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Suppose that an economy has $n$ agents, each with endowment $\omega_i$. Their preferences are represented by the quasiconcave and increasing function $u_i$. Let the set of Pareto efficient allocations be $E$ and let $C_i$ be the upper-contour set for $u_i(\omega_i)$ for all $i$.

My question is: can we define the core as the intersection of $E$ and all $C_i$, so $core = E \cap \left(\bigcap_{i = 1}^n C_i\right)$? I think this definition works when $n = 2$, but what about when $n > 2$?

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Assume there are three individuals $A, B, C$ and three goods $x, y, z$. Endowment for $A$ is $(1,0,0)$ for $B$ is $(0,1,0)$ and for $C$ is $(0,0,1)$.

Assume the utility function for the individuals are: $$ u^A(x,y,z) = xy + z, u^B(x,y,z) = x + yz, U^C(x,y,z) = xz + y $$ The utilities at the endowments are $$ u^A(1,0,0) = 0, u^B(0,1,0) = 0, u^C(0,0,1) = 0. $$ Now consider the allocation where $A$ gets $(0,0,0)$ for $B$ receives $(1,1,1)$ and for $C$ receives $(0,0,0)$. This gives utilities: $$ u^A(0,0,0) = 0, u^B(1,1,1) = 2, u^C(0,0,0) = 0, $$ So these are individual rational. It is also clear that this is Pareto efficient as any reallocation would decrease the utility of $B$.

Note however that the coalition $\{A, C\}$ has a profitable deviation. For example, they could switch their endowments, so $A$ consumes $(0, 0, 1)$ and $C$ consumes $(1, 0,0)$, giving both a utility equal to 1. So the above Pareto efficient and individual rational allocation is not in the core.

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