Paretian Ordering

Let us turn to the issue of evaluation. Within the setting of Lancasterian property space, economics offers just one way of defining uncontroversial progress: the Paretian.

Whenever one tool performs better in at least one dimension, without performing worse in any of the other, for instance being better as a knife, without being worse as a saw or a scraper, then we would say that it is better in a Paretian sense.

As indicated above, natural limits to a certain extent seem to favour efficiency in a specialized tool. Therefore, in the strict Paretian meaning just defined, we very seldom encounter a case of real progress. As was said above, most changes in history rather represent improvements in some dimensions at the expense of others.

The chance for such Paretian comparability of two objects, chosen at random in a group, increases with the number of objects in the group, and decreases with the dimension of the Lancasterian space in which they have to be represented.

We can even present a simple formal argument for the latter fact: In order to retain a certain degree of Paretian comparability, the number of objects has to increase geometrically when the dimension of the space increases arithmetically. The reason is that a point in n-dimensional property space divides the space into 2n orthants, among which only 2 are comparable in Paretian terms. Consequently, incomparability increases very fast with the dimension of the Lancasterian property space.

Very complex mass-produced goods, of which there are many examples in our high-tech society, may also need a property space of many dimensions for their representation, and comparisons of the different brands have basically the same characteristics as comparisons of different objects of art. Nevertheless, as long as there are many copies of each design it still makes some sense to define one dimension for each brand and to focus analysis on their numbers. In the case of artwork this is not meaningful, as there may be just one copy of each design.

In practice we expect the number of implements of every special group to more or less equal the number of dimensions in the property space. This is due to the possibility of combining them, just like the tools in a joiner's workshop. If the efficiency of combined performance were linear, we would even expect a strict equality to hold.

Too many tools would then imply that some of them are redundant, being dominated by a convex combination of the remaining ones, too few would only make it possible to span a subspace of the space of operations.

We can, of course, not expect things to be as simple as that. In particular, we have to consider that we are dealing with capital goods with long physical lifetimes, so at any moment we should expect to see vintages of tools which do not belong to the latest state of the technology.

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