Core Micro Lecture Preferences Utility and Choice

Core Micro Lecture 1 slides

1. Core Micro Lecture 1
Preferences, Utility, and Choice
S´everine Toussaert
October 2024
University of Oxford
Department of Economics & St John’s College

2. Welcome! :)
•Very nice to meet you all!
•Please call me Sev :)
•I am a behavioural/experimental economist.
•I work at the intersection of micro theory and data.
•Will try to find a good balance between formalism and examples
•The course is for you - any feedback welcome!
•Office Hours: Tuesdays Wk 1-3 @ 5:30pm in Office 276
•I will see you again in TT25 to talk about how to write a thesis...
•... and if you take Behavioural Economics next year (you should ;)
1

3. Aims of the Course
1.Learn and understand the microeconomics every academic economist
knows
2.Stimulate your interest in microeconomics as a field
3.Enable you to read theoretical papers and empirical papers that use
theory
4.Allow you to go to research seminars with reasonable expectation of
being able to understand them
5.Give you a taste of what topics you might encounter in more
advanced or field papers in Y2
2

4. Course Structure
Michaelmas Term ’24 (roughly)
Week 1: Preference and Utility (Sev Toussaert)
Week 2: Consumer Choice (Sev Toussaert)
Week 3: Choice under Uncertainty (Sev Toussaert)
Weeks 4-6: Producer Choice, General Equilibrium & Welfare Theorems
(Simon Cowan)
Weeks 6-8: Game Theory and Applications (Ines Moreno de Barreda)
3

5. Course Structure
Hilary Term ’25 (roughly)
Weeks 1-3: Game Theory, continued (Ines Moreno de Barreda)
Week 2: Auctions (Paul Klemperer)
Weeks 3-6: Information Economics (Meg Meyer)
Weeks 6-8: Collective Decision-Making and Welfare Economics (Alex
Teytelboym)
4

6. Books
Core Textbook(aka “The Bible”)
•Mas-Colell, A., M. Whinston and J. Green, Microeconomic Theory,
Oxford University Press. (Henceforth MWG)
Additional books
•Rubinstein, A., Lecture Notes in Microeconomic Theory: The
Economic Agent, Princeton University Press.(♡)
•Varian H. Microeconomic Analysis, Norton.
•Kreps, Microeconomic Foundations I: Choice and Competitive
Markets, Princeton University Press
•Dixit, A.K., Optimization in Economic Theory, OUP.
5

7. Practice Makes Perfect!
Problem Sets
•You need to submit all questions, unless otherwise indicated.
•A selection will be marked and covered in classes.
•You will get the solutions to all questions.
•Submission deadline is on Mondays by 4pm. Submit on Canvas.
•Please try - write down where you struggled in your reasoning and
ask questions in class!
6

8. Plan for Weeks 1-3 with Sev
Weeks 1 & 2
•Preference and Utility
•Consumer Choice
•Classical Demand Theory
•Welfare and Aggregation
See Chapters 1 - 4 of MWG and Chapters 1 - 6 of Rubinstein
Week 3
•Decision-Making under Risk
•Expected Utility Theory
•Risk Aversion
7

9. Preference and Utility

10. What’s Microeconomics?
(Yes, I will ask a lot of obvious questions)
Definition: (From Intro of Rubinstein)
“a collection of models in which theprimitivesare details about the
behaviorof units called economic agents. Microeconomic mod-
els investigate assumptions about economic agents’ activities and
about interactions between these agents.”
•Decision-making process as it manifests itself through choice
•Includes both decision theory and game theory
•Central assumption: rationality
8

11. What’s Microeconomics?
(Yes, I will ask a lot of obvious questions)
Definition: (From Intro of Rubinstein)
“a collection of models in which theprimitivesare details about the
behaviorof units called economic agents. Microeconomic mod-
els investigate assumptions about economic agents’ activities and
about interactions between these agents.”
•Decision-making process as it manifests itself through choice
•Includes both decision theory and game theory
•Central assumption: rationality
8

12. A First Look at the Behavior of Economic Agents
Today we will look at the following:
1.Preferences
2.Utility
3.Individual choice
What is the relationship between all three?
•How do they map onto each other?
•Do such mappings even exist? If so, when?
•What can we learn from studying these mappings?
9

