Lecture 3.1 Welfare

WelfareLecture 3.1 – Tom Holden

Intermediate Microeconomics Semester 2

http://micro2.tholden.org/

ECO2051 – Intermediate Microeconomics 1

http://micro2.tholden.org/

Readings

• Varian, Chapter 33

• Morgan, Katz and Rosen, Chapter 12


Welfare questions

• What is a social welfare function?

• How do we go about maximising a social welfare function?

• Why should we use one rather than another?

• Can society use voting to decide on a social welfare function?


Motivation

• 2nd fundamental theorem of welfare economics states:• Any Pareto efficient allocation can be achieved by a competitive equilibrium

with a suitable redistribution of initial resource endowments.

• (Given certain technical conditions.)

• This implies that society can choose over Pareto efficient outcomes, and leaves room to care about equity.• It is typically Pareto efficient for one person to consume everything, but is it

“fair”?


Utility Possibilities Frontier

Comparing utilities

• The concept of Pareto efficiency does not depend on the ability to measure and/or compare utilities.

• However, if we want to choose between Pareto efficient points then we need to start measuring utility.

• We do this using a social welfare function.

• Simple example:• Suppose for Aaron and Kees this takes the form 𝑊 = 𝑊(𝑈𝐾 , 𝑈𝐴).• 𝑊 is increasing in 𝑈𝐾 and 𝑈𝐴.• Note that the way that utility is represented is entirely arbitrary.• Such measurement and comparison may not be possible, but let’s assume

that it is.


Common formulations of the SWF

• Utilitarian: 𝑊 = 𝑈𝐾 + 𝑈𝐴

• More generally 𝑊 𝑈1, … , 𝑈𝑁 = 𝑖=1𝑁 𝑈𝑖.

• What’s important is the sum of total happiness.

• Weighted sum of utilities:• 𝑊 𝑈1, … , 𝑈𝑁 = 𝑖=1

𝑁 𝑎𝑖𝑈𝑖.

• Enables some individuals to have more weight than others.

• Rawlsian:• 𝑊 𝑈1, … , 𝑈𝑁 = min 𝑈1, … , 𝑈𝑁 .

• Society is as well off as its worst off member.

• These are all constant elasticity (of substitution) social welfare functions.• Have the same slope throughout.


Justifications

• Utilitarianism is justified by the kind of argument we gave last week.• “I could have been born as anyone in the world.”

• Perhaps weighted utilitarianism may be justified if we think we are more likely to be born as certain people.• “I know my parents, I could have been more like my brother, or more like my

sister, but I would never have been that different.”• In practice, weighted utilitarian objective functions are often used (implicitly)

by political parties. “Just make enough people happy to ensure you win the vote.”

• Rawlsianism is based on a misunderstanding of risk-aversion.• Don’t tell this to your political scientist friends.• May still be justified if we think people have a kind of “ambiguity aversion”,

so they’re particularly averse to uncertainty about who they are.


Social Indifference Curves/Isowelfare Curves


The slope of ICs represents value judgements about how much society cares about the utilities of the two individuals.At high levels of Aaron’s utility it can go down quite a lot with only a slight improvement in Kees level of U and SW stays constant.

As with a standard utility function, SWFs have corresponding ICs which show how the utility of agents can be traded off while keeping SW constant.

Maximising welfare


A particular Pareto-efficient allocation of resources need not be socially desirable. Point b is not Pareto efficient, yet it is preferred to point a, which is Pareto efficient.

However the social optimum is found when social welfare is maximised and Pareto efficiency is achieved.

Every PE point is a social optimum for some SWF.

The outcomes from different SWFs

11

K’s utility

A’s utility

45°

Rawlsian

Utilitarian

Cobb-Douglas

𝑈𝐴 Utilitarian

𝑈𝐴 Cobb-Douglas

𝑈𝐴 Rawlsian

Every point on the UPF is a welfare maximum for some SWF (although utilities may needto be weighted).

Every welfare maximum is a PEallocation and every PE allocationis a welfare maximum.

Simple example problems (see MKR, p. 451)

• Graph the social indifference curves which correspond to 𝑊 = 𝑈K +𝑈A, and 𝑊 = 2𝑈𝐾 + 𝑈𝐴.

• What types of SWF are these?

• How would you describe the relative importance assigned to the individuals’ respective well-beings, in each case?

• Given the following utility possibility frontier, show how the optimal allocation differs for the two social welfare functions.


𝑈𝐴

𝑈𝐾

The Bergson-Samuelson utility function

• Up to this point we haven’t said anything about what individual utilities depend upon.

• They could be a function of the entire distribution of consumption:• 𝑊 = 𝑊 𝑈𝐾 𝒙 , 𝑈𝐴 𝒙

• A common restriction is that individual utilities depend on individual consumptions, such that:• 𝑊 = 𝑊 𝑈𝐾 𝑥𝐾 , 𝑈𝐴 𝑥𝐴• This is called an individualistic social welfare function or a Bergson-

Samuelson welfare function. • In most of the computation exercises, this is the type of SWF that we will

deal with. • But in reality people often care about their friends’/partner’s consumption

too.


