Gravity model of trade volumes
 

In the otherwise good planetary design sequence in GURPS Space, there is a formula suggested for estimating the volume of trade (T) between any pair of planets with given economic outputs (V1 and V2), separated by a distance D. The formula is

T = k.V1.V2/D

where k is a constant set by the GM to reflect the specifics of his setting.

Anyone who attempts to map any reasonably large number of worlds and apply this formula pairwise is in for a nasty surprise. The amount that a given world trades with another world at distance D drops off with D^-1, but the number of worlds existing at about D rises with D^2. The result is that the total amount of trade that a given planet does at range D (so long as D is not larger than the radius of the settled part of space) is proportional to D. Ie. there will be more total trade at long ranges than at short. In any reasonably large setting k will have to be tiny to prevent all planets from having trade volumes far larger than their economies. And that will mean negligible trade volumes with neighbours.

The economists who use these models usually fix this problem by raising D to an exponent that is larger than the dimensionality of the space they are working in. When discussing transport economics on a world surface, for instance, they square D, producing a formula that shows you exactly why the term 'gravity' model is appropriate:

T = k.V1.V2/D-squared

To achieve the same fix in three-diensional space you would need to use a higher exponent, such as

T = k.V1.V2/D-cubed.

If you wanted to prevent the integral from diverging as the trade space expands indefinitely, it would be necessary to use an even higher exponent.


I would like to add that this formula would work better if you were to replace D (in parsecs or whatever) with C (cost in $/ton). Because the cost of getting goods into orbit in the first place, or up into orbit and out into the jump zone, can produce significant effects. Taking this into account will save the system from producing absurdly high figures for interplanetary (as contrasted with interstellar) trade volumes.

I don't suggest going this far in designing a game setting, but I will just add that any transport economist worth his or her salt would use not C (the freight and loading cost) but G, the 'generalised cost', which would include import and export duties, the interest cost on the capital value of the cargo for the transit and loading time, and possibly wastage and depreciation costs on perishable cargoes.

As for economists who are worth more than their salt, I didn't use gravity models myself, as being too crude. You wouldn't want to use a full network flow analysis, but I would suggest that a logit model would produce better results than a gravity model in this case.

Re: Gravity model of trade volumes
 

And such a model looks like?

Re: Gravity model of trade volumes
 

I am also pretty sure that the simplifications of the rules ion damage, injury and fatigue in GURPS are less than tolerable from a scientific point of view. But they work as an approximation in a game. ;-)

Re: Gravity model of trade volumes
 

Of course, that makes sense, for obvious reasons - unless T is not given in absolutes, which I do not recall at the moment - will have to look it up. (The ISW trade system at least does not work with a trade volume given in absolute amounts, but as an abstract value for use with the trade system's tables.)

But changing to a more detailed model with lots of factors and more complicated formulae does, in my opinion, not help at all.

Re: Gravity model of trade volumes
 

I have a feeling that this method is only going to be useful to someone who has actually practiced economics, or who is willing to put in a fair bit of time fiddling with the variables (including their guesstimates of sigma(1)) until they get some decent results (and then there's still a good chance the non-economist's results won't actually be 'reasonable'). IOW, I suspect you need to know what you're doing. :)

If there's a point to this post of mine, I'd be the question "Is the gravity model, once fixed, 'good enough' for guesstimation of trade volumes by non-economist universe desiogners?"

Re: Gravity model of trade volumes
 

You know if you over-analyze ANYTHING this way the end result is "too many numbers , not enough FUN! "

- E.W. Charlton
(Less talk, more Bloodwine!)

Re: Gravity model of trade volumes
 

Depends what you find fun. There are people like me for whom the analysis in itself is fun.

Re: Gravity model of trade volumes
 

I would argue that such a point still exists for you whswhs. Would you play in a system where you needed to do balistics equations everytime someone fired a projectile weapon? ;-)

Re: Gravity model of trade volumes
 

There's a wide gulf between engaging in speculative analysis on these forums or while preparing for a game and during play.

I love the former; the latter makes my head explode. (See the Slam rules.) ;)

Re: Gravity model of trade volumes
 

No, it is in absolute numbers. However, I find it plausible that the "squared" or "cubed" part was simply forgotten. A rather simple erratum, so to speak.

Question to JFZ: Is that the case?

Re: Gravity model of trade volumes
 

I would bet it was based on the gravity trade model in GT:FT, which is using a 2d map where 1/D makes sense. It's unclear what the exponent should be on a 3d map, we don't have any real-world models to look at (the terrestrial case is generally close enough to 2d that it can be called 2d).

