Multiplier effect Imagine that the government of a small

Problem 70E

Multiplier effect 
Imagine that the government of a small community decides to give a total of $W, distributed equally, to all of its citizens. Suppose that each month each citizen saves a fraction p of his or her new wealth and spends the remaining 1 − p in the community. Assume no money leaves or enters the community, and all of the spent money is redistributed throughout the community.

a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of $W increased? (Economists refer to this increase in the investment as the multiplier effect.)
b. Evaluate the limits p → 0 and p → 1 and interpret their meanings.

Solution:

Step 1

First month:
Money spent = W(1-p)
Money saved = Wp
Second month:
Money spent = Wp(1-p)
Money saved = Wp2
Third month:
Money spent = Wp2(1-p)
Money saved = Wp3

For nth month:
Money spent = Wpn(1-p)
Money saved = Wpn
Total money spent after n months = W(1-p)+Wp(1-p)+Wp2(1-p)+ …… Wpn(1-p)
                                                      = Wp(1-p)(1+p+p2+.....pn)
                                                      = Wp(1-p)

Hence the total money spent =