Economics 205 Final Examination Fall 2010Instructions.1. You have three hours to complete this examination.2. You may use scratch paper, but please write your final answers (including your completearguments) on these sheets.3. You may use one page of notes.4. Try to answer all eight problems. (Read all of the questions now and start on the ones thatseem easiest.)5. Do not be intimidated by the multi-part questions. In most cases, it is possible to answerthe later parts without answering the earlier parts.6. Think before you write. Doing so may save time and needless computations.7. Read each question carefully and answer the question that I ask. If you are uncertain abouthow to interpret the question, please ask me for clarification.8. Make your answers as complete and rigorous as possible. In particular, give reasons foryour computations, prove your assertions. You may use any result presented in class orsupplementary material to support your arguments, provided that state clearly the resultthat you are using and confirm that all necessary assumptions hold.9. Informal and intuitive arguments are better than no arguments.10. The table below gives the point values for each question, allocate your time appropriately.11. I plan to return the graded exams to your mailboxes on Monday afternoon (there is nochance that they will arrive earlier).Score PossibleI 150II 90III 150IV 170V 100VI 180VII 200VIII 160Exam Total 1200Course Total 1200Grade in Course1. In each part, determine at which points the derivative of the function h exists. When it doesexist, compute it. When it does not exist, explain why it does not exist.(a) h(x) = log(1 + log(1 + x)).(b) h(x) = (elog x)2.(c) h(x) =Rx0f(y)dy for a continuous function f.(d) h(x) =Rx0f(x)dy for a continuous function f .(e) h(x) =(0 if x ≤ 0x2log x if x > 02. Use integration by parts to computeR21x2log x dx.3. Consider the function f(x, y) = (1x2−1y) 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) = (1, 1, 0).(c) Decide whether or not f (·) is homogeneous. If f (·) is homogeneous, then determineits degree of homogeneity and explicitly verify Euler’s Theorem.(d) Find the directional derivative of f in the direction w =1√2(1, 1) at the point (x, y) =(.5, 1).(e) Let g(u, v) = (u2+v2+1, u+1). Use the chain rule to compute all partial derivativesof f ◦ g when u = v = 0.4. Consider the following matrices:(a) A =5 34 3(b) A =6 92 3(c) A =120 00 2 −10 −123.3.1 Say whether A−1exists and, if so, find A−1.3.2 State whether the quadratic form xtAx is positive definite, positive semi-definitive,indefinite, negative semi-definite, or negative definite.3.3 State whether the matrix is diagonalizable.3.4 For matrix b, find the eigenvalues and exhibit an invertible square matrix P and adiagonal matrix D such that A = PDP−1.5. A piece of cheese is located at (12, 10) in R2. A mouse is at (4, −2) and is running up theline y = −5x + 18. At the point (a, b) the mouse starts getting farther from the cheeserather than closer to it. What is a + b ?6. An agent has utility function u(x) where x is income. The agent has initial wealth w. Withprobability p the agent suffers a loss of l dollars, reducing her wealth to w − l. The agent’sexpected utility is pu(w − l) + (1−p)u(w). The agent’s certainty equivalent C is implicitlydefined by the equationu(C) = pu(w − l) + (1 − p)u(w). (1)(a) Show that if u is continuous, there exists a certainty equivalent. That is, there is asolution to equation (1).(b) Show that if u is strictly increasing, then there exists a unique certainty equivalent.That is, there is one and only one C that solves equation (1).(c) Suppose that for given values (p0, w0, l0) equation (1) has a solution C0. State con-ditions on u and its derivatives under which you can locally solve equation (1) forC as a differentiable function C = g(p, w, l) in a neighborhood of (p0, w0, l0) withg(p0, w0, l0) = C0. Write down a formula for Dg(p0, w0, l0).(d) State economically plausible conditions under which g is decreasing in p.7. A firm that uses two inputs to produce output has the production function 3x1/3y1/3, wherex is the amount of input 1 and y is the amount of input 2. The price of output is 1. The costof the inputs are wx and wy. The firm is constrained by the government to use no morethan 1000 units of input 1.(a) What is the profit maximizing input combination for the firm?(b) What is the most that the firm is willing to pay to have the right to increase the limiton input 1 by a tiny amount (from 1000 to 1000 + ∆ units), ∆ > 0?8. Decide whether each of the statements below is true. If the statement is true, then prove it.If the statement is false, then given a counterexample.(a) For any matrix A, AtA is a positive semi-definite matrix.(b) If A is a symmetric matrix that satisfies A3= I, then A = I.(c) If f : R → R is a continuous function that satisfies f (x) = −f(−x), then f(0) = 0.(d) If f : R → R is a twice continuously differentiable function that satisfies f (0) =f(1) = f (2), then there exists c ∈ [0, 2] such that f00(c) =