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Suppose that there exists 3 consumers and 2-type.
Type one has 2 consumers $x^{11} , x^{12}$ and they are not equally treated.
But is it possible that they are in core but not equally treated?

Thinking about edgeworth box,
they are in the same IC but their allocations(consumption bundle) are different.
It is possible that both points are in contract curve(which is essential to be core)?

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    – Community Bot
    Oct 10, 2021 at 11:50

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Usually, there won't be equal-treatment. This follows from the results in

Green, Jerry R. "On the inequitable nature of core allocations." Journal of Economic Theory 4.2 (1972): 132-143.

Green shows, as a simple corollary tp the classical equal treatment result of Debreu and Scarf, that if preferences are continuous, strictly monotone, and strictly convex, then equal treatment holds if each types occurs in a number with a common divisor larger than one. If there is no such common divisior and some additional differentiability asssumption holds, then for almost all (except for a closed set of Lebesgue measure zero) endowment allocation, the equal treatment property will fail. In your example with three consumers and two types, there will generically not be equal treatment.

Equal treatment means that consumers with the same type will receive the same commodity bundle. One might ask whether it is possible that this is not the case, but all consumers of the same type are still equally well off. This is not possible, under the assumption above.

Take any allocation in which two agents with the same type receive different but equally desirable commodity bundles. By strict convexity, both would be strictly better off if they would receive both the average of these two bundles, which is feasible. Since they are strictly better off and preferences are continuous and strictly monotone, they could give a bit away to everyone else afterward and still be better off. Then everyone is better off and the grand coalition can block the allocation.

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