Convex preferences: Difference between revisions

Revision as of 17:26, 23 October 2019

Revision as of 11:30, 17 November 2021

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2. Consider an economy with two commodity types, 1 and 2. Consider a preference relation represented by the following [[Leontief utility function]]:

:<math>u(x_1,x_2) = \min(x_1,x_2)</math>

This preference relation is convex. {{em|Proof}}: suppose ''x'' and ''y'' are two equivalent bundles, i.e. <math>\min(x_1,x_2) = \min(y_1,y_2)</math>. If the minimum-quantity commodity in both bundles is the same (e.g. commodity 1), then this implesimplies <math>x_1=y_1 \leq x_2,y_2</math>. Then, any weighted average also has the same amount of commodity 1, so any weighted average is equivalent to <math>x</math> and <math>y</math>. If the minimum commodity in each bundle is different (e.g. <math>x_1\leq x_2</math> but <math>y_1\geq y_2</math>), then this implies <math>x_1=y_2 \leq x_2,y_1</math>. Then <math>\theta x_1 + (1-\theta) y_1 \geq x_1</math> and <math>\theta x_2 + (1-\theta) y_2 \geq y_2</math>, so <math>\theta x + (1-\theta) y \succeq x,y</math>. This preference relation is convex, but not strictly-convex.

3. A preference relation represented by [[linear utility]] functions is convex, but not strictly convex. Whenever <math>x\sim y</math>, every convex combination of <math>x,y</math> is equivalent to any of them.