base year price level (which is 100). There is no inflationary pressure and the price level remains constant.

Now suppose the government increases its spending by Rs 600 crores by printing new currency notes [This method of financing government spending is known as deficit financing]. So the money income of the people will increase to Rs 2.000 crores (= Rs 1,400 crores + Rs 600 crores).

Now the government collects Rs 100 crores in taxes. So people’s disposable income now is Rs 1,900 crores, out of which people save another 200 crores. This means that net money income now available for expenditure is Rs 1,700 crores (= 2,000 – 100 – 200).

This represents an increase over the previously available income by Rs 300 crores (= 1,700 – 1,400). This excess of desired expenditure over full- employment output indeed measures what Keynes called the inflationary gap. This extra money income will exert an upward pressure on the general price level.

Since it is not possible to increase real output at full employment, only money income will rise, real income remaining the same. This excess of desired expenditure due to an increase in money income will cause the price level to rise to the extent necessary to equalise the current income with the value of full-employment output.

The money value of output will rise to Rs 1,700 crores, but the real value of output (= income) will remain constant at Rs 1,400 crores. Thus inflationary gap measures the excess of anticipated expenditure over available output at the base (i.e., pre-inflation) price level.

**The inflationary gap can be narrowed (if not completely eliminated) in two ways: **

(i) By reducing money incomes and

(ii) By increasing output.

**Deflationary Gap: **

An exactly opposite type of situation is known as deflationary gap. This gap is created when there is an excess of full-employment output over desired expenditure at the current price level. This will cause the price level to fall so as to equalise the current income with the value of full-employment output.

**Diagrammatic Illustration: **

These two concepts are illustrated in Fig. 6A.2 and Fig. 6A.3.

In Fig. 6A.2 we show real income on the horizontal axis and desired (anticipated) expenditure in the vertical axis. Here Y_{F} is the full-employment level, of output of Rs 1,600 crores. At this level of income aggregate saving = private investment + government spending. At this level of output income received 0Y_{F} = desired expenditure BY_{F}.

Now if aggregate demand increases and the AD curve shifts upward from (C + I + G)_{1} to (C + I + G)_{2} nominal output (income) will 10 doubt rise from Y_{F} to Y_{A}, but real output (income) will remain unchanged at the full-employment level (0Y_{F}). So the excess of nominal expenditure of AB over full-employment level of output will cause the price level to rise.

In Fig. 6A.3, we show an exactly opposite type of situation. The economy is supposed to reach full employment at point H, where output = income = .0Y_{F}. Here usual aggregate saving = investment + government spending.

However, if there is deficiency of demand due to unemployment (or equilibrium at less than full-employment as at point F), the desired expenditure falls short of this output as shown by the vertical distance HJ. This deficiency of desired expenditure at full-employment output (= 0Y_{F}) measures deflationary gap. This will cause the price level to fall.

It may also be noted that the decline in income is greater than the fall in demand caused by involuntary unemployment due to the working of the reverse multiplier process.

**Model 2: Income Determination with Government Expenditure and a Lumpsum Tax (Without Transfer): **

Since government expenditure is mainly financed by taxes, we have to take into account taxes also in our model of income determination. Let us suppose for simplicity that the government initially imposes a lumpsum tax (T), which does not rise or fall with income.

Now if government expenditure (G) is equal to lump sum tax (T), the budget of the government is said to be balanced (G = T). It however, G is greater than T, there will be a deficit in the budget. And if T > G, there will be a surplus in the budget.

Here we assume that the government budget is in balance (G = T). We also assume that the government does not make any transfer payments.

A lumpsum tax yields a fixed revenue at levels of national income, irrespective of the tax contribution of each individual. Suppose the government raises an extra revenue of Rs 20 crores by imposing such a tax. This will lead to a fall in the disposable income of the economy by the same amount. As a result people will reduce their consumption(C) and saving(S) at each level of national income.

However, consumption will not fall exactly by the amount by which disposable income falls. The reason is that a fall in disposable income will lead to a fall in both consumption and saving. In fact, the amount by which consumption falls will be equal to the MPC times the lumpsum tax. Suppose disposable income falls by ΔY_{d} due to tax.

