Casual Physics Enjoyer

Exploring time dilation and interest rates in Krugman's 'Theory of Interstellar Trade'

In university, I studied mathematics and theoretical physics. In most of my professional life, however, I've only worked in economics-related jobs. I am fascinated by the intersection of economics and physics. There are few better examples of this intersection than in 'The Theory of Interstellar Trade' (1978) by Paul Krugman¹. The paper explores how someone might think about trade and interest rates, provided they had access to spaceships that can travel near the speed of light.

In classical physics, time is treated as a fixed, ticking clock. However, Einstein's theory of special relativity successfully predicts that this is not how nature works and that time stretches depending on the observer. Krugman's paper explores how this affects the time value of money and, therefore, the opportunity costs of trading.

This post aims to explain some of the critical concepts of special relativity and flesh out some introductory of the content of Krugman's paper (since it can be hard to follow for a non-specialist reader).

Special Relativity and Inertial Reference Frames

Krugman's paper is centred around how the time value of money changes when traders experience a phenomenon called time dilation. Time dilation is a weird natural effect where two different observers experience time differently.

Here is an example of time dilation in action. Suppose I am standing still, and I wait for an hour. Then, a train whizzes past me near the speed of light, at a speed v. In the time it takes for an hour to elapse for me, a person on the train has only experienced a time of slightly less than an hour. Whilst this effect holds at all speeds, it only matters when the train travels insanely fast. We have a quantitative measure for this effect, shown in the equation below.

From this expression, if the train were travelling at a third of the speed of light, the person on the train would only experience a time of 56 minutes, whilst I experienced an hour. The difference in time is a bizarre result that is predicted by Einstein's theory of special relativity. To understand why this happens, we need to know a little bit about Einstein's theory of special relativity.

Modern physics cares a lot about the perspective of a given observer since your perspective dictates how you experience physical laws. In physics, this has a technical term: a reference frame. The reference frame is a loose word to refer to how a specific observer experiences or notates physical laws and is specific to the observer itself. For example, in a particular reference frame, an observer might have a different system of coordinates that they use to label where objects are, compared to another observer.

Consider the example of a moving train. Suppose I am waiting on the side of the tracks, and I watch the train whizz by. From my perspective, it looks like the train is moving, and I am still. However, from the perspective of a person on the train, it feels like they are still, whilst I am moving. Thus, there are an infinite amount of different reference frames to view a physical system. However, there is also no guarantee that physical laws are the same in different reference frames. If this is the case, then why is it possible to do physics at all?

There is a special type of reference frame called an inertial reference frame, and it is the only reasonable frame that we can do physics. An inertial reference frame is a frame that is 'stable'. By stable, it means that any object moving at a constant speed remains at constant velocity unless acted on by an external force. The principle of relativity states that physical laws stay the same in any inertial reference frame. To the best of our observations, the principle of relativity is empirically true.

If someone experiences a set of physical laws in an inertial reference frame on Earth, these physical laws would be the same as someone in an inertial reference frame far away on Neptune, another Galaxy, or my next-door neighbour. It doesn't matter where.

Einstein's theory of special relativity states that in an inertial frame, the speed of light is constant, denoted as c = 3 x 1⁰⁸ metres per second. The principle of relativity implies that in all reference frames, the speed of the light remains the same. Thus, for someone on the train or standing still on the platform, the speed of light is the same. However, constant speed is not necessarily true for moving objects that are not light. For example, consider the speed of someone cycling on a bike. The cyclist's measured speed is different for an observer on a train than an observer standing on a platform.

Explaining Time Dilation

There are remarkable consequences of the fact that the speed of light is constant in all reference frames. One of these consequences is that time is no longer a fixed ticking clock — it bends and dilates depending on your relative speed to other observers. I'll explain why this happens in this section.

Let's say we are standing still on a platform and observing a moving object at a constant speed. Let's also call the frame of reference 'OUR FRAME'. At some time t1, we observe that it is in space with coordinates (x1, y1, z1). Then, a short time after that, at time t2, let's say that it's at the point (x2, y2, z2). From this, we can construct a quantity called the 'invariant interval', which is the quantity shown below.

Now, suppose we were in a different inertial frame called 'OBJECT FRAME', observing the same process but with different labels. We would have different time labels t1' and t2' in this different reference frame instead of t1 and t2. The same would apply to our coordinates. In this new reference frame, we also have another invariant interval, given by the following expression.

