Let's look at some numbers: The average yearly income of 1920 was $3269.40citing the IRS. That implies about $62,87 per week without vacation or 10,90 per day for 300 work days (=sunday off+13 other days). Note though, that that is the average income in 1920 and the variance between factory workers and magnats was high. Looking at the cited source's source for the average wage, the IRS report, allows to make average wages for the different states by dividing the gross income by the number of returns.
Another list of wages per insustry was tabelerized by the Missouri libraries. One example for 1928 from there shows that a tutor at Yale university earns about $1500 to $2500, an assistant professor $3000 to $4000 and a professor $5000 to $8000. The principal of a junior high school with 30 classes would have earned $5500 to $6500. One of his teachers would have earned $2040 to $4200. A Fireman in Yale earned $1440 to $1500.
Taking these estimates as a baseline, and assuming that the investigators have a standard of living equivalent to an a university assistant or a young junior high teacher, they need to earn between $1500 and $3000 a year. This means about $30 to $60 a week. Assuming the 1920s common 6-day-week, this their expected income should be $5 to $10 a day.
However, detectivework and investigations are nothing that is available every day, so the rate to hire an Investigator should be about triple per day (ca 15 to 30$).
Of this income, the Investigator has to pay expenses. The two main expenses are food with about $461.5 to $521 in 1920 for an average household (estimated from retail prices) and 19.3% on rent.
This leaves us with the spendable "free" income of about 689.50 to 1529$ per annum for our teacher equivalent: That's not bad for that time, but it does not account for other, miscelanious expenses like going ot to eat or new clothes. A good estimate that at best half of this might be saved for other spending ($344.75 to $764.5).
One year's worth of salary saved up is added together between 3 and 5 years, depending on the exact saving behavior.
By the way:
Minimum wage was only invented in 1938 at $0.25 per hoursource.
And the Books!
The Good old 1920s Investigator Companion lists $1500 to $2500 per annum as "lower middle class"1920s Investigator Companion v2 (1994) p9 and tells it is a "modest apartment" in a decent area, possibly with a phone, and with some saving a car.
Ok, some time saving for a car? A new 1918 Buik 6-cylinder 170-cubic-inch costed $665 to $965here, a new 1915 Caddilac Eight V8 60HP 315-cubic-inch one costed $1975. That's about 2 to 6 years for a new car! Or, in 1927, a model Ts for $2801920s Investigator Companion v1 (1994) p32. Matches nicely, right?
I would assume the following:
Characters should have saved up for at least 5 years and in this time obtained one year's worth of income in moveable goods and other items.