Utility Theory

1
Utility Theory

• First Cardinal
• Second Ordinal
• Third Cardinal

2
In Essence the Principle of Utility has two routes
3
Jeremy Bentham

• February 15, 1748-June 6, 1832
• The philosopher and jurist Jeremy Bentham
  (1748-1832) was born in Spitalfields, London, on
  15 February 1748. He proved to be something of a
  child prodigy while still a toddler he was
  discovered sitting at his father's desk reading a
  multi-volume history of England, and he began to
  study Latin at the age of three. At twelve, he
  was sent to Queen's College Oxford, his father, a
  prosperous attorney, having decided that Jeremy
  would follow him into the law, and feeling quite
  sure that his brilliant son would one day be Lord
  Chancellor of England.

4
Jeremy Bentham

• Bentham, however, soon became disillusioned with
  the law, especially after hearing the lectures of
  the leading authority of the day, Sir William
  Blackstone (1723-80). Instead of practising the
  law, he decided to write about it, and he spent
  his life criticising the existing law and
  suggesting ways for its improvement. His father's
  death in 1792 left him financially independent,
  and for nearly forty years he lived quietly in
  Westminster, producing between ten and twenty
  sheets of manuscript a day, even when he was in
  his eighties.

5
Jeremy Bentham

• Bentham is often credited with being one of the
  founders of the University of London, the
  forerunner of today's University College London.
  This is not, in fact, true. Bentham was eighty
  years of age when the new University opened its
  doors in 1828, and took no part in the campaign
  to bring it into being. However, the myth of his
  participation has been perpetuated in a mural by
  Henry Tonks (1862-1937), in the dome above the
  Flaxman gallery in the main UCL library

6
Jeremy Bentham

• Yet although Bentham played no direct part in the
  establishment of UCL, he still deserves to be
  considered as its spiritual father. Many of the
  founders, particularly James Mill (1773-1836) and
  Henry Brougham (1778-1868), held him in high
  esteem, and their project embodied many of his
  ideas on education and society.

7
Jeremy Bentham
8
Jeremy Bentham

• The cabinet contains Bentham's preserved
  skeleton, dressed in his own clothes, and
  surmounted by a wax head. Bentham requested that
  his body be preserved in this way in his will
  made shortly before his death on 6 June 1832. The
  cabinet was moved to UCL in 1850.

9
Jeremy Bentham

• Not surprisingly, this peculiar relic has given
  rise to numerous legends and anecdotes. One of
  the most commonly recounted is that the Auto-Icon
  regularly attends meetings of the College
  Council, and that it is solemnly wheeled into the
  Council Room to take its place among the
  present-day members. Its presence, it is claimed,
  is always recorded in the minutes with the words
  Jeremy Bentham - present but not voting. Another
  version of the story asserts that the Auto-Icon
  does vote, but only on occasions when the votes
  of the other Council members are equally split.
  In these cases the Auto-Icon invariably votes for
  the motion.

10
Extract from Jeremy Bentham's Last Will and
Testament

• My body I give to my dear friend Doctor Southwood
  Smith to be disposed of in a manner hereinafter
  mentioned, and I direct ... he will take my body
  under his charge and take the requisite and
  appropriate measures for the disposal and
  preservation of the several parts of my bodily
  frame in the manner expressed in the paper
  annexed to this my will and at the top of which I
  have written Auto Icon.

11
Extract from Jeremy Bentham's Last Will and
Testament

• If it should so happen that my personal friends
  and other disciples should be disposed to meet
  together on some day or days of the year for the
  purpose of commemorating the founder of the
  greatest happiness system of morals and
  legislation my executor will from time to time
  cause to be conveyed to the room in which they
  meet the said box or case with the contents
  therein to be stationed in such part of the room
  as to the assembled company shall seem meet .

12
Jeremy Bentham

• February 15, 1748-June 6, 1832
• Introduction of Morals and
  Legislation (1789)
• Nature has placed mankind under the governance
  of two sovereign masters, pain and pleasure. It
  is for them alone to point out what we ought to
  do as well as to determine what we shall do
  (p.17)

13
Natural Harmony

• Remember that for Adam Smith the self would
  with the check of the market lead to economic
  progress
• Bentham did not see Natural Harmony
• Example is that fact that there is CRIME

14
Benthams Central Point

• Interest of the Individual must be identified
  with the general interest, and that it was the
  business of the legislatures to bring about this
  identification through direct intercession.
• Bentham is similar to the Greek Hedonism
  philosophy
• Differs The greatest happiness for the greatest
  number (of individuals)

15
Moral Arithmetic

• Influenced by Newton he not only his work to
  scientific but thought that if there was a
  possibility of measurement than legislatures
  could measure Social Welfare
• Pleasure are added at the individual level but
  multiplied by the number of individuals

16
The Felicific Calculus

• The Intensity of Pleasure or Pain
• Its duration
• Its Certainty or uncertainty

• Its propinquity or remoteness
• Its fecundity, or the chance it has of being
  followed by sensations of the same kind
• Pleasure Þ Pleasure
• Pain Þ Pain

17
The Felicific Calculus

• Its purity, or the chance that it has of not
  being followed by sensations of the opposite kind
• Pleasure Þ Pain
• Pain Þ Pleasure
• Its extent, that is, the number of people who are
  affected by it
• NOTE fecundity and purity are not inherent
  properties of pleasure or pain, thus, only matter
  in the aggregate of an event

18
Measure of Social Welfare on a Given ACT

• First for any given one person of those whose
  interest seem most immediately to affected by it
  and take account
• Of the value of each distinguishable pleasure
  which appears to be produced in the first
  instance
• Of the value of each pain which appears to be
  produced by it in the first instance

19
Measure of Social Welfare on a Given ACT
(cont.)

• Of the value of each pleasure which appears to be
  produced by it after the first. This constitutes
  the fecundity of the first pleasure and the
  impurity of the first pain
• Of the value of each pleasure which appears to be
  produced by it after the first. This constitutes
  the fecundity of the first pain and the impurity
  of the first pleasure

20
Measure of Social Welfare on a Given ACT
(cont.)

• Sum up all the values of all the pleasures on the
  one side, and those of all the pains on the
  other. The balance, if it be on the side of
  pleasure, will give the good tendency of the act
  upon the whole, with respect to the interests of
  that individual person if on the side of pain,
  the bad tendency of it upon the whole.

21
Measure of Social Welfare on a Given ACT
(cont.)

• Take the people that appear to be concerned find
  those that have a pleasure balance and add up
  their degrees of pleasure then take those that
  have a pain balance and add up their degrees of
  pain. Take a balance and that yields the social
  welfare impact

22
Finals Remarks on Bentham

• How to compare across individuals
• Problems in weighting for instance, which
  produces higher pleasure (or pain) those of the
  mind or the body.
• Fallacy of Composition
• what is true of the parts may not be true of the
  whole

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Arsine-JulesEmile-Juvenal Dupuit

• 1804-1866
• Born in Fossano, Italy which at the time was
  under French domination

24
Jules Dupuit

• Graduated from the School of Civil Engineering in
  Paris
• He was one of the great engineers of his time.
• In 1855, he was name Inspector-General of Civil
  Engineering
• Took the study of political economy more as an
  avocation rather than a profession

25
Dupuits Approach

• Combined three elements to produce analytical
  tools
• subjects of economic interest and importance
• relevant, observed facts and statistics from
  these subjects
• mathematical analysis-deductive logic and
  graphical depiction- to organize and reorganize
  relations suggested by these facts and statistics.

26
Marginal Utility and Demand

• Early work of Gregory King (1648-1712), which was
  refined by the work of Charles Davenant
  (1656-1714), found the inverse relationship
  between price and quantity.
• The work by Davenant is
• An Essay upon the Probable Method of Making a
  People Gainers in the Balance of Trade (1699)
• The King-Davenant law of Demand

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King-Davenants Law of Demand
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Marginal Utility and Demand(Jules Dupuit)

• Using the example of Water consumed in a city he
  argued that if it was difficult to obtain the
  water and they had to pay 50 Francs and they
  purchased it that it had to provide the household
  with at least that much utility

29
Dupuit and Marginal Utility

• However, he argued that as more water was
  introduced to the city the time would come when
  the households would not require more
• Consequently, the concept of the Law of
  Diminishing Mariginal Utility

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Dupuits Argument
MU Price
p1
p2
q1
q2
Quantity of Water
31
Dupuits Consumer Surplus
Quantity
Thus, if price is p1 consumer pays Or1p1 and the
consumer surplus is r1p1P
r1
r2
p
p1
p2
O
Price
32
Consumers Surplus, Monopoly, and Discrimination
Yield of the Toll
Tariff
of Passengers
Utility
Gross
Net
2 f _at_
100
445 425 391 352 316 276 234 192 144 99 69 36 0
0 -200
0 1 2 3 4 5 6 7 8 9 10 11 12
80 63 50 41 33 26 20 14 9 6 3 0
80 -80
126 0 150 50 164
82 165 99 156 104 140
100 112 84 81
63 60 48 33 27 0
0
33
William Stanley Jevons

• 1835-1882
• (drowned just short of 47)

34
W. S. Jevons

• Raised in a Unitarian Environment
• (educated but not academic)
• Financial problems led him to move to Australia
  at the age of 18
• He wrote a book in which he he made an analogy of
  coal to the industrial age much like corn in
  Malthus population theory

35
W. S. Jevons

• He also wrote about the business and solar cycles
  (The Solar Period and the Price of Corn - 1875)
• Also known for his first attempts at
  understanding inflation in On the Study of
  Periodic Commercial Fluctuations- 1862 and A
  Serious Fall in the Value of Gold - 1863

36
W. S. Jevons

• Utility and Marginal Analysis
• Jevons noted the work of Weber-Fechner
• Recognized the difficulty of a cardinal
  measurement and acknowledge that only an ordinal
  measurement could be found
• However, proceeded as if indirectly the cardinal
  measurement could be found

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Graphical Analysis
TU
Units of X
MU
Units of X
38
Utility

• The Equimarginal Principle
• MuX MUY
• The only difference with the Equimarginal
  Principle is that it assumes the PX PY 1

39
Theory of Exchange
MU Beef
MU Corn
MU Beef
MU Corn
40
Theory of Labor
Pleasure
Degree of Utility of Real Wages
Real Wages or Amount of Product
Net Pain of Labor Curve
At Arrows have same amount
Pain
41
Ordinal Utility
U
2
U
x1
0
x2
42
Expected Utility

• Expected Value
• E(X) p1O1 p2O2 p3O3
• Where p1 p2 p3 1
• Expected Utility
• U(X) p1U(O)1 p2U(O)2 p3U(O)3

43
Risk Averse vs. Risk Loving
Total Utility
Risk Neutral
Return on Investment
44
Risk Averse vs. Risk Loving
Total Utility
Risk Averse
Return on Investment
45
Risk Averse vs. Risk Loving
Total Utility
Risk Loving
Return on Investment