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丑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.)