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Say that two consumers, A and B, have the same utility function, just $u(x) = (x_1)^2 + (x_2)^2$ for simplicity. If consumer A has endowment $x_A = (4, 3)$, and consumer B has endowment $x_B = (3, 4)$, and the price of good 2 is $p_2 = 1$. I think they would definitely gain if they trade with each other, but what's a price $p_1$ where they will be willing to trade? If the consumers traded a marginal amount at that price, who will give up $x_1$ to receive $x_2$?

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Have a look at this pdf for a complete solution. –  ivan Sep 14 '12 at 9:14
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That seems like a weird utility function. Anyway, it would seem optimal for the two consumers to trade in a way such that consumer A will end up with endowment $(7,0)$ and comsumer B with endowment $(0,7)$. It is unclear to me what "price" means here and how that figure enters the problem. –  Rasmus Sep 14 '12 at 9:15

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