I use the phrase ‘clever argument’ deliberately: I have reached a conclusion that contradicts the usual wisdom around here, and want to check that I didn’t make an elementary mistake somewhere.
Consider a lottery ticket that costs $100 for a one-in-ten-thousand chance of winning a million dollars, expected value, $100. I can take this deal or leave it, and of course a realistic ticket actually costs 100+epsilon where epsilon covers the arranger’s profit, which is a bad deal.
But now consider this deal in terms of time. Suppose I’ve got a well-paid job in which it takes me an hour to earn that $100. Suppose further that I work 40 hours a week, 50 weeks a year, and that my living expenses are a modest $40k a year, making my yearly savings $160k. Then, with 4% interest on my $160k yearly, it would take me about 5.5 years to accumulate that million dollars, or 11000 hours. Also note that with these assumptions, once I have my million I don’t need to work any more.
It seems to me that, given the assumptions above, I could view the lottery deal as paying one hour of my life for a one-in-ten-thousand chance to win 11000 hours, expected value, 1.1 hours. (Note that leisure hours when young are probably worth more, since you’ll be in better health to enjoy it; but this is not necessary to the argument.)
Of course it is possible to adjust the numbers. For example, I could scrimp and save during my working years, and make my living expenses only 20k; in that case it would take me less than 5 years to accumulate the million, and the ticket goes back to being a bad deal. Alternatively, if I spend more than 40k a year, it takes longer to accumulate the million; in this case my standard of living drops when I retire to live off my 4% interest, but the lottery ticket becomes increasingly attractive in terms of hours of life.
I think, and I could be mistaken, that the reason this works is that the rate at which I’m indifferent between money and time changes with my stock of money. Since I work for 8 hours a day at $100 an hour, we can reasonably conclude that I’m *roughly* indifferent between an hour and $100 at my current wealth. But I’m obviously not indifferent to the point that I’d work 24 hours a day for $2400, nor 0 hours a day for $0. Further, once I have accumulated my million dollars (or more generally, enough money to live off the interest), my indifference level becomes much higher—you’d have to offer me way more money per hour to get me to work. Notice that in this case I’m postulating a very sharp dropoff, in that I’m happy to work for $100 an hour until the moment my savings account hits seven digits, and then I am no longer willing to work at all; it seems possible that the argument no longer works if you allow a more gradual change in indifference, but on the other hand “save to X dollars and then retire” doesn’t seem like a psychologically unrealistic plan either.
Am I making any obvious mistakes? Of course it may well be the case that the actual lottery tickets for sale in the real world do not match the wages-and-savings situations of real people in such a way that they have positive expected value; that’s an empirical question. But it does seem in-principle possible for an epsilon chance at one-over-epsilon dollars paid out right away to be of positive expected value after converting to expected hours of life, even though it’s neutral in expected dollars. Am I mistaken?
Edit: Wei Dai found the problem. Briefly, the 100 dollars added to my savings would cut more than 1.1 hours off the time I had to work at the end of the 5.5 years.
A clever argument for buying lottery tickets
I use the phrase ‘clever argument’ deliberately: I have reached a conclusion that contradicts the usual wisdom around here, and want to check that I didn’t make an elementary mistake somewhere.
Consider a lottery ticket that costs $100 for a one-in-ten-thousand chance of winning a million dollars, expected value, $100. I can take this deal or leave it, and of course a realistic ticket actually costs 100+epsilon where epsilon covers the arranger’s profit, which is a bad deal.
But now consider this deal in terms of time. Suppose I’ve got a well-paid job in which it takes me an hour to earn that $100. Suppose further that I work 40 hours a week, 50 weeks a year, and that my living expenses are a modest $40k a year, making my yearly savings $160k. Then, with 4% interest on my $160k yearly, it would take me about 5.5 years to accumulate that million dollars, or 11000 hours. Also note that with these assumptions, once I have my million I don’t need to work any more.
It seems to me that, given the assumptions above, I could view the lottery deal as paying one hour of my life for a one-in-ten-thousand chance to win 11000 hours, expected value, 1.1 hours. (Note that leisure hours when young are probably worth more, since you’ll be in better health to enjoy it; but this is not necessary to the argument.)
Of course it is possible to adjust the numbers. For example, I could scrimp and save during my working years, and make my living expenses only 20k; in that case it would take me less than 5 years to accumulate the million, and the ticket goes back to being a bad deal. Alternatively, if I spend more than 40k a year, it takes longer to accumulate the million; in this case my standard of living drops when I retire to live off my 4% interest, but the lottery ticket becomes increasingly attractive in terms of hours of life.
I think, and I could be mistaken, that the reason this works is that the rate at which I’m indifferent between money and time changes with my stock of money. Since I work for 8 hours a day at $100 an hour, we can reasonably conclude that I’m *roughly* indifferent between an hour and $100 at my current wealth. But I’m obviously not indifferent to the point that I’d work 24 hours a day for $2400, nor 0 hours a day for $0. Further, once I have accumulated my million dollars (or more generally, enough money to live off the interest), my indifference level becomes much higher—you’d have to offer me way more money per hour to get me to work. Notice that in this case I’m postulating a very sharp dropoff, in that I’m happy to work for $100 an hour until the moment my savings account hits seven digits, and then I am no longer willing to work at all; it seems possible that the argument no longer works if you allow a more gradual change in indifference, but on the other hand “save to X dollars and then retire” doesn’t seem like a psychologically unrealistic plan either.
Am I making any obvious mistakes? Of course it may well be the case that the actual lottery tickets for sale in the real world do not match the wages-and-savings situations of real people in such a way that they have positive expected value; that’s an empirical question. But it does seem in-principle possible for an epsilon chance at one-over-epsilon dollars paid out right away to be of positive expected value after converting to expected hours of life, even though it’s neutral in expected dollars. Am I mistaken?
Edit: Wei Dai found the problem. Briefly, the 100 dollars added to my savings would cut more than 1.1 hours off the time I had to work at the end of the 5.5 years.