13. Preferences

14. Intro
Two main alternative approaches to individual choice:
1. : preferences are the primitive concept
2. : choices are the primitive concept
The former typically represents flational choice” (to be defined), while
the latter need not.
In each case, the idea is that we consider certain properties (either of
preferences or choice behaviour) and analyse the consequences.
We will cover both approaches, starting with Approach 1.
10

15. Set of Alternatives
The first piece of the puzzle concerns thedomain of preferences:
•What type of objects are evaluated by the decision maker?
•How many of these objects are there?
X:=set of mutually exclusive alternatives from which to choose.
Can be modelled very flexibly to capture different types of choices:
•Meal for lunch:X={salad,burger}
•Lunch place to go to:X={{salad},{burger},{salad,burger}}
•Meal plan for the week:
X={(salad,salad, ...,salad),(salad,salad, ...,burger)}
•Salad meal frequency over the year:X= [0, 1]
We will denote byx,y,z∈Xthe generic elements ofX.
11

16. Expressing Preferences
•Now that we know the set of alternatives, we can ask the agent to
express a preference over these alternatives.
•These preferences could be expressed via questionnaires.
•The questionnaire asks questions of type
Q: “How do you personally evaluate optionsxandy?”
for various pairs of optionsxandy.
•We need to consider the design of the questionnaireQ(x,y):
•Presumably, we want closed-ended questions with a finite set of
response items.
•The concept of “preference relation” and its properties allows us to
formulate what are “legal” answers.
12

17. Preference Relations (1)
•A preference relation is a binary relation≿which compares any pair
of alternativesx,y∈Xand ranks them
≿:={(x,y)∈X×X|x≿y}
•Herex≿ymeans that “xis weakly preferred toy”
•MWG and others use the terminology “xis at least as good asy”
•Goodness is not equivalent to preference (two options could be bad
but rankable in terms of preference)
•Note that the expression of preference is weak here - it is not clear
whether the agent has a “clear” preference forx.
13

18. Preference Relations (2)
To refine the language of preferences, we can obtain two more relations
from any preference relation≿:
1.Thestrictpreference relation≻defined by:
x≻y⇔x≿ybut noty≿x
In words,x≻ymeans “xis strictly preferred toy”.
2.Theindifferencerelation∼defined by:
x∼y⇔x≿yandy≿x
In words,x∼ymeans “xis indifferent toy”.
14

19. Requirements on Preferences
What properties would we want to ask of the self-reported preferences for
them to be usable?
•Some answer is probably better than no answer.
•We would like answers to be coherent across pairs of alternatives in
some sense.
•We might want to be able to draw conclusions about comparisons of
some alternatives not contained in the questionnaire.
•If the agent declares strictly preferringxtoyandytoz, the answer
“I strictly preferztox” might make us feel uncomfortable.
15

20. Rational Preferences
Definition: Rational Preference Relation
The preference relation≿is said to berationalif it is complete
and transitive:
•Completeness: for allx,y∈X, we either havex≿yor
y≿x(or both).
•Transitivity: for allx,y,z∈X, whenever we havex≿yand
y≿zthen we havex≿z.
If≿is rational, we have the following restrictions on≻and∼(see Hw1):
1.≻is bothirreflexive(i.e.,x≻xcannot hold) and transitive.
2.∼is bothreflexive(i.e.,x∼xalways holds) and transitive.
3.Ifx≻yandy≿zthenx≻z.
16

21. What We’ve Ruled Out
Through the concept of rational preference, note that we’ve ruled out a
few things in the formulation of our questionnaire:
1.Lack of ability to compare (e.g., “I don’t know” or “I strictly prefer
xoveryand alsoyoverx)”
•Note that≻isasymmetric: Ifx≻ythen noty≻x(why?)
2.Dependence on other factors (e.g., “It depends on my mood”)
•Preference relation only takes(x,y)as input, not choice context.
3.Ability to express preference intensity (e.g., “I somewhat preferx” or
“I lovexand hatey”)
•Comparisons allow for a full ordinal ranking (by completeness and
transitivity), but no cardinal statement is made.
17

22. Discussion of Completeness
You might challenge completeness as a rationality requirement.
1.Some ethical dilemmas might call for no answer (though not
choosing is also a choice).
2.Many options are multi-dimensional and very difficult to compare.
3.“Too much choice kills choice”: choice overload may lead to
indecision.
⇒“Tyranny of choice”
18

