MATHEMATICS OF ENERGY SUBSTITUTION

New products and technologies replace old ones. But only in 1971 did General Electric researchers J.C. Fisher and R.H. Pry show that a newcomer penetrates the market and pushes out its predecessor with mathematical precision. If

'f' is the fraction of the market captured by the new product (a value that changes from 0% to 100% during the takover), the market share left to the old product is obviously 1- f.

Stated without mathematical complications, the Fisher-Pry rule simply says that the ratio of the two, i.e., f/(1-f), will plot on semilogarithmic paper as a straight line. (Semilogarithmic paper has a logarithmic vertical, but linear horizontal scale.) Only the slope of the line will differ for various takeovers and, of course, the theory cannot predict when a new product or technology will appear in the first place.

An example is shown at the top right the transition from Open Hearth to Basic Oxygen Furnace in steel manufacture. Japan made the transition earlier (position of line) and quicker (slope), but the transition law is the same. Fisher and Pry checked the rule on 17 such transitions (oil-based to water-based paint: soap to detergent; Bessemer to Open Hearth steel; etc.). Not only did it work for all 17, but it even worked for planned economies (see USSR above later and slower, but as straight as the capitalist lines).

In 1976, Cesare Marchetti of the International Institute for Applied Systems Analysis in Austria generalized the Fisher Pry rule to several (not just two) products competing for a market, and applied it to fuels wood, coal, oil, gas (AtE Ju1.77). The results were uncannily precise: By considering only the data prior to 1940, the rule as modified by Marchetti would predict what actually happened up to 1970 with an accuracy better than l%.

Marchetti's work, though brilliant, had nevertheless some deficiencies. His modifications of the rules for multiproduct competition were somewhat arbitrary; but more important, the rule remained entirely empirical it worked, but no one really knew why.

Czech scientist V. Peterka, working with Marchetti at IIASA, may now have provided the explanation. From simple mathematical assumptions, he derived general formulas that include the Fisher Pry Marchetti rules as special cases.

Many computer models are just programs bent and twisted until they produce what the programmer expects them to do (the Club of Rome's Limits to Growth is a drastic example); Peterka's model, on the contrary, merely implements the numerical computation of a formula derived mathematically from simple assumptions. Even the numerical constants (such as the dollar investment per installed kilowatt) were not "fitted," but were taken from the ledgers of the real world. The agreement with actually observed data (see figure on preceding page) is therefore of a quite different kind and far more impressive.

Macrodynamics of Technological Change: Market Penetration by New Technologies by V. Peterka asumes fairly advanced mathematics, including probability and nonlinear differential equations. Stock no. RR 77 22, $5.50 from IIASA Publications Dept., A 2361 Laxenburg, Austria. (US personal checks accepted, but allow 12 months for delivery.)