Financial requirements to sponsor relatives visiting the Schengen area [Germany]?

A sponsor who signs a Verpflichtungserklärung agrees to reimburse the German state for costs from (intentional or unintentional) overstays, medical costs not covered by insurance, etc. Some towns use a formula for income based on

• the family size of the sponsor,
• the number and age of the visitors,
• limits of exemption from execution and welfare payments.

The sponsor must be able to pay the minimum cost of living for all visitors while staying over the limit of exemption for his or her family. If the income is not high enough, a deposit may be possible. A deposit is different from simply showing savings in that the sponsor cannot use the deposit until the visitors depart again.

The law does not set amounts. This PDF is from Cologne, not from Stuttgart, it mentions income and deposits but not savings. Amounts are not spelled out, but it mentions using SGB II as a guideline. This page is from Berlin, it mentions savings.

I found some guidelines on berlin.de. It's intended for people residing in Berlin, not Stuttgart, but I assume the requirements are similar, if not identical (the form for Stuttgart can be found here but they don't provide as many details as Berlin does).

For two adults, the requirement is a net monthly income of €1506 or at least €18090 in savings, actually available at the time of travel (no loan, real estate, stocks, bonds, or other illiquid investments). I can't tell if either high income or large savings are sufficient or if you need both of them at the same time.

Without that, I think you may still apply for a visa but without a proper Verpflichtungserklärung, your own situation would be judged much more strictly.