Assume we have two planets, Earth and Trantor, separated by a large distance, the traversal of which necessitates travel at velocities comparable to the speed of light. Assume that Earth and Trantor are in the same inertial reference frame, i.e. they are not accelerating with respect to each other.
Assume that a spaceship traveling between the two planets travels at a constant .
Let’s say that from the perspective of an observer on one of the planets, the time it takes for a spaceship to make the trip is .
Then, the time it takes for a spaceship to make the trip from the perspective of someone on the spaceship is
(which is shorter than ). The factor on the right is a well-known result from relativity, derived mathematically from the Lorentz transformation, that gives us “time dilation” from traveling at relativistic speeds. Krugman demonstrates the above relation by representing the voyage in Minkowski space-time in a figure using imaginary axes.
be the price of Earth goods and the price of Trantorian goods on Earth, respectively.
be the price of Earth goods and the price of Trantorian goods on Trantor, respectively.
be the interest rates on Earth and Trantor, respectively.
The first question Krugman asks is what is the correct interest rate on a planet with regard to interstellar trade, since depending on whether you’re on a planet or on a spaceship engaging in trade, the passage of time is different. So putting ourselves on Trantor, we compare what happens to a Trantorian trader who can engage in investing in a Trantorian bond or engage in interstellar trade with Earth.
Interstellar trade with Earth would involve buying goods on Trantor, traveling to Earth, selling the Trantorian goods there and then buying Earth goods with those proceeds, and then traveling back to Trantor and selling the Earth goods there. Let be the cost of outfitting the spaceship. The cost of buying goods on Trantor is where we define as the quantity of Trantorian goods to be traded. Thus, the initial expenditure of the Trantorian trader while still on Trantor is
The trader then travels to Earth and then sells its (not his or her) goods at price . The trader now has money and will buy Earth goods with this money to bring back to Trantor. Earth goods cost so the quantity of Earth goods that can be bought is . Let us call this quantity of Earth goods to be brought over to Trantor . So we have
When the trader arrives back on Trantor, it will sell these Earth goods at price is , resulting in a revenue of
But the Trantorian trader also had the choice of investing its money in a Trantorian bond before leaving for Earth. The expenditure needed for the trading venture of invested into a Trantorian bond would:
after time , which is the time it takes for a round trip from the point of view of a trader who decided to stay on Trantor for that duration of time instead of go to Earth and back, become
after time , which is the time it takes for a round trip from the point of view of a trader who decided to actually make the round trip and thus be on a spaceship for that duration of time (maybe he puts some money into a Trantorian bond and some money into the trading venture that he personally goes on), become
So which perspective is right? The perspective from Trantor or from the spaceship? Krugman answers this by reminding us of the reasoning behind present value calculations, which is that of opportunity cost – any money that you choose not to possess today (by delay possessing it until the future for a larger amount) is money that you could have invested in a riskless bond today that would have also grown in value in the future. And so the way I’ve framed it here sort of answers the question in advance, which is that the correct value of the bond is from investing it risklessly with some riskless bond issuer, like the Trantorian government. The opportunity cost of spending money to outfit a trading venture to Earth is the lost opportunity of buying a Trantorian bond with that money and then receiving the proceeds by the time the trader would return from the venture, which according to the bond issuer, the Trantorian government, is time. Thus, we have Krugman’s First Fundamental Theorem of Interstellar Trade:
When trade takes place between two planets in a common inertial frame, the interest costs on goods in transit should be calculated using time measured by clocks in the common frame, and not in the frames of trading spacecraft.
Another way to think about it is that a bond earns interest because the bond issuer gets to possess cash now, invest it during some time, and afterwards will have more cash as a result. The location that this is occurring is at the bond issuer’s location, which is Trantor. The trader can always come back to Trantor and earn his bond interest and principal (assuming it’s a bond that automatically renews after maturing every time). All this occurs on Trantor, and so the bond’s time progression is according to Trantor’s time progression.
Now, for simplification Krugman assumes that perfect competition reduces the profits of Earth-trading to 0. In other words, we force the revenue earned from a trading venture to Earth that started with expenditure to equal the revenue earned from if we had invested that same expenditure into a Trantorian bond and waited the time that it takes to make a round trip to Earth and back:
Also for simplification, Krugman assumes away , the cost of outfitting a spaceship:
Krugman asks (or rather, asks what if someone asks) about a Trantorian trader who instead of going back to Trantor, decides to settle on Earth, thus making a one-way trading trip. Krugman writes that the Trantorian trader has two options: either buy Trantorian goods, bring them to Earth, and sell them there, or buy Trantorian bonds, bring them to Earth, and sell them there. While in a “realistic” situation, we would just need to assume that there is a liquid market for Trantorian bonds on Earth to make the second option possible, let us restrict our picture to just imagining one Trantorian in the whole universe who wants to migrate to Earth and then as many other traders as we want to imagine who all only intend to make round trips and live on their home planet and thus have no interest in the bonds of the other planet (thus, no Trantorian bond market on Earth). In this case, Krugman assumes that we just have at least one Earthling who wants to migrate to Trantor and thus may be interested in buying a Trantorian bond before making his or her one-way trip.
