[SPACE] World Trade

[hal]

confession time:

I'm not understanding the notation below:

T(1, 2) = V(1) / (1 + exp(k.G(1, 2) - l))


Could you guys break it down into something I can use in a step by step manner? Thanks :)

[Ulzgoroth]

Originally Posted by hal

confession time:

I'm not understanding the notation below:

T(1, 2) = V(1) / (1 + exp(k.G(1, 2) - l))


Could you guys break it down into something I can use in a step by step manner? Thanks :)

So T, V, and G are functions, indicated in the earlier post. k and l are some numbers, nobody's saying what. And for some reason the '.'s mean multiplication rather than using, for instance, *.

exp(x) means e to the power of x. Otherwise written as e^x.

[Agemegos]

Originally Posted by hal

confession time:

I'm not understanding the notation below:

T(1, 2) = V(1) / (1 + exp(k.G(1, 2) - l))


Could you guys break it down into something I can use in a step by step manner? Thanks :)

T(1, 2) is the volume of trade from world 1 to world 2. The formula actually works for any pair of worlds: I ought to have used subscript "i, j", but this markup language doesn't do subscripts, and I thought it would be clearer to use something that is clearly not another constant to estimate.

G(1, 2) is the generalised cost of freight from world 1 to world 2 (for example). It includes components that depend on the distance between the worlds, such as the actual freight charges, and components that are fixed, such as packing, handling, and launch-to-orbit costs, import duties, etc. If travel is slow or interest rates are high or traded goods are mostly stuff that is extremely expensive, then the interest you could have got during the transit time on the money invested in the cargos is also a component of generalised cost.

V(1) and V(2) are the economic volumes of worlds 1 and 2 respectively, and so forth for other worlds. These are figures that the world generation sequence in Space will provide, depending on population, tech level, resource value modifier, etc. Economic Volume in GURPS stands in for both GDP and GNP.

exp is the exponential function, with exp(x) also written as e^x, where e is Euler's Constant (approximately 2.71828).

k and l are parameters that depend on the characteristics of the economy in your setting, and that I can't work out because I don't know what the economy in your setting is like. k controls the how sensitive trade volumes are to transport costs, and it relates to the ratio of freight charges to the typical value of cargo. l controls the overall propensity to trade rather than self-subsist; it depends on the universalisation of production, the diversity of world's endowment of resources, labour, and technology, consumer's taste for the exotic or their preference to buy local, etc. etc..


Here's want you could do, in a step by-step fashion.

1. Knowing the pair-wise distances between all the worlds in your set, and the freight rate (include insurance) as a function of distance for the interstellar technology you are using, and the loading-and-launch costs for the ground-to-orbit-and-back-to-ground technology you are using, calculate the cost of transporting goods from each world to each other world. Ignore interest costs and taxes for this exercise; those are elegant refinements. If you are using a spreadsheet (which would be very much the easy way to do this) make a special case that generalised cost is zero in shipping from a world to itself: you don't have to pay the fixed launch costs. Store these in a square array (it will have a string of '0's down the diagonal); in future I'm going to refer to the cost of transport from world i to world j as G(i, j)
   
2. Guess values for k and l, constants that relate to the unknown ways that imports fit into the production functions of the worlds, consumer preferences, the relationship of freight rates to cargo values, and all sorts of things.
   
3. Using the Economic Volume figures that GURPS Space's generation supplies, and where the Economic Volume of world i is V(i), go through all the pairs of possible origin with possible destination, including all the cases where the origin is the destination. For every origin-destination pair (i is the origin, j is the destination), calculate the following temporary value:
   

   T(i, j) = V(j) / (1 + e ^ ( k * G (i, j) + l))

   Store those in a square array.
   
4. For each possible origin i, calculate the total of the T(i, j), and call the result ∑(i). That is, sum the rows.
   
5. For each origin-destination pair i, j calculate the proportion P of the output produced on i that is consumed on j, which is P(i, j) = T(i, j)/∑(i).
   
