In a two-person exchange economy, the core contains many allocations — the entire segment of the contract curve between the two agents' indifference curves through the endowment. What happens as you replicate this economy by adding more agents with identical preferences and endowments?
AThe core expands, because more agents means more possible mutually beneficial trades
BThe core remains the same size, because adding identical agents doesn't change the fundamental tradeoff
CThe core shrinks, because each agent now has better outside options and can block more allocations
DThe core disappears entirely once a third agent is added
As the economy is replicated, each agent has more potential trading partners and therefore better outside options. A coalition of, say, half the buyers and half the sellers can now arrange a better deal internally, blocking allocations that would have survived in the two-person case. In the limit of infinitely many agents, the core converges exactly to the competitive equilibrium allocations — this is the core equivalence theorem.
Question 2 Multiple Choice
An allocation is proposed in a 10-agent economy. A coalition of 4 agents realizes that by redistributing their own endowments among themselves, all 4 can be made strictly better off than under the proposed allocation. What can we conclude?
AThe allocation is Pareto efficient but not individually rational
BThe allocation is blocked by this coalition and therefore not in the core
CThe allocation is in the core, because the other 6 agents are unaffected
DThe allocation is a competitive equilibrium, since no outside agent is harmed
An allocation is blocked if any coalition can redistribute its own endowments to make all members at least as well off and at least one strictly better off. The proposed allocation fails this test — the 4-agent coalition can do better. That means the allocation is not in the core, regardless of what happens to the other 6 agents. The core only contains allocations that no coalition — of any size — can profitably deviate from.
Question 3 True / False
In a large replicated economy, the core converges to the set of competitive equilibrium allocations.
TTrue
FFalse
Answer: True
This is the core equivalence theorem (Edgeworth's conjecture, proven by Debreu and Scarf). As the economy grows, each agent's outside options improve because there are more potential coalition partners. More and more non-competitive allocations become blockable by some coalition. In the limit, only the competitive equilibrium allocations survive — no coalition can profitably deviate from them. This result provides a game-theoretic foundation for competitive equilibrium without assuming price-taking behavior.
Question 4 True / False
In a two-person Edgeworth box economy, the core is identical to the entire Pareto frontier (the full contract curve).
TTrue
FFalse
Answer: False
The core in a two-person economy is only the segment of the contract curve that lies between the two agents' indifference curves through the initial endowment point. Allocations on the contract curve outside this segment are Pareto efficient but not individually rational — at least one agent would be worse off than at their endowment, so that individual can block the allocation by simply refusing to trade. The core requires both Pareto efficiency and individual rationality.
Question 5 Short Answer
Why does the core shrink as the economy is replicated, and what does this convergence imply about the relationship between competitive equilibrium and cooperative game theory?
Think about your answer, then reveal below.
Model answer: As the economy is replicated, each agent gains more potential trading partners, which improves their outside options. Any allocation that deviates from competitive prices can be blocked by a coalition: for example, if a seller is receiving below the competitive price, a coalition of sellers and buyers can form and trade at a better price, leaving the deviating allocation blocked. In the limit, only competitive equilibrium allocations survive because no coalition can do better by breaking away. This implies that competitive equilibrium is not merely an artifact of assuming price-taking behavior — it is the outcome that emerges from free coalition formation in large economies, bridging cooperative game theory and the Walrasian tradition.
The core equivalence result is profound because it shows two very different analytical frameworks — cooperative game theory (which asks what coalitions can enforce) and Walrasian equilibrium theory (which asks what prices clear markets) — converge on the same answer when markets are thick. Competition 'disciplines' agents not because they are forced to take prices as given, but because in a large economy, any group that tries to deviate can be undercut by a competing coalition.