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.

core(N)↓{x∗}(N→∞)\text{core}(N) \downarrow \{x^*\} \quad (N \to \infty)

Part of these learning paths

Examples & Exercises

In the symmetric economy, compare the core intervals for N=1N = 1 and N=2N = 2 replicas, and verify that the core width shrinks.

N=1:[4.583, 5.417],width=0.833N=1: [4.583,\ 5.417], \quad \text{width} = 0.833
N=2:[4.792, 5.208],width=0.417N=2: [4.792,\ 5.208], \quad \text{width} = 0.417
N=1→2N = 1 \to 2: core width 0.833→0.4170.833 \to 0.417.
Show proof

  1. For N=1N = 1 the core is the individually rational segment of the contract curve (the diagonal), [4.583,5.417][4.583, 5.417].

  2. With N=2N = 2, new coalitions like (2,1)(2, 1) or (1,2)(1, 2) appear.

    For instance, x11=5.42x^1_1 = 5.42 was in the core for N=1N = 1 but is blocked for N=2N = 2 because coalition (1,2)(1, 2) can offer better.

    The remaining core [4.792,5.208][4.792, 5.208] is half as wide.

Show that the competitive allocation stays in the core for every NN, and explain why this is one direction of the Debreu–Scarf limit theorem.

x∗=(5,5)∈core(N)∀Nx^* = (5, 5) \in \text{core}(N) \quad \forall N
CE (5,5)(5, 5) stays in the core for all NN.
Show proof

  1. The competitive price is p1=1p_1 = 1 with allocation x∗1=x∗2=(5,5)x^{*1} = x^{*2} = (5, 5). If a coalition SS blocked with ySy^S, every member strictly improves, but x∗ix^{*i} is budget-optimal so p⋅yi>p⋅eip \cdot y^i > p \cdot e^i; summing contradicts feasibility (same argument as the Ch. 13 exercise).

  2. Hence x∗x^* is never blocked, for any NN. Since the core shrinks to {x∗}\{x^*\} as N→∞N \to \infty, core and competitive allocations coincide in the limit — the Debreu–Scarf theorem.

14. Foundations of Competitive Equilibrium