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The curse of dimensionality

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Formalised into concept of 'indifference curves' by neoclassical economists (see ... some commodities not even contemplated (sea slugs, anyone?) Income & Needs ... – PowerPoint PPT presentation

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Title: The curse of dimensionality


1
The curse of dimensionality
  • And the impossibility of utility maximisation

2
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

3
From indifference
  • Combinations A and B give same level of
    satisfaction

Biscuits
  • Combination C gives higher level than A or B

A
  • Combination D gives lower level than A or B
  • Just one wee problem

C
B
D
Bananas
  • Indifference curves no more observable than
    angels dancing on heads of medieval pins

4
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

5
To revealed preference
  • All points in rectangle necessarily preferred to
    A (non-satiation)

Biscuits
  • 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

A
Bananas
Biscuits
C
A
  • Appears scientific, but problems when
    verification attempted

B
Bananas
6
To experimental testing
  • In theory, consumer starts with complete
    preference set
  • (Preferences and budget completely independent)

Bananas
  • Imposes budget line
  • Works out point of tangency
  • Buys this combination

Biscuits
  • In experiment, easy to decide whether someone is
    rational or irrational according to theory

7
The neoclassical expectation
  • X Initial budget line

Budget Y A no longer best, but clearly better
than B
Bananas
  • Consumer chooses A when A B both affordable

X
  • A must lie on higher indifference curve
  • Rational consumer should always prefer A to B

A
B
Y
Biscuits
  • But in experiments they dont do this! Sometimes,
    they choose B instead of A

8
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

9
Putting some numbers on the graph
  • 121 combinations

Bananas
  • Some you ignore

10
9
  • Others you cant

8
7
  • 10 pairs
  • 10 additions
  • 10 comparisons
  • Easy!
  • But

6
5
4
3
2
1
Biscuits
0
0
1
2
3
4
5
6
7
8
9
10
10
Putting some numbers on the graph
  • Every additional commodity considered adds
    another dimension. With no more than 10 units of
    each
  • 2 commodities, 100 combinations(actually 121)
  • 3 commodities, 1,000 combinations
  • 4 commodities, 10,000 combinations
  • 30 commodities
  • how many combinations?

11
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) doesnt bear contemplation
  • let alone millions of products in modern economy
  • Instead, intelligent partitioning of commodity
    space vital

12
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
    (Maslows concepts)
  • Rolls Royces 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

13
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)
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