# 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