Fairness and Social Welfare Functions

Deriving the Utility Possibility Frontier (UPF)

We begin with the Edgeworth Box that starts with individual 1, and then adds individual 2.

The tangencies of indifference curves give the contract curve – the efficient allocations.The allocation A then translates into one way utility can be divided between the two individuals, as does allocation B and allocation C.Repeating this for each allocation on the contract curve then gives us this economy’s

utility possibility frontier.By plotting income or consumption on the axes, we could similarly illustrate income or consumption possibility frontiers.

First Best vs. Second Best UPFs

D

It’s easy to see that any “utility allocation” inside the UPF is inefficient. Consider, for instance, the allocation D – it is inefficient because we can make everyone better off.

It is not, however, clear that all utility allocations are achievable under real world constraints.

This is because the assumed mechanism for achieving them involves lump sum transfers while in the “real world” we rely on distortionary redistributive taxation.

Suppose that endowments are unequally distributed, with individual 1 initially owning everything – resulting in the initial allocation E.

E

If redistribution involves deadweight losses, the “real world” UPF would then lie inside the first- best UPF that assumes lump sum transfers …

… and it may in fact have an upward sloping part.

UPFs that arise through non-lump sum redistributions are called second-best UPFs – and these give rise to equity/efficiency tradeoffs.

first-best UPF

second-best UPF

Social Welfare Functions and First Best UPFs

A social welfare function (SWF) is a “utility function over utility allocations” that gives rise to social indifference curves over utility allocations. It is a tool often used in normative economics to evaluate which policies are “better” than others.

The Pareto SWF views individual utilities as perfectly substitutable for one another.

Other SWFs treat inividual utilities as less substitutable for one another …

… and the Rawlsian SWF treats individual utilities as perfect complements.

Under the Pareto SWF, the aim is to maximize to total utility …

… whereas under the Rawlsian SWF, the aim is to maximize to utility of the least well-off individual.

In our special case here, all these SWFs lead to the same social optimum.

Social Welfare and Second Best UPFs

But the different social values that are built into different SWFs lead to different evaluations of policies when UPFs take on less symmetric shapes.So long as the UPF is downward sloping, the Rawlsian SWF will choose an allocation on the 45-degree line, but the Pareto SWF will usually not.When SWFs have upward sloping parts, even the Rawlsian does not insist on full equality.

Toward Measurable Approaches

While we could apply the same tools to consumption or income possibility frontiers, more practical tools for characterizing the degree of inequality in a society have been developed.

One such tool is the Gini Coefficient derived from Lorenz Curves.

We begin with a graph of population share on the horizontal and income share on the vertical.

Point A implies that the lowest-earning 80% of the population earns 50% of all income, and B indicates that the lowest-earning 40% of the population earns 5% of all income.

The curve that plots the income share for every percentile of the population share is called the Lorenz Curve.

The shaded area divided by the entire area below the 45-degree line is a measure known as the Gini Coefficient.

Lorenz Curves and Gini Coefficients

A society in which the lowest-earning 80% and the lowest-earning 40% earn a greater share of all income would have less inequality …

… which implies a Lorenz curve that lies closer to the 45-degree line.

As income is more equally distributed, the Lorenz Curve therefore gets closer to the 45-degree line,

The shaded area furthermore shrinks with more equally distributed income …

… implying the Gini coefficient converges to 0 as the society converges to full equality.

Perfect inequality would imply one individual owns everything, …

with full equality implying aLorenz Curve on the 45-degree line.

… with the shaded area becoming equal to the area below the 45-degree line – and the Gini Coefficient equal to 1.

• An alternative approach (epitomized by Robert Nozick) emphasizes the “fairness” of the rules of the game over the “fairness” of the consequences of rules.

• It suggests that an outcome is “just” if we arrived at the outcome from a set of initial circumstances that are “just”.

• Such an approach would tend to emphasize policies aimed at such goals as equal educational opportunity rather than policies that redistribute income or wealth.

• It is particularly critical of an emphasis on outcomes in the presence of compensating wage differentials, and has no preconceived notions about the “right” level of inequality.

• In practice, however, the two philosophical approaches are often not as far apart as one might at first assume – with the consequentialist approach not necessarily focused solely on redistribution.

Other Approaches to Normative Economics

Consequentialist Normative Economics

Suppose individual 1 is endowed with 1 unit of leisure time, any fraction of which can be turned into dollars for consumption at a wage normalized to 1.

Letting leisure consumption be denoted by , private good consumption by individual 1 is then in the absence of any taxation.

Individual 2 is unable to work and can only consume if dollars are transferred from individual 1 to individual 2.

Suppose further that preferences for the two individuals can be described by

Suppose first that we can redistribute individual 1’s endowment in a lump sum way by reducing his leisure by a fraction T and giving the earnings from T to individual 2.

Individual 1’s endowment then shrinks to and her consumption to

Given the equal weight put on consumption and leisure in individual 1’s Cobb-Douglas utility function, individual 1 will optimize at and , with individual 2 receiving Thus,

First Best and Second Best UPFs

We have therefore derived the first-best utility possibility frontier.

Next, suppose we can only use a distortionary tax t levied on individual 1’s earnings

The consumption level of the two individuals will then depend on individual 1’s leisure choice, with

Solving individual 1’s utility maximization problem given the tax rate t, we get and implying a consumption level of for individual 2.

The resulting utility levels are then

and .

From , we can write and substitute it into our expression for the utility of individual 1 to get the second-best utility possibility frontier

Social Welfare Functions

Social preferences used to choose from utility possibility frontiers are then simply expressed as “utility functions defined over utility allocations”.

For instance, a social welfare function might take the Cobb-Douglas form

which implicitly sets the elasticity of substitution of utility allocations equal to 1.

Alternatively, the social welfare function might treat individual utilities as perfect substitutes, giving us the Benthamite social welfare function

which aims to maximize the total utility in society without paying attention to how utility is distributed.

On the other extreme of the spectrum lies the Rawlsian social welfare function

that aims to “maximize the utility of the least well-off”.

Social Optimum with Cobb-Douglas SWF

In our example, we calculated the first-best utility possibility frontier as

and the second-best utility possibility frontier as

Maximizing the SWF subject to each of these constraints, we then get the following for different a:

1

1

1/2