Summation ordering is continuous Lexicographic order is discontinuous(see the following diagram on R2+) (x1,x2)>(1,1) AXIOM 4A: (Strong Monotonicity)If x 2 y and x+y, then x>y AXIOM 4B: Weak Monotonicity)If xi 2y for all i, then x 2 Weak monotonicity says that "at least as much of everything is at least as good. If the consumer can costlessly dispose of unwanted goods, this assumption is trivial Strong monotonicity says that at least as much of every good, and strictly more of some good is strictly better. This is simply says assuming that goods are good If one of the goods is a"bad", such as garbage or pollution, then strong monotonicity will not be satisfied. But we can easily get around this problem by respecifying the good to be absence of garbage, or absence of pollution, which will normally lead to strong monotonicity AXIOM 5:(Local? Nonsatiation) Given any x in X and E >0, then there is some bundle y in X with lx-y <s such that y>x (An alternative definition: requiring this to hold over some set that contain the set defined by the relevant budget constraint. Local nonsatiation says that one can always do a little bit better, even if one restricted to only small change in the consumption bundle It can be shown that strong monotonicity implies local nonsatiation but not vice versa Key consequence of local nonsatiation rules out"thick"indifference curves The following two assumptions are often used to guarantee nice behavior of consumer demand AXIOM 6A:( Convexity)Given x, y, z E X such that x z and a z, then Ix+(1-Dy 2 zfor ll0≤t≤1. AXIOM 6B: ( Strict Convexity)Given x*y, z e X such that x2 z and y2 z, then x+(1-d)y>z for all 0<(< 1 Convexity implies that an agent prefers average to extremes Convexity is a generalization of the neoclassical assumption of"diminishing marginal rates of substitution Before we move on the functional representation of the preference relation, we must emphasize that the a preference relation is an ordinal, rather than cardinal, concept even though we have attempted to incorporate additional structures by imposing some of the above assumptions3 • Summation ordering is continuous. • Lexicographic order is discontinuous (see the following diagram on R2 +) AXIOM 4A: (Strong Monotonicity) If x  y and x  y, then x  y. AXIOM 4B: (Weak Monotonicity) If xi  yi for all i, then x y. • Weak monotonicity says that "at least as much of everything is at least as good." If the consumer can costlessly dispose of unwanted goods, this assumption is trivial. • Strong monotonicity says that at least as much of every good, and strictly more of some good, is strictly better. This is simply says assuming that goods are good. • If one of the goods is a "bad", such as garbage or pollution, then strong monotonicity will not be satisfied. But we can easily get around this problem by respecifying the good to be absence of garbage, or absence of pollution, which will normally lead to strong monotonicity. AXIOM 5: (Local? Nonsatiation) Given any x in X and  > 0, then there is some bundle y in X with || x - y || <  such that y  x. (An alternative definition: requiring this to hold over some set that contain the set defined by the relevant budget constraint.) • Local nonsatiation says that one can always do a little bit better, even if one is restricted to only small change in the consumption bundle. • It can be shown that strong monotonicity implies local nonsatiation but not vice versa. • Key consequence of local nonsatiation rules out "thick" indifference curves. The following two assumptions are often used to guarantee nice behavior of consumer demand functions. AXIOM 6A: (Convexity) Given x, y, z  X such that x z and y z, then tx + (1-t)y z for all 0  t  1. AXIOM 6B: (Strict Convexity) Given x  y, z  X such that x z and y z, then tx + (1-t)y  z for all 0 < t < 1. • Convexity implies that an agent prefers average to extremes. • Convexity is a generalization of the neoclassical assumption of "diminishing marginal rates of substitution." Before we move on the functional representation of the preference relation, we must emphasize that the a preference relation is an ordinal, rather than cardinal, concept even though we have attempted to incorporate additional structures by imposing some of the above assumptions. x2 {(x1, x2)  (1, 1)} 1 1 x1