$(20$ pts$)$ Consider a central authority who operates $J$ firms with differentiable convex cost functions $c_j(q_j)$ for producing good $l$ from the numeraire. Define $C(q)$ to be the central authority's minimized cost level for producing aggregate quantity $q$; that is$,$

$C(q)=mi{n}_{q_1,\text{d}\text{o}\text{t}\text{s}\text{,}{q}_J}0{\displaystyle \underset{j=1}{\overset{J}{}}}{c}_j(q_j),s.t.,{\displaystyle \underset{j=1}{\overset{J}{}}}{q}_{j}q$

$($a$)(10$ pts$)$ Derive the first$-$order conditions for this cost$-$minimization problem.

$($b$)(5$ pts$)$ Show that at the cost$-$minimizing production allocation $(q_1^{**},$dots,$q_J^{**}),C^{\text{'}}(q)=c_j^{\text{'}}(q_j^{**})$ for all $j$ with $q_j^{**}>0($i$.$e$.,$ the central authority's marginal cost at aggregate output level $q$ equals each firm's marginal cost level at the optimal production allocation for producing $q).$

$($c$)(5$ pts$)$ Show that if firms all maximize profit facing output price $p=C^{\text{'}}(q)($with the price of the numeraire equal to $1),$ then the consequent output choices result in an aggregate output of $q.$ Conclude that $C^{\text{'}}(*)$ is the inverse of the industry supply function $q(*).$