« Donald Tovey on Mozart | Main | Why I never believed James Frey »
The economics of relativity
An excerpt from a work in progress (by me) on the appropriate choice of discount rate:
Looking to physics, Einstein’s theory of relativity suggests there is no fact of the matter as to what time it is. Any measurement of time (when is “now”?) is relative to the perspective of an observer, and to the velocity of that observer, relative to the speed of light. In other words, if you are traveling very fast, you are moving into the future at an especially rapid rate. Yet it seems odd, to say the least, to discount the well-being of people as their velocity increases. Should we pay less attention to the safety of our spacecraft, and thus the welfare of our astronauts, the faster those vehicles go? 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 – because of positive discounting -- not give them enough fuel to make a safe landing? And if you decline to condemn them to death, how are they different from other “residents” in the distant future?
I am told that L. Ron Hubbard considered the related question of what a currency should be worth in a very fast spaceship. What about the term structure of interest rates? Uncovered interest parity? To be fully general, must we add a "speed" term in addition to the usual "risk premium"? Comments are open.
Posted by Tyler Cowen on January 27, 2006 at 06:42 AM in Economics | Permalink
TrackBack
TrackBack URL for this entry:
http://www.typepad.com/t/trackback/3576/4035118
Listed below are links to weblogs that reference The economics of relativity:
» The Second Coming from misplaiced
Hoo-wee, its been a while. I guess I fell through on most of my promises. But its been quite the few months.
Ill try to get back into the habit by referencing some interesting things Ive read recently. Ive spent the... [Read More]
Tracked on Jan 29, 2006 2:38:30 PM
» Relativistic Interest from A Voice in the Dark
Marginal Revolution asks some interesting questions about interest rates and currency values at near-light speeds.
Unfortunately, the answers to these questions are both obvious and markedly uninteresting. In both cases, you have two parties under cons... [Read More]
Tracked on May 10, 2006 7:37:43 PM
Comments
Here's a postulate worth considering, which strikes me as being intuitively obvious:
A utility function should transform as a scalar under the Lorentz transformation.
I don't have time to check the algebra rigourously, but I think this should imply that:
(Observed discount rate) * (Observed elapsed time) = (Proper discount rate) * (Proper elapsed time)
[Or r' t' = r t, for short]
Using the standard result from the Lorentz transformation that:
t' = t/sqrt(1-v^2/c^2)
we then get
r' = r * sqrt(1 - v^2/c^2)
[v and c, being the speed of the traveller and the speed of light, respectively]
So, in other words, as the astronaut's speed approaches the speed of light, his discount rate, as observed from earth, will approach zero.
Posted by: Andy Wood at Jan 27, 2006 7:43:58 AM
This discussion reminds me of a classic filk song called "Compound Interest": "Space is dark and space is deep/But so's the vault in which we keep/One thousand years of hazard pay/Where interest compounds every day/And accountants there will testify/That there's nought we see that we can't buy..."
I don't know if there's an actual story on which the filk is based, but the premise is simple: two astronauts are sent on the first interstellar mission. They have their pay put in a tax-free trust whose sole function is to research the fabled "hyperdrive" (an FTL drive: they are going on their mission the old-fashioned way). Once they discover the hyperdrive, of course, the trust is supposed to go fetch them, basically: meanwhile, since everyone has to license the hyperdrive from them...well, you get the point.
Of course, this overlooks a few trivialities, like whether the government would ever let this happen, or whether even if it did, the trustees could be counted on not to subvert the wishes of the ultimate absentee landlords...but it makes for an amusing thought experiment! :-)
Posted by: David Hecht at Jan 27, 2006 8:38:32 AM
Your comments are not quite correct. There is a fact of the matter as to what time is. It is another dimension. There is no fact of the matter as to when 'now' is. There is no fact of the matter as to whether events with a certain relationship happened before one another -- but there are events where it is true that no matter your speed you will agree that one happened before the other. (Two events which are timelike separate -- that is, when a line drawn between the two on a graph of spacetime could be the worldline of a particle -- are always seen as happening in the same order by any observers. Two events which are spacelike separated -- that is, when a line connecting the two points on a spacetime graph could not be the worldline of a particle or a photon -- are such that people can disagree on their temporal order based upon their frame of reference.) And the comment about "moving into the future faster" seems . . . very odd. The actual affect of inertial motion is that other clocks slow down, not speed up. The "fly away really fast and come back and everybody is really old" affect only happens because you have to accelerate really rapidly. If we lived in the Pac-Man universe (ie- some kind of 4-D donut such that you could go around the universe without accelerating), I'm not sure you'd have the affect then.