So pursuing option 2, what price could this Trantorian bond fetch on Earth? The Earthling has a choice of either buying and then bringing Earth goods to Trantor for sale or buying this Trantorian bond instead that can be redeemed when he or she arrives on Trantor. The value of a one “T$ (Trantorian Dollar) bond on arrival at Trantor will be in Trantorian Dollars. Thus, we want to know what value of Earth goods, when sold on Trantor, would net the same amount of Trantorian Dollars, for that is the Earth price of the Trantorian bond.
Suppose the Earthling buys amount of goods on Earth to bring to Trantor. The cost of that purchase is in Earth Dollars. Once on Trantor, those goods are sold for a total of Trantorian Dollars. Let this equal the amount that would have been earned if a Trantorian bond was redeemed instead. Thus, we have
The original Earth price for this transaction was
Thus, is the fair price in Earth Dollars for a Trantorian bond that can be redeemed in Trantor for .
If the Trantorian trader went with option 1, that means investing that one Trantorian Dollar in buying Trantorian goods to sell on Earth instead of buying a Trantorian bond and bringing it over. One Trantorian Dollar buys quantity of Trantorian goods, which on Earth will sell for a total of Earth Dollars. But from the “Relative goods prices” equation, we have that
Thus, one Trantorian Dollar invested in Trantorian goods and brought over to Earth can be sold for an amount of Earth Dollars that is the same as the fair price in Earth Dollars that a Trantorian bond will fetch on Earth. So as long as there is at least one Earthling who also wants to make a one-way trip to Trantor and is open to buying a Trantorian bond instead of buying and bringing over Earth goods to sell, the Trantorian trader is indifferent between bringing over Trantorian goods to sell or bringing over Trantorian bonds to sell – both will earn the same profit. This shows that the First Fundamental Theorem of Interstellar Trade holds as long as for one migrant going one way, we have another migrant going the other way. What we seem to have is that as long as there is an “effective” round trip (either made by one Trantorian trader, or one Trantorian and one Earthling migrant) and the assumption of no arbitrage (so on each leg of the trip, the traveler is indifferent between carrying goods or bonds), we have a relation between the Trantorian interest rate and the prices of goods, and that none of this challenges the statement that the Trantorian bond’s interest gains run according to Trantorian time (the First Fundamental Theorem). For Earth, we can construct the same scenarios except with the trip originating from Earth. Krugman writes that he proves the theorem in the presence of transportation costs in a paper from the future.
Krugman then asks whether interest rates on Earth and Trantor will be the same or not. The “Relative goods prices” equation (and its associated assumptions) are kept. Even though transportation costs, as in , the cost of outfitting a spaceship, was assumed away earlier, the “Relative goods prices” equation shows that there is an effective cost to transporting goods across stars that comes from the transportation time needed. (If you are on Earth and you desire Trantorian goods, even if it costs no money for the Trantorian trader to outfit its spaceship to make the journey to Earth and sell its goods to you, the profit that the Trantorian trader makes by the time it returns to Trantor after its round trip needs to match the profit it would have made from just investing in Trantorian bonds and sitting at home for the same duration of time. Thus, that gets built into the cost of Trantorian goods in planets that are interstellar distances away from Trantor.) Could interstellar distances cause differences in the planets’ interest rates?
We imagine a scenario where a Trantorian trader buys Trantorian goods, travels to Earth, sells the goods there, invests the proceeds into Earth bonds, spends time on Earth, redeems the bonds and buys Earth goods with that money, travels back to Trantor, and sells the goods there. This scenario involves the prices of goods and Earth’s interest rate; and there is always another option: invest in Trantorian bonds at the beginning, sit at home for time, and then redeem after that. Then by forcing a no arbitrage condition, we are then able to obtain a relation between the prices of goods, Earth’s interest rate, and Trantor’s interest rate.
In the first option, the trader buys (where is the quantity of Trantorian goods bought) worth of Trantorian goods, travels to Earth, sells these goods for , invests that in Earth bonds for time, earns (1+r)^k $ after that, buys quantity of Earth goods, travels back to Trantor, and then sells these goods for a total value of Trantorian Dollars. Forcing a no arbitrage condition, we require that this revenue equal the revenue from investing the initial expenditure in Trantorian bonds for the same amount of total time, giving us:
From earlier, we had
Putting the two equations together gives us:
Thus, we have the Second Fundamental Theorem of Interstellar Trade:
If sentient beings may hold assets on two planets in the same inertial frame, competition will equalize the interest rates on the two planets.
In his conclusion, Krugman writes
I have not even touched on the fascinating possibilities of interstellar finance, where spot and forward exchange markets will have to be supplemented by conditional present markets.