Consider a firm with a production function q = ∙z, 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 unconditional 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, z(w; p), for any output level q. (For now, we write the constraint of the CMP to be f(z) = 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)). = z(w, p) e) Shephard's lemma. Evaluate the CMP's objective function, w. z, at the conditional factor demand z(w; q), to obtain the cost function, that is, find c(w, q) = w. z(w, q). Differentiate the cost function with respect to w, and show that your result coincides with the conditional factor demand z(w; q).