December 16, 2005
Economics 617 Final Exam
1. This exam has four questions. You have 2 hours and 30 minutes to write the exam. 2. This is a closed-book exam. 3. If you find any question ambiguous, explain your confusion and make whatever assumptions you t
TA: Yankun Wang
Economics 617 Problem Set 5 Solution Key
1. [Distance] Let d denote the Euclidean distance function on Rn Rn . Define: g(x, y) = d(x, y)/[1 + d(x, y)] for all x, y in Rn Show that: (a) g(x, y) 0 for all x, y in Rn (b) g(x, y) = 0 if and on
TA: Yankun Wang Fall 2006 Economics 617 Problem Set 4 Suggested Solution 1. [Properties of Determinants] Suggested Proof: Since we are free to use any properties listed in the lecture notes, there are many ways to prove this problem. However, you are not
TA: Yankun Wang
Economics 617 Problem Set 3 Solution Key
1. [Matrix Multiplication and Transpose] Suppose A and B are symmetric n n matrices. (a) Under what conditions on A and B is the matrix AB also symmetric? Explain. Solution: What we know is A = A0 ,
TA: Yankun Wang Econ 617 Problem Set 2 Solution Key 1. [Linear Dependence and Independence] (a) Let S = cfw_e1 , e2 , e3 be the set of unit vectors in R3 . Let T = cfw_x1 , x2 , x3 be a set of vectors in R3 , defined by: x1 = e1 + e2 , x2 = e2 + e3 , x3
TA: Yankun Wang
Economics 617 Problem Set 7 Solution Key
1. [Convex Sets] (a) Define the set: n Rn :
i
i1
n
1
This set is called in the unit simplex in R n . Show that n is a convex subset of R n . Solution: Let , n , and 0, 1. We need to prove: 1 - n .
TA: Yankun Wang
Economics 617 Problem Set 9 Solution Key
1. [On Slater's Condition] Let X = R2 , and f, g 1 and g 2 be functions from X to R, defined by: + f (x1 , x2 ) = 2x1 + x2 g 1 (x1 , x2 ) = 1 - (x1 + x2 ) g 2 (x1 , x2 ) = (x1 + x2 ) - 1 M ax f (x1
TA: Yankun Wang
Economics 617 Problem Set 10 Solution Key
1. [Using the Kuhn-Tucker Necessity Theorem] Let a, c be arbitrary positive parameters, satisfying a > c. Define the following functions: F (x) = R - x a for x [0, a] x G(x) = 0 F (z)dz for x [0, a
TA: Yankun Wang
Economics 617 Problem Set 8 Solution Key
1. [Unconstrained Optimization: Sufficient Conditions for a Global Maximum] Suppose A is an open convex set in R n , and f : A R is twice continuously differentiable and quasi-concave on A. Suppose
TA: Yankun Wang Econ 617 Problem Set 1 Solution Key 1. The whole point of this problem is to understand the defition of continuity and to get used to the - argument.
(a): For f : R+ R+ , we say that f is continuous if: for every x R+ and for any > 0, we c
T. Mitra Fall, 2006
Economics 617 Problem Set 10
[For practice only; do not hand in solutions] 1. [Using the Kuhn-Tucker Necessity Theorem] Let a, c be arbitrary positive parameters, satisfying a > c. Define the following functions: F (x) = a - x for x [0
T. Mitra, Fall 2006
Economics 617 Problem Set 1
[Due on Wednesday, September 6]
1. (a)Show, by using the definition of continuity, that the following functions are continuous on R+ . (i) f : R+ R+ is defined by: f (x) = x for all x 0 (ii) f : R+ R+ is def
T.Mitra, Fall 2005
Economics 617 Second Exam
(October 26, 2005) 1. This exam has three questions. You have 1 hour and 15 minutes to write the exam. 2. This is a closed-book exam. 3. If you find any question ambiguous, explain your confusion and make whate
Q1:
You lost 1 point for not saying "no degenerate pivots";
you also lost a point for saying there were degenerate
pivots. You lost 2 points for not finding the degenerate
solution.
Q2
(a) if you use min c'x not max -c'x as required as standard equality