<p>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 <span class="math-container">$f$</span> where <span class="math-container">$f(x) = y$</span>.</p>

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

CC BY-SA 4.0
2019-01-18T18:25:40.537Z
2019-01-18T18:25:40.537Z
Giskard
<p>Let's say we have an utility function, <span class="math-container">$ U(x,y) = \sqrt{x \cdot y} $</span>. The indifference curve associated with this is <strong>convex</strong>, while the function itself is quasi concave (because it satisfies <span class="math-container">$ f_{xx} f_x^2 - 2 f_{12} f_1 f_2 + f_{yy} f_y^2 $</span>). </p>

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

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

(Asked by: username_0 on 2019-01-18)

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)?

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2019-01-18. Score: 3

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.


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