**Then fall in consumption (- ΔC) will be: **

– ΔC = T.MPC … (1)

Since a lumpsum tax leads to a fall in disposable income,

**we can write: **

T= -ΔY_{d} … (2)

**So equation (1) now becomes **

– ΔC = – ΔY_{d}.MPC

For example, if the government imposes a lumpsum tax of Rs 20 crore and MPC = 0.75, then consumption will fall by Rs 20 crore × 0.75 = Rs 15 crore and saving will fall by Rs 20 crore × 0.25 = Rs 5 crore.

The converse is also true. If the government reduces lumpsum tax by Rs 20 crore, consumption will increase by Rs 150 crore (ΔC = ΔY_{d}.MPC) and saving by Rs 5 crore.

In Fig. 6A.4, we show the effect of an increase in lumpsum tax on consumption. We see that the consumption line shifts downward from C_{0} to C_{1}. The magnitude of the shift is determined by multiplying the amount of the tax by the MPC. The converse is also true. A cut in lumpsum tax will shift the consumption line upward to C_{2}.

In Fig. 6A.4 we show the effect of lumpsum tax Rs 20 crore on the equilibrium value of national income. Due to the lumpsum tax the consumption line shifts downward from C_{0} to C_{1} in Fig. 6A.3. This gets reflected in Fig. 6A.4 where the aggregate demand curve initially intersects the aggregate supply curve at point E.

So the initial level of national income is 0Y_{0}. Now, after the imposition of the lumpsum tax, the aggregate demand curve shifts downward from C_{0} + I + G to C_{1} + I +G where C_{1} = C_{0 }– ΔC. Here ΔC = MPC.T = b.T.

Thus the aggregate demand curve shifts downward exactly by the amount by which the consumption function shifts downwards due to a fall in Y_{d} caused by the lumpsum tax. This is shown by the vertical distance EE’ which is equal to – ΔC = b.T where b = MPC. As a result, the economy reaches a new equilibrium point F. So national income falls from 0Y_{0} to 0Y_{1}.

By comparison, a reduction in taxes increases disposable income of an economy and thus raises consumption spending. As a result national income rises not by the same amount (ΔC) but by a multiple of it, depending on the value of the multiplier. We now discuss this point in detail.

**The Multiplier Effect: **

It is important to note here that the fall in the level of income (- ΔY) is not the same as the amount of lumpsum tax (T) imposed by the government, but a multiple of it. The extent to which income falls due to the imposition of a lumpsum tax depends on the value of the tax multiplier which is expressed as – MPC/(1 – MPC) = -b/(1 – b) where b is the MPC.

**Leakages and Injections: **

A related point may be noted in this context. While government expenditure increases national income, taxes reduce it. The reason is that government expenditure is an injection into the circular flow of income and taxes are a leakage from the flow. So in a three-sector model, there are two injections (I and G) and two leakages (S and T). National income attains its equilibrium value when the sum of injections is equal to the sum of leakages.

**The Equilibrium Condition Expressed Algebraically: **

**The equilibrium condition of national income may be expressed clearly in algebraic terms: **

Y = AD = C + I + G … (1)

We put bars over I and G to indicate their fixed values (levels). Now the condition is written as

C = a + bY_{d} … (2)

where Y_{d} = Y – T = disposable income i.e., the difference between national income (Y) and lumpsum tax (T). We know that a is the autonomous consumption and b is MPC.

**Now the consumption function is rewritten as: **

C = a + b (Y – T)

= a + bY – bT

If we now substitute the value of C in income equation (1), we have

Y = a + bY – bT + I̅ + G̅

or, Y – bY = a – bT + I̅ + G̅

or, Y ( 1 – b) = a – bT + I̅ + G

or, Y = 1/1 – b (a – bT + I̅ + G̅) = 1/(1 – b) (A̅) …(3)

where A̅ is fixed autonomous expenditure. Its value is constant since all the terms inside the bracket on the right hand side of equation (3) are constants.

Thus, once again, if we multiply fixed autonomous expenditure by the autonomous expenditure multiplier, we arrive at the equilibrium value of national income in a three-sector model with government expenditure (G) and lumpsum tax (T). The following example clearly illustrates the method of income determination.

