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ECON 5111 Mathematical Economics
Fall 2015
Test 1 October 2, 2015Answer ALL Questions Time Allowed: 1 hour
Instruction: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Re-member to put your name on the front page. You cankeep the question sheet after the test. Phones or calcu-lators are not required for the test.
1. Provide a brief explanation to justify each of the fol-lowing sentences is a logical statement or not.
(a) In a close system, entropy increases over time.
(b) You must vote for the Green Party to save theenvironment and to promote economic growth.
(c) In a market economy, the income e↵ect of a pricechange is always greater than the substitutione↵ect.
(d) This statement is false.
(e) The set of natural numbers N contains all thewisdom and power of the universe.
2. Let p and q be logical statements. Use a truth tableto verify that the following statement is true:
[(p _ q) ^ (⇠ q)] ) p.
3. In each part below, the hypotheses are assumed tobe true. Use the tautologies from the appendix toestablish the conclusion. Indicate which tautologyyou are using to justify each step.
(a) Hypotheses: q ) ⇠ p, r ) s, p _ r
Conclusion: ⇠ q _ s
(b) Hypotheses: r ) ⇠s, t ) u, s _ t
Conclusion: ⇠r _ u
4. Let n be an even integer. Prove or disprove: n
2 +2n+ 3 is odd.
5. Let A and B be sets.
(a) Define A \B in logical and set symbols.
(b) Let C = A\B andD = B\A. Prove or disprove:C and D are disjoint.
Appendix
Some Useful Tautologies
(a) ⇠ (p ^ q) , (⇠ p) _ (⇠ q)(b) ⇠ (p _ q) , (⇠ p) ^ (⇠ q)(c) ⇠ [8 x, p(x)] , [9 x 3 ⇠ p(x)](d) ⇠ [9x 3 p(x)] , [8 x, ⇠ p(x)](e) (p ) q) , (p ) p1 ) p2 ) · · · ) q2 ) q1 ) q)(f) (p ) q) , [(⇠ q) ) (⇠ p)](g) (p ) q) , [p ^ (⇠ q) ) c](h) p , (⇠ p ) c)(i) [p ) (q _ r)] , [(p ^ (⇠ q)) ) r](j) [(p _ q) ) r] , [(p ) r) ^ (q ) r)](k) [p ^ (p ) q)] , q
(l) ⇠ (p ) q) , [p ^ (⇠ q)](m) [(p _ q) ^ (⇠ q)] ) p
(n) [(p ) q) ^ (r ) s) ^ (p _ r)] ) (q _ s)
“Look, the new emoji are here.”
ECON 5111 Mathematical Economics
Fall 2015
Test 2 October 16, 2015Answer ALL Questions Time Allowed: 1 hour
Instruction: Please write your answers on the answer
book provided. Use the right-side pages for formal an-
swers and the left-side pages for your rough work. Re-
member to put your name on the front page. You can
keep the question sheet after the test. Phones or calcu-
lators are not required for the test.
1. Let S be a nonempty set.
(a) What is P(S), the power set of S?
(b) Let ⇢, meaning “is a proper subset of”, be a
relation on P(S). Prove or disprove: (P(S),⇢)
is a partial order.
(c) Is there a maximal element in (P(S),⇢)? If yes,
what is it?
2. Let A = R\{5} and B = R\{1}. Show that f : A !B defined by
f(x) =
x+ 1
x� 5
is a bijection.
3. Let S be a nonempty set.
(a) Define an ordering % on S.
(b) Suppose that a 2 S. What is %(a)?
(c) Show that for all a, b 2 S, if a % b, then
%(a) ✓ %(b).
4. Consider the metric space of the set of complex num-
bers C with the Euclidean metric. Let A ✓ C be
given by
A = {(x, y) : 0 x < 1, y = �x}.
(a) What is the diameter of A?
(b) What is the set of all limit points of A?
