Econ 700 Final Exam
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Econ 700 Final Exam
December 20, 2022
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. (60 points) Please state whether each of the following statements is true or false. Prove the true statements and disprove the false statements.
(a) Q (the set of the rationals) is not open and not closed.
(b) All Cauchy sequences are monotone.
(c) If f : R 一 R is a continuous function and {xn } is a Cauchy sequence in R, then the sequence {yn } defined by yn = f(xn ), An E N, is also a Cauchy sequence in R.
(d) Concavity is an ordinal property.
(e) v(x,y) = log x - logy is homothetic.
2. (20 points) Find all critical points off(x,y) = 2x4 +y2 -xy+1 and check the second order conditions at each critical point to determine what local maxima and minima exist for this function.
3. (20 points) Consider the following utility maximization problem, and an- swer the following questions:
c(m)xc1/2l1/2
subject to
c ≤ w(1 − l)
c ≥ 0
0 ≤ l ≤ 1
(a) Solve the utility maximization problem by checking the NDCQ, find- ing the complete set of first order conditions from the Lagrangian, and finding all critical points (including the value of all multipliers). What is the optimal choice of consumption (c) and leisure (l) as a function of the wage (w)?
(b) Are the utility-maximizing values of consumption and leisure (c* (w) and l* (w)) homogeneous as a function of w? Is the maximized utility u(w) homogeneous? If so, what is the degree of homogeneity in each case? How will each change if the wage is doubled?
4. (24 points) Consider the set (0, 1]:
(a) Using the definition of finite covering property directly, show that the set (0, 1] does not have the finite covering property.
(b) Without using the definition of finite covering property directly, show that the set (0, 1] does not have the finite covering property. (Hint: think about what other properties are equivalent to the FCP.)
5. (26 points) Define f : R2 → R by
f(x,y;a) = x2 + x + ay2 .
Consider the constrained optimization problem of maximizing f(x,y;a) on the constraint set x + y ≤ 1, x ≥ 0, and y ≥ 0.
(a) Briefly explain why a solution of the above constrained optimization problem exists.
(b) Let (x* (a), y* (a)) be the solution of the above constrained optimiza-tion problem given a. Find (x* (1), y* (1)) and f(x* (1), y* (1);1).
(c) Use your result from part (b) and a first order approximation to estimate f(x* (1.3), y* (1.3);1.3).
2023-12-18