ecg765-old midterms

ecg765-old midterms - Atsushi Inoue ECG 765: Mathematical...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Spring 1999 Midterm Exam I. Let x = ( x 1 ,x 2 ,...,x n ) be an n-dimensional column vector and A = { a ij } be an n n symmetric matrix. Define f ( x ) =- x > A x and g ( x ) = x > x . Both f and g are defined on < n . (a) Write f ( x ) and g ( x ) in terms of x 1 ,...,x n and { a ij } . (b) Calculate the gradient vector of f and that of g . (c) Compute the Hessian matrix of f . (d) Under what conditions on A , is f concave? (e) From this point on, consider the following maximization problem: max x f ( x ) subject to g ( x ) = c, where f and g are the same as before. Write the Lagrangian and the first-order necessary condition in terms of c , x and A . (f) State the second-order sufficient condition for constrained max- imum in terms of x and A . (You do not have to show that the second-order sufficient condition is satisfied at this point.) (g) Let c = 1, n = 2 and A = " 2 1 1 2 # . Find all the values of ( x 1 ,x 2 , ) which satisfy the first-order nec- essary condition, where denotes the Lagrange multiplier. (h) Determine whether each value of ( x 1 ,x 2 ) obtained in (g) satisfies the second-order sufficient condition. II. True or false. If you answer true, prove the following statement. If you answer false, provide a specific counterexample. Suppose that f : X < is differentiable and convex, where X is open and convex in < . If x * X is a local minimum point of f , then x * is a global minimum point of f . III. Let A be a nonsingular matrix. Prove that, if is an eigenvalue of A , then 1 / is an eigenvalue of A- 1 . 1 Atsushi Inoue ECG 765: Mathematical Methods for Economics Spring 2000 Midterm Exam ] 1 1. Let x = 1 2 3 , y = 4 5 6 , z = 7 8 9 . (a) Is the product xy defined? If your answer is yes, compute the product. (4 points) (b) Is the product x > y defined? If your answer is yes, compute the product. (4 points) (c) Is the product xy > defined? If your answer is yes, compute the product. (4 points) (d) Is the product x > z defined? If your answer is yes, compute the product. (4 points) (e) Is the product xz > defined? If your answer is yes, compute the product. (4 points) 2. Let A = 1 2 3 4 4 6 7 8 9 ....
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ecg765-old midterms - Atsushi Inoue ECG 765: Mathematical...

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