ECON6012 / 2125: Semester Two, 2024
Tutorial 2
Tutorial Assignment 1
This assignment involves submitting answers for each of the tutorial ques-tions, but not for the additional practice questions, that are contained on the tutorial 2 questions sheet (this document). You should submit your answers on the Turnitin submissions link for Tutorial Assignment 1 that is available on the Wattle site for this course (under the “In-Semester Assess-ment Items” block) by no later than 08:00:00 am on Monday 5 August 2024. If you have trouble accessing the Wattle site for this course or the Turnitin submission link, please submit your assignment to the course email address (which is [email protected] if you are a postgraduate student, and [email protected] if you are an undergraduate student). One of the tutorial questions will be selected for grading and your mark for this tuto-rial assignment will be based on the quality and accuracy of your answer to that question. The identity of the question that is selected for grading will not be revealed to students until some point in time after the due date and time for submission of this assignment.
A Note on Sources
These questions and answers do not originate with me. They have either been influenced by, or directly drawn from, other sources.
Key Concepts
Binary Relations, Potential Properties of Binary Relations, Weak Pref-erence Relations, Indifference Relations, Strict Preference Relations, Al-gebraic Structures, Metric Spaces, Properties of Distance Metrics, Open Sets, Closed Sets, Clopen Sets, Sets that are Neither Open Nor Closed, Continuity.
Tutorial Questions
Tutorial Question 1
1. Show that (Q, +) is a group when addition of two rational numbers is defined by
b/a + d/c = bd/ad + bc.
2. Show that (Q\ {0} , ×) is a group when multiplication of two rational numbers is defined by
b/a × d/c = bd/ac.
Tutorial Question 2
1. Show that the discrete metric is a valid distance metric for any set X.
2. Show that the Euclidean metric is a valid distance metric for Rn, where n ∈ N.
3. Show that the sum-of-absolute-differences metric is a valid distance metric for R
n
, where n ∈ N.
Tutorial Question 3
Let (A, d) be a metric space and suppose that a, b ∈ A with a = b. Prove by contradiction that B∈(a) ∩ B∈(b) = ∅ for any ∈ < d(a,b
2
)
. (Hint: You will need to use the triangle inequality.) (Note that B∈(x) is the “open epsilon-neighbourhood”, or “open epsilon-ball”, centred on the point x ∈ A and with a “radius” of ∈ > 0.)
Tutorial Question 4
Let (A, d) be a metric space and suppose that x ∈ A and s > 0. Given any a ∈ Bs(x), show that there exists some r > 0 such that Br(a) ⊂ Bs(x).
Tutorial Question 5
Prove that both (−∞, x] and [x, ∞) are closed subsets of R for all x ∈ R.
Additional Practice Questions
Additional Practice Question 1
Lexicographic Preferences on R
2
+: Suppose that a consumer has lex-icographic preferences over bundles of non-negative amounts of each of two commodities. The consumer’s consumption set is R
2
+. The consumer weakly prefers bundle a = (a1, a2) over bundle b = (b1, b2) if either (i) a1 > b1, or (ii) both a1 = b1 and a2 > b2. In any other circumstance, the consumer does not weakly prefer bundle a to bundle b. (Note that these preferences are not continuous. Furthermore, they cannot be represented by a utility function.)
1. Under what circumstances will bundle c = (c1, c2) be strictly pre-ferred to bundle d = (d1, d2)?
2. Under what circumstances will bundle c = (c1, c2) be indifferent to bundle d = (d1, d2)?
3. Are these preferences weakly complete? Explain why.
4. Are these preferences reflexive? Explain why.
5. Are these preferences strongly complete? Explain why.
6. Are these preferences transitive? Explain why.
7. Are these preferences rational? Explain why.
8. Show that these preferences are not continuous.
Additional Practice Question 2
Show that {(x, y) ∈ R2
: y > x} is open in R2.
Additional Practice Question 3
Let (A, d) be a metric space in which d is the discrete metric.
1. Show that every subset of A is clopen in this case.
2. Show that f : A → B is continuous for any metric space (B, r).