23. Ethical Dilemmas
19

24. Sophie’s Choice
•A powerful example is contained inSophie’s Choice, a 1979 novel by
William Styron.
•Adapted to the screen in 1982 with Meryl Streep featuring Sophie.
•Drama partly staged during WWII.
•Sophie sent to Auschwitz with her son Jan and daughter Eva.
Forced to choose who to send to the gas chamber. Sophie’s choice:
C({Jan dies,Eva dies}) =Eva dies
•But, clearly, neitherJan dies⪰Eva dies, norEva dies⪰Jan dies.
•Some choices are just impossible to make.
20

25. Conflicts Among Attributes
Sev’s choice of PhD programs:
Caltech New York
Amazing weather≻Variable weather
Turtle pond ≻No turtle pond
No macro ≻Macro
Small town ≺Big city
Busy faculty ≺Available faculty
How to aggregate preferences across all attributes when there is no clear
dominance relationship?
21

26. Tversky and Shafir, Psychological Science (1992)
•People tend to postpone their decision more (i.e., refrain from
expressing a preference) when facing conflicting alternatives.
•Choice of compact disk (CD) player: “popular SONY player for just
$99”, or Ψop-of-the-line AIWA player for just$159”.
Scenario 1:
•x. buy the AIWA player⇒27%
•y. buy the SONY player⇒27%
•z. wait until you learn more about the various models⇒46%
Scenario 2:
•y. buy the SONY player⇒66%
•z. wait until you learn more about the various models⇒34%
22

27. It also Happens to the Best of Us...
“A case in point was described to us by Thomas Schelling, who
some time ago had decided to buy an encyclopedia for his chil-
dren. To his chagrin, he discovered that two encyclopedias were
available in the bookstore. Although either one would have been
satisfactory, he found it difficult to choose between the two, and
as a result bought neither.”
In Tversky and Shafir, Psychological Science (1992)
23

28. Too Much Choice Kills Choice
•The rational man is never hurt by the presence of more options.
•Iyengar and Lepper (2000) conduct a study with shoppers shown
several varieties of gourmet jams that they could sample from.
•Two conditions:
•Small display: 6 varieties of jams on display
•Large display: 24 varieties of jams on display
•Shoppers more likely to stop at booth with large display but much
less likely to buy (3% purchase rate with large display vs. 30% with
small display).
•SeeThe Paradox of Choiceby Barry Schwartz
https://hbr.org/2006/06/more-isnt-always-better
24

29. Discussion of Transitivity
Transitivity may also fail for a number of reasons, such as:
1.Imperceptible Differences: Someone may be indifferent between two
slices of cake which weigh within 1 gram of each other, but not
indifferent between slices differing by more than 50 grams.
2.Endogenous Taste Change: For example, habit formation or
Ψemptation preferences”
3.Multi-attribute Decision-making: Choosing the option that
dominates on a majority of attributes can lead to cycles.
25

30. Transitivity and Attribute Aggregation
K. O. May, Econometrica (1954)
•62 subjects asked to rank three potential partners:x,yandx
x y z
Very intelligentIntelligentFairly intelligent
Plain lookingVery good lookingGood looking
Well-off Poor Rich
•Group preferences:x≻y,y≻zandz≻x
•Reason: subjects choose alternative which is superior in 2 out of 3
criteria
26

31. Transitivity and Social Choice
•Natural way of aggregating preferences is the majority rule:
x⪰
S
y⇐⇒ |{i∈Ns.t.x⪰
iy}| ≥ |{i∈Ns.t.y⪰
ix}|
•⪰
S
is complete but, as Condorcet showed, it is not transitive:
AliceBobCharlie
x y z
y z x
z x y
•We havex≻
S
y,y≻
S
zandz≻
S
x⇒famous “Condorcet cycle”
•Different rationality requirements might be reasonable depending on
the type of economic agents being analyzed. Transitivity is a lot to
ask of the rationality of groups.
27

32. What Happens when Transitivity Fails
•When transitivity is violated, there may be preference cycles and
hence no Φest” option among the set of choices.
•For example, ifX={x,y,z}andx≻yandy≻zandz≻x, then
there is nob∈Xsuch thatb≿cfor allc∈X.
•Rough measure of how rational someone is: number of cycles of an
individual in the data.
•Transitivity and completeness guarantee the existence of a
⪰-maximal (and minimal) element in any setXsuch that|X|<∞.
(Can be proven by induction on the size ofX; see Rubinstein, Ch2)
28

