Question

Imagine that the government of a small community decides to give a total of \(\$ W\), distributed equally, to all its citizens. Suppose that each month each citizen saves a fraction \(p\) of his or her new wealth and spends the remaining \(1-p\) in the community. Assume no money leaves or enters the community, and all the spent money is redistributed throughout the community. a. If this cycle of saving and spending continues for many months, how much money is ultimately spent? Specifically, by what factor is the initial investment of \(\$ W\) increased (in terms of \(p\) )? Economists refer to this increase in the investment as the multiplier effect. b. Evaluate the limits \(p \rightarrow 0\) and \(p \rightarrow 1,\) and interpret their meanings.

Short answer

Question: Determine the multiplier effect in terms of the fraction p of new wealth saved by each citizen, and interpret the results when p approaches 0 and when p approaches 1. Answer: The multiplier effect (M) is given by \(M = \frac{1}{1 - p}\). As \(p \rightarrow 0\), the multiplier effect is 1, meaning that when no money is saved and all money is spent, the initial investment does not increase. As \(p \rightarrow 1\), the multiplier effect approaches infinity, indicating that when all money is saved, the initial investment will have no limit. However, this is an unrealistic situation since there is no money in the community to spend. The results suggest that a balance between saving and spending is important for the overall wealth of the community.

Step-by-step solution

Understand the problem and identify the variables involved

Here, we have the following variables: 1. \(W\): The total amount of money distributed by the government 2. \(p\): The fraction of the money saved by each citizen 3. \(1-p\): The fraction of the money spent by each citizen We need to find out what happens to the total amount of money spent after many months (i.e., the multiplier effect).

Sum of the series

Each month, the community members spend a fraction of their previous money and the redistributed sum looks like a geometric series. The total money spent after many months can be obtained by summing up this geometric series. Let's assume \(S\) to be the total money spent in the community after many months. The sum of this geometric series can be given by the formula: $$S = \frac{W (1 - p^n)}{1 - p},$$ where \(n \rightarrow \infty\).

Find the limit of the sum as n approaches infinity

To find the multiplier effect, we need to determine the limit of the sum as \(n \rightarrow \infty\). Taking the limit of the sum as \(n \rightarrow \infty\): $$S = W \cdot \frac{1 - (1 - p)^n}{1 - p},$$ and noting that as \(n \rightarrow \infty\), we have that \((1 - p)^n \rightarrow 0\): $$\lim_{n \rightarrow \infty} S = W \cdot \frac{1 - 0}{1 - p}.$$

Calculate the multiplier effect

Now, we have the total amount of money spent after many months, we can calculate the multiplier effect (M): $$M = \frac{S}{W} = \frac{1}{1 - p}$$ This equation shows how much the initial investment of \(\$ W\) is increased by the factor \(M\).

Evaluate the limits as p approaches 0 and 1

We need to evaluate the multiplier effect (M) as \(p \rightarrow 0\) and \(p \rightarrow 1\): As \(p \rightarrow 0\): $$M = \frac{1}{1 - 0} = 1$$ This means when no money is saved by the community members (spend all their money), the initial investment does not increase (it remains the same). As \(p \rightarrow 1\): $$M = \lim_{p \rightarrow 1} \frac{1}{1 - p} = \infty$$ This means when all the money is saved by the community members (spend nothing), the initial investment will have no limit (an infinitely increasing multiplier effect). However, this is an unreal situation since all the money is saved, and there is no money in the community to spend.

Interpret the results

We can interpret the results as follows: 1. As more money is saved (p approaches 1), the multiplier effect increases with no limit. 2. When no money is saved (p = 0), the multiplier effect is 1, which means the initial investment remains the same. These results suggest that the balance between saving and spending is important for the overall wealth of the community, as spending leads to an increase in the initial investment (multiplier effect).

Key concepts

Geometric Series

When citizens in a community receive money and spend a part of it, the sequence of spending forms a geometric series. A geometric series is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number. In the context of community wealth, this means each month, people will spend a fraction of their money, circulating it among the citizens.
This sequential spending can be described as a geometric series because the amount spent every month is a fixed fraction of what was available at the start of that month. The key feature of a geometric series is its common ratio; in this case, it is the fraction of wealth being spent, denoted as \(1 - p\).

To find out how much money will ultimately be spent in the community, the entire series is summed up. The sum of an infinite geometric series can be calculated if the common ratio is between -1 and 1, using the formula:

• \(S = \frac{a}{1 - r}\)

Here, \(a\) is the first term (initial money spent), and \(r\) is the common ratio. This is crucial for calculating how much initial money increases and plays a foundational role in deriving the multiplier effect.

Limit Evaluation

When discussing the community's financial dynamics over a long period, it's essential to evaluate limits to understand extreme scenarios. In mathematics, taking a limit helps describe the behavior of a function as its input approaches a certain value.
For the multiplier effect, we consider two extreme cases: as the saving fraction \(p\) approaches 0 and 1. This helps assess the community's economic health under different saving behaviors.

• As \(p \rightarrow 0\), citizens spend all their money. The multiplier effect, calculated by \(M = \frac{1}{1 - p}\), will be \(M = 1\). This indicates no increase in total investment since the community uses all its money immediately, thus not generating any wealth cycles.
• As \(p \rightarrow 1\), citizens save all their money. Theoretically, \(M\) approaches infinity because nobody spends money, leading to no actual exchange. The investment multiplier can't be applied practically in such a scenario because economic activity halts without spending.

The limits reveal that both extremes are not optimal for community wealth. Instead, a balanced approach of saving and spending leads to sustainable growth.

Community Wealth

Community wealth represents the total financial health and prosperity of a group. It encompasses how funds circulate and grow within a closed economic system. Here, the concept of the multiplier effect relates directly to community wealth, indicating how much larger the original investment means becomes through economic transactions.
The multiplier effect is essentially the heart of community wealth growth. It describes how money initially injected into the community leads to additional rounds of spending and income. This cycle continually boosts the local economy over time.

The critical factor in determining community wealth through spending cycles is the balance of saving versus spending. If more money is spent (lower \(p\)), people rapidly circulate money, leading to quicker but potentially limited growth. Conversely, if too much is saved (higher \(p\)), the circulation diminishes, stifling potential economic benefits.

Key insights for maintaining and enhancing community wealth include:

• Regularly balancing spending and saving to optimize the multiplier effect
• Encouraging policies that favor sustainable economic activities
• Understanding that wealth isn't just about money, but also resources and opportunities exchanged within the community

This holistic view ensures the community not only grows but thrives in the long term.