Chapter 25
POLITICAL ECONOMICS
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
Social Welfare Criteria
• Analyzing the choice among feasible
allocations of resources is difficult
– it involves making choices about the utility
levels of different individuals
– in choosing between two allocations (A and
B) the problem arises that some individuals
prefer A while others prefer B
Social Welfare Criteria
• We can use the Edgeworth box diagram
to show the problems involved in
establishing social welfare criteria
– only points on the contract curve are
considered as possible candidates for a
social optimum
– along the contract curve, the utilities of the
two individuals vary, and these utilities are
directly competitive
Social Welfare Criteria
OJ
UJ1
UJ2
US4
UJ3
US3
UJ4
US2
US1
Contract curve
OS
Social Welfare Criteria
• If we are willing to assume that utility
can be compared among individuals, we
can use the contract curve to construct
the utility possibility frontier
Social Welfare Criteria
Jones’s utility
The utility possibility frontier shows those utility
levels for Smith and Jones that are obtainable
from the fixed quantity of goods available
OS
Any point inside the curve is
Pareto-inefficient

C
OJ
Smith’s utility
Equality Criterion
One possible criterion could
require complete equity giving
Smith and Jones the same level
of welfare
Jones’s utility
OS
A

UJA
This occurs at UJA and USA
Utility is equal in this
case, but the quantities of
X and Y may not be
45°
USA
OJ
Smith’s utility
Equality Criterion
XJA
OJ
UJ1
YSA
UJA
US2
A

UJ2
USA
YSA
US1
Contract curve
OS
XSA
Utilitarian Criterion
• A similar criterion would be to choose
the allocation on the utility possibility
frontier so that the sum of Smith’s and
Jones’s utilities is the greatest
– this point would imply a certain allocation
of X and Y between Smith and Jones
The Rawls Criterion
• This was first posed by philosopher
John Rawls
• Suppose that each individual begins in
an initial position in which no one knows
what his final position will be
– individuals are risk averse
– society will only move away from perfect
equality when the worst off person would
be better off under inequality than equality
The Rawls Criterion
Unequal distributions such as B
would be permitted when the
only attainable equal distributions
are below D
Jones’s utility
OS
B

Equal distributions that lie
between D and A are
superior to B because the
worse-off individual is better
off there than at B
 A

D
45°
OJ
Smith’s utility
Social Welfare Functions
• A social welfare function may depend
on Smith’s and Jones’s utility levels
such as
social welfare = W(US,UJ)
• The social problem is to allocate X and
Y between Smith and Jones as to
maximize W
Social Welfare Functions
The optimal point of social
welfare is where W is
maximized given the utility
possibility frontier
Jones’s utility
OS
UJE
E
This occurs at UJE and USE
W2
W1
USE
OJ
Smith’s utility
Social Welfare Functions
Note the tradeoff between
equity and efficiency
Jones’s utility
OS

