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RP versus EP: EP wins very time.
Ranking bundles of goods with n commodities an “exponential problem”. Number of comparision scales exponentially with number of commodities. Comparisions – (1 + UnitsBought)^numberCommodities.
In our example – buy or not buy one item in 50 commodity shop: Comparisions = 2^50 (10 million billion different potential bundles). Such problems inherently non-computable: Simply impossible for any program on any computer to find highest combination in finite time. “Consider all option” – Computing (and by inference deductive thinking) restricted to “polynomial problem”.
Definitive (optimum) programs must run in polynomial time: e.g. “bubble sot algorithm”: sort list of n number: - Select last (“pivot”); - Choose next to last (pre-pivot”) and another (“rand”) at random. If either larger than pivot – swap larger with pivot and move smaller to where larger was. Repeat till all before pivot smaller than it. Partition list into two and repeat.
Worst case: (List starts in reverse order) – algorithm takes n^2 steps where n is length of list: n = 10 – 100 steps; n= 1000 – 1,000,000 steps; n = 1,000,000 – 1,000,000,000,000 steps (still a lot but do-able in finite time).
Average case: (list starts in purley random order) – Takes nxlog(n) steps: n=10 – 10 steps; n=1000 – 3000 steps; n=1,000,000 – 6,000,000 steps. Best case list already sorted, just n steps. 34 steps in previous example – between 10^2= 100 and 10xlog(10) =10.
Simply isn’t possible to “be rational” as economists define it. At a billion comparisons a second a “Revealed Preference” shopping trip would take longer than the Age of the Universe times the Age of the Universe: Bottom line – Neoclassical theory of rational behavior falls over at the first step.
“Completeness” – Given any 2 bundles of commodities A & B, consumer can decide whether prefers A to B (A > B), B to A (B > A) or is indifferent between them (B = A).; “Transivity”; Non-satiation”; “Convexity”; All breached in practice because depend upon Completeness to work! – But people still manage to shop; So they do different rational things to shop in finite time;
Reality – Capacity to compare fails even with 8 goods in bundle; Computational overload means can’t compare available bundles in finite time. “Satisfice” – Choose satisfactory bundle; “Prioritise” – Consider most desirable item in bundle and ignore others; “Habit” Buy as always with some change; “Categorise” – Purchase within categories; Drastically reduces dimensionality of choice;
Even attempting to utility-maximise is irrational in a world with more than 20 commodities; Computational complexity overwhelms optimizing; “If the brain is performing computation, it should obey the laws of computational theory”; These results come from two areas, computability and complexity, and can be paraphrased as follows: 1: “You cannot compute nearly all the things you want to compute [Godel/Turing proof that most things can’t be proven – not discussed here]; 2 – The things you can compute are too expensive to compute (Ballard 200, p. 6) (i.. exact answers to anything complex are impossible to achieve; and even shopping is complex).
Can’t characterize that behavior using “indifference curves” and budget lines. Normal behavior must violate Revealed Preference model because Revealed Preference behavior is computationally impossible. True “rational behavior” for real-world consumers is – making a satisfactory decision in finite time.
Next – Even if revealed preference did work – Market demand curves can’t be downward sloping.
“Law ofDemand” applies to individual Hicksian-compensated demand curve – Reduce price, demand becessarily rises. Doest it apply to a market demand curve? NO! “we prove that every polynomial .. is an excess demand function for a specified commodity in some n commodity economy (Sonnenschien 1972, pp 549 – 550). That is a demand curve for a songle 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 nt 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;
Ancient technique to prove a mathematical proposition – Assume something is true (e.g. The square root of 2 is a rational number). Follow through the logic. Find a contradiction – Thuse prove that “the square root of 2 is not a rational number”. If the square roo of 2 is rational then there are intergers a and b which are the smallest numbers for which a/b = 2 square root. So we start with: condition that intergers 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 a even since 2 times any integer is an even number. Therefor b is divisible by 2… So a and b have 2 as a common factor!.
But we bagan with condition tha ab an b had no coomon factor – our assumptionthat the squre root of 2 is rational has been contradicted by series of logical steps. Therefore “proof by contradiction” that the assumption that the square root of 2 is a rational number must be false. Therefor the square root of 2 must be irrational. It canot 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 ecomomists 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 Denand” derived from Hiscksian compensated demand curve procedure. Take individual with well-behaved utility function. Vary price of one commodity while others constant and consumer income constant.
(2) Can change income and perfectly compensate for income effect of lwer price (Hicksian compensation). Outcome: Hicksian-compensated individual demand curve necessarily slopesdown: 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 the Coconut owner. Coconuts necessity, Bananas luxury; Friday higher preference for coconuts than Crusoe.
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