Tax Rebates and the Multiplier Effect, Computing a Lifetime Salary, Allowance (Geometric Sequences)

📌 Example 1: Tax Rebates and the Multiplier Effect
A tax rebate that returns a certain amount of money to taxpayers can have a total effect on the economy that is many times this amount. In economics, this phenomenon is called the multiplier effect. Suppose, for example, that the government reduces taxes so that each consumer has $2000 more income. The government assumes that each person will spend 70% of this (= $1400). The individuals and businesses receiving this $1400 in turn spend 70% of it (= $980), creating extra income for other people to spend, and so on. Determine the total amount spent on consumer goods from the initial $2000 tax rebate.


✍ Solution:
The total amount spent is given by the infinite geometric series
=1400+980+686 +⋯.

The first term is 1400: a1=1400. The common ratio is 70%, or 0.7: r=0.7. Because r=0.7, the condition that |r|<1 is met. Thus, we can use our formula to find the sum. Therefore,

This means that the total amount spent on consumer goods from the initial $2000 rebate is approximately $4667.

📌 Question 1: Computing a Lifetime Salary
A union contract specifies that each worker will receive a 5% pay increase each year for the next 30 years. One worker is paid $20,000 the first year. What is this person’s total lifetime salary over a 30-year period?
✍ Solution:
The salary for the first year is $20,000. With a 5% raise, the second—year salary is computed as follows:

Salary for year 2=20,000+20,000(0.05)=20,000(1+0.05)=20,000(1.05).

Each year, the salary is 1.05 times what it was in the previous year. Thus, the salary for year 3 is 1.05 times 20,000(1.05), or 20,000(1.05)2. The salaries for the first five years are given in the table.

Yearly Salaries

The numbers in the bottom row form a geometric sequence with a1=20,000 and r=1.05. To find the total salary over 30 years, we use the formula for the sum of the first n terms of a geometric sequence, with n=30.

Use a calculator.

The total salary over the 30-year period is approximately $1,328,777..

📌 Example 2 (Allowance). Danielle’s parents have offered her two different options to earn her allowance for a 9-week period over the summer. She can either get paid $30 each week or $1 the first week, $2 for the second week, $4 for the third week, and so on.
a. Does the second option form a geometric sequence? Explain.
b. Which option should Danielle choose? Explain.
✍ Solution:
a. Calculate the common ratio.

There is a common ratio of 2. So, the second option does form a geometric sequence.
b. Calculate how much Danielle would earn with each option.
Option 1

Option 2
1+2+4+8+16+32+64+128+256 or 511

In nine weeks, Danielle would earn $270 with the first option and $511 with the second option. So, she should choose the second option.