ECON 4117/5111 Mathematical Economics

Fall 2007

Test 1 September 28, 2007Answer ALL Questions Time Allowed: 1 hour

Read Me: 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. Donot forget to put your name on the front page.

1. Use a truth table to show that the following state-ment is a tautology:

[! q " (p # q)] # ! p.

2. In each part below, the hypotheses are assumed tobe true. Use tautologies from Figure 1 to establishthe conclusion. Indicate which tautology you areusing to justify each step.

(a) Hypotheses: p # ! q, r # s, r $ qConclusion: s $ ! p

(b) Hypotheses: p # ! r, s # ! r, p $ s,! q # rConclusion: q

3. Let X be a set.

(a) Define the power set of X, P(X).(b) Consider the binary relation % on P(X).

Prove that % is transitive.(c) Is % a weak order on P(X)? Explain.

4. Let S be the set of all lines in the plane and letR be the relation “is parallel to.” Determine (withexplanation) whether the relation R on S is

(a) complete,(b) reflexive,(c) symmetric(d) transitive,(e) antisymmetric.

Figure 1: List of Tautologies for Question 2

5. Let A, B, and C be subsets of a universal set U .Prove that

A \ (B & C) = (A \B) ' (A \ C).

ECON 4117/5111 Mathematical Economics

Fall 2007

Test 2 October 12, 2007Answer ALL Questions Time Allowed: 1 hour

Read Me: 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. Donot forget to put your name on the front page.

1. Let X be an infinite set. For x, y ! X, define

!(x, y) =!

1 if x "= y,0 if x = y.

(a) Prove that (X, !) is a metric space.(b) Find the open ball B1(x) for any x ! X.

2. Consider the Euclidean space R2. Let A = (a, b),that is, A is an interval on the horizontal axis. De-termine and explain whether A is

(a) open,(b) closed,(c) perfect,(d) compact.

Find b(A).

3. Let 0 be the zero vector in a vector space V . Usingthe axioms listed in the Appendix, prove that forevery vector x ! V ,

(a) "0 = 0 for all " ! R,(b) if "x = 0 then either " = 0 or x = 0.

Indicate which axioms you use in each step of theproof.

4. Let V be a vector space.

(a) Define a subspace S # V .(b) Show that 0 ! S.(c) Show that for all x ! S, $x ! S.

5. Consider a function f : E % R where E =[$#/2,#/2]. Explain the following:

(a) Let f(x) = sin x. Is f bijective?(b) Let f(x) = cos x. Is f bijective?

Appendix — Axioms for Vector Spaces

Let V be a vector space. For all x,y, z ! V and ", $ ! R,there exists vectors x + y,y + z ! V such that

1. x + y = y + x

2. (x + y) + z = x + (y + z)

3. &0 ! V ' x + 0 = x

4. (x ! V, & $ x ' x + ($x) = 0

5. ("$)x = "($x)

6. 1x = x

7. "(x + y) = "x + $y

8. (" + $)x = "x + $x

ECON 4117/5111 Mathematical Economics

Fall 2007

Test 3 October 26, 2007Answer ALL Questions Time Allowed: 1 hour

Read Me: 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. Donot forget to put your name on the front page.

1. Let u : X ! R be a utility function on an orderedset (X, !). Let g : R ! R be a strictly increasingfunction. Show that g " u is also a utility functionrepresenting the order !.

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

f(x, y) =!

x2 + y2 for (x, y) #= (0, 0)2 for (x, y) = (0, 0) .

(a) Define the graph of f .(b) Is f upper semi-continuous? Explain.(c) Is f lower semi-continuous?

3. Let f : V ! W be a linear transformation from Vinto W .

(a) Show that f(0) = 0.(b) Define the kernel of f .(c) Show that the kernel is a subspace of V .

4. Define a function f : R2 ! R by

f(x, y) = |x| + |y|.

Is (R, f) an inner product space? Explain.

5. Let V be an inner product space. For all x,y $ V ,define the norm as %x% = (xTx)1/2.

(a) State the Cauchy-Schwarz Inequality.(b) Prove that

%x + y% & %x%+ %y%.

Happy Halloween

ECON 4117/5111 Mathematical Economics

Fall 2007

Test 4 November 9, 2007Answer ALL Questions Time Allowed: 1 hour

Read Me: 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. Donot forget to put your name on the front page.

1. Let the matrix representation of a linear operator fon a vector space V with respect to a basis be

A =

!

"3 4 04 !3 00 0 2

#

$ .

Find the eigenvalues and normalized eigenvectors off .

2. Suppose that f is a symmetric linear operator on afinite-dimensional vector space V .

(a) Let ! be an eigenvalue of f . Show that theset of eigenvectors corresponding to !, togetherwith 0, form a subspace of V .

