Problem 128 - Precalculus with Limits

Question

In Exercises 127 - 130, use the following information. A tax rebate has been given to property owners by the state government with the anticipation that each property owner will spend approximately \( p\% \) of the rebate, and in turn each recipient of this amount will spend \( p\% \) of what they receive, and so on. Economists refer to this exchange of money and its circulation within the economy as the multiplier effect. The multiplier effect operates on the idea that the expenditures of one individual become the income of another individual. For the given tax rebate, find the total amount put back into the states economy, if this effect continues without end. Tax rebate \( \$250 \) \( p\% \) \( 80\% \)

Step-by-Step Solution

Answer

If this effect continues without end, the total amount of $1250 would be put back into the state economy.

Step 1: Convert the percentage into a decimal

First off, in order to use the percentage in calculations, convert it into decimal form. The percentage to be spent from each transaction is \(p\% = 80\%\). In decimal form, this is \(0.8\).

Step 2: Identify the first term and common ratio of the geometric series

Here, the first term of the geometric series, which is the initial amount or tax rebate, is $250. The common ratio, which is the fraction of the previous amount that is spent in each transaction, is the converted percentage i.e. \(0.8\). Thus, we have: \nFirst Term \(a = \$250\)\nCommon Ratio \(r = 0.8\)

Step 3: Calculate the total amount using the sum formula for geometric series

Because the question considers an infinite number of transactions, an infinite geometric series is formed. The sum \(S\) of an infinite geometric series can be calculated using the formula \[S = \frac{a}{1 - r}\] where \(\(a\) is the first term and \(r\) is the common ratio. Plugging in the values from Step 2, we have \[S = \frac{\$250}{1 - 0.8}\].

Step 4: Perform the computation

Performing the computation we get \[S = \frac{\$250}{0.2} = \$1250\]. So, if the multiplier effect continues indefinitely, a total of $1250 would be put back into the state's economy.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in a series that starts with 1 and has a common ratio of 2, the terms would be 1, 2, 4, 8, and so on. In the context of economic scenarios like the tax rebate case, it reflects a chain of spending where each recipient spends a portion of what they receive, creating subsequent terms in the series. The formula to find the sum of the first n terms in a geometric series is given by ewline ewline \[ S_n = a \left(\frac{1 - r^n}{1 - r}\right) \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula can be applied in various contexts to determine cumulative outcomes of processes that have a multiplicative nature.

Infinite Series

An infinite series is a summation of an infinite sequence of terms. The concept can seem abstract, but it has practical implications. Not all infinite series converge to a finite value, but those that do, such as when the absolute value of the common ratio is less than 1, are particularly important in economics and finance. The sum of an infinite geometric series is given by ewline ewline \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) the common ratio. The analysis of infinite series allows us to estimate the total impact of ongoing processes, like the continuous circulation of money in an economy due to the multiplier effect implied by a tax rebate.

Economic Impact of Tax Rebates

When governments issue tax rebates, they intend to stimulate economic activity. This is based on the assumption that recipients of the rebate will spend it, leading to an increase in consumption. This spending, in turn, becomes someone else's income, leading to more spending and so on. This chain reaction is called the multiplier effect, a critical concept in macroeconomics which illustrates how initial government spending can lead to a greater final impact on the total economic output. The size of the multiplier effect depends on the propensity to consume, or the percentage of additional income that individuals will spend rather than save.

Percentage to Decimal Conversion

Converting percentages to decimals is a fundamental skill in mathematics and economics, as it simplifies calculations. To convert a percentage to a decimal, divide the percentage value by 100. This moves the decimal point two places to the left. For instance, when given an 80% spending rate from the tax rebate example, it is converted to decimal as follows: ewline ewline \[ 80\% = \frac{80}{100} = 0.8 \] Understanding this conversion is critical when applying percentages in formulas, especially those related to financial transactions, interest rates, or economic models like the multiplier effect.