**Example: **

**Problem: **

**Suppose we have the following data about a simple economy: **

C = 10 + 0.75Y_{d}

I = 50

G = T = 20

where C is consumption, I is investment, Y_{d} is disposable income, G is government expenditure and T is lumpsum tax.

(a) Find out the equilibrium level of national income.

(b) What is the size of the multiplier?

**Solution: **

**Substituting the values of C, I and G in Y we have **

Y = a + bY_{d} + I + G

= 10 + 0.75 (Y – 20) + 50 + 20

= 10 + 0.75 Y- 15 + 50 + 20

or, Y – 0.75 Y = 65

or, Y (1 – 0.75) = 65

or, 0.25 Y = 65

or, Y = 65 × 4 = 260.

This is indeed the equilibrium value of Y in this model. The value of the autonomous expenditure multiplier in this case is = 1/(1 – MPC) = 1/(1 – b) = 1/(1 – 0.75) = 1/0.25 = 4

**Model 3: The Determination of National Income in a Three-Sector Economy with Taxes and Transfers: **

So long we have examined how national income is determined in a three-sector model with lumpsum taxes but without transfer payments. Now we extend the model to take into consideration the effect of transfer payments.

We have already introduced the concept of transfer payments in our discussion of national income. Recall that transfer payments are basically welfare transfers. These are payments made by the government to certain-sections of society at the cost of the taxpayers.

These are one-sided payments since those who receive such payments do not provide anything to the government in exchange. In a modern mixed economy such payments are made to the unemployed, needy families, to the poverty-stricken people, and the retired persons.

Transfer payments are exactly the opposite of taxes. Just as taxes reduce the disposable income of the people, transfer payments increase it. So people have more money to spend on consumption of goods and services.

An important point has to be noted in this context. Transfer payments may be financed without imposing taxes. In this case such payments are autonomous like government purchase of goods and services.

But if transfer payments are financed by imposing a lumpsum tax, then transfer payments become a part of government expenditure (G). This means that if the government imposes an additional lumpsum tax to finance transfer payments, then the additional tax becomes a part of lumpsum tax (T) and disposable income will fall. In this case the process of income determination will be same as the one presented in Model 2 before.

However, here we treat transfer payment as autonomous expenditure and see how the analysis of income determination is modified.

Since transfer payments increase the disposable income of the people, such payments will lead to an increase in C. The extent to which C increases due to transfer payment depends on MPC.

**So we can now present our new model in terms of the following modified equations: **

C = a + bY_{d} … (1)

and Y_{d}= Y – T + TR … (2)

where T is lumpsum tax and TR is transfer payments.

**Now by substituting the value of Y _{d} from (2) in (1), we get the following consumption function: **

C = a + b (Y – T + TR)

**Now the national income equation becomes: **

Y = C + 1 + G

= a + b (Y – T + TR) + I + G

= a + bY – bT + bTR + I + G

or, Y – bY = a – bT + bTR + I + G

or, Y (1 – b) = a – bT + bTR + I + G

or, Y = 1/(1 – b) (a – bT + bTR + I + G) … (3)

**Government Spending Multiplier****: **

It may apparently seem that government spending multiplier and transfer payment multiplier will have the same value since both are injections into the circular flow. The reason is that if G increases by Rs 20, and MPC = 0.75, Y will increase by Rs 80.

This means that an increase in government spending directly increases aggregate demand and by the full amount, i.e., aggregate demand rises exactly by the amount by which government spending increases. But if disposable income increases due to transfer payments, consumption demand does not increase by the full amount.

It increases by less than the amount of increase in disposable income since MPC is less than 1. This point may be explained clearly by referring to equation (3). Let us suppose government spending increases by AG. As a result income increases.

**The new level of income is written as: **

Y + ΔY = 1/1 – b (a – bT + bTR + I + G + ΔG) … (4)

**If we now subtract equation (3) from equation (4), we find out the absolute change in income which is written as: **

ΔY = 1/1 – b (ΔG)

or, ΔY/ΔG = 1/1 – b

This is indeed the government spending multiplier which is the same as investment multiplier in the two-sector model.

The reason is that according to Keynes, any increase in autonomous spending will have a multiplier effect. Here it suffices to note that there are two ways of interpreting this autonomous expenditure multiplier, or simply the multiplier.