Even though point F is
Pareto-inefficient, it is still
preferred to point D
D

F
W2
W1
OJ
Smith’s utility
Equitable Sharing
• A father arrives home with an 8-piece
pizza and must decide how to share it
between his two sons
• Teen 1 has a utility function of the form
U1  2 X1
• Teen 2 has a utility function of the form
U2  X 2
Equitable Sharing
• The least resistance option would be to
give each teen 4 slices
– U1 = 4, U2 = 2
• The father may want to make sure the
teens have equal utility
– X1 = 1.6, X2 = 6.4, U1 = U2 = 2.53
• The father may want to maximize the
sum of his sons utility
– X1 = 6.4, X2 = 1.6, U1 = 5.06, U2 = 1.26
Equitable Sharing
• Suppose the father suggests that he will
flip a coin to determine who gets which
portion listed under the three allocations
• The expected utilities of the two teens
from a coin flip that yields either 1.6 or
6.4 slices is
E(U1) = 0.5(2.53) + 0.5(5.06) = 3.80
E(U2) = 0.5(2.53) + 0.5(1.26) = 1.90
Equitable Sharing
• Given this choice, the teens will opt for
the equal distribution because each
gets higher expected utility from it than
from the coin flip
Equitable Sharing
• If the father could subject the teens to a
“veil of ignorance” so that neither would
know his identity until the pizza is
served, the voting might still be different
– if each teen focuses on a worst-case
scenario, he will opt for the equal utility
allocation
• insures that utility will not fall below 2.53
Equitable Sharing
• Suppose that each teen believes that he has
a 50-50 chance of being labeled as “teen 1”
or “teen 2”
• Expected utilities are
X1 = X2 = 4
E(U1) = 0.5(4) + 0.5(2) = 3
X1 = 1.6, X2 = 6.4
E(U1) = 0.5(2.53) + 0.5(2.53) = 2.53
X1 = 6.4, X2 = 1.6
E(U1) = 0.5(5.06) + 0.5(1.26) = 3.16
• The teens will opt for the utilitarian solution
The Arrow Impossibility
Theorem
• Arrow views the general social welfare
problem as one of choosing among
several feasible “social states”
– it is assumed that each individual can rank
these states according to their desirability
• Arrow raises the following question:
– does there exist a ranking on a societywide
scale that fairly records these preferences?
The Arrow Impossibility
Theorem
• Assume that there are 3 social states
(A, B, and C) and 2 individuals (Smith
and Jones)
– Smith prefers A to B and B to C
• A PS B and B PS C and A PS C
– Jones prefers C to A and A to B
• C PJ A and A PJ B and C PJ B
The Arrow Impossibility
Theorem
• Arrow’s impossibility theorem consists of
showing that a reasonable social ranking
of these three states cannot exist
• Arrow assumes that any social ranking
should obey six seemingly
unobjectionable axioms
– “P” should be read “is socially preferred to”
The Arrow Axioms
• It must rank all social states
– either A P B, B P A, or A and B are equally
desirable (A I B) for any two states A and B
• The ranking must be transitive
– if A P B and B P C (or B I C), then A P C
• The ranking must be positively related
to individual preferences
– if A is unanimously preferred by Smith and
Jones, then A P B
The Arrow Axioms
• If new social states become feasible, this
fact should not affect the ranking of the
original states
– If A P B, then this will remain true if some
new state (D) becomes feasible
• The social preference function should
not be imposed by custom
– it should not be the case that A P B
regardless of the tastes of individuals in
society
The Arrow Axioms
• The relationship should be
nondictatorial
– one person’s preferences should not
determine society’s preferences
Arrow’s Proof
• Arrow was able to show that these six
conditions are not compatible with one
another
– because B PS C and C PJ B, it must be the
case that B I C
• one person’s preferences cannot dominate
– both A PS B and A PJ B, so A P B
– transitivity implies that A P C
– this cannot be true because A PS C but C PJ A
Significance of the
Arrow Theorem
• In general, Arrow’s result appears to be
robust to even modest changes in the set
of basic postulates
• Thus, economists have moved away
from the normative question of how
choices can be made in a socially optimal
way and have focused on the positive
analysis of how social choices are
actually made
Direct Voting
• Voting is used as a decision process in
many social institutions
– direct voting is used in many cases from
statewide referenda to smaller groups and
clubs
– in other cases, societies have found it
more convenient to use a representative
form of government
Majority Rule
• Throughout our discussion of voting, we
will assume that decisions will be made
by majority rule
– Keep in mind though, that there is nothing
particularly sacred about a rule requiring
that a policy obtain 50 percent of the vote
to be adopted
The Paradox of Voting
• In the 1780s, social theorist M. de
Condorcet noted that majority rule
voting systems may not arrive at an
equilibrium
– instead, they may cycle among alternative
options
The Paradox of Voting
• Suppose there are three voters (Smith,
Jones, and Fudd) choosing among
three policy options
– we can assume that these policy options
represent three levels of spending on a
particular public good [(A) low, (B) medium,
and (C) high]
– Condorcet’s paradox would arise without
this ordering
The Paradox of Voting
• Preferences among the three policy
options for the three voters are:
Smith
Jones
Fudd
A
B
C
B
C
A
C
A
B
The Paradox of Voting
• Consider a vote between A and B
– A would win
• In a vote between A and C
– C would win
• In a vote between B and C
– B would win
• No equilibrium will ever be reached
Single-Peaked Preferences
• Equilibrium voting outcomes always
occur in cases where the issue being
voted upon is one-dimensional and
where voter preferences are “singlepeaked”
Single-Peaked Preferences
We can show each voters preferences in
terms of utility levels
Utility