33. The Money Pump Argument
Consider objects and money,X×R. Endowment(z,m)
1.Ify≻z, the individual is willing to pay some money to gety:
(y,m−t)∼(z,m)
2.Ifx≻y, the individual is willing to pay some money to getx:
(x,m−t−t
′
)∼(y,m−t)
3.Ifz≻x, the individual is willing to pay some money to getz:
(z,m−t−t
′
−t
′′
)∼(x,m−t−t
′
)
Money pump: Our individual ended up with the original object,z, but
lost money,t+t
′
+t
′′
.
29

34. Implicit Assumptions
•Implicit in rational approach to choice: assumption that how options
areframedand their order of presentation do not matter.
•But choice can be affected by framing or presence of “irrelevant
alternatives” e.g., see the “attraction effect” (or “decoy effect”).
•“Peer effects” also influence choice e.g., charitable donations are
affected if donors know how much previous donors gave.
•That said, the basic rational choice framework is very flexible and
can often be generalised to account for such effects e.g., see
Rubinstein Ch3.
30

35. Tversky & Kahneman, Science (1981)
Two programs to fight a disease outbreak - which do you choose:
Program A: 200 people will be saved.
Program B: 600 people will be saved with 1/3 probability; no one
will be saved with 2/3 probability.
What would you choose now:
Program C: 400 people will die.
Program D: nobody will die with 1/3 probability; 600 people will
die with 2/3 probability.
•Majority selects A and D
•But the problems are the same; if you chose A, you should have
chosen C⇒”preference reversal”
31

36. Tversky & Kahneman, Science (1981)
Two programs to fight a disease outbreak - which do you choose:
Program A: 200 people will be saved.
Program B: 600 people will be saved with 1/3 probability; no one
will be saved with 2/3 probability.
What would you choose now:
Program C: 400 people will die.
Program D: nobody will die with 1/3 probability; 600 people will
die with 2/3 probability.
•Majority selects A and D
•But the problems are the same; if you chose A, you should have
chosen C⇒”preference reversal”
31

37. Tversky & Kahneman, Science (1981)
Two programs to fight a disease outbreak - which do you choose:
Program A: 200 people will be saved.
Program B: 600 people will be saved with 1/3 probability; no one
will be saved with 2/3 probability.
What would you choose now:
Program C: 400 people will die.
Program D: nobody will die with 1/3 probability; 600 people will
die with 2/3 probability.
•Majority selects A and D
•But the problems are the same; if you chose A, you should have
chosen C⇒”preference reversal”
31

38. Utility

39. Utility
•We often represent a preference relation by means of what is called
autility function:
Definition: Utility Function
A functionu:X→Ris a utility function representing the
preference relation≿if, for allx,y∈X,
x≿y⇔u(x)≥u(y)
•In words, a utility functionu(x)assigns a scalar numerical value to
each elementx∈Xsuch that more-preferred elements get assigned
bigger numbers than less-preferred elements.
•Extremely useful: summarizes a rather nebulous-sounding thing
(“preference”) by a real number.
32

40. Utility is an Ordinal Concept
•Note thatuis not unique, e.g, ifu(x)represents⪰, so does
ˆu(x) =u(x) +kfor anyk∈R.
•In fact, the concept of utility function is “ordinal”, not “cardinal”.
Definition: Ordinal property
A property of a function is ordinal if it is preserved under any
strictly increasing transformation of this function.
•Claim: Ifurepresents⪰andf:R→Ris a strictly increasing
function, thenv(x) =f(u(x))represents⪰as well.
Proof:x≿y⇔u(x)≥u(y)(sinceurepresents⪰)
⇔f(u(x))≥f(u(y))(fstrictly increasing)
⇔v(x)≥v(y)
•Thus, “preference representation function” would be a much more
accurate and useful (albeit clunky) term than “utility function”.
33

41. Interpretation of Utility (1)
•In everyday English “utility” means usefulness - a property of an
object.
•Jeremy Bentham used it to mean a particular sort of usefulness:
“By utility, is meant that property in any object, whereby it tends
to produce benefit, advantage, pleasure, good, or happiness ...
to the party whose interest is considered”.
An Introduction to the Principles of Morals and Legislation, 1823.
34

42. Interpretation of Utility (2)
But over time, its meaning shifted away:
•From the tendency of anobjectto produce Φenefit, advantage,
pleasure, good, or happiness”...
•... to the Φenefit, advantage, pleasure, good, or happiness”itself.
•Shift in meaning causing difficulties of interpretation:
“As used by economists, the term ”utility” has become so am-
biguous as to cause immense confusion. It should be used less,
not more.”
John Broome, ”Utility”,Economics and Philosophy7 (01):1-12 (1991)
35

