The dynamics of crime and punishment
This article analyzes crime development which is one of the largest threats in today's world, frequently referred to as the wax on crime. The criminal commits crimes in his free time (when not in jail) according to a non-stationary Poisson process which accounts for fluctuations. Expected values and variances for crime development axe determined. The deterrent effect of imprisonment follows from the amount of time in imprisonment. Each criminal maximizes expected utility defined as expected benefit (from crime) minus expected cost (imprisonment). A first-order differential equation of the criminal's utility-maximizing response to the given punishment policy is then developed. The analysis shows that if imprisonment is absent, criminal activity grows substantially. All else being equal, any equilibrium is unstable (labile), implying growth of criminal activity, unless imprisonment increases sufficiently as a function of criminal activity. This dynamic approach or perspective is quite interesting and has to our knowledge not been presented earlier. The empirical data material for crime intensity and imprisonment for Norway, England and Wales, and the US supports the model. Future crime development is shown to depend strongly on the societally chosen imprisonment policy. The model is intended as a valuable tool for policy makers who can envision arbitrarily sophisticated imprisonment functions and foresee the impact they have on crime development.
crime punishment non-stationary poisson process imprisonment differential equations expected utility model drive theory operational learning prediction