ECON 5111 Mathematical Economics - Lakehead...

ECON 5111 Mathematical Economics

Fall 2014

Test 1 September 26, 2014

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.

1. (a) Construct a truth table for the statement

(⇠p ^ ⇠(q _ p)) _ p.

(b) Can you find a simpler statement which is equiv-alent to the above statement?

2. 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: ⇠r, (⇠r ^ s) ) r

Conclusion: ⇠s

(b) Hypotheses: r ) ⇠s,⇠r ) ⇠ t,⇠ t ) u, v ) s

Conclusion: ⇠v _ u

3. Prove or disprove: Ifp2x is an irrational number,

then x is also an irrational number.

4. Let p be the statement

“If A ✓ B, then P(A) ✓ P(B).”

(a) Prove or disprove the above statement.

(b) State the contrapositive of p.

5. Let An = (�1/n, 1 + n], n 2 N. Find \n2NAn and[n2NAn.

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)

“The United States wants you to become a democracy.”

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Fall 2014

Test 2 October 10, 2014Answer 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.

1. For all m,n 2 N, define the relation m|n as m divides

n.

(a) Explain whether the relation is a partial order.

(b) Is it a linear order? Explain.

(c) What is the inverse relation?

2. Let A be a nonempty bounded subset of a linear order

(S,%).

(a) Define the set of lower bounds for A.

(b) Define inf A.

(c) Show that inf A is unique.

3. Suppose that

A = {a, b, c, . . . , z},B = {0, 1, 2, 3, . . . , 9}.

Let f be a function which maps A to B according to

the phone pad as shown.

Provide explanations to your answers below:

(a) Is the function f everywhere defined?

(b) Is f one-to-one?

(c) Is f onto?

(d) Is f a bijection?

(e) What is f�1({8, 0, 7})?

4. Consider the Euclidean metric space R2. Let

A = {x 2 R2: ⇢(x,0) > 1}.

(a) What is the diameter of A?

(b) Is A bounded?

(c) What is b(A)?

(d) What is

¯A?

(e) Is A open, closed, neither, or both?

5. Suppose that A is an open set in a metric space (X, ⇢).Show that Ac

is a closed set.



Fall 2014

Test 3 October 24, 2014Answer ALL Questions Time Allowed: 1 hour






1. Suppose that f : X ! R is continuous. Prove that

%f (a) = {x : f(x) � a} is closed.

2. Show that

fn(x) =sinnxp

n

converges uniformly to a limit function.

3. Let f : R ! R be defined by

f(x) =

(x

2if x 6= 1,

10 if x = 1.

Show that f is upper semi-continuous.

4. The Appendix lists the axioms of a vector space V .

Use the axioms to justify each step in the following

proofs: For every x 2 V and ↵ 2 R,

(a) ↵0 = 0.

(b) If ↵x = 0 and x 6= 0, then ↵ = 0.

5. Let Mn⇥nbe the vector space of all n ⇥ n square

matrices. A symmetric matrix A is a square matrix

equal to its transpose, that is, A = A

T. In other

words, its elements aij = aji for all i, j = 1, . . . , n.

Prove that the set of all n⇥ n symmetric matrices is

a subspace of Mn⇥n.

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

“O.K., trick or treat, please. Don’t mess with me, lady—I’m on a sugar high.”



Fall 2014

Test 4 November 7, 2014







1. Let V be an inner product space. Show that for any

x 2 V, x

T0 = 0, where 0 is the zero vector in V .

2. Let S be a subspace of an inner product space V .

(a) Define the orthogonal complement S?.

(b) Show that S?is a subspace of V .

3. Suppose that a linear operator f on R3is represented

by the matrix 0

@1 0 0

0 3 1

0 1 2

1

A

relative to the standard basis.

(a) Show that f is a bijection.

(b) Find the matrix representation of f�1relative

to the standard basis.

4. Let V be an inner product space.

(a) Define a symmetric linear operator f on V .

(b) Suppose that �1 and �2 are two distinct eigen-

values of f . Show that the corresponding eigen-

vectors x1 and x2 are orthogonal.

5. Suppose that a linear operator f on R3is represented

by the matrix 0

@5 0 0

4 0 0

3 2 1

1

A

relative to the standard basis.

(a) Find the eigenvalues of f .

