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Find Pareto optimal allocations and the core for the following economies.

There are two consumers and two goods. Utility functions are $u_1(x_1,y_1)= 10x_1-(y_1-2)^2$ and $u_2(x_2,y_2) = 10y_2 − (x_2 − 2)^2$. The endowments are (0,4) for agent 1 and (4,0) for agent 2.

I have its intuitive solution but I don’t understand this.

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Please help me to provide a solution by calculation. Thanks a lot.

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In the economy described above, set of efficient allocations is given by the red curve. Just do the slope of ICs comparisons at the boundaries and you will get that.

Set of efficient allocations

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    $\begingroup$ Many Thanks dear Amit☺️ $\endgroup$ – 1190 Jan 25 at 15:15

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