Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

Multiplier effect A manufacturing company that has just located in a small community will pay two million dollars per year in salaries. It has been estimated that \(60 \%\) of these salaries will be spent in the local area, and \(60 \%\) of the money spent will again change hands within the community. This process, called the multiplier effect, will be repeated ad infinitum. Find the total amount of local spending that will be generated by company salaries.

Step-by-Step Solution

Answer

The total local spending generated will be $5,000,000.

Step 1: Define the Initial Spending

The manufacturing company pays $2,000,000 in salaries annually. Initially, $2,000,000, or 100%, enters the local economy.

Step 2: Determine the First Cycle of Local Spending

Of the $2,000,000, 60% is spent locally, which is $1,200,000. This is the amount that stays and circulates within the local economy in the first cycle.

Step 3: Calculate Successive Spending Cycles

Every successive cycle continues to spend 60% of the amount from the previous cycle within the community. For the second cycle, 60% of the $1,200,000 from the first cycle is spent locally, which is $720,000.

Step 4: Establish the Infinite Geometric Series

Each cycle follows a geometric series, where the amount spent locally is 60% of the previous cycle's spending. The spending chain is represented by: \[ 2,000,000 + 1,200,000 + 720,000 + ext{...} \]

Step 5: Find the Sum of the Infinite Geometric Series

The series formed is a geometric series with the first term \(a = 2,000,000\) and a common ratio \(r = 0.6\). The formula for the sum \(S\) of an infinite geometric series is \[ S = \frac{a}{1 - r} \]Substituting the values, we get \[ S = \frac{2,000,000}{1 - 0.6} = \frac{2,000,000}{0.4} = 5,000,000 \]

Step 6: Interpret the Result

The total local spending that will be generated by the company salaries, taking into account the multiplier effect, is $5,000,000.

Key Concepts

Geometric Series

A geometric series is a sum of terms that have a constant ratio between successive terms. In our context, this ratio is the percentage of the company salaries that is repeatedly spent within the local economy. Initially, \(2,000,000 is introduced, and after that, each spending cycle represents 60% of the previous cycle's amount.
This forms a classic infinite geometric series, where the first term is the initial spending or \)2,000,000, and each subsequent term is 60% of the preceding term. Thus, this can be mathematically represented as:

• First term ( a ) is 2,000,000
• Common ratio ( r ) is 0.6

The formula to find the sum of an infinite geometric series is:\[ S = \frac{a}{1 - r} \]. This formula allows us to calculate the sum of all terms in the series, providing the total spending generated by the multiplier effect in this scenario.

Local Economy

The local economy refers to the financial system within a defined geographic area, involving local businesses, employees, and services. When a company introduces new wages, a portion of that money is expended within the local community. This company's expenditures create a chain reaction of new economic activity.
In our example, 60% of every salary dollar stays in the community, representing money spent on local goods, services, and possibly other businesses. Each layer of spending locally fortifies the financial health of the community. Local businesses benefit from increased sales and services, providing jobs and income to more people in the community.
A healthy local economy leads to:

• Job creation and maintenance
• Increased tax base for community services
• Strengthened neighborhood infrastructure and growth

Spending Cycles

Spending cycles occur each time money changes hands in the economy. Each cycle represents a new generation of economic activity triggered by the initial spending. This is essential in understanding how initial company salaries feed into further local financial interactions.
Here's how the cycles play out:

• First, 60% of $2,000,000, which is $1,200,000, is spent locally.
• In the next cycle, 60% of this $1,200,000 (or $720,000) circulates.
• This proceeds as money keeps circulating through successive cycles.

The long-term effect of these cycles is more comprehensive local spending than the original amount paid in salaries. This continuous cycle and eventual additional economic activity exemplify the multiplier effect, boosting the community's financial dynamics.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. In this context, the series represents all the spending going on due to the salaries of the company employees. As money continues to change hands through numerous cycles of spending, it creates an infinite series.The geometric series sum \[ S = \frac{a}{1 - r} \] delivers the solution to this infinite problem by providing a finite result. In our scenario:

• The first term (a) is \(2,000,000.
• Common ratio (r) is 0.6.

Thus, this results in a total local spending of \)5,000,000. This method shows despite the infinite nature of the cycles, the impact on the local economy is quantifiable and helps in understanding economic contributions effectively.