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Econ 700 Final Exam

December 20, 2021

Answer the following questions to the best of your ability. Do not provide irrelevant information. If a question seems ambiguous to you, state the most reasonable assumptions to resolve the ambiguity and proceed on that basis. Show your work and explain your reasoning. Use the definitions of concepts that we have developed in class. If you would like to use an alternate definition, you should be prepared to show that your definition is equivalent to the one we have used. If we have stated a theorem in class, you may use it without proving it unless I explicitly ask you to prove it. If a theorem was stated only in homework, you will need to prove it in order to use it. No notes are allowed. Label clearly which question you are answering. There are 150 total points available.

1. (26 points) Please state whether each of the following statements is true or false. Prove the true statements and disprove the false statements.

(a) R m and Ø (the empty set) both satisfy the definitions both of an open set and of a closed set in Rm.

(b) If xn is a sequence in R such that |xn − xn+1| ≤ 1/4 |xn−1 − xn|, then xn is a Cauchy sequence.

2. (20 points) Which of the following are the MRS (marginal rate of substi-tution) of a homothetic utility function? Why or why not?

3. (20 points) For each of the following functions, determine whether the function is convex, concave, quasiconvex, or quasiconcave.

4. (21 points) Let U be a convex subset of R n. Let g : U → R be a C1 function. Show that, if U is open and if ∇g(x) ≠ 0 for all x ∈ U, then g is pseudoconcave on U if and only if g is quasiconcave on U.

5. (13 points) Fix x0 > 0. Use the definition of continuity to show that f(x) = x3 is continuous at x0.

6. (30 points) Consider the problem of minimizing 2x2 + 2y2 − 2xy − 9y on the constraint set

4x + 3y ≤ 10, x ≥ 0, y ≥ 0

(a) Prove that a solution exists.

(b) Find all the points that fail the NDCQ.

(c) Find the conditions for a critical point.

(d) Find all critical points and the solution of this optimization problem.

7. (20 points) For each of the following quadratic forms, determine its defi-niteness on the given constraint set: