4. Consider a firm's cost function C(w, r, q) = q war - where q is the level of output, w is the price of the labour input L, and r is the price of the capital input K. (a) Derive the conditional input demand functions for L and K. (b) Show that these demand functions are homogeneous of degree zero in input prices. Is this property peculiar to demand functions derived from this specific cost function? (c) Are these demand functions the same as those derived under the assumption of profit maximization? Why? (d) Let y = 1, a = 1/4, w = 16 and r = 1, and suppose the firm acts as a perfectly competitive price taker facing the demand func- tion Q = 100 - P, where Q is aggregate demand. Calculate the equilibrium price and output