If you have one day where all crime becomes legal, you remove crime from being a possibility, ergo your crime rate drops to zero for the day.
Therefore, of course the crime rate would drop. The act of legalizing crime is probably the most direct way of reducing the crime rate.
Just think about it. If you legalize marijuana, you're going to have less people arrested for being in possession of marijuana.
Much crime today often happens in the spur of the moment. When discussing crime rate first there are two points:
1) Crime that would happen on this day will never be solved.
The point of lowering the crime rate is to increase justice. By making a day in which it is then legal and never solved is the great injustice to our justice system. Riots, murder, rape; numerous crimes never accounted for that can easily spill over into the next day. This then allows people to commit acts when they themselves will never get caught, thus not helping justice, but simply hindering it in the first place.
2) One day does not compensate for a year.
Even if this day of legalized crime existed, crime around the nation would still happen. Drug trade, turf war, even murder will still occur because crime is often a spur of the moment situation. One day does not give some kind relief for people from a whole year. That "relief" would only really apply to people who are emotional unstable
If nothing was illegal and you could do whatever, just think. If you could possess drugs, there'd be more people using drugs. Murder, population would be threatened. Friends, family would all be killed and the deceased wouldn't get justice because well, its legal. Robbery? Stores would be out of business sooner than not. So, they'd close down. Where would you buy things? We have a strong governments and law enforcement for reasons. It'll be like the purge except 24/7.. Wow.
Just because its legal for one day wont stop it from ever happening any other days. If someone wants to commit a crime badly enough they wont wait for that one day. So on top of the illegal crime there will be ten times that amount on the day that crime is actually legal. The crime rate will skyrocket and the population will decrease. Giving people one day to do anything they want will allow them to channel all their anger stress and problems and frustration to that one day and go on a crime or killing spree, doing whatever they feel that will alleviate them of all their problems and worries.
Not only will people be killing each other but there will be robberies everywhere; especially banks. Every bank within a five mile radius will have an attempted or successful robbery and eventually with people stealing from everywhere possible the economy will crash. Starting with the elimination of the lower class due to lack of money to protect themselves from these crime days (because once the crime is legal people will be aiming for the easy targets which are the week and defenseless lower class). Eventually the only people left standing will be the upper class . And soon enough they wont even be able to keep themselves up because the larger bushiness and corporations depend on the smaller businesses, which are the lower and middle class. And the economy could crumble and fall into a depression.
Crime would still continue to happen anyway. We are outnumbered when it comes to the amount of criminals to the amount of police officers in the United States and other countries around the world. If there was a day where people could commit crimes and not get in trouble, there would be many innocent lives taken by guilty people.