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Example: The Multiplier Effect as an Infinite Geometric Series
A government gives a $1,000 tax rebate to each household. Statistics indicate that each recipient will spend 80% of the money on goods and services, and each subsequent recipient of that spending will also spend 80% of what they receive. What is the total effect of the rebate on the economy?
Each round of spending forms a geometric pattern:
This is an infinite geometric series with and . Because , the series converges and its sum can be computed:
The total economic impact of the $1,000 rebate is $5,000 per household. This result — known as the multiplier effect — shows that when each recipient re-spends a fixed percentage of what they receive, the cumulative impact on the economy is a multiple of the original amount. The infinite geometric series formula provides a concise way to calculate that total.
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Example: The Multiplier Effect as an Infinite Geometric Series
In financial planning and engineering, we often calculate the total value of a process that decreases by a constant percentage over time. Match each component of the infinite geometric series model to its correct description.
In a corporate financial model, a manager is estimating the total long-term value of a service contract that generates decreasing annual returns. If the initial revenue in the first year is and each subsequent year's revenue is a fraction of the previous year, which of the following formulas correctly represents the total infinite sum , assuming that ?
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Formula for the Sum of an Infinite Geometric Series
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A local economic development board awards a 5000 grant to a small business incubator to stimulate the community. The board uses the infinite geometric series formula S = rac{a_1}{1 - r} to estimate the 'multiplier effect'—the total cumulative economic activity generated as that grant money is spent and re-spent throughout the city. In this specific model, what does the variable represent?
A corporate financial analyst is using an infinite geometric series to estimate the total economic impact of a 1,000 dollar employee bonus program. Match each component of the economic model with its correct definition.
Mathematical Condition for the Multiplier Effect
A regional healthcare system provides a stipend to its nurses to support local businesses. Arrange the following steps in the correct order to represent how this 'multiplier effect' is modeled using an infinite geometric series.
A corporate financial planning team is modeling the cumulative economic impact of a new local business grant. In their infinite geometric series model, the initial grant amount is the first term (). The fixed percentage of the money that each recipient subsequently re-spends in the local economy represents the common ________ of the series.