Consider a competitive market, X. Firms in that market use two inputs, A and B. Input A is produced by a monopolist firm and input B is supplied by a competitive market. Show that if A and B are used in fixed proportions in producing X, the upstream monopolist cannot increase its profits by vertically integrating into the downstream industry, but if inputs A and B are used in variable proportions (like, for example, as in the case of the Cobb-Douglas production function) vertical integration is profitable. Use a numerical example to prove your case.