# If a utility function is quasi-concave, can we say that the IC curve associated with it is convex?

Let's say we have an utility function, $$U(x,y) = \sqrt{x \cdot y}$$. The indifference curve associated with this is convex, while the function itself is quasi concave (because it satisfies $$f_{xx} f_x^2 - 2 f_{12} f_1 f_2 + f_{yy} f_y^2$$).

So, can we say that an utillity function is quasi-concave if the indifference curve is convex (and vice-versa)?

## 1 Answer

In your example the IC curve is not convex in the usual meaning of the word convex when applied to sets. What you probably mean is that the IC curve implicitly defines a convex function $$f$$ where $$f(x) = y$$.

If this is indeed what you mean then as long as you also assume that $$U$$ represents monotone preferences, then the quasi-concavity of $$U$$ will imply the 'convexity' of the IC curve and vice-versa. In this case the upper contour set of any level of $$U$$ will be a convex set, the lower border of which is the 'convex' IC you mention.