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 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. (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 to be true. Use the tautologies from the appendix to establish the conclusion. Indicate which tautology you 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: If p 2x 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 A n =(-1/n, 1+ n],n 2 N. Find \ n2N A n and [ n2N A n . 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 ) p 1 ) p 2 ) ··· ) q 2 ) q 1 ) 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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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.
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.