Can a quasiconcave function be convex?

The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity.

Are all quasiconcave functions concave?

Thus every upper level set is convex and hence f is quasiconcave. The converse of this result is not true: a quasiconcave function may not be concave.

Can a function be neither convex or concave?

Note that it is possible for f to be neither convex nor concave. We say that the convexity/concavity is strict if the graph of f(x) over the interval I contains no straight line segments.

Are linear functions quasiconcave?

* A function that is both concave and convex, is linear (well, affine: it could have a constant term). Therefore, we call a function quasilinear if it is both quasiconcave and quasiconvex. Example: any strictly monotone transformation of a linear aTx.

Why is a utility function quasiconcave?

Definition: A function is quasiconcave if all of its upper contour sets are convex. Definition: A function is quasiconvex if all of its lower contour sets are convex. So in most of the economics you do, the assumption you will see is that utility functions are quasi-concave.

Can a function be both concave and convex?

That is, a function is both concave and convex if and only if it is linear (or, more properly, affine), taking the form f(x) = α + βx for all x, for some constants α and β. Economists often assume that a firm’s production function is increasing and concave.

Is the exponential function convex?

The exponential function f(x)=ex is convex. It is also strictly convex, since f″(x)=ex>0, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function g(x)=ef(x) is logarithmically convex if f is a convex function.

Is a linear function quasiconcave?

Is Cobb Douglas strictly quasiconcave?

Now, let us apply a monotonically increasing transformation to G – the exponential function: exp{G(x,y)} = Axayb = F(x,y). Thus, we can write any such Cobb-Douglas function as a monotonic transformation of a concave (also Cobb-Douglas) function, which proves that the function is quasiconcave.