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What are the differences between convex preferences and covex utility function? Why are convexity preferences usually represented by the quasi-concave function and not the convex function?

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Q1:

What are the differences between convex preferences and covex utility function

They have different definitions, which imply different things. From Wikipedia

Formally, a preference relation $\succeq$ on the consumption set $X$ is called convex if whenever $x, y, z \in X$ where $y \succeq x $ and $z \succeq x $, then for every $\theta\in[0,1]$: $$\theta y + (1-\theta) z \succeq x. $$

while a function $f$ is convex, if, again from Wikipedia,

For all $0 \leq t \leq 1$ and all $x_1, x_2 \in X$: $$f\left(t x_1 + (1-t) x_2\right) \leq t f\left(x_1\right) + (1-t) f\left(x_2\right)$$


Q2:

Why are convexity preferences usually represented by the quasi-concave function and not the convex function?

The word convex appears in both, but that does not make it the same property, the definitions are different.

Just looking at the preference relation $$\theta y + (1-\theta) z \succeq x $$ from before, we get the utility equation $$U(\theta y + (1-\theta) z) \geq U(x), $$ which is not very similar to the definition of a convex function.

An example where a convex utility function represents a non-convex preference relation:

The utility function $U(x,y) = x^2 + y^2$ is convex (validate it using the definition). Then $U(6,0) = U(0,6) = 36$, while $U(4,4) = 32$ and $U(3,3) = 18$. $(6,0)$ and $(0,6)$ are preferred to $(4,4)$, as they have a higher utility. The point $(3,3)$ is a convex combination of $(6,0)$ and $(0,6)$. If the preference represented by this utility function was convex, according to the definition $(3,3)$ would also be preferred to $(4,4)$, but it is not, it has a lower utility value.

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