Microeconomics,, 16th Canadian Edition

Dividing through by gives

and thus

30. The total expenditure over all rounds is the sum of an infinite
    series. If we let A stand for autonomous expenditure and z for the
    marginal propensity to spend, the change in autonomous
    expenditure is in the first round, in the second,
    in the third, and so on. This can be written as

If z is less than 1, the series in parentheses converges to

as n approaches infinity. The total change in expenditure is thus

. In the example in the box, ; therefore, the
change in total expenditure is

ΔYD=ΔC+ΔS

ΔYD

ΔΔYYD = +

D

ΔC

ΔYD

ΔS

ΔYD

1 =MPC+MPS

ΔA z⋅ΔA z^2

ΔA⋅( 1 +z+z^2 +...+zn)

1 /( 1

ΔA/( 1 −z) z=0.80

1 Δ−Az =Δ0.2A =5.ΔA