1. The multiplier is the number by which an increase in the autonomous spending (in this case AG) has to be multiplied to find out the resulting increase in Y.

2. Alternatively, the multiplier is the ratio of two absolute changes, i.e., ΔY/ΔG

Now we consider transfer payments multiplier.

**The Transfer Payments Multiplier: **

Now suppose the government increases its transfer payments by ΔTR (rather than its spending on goods and services).

**So we get the following income equation: **

Y + ΔY = 1/1 – b (a – bT + bTR + bΔTR + G) … (5)

Now if we subtract equation (3) from equation (5), we get equation (6)

ΔY = 1/1 – b (bΔTR) … (6)

or ΔY/ΔTR = b/1 – b … (7)

**This is indeed the transfer payments multiplier. Now, if we take the difference between the two multipliers, i.e., the government spending multiplier and transfer payments multiplier, we have **

1/(1 – b) -b/(1 – b) = (1 – b)/(1 – b) = 1

Suppose ΔG = 20, b = 0.75.

So ΔY = 1/1 – 0.75 (20) = 4(20) = 80

If ΔTR = 20, b = 0.75, then

ΔY = 0.75/1 – 0.75 (20) = 3(20) = 60.

Thus we see that the transfer payments multiplier is 3, while government spending multiplier is 4. So we arrive at an interesting result— the transfer payments multiplier is one less than the government spending multiplier.

A simple example will make the point clear.

**Problem: **

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

C = 50 + 0.75 Y_{d }

I = 100

G = T = 50

TR = 40.

Where all the terms have their usual meaning.

(i) Find out the equilibrium value of Y.

(ii) Calculate the government spending multiplier and the transfer payments multiplier. What is the difference between the two multipliers?

(iii) If the economy seeks to achieve a target income of Rs 1,000 crore, find out the increase in government spending needed to reach the target.

**Solution: **

**(i) We know that **

Y = C + I + G

= a + bY_{d} + I + G

where Y_{d} = disposable income = Y – T + TR.

So the income equation has to be written as

Y = a + b(Y – T + TR) + I + G

**Substituting the values of the parameters and variables, we have: **

Y = 50 + 0.75 (F – 50 + 40) + 100 + 50

or, Y = 50 + 0.75F- 7.5 + 150

or, Y – 0.75Y= 200 – 7.5

or, Y (1 – 0.75) = 192.5

or, 0.25Y = 192.5

or, Y = 192.5 × 4 = 770

which is indeed the equilibrium value of Y.

(ii) In this case government spending multiplier (m_{G}) = 1/1 – b = 1/1 – 0.75 = 1/0.25 = 4.

Transfer payments multiplier (m_{TR}) b/1 – b = 0.75/1 – 0.75 = 3.

The difference between the two multipliers is equal to 1 as noted above [You can check this interesting result for any value of the MPC]

(iii) The equilibrium level of Y is 770. But the target level of Y is 1,000. So Y has to be increased by 230. We know that the increase in income (ΔY) is government spending multiplier (m_{c}) times the increase in government expenditure (ΔG).

**So we can write: **

ΔY = m_{G} × ΔG

or, ΔG = ΔY/m_{G}, where^{ }m_{G} is the government spending multiplier.

= 230/4 = 57.5

This means that government expenditure has to be increased by Rs 57.5 crore if the objective of the government, is to increase Y by Rs 230 crore. This target level of Y (i.e., Rs 1,000 crore) may be taken as the full employment level of Y.

**Proportional Income Tax and the Equilibrium Level of Income****: **

We now assume that instead of a lumpsum tax which collects the same amount of money (in absolute terms), the government imposes a proportional income tax. Under the proportional income tax, the government collects a fixed proportion say 20% of the income.

So if national income increases from say Rs 100 crore to Rs 110 crore within a year, the revenue of the government will increase from Rs 20 crore to Rs 22 crores. In this case consumption spending will not fall by the same amount at all levels of income, but by the same percentage of the level of income.

As a result the consumption function rotates downwards from C_{1} to C_{2}, with the intercept a (showing autonomous or income-independent consumption) remaining the same. The reason is that there is no parallel downward shift of the consumption function in this case.

