Why Just For-Profits?

The New York Times editorializes in favor of competitive bidding in government contracts awarded to for-profit companies. If competitive bidding is a good idea for for-profits that want money from the government, why not extend the principle to apply to non-profits, too? There may be some good reason why, but the Times doesn't explain, leaving one to wonder if there's a general suspicion or bias against those motivated by a desire for profit.