(c) Explain whether A a closed set and/or an open
set.
(d) Explain whether A is compact.
(e) Is A convex?
5. Consider the functional f : R ! R, given by
f(x) = x
2.
(a) What is the lower contour set -f (2)?
(b) What is the hypograph of f?
(c) Is f a monotone function? Explain.
(d) Find all the fixed points of f .
An Equivalence Class
ECON 5111 Áathematical Economics
Fall 2015
Test 3 October 30, 2015Answer ALL Questions Time Allowed: 1 hour
Instruction: Please write your answers on the answer
book provided. Use the right-side pages for formal an-
swers and the left-side pages for your rough work. Re-
member to put your name on the front page. You can
keep the question sheet after the test. Phones or calcu-
lators are not required for the test.
1. Let
xn ! x and yn ! y
in R. Show that the sequence {xn + yn} converges to
x+ y.
2. Consider the geometric series
1X
n=0
ax
n,
where a 6= 0 and 0 < x < 1.
(a) Show that the series converges.
(b) Find
1X
n=0
✓r
1 + r
◆n
,
where r is the market interest rate.
3. Let f be a continuous function which maps a metric
space (X, ⇢X) into a metric space (Y, ⇢Y ). Suppose
that {xn} is a sequence in X which converges to a
point x. Show that the sequence {f(xn)} converges
to f(x) in Y .
4. Show that
fn(x) =cosnxp
n
converges uniformly to a limit function.
5. Let S be the set of all sequences in R. For all
{xn}, {yn} 2 S and ↵ 2 R, define
{xn}+ {yn} = {xn + yn},
and
↵{xn} = {↵xn}.Use the Appendix to show that S is a vector space.
Appendix — Axioms for Vector Spaces
Let V be a vector space with vector addition and scalar
multiplication. For all x,y, z 2 V and ↵,� 2 R,
1. x+ y = y + x
2. (x+ y) + z = x+ (y + z)
3. 9 0 2 V 3 x+ 0 = x
4. 9 � x 2 V 3 x+ (�x) = 0
5. (↵�)x = ↵(�x)
6. 1x = x
7. ↵(x+ y) = ↵x+ ↵y
8. (↵+ �)x = ↵x+ �x
ECON 5111 Mathematical Economics
Fall 2015
Test 4 November 13, 2015
Answer ALL Questions Time Allowed: 1 hour
Instruction: Please write your answers on the answer
book provided. Use the right-side pages for formal an-
swers and the left-side pages for your rough work. Re-
member to put your name on the front page. You can
keep the question sheet after the test. Phones or calcu-
lators are not required for the test.
1. Let f be a linear operator on an n-dimensional vector
space V .
(a) Show that the image of the zero vector is the
zero vector.
(b) Show that the kernel of f is a subspace of V .
(c) Suppose that the nullity of f is equal to m. Find
the rank of f . Justify your answer.
2. Suppose that L(V ) is the set of all linear operators on
an n-dimensional vector space V . Let D : L(V ) ! Rbe the determinant function. Explain whether D is a
linear functional.
3. Suppose that A and B are both n⇥ n invertible ma-
trices. Show that
(a) (AB)
�1= B�1A�1
,
(b) |A�1| = 1/|A|.
4. Suppose that u,v and w are non-zero vectors in a
vector space V . Explain whether the following state-
ments are true or false.
(a) If u+ v = v +w, then u = w.
(b) If
span({u,v}) = span({v,w}),
then
span({u}) = span({w}).
(c) If u ? v and v ? w, then u ? w.
5. Suppose that the matrix representation of a linear
operator f on a vector space V relative to a basis is
given by
A =
0
@1 0 0
2 1 3
1 2 2
1
A .
(a) Find the eigenvalues of f .
(b) What is the determinant of f?
(c) What is the rank of f?
(d) Find the normalized eigenvectors of f .
“You’d better talk—if this doesn’t work, we make you
take linear algebra.”
