Three additional assumptions are made the probability of

Three additional assumptions are made: the probability of apprehension is assumed to increase monotonically with the spending in private protection (i.e. p=p(e), where p′>0); victims and criminals are considered as risk-neutral; criminals are assumed to have complete information in that they observe their victim\'s wealth and are able to correctly infer their risks of being apprehended. In this context, a criminal will attempt to victimize citizen i who possesses a wealth of and have spent e in private protection as long as the following inequality holds
Since all citizens are potential victims for criminals, a given citizen i can avoid becoming a victim if he or she spends at least h on private protection, where h indicates the spending on protection that would make a criminal indifferent between attempting to steal from i doing it because the risk involved is very high. In short, where p(−1) is the inverse of function p that links private spending in protection to the probabilities of punishing a criminal. Eq. (2) gives, for each wealth level, the minimal spending on private protection required to prevent crime by deterring criminals. Therefore, citizens must decide whether they will spend h on their own protection or do not invest at all. They will spend h only if it does not exceed the prospective losses of being victimized. That is, if
Wealthier persons would need, ceteris paribus, greater spending in private protection to avoid victimization. This is the conclusion reached based on the first derivative of sglt-2 (2) with respect to ,
But are wealthier persons willing to spend more on protection to prevent being victimized? Or will they instead prefer to bear some risk? The answer depends on the second derivative of h with respect to ,
Eq. (5) will be negative, unless the second derivative of p is both negative and large in absolute value. So the wealthy will routinely spending in private security in order to avoid being victimized unless p exhibits sharp diminishing returns to scale. In summary, according to the approach adopted by Gaviria and Pagés (2002), the wealth of individuals determines both their economic attractiveness as victims and their capacity to protect themselves from criminals by paying for their protection. According to the findings of Becker (1968), Ehrlich (1973), Cohen et al. (1981) and Gaviria and Pagés (2002) it is to be expected that, given the opportunity cost of crime, the likelihood of failure determined by government spending on public safety, the penalties provided for in the law, and the costs involved in planning and committing a crime, criminals will pick their victims based on their evaluation of those who are more economically attractive for the criminal act. In this subjective evaluation, criminals take into account both the wealth of potential victims and the likelihood of failure determined by how much they spend on their own protection. By doing this, criminals optimize the expected return on crime. Therefore, the behavior of potential victims has a direct bearing on the optimization process that is implicit in the rational choice of a criminal. Thus, if the principle of economic rationality on the part of criminals holds, the risk of victimization increases with wealth. However, as pointed out by Gaviria and Pagés (2002), wealthier individuals have stronger reasons and more money to spend on their own security to protect themselves from criminals. On the other hand, poorer individuals lack the financial means to pay for their protection to avoid being victimized, but are less economically attractive potential victims to criminals than wealthier individuals.
Empirical modelling The data set used in the estimations of these models is made up of a pooled cross section of two victimization surveys conducted in São Paulo city in 2003 and 2008 by the Future Brazil Institute and the company Ipsos Public Affairs, which carried out the sampling process and interviews and set up the database. In those surveys, the period analyzed with respect to victimizations was a one-year period.