A

 Fudd
 Jones


B
C
For Smith and Jones,
preferences are singlepeaked
Fudd’s preferences have
two local maxima
Smith
Quantity of
public good
Single-Peaked Preferences
If Fudd had alternative preferences with a
single peak, there would be no paradox
Utility



A

 Fudd

 Jones
Option B will be chosen
because it will defeat
both A and C by votes 2
to 1


B
C
Smith
Quantity of
public good
The Median Voter Theorem
• With the altered preferences of Fudd, B
will be chosen because it is the
preferred choice of the median voter
(Jones)
– Jones’s preferences are between the
preferences of Smith and the revised
preferences of Fudd
The Median Voter Theorem
• If choices are unidimensional and
preferences are single-peaked, majority
rule will result in the selection of the
project that is most favored by the
median voter
– that voter’s preferences will determine
what public choices are made
A Simple Political Model
• Suppose a community is characterized
by a large number of voters (n) each
with income of Yi
• The utility of each voter depends on his
consumption of a private good (Ci) and
of a public good (G) according to
utility of person i = Ui = Ci + f(G)
where fG > 0 and fGG < 0
A Simple Political Model
• Each voter must pay taxes to finance G
• Taxes are proportional to income and
are imposed at a rate of t
• Each person’s budget constraint is
Ci = (1-t)Yi
• The government also faces a budget
constraint
n
G   tYi  tnY A
i 1
A Simple Political Model
• Given these constraints, the utility
function of individual i is
Ui(G) = [YA - (G/n)]Yi /YA + f(G)
• Utility maximization occurs when
dUi /dG = -Yi /(nYA) + fG(G) = 0
G = fG-1[Yi /(nYA)]
• Desired spending on G is inversely
related to income
A Simple Political Model
• If G is determined through majority rule,
its level will be that level favored by the
median voter
– since voters’ preferences are determined
solely by income, G will be set at the level
preferred by the voter with the median level
of income (Ym)
G* = fG-1[Ym/(nYA)] = fG-1[(1/n)(Ym/YA)]
A Simple Political Model
• Under a utilitarian social welfare
criterion, G would be chosen so as to
maximize the sum of utilities:
n
SW   Ui   [(Y A  G / n )Yi / Y A  f (G )]  nY A  G  nf (G )
i 1
• The optimal choice for G then is
G* = fG-1(1/n) = fG-1[(1/n)(YA/YA)]
– the level of G favored by the voter with
average income
Voting for Redistributive
Taxation
• Suppose voters are considering a lumpsum transfer to be paid to every person
and financed through proportional
taxation
• If we denote the per-person transfer g,
each individual’s utility is now given by
Ui = Ci + g
Voting for Redistributive
Taxation
• The government’s budget constraint is
ng = tnYA
g = tYA
• For a voter with Yi > YA, utility is
maximized by choosing g = 0
• Any voter with Yi < YA will choose t = 1
and g = YA
– would fully equalize incomes
Voting for Redistributive
Taxation
• Note that a 100 percent tax rate would
lower average income
• Assume that each individual’s income
has two components, one responsive to
tax rates [Yi (t)] and one not responsive
(Ni)
– also assume that the average of Ni is zero,
but its distribution is skewed right so Nm < 0
Voting for Redistributive
Taxation
• Now, utility is given by
Ui = (1-t)[Yi (t) + Ni] + g
• The individual’s first-order condition for a
maximum in his choice of t and g is now
dUi /dt = -Ni + t(dYA/dt) = 0
ti = Ni /(dYA/dt)
• Under majority rule, the equilibrium
condition will be
t* = Nm /(dYA/dt)
Representative Government
• In representative governments, people
vote for candidates, not policies
• Politicians’ policy preferences are
affected by a variety of factors
– their perceptions of what their constituents
want
– their view of the “public good”
– the forcefulness of “special interests”
– their desire for reelection
Probabilistic Voting
• Assume there are only two candidates
for a political office
– each candidiate announces his platform (1
and 2)
– also assume that the candidate, once
elected, will actually seek to implement the
platform he has stated
• Each of the n voters observe the two
platforms and choose how to vote
Probabilistic Voting
• The probability that voter i will vote for
candidate 1 is
i = fi [Ui(1) - Ui(2)]
where f’ > 0 and f’’< 0
• The probability that voter i will vote for
candidate 1 is 1 - i
The Candidate Game
• Candidate 1 chooses 1 so as to
maximize the probability of his election
n
n
i 1
i 1
expected vote  EV1   i   fi [U i (1 )  U i (2 )]
• Candidate 2 chooses 2 so as to
maximize his expected votes
n
expected vote  EV2   (1  i )  n  EV1
i 1
The Candidate Game
• Our voting game is a zero-sum game
with continuous strategies (1 and 2)
• Thus, this game will have a Nash
equilibrium set of strategies for which
EV1(1,2*)  EV1(1*,2*)  EV1(1*,2)
– Candidate 1 does best against 2* by
choosing 1*
– Candidate 2 does best against 1* by
choosing 2*
Net Value Platforms
• A “net value” platform is one in which a
candidate promises a unique dollar
benefit to each voter
• Suppose candidate 1 promises a net
dollar benefit of 1 to each voter
• The candidate is bound by a government
budget constraint:
n