For a particularly good discussion of the topic, you should look at "General Relativity from A to B" by Geroch. (http://www.amazon.com/gp/product/0226288641/sr=1-3/qid=1138370133/ref=sr_1_3/103-6951743-2955027?%5Fencoding=UTF8)
Posted by: Joel Fried at Jan 27, 2006 9:00:19 AM
Assuming that funds can be "wired" at the speed of light from A to B, doesn't having money worth different amounts at different relative speeds permit huge arbitrage opportunities?
Posted by: Richard Bellamy at Jan 27, 2006 10:26:46 AM
When we are deciding what near-light-speed missions to fund, we're going to make decisions using our earthbound frame utility function. Interstellar trade is not an attractive prospect for a quarterly earnings-driven company, whatever unfathomable magnitude the centuries-later payoff might have. If a spaceship company did its business on up-front payments from passengers, one would indeed not expect them to worry particularly about safety at the end of the journey. Talk about agency costs!
The people going on the mission will discount the payoff from the journey differently from the spaceship company, of course, but they surely can't realistically predict the payout beforehand--for example, any destination would be perceived at a lag of many light years when they're making the decision whether to embark. Maybe aliens just blew up the planet.
I expect if the travelers plan to stay on uninhabited Planet X, they can use a discount rate equal to their current discount rate, from the time expected to spend in their reference frame (presumably not much time at all) and their expectation of the awesomeness of Planet X.
But if they're planning on coming back to Earth, many centuries later, they'll need to consider what their journey will net them on Earth--and it's hard to guess what goods Earth will pay for a thousand years from now. Then there's the hard-to-guess probability that Earth gets blown up before they get back.
Hey, maybe in the future humanity will have gone socialist and the returning travelers will be able to share in the plenty for no charge! More plausibly, they can work part-time in a museum and earn a decent living--assuming returning near-light-speed travelers are rare at any point in far-future time. But the market would be responsive to the demands of exotic 80th century consumers, not 22nd century relics, so I wouldn't bet on outrageously increasing expected utility by time-shifting to the future. (Even a mere 2.5 centuries is no sure thing--would Mozart enjoy being zapped to the present? Well, possibly. He could surely become a top hip-hop producer; I suspect he might find the lifestyle to his taste.)
Basically, I am not bullish on interstellar trade.
Posted by: gundryggia at Jan 27, 2006 10:36:19 AM
"Assuming that funds can be "wired" at the speed of light from A to B, doesn't having money worth different amounts at different relative speeds permit huge arbitrage opportunities?"
Maybe there's an arbitrage and maybe there isn't--I'd have to do the math--but arbitrage trades usually suck and I suspect this is no exception. Assume our interstellar travelers keep the same currency and on-ship monetary policy (unlikely, but hey). If you're traveling at an extremely close to light speed velocity (something like 0.999c), a speed-of-light signal from your interest-bearing Earth trust fund is not going to make up a ton of ground on you. So maybe investing in the parallel spaceship economy is not so bad after all.
Posted by: gundryggia at Jan 27, 2006 10:55:55 AM
Only the least risk averse among us would trust the stability of everything from the individuals to a system of governance while millions of miles away for thousands of years.
Those that effectively travel to the future have little to give to that future, especially considering the trends of technological progress.
Such a mission is selfish (without the pejorative).
Posted by: Ivan Kirigin at Jan 27, 2006 12:17:56 PM
I really do know that this work is bound for some sort of economics journal, and not, as my forthcoming comments might suggest, a publication on the physics of relativity. I'm not a physicist either -- but I am a know-it-all, so I can't resist a quibble or two:
1. "relative to the speed of light"
This is what started the whole relativity thing - the speed of light is not relative, and it would be in an intuitive universe. In fact, the speed of light is constant in every reference frame. Two observers will always measure light to be moving at the same speed, even if they are moving in opposite directions. What you want to say is "relative to their surroundings".
2. "In other words, if you are traveling very fast, you are moving into the future at an especially rapid rate"
Actually, it's the other way around. "if you are traveling very fast, are moving into the future at a reduced rate" I'd explain this if I could do it briefly and I remembered by relativistic physics course. I do remember reading that photons -- which travel at c, don't experience the passage of time at all.
Posted by: John Deszyck at Jan 27, 2006 12:28:03 PM
In Orson Scott Card's Ender novels, one of the main characters grows very rich and lives for thousands of years by spending most of his time in interstellar travel at relativistic speeds. It seemed to me like it was so easy to earn fantastic sums through compound interest that more people would adopt that tactic and interest rates would plummet so that it no longer worked. The cost of course is that of travelling into the future and thus being left behind by social change, having your friends and family age and die, etc. An interesting item to ponder.
