2, (10 points) Consider the utility maximization problem max z 4+ y s.t. pz +y = m, where the three parameters p, q, and m are positive, and the constant a satisfies 0 st. g(z,y)=2>+2 =44 (a) Solve the problem. (b) What is the approximate change to the optimal value of f(z,y) if 44 is changed to 45? 5, (5 points) Consider the maximization problem maxzy st.z+y=2 (a) Reduce the problem to a one-variable problem of maximization z(2 z), and show that (z,y) = (1,1) is the only possible solution. (b) Check that the solution from part (a) satisfies the first-order condition for the constrained optimization problem, with Lagrangian multiplier A = 1. (c) Show that the solution (1,1) however does not maximize the Lagrangian function L(z,y)=ay1-(z+y2). 6, (10 points) Consider the problem max f(z,y) =2z +3y st g(z,y) =V +y=>5 (a) Show that the Lagrangian multiplier method suggests that the solution is (z,y) = (9, 4). Show that this does not solve the constrained optimization problem, as f(9,4) = 30 yet f(25,0) = 50. (b) Find the true solution to the problem by studying the level curves of f(z,y) = 2z + 3y, along with the graph of the constraint equation. (c) Which assumption of Lagrangian's Theorem (Theorem 18.4.1 of EMEA) is violated here