Re: Gravity model of trade volumes
 

Can't be, since the gravity trade model in GT:FT uses 1/D^2 (although it is concealed by the logarithms used in the actual tables).

It looks like a simple omission -- there should be another variable as an exponent on the distance factor.

When I asked Jim Maclean (the professional economist behind GT:FT, for those who don't know) he suggested that 1/D^3 was probably close enough.

Part of the problem with this subject is that the gravity trade model is an empirical observation, not a theoretical result. There are a number of competing explanations for why it works across such a wide range of applications -- and thus, at least as many possible ways to extend it to three dimensions.

The actual exponent for international trade is more like 1.9 than 2.0, by the way, but that's close enough to the square to make "gravity model" a reasonably accurate description.

Re: Gravity model of trade volumes
 

Good point. It might be simple reading on the gravity trade model, most of which seem to use an exponent fairly close to 1. I suspect the real problem is that remoteness is ignored and that most interstellar polities are rather limited in size.

Re: Gravity model of trade volumes
 

No, but I'm not talking about doing it in a game. I have in fact worked out interplanetary orbital trajectories for pleasure, and that's a lot like doing firearm ballistics. But doing it in a game would be a mistake, because the actual play of games is devoted to other sorts of fun.

Re: Gravity model of trade volumes
 

So... has anyone submitted this as an erratum yet?

Re: Gravity model of trade volumes
 

I'm over a thousand miles away from my notes, and I don't recall enough to know whether there's just a dropped exponent there, but I suspect there is. What we've really got here is the economic equivalent of Olbers' paradox - and just as the sky is dark at night, you really don't want your model predicting an infinite amount of trade coming in to every world in the galaxy.

If you lot can agree on a reasonable value for the exponent in a three-dimensional universe, and submit an erratum, I'd be happy to nod my head and say "yes" when the buck gets passed to my desk.

Re: Gravity model of trade volumes
 

The missing factor on trade models is the fact that the larger the network of potential trade partners, the lower the trade with any given partner. That's what would actually prevent Olber's Paradox in this case, though adjusting the exponent for range may not be a bad idea regardless.

Re: Gravity model of trade volumes
 

But still if you know better on a subject, it's hard to sometimes let the grossly unrealistic stuff just fly. Myself, I just picked up Business for Dummies, so this is all news to be.

Re: Gravity model of trade volumes
 

For 3D maps, that is. Maybe one could add "^(number of dimensions on the map)" to the formula in Space. That way, some Infinite Worlds in Space version with 3+1 dimenions would be covered, too, as would be Traveller-like settings with some kind of two-dimensional "jump space structure".

Re: Gravity model of trade volumes
 

A real man generates all the systems randomly, by hand. Then he dies of old age.

Re: Gravity model of trade volumes
 

I don't know if you know this, but you actually have to send an email to errata@sjgames.com.

Re: Gravity model of trade volumes
 

For some people it is. I know enough about physics to know that the idea of an FTL drive that doesn't play merry hell with cauality (people arriving before they leave etc.) makes no scientific sense, but can play in FTL sttings without blinking.

I can list at great length the utter absurdities not only physical, but social with a standard Super-Hero world, and yet supers is still one of my favorite genres.

Re: Gravity model of trade volumes
 

For most applications, that's fair enough. It's just that "number of dimensions of the map, normally 3 for a space map" would be even more accurate.

Re: Gravity model of trade volumes
 

"Number of dimensions".

Re: Gravity model of trade volumes
 

Hear, hear.

Re: Gravity model of trade volumes
 

Okay, I have written to errata@sjgames.com as follows:

Re: Gravity model of trade volumes
 

Well, not anyone. I for example, assumed that no numerically significant trade route was longer than about 15 light years in distance (with trade at longer ranges passing through economically significant planets and thus being subsumed into the trade of those planets) . Thus, it wasn't "any pair of planets". It was "any pair of planets between 15 light years". And, since there would be only about four economically significant planets at most in that range, the problem simply never arose.

Re: Gravity model of trade volumes
 

Holy thread necromancy.

Re: Gravity model of trade volumes
 

Seriously.

Re: Gravity model of trade volumes
 

Necromancy is traditionally considered unholy.

Re: Gravity model of trade volumes
 

Very interesting (if ancient) discussion. Are there any ideas for a non-gravity model, for cases like high surface-to-orbit but low orbit-to-orbit costs?

Re: Gravity model of trade volumes
 

And considering almost none of the new posts add anything to the thread I'll let it go back to its rest. Closed.