Discounting the relativistic future

Tyler Cowen’s new book, “Stubborn Attachments,” contains a quote about why we should use low discount rates. He’s right about the conclusion of wanting low discount rates, but I think the example doesn’t quite make the point. (H/t for pointing out the quote goes to Robert Wiblin.)

…it seems odd, to say the least, to discount the well-being of people as their velocity increases. If, for instance, we sent off a spacecraft at near the velocity of light, the astronauts would return to earth, hardly aged, many millions of years hence. Should we pay less attention to the safety of our spaceship… the faster those vehicles go?

As I responded on Twitter, I’m fairly sure this is conceptually wrong because economists are used to thinking about time in Newtonian terms. If we use a proper spacetime metric, the problem, I argue, goes away — and so do some other things.

Let’s work through Tyler’s example. An astronaut leaves earth and immediately accelerates to 0.99c, crushing him into a pulp in a way that is mathematically conveniently for us. As economically rational agents, assuming his spaceship conveniently resurrects him ,should we care about his safety? [Note: The assumption of economically rational agents is obviously ridiculous, but it’s only slightly more of an exaggeration than the other parts of our story.]

So let’s look forward in time. When it’s a year later on earth, how much do we care about the astronaut? Using a typical discount rate, of say, 5%, we care about him 95% as much.

He, however, has had only about 0.02 years of time pass, and cares a bit more. But when he lets a year pass in his reference frame, he cares 95% as much about future him, but us earthbound people need to wait 50 years for that to happen, and we care about him 50 years from now about 8.7% as much as we did when he launched.

But where is he? About 10¹² kilometers away. Americans can’t be bothered to think about poor people in Africa, so why should they care about this guy who is about 100,000,000 times as far away? But Tyler Cowen agrees with Peter Singer in his moral objections to distance based discounting, so after we’ve spend the next 50 years avoiding existential risks and solving poverty in some economically efficient way, we need to decide how much value we should have initially placed on our astronaut.

Even if we don’t want to discount distance in space, unless using a discount rate of 0%, these post-Einstein sophisticates need to discount distance in space-time. Our astronaut travelling at 0.99c is about already a light year away, and using a handy-dandy space-time distance calculator, that means he’s just about 51 years away, and we think he’s worth about 8% of what he was when we launched him.

Let’s say he turns around, once again suddenly changing velocity, getting crushed to a pulp, and being resurrected by his ship. On the centennial of his launch, he comes back, 2 years older. Unless we’re doing something really complicated with our intergenerational discounting, we should initially have discounted this future by that same 5% yearly, and our future returnee is worth 0.76% of a person. That has nothing to do with space travel, it just means people don’t care about the future. [Note: Yes, high discount rates might be bad if you’re hoping to live to 100, because it means decision-makers now should trash the future for present gain. As if they aren’t already. But we’ll get back to that.]

Our prospective astronaut, however, has a higher self-valuation, and thinks this future is worth about 90% as much as the present. That makes sense — he’s only lived 2 years. [Note: If you’ve got a fast enough spaceship, you’re gonna be able to find a hell of an IRR for your investments. Just make sure you figure out that whole not getting crushed to death thing.] But different people always have different discount rates — we’re just saying that high-speed relativistic astronauts should hope that society cares about the long term future.

So we conclude that the people on earth care about events happening in a century very little, but people who travel really fast care quite a bit more. And we conclude that people who are really far away are absolutely worth less than people nearby, if only because they can’t get back here until the far future. But if we want to put someone on a spaceship, they better realize that they care about their safety a lot more than we do.

The conclusion is inescapable; we need to launch political decision makers away from earth as fast as we can possible make them go. We don’t even need to make sure the spaceship is safe , because in our reference frame, it’ll be a long time until it gets back. This way, they might start to care a little bit more about the far future. [Note: Or at least they’ll care a bit more about engineering standards.] The problem goes away, and so do the politicians.