(b) What is the determinant of f?

(c) What is the rank of f?

(d) What is the nullity of f?

(e) What is the definiteness of f?

“O.K., fellas—calm down. Does it really matter whetherthe statement is true or false?”



Fall 2014

Test 5 November 21, 2014







1. Suppose that f : R2 ! R is given by

f(x, y) = 2x

2+ xy � 3y

2+ 7.

(a) Find the gradient of f at the point a = (2, 3).

(b) Find the directional derivative of f at a in the

direction of b = (1, 1)

2. Suppose that f : R2 ! R2is given by

f(x, y) = (x cos ✓ � y sin ✓, x sin ✓ + y cos ✓),

where ✓ 2 [0, 2⇡).

(a) Find the derivative of f .

(b) Find the Jacobian of f .

(c) Find the subset in R2where f has an inverse

function.

(d) Find the inverse function f

�1and verify the in-

verse function theorem.

3. Suppose that f : R2 ! R is given by

f(x1, x2) = x1 sinx2.

Find the second-order Taylor approximation of f at

the point x

⇤= (0, 0).

4. Let V be a vector space.

(a) Define an a�ne set M ✓ V .

(b) Define the translation of M by a vector b.

(c) Show that the translation of M by b is an a�ne

set.

5. Suppose that f is a linear transformation which maps

a vector space V into a vector space W . Let S be a

convex set in W . Show that f

�1(S) is a convex set.

“Can you swim out there and charge my phone?”



Fall 2014

Final Examination December 15, 2014

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. Starta new page for each question. Remember to put yourname on the front page. You can keep the question sheetafter the test.

1. Suppose that

f(x) =1

4x

4 +1

2ax

2 + bx,

where a and b are parameters.

(a) Find the set of critical points of f .

(b) Apply the implicit function theorem to the nec-essary condition in part (a) to find the rate ofchange of the critical point x with respect to a

and b.

(c) Find the set of points (a, b) such that the implicitfunction theorem fails to apply.

2. Let L(V ) be the set of all linear operators on a finite-dimensional vector space V . Prove or disprove: Theset of all invertible operators in L(V ) is a convex set.

3. Let f : Rn

+ ! R be a C1 homogeneous function ofdegree k. Show that

kf(x) = rf(x)Tx.

4. A swimming pool is to be constructed with reinforcedconcrete. The pool will have a square base withlength and width x and height h. The cost of theconcrete, including labour and materials, forming thebase is p per square metre. The concrete for the sidewalls costs q per square metre. The budget for theconstruction cost of the pool is M .

(a) Setting up a constrained-optimization problemto maximize the volume of the pool.

(b) Find the Lagrangian function.

(c) Find the e↵ect on the optimized volume with re-spect to a change in the cost of the base materialp.

5. Consider the linear model in statistics, with one de-pendent variable, k independent variables, and n ob-servations. The model can be presented in the follow-ing matrix form:

y = X� + ✏

where y 2 Rn, X is a n⇥ k matrix, and ✏ 2 Rn is thevector of random variables caused by measurementerrors. The error term of each observation ✏

i

is as-sumed to have the same distribution with zero meanand variance �

2. Also, assume that E(✏i

✏

j

) = 0 fori 6= j.

(a) Derive an expression for ✏T✏.

(b) Find the stationary point of ✏T✏ with respect to�. Assuming X have rank k, derive the leastsquare estimator for �.

(c) Find the Hessian of ✏T✏. Show that it is pos-itive definite so that you have indeed found aminimum point.

6. A consumer who buys two goods has a utility functionU(x1, x2) = min{x1, x2}. Given income y > 0 andprices p1 > 0 and p2 > 0. Since the utility functionU is increasing, the budget constraint is binding.

(a) Set up the utility maximization problem as anequality-constrained optimization problem.

(b) Does theWeierstrass theorem apply to this prob-lem?

(c) Can the Lagrange multiplier theorem be used toobtain a solution?

(d) Solve the maximization problem.

7. An inequality-constrained optimization problem isgiven by

maxx,y

x

2 � y

2

subject to x

2 + y

2 9,

y 0,

y x.

(a) Derive the Kuhn-Tucker conditions by assumingthat only the first constraint is binding. Deter-mine whether a solution exists with this assump-tion.

(b) Find the solution of the problem.

2

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