ECON 5111 Mathematical Economics - Lakehead …flash.lakeheadu.ca/~kyu/E5111/Tests2015.pdf3. Suppose...

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?”