Question

) Suppose that, instead of a fixed level of taxes, we had an income tax so that T = T0 + T1Y Where was the income tax rate. Derive an expression for equilibrim income for this case in which the level of tax collections depends on income. What are the expressions for government expenditure multiplier, autonmous tax multiplier and balanced budget multiplier for this case of an income tax? Do the multipliers fall or rise? Explain

Answer 1

Answer

Aggregate Expenditure is given by:

AE = C + I + G + NX

where C = Co + c(Y - T) where Co = Autonomous Consumption

At Equilibrium Y(Income) = AE and If T is independent of Income then

Y = Co + c(Y - T) + I + G + NX

=> Y = (1/(1 - c))(Co - cT + I + G + NX)

where c = MPC

Government expenditure multiplier = $\Delta$ Y/$\Delta$G , Autonomous tax multiplier = $\Delta$ Y/$\Delta$T , Balance Budget multiplier = Government expenditure multiplier + Tax multiplier

Government expenditure multiplier = (1/(1 - c)) , Autonomous tax multiplier = -c/(1 - c) and , Balance Budget multiplier = Government expenditure multiplier + Tax multiplier = 1/(1 - c) -c/(1 - c) = 1

Now Of T depends on Income (Y) then Equilibrium Income is given by:

Y = Co + c(Y - (T₀ + T₁Y)) + I + G + NX

=> Y = (1/(1 - c + cT₁))(Co - cT₀+ I + G + NX)

Hence,

Government expenditure multiplier = (1/(1 - c + cT₁)) , Autonomous tax multiplier = -c/(1 - c + cT₁) and , Balance Budget multiplier = Government expenditure multiplier + Tax multiplier = 1/(1 - c + cT₁) -c/(1 - c + cT₁)

= (1 - c)/(1 - c + cT₁)

As both c amd T1 are greater than 0. Hence for All of them i.e. Govt Expenditure multiplier , Autonomous tax multiplier and Balance Budget Multiplier, Denominator increases and Numerator remains same. Hence All of them i.e. Govt Expenditure multiplier , Autonomous tax multiplier and Balance Budget Multiplier decreases