43. Interpretation of Utility (3)
Unfortunately, the confusion persists even in textbooks.
“Economists use the term utility to represent a measure of the
satisfaction or happiness that individuals get from the consump-
tion of goods and services”
Pindyck & Rubinfeld,Microeconomics, p. 81
“Economists have abandoned the old-fashion view of utility as
being a measure of happiness”Varian,Intermediate Microeconomics, p. 54
36

44. Existence of Utility Representations
•Basic question of utility theory: What conditions (on preferences
and choice sets) guarantee the existence of a utility representation?
•Is rationality of the preference relation (i) a necessary condition for
existence? (ii) a sufficient condition?
•Claim: If a preference relation⪰can be represented by some utility
u:X→R, then⪰must be rational:
•⪰is complete: Ifx,y∈X, then eitheru(x)≥u(y)oru(y)≥u(x)
i.e., eitherx⪰yory⪰x.
•⪰is transitive: Ifx,y,z∈Xandx⪰yandy⪰z, thenu(x)≥u(y)
andu(y)≥u(z), so thatu(x)≥u(z)i.e.,x⪰z.
•In other words, rationality of the preference is a necessary condition
(i.e., an implication) of existence of utility representation.
37

45. Existence of Utility Representations
•Is the converse also true? If⪰is rational, can we always find a
utility function that represents it?
•Otherwise stated, is rationality a sufficient condition for⪰to be
representable by someu?
•The answer in general is Ωo” :
•IfXis uncountable (e.g.,X=R), completeness and transitivity are
not enough (see Rubinstein Ch2: “Lexicographic preferences”, ).
•IfXis finite or countable, a utility representation always exists. We
can construct it inductively, using the natural numbers. (See Hw1 for
one proof when|X|<∞)
•Conclusion: Existence of a utility function is not the same as
rationality, but it’s close.
38

46. Summary: Preference and Utility
•Preferences can be generally described using a preference relation.
•Typically assume preferences are rational: complete and transitive.
•Completeness and transitivity do not always hold.
•If we can find a utility function to represent⪰, it must be rational.
•Partial converse is also true.
39

47. So why should we care?
Being able to represent preferences is important in many scenarios:
•Should taxes be increased to pay for a new road?
•Would people prefer to be enrolled in a pension scheme or not?
More broadly, formulating and representing preferences allows us to:
•Description: Provide an explanation for an agent’s behaviour and
assess its plausibility (models as stories)
•Prediction: Consider counterfactuals (derive comparative statics)
•Normative benchmark: Interrogate the coherence of an agent’s
choices; evaluate situation relative to first best
40

48. Choice

49. Starting from Choice
•So far, we assumed we could run a questionnaire, map out the
respondent’s preferences and, provided they are rational, possibly
represent them with a utility function.
•But what if we can’t run a questionnaire?
•Often we can rely on choice data e.g., purchases under different
prices, meals from different menus, etc.
•When are these choices consistent with the maximization of a
rational preference relation?
41

50. Choice Structures
Taking choice as a primitive, let’s introduce the notion of “choice
structure” with(B,C(·))
•Family of choice setsB, whereB⊆Xfor allB∈ B →all possible
choice experiments
•Choice rule (or correspondence)C(·)selecting a non-empty subset
of elements from each setBi.e.,∅̸=C(B)⊆Bfor allB∈ B.
Examples withX={x,y,z}andB={{x,y},{x,y,z}}:
•(B,C1(·))withC1({x,y}) ={x}andC1({x,y,z}) ={x}.
•(B,C2(·))withC2({x,y}) ={x}andC2({x,y,z}) ={x,y}.
In the second case,xandyare “acceptable” options.
42

51. Choice Consistency
•What restrictions might we want to impose on an agent’s choices to
consider them coherent across choice experiments?
•If the agent chosexoveryin one choice set (revealing a “proclivity”
forx), we might be surprised to seeychosen overxin another set.
•Samuelson (1947) proposed the weak axiom of revealed preference
(WARP), which captures this idea.
Definition: WARP
A choice structure(B,C(·))satisfies WARP if wheneverx∈
C(A)for someA∈ Bandy∈C(B)for someB∈ Bsuch
thatx,y∈A∩B, we also havex∈C(B).
•In words, ifxis ever chosen whileywas available, there cannot be
any choice set where both are available andyis chosen butxis not.
43

