On the Existence of Competitive Equilibrium with Chores

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" summary

Shows that deciding whether a competitive equilibrium exists with divisible chores is NP-hard to decide existence, gives two simple sufficient conditions that guarantee existence — no subset of agents only do a strict subset of chores they own and disutility graph is disjoint union of bicliques — and proves existence under those conditions via a nested fixed-point proof (Brouwer then Kakutani), plus demonstrates a one-edge-from-biclique counterexample where equilibrium fails.

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" chapters (15)

1. 0:00Intro
2. 0:16Allocation in the Exchange model
3. 0:47Exchange Markets (Exchange Setting with Goods)
4. 5:56CE in Exchange Setting with Chores
5. 8:54Existence of CE in the Exchange Setting
6. 15:46Preliminaries- Brouwer's Fixed Point Theorem
7. 16:15Preliminaries-Kakutani's Fixed Point Theorem
8. 17:25Proof of Existence-Goods Case
9. 18:48Defining the Correspondence (Goods Case)
10. 20:35Fixed Point to CE
11. 24:06Generalizing the Approach to Chores (Problem)
12. 26:47Defining the new Correspondence
13. 28:02Nested Fixed Point Theorems
14. 29:09Summary
15. 29:50Future Directions