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For the production function q=KaLb, answer the following questions: a. Write the constrained profit maximization problem, and solve for the profit maximizing levels of K and L, and the supply function as functions of p, w, and r. b. Demonstrate that the supply function is upward sloping. c. Show that the supply function and the demands for K and L are homogeneous of degree zero in p, w, and r. d. Demonstrate that the profit function is homogeneous of degree one in p,w, and r. e. Show that the profit function is non-decreasing in p and non-increasing in w and r. f. Prove that the profit function is convex in p (itself); and also convex in w and r (jointly - via the Hessian). g. Demonstrate Hotelling's Lemma, that d(profit)/dw =L and d( profit )/dr=K h. Demonstrate the following symmetries: I. K/w=L/r II. K/p=q/r III. L/p=q/w