Convex Preferences and optimal solutions. Let denote the consumer's preference relation on C =R2. We say...

 

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Convex Preferences and optimal solutions. Let denote the consumer's preference relation on C =R2. We say a preference relation is convex (re- spectively, strictly convex) if for any two bundles x and y such that ry (i.e., x and y indifferent), then for any A € [0, 1] (respectively, A € (0,1)), and z = Xr+ (1-X)y, zry (respectively, zx y). We say a preference relation is continuous if the two sets: weakly less preferred: WLP(x) = {y € C\x ≥ y, C} and weakly preferred: WP(x) = {y Cyx, xe C} are "closed" (i.e., contain their boundaries. See discussion in class. Answer the following questions. Let the consumption set be C =R² (a) Show if is convex, WP(x) convex. (b) Show if is strictly convex, WP(x) is strictly convex. (c) Is WLP(x) convex? Consider a consumer facing a budget set B(p, m) = {x € Cpx ≤m} for p >> 0. Define the best choice set X* (p.m) = {r* € C\x*x for all x = B(p, m)} (d) Show if is convex, X*(p, m) might have many elements (i.c., many optimal demand choices). (e) Show if is strictly convex, X* (p, m) is a unique for each price-income pair. Convex Preferences and optimal solutions. Let denote the consumer's preference relation on C =R2. We say a preference relation is convex (re- spectively, strictly convex) if for any two bundles x and y such that ry (i.e., x and y indifferent), then for any A € [0, 1] (respectively, A € (0,1)), and z = Xr+ (1-X)y, zry (respectively, zx y). We say a preference relation is continuous if the two sets: weakly less preferred: WLP(x) = {y € C\x ≥ y, C} and weakly preferred: WP(x) = {y Cyx, xe C} are "closed" (i.e., contain their boundaries. See discussion in class. Answer the following questions. Let the consumption set be C =R² (a) Show if is convex, WP(x) convex. (b) Show if is strictly convex, WP(x) is strictly convex. (c) Is WLP(x) convex? Consider a consumer facing a budget set B(p, m) = {x € Cpx ≤m} for p >> 0. Define the best choice set X* (p.m) = {r* € C\x*x for all x = B(p, m)} (d) Show if is convex, X*(p, m) might have many elements (i.c., many optimal demand choices). (e) Show if is strictly convex, X* (p, m) is a unique for each price-income pair.