[SOLVED] ECON6012 / ECON2125 Semester Two 2024 Tutorial 3 Questions SQL

ECON6012 / ECON2125: Semester Two,

2024

Tutorial 3 Questions

A Note on Sources

These questions and answers do not originate with me.  They have either been in uenced by, or directly drawn from, other sources.

Key Concepts

Compact Sets, The Heine-Borel Property, The Heine-Borel Theorem.

Tutorial Questions

Tutorial Question 1

Use the Heine-Borel Theorem to show that

S2  ={(x, y) ∈ R2  : d((x, y), (0, 0)) = 1}

is compact in R2 .

Tutorial Question 2

Use the Heine-Borel Theorem to show that [—1, 1] × [—1, 1] is compact in R2 . (You may use the fact that the functions fi  : R2  —→ R defined by f1 (x, y) = x and f2 (x, y) = y are both continuous.)

Tutorial Question 3

Consider a consumer whose preferences are defined over the consump- tion set X  = R2+.  This consumption set consists of bundles of non- negative quantities of each of two commodities. Denote atypical con- sumption bundle by (x1 , x2), where x1  is the quantity of commodity one in the consumption bundle and x2  is the quantity of commodity two in the consumption bundle. Suppose that this consumer faces a budget constraint of the form. p1 x1  + p2 x2 ≤ y, where p1 > 0 is the linear price per unit for commodity one, p2 > 0 is the linear price per unit for commodity two, and y >  0 is the consumer’s income. The consumer also faces non-negativity constraints on his or her con- sumption of each commodity. This means that x1 > 0 and x2 > 0.

1. What is the consumer’s constraint set?

2. Is the consumer’s constraint set a subset of his or her consump- tion set?

3. Is the consumer’s constraint set a proper subset of his or her consumption set?

4. Is the consumer’s constraint set non-empty?

5. Is the consumer’s constraint set a compact set?  Justify your answer.

Additional Practice Questions

Additional Practice Question 1

Use the Heine-Borel Theorem to show that

S3  ={(x,y, z) ∈ R3  : d((x,y, z), (0, 0, 0)) = 1}

is compact in R3 .

Additional Practice Question 2

Use the Heine-Borel Theorem to show that [—1, 1]n  is compact in Rn. (You may use the fact that the functions fi  : Rn  —→  R defined by f1 (x, y) = xi  for each i ∈ {1, 2, · · · , n} are all continuous.)