Non-graphical example

• Suppose:• Individual one gets income 𝑌1 and individual two gets income 𝑌2.

• Along the Pareto frontier of the economy 𝑍 = 𝑌1 + 𝑌2.

• Social welfare is given by 𝑊 = 𝑌1 + 𝑌2. May be interpreted as a utilitarian

social welfare function in which people’s utility is 𝑌, or an inequality averse social welfare function in an economy in which people’s utility is just 𝑌.

• May be solved by any of the techniques you’ve seen before.

• Equating the MRS and the slope of the budget line. (MRS =

𝜕𝑊

𝜕𝑌1𝜕𝑊

𝜕𝑌2

. Budget line:

𝑍 = 𝑌1 + 𝑌2.)

• Lagrangians: ℒ = 𝑌1 + 𝑌2 + 𝜆 𝑍 − 𝑌1 − 𝑌2

• Direct substitution: 𝑌1 + 𝑍 − 𝑌1


Comments on welfare economics

• Social welfare depends on individual utilities, it is these that matter.• Any other objectives of society as a whole are not considered.

• Society is only the sum of its individuals.

• Welfare economics is anti-paternalistic:• We assume that individuals know what is good for them.

• No room for nudging.

• Some thinkers worry that welfare economics lacks a moral core.• Do the means matter as well as the ends?

• Others emphasise the importance of freedom and “rights”.


Voting to decide on a social welfare function

• We have seen that different SWFs take different stances on inequality.

• We might imagine that society could use the democratic process to decide on its SWF.

• But there is a problem here, all the voting procedures we could adopt fail in some way.

• Arrow’s impossibility theorem sets up a number of reasonable criteria and shows that all decision making rules fail in one way or another.


Majority voting fails

• One approach is to say that x is socially preferred to y if a majority of individuals prefer x to y.

• But this approach does not always lead to a transitive ordering, despite rational individual preferences.

• x preferred to y.

• y preferred to z.

• Transitivity implies that x preferred to z, but voting gives the opposite outcome.

• The winner depends on the order the votes are taken in.


Person A Person B Person C

x y z

y z x

z x y

Rank-order (Borda count) voting fails

• Another approach is that voters give outcomes a score, which depends on how they rank them.

• The winner is the outcome with the lowest total score.

• As it stands, y is a clear winner, as y scores 3 and x scores 4.

• If z were excluded, then both y and x would have a total score of 3.

• Does it seem right that the social preference between x and y depends on whether z is an option?


Score Person A Person B

1 x y

2 y z

3 z x

Arrow’s Impossibility Theorem

• Take some reasonable criteria:1. Both individual and social preferences are complete, reflexive and

transitive.

2. If everybody prefers x to y then the social preference must be for x not y.

3. The preferences between x and y should depend only on how people rank x versus y, and not on how they rank other alternatives.• Referred to as the “independence of irrelevant alternatives” or IIA.

• AIT: If a social decision mechanism satisfies 1, 2 and 3, then it must be a dictatorship:• All social rankings reflect the rankings of one individual.

• To make social choices we must drop IIA (3 above) and replace it with a weaker condition.


A reasonable alternative to IIA

• Suppose that if voters were asked to vote on a choice between A and any other specific option (e.g. B,C,D,…) they would always choose A.• So maybe if people were asked to vote on the LibDems versus the Tories,

they’d always vote LibDem, and if they were asked to vote on the LibDemsversus Labour they’d also vote LibDem.

• Then it seems reasonable to hope that A would be chosen by the aggregation method.• This is called the Condorcet criterion, and aggregation methods satisfying it

are called Condorcet methods.

• There are many Condorcet methods, but neither “First Past the Post” or “Instant Run-Off Voting” (AKA “Alternative Vote”) satisfy the criteria.


A stronger condition

• Suppose there is some set of options S, such that if voters were given a choice between a specific option in the S and any other specific option not in the set S, then they would always choose the option in the set S.• So maybe the set S could be {Labour, LibDem, Tory}. Minor parties e.g. UKIP

might be outside the set.

• Then it seems reasonable to hope that an option in the set S would be chosen by the aggregation method.• This is called the Schwartz criterion.

• Unlike the Condorcet criterion, there is a unique aggregation method that satisfies it, the Schulze method.

• This is what your student union (and your national government) ought to be using (IMO)…


Conclusions

• Even if the economy generates a Pareto-efficient allocation of resources, government intervention may be necessary to achieve an acceptable distribution of utility.

• This is an attempt to optimise society’s welfare.

• But we need to find some way to find out what these preferences are.

• Arrow’s impossibility theorem demonstrates that aggregating society’s preferences is not straightforward.


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Lecture 3.1 Welfare