Economics 205, Fall 2002: Final ExaminationInstructions.1. Try to answer all nine problems. (Read all of the questions now and starton the ones that seem easiest.)2. You have three hours to do the exam.3. Think before you write. Doing so may save time.4. Read e ach question carefully and answer the question that I ask.5. Make your answers as complete and rigorous as possible. In particular,give reasons for your computations, prove your assertions. You may useany theorem to support your arguments, provided that you state the the-orem and verify that its assumptions hold. Informal and intuitive argu-ments are better than nothing.6. The table below gives the point values for each question, allocate yourtime appropriately.7. Joel Watson, Room 310, will be available to answer questions between 10and noon.Score PossibleI 62II 50III 40IV 40V 20VI 65VII 60VIII 55IX 40Exam Total 432Course Total 432Grade in Course11. Consider the function f (x, y) = −{1x+1y} defined on {(x, y) : x, y > 0}.(a) Graph {(x, y) : x, y > 0, f (x, y) = −1}.(b) Find an equation of the hyperplane tangent to the graph of f(x, y) =z at the point (x, y, z) = (2, 2, −1).(c) Decide whether or not f (·) is homogeneous. If f(·) is homogeneous,then determine its degree of homogeneity and explicitly verify Euler’sTheorem.(d) State the Mean-Value Theorem as it applies to functions from R2→R.(e) Explicitly verify the Mean-Value Theorem by expressing f(2, 2) interms of f (1, 1) and the derivative of f(·).(f) Let g(u, v) = (u2+ v2+ 1, 10). Use the chain rule to compute allpartial derivatives of f ◦ g(·) when u = v = 10.22. The s ets x + 2y + z = 4 and 3x + y + 2z = 3 intersect in a straight line.(a) Find the equation of the line of intersection.(b) Find the equation of the plane perpendicular to the line you foundin part a and the point (0, 0, 0).(c) Find the point on the line of intersection found in part a that isclosest to the point (0, 0, 0).33. State which of the matrices A below are diagonalizable. You need notdiagonalize the matrices, but you must justify your answer.(a) A =3 0−1 3.(b) A =0 41 0.(c) A =4 11 −2.(d) A =4 0 00 4 20 2 4.4. For each of the symmetric matrices above, state whether A is positive-definite, negative-definite, positive semi-definite, negative semi-definite,or indefinite.5. Pick one of the diagonalizable matrices above and exhibit an invertiblematrix P such that P−1AP = D, for a diagonal matrix D.46. Consider the optimization problem:max −4x2+ 2y subject to x ≥ 0; y ≥ 0; and − 4x + 2y ≤ −1(a) Graph the feasible set. (That is, {(x, y) : x ≥ 0; y ≥ 0; and − 4x + 2y ≤ −1} .)(b) Graph a level set of the function f (x, y) = −4x2+ 2y.(c) Using the graphs from parts a and b, identify the solution to theoptimization problem.(d) Solve the optimization problem using calculus techniques.57. The demand for a good is a function D(p, w) of price, p, and income,w. The supply of the good is a function S(p) of price only. For fixedw, an equilibrium price is a value P such that D(P, w) = S(P ). Assumethat D and S are differentiable functions. Identify economically sensibleconditions that guarantee that D(P, w) = S(P ) implicitly defines P asa function of w. Find an expression for P0(w) and interpret the sign ofP0(w).68. Let g(x) = ex− 1 − x −x22. Ass ume x > 0.(a) Prove that g(x) > 0 for all x > 0.(b) Let K > 0. Prove that there exists a unique solution to the equationg(x) = K.79. Prove or give a counterexample to the statements below. In each part,assume that the function f : Rn→ R is continuous and S ⊂ Rn.(a) If x∗solves: max f(x) subject to x ∈ S, then x∗solves max ef(x)subject to x ∈ S.(b) If x∗solves: max f(x) subject to x ∈ S, then x∗solves min −f(x)subject to x ∈ S.(c) If x∗solves: max f (x) subject to x ∈ S, then x∗solves min f (x)subject to −x ∈ S.(d) If S ⊂ [0, 1], then the problem: max f (x) subject to x ∈ S has