Searched for: "private detective fees 1930s", "cost to hire a private detective 1930s", other variations along those lines; also went through the tags here
Frustratingly, I have found every other cost-of-living item I need to know for 1931 fairly easily. There are oodles of information out there on what people paid for food, utilities, rent, etc., what wages were for various occupations with standardized wages, and so forth. But this one plot-necessary item is proving to be elusive. My search terms have returned me lots of information on hiring private detectives in the modern day, or on famous 1930s fictional detectives, none of which is useful.
This is a detective in business for himself rather than working for a big well-known agency, which I assume would make a difference in what he's able to charge.