Income determination in a Three and Four Sector Economy, Fiscal and Foreign Trade Multiplier

Income determination in a three sector economy

The three sector model is formed by the addition of government sector in the two-sector model i.e. the three sector model consists of household, business, and government sectors. It is the extension of the two-sector model where two variables: government spending and income tax are added. The income determination in three sector economy is based on following assumptions:

• There are no transfer payments.
• There is only one form of tax i.e. lump sum income tax.
• The government expenditure is exogenously determined.

The mathematical expression of this at equilibrium can be written as

Y = C + I + G ----------- (i)

In three sector model, C = a + bY_d -------------- (ii)

where Y_d = Y – T (disposable income) and T is tax. Substitution this value of Y_d in equation (ii), we get

C = a + b(Y-T) -------------- (iii)

Now, from equation (i) and (iii), we get

Y = a + b(Y – T) + I + G

Y = a + bY – bT + I + G

Y(1 – b) = a – bT + I + G

Y = 1 / (1-b) * (a- bT + I + G) ---------------- (iv)

The equation (iv) gives the equilibrium level of national income in three sector economy.

Graphical representation

In the figure above, point E is the equilibrium point where AD curve (C + I + G) intersect the 45 degree line. The equilibrium level of income is OY.

Tax Multiplier

The tax creates a withdrawal effect in the circular flow of income and expenditure. So, it has a negative effect on equilibrium level of national income. Tax multiplier refers to the negative multiple effects of a change in tax on equilibrium level of national income.

In order to find the impact of change in tax, let’s introduce ΔT into the equilibrium equation. A change in tax by ΔT causes the change in national income by ΔY. Thus the equilibrium equation i.e. equation (iv) becomes:

Y + ΔY = 1 / (1-b) * (a- b (T + ΔT) + I + G) --------------------- (v)

Y + ΔY = 1 / (1-b) * (a- bT +b ΔT + I + G)

The effect of ΔT on national income i.e. ΔY can be obtained by subtracting equation (iv) from equation (v).

ΔY = 1 / (1-b) * (- bΔT)

ΔY = - bΔT / (1-b) ------------------- (vi)

Now, tax multiplier can be obtained by dividing equation (vi) by ΔT.

ΔY / ΔT = -b / (1 - b)

Hence, increasing tax by ΔT has a negative impact on the level of national income.

Government Expenditure Multiplier

In order to find the government expenditure multiplier, let us analyze the impact of the change in government expenditure on national income. The assumptions here are:

• The government spends its money on goods and services only.
• All other variables (I, G & T) remain constant.

Let the government increase its expenditure by ΔG. Increase in government expenditure causes increase in aggregate demand and it increases in national ΔY income by. Thus, the equilibrium level of national income from equation (iv) becomes;

Y + ΔY = 1 / (1 - b) * (a – bT + I + G + ΔG) -------------------- (vii)

Subtracting equation (iv) from equation (vii), we get

ΔY = 1/ (1-b) * ΔG

ΔY = ΔG / (1 – b) ---------------- (viii)

Government expenditure multiplier can be obtained by dividing equation (viii) by ΔG.

ΔY / ΔG = 1 / (1 –b)

Balanced Budget Multiplier

Balanced budget multiplier examines the balanced budget policy of the government on national income. During balanced budget policy, the government spends only as much as it collects through taxation. That is G = T and ΔG = ΔT. The effect of a balanced budget on national income is explained by a balanced budget multiplier or balanced-budget theorem which states that balanced budget multiplier is equal to one. The theorem can be proved as follow:

Let us incorporate ΔG and ΔT (ΔG = ΔT) and resulting change in income by ΔY, in the equation (iv)

Y + ΔY = 1 / (1-b) * (a – b(T + ΔT) + I + G + ΔG) ----------------------(ix)

Subtracting equation (iv) from equation (ix), we get

ΔY= 1/ (1-b) * ( -bΔT + ΔG)

Since, ΔT = ΔG. Hence,

ΔY= 1/ (1-b) * (- bΔG + ΔG)

ΔY (1 –b) = - bΔG + ΔG

ΔY (1 –b) = ΔG (1 – b)

ΔY = ΔG --------------------- (x)

The balanced budget multiplier can be obtained by dividing equation (x) by ΔG.

ΔY / ΔG = ΔG / ΔG = 1

Income determination in a Four sector Economy

Four sector economic model is formed by adding foreign sector i.e. four-sector economy consists of: household, business, government and foreign sectors. The four-sector economy is also called the open economy which involves the transaction of foreign sectors.

The aggregate demand function in open economy is

Y = C + I + G + (X – M) --------------------- (xi)

Where, C = a + b(Y – T)

M = + mY, is constant, autonomous import and m I marginal propensity to import.

Thus equilibrium level of national income can be obtained from equation (xi) as

Y = a + b(Y – T) + I + G + (X - = mY)

Y = 1 / (1 – b + m) * (a – bT + I + G + X - ) ----------------------- (xii)

The equation (xii) gives the equilibrium level of income in four sector economy.

Graphical representation

The graphical representation of income determination in the four-sector economy is presented in the graph above. The income determination in four sectors is based on following assumptions: (i) there is no transfer expenditure and (ii) X is autonomous.

The economy is at equilibrium on point E_2, without foreign sectors. When a foreign sector is included in the model (assuming M > X), the AD3 schedule shifts downward to AD₂ with equilibrium point shifting to E₁. The inclusion of foreign sector (with M > X) causes a reduction in national income. But when X > M, the AD₂ shifts upward to AD₃.

Foreign trade multiplier

In the equation (xii), let export increase by which causes and increase in national economy by ΔY. Other factors remaining constant, the equation (xii) becomes:

Y + ΔY = 1 / (1 – b + m) * (a – bT + I + G + X + ΔX - M)

This equation can also be written as

Y + ΔY = 1 / (1 – b + m) * (a – bT + I + G + X – M) + 1 / (1 – b + m) * ΔX ----------------- (xiii)

Subtracting equation (xii) from equation (xiii), we get

ΔY = 1 / (1 – b + m) * ΔX

Dividing above equation by ΔX

ΔY / ΔX = 1 / (1 – b + m) ---------------- (xiv)

The equation (xiv) is called the foreign trade multiplier.

References

Dwivedi, D. N. (2010). Macroeconomic theory and policy. New Delhi: Tata McGraw-Hill Education.