Posted by: Noah Yetter at Jan 27, 2006 12:43:59 PM
Spaceships are highly inflationary 'countries'.
If I took $100 in currency with me for a 10-year space flight that, due to the warped space-time continuum, takes 100 'earth years' (I am making the numbers up for simplicity's sake) I would demand a HUGE return on my 'spaceship money'. Since I could invest that same $100 in government bonds with a real return of, say 2% per year for 100 years, my investment would have to compound at a much higher rate over the 10 'ship years' (21.9% per year, REAL return). Inflation compounds the problem. Say 'earth inflation' is 3%/year for 100 years. The nominal return on my government bond would therefore be 5%/year. In order to match that, the nominal return on 'spaceship bonds' would have to be 63% per year for 10 'spaceship years'. What would likely happen, is that paper currency would not be used, and real goods (such as gold) would be used instead. At least gold would keep up with inflation. (Although its long-term expected REAL return is close to 0%/year). Another alternative would be to use 100 year zero coupon 'earth bonds' as currency - the value would be adjusted upward every 'earth year'.
Posted by: Doug at Jan 27, 2006 1:23:17 PM
Re: Doug
Well gold would likely be a real loser, there's lots of the stuff floating around in asteroids and such. Uranium would probably be a better bet, but it seems to me that more than likely most of what a space-based economy would "need" is energy, and since there's no real way to carry more energy into the future in any reliable sense, that seems to be a real loser of a proposition. Especially if said economy harvests lots of power from interstellar bodies like our sun.
Posted by: UberIcarus at Jan 27, 2006 2:37:47 PM
Probably better than minerals would be quaint souvenirs of the present. I assume there would be quite a market in the 50th century for authentic 22nd century antiques in perfect condition. Impossible to know for sure what would be likely to appreciate most in value. Classic arts-and-crafts? Vintage technology?
Posted by: gundryggia at Jan 27, 2006 2:53:15 PM
It is odd that the comments today about relativistic astronauts are so negative compared to the posts on cryonauts the other day. Is it because we think the cryonauts will wake in 100 years and the relativistic astronauts will return in 10,000? is that rational? it is just as easy, indeed, easier for the astronauts to return in 100 years than in 10,000.
I'd rather fly to the future. The view of the universe will be better than the view inside your freezer.
Posted by: DK at Jan 27, 2006 3:09:19 PM
Of course it would ben sensible for us on the earth side not to bother to fund fuel for landing our astronauts safely millions of years in the future because....well the discounted value of a benefit millions of years hence is basically zero. But for the astronauts themselves the value is very great. So if we were building spaceships for ourselves it would be sensible to invest in landing precautions because that future is only a few days away!
Likewise when we considering disposing of nuclear waste we don't pay much attention to projected cancer rates greater than 100,000 years in the future but we do pay more attention to projected rates 100 years into the future!
Posted by: Boonton at Jan 27, 2006 3:38:30 PM
I don't quite understand your comment: "In other words, if you are traveling very fast, you are moving into the future at an especially rapid rate. Yet it seems odd, to say the least, to discount the well-being of people as their velocity increases." No one else here seems to have suggested a similar idea.
The twin paradox says that someone in a spaceship which returns to earth will be younger than his twin who stayed behind. I don't see how you would get from this that we would discount the well-being of the traveller.
Well, here is a sense in which it might be true. Suppose we could give one of them comfortable conditions and require the other to endure discomfort. Then it would make more sense to have the traveller be the one who is uncomfortable, since he will have to endure it for a shorter subjective period. In that sense we might be said to discount his well-being. Is that what you had in mind?
In other comments here, some have suggested that astronauts could be thought of as having low (temporal) discount rates, in that they care about the future much more than we stay-behinds do. This could be seen as an advantage to them - they could invest a few dollars and come back from their trip to find it worth millions. In this way, having a low discount rate is seen as advantageous.
Indeed, some philosophers argue that we should not discount the future at all, on rational and even moral grounds. You can make a pretty good case that your total life happiness will be optimal if you have a very low discount rate.
If you could get a brain implant or take a drug to lower your discount rate, would you do so? Change yourself so that future happiness was as valuable as happiness today? Saving money for the future would be easy, or putting in time to improve your skills for future benefit. It would arguably make your expected total lifetime happiness higher, for the typical person.
