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Unformatted text preview: there exists some x ∈ such that ∑ =1 ≤ Therefore x ∈ ( p , ) which is nonempty. Let ˇ = min be the lowest price of the goods. Then ( p , ) ⊆ ( ,/ ˇ ) and so is bounded. (That is, no component of an affordable bundle can contain more than / ˇ units.) To show that ( p , ) is closed, let ( x ) be a sequence of consumption bundles in ( p , ). Since ( p , ) is bounded, x → x ∈ . Furthermore 1 1 + 2 2 + ⋅⋅⋅ + ≤ for every This implies that 1 1 + 2 2 + ⋅⋅⋅ + ≤ so that x → x ∈ ( p , ). Therefore ( p , ) is closed. We have shown that ( p , ) is a closed and bounded subset of ℜ . Hence it is compact (Proposition 1.4)....
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This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.
- Fall '10