CHOOSING IN GROUPS
MUNGER AND MUNGER
Slides for Chapter 4
The Analytics of Choosing in Groups
Outline of Chapter 4
 Definitions
 Preferences, weak orders, and utility functions
 Political preferences
 Public goods and market failures
 Consumer problem
 Private goods
 Public goods
 Lindahl equilibrium
 Representation of public preferences
 Aggregation problem
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
2
Definitions
 Preferences: Ordering of all alternatives; ties allowed
 Rational if complete and transitive
 Utility theory: Use of a mathematical function where utility depends on goods or
state of the world
 Under certain circumstances, utility can represent preferences
 Representation: Mapping of preference profile onto family of utility functions
 Binary relation: An ordered triple (X,X,S) in which X is mapped to S
 Social choice function: A mapping of >1 preference profiles into a social utility
function
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
3
Preference
 Define a binary relation
 “A
B” means “A is at least as good as B” from the chooser’s perspective
 A and B are both elements of X, the set of alternatives

(strict preference): If A

(strict indifference): If A
B then A
B then A
B but B
B and B
A
A
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
4
Preference (2)
 Reflexivity: The relation can be applied to one element, A

and
are reflexive (A
 Symmetry: If A

and
B, then B
A and A
A), but
A
is not
A
are not symmetric;
is symmetric
 Transitivity:
 Weak: if A B and B C, then C A
 Strong: if A B and B C, then A C
 We generally assume ,
and
are transitive for individuals
 Completeness: For any A and B in X, A
B or B
A
 Any two alternatives can be compared
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
5
Weak order
 A weak order is a binary relation
that is complete and transitive, defined over a
set of alternatives and representing the preferences of a given individual.
 We use weak order to represent preferences
 Table 4.2 below corresponds to the ordering A
B
C
D
E
Table 4.1: General Pattern of Preferences
A?B
A?C
A?D
A?E
B?C
B?D
B?E
C?D
C?E
D?E
A≂ A
B≂ B
C≂ C
D≂ D
E≂ E
Table 4.2: A Possible Set of Preferences
A≻ B
A≻ C
A≻ D
A≻ E
B≻ D
B≻ E
C≻ D
C≻ E
D≻ E
A≂ A
B≂ B
C≂ C
D≂ D
E≂ E
Slides Produced by Jeremy Spater, Duke University.
B≂ C
All rights reserved.
6
Weak order (2)
 The preferences in Table 4.3 correspond to A
B
C
D
A
 These preferences are not valid, because they are intransitive
 Another example of intransitive preferences: A
B and B
C, but A
C
Table 4.3: An Alternative Set of Preferences
A≻ B
A≻ C
D≻ A
A≻ E
B≻ D
B≻ E
C≻ D
C≻ E
D≻ E
A≂ A
B≂ B
C≂ C
D≂ D
E≂ E
Slides Produced by Jeremy Spater, Duke University.
B≂ C
All rights reserved.
7
Utility functions to represent preferences
 A function
can be said to represent
if:
 This notion of preferences is valid for considering private goods
 Utility function exists, assuming diminishing marginal utility, non-satiety, compactness, and
continuity of preferences
 We need a different conception for considering public goods (political preferences)
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
8
Political preferences
 Failure of completeness
 Rational ignorance
 Self-interested voters have limited incentives to learn about public good
 Information is limited and costly
 Politicians can mislead
 Voters likely don’t know about all possible alternatives
 Relationship between candidates and policy outcomes is ambiguous
 Completeness is more plausible in committee voting than in mass elections
 Failure of non-satiety
 It makes sense to assume people always (weakly) prefer more goods
 This assumption does not make sense with respect to public goods
 “More is better” does not apply, because more services mean more taxes
 For ideological issues, “more” does not always have a clear meaning
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
9
Public goods
 Excludable: How expensive is it to keep someone from using a good?
 Rival: Does another person’s use of a good decrease my use of it?
 Four types of goods, based on these two dimensions
Table 4.4: Four Types of Goods
Rivalrous Consumption
Excludable
Non-excludable
Private Goods
Common Pool Resources
( Apples,
(Fisheries, Atmosphere)
Oranges)
Non-Rival Consumption
Club Goods
Public Goods
(Swimming Pools)
(National Defense)
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
10
Public goods (2)
 Political choice:
 Who (individual or group) determines supply and allocation of different types of goods?
 Sometimes public goods are determined privately, and vice versa
 Considerable variation exists among nations
 Market failures
 Club goods: Acting individually, people supply less than the optimal amount
 Common pool goods: “Tragedy of the commons”: resources are overused
 Public goods: “Free rider problem”: Individuals prefer to let someone else produce the good
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
11
Public goods (3)
