3. Convex Preferences and optimal solutions. Let > denote the consumer\'s prefer

Question

3. Convex Preferences and optimal solutions. Let > denote the consumer's preference relation on C -RZ. We say a preference relation is convex (re- spectively, strictly convex) if for any two bundles x and y such that xy (i.e., x and y indifferent), then for any [0,1] (respectively, X E (0,1)), and z = Xx4(1-)y, z-x~y (respectively, z x). We say a preference relation is continuous if the two sets: weakly less preferred: WLP(x)-{y E C| y, x C} and weakly preferred: WP(x) = {y E Cly x, x C} are "closed" (i.e., contain their boundaries. See discussion in class Answer the following questions (a) Show if is convex, WP(x) convex (b) Show if is strictly convex, WP(a) is strictly convex (c) Is WLP(x) convex? . Let the consumption set beC-R

Explanation / Answer

(a) For convexity to hold, x WP to z & y WP to z, then it follows that lambda*x + (1- lambda)y WP to z for all lambda lies between 0 to 1.Convexity implies that an agent prefers averages to extremes and convex preferences may have ICs that exhibit flat spots.

(b) Given x not equal to y and if x WP to z & y WP to z then lambda*x + (1- lambda)y WP to z for all lambda lies between 0 to 1.This is the case of strict convexity. Strictly convex preference have ICs that are strictly rotund.

(c) WLP(x) is a convex set. A convex set has the property that if you take any two points, that line segment lies entirely in the set.

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