6. If you don't like the look of the figures, adjust k and l and repeat steps 2 – 5. Increasing k will reduce trade, long-distance trade being more affected than short-distance. Increasing l will reduce all trade without affecting long-distance more than short distance.
   
7. When you are happy with the pattern of trade that results, find that actual value of exports from each origin i to each destination j by multiplying P(i, j) by V(i).

[Agemegos]

Originally Posted by Ulzgoroth

And for some reason the '.'s mean multiplication rather than using, for instance, *.

The reason would be that I'm a cave-man, who learned mathematics before everyone had Excel.

[ericthered]

I think I have some meaning to give to those numbers:


(Resistance to Trade) = (1+exp[ -(Reluctance to Travel) * (Cost to Travel) - (Relectance to Exchange)])
(Trade Volume Between A and B) = [(Economic Volume of A) * (Economic Volume of B)]/(Resistance to Trade)

These numbers are proportional, and meaningful even if you run it on itself, as it gives a measure of how much trading it does compared with how much it produces itself.

The trick there then becomes that this gives you two different total trade numbers for the relation. I would guess taking the average between the two won't be a bad number.

The other trick is you have to build the entire spread sheet. The model isn't about a single network, its about the whole.

[Anthony]

Originally Posted by Agemegos

The reason would be that I'm a cave-man, who learned mathematics before everyone had Excel.

Nah, using * for multiplication is a computer-programming thing that just propagated into spreadsheets due to lack of a middle-dot key on a standard keyboard. If you really want to keep old habits, use · (alt-0183 on windows, unicode U+2219).

[Agemegos]

Originally Posted by Anthony

Nah, using * for multiplication is a computer-programming thing that just propagated into spreadsheets due to lack of a middle-dot key on a standard keyboard. If you really want to keep old habits, use · (alt-0183 on windows, unicode U+2219).

Yeah, the splat for multiplication goes back at least to FORTRAN in 1957. But the lecturers still used the dot in my maths classes 1982–1984 and my statistics/econometrics classes 1986–1989. Whereas a splat appears instead of an × on the multiplication keys of some calculators now. I think it was Excel that made it popular beyond the world of programming. At a guess, anyway.

I'm a cave-man, nevertheless.

[Anthony]

Originally Posted by Agemegos

I think it was Excel that made it popular beyond the world of programming. At a guess, anyway.

I doubt it was any one product, if you've ever done math on a computer keyboard you'd have run into * as multiplication and / as division. I do sometimes wind up using · when I'm writing up equations, but . is too easy to confuse for something else.

[Agemegos]

Originally Posted by ericthered

The trick there then becomes that this gives you two different total trade numbers for the relation.

For trade in the opposite directions? They ought to come out the same unless transport costs are assymetric.

The other trick is you have to build the entire spread sheet. The model isn't about a single network, its about the whole.

That's true, and could be a problem with a really large setting such as Traveller's Imperium with its 11,000 member worlds. However, trade volumes per partner are dropping off exponentially with travel cost in the limit, so unless you are trading on a scale-free network (ughh!) trade-per-partner drops off with distance faster than number-of-partners rises with distance, and most trade is local. You can take a workable locality in a typical region, large enough that you are confident that trade from the centre to the outside of it will be small enough to neglect, "calibrate" your parameters k and l.

[ericthered]

Originally Posted by Agemegos

For trade in the opposite directions? They ought to come out the same unless transport costs are assymetric.

yes, but I was detailing using a normalizing method: totaling the expected trade from each source (including yourself), dividing each element by the total, and then multiplying it by your total economic product. That most certainly doesn't give you identical results.

Think of three worlds, A,B, and C, arranged in a line, with C much larger than A and B. A and B won't match in this example because B does so much trade with C that the total we're normalizing against much higher for B than A. Yes, this refers to my extension of your equation. If you have an alternative method that won't give us numbers like 168%, please outline it.