Editing Logistic function (section)

=== In economics and sociology: diffusion of innovations ===
The logistic function can be used to illustrate the progress of the [[Diffusion of innovations|diffusion of an innovation]] through its life cycle.

In ''The Laws of Imitation'' (1890), [[Gabriel Tarde]] describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with <math>f(x)=2^x</math>; finally, the third stage is logarithmic, with <math>f(x)=\log(x)</math>, and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote.

In a [[sovereign state]], the subnational units (constituent states or cities) may use loans to finance their projects. However, this funding source is usually subject to strict legal rules as well as to economy [[scarcity]] constraints, especially the resources the banks can lend (due to their [[Equity (finance)|equity]] or [[Basel III|Basel]] limits). These restrictions, which represent a saturation level, along with an exponential rush in an [[Competition (economics)|economic competition]] for money, create a [[public finance]] diffusion of credit pleas and the aggregate national response is a [[sigmoid curve]].<ref>{{Cite journal|last1=Rocha|first1=Leno S.|last2=Rocha|first2=Frederico S. A.|last3=Souza|first3=Thársis T. P.|date=5 October 2017|title=Is the public sector of your country a diffusion borrower? Empirical evidence from Brazil|journal=PLOS ONE|language=en|volume=12|issue=10|pages=e0185257|doi=10.1371/journal.pone.0185257|issn=1932-6203|pmc=5628819|pmid=28981532|arxiv=1604.07782|bibcode=2017PLoSO..1285257R|doi-access=free}}</ref>

Historically, when new products are introduced there is an intense amount of [[research and development]] which leads to dramatic improvements in quality and reductions in cost.  This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs, [[electrification]], cars and air travel. Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated.

Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis ([[IIASA]]). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989).<ref>{{cite web
 | last1 = Ayres
 | first1 = Robert
 | author1-link = Robert Ayres (scientist)
 | title = Technological Transformations and Long Waves
 |date=February 1989
 | url = http://www.iiasa.ac.at/Admin/PUB/Documents/RR-89-001.pdf
 |website=International Institute for Applied Systems Analysis 
 | access-date = 6 November 2010
 | archive-date = 1 March 2012
 | archive-url = https://web.archive.org/web/20120301220936/http://www.iiasa.ac.at/Admin/PUB/Documents/RR-89-001.pdf
 }}</ref> Cesare Marchetti published on [[Kondratiev wave|long economic cycles]] and on diffusion of innovations.<ref>{{cite web
 |last1       = Marchetti
 |first1      = Cesare
 |title       = Pervasive Long Waves: Is Society Cyclotymic
 |year        = 1996
 |url         = http://www.agci.org/dB/PDFs/03S2_CMarchetti_Cyclotymic.pdf
|website=Aspen Global Change INstitute
 |archive-url  = https://web.archive.org/web/20120305095553/http://www.agci.org/dB/PDFs/03S2_CMarchetti_Cyclotymic.pdf
 |archive-date = 5 March 2012
}}</ref><ref>{{cite web
 | last1 = Marchetti
 | first1 = Cesare
 | title = Kondratiev Revisited-After One Cycle
 | year = 1988
 | url = http://www.cesaremarchetti.org/archive/scan/MARCHETTI-037.pdf
 | website = Cesare Marchetti
 | access-date = 6 November 2010
 | archive-date = 9 March 2012
 | archive-url = https://web.archive.org/web/20120309092521/http://www.cesaremarchetti.org/archive/scan/MARCHETTI-037.pdf
 | url-status = dead
 }}</ref> Arnulf Grübler's book (1990) gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves.<ref name="Grubler1990">{{cite book
 | last1 = Grübler
 | first1 = Arnulf
 | title = The Rise and Fall of Infrastructures: Dynamics of Evolution and Technological Change in Transport
 | year = 1990
|publisher=Physica-Verlag
|location= Heidelberg and New York
 | url = http://pure.iiasa.ac.at/id/eprint/3351/1/XB-90-704.pdf
 }}</ref>

Carlota Perez used a logistic curve to illustrate the long ([[Kondratiev wave|Kondratiev]]) business cycle with the following labels: beginning of a technological era as ''irruption'', the ascent as ''frenzy'', the rapid build out as ''synergy'' and the completion as ''maturity''.<ref name="Perez2002">{{cite book |title= Technological Revolutions and Financial Capital: The Dynamics of Bubbles and Golden Ages
|last1=Perez
|first1= Carlota
|year=2002 |publisher= Edward Elgar Publishing Limited |location=UK  |isbn=1-84376-331-1  |url=https://archive.org/details/technologicalrev00carl|url-access= registration
}}</ref>