52. Revealed Preference Relation
Another way of stating WARP is by defining a binary relation⪰
∗
from
observed choice behaviorC(·):
Definition: Revealed Preference Relation
x⪰
∗
y⇐⇒there is someB∈ Bsuch thatx,y∈Band
x∈C(B).
•x⪰
∗
ycan be read as “xrevealed at least as good asy”
•Note that⪰
∗
does not have to complete or transitive.
•We could also readx∈C(B)andy/∈C(B)forx,y∈Bas “x
revealed preferred toy”
•Restatement of WARP: “Ifxis revealed at least as good asy, then
y cannot be revealed preferred tox”.
44

53. Linking Preference Relations and Choice Rules
1.If an agent makes choices that maximize a rational preference, does
the choice structureinduced by⪰satisfy WARP? Formally:
DoesC
∗
(B,⪰) ={x∈B:x⪰yfor ally∈B}satisfy WARP?
Answer: Yes! (WARP necessary)
2.If a choice structure(B,C(·))satisfies WARP, is there a rational
preference relation⪰that can generate the observed choices?
Formally, can we construct a rational preference⪰such that
C(B) =C
∗
(B,⪰)for allB∈ B?
Answer: Only under some conditions (WARP not sufficient)
⇒If so, we say that⪰rationalizesC(·)with respect toB
⇒Choice behaviour isas ifit was derived from the maximization of
a rational preference.
45

54. WARP is necessary
Claim: If the agent’s preferences⪰are rational, then the choices
induced by⪰,C
∗
(·,⪰), must satisfy WARP.
Proof: Suppose there isx,y∈A∩Bsuch thatx∈C
∗
(A,⪰)and
y∈C
∗
(B,⪰). We want to show thatx∈C
∗
(B,⪰), i.e.,x⪰zfor all
z∈B.
Ifx∈C
∗
(A,⪰), thenx⪰zfor allz∈Aand sincey∈A,x⪰y.
Ify∈C
∗
(B,⪰), theny⪰zfor allz∈B.
Since⪰is transitive,x⪰yandy⪰zfor allz∈Bimpliesx⪰zfor all
z∈B, i.e.,x∈C
∗
(B,⪰).
46

55. WARP is not sufficient
A preference relation that rationalizes choice with respect to someBis
not guaranteed to be rational even if it satisfies WARP onB.
Example: Consider(B,C(·))withB={{x,y},{y,z}{x,z}}where
C({x,y}) ={x},C({y,z}) ={y}andC({x,z}) ={z}.
Note that WARP is trivially satisfied as there are no other sets containing
each pair that would allow us to falsify the axiom. But any preference
rationalizingC(·)onBwould need to satisfyx≻yandy≻z, which by
transitivity would implyx≻z. This would contradictC({x,z}) ={z}.
What is key: The set of choice experiments,B, needs to be rich enough
i.e., it needs to include at least{x,y,z}. In this case, WARP provides
sufficient restrictions to avoid transitivity violations.
47

56. Conditions for WARP Sufficiency
Claim(Arrow, 1959): If a choice structure(B,C(·))is such that
1.WARP is satisfied
2.Bincludes all subsets ofXup to 3 elements
then there is a preference relation⪰that rationalizesC(·)relative toB,
i.e.,C(B) =C
∗
(B,⪰)for allB∈ B. Furthermore, there is no other
preference relation that does so.
Proof sketch(assuming|X|<∞)
1.Use the revealed preference relation⪰
∗
as the rationalizing
preference.
2.Show it is complete and transitive. This is guaranteed by WARP +
richness ofB
3.Show that ifx∈C(B)thenx∈C
∗
(B,⪰)and vice-versa.
4.Uniqueness comes from availability of all sets with two elements.
48

57. What Have we Learned?
•The principle of flevealed preference” is powerful.
•By observing choices in different environments, we are offered a
chance to learn about an agent’s possible preferences.
•If choice can be rationalized by some preference, we can start asking
about individual welfare.
•But remember the assumptions e.g., inferences work only if choice is
not influenced by frames or other factors.
•Frame dependence poses challenges for welfare analysis - see
Bernheim and Rangel (2009).
•Sometimes we observe choices, but not the choice sets they came
from. The informational requirements are not trivial.
49