Does true generosity require sacrifice?

Is it truly generous to give someone a few dollars when you are yourself a millionaire?

I wonder if someone is generous when they give of themselves — when it's sometimes inconvenient to pick up a loved one from the airport, or when it's a little tight on money but you help out someone you care about.

If you have all the resources in the world, being generous may be out of convenience, or out of a desire to make oneself more morally acceptable by their peers or themselves.