©2005, Southwestern

Slides by Pamela L. Hall

Western Washington University

Welfare Economics

Chapter 20

2

Introduction To include society’s value of commodities under alternative

resource allocations directly involves welfare economics Study of all feasible allocations of resources for a society Establishment of criteria for selecting among these allocations

Public Choice Theory Attempts to understand and explain society’s actual choice for

resource allocation• Choice is based on normative economics

Involves value judgments Since various agents have conflicting value judgments, it is difficult to

establish a socially optimal allocation Even if these differing value judgments prevent a socially optimal allocation

Theory of welfare economics provides a method for delineating important conceptual issues facing all societies

3

Introduction Aim in this chapter is to investigate how economic theory

attempts to reconcile individual decentralized resource allocation with overall social values of a society May be accomplished with a social-welfare function

• Requires a cardinal measure of individual consumer preferences

We maximize welfare function subject to a utility possibilities frontier based on individual consumers’ preferences Then we specify and compare alternative egalitarian social-welfare

functions

We discuss Arrow’s Impossibility Theorem Indicates that a social-welfare function is impossible given

consumers’ ordinal ranking of utility and based on some reasonable assumptions concerning society’s social rankings

4

Introduction Because we cannot determine a social ranking based on individual

consumers’ ordinal preferences We evaluate idea of majority voting as a second-best Pareto-optimal allocation

We discuss causes of market failure Such as monopoly power, externalities, public goods, and asymmetric information

• As potential constraints on improving social welfare

We demonstrate Theory of the Second Best by showing how any policy designed for improving social welfare that only corrects some constraints may not result in social welfare improvement Because economists are unable to specify a social-welfare function, an army of

applied economists is required to develop and direct mechanism designs for filling in economic gaps resulting from missing markets

• Objective of each mechanism is to yield an incremental improvement in social welfare

• Tâtonnement process will move a society toward maximum social welfare

5

Social-Welfare Function Using broad definition of social welfare as a level of happiness for

society as a whole Measurement for this happiness is needed to determine socially optimal

allocation of resources• Such a measurement for determining how well off agents are in a society requires

a set of welfare criteria

Much of research on formulation of welfare criteria and their implications for economic policy has relied on Pareto-allocation criterion A Pareto criterion is a value judgment based on unanimity rule If one agent could be made better off without reducing welfare of others

• Social welfare could be improved by allocation that makes this one agent better off

Since no one agent is made worse off and at least one agent is made better off It is assumed, given independence of utility functions, that all agents would

support Pareto criterion

6

Social-Welfare Function Pareto-optimal allocation yields an efficient allocation of resources and

thus is a necessary condition for a social optimum However, many decisions on allocation result in an improvement of one agent’s

utility at expense of other agents• For example, a redistribution of endowments from taxing rich households and

providing subsidized housing for poor households may increase social welfare But cannot be justified by Pareto criterion

Fundamental inadequacy of Pareto criterion is its inability to yield a complete ranking of all social states within an economy

• Pareto criterion is useless in context of many policy propositions, so additional welfare criteria are necessary to determine if these policies will improve social welfare

To investigate a social-welfare function, a comparison of individual consumers’ utilities is generally required Assumed that utility functions can be measured on a cardinal scale

• Under this assumption, taking a monotonic transformation of utility function will change preference relationships

7

Pure-exchange Economy Consider pure-exchange economy developed in Chapter 6 Two-consumer (Friday and Robinson), two-commodity (q1 and

q2) economy is illustrated in Figure 20.1 Only points on contract curve can be considered as possible

candidates for a social optimum For example, points P1, P2, and P3 represent tangencies of Friday’s

and Robinson’s indifference curves• Any point off this contract curve is not Pareto efficient

Possible to increase welfare of one consumer without reducing welfare of the other

From contract curve in Figure 20.1,we can derive a utility possibilities frontier Theoretically similar in construction to production possibilities frontier

