Gravity model of trade volumes

[Agemegos]

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.

[Anders]

Originally Posted by Agemegos

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.

And such a model looks like?

[DrTemp]

Originally Posted by Agemegos

[...]
As for economists who are worth more than their salt, I didn't use gravity models myself, as being too crude.
[...]

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. ;-)

[DrTemp]

Originally Posted by Agemegos

Just so. And a gravity model for trade volumes will work as an approximation in a game so long as you put an exponent on the distance (or better, cost) that is higher than 2. [...]

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.

[Rupert]

Originally Posted by Agemegos

Actually calculating all the sigmas for an extensive setting would be a task for a computer, and one that you could only accomplish if all the planets were enumerated, which is neither practical nor desirable for a game setting. The practical alternative would be to guesstimate sigma(1) from planet 1's position in the general shape of settled space and the cost characteristics of the stardrive in use.

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?"

[Qoltar]

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!)

[whswhs]

Originally Posted by Qoltar

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

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

[zogo]

Originally Posted by whswhs

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

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? ;-)

[Brandy]

Originally Posted by zogo

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? ;-)

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.) ;)

[DrTemp]

Originally Posted by DrTemp

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. [...]

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?