Useful links:
Chicago Rent Inflation
US CPI YOY
Chicago House Price Index (by region)
This calculator compare the following two scenarios. Say you start with net worth X, and in each future year, you have after-tax income Y. You can either:
A) Rent, and pay rent using future income, keep your X in stock market and let it compound.
B) Buy, and make down-payment using your net worth, and cover future maintenance cost, mortgage payment, and property tax bills using future income.
In either case, the surplus future income each year will be invested in the stock market. There is house appreciation, which causes maintenance cost and property tax bills increase. And there is rent increase.