 Types of Goods, and Types of Choices
 Different types of choices are suitable for different types of goods
Table 4.5: Types of Goods, and Types of Choices
Type of Good
Choice Type:
Private
Club
Commons
Public
Private
Consumer
Market Failure:
Market Failure:
Market Failure:
Sovereignty
Underprovided
Overprovided
Underprovided Goods,
Too Much Pollution
Collective
Government Failure:
Private Clubs
Fix Property Rights
Classic Public Good
Tyranny of the Group
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
12
Consumer problem
 Two private goods, Apples (A) and Broccoli (B). Income I
 Rewrite with Lagrangian multiplier and take first-order conditions
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
13
Consumer problem (2)
 Consumer’s problem continued
 Re-arrange to obtain familiar condition for maximizing consumer welfare
𝜕𝑓
𝜕𝐴 = 𝑃𝐴
𝜕𝑓 𝑃𝐵
𝜕𝐵
 Last dollar spent on Apples has same benefit as last dollar spent on Broccoli
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
14
Walrasian auctioneer and equilibrium
 The total quantity demanded of each good depends on the prices
 “Walrasian auctioneer”: Hypothetical concept to explain markets
 “Tatonnement”: Trial and error
 Try different prices until total quantity demanded equals total supply available
 Auctioneer calls out prices; consumers respond with their supply demanded
 If too much is demanded, raise price; if too little, lower price
 Iterative process until correct prices are reached
 Process is applied to both supply and demand sides of economy
 Under certain conditions, equilibrium exists
 Resources are allocated from less-valued to more-valued uses
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
15
Consumer/citizen problem for public goods
 No “tatonnement” process
 Public goods are nonrival and nonexcludable
 Everyone consumes the same amount
 Individual consumption not related to how much each individual pays
 Transformation function:
 Subject to:
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
16
Consumer/citizen problem for public goods (2)
 Social welfare function (ui is each citizen’s utility):
 Objective function:
 Write down Lagrangian:
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
17
Consumer/citizen problem for public goods (3)
 Take first-order conditions of Lagrangian:
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
18
Consumer/citizen problem for public goods (4)
 The term
is the weight of each individual in the SWF
 1/n is a plausible weight for a democracy.
 For a dictatorship, perhaps the dictator’s weight is 1 and everyone else’s is 0.
 Individual conditions for private goods (w/ui terms cancel!):
 Individual conditions for public goods (w/ui terms don’t cancel!):
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
19
Consumer/citizen problem for public goods (5)
 Problems with representing public and private goods in same setting:
 Weights of SWF with respect to individuals don’t cancel out!
 Group constitution matters
 Implied equilibrium levels depend on personal appraisals of the public good
 No purely individual component; it depends on personal political beliefs
 How do we determine who pays how much?
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
20
Lindahl equilibrium
 Walrasian auctioneer process for public goods
 Announce a level of public goods, and ask citizens what they would pay
 If total payments > cost, raise level of public good (or vice versa)
 Tatonnement process for public goods
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
21
Lindahl equilibrium (2)
 Problems with Lindahl equilibrium:
 Demand revelation: Everyone under-reports their own willingness to pay
 Similar to free rider problem
 End up with lower than optimal level of public good
 Impossibility theorems (Arrow and others):
 There is no solution for a robust and moral SWF that represents public preferences
 Public goods preference revelation problem is impossible to solve
 Failure of “induced” public sector preferences
 Single utility function representing public and private preferences does not exist
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
22
Ways to represent public preferences
 Use weak ordering directly
 This avoids the representation problem entirely
 Model collective choice preferences directly
 Drop economists’ method of modeling “induced” preferences
 Model collective choice rather than public goods
 Deal directly with institutions and consequences of choosing in groups
 Extension to multiple dimensions
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
23
Problem of aggregation
 Core of social choice problem:
 Aggregation of individual choices into legitimate and accepted consensus
 Plott’s fundamental “equation” of politics:
 Outcome can change when institutions change (with constant preferences)
 Outcome can change when preferences change (with constant institutions)
Slides Produced by Jeremy Spater, Duke University.
All rights reserved.
24