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 .
Social Welfare Functions
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 incomeor consumption possibility frontiers.
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 makeeveryone 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.
If redistribution involves deadweight losses, the “real world” UPF would then lie inside the first- best UPFthat 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.
A social welfare function (SWF) is a “utility function over utility allocations” that gives rise to social indifference curvesover 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.
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 SWFwill choose an allocation on the 45-degree line,
but the Pareto SWFwill usually not.
When SWFs have upward sloping parts, even the Rawlsiandoes not insist on full equality.
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 Aimplies that the lowest-earning 80% of the population earns 50% of all income, and Bindicates 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.
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,
with full equality implying a
Lorenz Curve on 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 the shaded area becoming equal to the area below the 45-degree line – and the Gini Coefficient equal to 1.
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
We have therefore derived the first-best utility possibility frontier.
Next, suppose we can only use a distortionary tax tlevied 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
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 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”.
In our example, we calculated the first-best utility possibility frontier as
and the second-best utility possibility frontier as
Maximizing the SWF subjectto each of these constraints, we then get the following for different a: