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)