8

Figure 20.1 Contract curve for a two-consumer, two-commodity pure-exchange

economy

9

Pure-exchange Economy Utility possibilities frontier

Mapping of Pareto-efficient utilities for Robinson, R, and Friday, F, corresponding to each point on contract curve

For P1, P2, and P3, in Figure 20.1, corresponding utility levels for Robinson and Friday are plotted on horizontal and vertical axes, respectively, in Figure 20.2

Points on utility possibilities frontier correspond to tangency of indifference curves along contract curve in Figure 20.1

• Utility combinations associated with P1, P2, and P3 are same for both figures

Every point inside this utility possibilities frontier is a feasible allocation Corresponding to points inside Edgeworth box of Figure 20.1

Boundary of utility possibilities frontier represents efficiency locus (contract curve) in Figure 20.1

10

Pure-exchange Economy For a given amount of q1 and q2, utility possibilities frontier

indicates combination of UR and UF that can be obtained An increase in amount of q1 and q2 will result in utility possibilities

frontier shifting outward

With increasing opportunity cost (which yields a concave utility possibilities frontier) Sacrifice in Friday’s utility increases for an additional unit increase in

Robinson’s utility• However, although a monotonic transformation of an agent’s utility

function preserves preference ordering It can result in opportunity cost switching from increasing to decreasing

One basis for assumption of measuring utility on a cardinal scale

11

Figure 20.2 Utility possibilities frontier

12

Production and Exchange Economy We can also derive a utility possibilities frontier in a general-equilibrium context by

considering production Efficiency condition is based on a given level of utility for Friday

Illustrated in Figure 20.3

Changing this level of utility for Friday will result in alternative combinations of q1 and q2 produced and allocated between Robinson and Friday

As illustrated in Figure 20.4, maximizing Robinson’s utility given UFas Friday’s level of satisfaction results in Pareto-efficient allocation of (qR

1, qR2, qF

1, qF2) with q*1 and q*2

efficiently produced With an alternative level of satisfaction for Friday, say UF' maximizing Robin’s utility will result

in an alternative Pareto-efficient allocation, (qR'1, qR'

2, qF'1, qF'

2) with q*1 and q*2 produced

In general, considering all possible Pareto-efficient allocations (MRSR = MRSF = MRPT) We obtain a collection of Pareto-efficient utility levels for both Robinson and Friday

• By varying Friday’s utility from zero to level where Robinson’s utility would be zero

• Plotting these Pareto-efficient utility combinations yields utility possibilities frontier in Figure 20.2

13

Figure 20.3 Efficiency in production and exchange for a two-consumer economy

14

Figure 20.4 Efficiency in production and exchange for alternative utility levels

15

Production and Exchange Economy

For an economy with production, every utility bundle on this frontier represents a Pareto-efficient allocation Where MRSR = MRSF and MRSR = MRSF = MRPT

A utility bundle in interior of frontier, say point A, is not Pareto optimal It is possible to increase either Robinson’s or Friday’s utility without decreasing

other’s utility

In contrast, on the frontier, say at point P1, Friday’s utility cannot be increased without reducing Robinson’s utility At P1 utility combination and any other utility bundle on frontier are Pareto optimal

Initial endowment of resources held by Robinson and Friday will determine agent’s location on frontier If Robinson has a proportionally larger share of initial resources, utility bundle P1

may result A reversal of endowments may yield a higher utility level for Friday, such as

bundle P3

16

Maximizing Social Welfare Even after eliminating all Pareto-inefficient allocations, there remains an

infinite number of efficient allocations Represented by infinite number of points on utility possibilities frontier

First Fundamental Theorem of Welfare Economics A perfectly competitive equilibrium will result in a Pareto-efficient allocation Depending on initial distribution of endowments, a perfectly competitive

equilibrium can occur at any point on utility possibilities frontier However, from a society’s point of view, allocation resulting from a perfectly

competitive equilibrium may not be equitable• Society may redistribute income (initial endowments) among consumers in an

effort to achieve equity May take form of redistributing income

Taxing wealthy and giving tax revenue to poor Providing commodities to poor (for example, Medicare or surplus food from

agricultural support programs) Market regulation (for example, rent control or agricultural price supports)

