代写ECON6012 / ECON2125: Semester Two, 2024 Tutorial 3 Questions代做留学生SQL语言程序

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.)