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## Individual Rationality and the Core

Individual Rationality and the Core:• The Pareto principle does not ask how we get to an allocation, only whether there could be gains to everyone from moving away from it. Allocation D is Pareto optimal, although it would not be individually rational for Bob to trade from A to D.

• The First Welfare Theorem guides us in our hunt for a competitive equilibrium: restrict our search to the Pareto set.

• The search can be further restricted by limiting our attention to individually rational allocations: an allocation is individually rational if each agent does at least as well as at this endowment: For every agent construct the set of allocations that are weakly preferred to the endowment allocation. As well as intersection of these sets is the region of mutual advantage.

• Competitive equilibria should be individually rational as well as Pareto optimal.

• In he two-agent economy, agents would bargain bilaterally rather than behave competitively, but the equilibrium will be individually rational and Pareto optimal. With two agents, this corresponds to the core.

• Feasible allocation x is an element of the core of an economy if no group (i.e. breaking coalition) can do better alone.

⇒ Theorem: an aggressive equilibrium allocation is a core allocation.

• Suppose we replicate the 2-agent (i.e. Alice and Bob) endowment economy, so there are r agents of each type: now competition is additionally sensible.

⇒ Replication Theorem- The competitive equilibrium doesn’t depend on r.

⇒ Equal treatment Theorem- Agents of the same type receives the same core allocation.

⇒ Core Convergence Theorem- As r grows; the core converges to the competitive equilibrium allocation.

• These results permit us to use the Edge worth box to characterize the equilibrium with many agents of each type (core is generally smaller than the set of individually rational Pareto optima).

(Individual Rationality and the Core)

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