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Consider an Edgeworth box economy with endowments eA = (1, 0) and eB = (0, 1) and utility functions uA(x1,x2) and uB(x1, x2), where uA and uB are continuous and strictly quasi-concave.
(a) Can an equilibrium exist if both utility functions are everywhere strictly decreasing with respect to x1 and x2?
(b) Does an equilibrium necessarily exist if both uA and uB are decreasing with respect to x1 and increasing with respect to x2?
(c) Does an equilibrium necessarily exist if uA is decreasing with respect to x1 and increasing with respect to x2, and uB is increasing with respect to x1 and decreasing with respect to x2?
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