The Market Demand Curve?
Last week showed that revealed preference (a) doesn’t work and (b) doesn’t describe rational behavior). But let’s assume (i.e. pretend) that RP it does work… How to go from a consumer to a market demand curve?
Key assumption in deriving individual demand curve is – Changing relative prices doesn’t change income. But can’t assume this in general model – Prices determine incomes.
This operation: Assumes changing relative prices doesn’t alter incomes.
Ok for a single consumer but can’t be assumed for > 1 consumer. Prices are sources of income in neoclassical model: Changing relative price of bananas alters distribution of income.
The problem – With many consumers and many goods, a change in price for one commodity changes real income for all consumers. So, income bit of budget constraint doesn’t stay still when relative process change…(Neoclassical theorists (Gorman, Sonnenschien, Mantel, Debreu, Shafer..) set themselves the problem..
“Under what conditions will a market demand curve obey all the properties of an individual demand curve?”
Answer called “Sonnenschien-Mantel-Debreu Conditions”
Law of Demand applies to individual Hicksian compensated demand curve. Reduce price, demand necessarily rises. Does it apply to a market curve? NO! We prove that every polynomial is an excess demand function for a specified commodity in some n commodity economy. (Sonnenschien 1972). That is a demand curve for a single market can have any (polynomial) shape at all: Even study of a single market demand curve can’t be reduced to study of a demand curve derived from a single utility-maximising agent.
SMD Conditions (Sonnenschien 1973; Shafer and Sonnenschien 1993) Market demand curves do not obey the Law of Demand. Even if summing well behaved individual demand curves. An accidental Proof by Contradiction: Assume market demand curves do obey Law of Demand. Derive conditions under which this is true. These contradict initial assumptions. There for market demand curves don’t obey the Law of demand.
Ancient technique to prove a mathematical proposition. Assume something isture (e.g. the square root of 2 is a rational number) Follow through the logic. Find a contradiction. Thus prove that the square root of 2 is not a rational number. If the square root of 2 is rational then there are integers a and b which are the smallest numbers for which a/b = 2 square root. So we start with: Condition that integers a and b have no factors in common (except 1) and the assumption that a/b = 2 square root.
Now we square both sides to yield a^2/b^2 = 2. Rearrange to get a^2 = 2b^2. Can now deduce that a must be an even number: 2 times any integer (odd or even) is an even number. So we can express a as 2 times some other integer c: a = 2c.
So a squared is a^2 = (2c)^2 = 4c^2. Now substitute this into equation for a squared above: a^2 (=4c^2) = 2b^2. Divide last bit by 2 to yield 2c^2 = b^2. Which shows that b must also be even since 2 times any integer is an even number. Therefore b is divisible by 2. So a and b have 2 as a common factor!.
But we began with the condition that a and b had no common factor – our assumption that the square root of 2 is rational has been contradicted by series of logical steps. Therefor proof by contradiction that the assumption that the square root of 2 is a rational number must be false. Therefore the square root of 2 must be irrational. It cannot be equal to the ratio of two integers. This is how Pythagoreans discovered irrational numbers. Didn’t like it – began with belief that all numbers were rational – but forced to accept it by logic. Neoclassical economists instead resist a similar result. Conditions needed to derive downward-sloping market demand curve contradict assumption of different consumers with different goods.
Logic: Law of Demand derived from Hicksian compensated demand curve procedure. Take individual with well-behaved utility function. Vary price of one commodity while keeping others constant and consumer income constant.
Key assumptions:
(1) Can vary Price without altering income. Pivot point does not move.
(2) Can change income and perfectly compensate for income effect of lower price (Hicksian compensation)
Outcome: Hicksian compesnsated individual demand curve necessarily slopes down. The law of Demand. Motivation behind SMD research. Does this result survive aggregation to market demand? Answer: No!
Logic – Individual demand curve model ignores impact of price changes on income. But price changes will change income distribution. In two (or more) consumer model each must have: Different income sources and Different tastes. Otherwise there’s only one consumer. Tastes must change with income. Otherwise there’s only one commodity. Consider 2-consumer, 2-commodity world. Crusoe and Friday – Coconuts and Bananas; Crusoe the Banana owner; Friday Coconut owner; Coconuts necessity, Bananas luxury. Friday higher preference for coconuts that Crusoe.
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