Homework Assignment #6
16.2 Let Q(x) = xT Ax be a quadratic form on Rn . By evaluating Q on each of the coordinate axes in Rn ,
prove that a necessary condition for a symmetric matrix to be positive denite (positive semidenite)
is that all the diagonal en
Homework Assignment #4
13.12 Write the following quadratic forms in matrix form:
a) x2 2x1 x2 + x2 .
1
2
b) 5x2 10x1 x2 x2 .
1
2
c) x2 + 2x2 + 3x2 + 4x1 x2 6x1 x3 + 8x2 x3 .
1
2
3
Answer: If we require the matrices to be symmetric, the solutions are:
1
2
Mathematical Economics Final, December 9, 2014
1. A consumer has the unusual utility function u(x, y) = x2 +y2 . The consumer consumes
non-negative quantities of both goods, subject to the budget constraint: 2x + 3y 6.
Find (x , y ) that maximizes utility
Mathematical Economics Exam #1, September 26, 2011
1. In R3 ,
a) Find all vectors that are perpendicular to
1
2
1 and x2 = 0 .
x1 =
1
1
Answer: These vectors z obey x1 z = 0 and x2 z = 0. In matrix form, this becomes
z1
0
111
.
z2 =
0
201
z3
The augmente
Mathematical Economics Exam #1, September 26, 2011
You have until 2:50 to complete this exam. Answer all four questions. Each question is worth 25 points, for
a total of 100 points. Good luck!
1. In R3 ,
a) Find all vectors that are perpendicular to
1
2
Mathematical Economics Final, December 11, 2003
1. Given a prices px = 3 and py = 4, minimize expenditure px x + py y under the constraint
u(x, y) 25 where u(x, y) = x1/3 y1/3 . Be sure to check constraint qualication and the
second-order conditions.
2. L
Mathematical Economics Midterm #2, November 6, 2003
1. Are the following sets open in R2 ? Closed? Neither? Explain.
a) A = cfw_(x, y) : x2 + y2 = 1.
Answer: Let f(x, y) = x2 + y2 . Because f is a quadratic polynomial, it is continuous. Now A = f1 (cfw_1)
Mathematical Economics Midterm #1, October 2, 2003
1. Let A and B be n n matrices. Suppose that (A B )2 = A2 2AB + B 2 . Show that AB = BA.
We compute (A B )2 = (A B )(A B ) = A(A B ) B (A B ) = A2 AB BA + B 2 . Equating
to A2 2AB + B 2 , we nd A2 2AB + B
Mathematical Economics Final, December 10, 2002
1. Consider the function f(x, y) = x4 2x2 y + 2y2 + 3. Find and classify (maximum, minimum, saddlepoint)
all critical points of f.
Answer: The rst-order conditions are
0 = 4x3 4xy
0 = 2x2 + 4y
Substituting t
Mathematical Economics Midterm #2, November 12, 2002
1. Consider the sequence xn = (1)n + n/(n2 + 1).
a) Does it converge? If so, what is its limit?
b) If it doesnt converge, does it have any convergent subsequences? If so, identify
one of them and comput
Mathematical Economics Midterm #1, October 3, 2002
You have until 4:45 to complete this exam. Answer all ve questions. Each question is worth
20 points, for a total of 100 points. Good luck!
3
1. Consider the vector x = 2 . Find the co-ordinates of x in t
Mathematical Economics Final, December 7, 2012
1. Let A =
3
1
. Find
A.
1
3
Answer: The characteristic equation is 2 6 + 8 = 0, yielding eigenvalues (A) =
cfw_2, 4. The corresponding eigenvectors are v2 = (1, 1)T and v4 = (1, 1)T . Let P =
[v2 , v4 ]. We
Mathematical Economics Exam #2, November 9, 2012
1. [Corrected] Let f (x, y, z ) = x2 z 2 + 2xy + 3x 5. The point (1, 1, 0) satises the equation
f (x, y, z ) = 0. Can x be written as a C 1 function of (y, z ) near (1, 1, 0)? If so, let g be
the function w
Mathematical Economics Exam #2, November 4, 2013
t2
1. Consider the function f(t) =
.
t
a) Compute the tangent vector of f at any t.
b) Give an equation for the tangent line at the point
4
.
2
Answer:
a) The tangent vector is given by the derivative
2t
df
Mathematical Economics Exam #1, September 30, 2014
1. Let a = (1, 2) and consider S = cfw_x R2 : x a 2 and x a > 1. Determine
whether S is a closed set, open set, or neither. Justify your answer.
