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The “curse of dimensionality”
And the impossibility of utility maximisation
From indifference…
• Concept of subjective utility dates back at least to
Aristotle; made central tenet of economics and
philosophy by Jeremy Bentham (not Adam Smith!)
• Formalised into concept of “indifference curves” by
neoclassical economists (see Chapter 2, Debunking
Economics)
– Many different combinations of goods capable of
giving consumer same level of subjective utility
– Line linking points akin to “isoquant” on map, linking
points of equivalent altitude
– Christened “indifference curves”:
From indifference…
Biscuits
• Combinations A and B give
same level of satisfaction
• Combination C gives higher
level than A or B
• Combination D gives lower
level than A or B
• Just one wee problem…
A
C
B
D
Bananas
• Indifference curves no more observable than angels
dancing on heads of medieval pins…
To revealed preference…
• Samuelson proposed solution: “revealed preference”
• Indifference curves could be inferred from behaviour of
consumer, provided 4 postulates of rationality met:
– Completeness: consumer knows own subjective ranking
of all combinations of goods
– Transitivity: If combination A preferred to
combination B, and B to C, then A preferred to C
– Non-satiation: More is always preferred to less.
– Convexity: additional utility a consumer gets from
extra units of each commodity falls
– Preferences and budget completely independent
• Applying this:
To revealed preference…
Biscuits
• All points in rectangle
necessarily preferred to A
(“non-satiation”)
A
Bananas
Biscuits
C
A
B
Bananas
• If A purchased when both B
and C affordable, then A
necessarily on higher
indifference curve than B and
C
• Move budget line around, see
responses, can eventually
infer indifference map from
observed behaviour
• Appears scientific, but
problems when verification
attempted…
• In theory,
consumer
starts with
complete
preference
set
• Imposes
budget line
• Works out
point of
tangency
• Buys this
combination
Bananas
To experimental testing…
• (Preferences
and budget
completely
independent)
Biscuits
• In experiment, easy to decide whether someone is
rational or irrational according to theory:
• X Initial
budget line
• Consumer
chooses A
when A & B
both
affordable
• Rational
consumer
“should”
always
prefer A to
B
Bananas
The neoclassical expectation…
Budget Y: A no longer best,
but “clearly” better than B
X
• A must lie on higher
indifference curve
A
B
Y
Biscuits
• But in experiments they don’t do this!
Sometimes, they choose B instead of A
The neoclassical explanation…
• Consumers “irrational”
• But in reality, economists definition of “rational” is
irrational
– Many weaknesses
• but key one the unwitting assumption that
consumers have near infinite information
processing ability: the “curse of dimensionality”
– issue easily illustrated by putting numbers on
axes of economists’ graphs:
• 121
combinations
• Some you 10
ignore
9
• Others
8
you can’t… 7
• 10 pairs
6
• 10 additions 5
4
• 10
comparisons 3
2
• Easy!
1
• But…
0
Bananas
Putting some numbers on the graph…
Biscuits
0
1
2 3 4 5 6 7 8 9 10
Putting some numbers on the graph…
• Every additional commodity considered adds another
dimension. With no more than 10 units of each:
• 2 commodities,
• 3 commodities, ~1,000
~100 combinations
combinations
(actually 121)
• 4 commodities,
10,000 combinations
• 30 commodities…
• how many combinations?
Putting some numbers on the graph…
• 10,000,000,000,000,000,000,000,000,000,000!
• If budget obviously ruled out 99.9% of these;
• If each evaluation took 1 billionth of a second…
– Process would complete after 32 billion years
– Maximum (est.) age of universe 20 billion years
– Individual would take 1.6 times age of known universe
to make “utility maximising” choice of just 30
commodities
– Maximising utility in typical supermarket (1,000+
different items) doesn’t bear contemplation
• let alone millions of products in modern economy
• Instead, intelligent partitioning of commodity space
vital…
Utility maximising individuals?
• Sources of partitioning
– Culture
• some commodities not even contemplated (sea slugs,
anyone?)
– Income & Needs
• Fulfil hierarchy of wants from basic to ethereal
(Maslow’s concepts)
– Rolls Royce’s not part of “utility set” of poor
– Baked beans not part of “utility set” of rich
– Habit
• Buy mainly what you bought yesterday
– Change in consumption
• Evolutionary rather than “rational” process
Utility maximising individuals?
• Individual tastes no longer a “given” but vital economic
issue
– Explains individual partitioning of commodity space
– Selling new products requires movement of this space
• Marketing, advertising thus essential “economic”
activities if new products are to be sold
– Co-evolution of products and tastes an essential
aspect of economic development
• Consumers do not “utility maximise” but instead
“satisfice” (as per Herbert Simon)
• Demand curve for individual consumer becomes
meaningless, let alone neoclassical idea of market
demand curve (see Debunking Economics, Chapter 2)