Consider a firm with production function q = , using one input (e.g., labor) to produce units of output q. The price of every unit of input is w > 0, and the price of every unit of output is p > 0. (a) Set up the firm's profit-maximization problem (PMP), and solve for its uncondi- tional factor demand z(w, p). (b) What is the output level that arises from using the amount of inputs z(w,p)? Label this output level q(w). (c) Set up the firm's cost-minimization problem (CMP), and solve for its conditional factor demand (w, q) for any output level q. (For now, we write the constraint of the CMP to be (2) q, where the output level q that the firm seeks to reach does not necessarily coincide with that found in part (b), q(w).) (d) Evaluate the conditional factor demand z(w, q) at output level q = q(w), to obtain z(w, q(w)). Show that it coincides with the unconditional factor demand z(w,p) found in part (a), that is, z(w, q(w))=2(w,p)