Approximating Competitive Equilibria: A History
summary
A clear, historical account of how Walrasian tâtonnement is modeled as a price-adjustment differential equation, proving global convergence under two explicit conditions: gross substitutes (off-diagonal Jacobian entries positive, unique equilibrium via a Lyapunov function) and a dominant diagonal after fixing a numéraire, while citing Scarf and Gale counterexamples that show nonconvergence otherwise. Tools are Jacobian derivatives, Lyapunov and quasi-Newton methods, and fixed-point existence (Kakutani/Nash); limits include aggregate demand freedom, complementarities and incomplete markets.
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chapters (11)
- 0:00Start
- 12:27Budget Constraint
- 15:26Valrassa's Law
- 16:14Mantel Theorem
- 19:52Abraham Walt
- 23:42The Gauss-Seidel Method of Solving the System of Equations
- 30:18Gross Substitutes
- 33:32The Dominant Diagonal
- 36:13Complementarities
- 39:36Alternative Approaches
- 42:54Does the Market Work
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