Transcript swf and fairness slides

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
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.
If redistribution involves deadweight losses, the
“real world” UPF would then lie inside the firstbest 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
D
second-best UPF
E
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, with full equality implying
Lorenz Curve onathe 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.
Other Approaches to Normative Economics
• 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.
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
1
and the second-best utility possibility frontier as
1/2
Maximizing the SWF
subject to each of these
constraints, we then get the following for different a:
1