(b) Let !1 and !2 be two distinct eigenvalues off . Show that their corresponding eigenvectorsare orthogonal.

3. Suppose that f is a symmetric linear operator on avector space V .

(a) Define the quadratic form Q.(b) Show that for every x,y " V ,

xTf(y) =14Q(x + y)! 1

4Q(x! y).

4. Let

A =

!

"0 0 00 !3 00 0 2

#

$ , B =

!

"1 0 00 3 00 0 0

#

$ ,

and

C =

!

"!3 0 00 !1 00 0 !2

#

$ .

(a) Determine the definiteness of A,B, and C.(b) Determine the definiteness of A + B + C.

5. Let S be a convex cone in a vector space V .

(a) Define a linearly homogeneous function f :S # R.

(b) Define a homothetic function.(c) Let g : R # R be a strictly increasing function.

Is g $ f homothetic? Explain.

ECON 4117/5111 Mathematical Economics

Fall 2007

Test 5 November 23, 2007Answer ALL Questions Time Allowed: 1 hour

Read Me: 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. Donot forget to put your name on the front page.

1. Let f : Rn ! R be a C1 function.

(a) Define the directional derivative of f in the di-rection of a unit vector u.

(b) Suppose that f(x, y) = xy. Find the direc-tional derivative at the point (2, 3) in the di-rection of (1, 2).

2. Let f : R2 ! R2 be given by

f(x) = (x1 cos x2, x1 sinx2).

(a) Find the Jacobian of f at any point x.

(b) Find the set of points that the inverse of f doesnot exist.

3. Consider the unconstrained maximization of the ob-jective function f(x, !) where x " Rn is the controlvariable and ! " Rl is the parameter vector. Thesolution can be expressed as x! = "(!).

(a) What is the necessary condition for x! to be alocal maximum?

(b) What is the su!cient condition for x! to be aglobal maximum?

(c) Find the impact of a change in ! on x!, thatis, find D"(!).

4. Let f(x, y) = xy + 3y # x2 # y2.

(a) Find the stationary point(s) of f .

(b) Determine whether you have found a maxi-mum, a minimum, or neither.

5. Find the second order Taylor approximation of thefunction f(x, y) = ex+y about the origin.

“This is only linear algebra. The calculus part will be waytougher.”

ECON 4117/5111 Mathematical Economics

Fall 2007

Final Examination (RC3014) December 5, 2007Answer ALL Questions 1:00 – 3:00 PM

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. Also,start each question on a new page. Read the questionscarefully and provide answers to what you are askedonly. Do not spend time on what you are not asked todo. Remember to put your name on the front page.

1. A consumer has a utility function on two goodsgiven by

U(x1, x2) = x1/31 x2/3

2 .

Suppose the market prices of the goods are p1 andp2 respectively. Total expenditure is M .

(a) Write down the budget constraint as an equal-ity.

(b) Find the demand functions of the two goods.(c) Find the indirect utility function.

2. Solve the problem

minx,y,z x2 + y2 + z2

subject to 2x! 3y + 5z ! 19 = 0.

Find the bordered Hessian.

3. Let f : S " R where S # Rn is a convex set.Suppose f is a C1 concave function and $f(x!) =0. Show that x! % S is a global maximum for fover S. (Hint: see Appendix)

4. Let U : Rn+ " R be the utility function of a con-

sumer. For u % f(Rn+) and p % Rn

++, define theconsumer’s expenditure function as

E(p, u) = minx

{pTx : U(x) = u}.

Prove that E satisfies the following properties:

(a) E(!p, u) = !E(p, u) for ! > 0;(b) E is increasing in p, that is,

$pE(p, u) & 0.

5. Refer to Question 4 above. Show that E(p, u) isconcave in p for fixed u.

6. Refer to Question 4 above. Prove that

(a) $pE(p, u)Tp = E(p, u),(b) $2

pE(p, u)p = 0.

7. Solve the following optimization problem:

maxx,y x + 5 log ysubject to 2x + y ' 10,

x & 0,y & 0.

Appendix: Concave Functions

The following is a list of some useful characterizationsof a concave function:

1. The hypograph of f , {(x, z) % S (R : f(x) & z} isconvex.

2. For every x,y % S and ! % [0, 1],

f(!x + (1! !)y) & !f(x) + (1! !)f(y).

3. For every x,y % S,

($f(x)!$f(y))T(x! y) ' 0.

4. For every x,x0 % S,

f(x) ' f(x0) +$f(x0)T(x! x0).

5. For every x % S, the Hessian $2f(x) is negativesemi-definite.