Mathematic Economics Plz help

Part B: Please answer ALL of Question 4. 4. A consumer has preferences that can be represented by the utility func- tion u : R2 - R, where u(x, y) = x + 2 vy. The consumer faces prices px > 0 and py > 0 for the two goods, and has money income m > 0. The goods cannot be bought in negative quantities, so x 2 0 and y 2 0. (a) Write down the consumer's constrained optimization problem. [4] (b) Is the objective function concave? Why, or why not? [6] (c) Confirm that the constraint set is convex, and draw it in a graph. [4] (d) Why will a non-degenerate constraint qualification (NDCQ) hold? [4] (e) Write down the Lagrangian L for this problem. Compute the first- [6] order conditions. Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions. (f) Briefly explain why the solution to the equations in your answer to [2] part (e) will generate the solution to the problem in part (a). (g) Find the optimal demand functions x * (px, py, m) and y* (px, py, m). [8] What is the marginal utility of income at the optimum? (h) Show that the multiplier on the constraint x 2 0 is given by [6] max PX PYm - 1 , 0 Provide an economic interpretation of this Lagrange multiplier