Billy Rich And Michael Million Are Two Wealthy Elderly
Billy Rich And Michael Million Are Two Very Wealthy Elderly Men Sinc
Billy Rich and Michael Million are two wealthy elderly men with no heirs, planning to give away all but $1 of their fortunes before they die. Billy Rich has $1,340,000 and gives away one-third of his remaining money each year. Michael Million has $980,000 and gives away one-quarter of his remaining money annually. The problem asks who will reach their final dollar first and how many years it will take each of them to deplete their fortunes.
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Introduction
The process of charitable giving and wealth depletion among the affluent has been a subject of interest for economists and social scientists. In particular, understanding the timeline for wealth depletion when individuals donate a fixed proportion of their remaining assets annually provides insights into the sustainability of such philanthropic strategies. This analysis compares two wealthy elderly men, Billy Rich and Michael Million, who each plan to give away nearly all of their fortunes—leaving just one dollar—through annual proportional donations. The key questions are: who will exhaust their wealth first and after how many years?
Background and Assumptions
Billy Rich starts with a net worth of $1,340,000 and donates exactly one-third of his remaining wealth each year. Michael Million begins with $980,000 and donates one-quarter of his remaining wealth annually. Notably, each donor uses a proportional donation approach, meaning the actual dollar amount given away varies annually according to their remaining wealth, which diminishes over time.
The core assumptions are:
- Each individual continues this process indefinitely until their remaining fortune is reduced to $1.
- The process of donation occurs yearly at the specified percentage.
- The remaining amount after each donation is used as the starting amount for the next cycle.
- They are both committed to reducing their wealth to exactly $1, although in practice, the process approaches this asymptotically.
Mathematical Model
The wealth depletion process can be modeled using the concept of geometric progression. For each person, the amount remaining after each year can be represented by recursive formulas:
- For Billy:
\[
W_{n+1} = W_{n} - \frac{1}{3} W_{n} = \frac{2}{3} W_{n}
\]
- For Michael:
\[
W_{n+1} = W_{n} - \frac{1}{4} W_{n} = \frac{3}{4} W_{n}
\]
Since the amount remaining each year is multiplied by a constant factor (two-thirds for Billy and three-quarters for Michael), the general form for the wealth after \( n \) years is:
\[
W_{n} = W_0 \times \left(\frac{2}{3}\right)^n
\]
for Billy, and
\[
W_{n} = W_0 \times \left(\frac{3}{4}\right)^n
\]
for Michael, where \( W_0 \) is the initial wealth.
The process terminates when remaining wealth drops to $1:
\[
W_{n} \leq 1
\]
This leads to solving for \( n \):
\[
W_0 \times \left(\frac{2}{3}\right)^n \leq 1
\]
for Billy, and
\[
W_0 \times \left(\frac{3}{4}\right)^n \leq 1
\]
for Michael.
Applying logarithms:
\[
n \geq \frac{\ln \left( \frac{1}{W_0} \right)}{\ln \left( \frac{2}{3} \right)} \quad \text{(for Billy)}
\]
and
\[
n \geq \frac{\ln \left( \frac{1}{W_0} \right)}{\ln \left( \frac{3}{4} \right)} \quad \text{(for Michael)}
\]
Substituting initial wealth values:
- For Billy:
\[
n_{Billy} \geq \frac{\ln (1/1,340,000)}{\ln (2/3)}
\]
- For Michael:
\[
n_{Michael} \geq \frac{\ln (1/980,000)}{\ln (3/4)}
\]
Note that \(\ln (2/3)\) and \(\ln (3/4)\) are negative, so the inequality direction remains the same.
Calculations
Calculating the number of years:
- For Billy:
\[
n_{Billy} \geq \frac{\ln (1/1,340,000)}{\ln (2/3)}
= \frac{-\ln (1,340,000)}{-0.4055}
= \frac{\ln (1,340,000)}{0.4055}
\]
\[
\ln (1,340,000) \approx 14.113
\]
\[
n_{Billy} \approx \frac{14.113}{0.4055} \approx 34.8 \text{ years}
\]
- For Michael:
\[
n_{Michael} \geq \frac{\ln (1/980,000)}{\ln (3/4)}
= \frac{-\ln (980,000)}{-0.2877}
= \frac{\ln (980,000)}{0.2877}
\]
\[
\ln (980,000) \approx 13.79
\]
\[
n_{Michael} \approx \frac{13.79}{0.2877} \approx 47.9 \text{ years}
\]
Based on these calculations, it would take approximately 34.8 years for Billy to deplete his wealth to $1, while it would take approximately 47.9 years for Michael.
Conclusion
The analysis indicates that Billy Rich will deplete his fortune to the final dollar before Michael Million, taking roughly 34.8 years compared to Michael’s 47.9 years. This is primarily due to the larger initial wealth and higher rate of annual donation (one-third versus one-quarter) which accelerates his wealth depletion.
This result highlights how proportional donation strategies significantly impact the timeline of wealth reduction, with higher donation percentages and larger initial amounts leading to faster depletion. Such insights are valuable for philanthropy planning and understanding the long-term effects of fixed-percentage giving strategies.