17

Maximizing Social Welfare Efforts by governments to achieve a more equitable allocation are

costly in terms of possibly generating inefficiencies within an economy For example, government playing Robin Hood dampens incentive to work

and invest• Often directs resources toward tax avoidance

Can use concept of a social-welfare function as method for determining socially optimal allocation among points on a utility possibilities frontier With a social-welfare function, can determine point that maximizes social

welfare in terms of both equity and efficiency criteria Assuming government is not paternalistic, this function would generally

depend on welfare (utility) of agents within an economy• Government would then maximize social welfare subject to utility possibilities

frontier

18

Maximizing Social Welfare For example, consider following social-welfare function, U, for an economy

consisting of two consumers (Robinson, R, and Friday, F)

Assuming a diminishing marginal rate of substitution between consumer utilities, we can determine convex social indifference, or isowelfare, curves Assumption implies that society has inequality aversion

• Where (holding social welfare constant) the more satisfaction Robinson has the less society is willing to give up Friday’s utility for one more unit of Robinson’s utility

As illustrated in Figure 20.5, tangency between a social indifference curve and utility possibilities frontier results in maximum level of social welfare

• Point P2 is only point on utility possibilities frontier where there is no other point preferred to it For example, point P3 is Pareto efficient but there are points that are preferred to P3

Even though point A is Pareto inefficient, society prefers it over Pareto-efficient point P3

Using maximum level of social welfare, point P2, we determine optimal allocation of commodities in Edgeworth box (Figure 20.1) for a pure-exchange economy

• Or in production possibilities frontier (Figure 20.3) for a production and exchange economy

19

Figure 20.5 Maximizing social welfare

20

Shapes of Isowelfare Curves A social-welfare function represents society’s preferences for

particular Pareto-efficient points on a utility possibilities frontier Various social preferences may be represented by social indifference

curves taking on various shapes• These shapes (and thus social preferences) are generally based on some

equitable allocation among Pareto-efficient allocations

Comparison of alternative Pareto-efficient points requires value judgments concerning trade-off among consumer utilities Can be no one definition for equity

Social indifference curves will take on a number of forms Depending on which criterion (value judgment) is employed for

determining equitable allocation• Two criteria—egalitarian and utilitarian

21

Egalitarian Egalitarianism can take two forms

Allocate each consumer an equal amount of each commodity In terms of our Robinson and Friday two-commodity economy, this

egalitarian criterion sets qR1 = qF

1 and qR2 = qF

2

In a pure-exchange economy, Robinson and Friday would split total endowment of each commodity in half Unless Robinson and Friday have identical utility functions, level of utility

achieved by them will not be the same• But their utility levels are not a factor in this egalitarian equity

In terms of a social-welfare function, social preferences for Robinson’s or Friday’s utilities are identical Are perfect substitutes as long as commodities are allocated equally

between them

• Maximizing welfare function with additional constraint that it be Pareto-efficient in terms of a utility possibilities frontier will result in maximizing social welfare

22

Egalitarian Second type of egalitarian criterion is an allocation of commodities

Resulting in equality of utilities across all consumers

For Robinson and Friday, this criterion sets UR = UF

A social-welfare function resulting in equality of utilities is Rawlsian criterion Most equitable allocation maximizes utility of least-well-off consumer in

society For Robinson and Friday, Rawlsian criterion is

Maximum level of social welfare given a specific utility possibilities frontier is on Pareto-efficient utility possibilities frontier (Figure 20.6) Unless Robinson and Friday have the same utility functions

• Equality of utilities will not result in Robinson and Friday each receiving the same commodity allocation

