Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Equity vs. efficiency
Public Economics
Carlo Fiorio
Bocconi University
[email protected]
http://www.econpubblica.uni-bocconi.it/fiorio
Academic Year 2006/2006
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Today’s questions
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Should gov’t care about poverty and inequality?
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Shall we care more about inequality or about efficiency?
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Is more equity always counterbalanced by less efficiency?
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Can we construct a function to choose the best among a set
of alternative social states?
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
When equity increases efficiency
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Equity can produce externalities
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altruism
crime
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
UPF with altruism
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Link b/w poverty and crime
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
When equity increases efficiency
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Equity and incomplete market failures
I
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if some risks are not covered because a market does not exists,
social insurance can increase equity and increase efficiency
Equity and imperfect capital markets
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I
I
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access to capital markets can be inefficiently restrictive
(asymmetric information)
redistribution effects can also be spread across time
redistribution can increase productivity, hence efficiency
redistribution can produce disincentive effects, though.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Social choice
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Can we construct a social choice function that summarizes
social preferences as utility function does?
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“if the queen is happy, the society is happy”: is this a social
choice function? are you happy with it?
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Social choice in democratic societies
Arrow defined axiomatically the minimum criteria a social choice
function must have:
P Pareto principle: if at least one individual strictly
prefers x to y and no-one strictly prefers y to x, then
society prefers x to y ;
U Universal domain: the social choice function should
work for any set of individual preferences;
D Dictator-free: the social choice function should not
depend solely on the rankings of one individual;
I Independence of irrelevant alternatives: the social
ranking of x and y should only depend on individual
rankings of x versus y and not depend on other
choices, as for instance z.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Arrow’s impossibility theorem
Arrow’s impossibility theorem
There is no complete and transitive social choice function
satisfying axioms P, U, D, I
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What? No social orderings of alternatives?
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How to prove it?
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What can be done then?
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Arrow’s impossibility theorem: “proof”
To prove Arrow’s impossibility theorem, let’s use a
counterexample: the Condorcet’s paradox
A
B
C
x
1
2
3
y
2
3
1
z
3
1
2
Note: with majority rule, preference cycle here. x  y  z  x.
Coherent social choice is impossible.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
What shall we do, then?
We have two choices for (partly) overcoming Arrow’s theorem
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relax some of the assumptions of the Impossibility Theorem
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have some common scale for measuring well-being, i.e. make
interpersonal comparison possible.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Relaxing condition I, w/ utilitarian rule
A
B
C
A+B+C
I
x
15
8
3
26
y
10
6
20
36
z
5
10
17
32
Can utility be cardinal?
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Relax I, w/ utilitarian rule (2). Relax U
A
B
C
A+B+C
x
30
4
3
37
y
11
3
20
34
z
10
5
17
32
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Same conclusions as previous slide?
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The solution would be to constraint preferences to be on the
same scale: this means make interpersonal comparison of
well-being possible.
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Other possibile solution: restriction of U. If some solution are
ruled out (multiple peaks) a solution might be possible.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
The social welfare function (SWF)
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Once we allow interpersonal comparisons of well-being a
social welfare function (SWF) can be constructed.
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A SWF relates the welfare of a society as a whole to the
welfare of individuals: W = f (U 1 , U 2 , ..., U N ), where U i is
utility of individual i, i = 1, ..., N.
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Three properties are assumed for f :
P Pareto principle: if U i rises, then W rises.
N Individualism: SWF is a function of individual
welfare
I Inequality aversion: f is concave, i.e. averages
are preferred to extremes and welfare is higher
when inequality is lower.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Inequality aversion and SWF
Aversion to inequality can differ hugely among individuals and give
rise to different shape of SWF:
P i
I Utilitarian: W =
iU
I
Rawlsian: W = min{U 1 , U 2 , ..., U N }
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Isoelastic: W =
aversion.
P
I
I
i )1−e
(1−e) .
i (U
Where e is the inequality
If e = 0:Utilitarian SWF
If e = ∞:Rawlsian SWF
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
SWF and social optimum
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
What value of e is correct?
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Utilitarians: U is a measure of individual happiness. Would
you support this view or a social justice principle? What if a
poor is depressed?
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Rawls argues in favor of e = ∞ suggesting to think about a
“veil of ignorance”: individuals are infinitely risk averse
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Horizontal and vertical equity
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Measures of social welfare may fail to capture aspects of
social justice.
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Vertical equity is reached if richest give to poorest
(redistribution)
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Horizontal equity is reached if equals are treated equally
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SWFs are generally concerned with vertical inequality only
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Critique to “consequentialism”
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Consequentialism is the believe that social choices should be
evaluated by their outcomes and not by the processes by
which they are reached.
Nozick (1970) advocates anti-consequentialism: only
processes are just, not outcomes.
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transfers from rich to poor are unfair because they imply
taxation, which is involuntary exchange.
Past wrongs could be corrected (“rectification principle”) but
it would be too complicated.
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Measuring the equity-efficiency trade-off
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Redistribution has to be financed.
There are two main possible taxes: lump-sum and
distortionary
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BC with lump-sum tax: M − T = px + y (Note: price of y set
to 1)
BC with distortionary tax: M = (p + t)x + y
What are their effects?
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Measuring the equity-efficiency trade-off
Assume that a revenue is needed for some public investment. It
can be financed either by:
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lump-sum tax, or
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disortionary taxation on the price of pizza
Now, assume that the revenue is the same in the two cases.
How much should be transferred to compensate for the new
taxation?
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Marginal cost of public funds (MCPF): the efficiency cost
of raising e1 extra of gov’t revenue. In the graph that follows
is T + AB/T .
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Excess burden of distortionary taxation: MCPF
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Measuring the equity-efficiency trade-off
Now, assume that you want to compare the distortionary with
lump-sum taxation holding constant the level of utility.
Which one is providing the largest revenue?
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Excess burden (or deadweight loss): a measure of the
efficiency cost of distortionary taxation
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
Excess burden of distortionary taxation (cont.)
Carlo Fiorio
Equity vs. efficiency
Should we care about equity and inequality?
Arrow’s impossibility theorem
The Social Welfare Function
Measuring the equity-efficiency trade-off
References
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C&M, Ch. 3
Carlo Fiorio
Equity vs. efficiency