What is a tasty but more economical cut of beef to use instead of short ribs in mushroom barley soup

Beef shanks or shins also make a nice replacement for short ribs in soups. You can braise them in the oven or on the stovetop for 4-6 hours, or shorten up the time with a pressure cooker.

If you have an electric pressure cooker, use chuck roast or chuck eye roast and cook the beef first. Here is a recipe you could use as a starting point: https://cooking.nytimes.com/recipes/1020096-pressure-cooker-beef-stew-with-maple-and-stout If you don't have a pressure cooker, use the same cut of beef, but braise it, preferably in wine. That will take much longer, but is essential in making the beef more tender. Then use the bits (and the broth!) in the soup. Another tip: cook up the barley on its own first, too, as it will soak up a lot of liquid. The soup will taste better if the barley is pre-cooked in salted water (use the pressure cooker / Instant Pot for that as well). Both the meat and the barley can be cooked in advance, of course.