Components of aggregate demand in a closed economy are consumption(C), investment(I), and government expenditure(G).
So, AD = C + I + G
And consumption is a function of disposable income(Yd) which means
C = C1 + b(Yd)
Where disposable income Yd = (Y - T ). So, disposable income is the remaining income after paying the tax to the government.
So, the AD = C1 + b(Y - T) + I + G. For simplicity we can consider investment and government expenditure as constant.
At equilibrium Y = AD so, Y = C1 + b(Y - T) + I + G.
At period 0, when tax increases ΔT amount then it will decrease consumption at b* ΔT amount. So for any given level of Y, planned expenditure is now lower than income. So initially equilibrium Y decreases by b* ΔT amount. In period 1 this change in income will decrease the consumption at a rate of b(b* ΔT) amount. So Y also decreases by amount b2 * ΔT amount and so on.
So the total change in Y is = b* ΔT + b2 * ΔT + .....
= b* ΔT ( 1 + b + b2 + .....)
= b* ΔT [ 1/ (1- b)]
So, ΔY = b* ΔT [ 1/ (1- b)]
or, ΔY/ ΔT = b / (1-b)
So even if tax is not a direct component of aggregate demand it has a multiplier effect on income. If tax decreases then it will increase disposable income and so as the consumption. An increase in consumption has a positive effect on income in that period and that increased income has a positive effect on consumption in the next period and so as the income. This process will go on.
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