With reasonably simple assumptions, Einstein showed that these two invariant intervals should be the same, no matter what the inertial frames OUR FRAME and OBJECT FRAME are. This means that we can equate them. In particular, if the reference frame OBJECT FRAME is chosen to move with the object, then it looks like the object isn't moving at all from a spatial perspective. If that's the case, then x1' — x2' is just zero, along with the other spatial coordinates. So, we have a sequence of equations that looks like the following

In the last line, I use the letter d to denote a small change in the quantities. From the previous equation, if we divide both sides by c² and then take the square root of the equation, then we get the time dilation formula that I mentioned at the beginning of this article.

The term in the square root is the velocity v of the moving object itself. Hence, this equation explicitly shows that the time experienced by an observer moving at high-speed relative to another observer is slower.

What is a simple model for interplanetary trade?

Let's now apply this to some economic theory. In Krugman's paper, he looks at trade economics between two planets called Earth and Trantor. To keep things relatable, I will reframe this to a problem between Earth and Mars.

The setup of the problem is relatively simple. Suppose we have two sets of goods called Earth goods and Martian goods. Let's also assume that these goods have fixed prices depending on the planet they are sold in, and Earth and Mars have a single currency (the US dollar).

In economics, we typically say there is a 'risk-free' rate where economic agents can invest some cash and get more money in return. The risk-free cash generally is in the form of government bonds that people can buy. A government bond is a debt instrument that you can purchase and receive yearly coupon payments at some interest rate r until the bond expires. We assume that the government is liquid enough payout these cash payments annually without pretty much no risk of default, so the interest rate r is typically considered the risk-free rate.

In particular, if you park K dollars in government bond at the risk-free interest rate r, then due to compound interest, you can expect to have multiplied your cash by a small factor after leaving it there for n years.

Because of this, there is an opportunity cost to time to doing business! If you were a businessman or businesswoman, and your rate of return of a new endeavour was less than r, then you would be better off just parking your cash in government bonds.

So now, let's examine the profitability of trading goods back and forth between Earth and Mars. Suppose we are on Earth and want to start up the following business model as intergalactic merchants:

1. Purchase some Earth goods on Earth

2. Travel to Mars on your spaceship (which will cost you)

3. Sell those Earth goods on Mars at Mars prices, then buy some Mars goods

4. Travel back to Earth on your spaceship (which will cost you even more)

5. Sell the Mars goods at Earth prices

What is the profitability of this business? How much revenue will you make? Well, let's go through the steps one by one. The initial cost of doing steps 1. and 2. are the shipping costs, as well as buying up the goods you need to sell Earth goods you need to sell on Mars. So, the cost of doing this will be

The first term on the right-hand side is just the cost of shipping, and the second term on the right-hand side is the price of the Earth goods we have to buy on Earth, multiplied by the total quantity we're going to buy. Now, suppose we did make it to Mars, and we have a quantity q of Earth goods. If we sold all of them on Mars at Mars prices, we would get some cash. With this cash, we would then buy some Mars goods. So, we would have that the total Mars goods we would obtain at step 3. is

Once we come back to Earth, we would then sell Mars goods at Earth prices. This would mean that the total revenue we earn from the whole round trip would be

Where does time dilation play a role?

Time dilation plays a role in deciding when the trade becomes profitable. Why is this the case? Well, to earn this revenue, we initially had to pay up the capital costs, which could have been invested in government bonds in the time it takes to make the trip.

This is where it gets weird.

Suppose that from the perspective of Earth, it takes N years to make the trip to Mars. Then, from our perspective, the merchant takes 2N years to go there, make his sales, then come back. However, from the perspective of the merchant, due to time dilation, it takes less time!

If the government stayed on Earth and measured time in that reference frame, he could have just invested his capital costs in bonds instead of making the trade. The condition for the transaction to be profitable would depend on the revenue being bigger than the opportunity forgone of investing cash in the government for 2N years. So, on the one hand, we require that

The right-hand side of the equation is what the merchant could have earned had he just invested his startup costs into bonds.

However, if the government officials or Central Bank, accompanied the merchant on the journey instead, they too would experience time dilation. As a result, they would pay him returns compounded over slightly less time! So, the condition for trading to be profitable is relaxed somewhat since the opportunity cost becomes cheaper.

The factor next to 2N is added there due to time dilation. As you can see, there is a discount factor if time was measured from the merchant's perspective. The discount factor is increased the faster you go! In particular, this means that government could promote intergalactic trade by measuring the time elapsed from the merchant's perspective, not from Earth's perspective.

In Krugman's paper, there is an assertion that the former equation is the correct one to go by, where he uses an arbitrage argument. I'll explain this more in a later post.

Wrap up

I hope this was a quick crash course in some relativistic physics and trade economics. There is more to the paper, and I hope to write about it in another article. Stay tuned!

References

[1] Paul Krugman, 2010. "The Theory Of Interstellar Trade," Economic Inquiry, Western Economic Association International, vol. 48(4), pages 1119–1123, October.