Convex preferences: Difference between revisions

Revision as of 19:42, 29 November 2015

Revision as of 19:56, 29 November 2015

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== Examples ==

1. If there is only a single commodity type, then any weakly-monotonically-increasing preference relation is convex. This is because, if <math>y \geq x </math> and <math>z \geq x </math>, then every weighted average of ''y'' and ''zס'' is also <math>\geq x </math>.

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. 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 imples <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>.

== Relation to indifference curves and utility functions ==