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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 blue curve. Just do the slope of ICs comparisons at the boundaries and you will get that.

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