i 1
1i
0
Net Value Platforms
• The candidates’ goal is to choose 1 that
maximizes EV1 against 2*
• Setting up the Lagrangian yields
 n

L  EV1    1i 
 i 1 
n


*
L   fi [U (1i )  U (2 )]    1i 
 i 1 
Net Value Platforms
• The first-order condition for the net
benefit promised to voter i is given by
L/1i = fi’Ui’ +  = 0
• If the function fi is the same for all voters,
this means that the candidate should
choose 1i so that Ui’ is the same for all
voters
– a utilitarian outcome
Rent-Seeking Behavior
• Elected politicians perform the role of
agents
– choose policies favored by principals
(voters)
• A perfect agent would choose policies
that the fully informed median voter
would choose
– are politicians so selfless?
Rent-Seeking Behavior
• Politicians might engage in rent-seeking
activities
– activities that seek to enhance their own
welfare
• This would create an implicit tax wedge
between the value of public goods
received by voters and taxes paid
Rent-Seeking Behavior
• Extraction of political rent r would
require that the government budget
constraint be rewritten as
G = tnYA - r
• Voters would take such rent-seeking
activities into account when deciding on
public policies
– would likely reduce G and t
Rent-Seeking Behavior
• Whether political rents can exist in an
environment of open electoral
competition is questionable
– Candidate A announces policy (G,t)A
– Candidate B can always choose a policy
(G,t)B that is more attractive to the median
voter by accepting a smaller rent
• Only with barriers to entry or imperfect
information can positive rents persist
Rent-Seeking Behavior
• Private citizens may also seek rents for
themselves by asking politicians to grant
them favors
• Thus, economic agents engage in rentseeking activities when they use the
political process to generate economic
rents that would not ordinarily occur in
market transactions
Rent Dissipation
• If a number of actors compete in the
same rent-seeking activity, it is possible
that all available rent will be dissipated
into rent seekers’ costs
• Suppose a monopoly might earn profits
of m and a franchise for the monopoly
can be obtained from the government
for a bribe of B
Rent Dissipation
• Risk-neutral entrepreneurs will offer
bribes as long as the expected net gain
is positive
• If each rent seeker has the same
chance of winning the franchise, the
number of bribers (n) will expand to the
point at which
B = m /n
Important Points to Note:
• Choosing equitable allocations of
resources is an ambiguous process
because many potential welfare
criteria might be used
– in some cases, achieving equity
(appropriately defined) may require
some efficiency sacrifices
Important Points to Note:
• Arrow’s impossibility theorem shows
that, given fairly general
assumptions, there is no completely
satisfactory social choice mechanism
– the problem of social choice theory is
therefore to assess the performance of
relatively imperfect mechanisms
Important Points to Note:
• Direct voting and majority rule may
not always yield an equilibrium
– if preferences are single-peaked,
however, majority rule voting on onedimensional public questions will result
in choosing policies most favored by the
median voter
• such policies are not necessarily efficient
Important Points to Note:
• Voting in representative governments
may be analyzed using the tools of
game theory
– in some cases, candidates’ choices of
strategies will yield Nash equilibria that
have desirable normative consequences
• Politicians may engage in opportunistic
rent seeking, but this will be constrained
by electoral competition