Posted by: Hal at Jan 27, 2006 7:01:06 PM
What about the uncertainty principle? If there is a particle, such as an electron, moving through space, one can characterize its motion by telling you where it is (its position) and what its velocity is (more precisely, its momentum), but not both at the same time. Thus that object is indeterminate. Philosophically (and Quantumly), one could say that observing an object's position rules out observing its velocity. You are effectively determining the object as one thing and not the other simply by oberving it. Apply this to the study of markets and economics. If a market takes into account all the actions of all those within the market's scope, and a person trys to use "knowledge of the market" to make decisions about say investing or predictions about market behavior, the market will adjust to his actions, thus changing from the state where it was when that person made that decision? So, aren't financial markets inherently indeterministic because the act of determining causes the market to adapt to that determination?
Posted by: James at Jan 27, 2006 9:34:55 PM
I think you should have a physicist check up on your paper, it is easy to go wrong when discussing relativity. Anyway, here are the comments I have as an enginering student who has read a little but not much special relativity.
Your statement that you travel towards the future with an increased speed when you go fast is wrong. If I see you pass by with great speed I will observe that time is going slower for you. In the same way, you will observe that time is going slower for me. (It is important not to confuse the effects of acceleration with the effects of speed. You and I both agree whom is accelerating of us but not who is moving at what speed.)
Also I am sceptical of the first comment. In particular what is meant by the observed discount rate should be specified.
Posted by: Johan Richter at Jan 28, 2006 9:19:16 AM
"Also I am sceptical of the first comment. In particular what is meant by the observed discount rate should be specified."
I thought it would have been obvious, but obviously not.
The discount rate is observed by observing the astronaut's behaviour. Thus, if, as observed in his own reference frame, he is indifferent between an extra dollar's worth of consumption now and an extra $1.05 worth of consumption in a year's time, then he has a proper discount rate of 5%/year.
However, as observed from earth, which is moving with a speed of, say, 0.866c relative to the astronaut, the astronaut is indifferent between an extra dollar's worth of consumption now and an extra $1.05 worth of consumption in two year's time.
Therefore, an earthbound observer would conclude that the astronaut has a discount rate of only 2.5%/year.
Does that make things clear?
Posted by: Andy Wood at Jan 28, 2006 10:06:30 AM
I am afraid you are not taking into account the relatavistic changes in the marginal propensity to consume (If the number of Angels dancing is (Ax) and the circumference of the pinhead is (Hubbardx)then the equation is (Ax)=(MHubbardx)squared.
Posted by: Kyle N at Jan 28, 2006 10:29:58 AM
I think Andy Wood nailed it. To the interstellar astronaut, the discount rate for Earthbound investments approaches zero. TIPS would be a perfect investment. But to the rest of us, the discount rate for onboard investments would approach infinity. Never lend money to an interstellar astronaut.
This theory puts quite a damper on interstellar commerce. If there was some good that was only obtainable through interstellar travel--dilithium crystals perhaps--even a thousand earth-year voyage to gather some would never be profitable. Using a previous poster's 2% rate of return each dollar invested would have to return 400 million dollars. A merchant 10 light-years away would have to wait 20 years (plus 30 days) to have his invoice paid; any interstellar good would be fantastically expensive given all the working capital tied up in interstellar wires.
This theory also supports the notion that extraterrestrials who visit are more likely to have hostile intent. Whether it's Lord Dark Helmet stealing the air from Planet Druidia, or Kanamit arrivng at the U.N. with his book, "To Serve Man," each could point to the usurious exchange rate with their home currency as justification for their actions.
Stuff like this is why I love this blog. Thanks.
Posted by: David at Jan 28, 2006 11:10:05 PM
To Andy Wood: Yes that makes things clear. Thank you for the explanation.
Posted by: Johan Richter at Jan 29, 2006 7:19:35 AM
smeris actie in bad ^^^ grietje groepssex thuis ^^^ live show mpeg ^^^ giovane fotti in casa ^^^ parfait etudiante pipes ^^^ epatant mediocre ^^^ defoncer trente trois ^^^ limite vingt quatre ^^^ piu carino segretaria strip ^^^ deciso giovane orale fotti ^^^ gulligast tonaring ejakulation ^^^ kryp tonaring alska ^^^ intrepido lesbiche gruppo ^^^ agente di polizia amore sulle scale ^^^ hanlig mams ^^^ perfekt mpeg ^^^ lattervekkende spillefilm ^^^ kortvarig 3kant ^^^ mathitis gamisi sto krevati ^^^ asiatis omadiko stin ethousa ^^^ papa film ^^^ maman sexy ^^^
Posted by: levan at Sep 6, 2006 2:36:26 AM
Posted by: at Oct 14, 2008 12:16:39 AM
