Chapter 25

POLITICAL ECONOMICS

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

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 CriteriaOJ

OS

UJ4

UJ3

UJ2

UJ1

US4

US3

US2

US1

Contract curve

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

Smith’s utility

Jones’s utility

OJ

OS

The utility possibility frontier shows those utility levels for Smith and Jones that are obtainable from the fixed quantity of goods available

Any point inside the curve is Pareto-inefficient

C

Equality Criterion

Smith’s utility

Jones’s utility

OJ

OS

One possible criterion could require complete equity giving Smith and Jones the same level of welfare

45°

Utility is equal in this case, but the quantities of X and Y may not be

This occurs at UJA and US

A

USA

UJAA

Contract curve

Equality CriterionOJ

OS

UJ2

UJA

UJ1

US2

USA

US1

A

XSA

XJA

YSA

YSA

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

Smith’s utility

Jones’s utility

OJ

OS

Unequal distributions such as B would be permitted when the only attainable equal distributions are below D

45°

D

B

A Equal distributions that lie between D and A are superior to B because the worse-off individual is better off there than at B

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

The optimal point of social welfare is where W is maximized given the utility possibility frontier

W1

W2

Social Welfare Functions

Smith’s utility

Jones’s utility

OJ

OS

E This occurs at UJE and US

E

USE

UJE

W1

W2

Social Welfare Functions

Smith’s utility

Jones’s utility

OJ

OSEven though point F is Pareto-inefficient, it is still preferred to point D

Note the tradeoff between equity and efficiency

D

F

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

11 2 XU

• Teen 2 has a utility function of the form

22 XU

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 theArrow 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

Smith Jones Fudd

A B C

B C A

C A B

• Preferences among the three policy options for the three voters are:

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 “single-peaked”

Single-Peaked Preferences

Quantity ofpublic good

Utility

A B C

Smith

We can show each voters preferences in terms of utility levels

Jones

Fudd

For Smith and Jones, preferences are single-peaked

Fudd’s preferences have two local maxima

Single-Peaked Preferences

Quantity ofpublic good

Utility

A B C

Smith

Jones

Option B will be chosen because it will defeat both A and C by votes 2 to 1

If Fudd had alternative preferences with a single peak, there would be no paradox

Fudd

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 isCi = (1-t)Yi

• The government also faces a budget constraint

An

ii tnYtYG

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

i

AAi

Ai GnfGnYGfYYnGYUSW

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 lump-sum 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 byUi = (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

i

n

iiiii UUfEV

1 1211 )]()([ vote expected

• Candidate 2 chooses 2 so as to maximize his expected votes

n

ii EVnEV

112 )1( vote expected

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

ii

11 0

Net Value Platforms• The candidates’ goal is to choose 1 that

maximizes EV1 against 2*

• Setting up the Lagrangian yields

n

iiEV

111L

n

iiii UUf

1121 )]*()([L

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 rent-seeking 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 one-dimensional 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