ECON 411 7/ 51 11 Mathema tical Ec onomics F all 200 4flash.lakeheadu.ca/~kyu/E5111/Tests2004.pdfT es t 2 O ctob er 8, 2004 Answ er ALL Ques tions Time Allo w ed : 1 h our Read Me:

ECON 4117/5111 Mathematical EconomicsFall 2004

Test 1 September 24, 2004Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answer book provided. Use the right-side pages for formal answers and the left-side pages for your rough work. Do notforget to put your name on the front page.

1. Determine whether the following statements are true or false. Explain youranswers.

(a) If Professor Moazzami is a woman then every student will get 100% inEconometrics.

(b) [x, y ! R] " [# y $ % x, y > x].

(c) All prime numbers are odd.

(d) This statement is both true and false.

2. Construct a truth table to prove each of the following tautologies (p, q, r arestatements):

(a) [p & (q " r)] ' [(p & q) " (p & r)]

(b) [( q " (p ) q)] ) ( p

3. Consider the compound statement: If f is di!erentiable on an open interval(a, b) and if f assumes its maximum or minimum at a point c ! (a, b), thenf !(c) = 0.

(a) Express the statement in four simple statements p, q, r, and s with logi-cal symbols ),%,#,$,&,", etc. (You have to define each of the simplestatement.)

(b) Find the negation of the statement in logical symbols.

4. If limx"a f(x) *= f(a), then f is not continuous at a. Also, f is di!erentiableat a implies that f is continuous at a. Prove that f is not di!erentiable at awhenever limx"a f(x) *= f(a).

5. Prove that for every positive integer n, n2 + 3n + 8 is even.


Test 2 October 8, 2004Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answer book provided. Use the right-sidepages for formal answers and the left-side pages for your rough work. Do not forget toput your name on the front page.

1. Let X be an infinite set. Suppose that A,B ! X and A " B #= $. Use a Venndiagram to illustrate each of the following sets:

(a) A "B,(b) A \ B.

Prove that A "B = B \ (B \ A). [Note: a Venn diagram is not a formal proof.]John Venn(1834-1923)2. A fair coin has two faces, head (H) and tail (T). In a random experiment a coin is

tossed three times.

(a) List all the elements in the sample space S. (Hint: S is a product set.)(b) What is |S|?(c) Define event E as “exactly two tosses having the same face.” What is E?

3. Let X be the set of all students at Lakehead University. Define a relation R on Xas “is a friend of”.

(a) Is R complete?(b) Is R an equivalence relation?

Give brief explanations to your answers.

4. Let Z = {1, 2, 3, . . .} be the set of natural numbers. Let ! be the relation “is amultiple of”. Define the following sets:

(a) ! (10),(b) " (10),(c) [3, 15)

Is there a worse (first) element in " (10)?

5. Define the following terms:

(a) limit point,(b) interior point, and(c) open set.

Is the set of rational number Q open in R?


Test 3 October 22, 2004Answer ALL Questions Time Allowed: 1 hour

Read Me: Please write your answers on the answer book provided. Use the right-sidepages for formal answers and the left-side pages for your rough work. Remember to putyour name on the front page.

1. Let x,y, z be vectors in a vector space V and !, " ! R. Determine whether thefollowing statements are true or false. If it is true provide a brief explanation,otherwise give a counter example.

(a) y + x = z + x implies that y = z.

(b) !x = !y implies that x = y.

(c) !x = "x implies that ! = ".

(d) (! + ")(x + y) = !x + !y + "x + "y.

2. Show that S = {(x, 0) : x ! R} and T = {(0, y) : y ! R} are subspaces of R2.

(a) Is S " T a subspace of R2?

(b) Is S + T a subspace of R2?

3. Define a basis of a vector space. Let M2!2 be the vector space of 2 # 2 matrices.What is the standard basis of M2!2? What is the dimension of M2!2?

4. Let S and T be convex subsets of a vector space V .

(a) Prove that S $ T is convex.

(b) Is S " T convex?

