Fixed costs and other economies of scale result in the non-convexity of the production function. To what extent does this affect the traditional welfare theorems? Suppose we convexify the production set P to a set P’. Then traditional welfare theorems go through. Suppose that the (a) resulting equilibrium is in a section of P’ that was locally convex as part of P. Then this equilibrium clearly would exist even in the original situation with P. Thus we retain equilibria that are part of P.
How do we work with fixed costs. Clearly they necessitate - and result in - imperfect (oligopolistic competition).