23

Figure 20.6 Rawlsian social-welfare function

24

Utilitarian Maximizes sum of consumers’ utility Criterion was formally developed by Bentham and provided initial impetus to

utility theory For Robinson and Friday, criterion is

Called classical utilitarian, or Benthamite welfare function Maximized subject to a utility possibilities frontier (Figure 20.7)

Under utilitarian criterion, increases or decreases in individual consumers’ utility results in identical changes in social welfare Only total utility is relevant, so utilitarian criterion does not consider distribution of

utility• As long as social gain is greater than social loss, it makes no difference that consumer who

gains in utility may already be happier than the other consumer

Unless utility functions of individual consumers are close to being identical Utilitarian criteria can result in substantial differences in consumers’ utility

Although ethics teaches that virtue is its own reward, classical utilitarian function teaches that reward is its own virtue Only total level of utility is important

25

Figure 20.7 Benthamite (classical utilitarian) social welfare function

26

Utilitarian By incorporating some virtue into classical utilitarian function, we get a

generalization of this function Weighted sum of utilities

• Weights (R, F) indicate how important each consumer’s utility is to overall social welfare

For example, utility of an individual such as Mother Teresa will be weighted higher than that of a child sex offender

In Figure 20.7, utilitarian social welfare optimal allocation is tangency between social indifference curve and utility possibilities curve Depending on weights associated with individual consumers’ utility

• Any Pareto-efficient point on utility possibilities frontier could be a social-welfare maximum

The more egalitarian a society is, the more its social indifference curves will approach right angles Indicating society is concerned with equity issues of distribution

• For a utilitarian society that is indifferent to distribution, curves are more linear Showing society simply maximizes output

27

Arrow’s Impossibility Theorem A problem in maximizing social welfare is how to establish this social-

welfare function A welfare function based on individual consumer preferences would be a

desirable• Assuming social welfare is to reflect some aggregate consumer preferences

However, because preference ranking by consumers is generally only ordinal

• There is not sufficient information to determine a reasonable social preference ranking of choices

Numerous examples where, due to ordinal preference ranking among individuals, an aggregate ranking is impossible One example is Battle of the Sexes game discussed in Chapter 14

• Couple cannot jointly (socially) rank their preferences for opera or fights

28

Arrow’s Impossibility Theorem Arrow’s Impossibility Theorem

Impossible to establish a reasonable social preference ranking based solely on individual ordinal preference rankings

Suppose there are several feasible social states It is assumed each individual in society can ordinally rank these states

as to their desirability To derive a social-welfare function, there must exist a ranking of these

states on a society-wide scale that fairly considers these individual preferences

• Let’s consider just three possible social states (A, B, and C) For example, these states could be sending a human to Mars, building and

equipping a new aircraft carrier, or curing cancer Arrow’s Impossibility Theorem says a reasonable social ranking of these

three states cannot exist based only on how individual agents ordinally rank these states

29

Arrow’s Impossibility Theorem A reasonable social ranking may be stated with the following axioms relating

individual consumers’ preferences Axiom 1: Completeness—Social ranking must rank all social states

• Either A > B, B > A, or A B for any two states Identical to Completeness Axiom for individual preference ordering

Axiom 2: Transitivity—Society’s social ranking must be transitive• Given three social states, A, B, and C, if A > B and B > C, then A > C

Identical to Transitivity Axiom for individual preference ordering

Axiom 3: Pareto—If every consumer prefers A to B, then A is preferable in a social ranking

• This also holds for the other two pairs (A, C) and (B, C) Identical to a Pareto improvement

Axiom 4: Nondictatorial—One consumer’s preferences should not determine society’s preferences

• No agent paternalism Axiom 5: Pairwise Independence—Society’s social ranking between A and B should

depend only on individual preferences between A and B• Not on individual preferences for some other social state, say state C

Identical to Independence Axiom for individual preference ordering of states of nature