5. Let f : R % R be the function f(x) = x3.

(a) Is f one-to-one?

(b) Is f onto?

(c) What is f"1([&27, 27])?

(d) Find all the fixed points of f .


Test 4 November 5, 2004Answer ALL Questions Time Allowed: 1 hour


1. Let X, Y , and Z be ordered sets, with orders !X ,!Y , and !Z respectively.

(a) Define an increasing function f : X ! Y .

(b) Suppose that f : X ! Y and g : Y ! Z are increasing functions. Prove thatg " f is increasing.

(c) Suppose that f : X ! R is increasing. Prove that #f is decreasing.

2. Suppose that f : X ! R is continuous. Prove that !f (a) = {x : f(x) $ a} isclosed.

3. Let f : [0.1] ! [0, 1] be continuous. Prove that f has at least one fixed point.

4. Show thatf(x1, x2, . . . , xn) = p1x1 + p2x2 + · · · + pnxn

is a linear functional. What is the matrix representation of f?

5. Let f : X ! Y be a linear function of vector spaces X and Y . Define the followingterms:

(a) rank f ,

(b) kernel of f ,

(c) nullity of f .

State the dimension (rank) theorem.


Test 5 November 19, 2004Answer ALL Questions Time Allowed: 1 hour


1. Let x and y be vectors in an inner product space.

(a) State the Cauchy-Schwarz Inequality(b) Define !, the angle between x and y, as

cos ! =xTy!x!!y!

for 0 " ! " ". Show that #1 " cos ! " 1.(c) Show that x $ y if ! = "/2.

2. Let the matrix representation with respect to the standard basis of the linear oper-ator f : R2 % R2 be

A =!

2 33 4

".

(a) Show that f is invertible.(b) Show that f!1 is represented by the matrix

A!1 =!#4 33 #2

".

(c) Is it true that |A| = 1/|A!1|?

3. Find the eigenvalues and the normalized eigenvectors of the linear operator f inQuestion 2. Show that the eigenvectors are orthogonal to each other.

4. Let f be a linear operator on an n-dimensional vector space V .

(a) State the condition for f to be symmetric.(b) Show that symmetry in f implies that the matrix representation has the prop-

erty A = AT.(c) Let #1,#2, . . . ,#n be the eigenvalues of f . Show that the eigenvalues of f & f

is #21,#

22, . . . ,#

2n.

5. (a) Give a definition of a convex function.(b) Let f : X % R be a convex function on a metric space X and g : R % R be a

convex and increasing function. Show that g & f is convex.


Final Examination December 7, 2004Answer ALL Questions 9:00 – 11:00 AM


1. Let f : R2 ! R2 be given by

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

(a) Find the Jacobian of f at the point a = (1, 0).(b) Is f invertible at a? If no, explain. If yes, find Df!1(b) where b = f(a).

2. Consider the following Keynesian model in macroeconomics:

Y = C[(1" t)Y ] + I(r, Y ) + G,

M/P = L(Y, r).

In this model, Y (output) and r (interest rate) are endogenous and P (price), G(government expenditure), t (tax rate) , and M (money demand) are exogenous.C, I, and L are functions for consumption, investment, and money demand respec-tively.

(a) What assumptions do you have to make in order for the model to have asolution for Y and r?

(b) During a recession, the government increases its expenditure. What is thee!ect of the increase on the interest rate?

3. Consider the functional

f(x, y) = xy + 3y " x2 " y2.

(a) Find the stationary point(s) of f .(b) Is it a maximum, minimum, or neither? Explain.

4. Solve the following maximization problem:

Maximize xysubject to x + 2y # 5, x $ 0, y $ 0.

5. Let f : Rn ! R be a linearly homogeneous C2 functional. Show that for all x % Rn++,

&2f(x)x = 0.

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ECON 411 7/ 51 11 Mathema tical Ec onomics F all 200 4flash.lakeheadu.ca/~kyu/E5111/Tests2004.pdfT es t 2 O ctob er 8, 2004 Answ er ALL Ques tions Time Allo w ed : 1 h our Read Me: