14. Foundations of Competitive Equilibrium

Why is the price mechanism justified — the cooperative foundations of competitive equilibrium, via the shrinking core and blocking coalitions.

• Shrinking of the Core (Debreu–Scarf) — Copy the same economy and grow its scale. Watch the divisions everyone accepts narrow down to one. When the economy is small, a few agents can break away and do better alone. As 🧑 people multiply, those escape routes get sliced away. Exactly one division survives in the end. It is the very division the market picks. The core shrinks to a point, as in the formula below. For example, splitting 10 🍎 apples between 2 friends, almost any split works. Grow the group to 100 friends and the story changes. Unhappy friends can team up and cook up a new split. In the end only one split survives — one nobody can break away from.
• Blocking Coalitions — Drag the division around and watch who objects. If some group can object, the division falls. If one 🧑 person ends up worse than going alone, that person walks out. If a group can split things better together, it falls too. Only divisions nobody can walk away from survive. That surviving set is the core. For example, three friends split 10 🍬 candies. If one friend gets only 1 candy, that friend walks away. If two friends secretly find a better split, the original split collapses. Only a split nobody wants to leave stays standing.