There are N periods in the economy. Suppose that we have expense need of E 0 ,
Question:
There are N periods in the economy. Suppose that we have expense need of E0, E1,..., EN, where En is your expense on date = n. Let r be interest rate and Sn, n=0,...,N be the amount you deposit into your saving account on date = n. Initially, there is no money in the account. You finance the spending by withdrawing the same amount from the saving account. Therefore,
the account balance at the end of date 0 = S0 - E0
the account balance at the end of date 1 = (S0 - E0)(1+r)+ S1 - E1 , ..., and so forth.
In order to find a saving schedule, S0, S1,..., SN, that is just enough for financing all of your expense need E0, E1,..., EN, you want to set the account balance at the end of date N (ending balance) equal to zero. [If the balance is negative, you don't have enough money to pay all of your expenses; You are broke. If it is positive, you have saved too much and gotten extra money.]
Show that "the ending balance =0" is equivalent to "PV of saving = PV of expense."