Crime is a Function of Propensity and Sanctioning
The first step in solving the criminological puzzle is to understand that crimes are a type of harm. The level of harm varies from crime to crime, and the harm can be real or imagined. The focus of the harm can be other individuals or the person could be harming themselves. The second step is to recognize that crime rates are a function of two separate processes. The first process creates individual variation. Each individual has a property called “criminal propensity” that governs their probability of creating real or potential harm to others or themselves. Criminal propensity is normally distributed in the population. The second process is societal. Each society defines which behaviors are going to be called crimes and then determines when to prosecute particular individuals for those behaviors. The societal selection process is asymmetric, or one-sided. Societies tend to pick only the most harmful behaviors to define as crimes. The crime rate is a mathematical function whose solution is determined by the level of individual variation (criminal propensity) and the strength of efforts to control the harm caused by individuals (societal sanctioning). It will be argued that because criminal propensity is normally distributed in the population, and societal selection is asymmetric, the function relating criminal propensity, societal sanctioning, and crime rates is a nonlinear sigmoid function. The literature in this area is relatively sparse. The following works represent some of the critical readings regarding criminal propensity, societal sanctioning, and crime rates.

Cesare Becarria and Jeremy Bentham images were downloaded from Wikipedia
Cesare Beccaria and Jeremy Bentham
Some of the first discussions of criminal propensity and societal sanctioning are found in the works of Cesare Beccaria (1764) and Jeremy Bentham (1789). These authors suggested that everyone has a propensity for crime. In their model, the propensity for crime is a uniform property that every person has. The propensity for crime can be partially reduced by the proper use of societal sanctioning. If crimes are punished, the propensity for crime should be reduced.

Alphonse de Candolle image was downloaded from Wikipedia
Alphonse de Candolle
de Candole (1830) was one of the first writers to stress the importance of considering the effects of both criminal propensity and societal sanctioning when examining crime rates. He noted that both propensity and sanctioning can vary. If both effects are not taken into account, a higher crime rate cannot tell us whether a particular group has a higher level of criminal propensity or whether the society in which these people live has a higher level of societal sanctioning.

Adolphe Quetelet image was downloaded from Wikipedia
Adolphe Quetelet
Adolphe Quetelet (1833/1984) had suggested that there was a linear (ratio) relationship between criminal propensity and crime. Quetelet (1848, 1869) later went on to suggest that the distribution of criminal propensity was normal, with time varying covariates. He even recognized that societal sanctioning was asymmetric. He failed to recognize that the mathematical relationship between criminal propensity, societal sanctioning, and crime rates must be a nonlinear sigmoid relationship.

Émile Durkheim image was downloaded from Wikipedia
Émile Durkheim
Émile Durkheim (1895) also deserves recognition for noticing that criminal propensity varies in the population. Although he stated that “crime is normal,” it could be argued that he was writing about the propensity for crime. Durkheim recognized that variation in criminal propensity was a normal and even healthy characteristic of the population. Durkheim (1895) also noted that societal sanctioning is normal as well, and can vary from one society to another. Although his discussions did not fully account for the nonlinear mathematical relationship between criminal propensity, societal sanctioning, and crime, he did see that criminal propensity and societal sanctioning both covary.

Notable Contemporary Discussions
Notable Contemporary Discussions
There does not seem to be a large body of contemporary literature that provides a discussion of the normal distribution of criminal propensity with its relationship to an asymmetric societal sanctioning. The model produced by Wilkins (1965) seems to come closest to bivariate probit model discussed in the theory presented on this web site. Cavan (1961), Rowe, Osgood, and Nicewander (1990), Clarke and Weisburd (1990), Clarke (1996) and Wikström (2009) all discuss the possibility that criminal propensity may be normally distributed in the population. These works would appear to be the most notable contemporary theoretical and empirical explorations of the distribution of criminal propensity with possible links to societal sanctioning.

References
Beccaria, Cesare (1764/1767). An Essay on Crimes and Punishments. Translated from the Italian; With a Commentary, Attributed to Mons. De Voltaire. Translated from the French. Translator unknown. London: J. Almon.
Bentham, Jeremy (1789/2000). An Introduction to the Principles of Morals and Legislation. Ontario, CA: Batoche Books: Kitchener.
Cavan, Ruth (1961). The Concepts of Tolerance and Contraculture as Applied to Delinquency. The Sociological Quarterly, 2(4): 243-258.
Clarke, Ronald V. (1996). The distribution of deviance and exceeding the speed limit.” British Journal of Criminology 36(2): 169-181.
Clarke, Ronald V., & Weisburd, David L. (1990). On the Distribution of Deviance, In Don M Gottfredson and Ronald V Clarke, eds., Policy and Theory in Criminal Justice, pp 10-27, Aldershot, England: Gower Publishing Company.
De Candolle A. (1830). Considérations sur la statistique des délits. Reprinted in: Déviance et société. 1987 – Vol. 11 – N°4. pp. 352-355.
Quetelet, Adolphe. (1833/1984). Research on the Propensity for Crime at Different Ages. Translated and introduced by Sawyer F. Sylvester. Cincinnati: Anderson.
Quetelet, Adolphe (1848). Du système social et des lois qui le régissent. Paris: Guillaumin et Cie Libraires.
Quetelet, Adoplphe (1869). Physique Sociale ou Essai sur le Development des Facultés de L’Homme. (Tome II). Bruxelles: C-Muquardt, Libraire-Editeur.
Rowe, David C., Osgood, D. Wayne, & Nicewander, W. Alan (1990). A latent trait approach to unifying criminal careers. Criminology 28(2): 237-270.
Wikström, Per-Olof H. (2009). Crime Propensity, Criminogenic Exposure and Crime Involvement in Early to Mid Adolescence. Monatsschrift für Kriminologie und Strafrechtreform, 92:253–266.
Wilkins, Lelsie T. (1965) Social Deviance: Social Policy, Action, and Research. Englewood Cliff, NJ: Prentice Hall.