ECON 5111 Mathematical Economics
Fall 2015
Test 5 November 27, 2015
Answer ALL Questions Time Allowed: 1 hour
Instruction: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Re-member to put your name on the front page. You cankeep the question sheet after the test. Phones or calcu-lators are not required for the test.
1. Suppose that � and ⌘ are two distinct eigenvaluesof a symmetric linear operator f on an inner prod-uct space V , with eigenvectors x and y respectively.Show that x and y are orthogonal vectors.
2. Suppose that f : Rn ! R is a C2 function.
(a) What is the second-order Taylor approximationof f about a point x0?
(b) Find the gradient of the approximation in part(a).
(c) Find the Hessian of the approximation in part(a).
3. Suppose that f : R2 ! R is given by
f(x, y) = 5 log x+ 2y.
Let the vectors x = (5,�2) and u = (3, 4).
(a) Find the directional derivative of f at the pointx in the direction of u.
(b) Find the directional derivative of f at the pointx in the direction which the value of the functionincreases at the fastest rate.
4. Consider the Henon map f : R2 ! R2 defined by
f(x) = (1� 2x2 � x
21, x1).
(a) Find the Jacobian Jf (x) at the point x 2 R2.
(b) Is f a bijection? Explain.
(c) If yes, find the derivative of f
�1 at the pointy = (1, 1).
5. Suppose that S is a subset of a vector space V .
(a) Show that if S is a subspace of V , then it is aconvex cone.
(b) Prove or disprove the converse of the statementin part (a).
“I can’t wait to lie down on a warm beach and forget
everything I learned this semester.”
ECON 5111 Mathematical Economics
Fall 2015
Final Examination December 18, 2015
Answer ALL Questions Time Allowed: 2 hours
Instruction: Please write your answers on the answerbook provided. Use the right-side pages for formal an-swers and the left-side pages for your rough work. Re-member to put your name on the front page. You cankeep the question sheet after the test. Phones or calcu-lators are not required for the exam.
1. Let S be a convex cone in Rn
+. Suppose that h : S !R is a linearly homogeneous function and g : R ! Ris a strictly increasing function. Show that f = g � his homothetic.
2. Suppose that f : Rn
+ ! R is a linearly homogeneousC2 functional. Show that for all x 2 Rn
++,
(a) f(x) = rf(x)Tx,
(b) r2f(x)x = 0.
3. Consider the functional
f(x, y) = �2x2 � y
2 + 4x+ 4y � 3.
(a) Find the stationary point(s) of f .
(b) Is it a maximum, minimum, or neither? Explain.
4. In the constrained optimization problem
maxx2G(✓)
f(x, ✓),
suppose that f is strictly quasi-concave in x and G(✓)is convex. Prove that a local maximum is a strictglobal maximum.
5. Suppose that a consumer’s preference relation is rep-resented by an increasing and strictly quasi-concaveC2 utility function U(x), where x 2 Rn
+ is a con-sumption bundle. The market prices of the bundleare p 2 Rn
++. Let the consumer’s income be y > 0.
(a) Set up the utility maximization problem withequality constraint.
(b) What is the Lagrangian function?
(c) Find the bordered Hessian B.
(d) What condition does B have to satisfy so thatthe optimal bundle x
⇤ is a local maximum?
6. Let f(x) be the C2 production function of a firm,where x 2 Rn
+ is an input bundle with price vectorp 2 Rn
++. The firm is trying to minimize the totalcost pT
x given a particular output level f(x) = y.
(a) Set up the cost minimization problem.
(b) What is the Lagrangian function?
(c) Show that p1 p
2 implies C(p1, y) C(p2
, y).
7. Solve the following optimization problem:
maxx,y
[�(x� 4)2 � (y � 4)2]
subject to x+ y 4,
x+ 3y 9.
Ten years later . . .
“What was the definition of an open ball?”