Convexity Postulate

Chapter 4, Section 5: Convexity Postulate

We can safely install another constraint to behaviors. That is, an indifference curve must curl inwards (concaving towards the lower left), like the bow weapon of Hua Rong the Little Li Guang, a fiction character in Water Margin, which "bends to a full moon". (Well, this is a joke, indifference curve doesn't have to curl that heavily.) This constraint (indifference curve neither is straight nor curls outwards) is called the convexity postulate, or the postulate of diminishing marginal rate of substitution.

Intention of the constraint is obvious. If utility stays unchanged (on the same indifference curve), the more A goods one has, the less willing he must be to substitute his B goods for more of A. This postulate is safe, if only the substitute happens on a same indifference curve. If the wealth or income of this man increases and he jumps to a higher indifference curve, his marginal rate of substitution will be totally changed. This is a great obstacle for the application of utility analysis to behavior prediction, and disables an important constraint to behaviors. Later we will get back to this.

Well, by the same indifference curve, the convexity postulate has a conclusion, which is not so useful. The conclusion says, if the price of a good decreases, the quantity demanded for this good on a same indifference curve must increase. That's because a price is always relative, when the price of a good A decreases it actually means the sacrifice of other goods needed for same amount of A reduces. In that way, the decrease in marginal rate of substitution would enlarge the quantity demanded for this price-lowered good.

The difficulty is, indifference curves and corresponding utilities are castles in the air, imagined by economists, and don't actually exist in reality. We cannot make sure whether one's choices would remain on the same indifference curve when the price of some good decreases. The logic reasoning for that is: when the price decreases, a consumer's real income would relatively increase, so he will jump to another higher indifference curve. One step further, when the marginal rate of substitution changes due to some other reason, what can we do?