I do not want to put many constraints on the structure of the problem, except by defining a competitive equilibrium as such where firms are price takers in the factor and good(s) market, and, perhaps (?), that firms make no profits.
For example, is it possible to have something like "competitivistic" competition, where consumers have love for variety and firms produce differentiated goods, yet, instead of monopolistic competition, firms do not charge a markup, because there is free entry, so that firms end up producing at zero profit?
Or for example, a model of consumers geographically immobile, and where differentiated transportation costs across regions mean firms need to charge a markup to pay for that cost, and where, again, free entry deter firms from exploiting consumers?