When someone gets robbed there are some direct costs and benefits. Firstly, the perpetrator receives the goods and or services, and may also derive some pleasure from the activity itself. Secondly the victim suffers the loss of goods and services, and is hurt by the fact they have become a victim.
If this crime is reported to the police and solved successfully then there are a number of losers. Society has to pay the police to solve the crime, the criminal loses any goods and has to pay/go to jail, and the victim loses a significant amount of time making statements etc. In some cases, the time cost to the victim turns out to be greater than the benefit from having the crime solved (these are the people that regret having taken their case to the police).
In some cases, this type of outcome is quite common (eg for small traffic accidents), and the only reason people initially go to the police is because they don’t realise how large the effort and time cost will be (often as a result of mis-information). If this is the case, then everyone loses from a crime being reported and solved, implying that it would have been better if the crime was never reported.
However, this logic feels like it has something missing. Why would police be trained to under-sell the costs associated with reporting these crimes unless there was some good reason for it?
In fact I can think of two things that could make policing optimal in such a case:
Social equity concerns. It is important to society that crimes are solved, implying that their is an additional social benefit from solving the crime
The social benefit from the impact on criminal behaviour. If criminals realise that they are going to get caught, they are less likely to commit crimes. If the net social impact of the initial crime is negative (which I suspect it generally is) and the probability of being caught rises with the probability that the victim will report the crime, then this will be welfare enhancing.
Although this explains laws that are enforced even though it is not in the interest of any of the agents that are involved directly, there are other crimes that aren’t policed (e.g. obtaining some types of illegal substances). According to our little discussion the full social costs of enforcing these laws is greater than the social benefits. This raises two questions: Is the decision over what laws to enforce made efficiently? If the law is not going to be enforced, why does it exist?