Economic

The Australian National University ECON8013: Semester One, 2021

Tutorial 2 Questions

Dr Damien S. Eldridge

To be Discussed in Week 3 Tutorials. (Assignment Week)

(Due: 8:00 am on Tuesday 9 March 2021.)

A Note on Sources

These questions 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

a

b + c

d = ad + bc

bd .

2. Show that (Q�{0} ,×) is a group when multiplication of two rational numbers is defined by

a

b × c

d = ac

bd .

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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 Rn, where n ∈ N.

Tutorial Question 3

Let (A,d) be a metric space and suppose that a,b ∈ A with a 6= b. Prove by contradiction that B�(a) ∩B�(b) = ∅ for any � <

d(a,b) 2

. (Hint: You will need to use the triangle inequality.)

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 R2+: 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 R2+. 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)?

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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).

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