Reconsidering Revealed Preference

“Rational” consumer assumed to obey these rules: Completeness, Transitivity, Non-satiation and Convexity

Consider “Completeness”: Given any 2 bundles of commodities A and B, consumer can decide whether prefers A to B (A > B), B to A (B > A), or is indifferent between them (B = A).

Looks easy on 2-dimensional graph: Each bundle contains just two items – (1, 4): 1 biscuit, 4 bananas

100 combinations, some you can ignore, others you can’t, 10 pairs, 10 budget sums, 10 utility comparisons, easy!, But what about when you add another good?

How to represent additional good on indifference map? Have to add an additional axis. Every additional commodity adds another dimension. With no more than 10 units of each: 2 commodities – 100 combinations; 3 commodities – 1000 combinations; 4 commodities – 10,000 combinations; How many combinations in Sippel’s experiment? 8 commodities so 8 dimensions.

Even if discrete choice and consider 5 combinations per good (0, 15, 30, 45, 60 minutes of video etc.). There are 5^8 combinations to consider: 5x5x5x5x5x5x5x5 – 390,624 different combinations!

Combo 1: 15 min video, 30 min game, 45 min magazine, 500g cola, 250g orange juice, 500g coffee, 1kg Haribo, 200g snacks.

Combo 2: 30 min video, 45 min game, 0 min magazine, 1 litre cola, 500 g orange juice, 0 coffee, 500g Haribo, 500g snacks.

Which do you prefer? Impossible to differentiate finely – instead tend to consider one or two items you like and ignore the rest.

Is this irrational? According to revealed preference/utility theory, yes. In real life, no! Reality is bewildering array of choices. Difficulty is not choosing best option, but making satisfactory choice in finite time.

Consider simple shopping trip: Say 100 items you could but at supermarket; But either 0 or 1 units of each; How many different combinations to compare? 2^100 – That’s one million trillion trillion different combinations.

Revealed preference/Indifference curves a “toy” model – Looks good on paper – Can’t possibly scale to reality. Consumption an “exponential complexity” problem: Number of combinations scales exponentially as additional commodities considered. To buy or not to buy decision a 2^n problem: 2 choices, zero or one unit, n combinations for n commodities, Put revelaed preference function in computer, Program it to find highest utility combination, If calculating utility of a bundle takes 10^-7 sec.

What about a human “computer”? More to brain than neurons but: Brain has 10^11 neurones (100 billion); Each neurone connects to 1000 others; Signalling between neurons basic operation in thinking, learning, deciding, acting; Signals transmitted by voltage spikes: Neurone takes 1 millisecond (10^-3) to generate a spike; Like computer transferring one bit of data from one register to another; Actual decision by computer might take 100 such steps.

50 – 100 milliseconds shortest time for actual perception (“That’s a tube of toothpaste”); 100 such perceptions would take at least 5 seconds; So, if brain acted as massively parallel HRCP (“Human Computer Revealed Preference”) machine (which it doesn’t), and if every decision took 5 seconds then “Human Computer” would operate at 5x10^-11 seconds per RP decision.

2252 seconds to shop in a 50 commodity store. “What if” each decision between bundles took minimum human perception time (50 ms) in massively parallel processing (10^11 neurons), regardless of number of commodities in a bundle?

Decision speed then 0.5x10^-12: “TO buy or not to buy” (o or 1 of ach commodilty) RP shopping trip in 100-commodity store would take 80,000,000,000 years (6 times estimated age of universie (13.7 billion years)