what does the axiom of strict convexity of imply about prefe

what does the axiom of strict convexity of imply about preferences??

Solution

Economics: Situation in which a combination of two or more items is preferable to any one of the individual items. If, for example, someone prefers one slice of bread and half a glass of milk than either a whole bread or a whole glass of milk, he or she has convex preference. Similarly, if it is easier to make a product using two men and two machines than it is using four machines and no men, it is a convex technology.
Convexity: if x\'>=x\'\' then tx\'+(1-t)x\'\'>=x\'\' for all t [0,1]
Strict Convexity: if x\' different from x\'\' and x\'>x\'\' then tx\'+(1-t)x\'\'>x\'\'

Convex Preferences Take one particular consumption bundle x : the at least as good as set (x) is the set of all consumption bundles which the person finds at least as good as the “reference bundle” x. In other words, (x) is the set consisting of all the consumption bundles on the indifference curves through x, or on better indifference curves. Preferences are said to be convex if, for any reference bundle x X, the at least as good as set (x) is a convex set.

An (equivalent) alternate definition : preferences are convex if : for any consumption bundle x, if x 1 x, and if x 2 x, and if x 3 tx 1 + (1 t)x 2 where t is any number between 0 and 1, then x 3 x. That is, if x 1 and x 2 are both at least as good as x, then any convex combination of x 1 and x 2 is at least as good as x. A convex combination of two vectors is defined as any point between the two vectors, on the line connecting the two vectors : every point on the line connecting x 1 and x 2 can be expressed as some fraction of the way along that line, tx 1+(1t)x 2 , where t is the fraction of the way along the line that the point is.

Strictly Convex Preferences

Strict convexity of preferences is a stronger property than just plain convexity. Preferences are strictly convex if : for any consumption bundle x, if x 1 x, and if x 2 x, (with x 1 6= x 2 ) then for any 0 < t < 1, tx 1 + (1 t)x 2 ­ x So, in two dimensions, with strictly monotonic preferences, strict convexity says that if two consumption bundles are each on the same indifference curve as x, then any point on a line connecting these two points (except for the points themselves) will be on a higher indifference curve than x. In two dimensions, if indifference curves are straight lines, then preferences are convex, but not strictly convex.