strict convexity

More is better and strict convexity:

In Figure shown below was (implicitly) assumed that goods to the north-east of the indifference curve through A are always better (or no worse) than A, which means that the consumer is a greedy type who always prefers more to less of each good. This, however, is not something which we must assume. It is perfectly possible that the consumer have had enough of both goods at some point and beyond that point they actually are not “goods” but “bad”, and the consumer would never pay anything to acquire more of these goods (in fact he/she must be paid in order take them on). If our consumer has reached such a point we say that he/she is satiated with both goods. Basically, as long as the indifference curves are negatively sloped, and we have not yet reached the satiation point, both goods are actually “good” in the eyes of the consumer. If the indifference curve is positively sloped, the consumer is satiated with one good, but not with the other.

70_indifference set curve.jpg

If our consumer is one of these greedy types which never get enough (cf. the American credo: “too much is never the enough”) we say that her/his preferences are monotonic. This is a very stringent assumption to make, so most economist prefer the more modest assumption of local – nonsation. This means that, over the relevant set of bundles of goods we’re considering (the one within reach of the consumer), the consumer is not satiated with either. We should note that it is less likely that a person is actually satiated with both goods the broader are the categories of goods that we consider. For example if good 1 is bananas and good 2 are all other goods, or simply money income, it is quite likely that one gets satiated after eating, say ten bananas, per day, but will you be satiated with 20.000 or 30.000kr of monthly income? (I’m not, I want a raise!)

The “at least as good as A” set in figure below was drawn as a strictly convex set. This means that if we pick out any two (different) elements (bundles) in the set and draw a straight line between these two bundles, all the bundles on the line must also be in the same set (try it). The economic meaning of the assumption that this set (or the indifference curves) is (strictly) convex is that the consumer usually prefers bundles which contains some of both goods over bundles consisting of only one of the goods. If I had drawn the set the other way around the indifference curve would have been concave and this means that the consumer usually prefers bundles which consist of only one of the goods. It is also possible that the indifference curve could have a big “bulge” in the middle, or meander back and forth. These would be examples of non-convex preferences.

In general we should not rule out any shape of form of the preference sets, a priori, however, life would be easier for us if we could be certain that the sets have the form as in figure shown. The reason for this will become clear later on, but at present we will just hint that if preferences have this shape, the demand curves for both goods will be smooth curves without any annoying holes, or jumps, in them, i.e., they would look the way that we are used to draw them. Since we like to deal with convex preferences, we say that this type of preferences are “well-behaved”.

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