As shown in Fig. 6A.6 the original consumption function is C, (before the imposition of the tax). The after-tax consumption function is C_{2} which is flatter than C_{1}. This means that the slope of the consumption function falls, i.e., the MPC falls.

In Fig. 6A.6 the vertical distance between that two consumption functions C_{1} and C_{2} shows the fall in consumption after the imposition of proportional income tax. The question now is how much does consumption fall in this case.

The answer is that, at each level of income, consumption falls by the tax revenue (T) times the MPC (b). Now the tax revenue is the tax rate (t = 20%, in our example) times national income i.e., T = tY. So consumption falls by AC = t. Y.b.

If Y = Rs. 100 crore, t = 20%, and b = 0.75, then AC = (Rs. 20 crore). (0.75) = Rs. 16.5 crore. And if Y = Rs. 80 crore, then consumption will fall by ΔC = (Rs. 16). (0.75) = Rs. 12 crore. This means that if Y increases (falls) the government collects more (less) revenue but consumption falls (rises).

It may also be noted that if the tax rate is increased from 20% to 30% consumption will fall at each level of income. If Y = Rs. 100 crore, t = 30% and b = 0.75, then ΔC = (Rs. 30 crore). (0.75) = Rs. 22.5 crore. The converse is also true.

However, the revenue of the government will rise from T= t.Y = 20% (Rs. 100 crore) = Rs. 20 crore to T – t’.Y = 30%. (Rs. 100 crore) = Rs. 30 crore. But if a lumpsum tax of Rs. 20 crore is imposed by the government, its tax revenue will remain the same, at all levels of income, although consumption demand will increase (fall) when income increases (falls).

In Fig. 6A.7 we show the effect of proportional income tax on the equilibrium level of national income. We see that the economy is initially at point E, where aggregate demand (C_{1} + I + G) is equal to aggregate supply and equilibrium national income is Y_{1}. Now as the consumption function rotates downward after the imposition of the proportional tax, the aggregate demand function also rotates downward (with point A = a + I + G as a pivot).

Now the aggregate demand curve C_{2} + I+ G intersects the aggregate supply curve at point E’. As a result national income falls from 0Y_{1} to 0Y_{2}. So our main prediction here is the following:

All other things remaining the same, the higher the rate of proportional income tax, the greater is the fall in consumption and the larger is the fall in the equilibrium level of income.^{ }

**Algebraic Treatment of Model 3: **

We have noted that under the proportional tax, the revenue of the government is expressed as T = i. Y, where t is the tax rate and Y is national income. However, in reality the government may impose both a lumpsum tax and a proportional tax on income.

**Thus the tax function becomes: **

T = T + tY … (1)

where T is the autonomous component of the tax function since it is not related to national income

Now the equilibrium condition of national income has to be modified since the consumption function now takes the following form

C = a + bY_{d}

= a + b (Y – T – tY + TR) … (2)

**Now if we substitute the value of C in the income equation we get: **

A simple example will be useful at this stage because we are gradually introducing some doses of realism in our analysis.

**Problem: **

**Suppose a closed economy is described by the following equations: **

C = 20 + 0.75 Y_{d }

I = 50

G = 100

TR = 60

t = 0.2.

(a) Write down the reduced form of the income equation.

(b) What is the numerical value of equilibrium income?

(c) What is the value of tax multiplier?

**Solution: **

(a) The reduced form equation is one which is expressed only in terms of exogenous variables and parameters and not in terms of any endogenous variable.

**The reduced form of equation of national income is: **

Y = 1/1 – b + bt (a + bTR + I + G)

Here t, TR, I and G are all exogenous variables and a and b are parameters. No endogenous variable has appeared on the right hand side of this equation.

**(b) If we substitute the values of the parameters and exogenous variables in the above reduced form equation, we get: **

Y = 1/1 – 0.75 + 0.75 × 0.2 (20 + 0.75 × 60 + 50 + 100)

= 1/1 – 0.60 (225)

= 1/0.40 (225)

= 225 × 100/4 = 562.5 (approx)

This is the equilibrium value of Y.

**(c) The tax multiplier now is: **

mt = 1/1 -b + bt = 1/1 – 0.75 + 0.75 × 0.2

1/1 – 0.60 = 1/0.40 = 2.5.