Answer: The set S is neither open nor closed. To see it is
Homework Assignment #4
13.15 Prove that a linear function from Rk to Rm sends a line in Rk to a point or a line in Rm .
Answer: A line can be written as = cfw_x0 + tv : t R. Let L be a linear function from Rk Rm .
Then L(x0 + tv) = Lx0 + tLv. If Lv = 0, t
Homework Assignment #1
6.3 The economy on the island of Bacchus produces only grapes and wine. The production of 1 pound of
grapes requires 1/2 pound of grapes, 1 laborer, and no wine. The production of 1 liter of wine requires
1/2 pound of grapes, 1 labo
Homework Assignment #2
8.4 If you choose four numbers at random for the entries of a 2 2 matrix A, and four others for another
2 2 matrix B , AB will probably not equal BA. Carry out this proceedure a few times.
Answer: I will only give you one example. L
Homework Assignment #3
11.3 Determine whether or not each of the following collections of vectors in R4 are linearly independent
1
1
0
0 0 0
a) , , ;
1 0 1
0
1
1
1
1
0 0 0
b) ,
1 1 , 0 .
1
0
0
0
Answer: Label the vectors v1 , v2 , v3 . In case (
Mathematical Economics Exam #1, September 26, 2012
1. Let Z = cfw_z1 , . . . , zm be a collection of vectors in Rn . Dene W = cfw_x Rn : x zi = 0,
for all zi Z.
a) Show that W is not empty.
Answer: The vector 0 is in W because 0 z = 0 for every vector z,
Mathematical Economics Final, December 7, 2011
1. Let p 0 and m > 0. Let B (p, m) = cfw_x R2 : p x m.
+
a) Is B (p, m) a closed set? Justify your answer.
b) Is B (p, m) a bounded set? Justify your answer.
c) If u is a continuous function from R2 into R, d
Mathematical Economics Exam #1, September 26, 2011
1. In R3 ,
a) Find all vectors that are perpendicular to
1
2
1 and x2 = 0 .
x1 =
1
1
Answer: These vectors z obey x1 z = 0 and x2 z = 0. In matrix form, this becomes
z1
0
111
.
z2 =
0
201
z3
The augmente
Mathematical Economics Exam #2, November 9, 2011
1. Let f (x, y, z ) = (x + y )z + y . The point (1, 1, 0) satises the equation f (x, y, z ) = 1. Can
x be written as a C 1 function of (y, z ) near (1, 1, 0)? If so, let g be the function with
f (g (y, z ),
Mathematical Economics Final, December 13, 2001
1. Consider the function f (x, y ) = x3 + xy + y 2 x/2 + y . Find and classify (maximum, minimum,
saddlepoint) all critical points of f .
Answer: The derivative is df = (3x2 + y 1/2, x +2y +1). Setting this
Mathematical Economics Midterm #2, November 8, 2001
1. Consider the sequence xn = 2n + (2)n + 1/n2 .
a) Does the sequence converge? If so, what is its limit?
Answer: The sequence does not converge. In fact, |xn xn+1 | > 2n , so the terms
get farther apart
Mathematical Economics Midterm #1, October 4, 2001
1. (Eigenvalues and eigenvectors) Let A =
9
5
2
5
2
5
6
5
.
a) Find the eigenvalues of A.
9
(5
The eigenvalues are found by solving det(A I ) = 0 for . Now det(A I ) =
6
4
)( 5 ) 25 = 2 3 + 2 = ( 2)( 1).
Homework Assignment #5
15.1
a) Prove that the expression x2 xy 3 +y 5 = 17 is an implicit function of y in terms of x in a neighborhood
of (x, y ) = (5, 2).
b) Then, estimate the y value which corresponds to x = 4.8.
Answer:
a) Let f (x, y ) = x2 xy 3 + y
Homework Assignment #7
18.4 Find the general expression (in terms of all the parameters) for the commodity bundle (x1 , x2 ) which
maximizes the Cobb-Douglas utility function U (x1 , x2 ) = kxa x1a on the budget set p1 x1 + p2 x2 = I .
12
Answer: We assum
Homework Assignment #8
23.5 Suppose that A is an invertible matrix. Show that (A rI )v = 0 implies that (A1 1 I )v = 0.
r
Conclude that for an invertible matrix A, r is an eigenvalue of A if and only if 1/r is an eigenvalue of
A 1 .
Answer: Since A is inv