30

Arrow’s Impossibility Theorem Can now state Arrow’s Impossibility Theorem more formally

A social preference ranking satisfying these five axioms is impossible, given an ordinal ranking of individual agent preferences

• Implies that there is no way to aggregate agents’ ordinal preferences into a social preference ranking without relaxing at least one of these axioms

Axioms may seem a reasonable set of conditions for democratically choosing among social states However, Arrow demonstrated that it is impossible to socially choose

among all possible sets of alternatives without violating at least one of the axioms

Thus, social choice must be unreasonable if it is based on agents’ ordinal preference ranking

31

Majority Voting To see that Arrow’s Impossibility Theorem holds, let’s consider majority

voting Important social preference mechanism design

• Set of rules governing procedures for social [collective] choice

Majority voting satisfies both Pareto Axiom and Nondictatorial Axiom Sensitive to each individual agent’s preferences

Majority voting is symmetric among agents Treats all agents the same and all agents have just one vote

It is also neutral among alternatives By not making a distinction among alternatives a priori

However, majority rule can lead to a pattern of social choices that is not transitive Even though every voter has ordinal and transitive preferences

• Thus, it violates Axiom 2

32

Majority Voting Consider ballot in Table 20.1 among three

voters, Robinson, Friday, and Simpson Voters’ preferences are as follows

• Robinson and Simpson prefer alternative A to B• Robinson and Friday prefer alternative B to C• Friday and Simpson prefer alternative C to A

Majority (two) prefers A to B and B to C, but majority also prefers C to A

Thus majority voting results in a cyclical pattern that is intransitive

Called Condorcet Paradox Presents a major problem for group decision making

33

Table 20.1 Condorcet Paradox

34

Majority Voting Next let’s consider case in which each voter must

vote for just one alternative As illustrated in panel (a) of Table 20.2, ordinal

preference ranking in Table 20.1 results in a three-way tie

• All three alternatives receive equal votes

However, if one alternative is removed, a clear winner results

• As illustrated in panel (b), when alternative C is removed, alternative A receives majority vote

Here, Axiom 5 is violated We see this violation of Axiom 5 often in U.S. presidential elections

Where a third-party candidate has determined the outcome

35

Table 20.2 Pairwise Independence

36

Majority Voting Development of a social-welfare function requires more than just an ordinal

ranking of individual consumer preferences Requires a comparison of utilities across consumers on a cardinal scale

• For example, one reason a third party can influence results of an election is that no weight is given to intensity of voters’ desires

However, intensity of desires is a utility measure that can only be measured on at least a cardinal scale

Magnitude or intensity of an individual voter’s desires is not known when she votes

However, allowing voters an ordinal preference ranking (Table 20.1) instead of just one vote (Table 20.2) does elicit additional information on voter’s preference May result in a social ranking more consistent with a majority of electorate

• New voting machines, being put into place after 2000 presidential election, have capability to allow voters to ordinally rank candidates

Called instantaneous voting, procedure has not yet been widely adopted But offers potential of further revealing voters’ preferences and mitigating any strategic

voting

37

Strategic Voting A problem with allowing ordinal ranking of candidates (or any other choices) is

possibility of strategic voting Where an agent does not reveal her true preferences but instead votes to enhance

outcome in her favor A game-theory strategy

Particularly effective when number of voters is relatively small or when a strategic-voting coalition can be formed

One form of strategic voting is for an agent, say Friday, to rank her first choice highest Then rank other alternatives inversely to expected outcome

• Thus, Friday would rank alternative expected to be in close competition with her first choice last, suppressing competitive threat

Strategic voting is illustrated in Table 20.3 for determining social ranking of four alternatives In panel (a), alternative A, which was not Friday’s top choice, comes out on top

• However, as illustrated in panel (b), Friday can change outcome by ranking alternative A low (strategic voting)

Now Friday’s top choice, alternative B, comes out on top as the social choice Judges in Olympic games have been accused of practicing this type of strategic voting

