Not sure if this is the crux of the issue, but if the lottery fails to pay off, you lose the interest on the $100 in addition to the $100 itself, and in order to get to $1 million, you’d have to make up for it by working more than 1.1 hour at the end of 5.5 years. Is there a way to change the numbers to make your argument work out even after taking this into account?
That seems like a possible candidate for the problem: Perhaps my round figure 5.5 years was a bit too round. Let us do the calculations. For simplicity I’m going to assume that interest works like this: On December 31st of each year, I put my earnings for the year into a savings account, which then grows by a factor 1.04. This is not realistic, of course, but it does seem that the answer ought not to depend on details of the model of interest. So, consider how much I have to work in the case where I buy the ticket and don’t win. At the end of the first year, I put $159900 in my account, which grows to $166296 by the magic of 4% interest. Then it grows thus:
End of 2nd year: 339347.84
End of 3rd year: 519321.75
End of 4th year: 706494.62
End of 5th year: 901154.41
and I need to work another 988.46 hours to reach the million. Now consider the case where I didn’t buy the ticket. Then I have 901276.07 at the end of the fifth year, and must work another 987.24 hours. The difference is 1.22, which as you suggested is larger than the 1.1.
A cross-check: If I buy the ticket, then there’s a 99.99% chance that I have to work those (5 years + 988.46 hours), which is 10987.36 expected hours. If I don’t, there’s a 100% chance that I have to work (5 years + 987.24) hours, or 10987.24 expected hours. So the ticket is costing me 0.12 hours, the same as we found above.
I repeated the calculation on the assumption that my expenses are $35k and $45k a year, and both are even worse for the lottery-buyer, which surprises me a bit. I don’t see an obvious way to change the numbers, then; although I could still fall back on the argument that leisure hours at age 30 are worth more than at age 35. It presumably does not take a whole lot of weighting to overcome the 0.12-hour disadvantage of the lottery. But my initial numbers were clearly flawed due to not calculating exactly enough what was needed at the end of the period. Thanks for clearing up my confusion.
Not sure if this is the crux of the issue, but if the lottery fails to pay off, you lose the interest on the $100 in addition to the $100 itself, and in order to get to $1 million, you’d have to make up for it by working more than 1.1 hour at the end of 5.5 years. Is there a way to change the numbers to make your argument work out even after taking this into account?
That seems like a possible candidate for the problem: Perhaps my round figure 5.5 years was a bit too round. Let us do the calculations. For simplicity I’m going to assume that interest works like this: On December 31st of each year, I put my earnings for the year into a savings account, which then grows by a factor 1.04. This is not realistic, of course, but it does seem that the answer ought not to depend on details of the model of interest. So, consider how much I have to work in the case where I buy the ticket and don’t win. At the end of the first year, I put $159900 in my account, which grows to $166296 by the magic of 4% interest. Then it grows thus:
End of 2nd year: 339347.84 End of 3rd year: 519321.75 End of 4th year: 706494.62 End of 5th year: 901154.41
and I need to work another 988.46 hours to reach the million. Now consider the case where I didn’t buy the ticket. Then I have 901276.07 at the end of the fifth year, and must work another 987.24 hours. The difference is 1.22, which as you suggested is larger than the 1.1.
A cross-check: If I buy the ticket, then there’s a 99.99% chance that I have to work those (5 years + 988.46 hours), which is 10987.36 expected hours. If I don’t, there’s a 100% chance that I have to work (5 years + 987.24) hours, or 10987.24 expected hours. So the ticket is costing me 0.12 hours, the same as we found above.
I repeated the calculation on the assumption that my expenses are $35k and $45k a year, and both are even worse for the lottery-buyer, which surprises me a bit. I don’t see an obvious way to change the numbers, then; although I could still fall back on the argument that leisure hours at age 30 are worth more than at age 35. It presumably does not take a whole lot of weighting to overcome the 0.12-hour disadvantage of the lottery. But my initial numbers were clearly flawed due to not calculating exactly enough what was needed at the end of the period. Thanks for clearing up my confusion.