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Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier. Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here. Here ...


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Without convexity, we'd lose the convenience of using the first order condition (or the tangency condition) to identify the optimal consumption bundle. As shown in the following figure, where the red-shaded region is a non-convex budget set, the tangency at point $E$ is not the optimal consumption point; rather, the optimum is a corner solution, where only $...


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