38

Table 20.3 Strategic Voting

39

Strategic Voting A method for removing this potential of strategic voting is

sequential voting Lowest-ranking alternative after each vote is dropped and another

vote is then taken on remaining alternatives

In panel (b) of Table 20.3, alternative C only received a rating of 5 Dropping this alternative from list yields individual preference ranking

for the three alternatives listed in panel (a) of Table 20.4• Now alternative D receives lowest ranking

• Dropping alternative D and re-voting on alternatives A and B yields outcome in panel (b)

From panel (b), alternative A is still selected even given strategic voting by Friday

40

Table 20.4 Sequential Voting

41

Strategic Voting Sequential voting is used to elect Speaker of the House in

U.S. House of Representatives Employing sequential voting also allows for a social ranking

of alternatives based on Pairwise Independence Axiom Implementing such a process for U.S. presidential elections

would probably have changed a number of outcomes By adopting instantaneous voting, where voters rank their choices

• Low-ranking alternatives could be automatically dropped until only two alternatives are left

Given these two remaining alternatives, a president with majority of support would then be elected

42

Strategic Voting Illustrates that a confederation of individuals forming a

society should not be expected to behave with same coherence as would be expected from an individual Arrow’s Theorem implies that institutional detail and procedures of a

political process (mechanism design) cannot be neglected• Thus, it is not surprising that academic disciplines that complement

economics, such as political science and psychology Have evolved to address process of group choice

Attempt to determine intensities of individual and group desires Formulate policies and rules for group choice and actions

As demonstrated by Condorcet Paradox and quid pro quo example in Chapter 14 An agenda that determines which alternatives are first considered

will affect social choice

43

Market Failure Suppose some process for group decision does exist for

determining optimal social choice A naive solution, based on Second Fundamental Theorem of

Welfare Economics• Would advocate allowing markets freedom to obtain this social optimal

given a reallocation of endowments

Unfortunately, this solution is based on properties of a perfectly competitive equilibrium

• Extreme theoretical case of resource allocation Does not generally hold for any society

When properties of a perfectly-competitive equilibrium do not hold Resulting equilibrium is not efficient, so market failure exists

44

Market Failure In general, conditions causing market failure are classified into four categories

Monopoly power• Exists when one or a number of agents (suppliers or demanders of a commodity) exert some

market power in determining prices

Externalities• An interaction among agents that are not adequately reflected in market prices—effects on agents

are external to market Air pollution is classic example of an externality

Public goods• One individual’s consumption of a commodity does not decrease ability of another individual to

consume it Examples are national defense, income distribution, and street lights

Asymmetric information• When perfectly competitive assumption of all agents having complete information about

commodities offered in market does not hold Incomplete information can exist when cost of verifying information about a commodity may not be universal

across all buyers and sellers For example, sellers of used automobiles may have information about quality of various automobiles

that may be difficult (costly) for potential buyers to acquire

• When there is asymmetry in information buyers may purchase a product in excess of a given quality

45

Market Failure Existence of monopoly power, externalities, public goods,

and asymmetric information are justification for establishment of governments to provide mechanisms to address resulting market failures Governments can regulate firms with objectives of mitigating

monopoly power and negative externalities Governments can provide for public goods either by direct

production or private incentives Governments can generate information, aid in its dissemination, and

mandate that information be provided in an effort to reduce asymmetric information

• The more a government must intervene in marketplace to correct these failures

The less dependent will the economy be on freely operating markets

46

Market Failure In some societies these market failures appear quite

large and, thus, freely operating markets are severely limited True in many centrally-planned economies

• Where government determines what and how to produce as well as who should receive commodities produced

Even within U.S., which generally relies on free markets to allocate resources and outputs, there is always the question concerning level of government intervention

• For example, many environmental regulations directly limit inputs firms can use in their production decisions

For example, local zoning ordinances may restrict a firm’s use of inputs that generate noise, smoke, or odors