Many are probably aware of how discounting works, but I’ll give a brief summary first:
Humans have time preferences, which is to say that we prefer to have money (or any item of utility) sooner rather than later, all else equal. One way of capturing this is by converting the present value of cash to an equivalent value in the future with a discount function. Studies show that humans tend to use a hyperbolic discount function. This steeply discounts gains in the near future and mildly discounts in the distant future, which leads to preference reversals. For example, committing to begin saving money a year from now seems tempting, but committing to begin right away seems daunting. Suppose you commit to the former. After a year passes, you’ll begin to regret the commitment since your scenario will now match the latter.
There are exactly two ways to avoid these preference reversals: not discounting at all (which requires a different treatment of time preference), or discounting exponentially. Exponential discounting gives you a constant conversion rate d. So that seems to settle it. Exponential discounting is the way to go, right?
I argue that it has a serious setback in that it doesn’t account for our mortality. To be fair, neither does hyperbolic discounting! But just to use exponential discounting as an example, I’ll show how you can be turned into a money pump: Let d be your annual discount rate, and suppose we know that you will live for at most t years. According to exponential discounting, you value one dollar now the same as d−t dollars in t years. I will generously offer you d−t+1 dollars in t years if you lend me one dollar now. Who cares that you’ll be dead by the time you would collect it?
You might protest that your remaining lifespan is a random variable, but the argument holds so long as it’s bounded. You can let t be large enough that the universe will have succumbed to heat death. You may also protest that the (zero) probability of living until the collection date ought to be involved in the calculation somehow. But the uncertainty of payment is one of the motivations for discounting in the first place! The whole point is that the discounting method fails to do that here.
Mortality requires the discount function to reach 0 in finite time. As no exponential function does this, any discounting method must either neglect mortality or allow preference reversals. My tentative conclusion is that discounting is perhaps not the “ideal” way to express time preferences, but I am open to suggestions.
Mortality and Discounting
Many are probably aware of how discounting works, but I’ll give a brief summary first:
Humans have time preferences, which is to say that we prefer to have money (or any item of utility) sooner rather than later, all else equal. One way of capturing this is by converting the present value of cash to an equivalent value in the future with a discount function. Studies show that humans tend to use a hyperbolic discount function. This steeply discounts gains in the near future and mildly discounts in the distant future, which leads to preference reversals. For example, committing to begin saving money a year from now seems tempting, but committing to begin right away seems daunting. Suppose you commit to the former. After a year passes, you’ll begin to regret the commitment since your scenario will now match the latter.
There are exactly two ways to avoid these preference reversals: not discounting at all (which requires a different treatment of time preference), or discounting exponentially. Exponential discounting gives you a constant conversion rate d. So that seems to settle it. Exponential discounting is the way to go, right?
I argue that it has a serious setback in that it doesn’t account for our mortality. To be fair, neither does hyperbolic discounting! But just to use exponential discounting as an example, I’ll show how you can be turned into a money pump: Let d be your annual discount rate, and suppose we know that you will live for at most t years. According to exponential discounting, you value one dollar now the same as d−t dollars in t years. I will generously offer you d−t+1 dollars in t years if you lend me one dollar now. Who cares that you’ll be dead by the time you would collect it?
You might protest that your remaining lifespan is a random variable, but the argument holds so long as it’s bounded. You can let t be large enough that the universe will have succumbed to heat death. You may also protest that the (zero) probability of living until the collection date ought to be involved in the calculation somehow. But the uncertainty of payment is one of the motivations for discounting in the first place! The whole point is that the discounting method fails to do that here.
Mortality requires the discount function to reach 0 in finite time. As no exponential function does this, any discounting method must either neglect mortality or allow preference reversals. My tentative conclusion is that discounting is perhaps not the “ideal” way to express time preferences, but I am open to suggestions.