Consider a central authority who operates J firms with differentiable convex cost functions c,(q;) for producing good I from the numeraire. Define C(q) to be the central authority's minimized cost level for producing aggregate quantity q; that is: C(q) = Min (91,...,93)20 s.t. (13) J j=1 1. Derive the first-order conditions for the cost-minimization problem. 2. Show that at the cost-minimizing production allocation (qi,...,9;), C'(q) = c; (q; )Vj with q; > 0 (i.e. the central authority's marginal cost level at the aggregate output level q equals the firm's marginal cost level at the optimal production allocation for producing q). 3. Show that if firms all maximize profit facing output price p = C (q) (with the price of the numeraire equal to 1), then the consequent output choices result in an aggregate output of q. Conclude that C (.) is the inverse of the industry supply function q(.)