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I have been struggling trying to understand the difference between a quasiconcave and a convex utility function. As far as I understand a function can be both at a certain point, but is not clear to me why sometimes is said that the function is quasiconcave rather than just convex.

Can someone please tell me what are the implication of

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Strict quasiconcavity implies single-peakedness, i.e. any strictly quasiconcave function has a unique supremum (or maximum if the domain is compact). Hence, any strictly increase convex function is also strictly quasiconcave.

Here are a couple figures to illustrate the difference. The function on the right is not quasiconcave because it has two local maxima.

enter image description here

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Quasiconcavity is a generalization of concavity. And quasiconvexity is a generalization of convexity. There are functions which are quasiconcave and convex at the same time, for example a linear function. However, sometimes when economists talk about a convex utility function they don't mean that the function is convex, rather the preferences are convex, and this mathematically means that the upper contour sets are convex sets. In consumer theory, the upper contour sets can be interpreted as "the set of all goods bundles that are viewed as being at least as desired". It turns out that convexity of upper contour sets is the definition of a quasiconcave function. So convexity of preferences is equivalent to quasiconcavity of the utility function.

The conclusion is that we have to be very careful when we refer to preferences (convex) and to utility (quasiconcave).

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