# Convex preferences

Summary

Convex preferences

• $$\left( x_1,x_2 \right)$$ and $$\left( y_1,y_2 \right)$$ are two different arbitrary bundles where $$\left( x_1,x_2 \right)~\left( y_1,y_2 \right)$$ .
• If $$\left( z_1,z_2 \right)≽\left( x_1,x_2 \right)$$ for all a bundles $$\left( z_1,z_2 \right)$$ located on a straight line connecting $$\left( x_1,x_2 \right)$$ and $$\left( y_1,y_2 \right)$$ then preferences are said to be convex .
• If $$\left( z_1,z_2 \right)≻\left( x_1,x_2 \right)$$ for all a bundles $$\left( z_1,z_2 \right)$$ located on a straight line connecting $$\left( x_1,x_2 \right)$$ and $$\left( y_1,y_2 \right)$$ , with the exception of $$\left( x_1,x_2 \right)$$ and $$\left( y_1,y_2 \right)$$ , then preferences are said to be strictly convex .

• Given
• A two goods model
• Weak preferences which are totally ordered, strictly monotonic and (strictly) convex
• $$x_2=f\left( x_1 \right)$$ is the equation for a given indifference curve
• Then
• $$f$$ is strictly decre a sing and (strictly) convex .
• If $$f$$ is differentiable then $$f'\left( x_1 \right)<0$$ and $$f^{''}\left( x_1 \right)≥0$$ for all $$x_1$$
• We say that preferences are well-behaved if weak preferences are totally ordered, strictly monotonic and strictly convex