Let's add taxes and government spending into the multiplier formula!

First, we begin with:

Y = C + I + G (1)

Then we take our consumption function:

C = a + bY_d (2)

Only now we have to account for the fact that Y and Y_d are not equal

Y_d = Y - T (3)

because disposable income is aggregate income less taxes. Since taxes can be

determined by the tax rate times aggregate income:

T = tY (4)

Then:

Y_d = Y - tY (5)

Substituting equation (5) into the consumption function:

C = a + b(Y - tY) (6)

And substituting equation (6) into equation (1):

Y = a + b(Y - tY) + I + G (7)

We then solve for Y:

Y = a + bY - btY + I + G (8)

Y - bY + btY = a + I + G (9)

Y(1 - b + bt) = a + I + G (10)

_{Y = 1 (a + I + G)} (11)

^{1- b + bt}

So the multiplier with taxes is:

¹/_{(1 - b + bt)} (12)

And the multiplier times total injections (a + I + G) will give us the

equilibrium level of output and income.