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Here's the definitions I'm using, just in case.

Let $E$ be the exchange economy given by agents $A,B$, starting allocations $x_A=(0,1)$, $x_B=(1,0)$ and utility functions given by $u_A=x+y$ and $u_B=x+\sqrt{y}$. It's easy to check that the core of this economy is comprised of any allocation $x_A=(t, 1/4)$, where $t\in[3/4, 1-\sqrt{3/4}]$ and that the only competitive equilibrium is the one given by $x_A=(3/4,1/4)$ together with the price vector $v=(1,1)$.

How can I prove that for every non-competitive core allocation $x$, the 2-fold replica of $x$ is not in the core of the 2-fold replica of $E$?

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It's probably too late for your homework, but you may want to check you computations before you try to answer the 2-replica question.

You claim for instance that the competitive allocation is $ \Big( x_A = (3/4, 1/4), x_B= (1/4,3/4) \Big)$, and is in the core, as any competitive allocation. Yet, the grand coallition can improve upon (there exists pareto improvement at this allocation), in contradiction with the definition of the core (and the properties of competitive equilibria).

You can for instance have $A$ giving $1/4$ of good one to $B$, and receiving $1/4$ of good two, leading to allocation $x'= \Big( x_A = (1/2, 1/2), x_B= (1/2,1/2) \Big)$.Then you would have:

  • $u_A (x') = u_A (x) = 1$
  • $u_B(x') = 1/2 + \sqrt{1/2} \approx 1.20 > u_B (x) = 1/4 + \sqrt{3/4} \approx 1,11$.
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