When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending D dollars. Suppose that each recipient of spent money spends 100c% and saves 100s% of the money that they receive. The values c and s are called the marginal propensity to consume and the marginal propensity to save and, of course, c + s = 1.

Note: The federal government uses this principle to justify deficit spending. Banks use this principle to justify lending a large percentage of the money that they receive in deposits.

(a)

Let

S_n

be the total spending that has been generated after n transactions. Find an equation for S_n in terms of D and c.S_n =

(b)

Show that

lim n S_n = kD,

where k =

1

s

. The number k is called the multiplier.

From the result of part (a), and since 0 < c < 1, the limit can be evaluated in terms of D and c as follows.

lim n S_n =

lim n (1 cⁿ) =

Since c + s = 1, write this expression in terms of D and s.

Since k =

1

s

, in terms of D and k this expression is equal to

. Thus,

lim n S_n = kD.

(c)

What is the multiplier if the marginal propensity to consume is 30%?

k =