To economize is to be frugal — to be careful not to waste anything, including money. You may need to economize on heating oil, keeping the temperature in your apartment on the nippy side during the winter.
Giving a kid an allowance is one way to teach her to economize, spending it carefully in an attempt to save some and make the rest last. Unless money is no object, most people need to economize to some extent, being frugal and avoiding waste. In the 17th century, to economize was "to govern a household," from economy, at the time "household management," from the Greek oikonomia, "household management or thrift."