Theories attempting to relate economic value to a single factor (such as labor or energy) have a long and somewhat disreputable history in economics. It must be emphasized at the outset that no such notion is contemplated here. To be sure, I do argue that labor skills, capital, available energy and technology are all more or less embodied - or "condensedn - forms of information. It does not follow that the market price of a given product or service is (or should be) simply or directly related to its numerical information content. In particular, there is no justification for confusing thermodynamic and morphological information in this
regard.
A far more plausible possibility is that condensed or embodied information of a given kind has a relatively well-defined cost per unit. Two examples will be presented briefly in this section. The first example pertains to the cost of physi- cal separation of pure substances from mixtures or solutions. In this case, the
process is metabolic, and the major cost element is energy. I also show that the cost tends to be a linear (or near-linear) function of the equivalent information added by the separation process. The second example pertains to the cost of accuracy in manual machining, a very labor-intensive process. Evidence is set forth suggesting that the cost of increasing precision is a highly nonlinear func- tion of the equivalent morphological information embodied.
E x a m p l e 1: C o s t of s e p a r a t i o n - c o n c e n t r a t i o n
As stated in Appendiz 1.A, the information embodied per mole by concentration, or lost by diffusion, can generally be approximated by Boltzmann's ideal gas approximation:
Hc = R ln(Xc/Xo) = R lnc
where Xc is the mole fraction in the concentrated state, Xo is the mole fraction in the diffused state, and R is the ideal gas constant (-2 cal/mole). The ratio of mole fractions is equal to the concentration ratio c.
It is important to note that the incremental information added by concen- tration depends on the starting point. Not much is gained by starting from a highly concentrated source. However, if we are interested in comparing the information embodied in different materials in absolute terms, the best way is to calculate the information that would be lost if the material were completely dispersed (diffused) into the environment. (The appropriate definition of
"environmentn would be the earth's crust for most solids, oceans for water- soluble salts or liquids, or the atmosphere for gases.) The difference between information lost by diffusion, and information added by concentration from high-quality natural sources, is, of course, a gift from nature.
It has been suggested, e.g., by Sherwood (1978), that costs (or prices) of many materials are inversely proportional to their original concentrations in their uore" or original form, and therefore proportional to the concentration fac- tor needed to purify them. This relationship implies a linear relationship between the logarithms of price (cents/lb) and concentration factors. Such a relationship is indeed observed for many materials, particularly where several different initial concentrations. This applies to various minerals, such as vermi- culite, diatomite, graphite, asbestos, sheet mica, and gold. It also applies to atmospheric gases (oxygen, nitrogen, argon) and to various chemicals found in brine. See Figure 1.2.
It must be noted that a linear relationship between cost-price and concen- tration factor is not a linear relationship between cost-price and embodied infor- mation. In fact, it is the logarithm of cost that is proportional to embodied information. In other words, the cost-price seems to be, on average, an exponen- tial function of embodied information associated with physical concentration:
I
Mica (sheet I* 00-
Graphite7 /' Bromine
~ l // ~ *Asbestos ~ *Argon ~ i ~ ~ ~
0
Figure 1.2. Cost of separation versus information added.
C 5 exp H,
In Figure 1.2, I have compared only materials requiring physical separation, which eliminates one of the complicating factors. Consider for instance, the extremely complex multi-stage process for refining platinum group metals from their ores versus the extraordinarily simple process for refining mercury from its ore (simple low-temperature retorting). Consider also the vast differences between by-products, such as arsenic, and primary products in this regard. Still it is more than a little surprising to observe an apparent clear relationship between cost-price and concentration, in view of the enormous differences, for instance, in scale of production or use among different substances. Further,
there are differences in inherent utility from one material to another. As an example, consider the great inherent utility of platinum as a catalyst compared to osmium, an equally rare metal of the platinum group with no known uses whatever.
Example 2: Cost of increasing precision
In machining operations, the information H(t) required to achieve a tolerance t can be written as
where K is a constant determined by the unit of information (e.g., bits) and t is usually defined for convenience as the maximum allowable machining error per unit (inch) of linear tool travel on the workpiece.
Table 1 . 4 . Cost-tolerance relationship.
W t ) Relative cost Tolerance t (bits) (Boltz, 1976)
.064--2-4 4 0.75
.048--.05 1 .O
.040 1.2
.032=2-' 5 1.5
,024 2.0
.020 2.4
.016 6 3.0
.012 4 .o
.008 7 6.0
.006
.004 8 12.0
.003
.002 9 24.0
.0015 lo
.001=2- 10 48.0
A cost-tolerance relationship taken from a standard engineering handbook (Boltz, 1976) is given in Table
1.4.
While the relative cost figures given are only approximate (taken from a graph - see Figure 1.9), it is clear that the relative cost is not a simple linear function of information content. In fact, in the normal range of tolerances from 2-5 to 2-lo where precision increases by a factor of Z5= 32, the information content only doubles. This implies that relative cost increases as the fifth power of relative information content or precision, viz.,
Tolerances (plus or minus)
Figure 1.3. Cost versus tolerance relationship. Source: Boltz (1976).
This relationship presumably reflects a body of experience with manually operated machine tools. Therefore, it can be interpreted as an average relation- ship between skilled machinists' time inputs a s a function of information actually embodied in the workpiece. Another way to look a t it is to note that embodied information is a very small fractional power of cost, viz.,
The time-information relationship for human labor is discussed in the second part of this Research Report. One is roughly proportional to the other.
That is, the amount of visual or tactile information required for motion control determines the time required for an elementary motion. Why, then, should the amount of information embodied in the product not be simply proportional to labor time? No definitive answer can be given here. However, it would seem likely that the nonlinearity of the cost-precision function results from the onset of mental overload as a human machinist approaches the limits of his natural sensory discrimination ability. As is discussed in more detail in the second part
of this Report, each successive manual correction becomes more and more difficult as the discrimination limit is approached. Therefore, it takes longer.
A numerically controlled machine tool with internal feedback control might also be ultimately limited by the inherent discrimination ability of its detectors or built-in "sensesn, but specialized electronic devices can be designed to be much more sensitive in a particular domain than the general purpose human sen- sory system. This is why NC or CNC (computer numerical control) machines can perform high-precision machining operations far faster than humans, although they offer little benefit in the case of simpler operations or lower preci- sion. One would therefore expect the relationship between control information input and information embodied in output to be much more nearly linear for NC or CNC machines.