T.Mitra, Fall, 2008
Economics 6170
Problem Set 9
(Due on Monday, December 1)
1. [Unconstrained Optimization: First-Order Conditions]
Here is a statement of the ﬁrst-order condition for a maximum of real
valued functions of a real variable.
Theorem:
Let f
Problem Set 6 Solutions
Economics 6170
TA: Christopher Handy
October 27, 2008
Problem 1
(a) Consider an arbitrary x S T . Now, since x S and S is open in Rn , there
exists rS > 0 such that B (, rS ) S . And since x T and T is open in Rn ,
x
there exists r
T.Mitra, Fall, 2008
Economics 6170
Problem Set 6
[For practice only; do not hand in solutions]
1. [Open Sets]
(a) Let S and T be open sets in Rn . Show that S ∩ T is also an open set
in Rn .
(b) Let A be the set deﬁned by:
A = {(x1 , x2 ) ∈ R2 : x1 > 0, x
Problem Set 7 Solutions
Economics 6170
TA: Christopher Handy
November 14, 2008
Problem 1
The proofs of parts (a) and (b) are very similar, so lets construct and prove the more
general result requested in part (b). Then part (a) will follow for the special
T. Mitra
Fall, 2008
Economics 6170
Problem Set 7
[Due on Wednesday, November 12]
1. [Converse of Euler’s Theorem]
(a) Let f : Rn → R+ be a continuously diﬀerentiable function on its
+
domain, which satisﬁes
x f (x) = f (x)
for all x ∈ Rn . Show that f is
T. Mitra
Fall, 2008
Economics 6170
Problem Set 8
[Due on Wednesday, November 19]
1. [Intermediate Value Theorem]
Here is a statement of the Intermediate Value Theorem for continuous
real valued functions of a real variable.
Theorem:
Let f be a continuous
Problem Set 9 Solutions
Economics 6170
TA: Christopher Handy
December 5, 2008
Problem 1
(a) Since x C is a point of local maximum of F , there is some > 0 such that for all
x C satisfying d(x, x) < , we have F (x) F (). And since C is open, there is
x
som
First Exam Solutions
Economics 6170
TA: Christopher Handy
September 29, 2008
Problem 1
(a) Since S R2 , we know that rank(S ) 2. So, to show that rank(S ) = 2, we only
need to nd two linearly independent vectors in S . Consider d1 , d2 S dened by
d1 = 4e1
Problem Set 8 Solutions
Economics 6170
TA: Christopher Handy
November 21, 2008
Problem 1
(a) Note that an arbitrary x1 B and x2 B are given in the hypothesis of the Corollary.
Now, let A = [0, 1]. Dene f : A R by
f (t) = F (tx1 + (1 t)x2 )
for all t A
Not
T.Mitra, Fall, 2008
Economics 6170
Problem Set 5
[Due on Wednesday, October 15]
1. [Non-Symmetric Matrices and Real Eigenvalues]
Let A be the 2 × 2 matrix, given by:
A=
a11 a12
a21 a22
where a12 = a21 . Let p be the trace of A, and let q be the determinan
Problem Set 5 Solutions
Economics 6170
TA: Christopher Handy
October 17, 2008
Problem 1
(i) Note that p = tr(A) = a11 + a22 and q = det A = a11 a22 a12 a21 . Then we want to
solve
0 = f () = det(A I ) = (a11 )(a22 ) a12 a21
= 2 (a11 + a22 ) + (a11 a22 a12
Problem Set 10 Solutions
Economics 6170
TA: Christopher Handy
December 10, 2008
Problem 1
(a) Dene the set X = R4 and dene the functions f , g 1 , and g 2 , each from X to R,
+
by
1
1
for all x X
2
2
f (x) = x1 + x2
1
g (x) = px3 + x4 px1 x2
2
g (x) = 1
T. Mitra, Fall, 2008
Economics 6170
Problem Set 10
[For practice only; do not hand in solutions]
1. [Using Kuhn-Tucker Suﬃciency Theory by Contracting the Domain]
Let p be an arbitrary positive real number. Consider the following constrained optimization
Problem Set 1 Solutions
Economics 6170
TA: Christopher Handy
September 15, 2008
Problem 1
1
Applying the denition of continuity, we can say that f (x) = 1+x is continuous on R+ if
for any choice of x 0, we have limxx f (x) = f (). Appealing now to the den
T. Mitra, Fall 2008
Economics 6170
Problem Set 1
[Due on Wednesday, September 10]
1. [Limit and Continuity]
Let f : R+ → R+ be deﬁned by:
f (x) = 1/(1 + x) for all x ≥ 0
Use the deﬁnition of continuity of a function to show that f is continuous
on R+ .
2.
Problem Set 2 Solutions
Economics 6170
TA: Christopher Handy
September 19, 2008
Problem 1
(a) We will show that the vectors x1 , . . . , xn are linearly dependent when n is even and
linearly independent when n is odd. Consider a linear combination of the
T.Mitra, Fall 2008
Economics 6170
Problem Set 2
[Due on Wednesday, September 17]
1. [Linear Dependence and Independence]
Let S = {e1 , e2 , . . . , en } be the set of unit vectors in Rn .
(a) Let T = {x1 , x2 , . . . , xn } be a set of vectors in Rn , deﬁ
T.Mitra, Fall 2008
Economics 6170
Problem Set 3
(For practice only: do not hand in solutions)
1. [Matrix Multiplication]
Suppose A is an m × n matrix, B is an n × r matrix, and C is an r × s
matrix. Verify that:
A(BC) = (AB)C
2. [Transpose of a Matrix]
Su
Problem Set 4 Solutions
Economics 6170
TA: Christopher Handy
October 10, 2008
Problem 1
We can write the system (1) as Ax = b, where
24
8
x1
A = 3 3 , x =
, b = 9 ,
x2
23
7
248
Ab = 3 3 9
237
(a) To show that Ax = b has a solution, we must show that rank
T.Mitra, Fall, 2008
Economics 6170
Problem Set 4
[Due on Wednesday, October 8]
1. [System of Linear Equations: Existence and Uniqueness of Solutions]
Consider the following system of linear equations:
2x1 + 4x2 = 8
3x1 + 3x2 = 9
2x1 + 3x2 = 7
(1)
(a) Show
Second Exam Solutions
Economics 6170
TA: Christopher Handy
October 31, 2008
Problem 1
(i) First, note that Cramers Rule can be used because A is square and non-singular (its
determinant is nonzero). Then we have
x1 =
x2 =
1 a12
0 a22
=
=
a11 a12
a21 a22
a