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Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Inequality aversion and risk aversion
Christopher P. Chambers
Conference on Inequality and Risk, Paris, June 26, 2010
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Introduction
Risk attitudes for groups or households
Samuelson’s social welfare justification of representativeconsumer hypothesis
A society allocating resources to maximize social welfarebehaves as a single agent
Optimal allocation of aggregate risk in a household
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Samuelsonian aggregation in this context: risk sharing/riskallocation
Social welfare functions can be compared with respect toinequality aversion
Is there a general relationship between inequality aversionand household risk aversion?
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
To talk about inequality aversion, need interpersonallycomparable notion of utility
In the risk case, a natural benchmark is the certaintyequivalent
Certainty equivalent is calibrated so that utility of ariskless prospect is the value of that prospect
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Simple intuition suggests more inequality averse socialwelfare functions imply more risk averse households
Intuition is confirmed in some basic cases
In comparing “absolutely inequality averse” social welfarefunctions
But the intuition does not extend more generally
In fact, inequality neutrality generates the least risk aversehouseholds of all
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Basic setup
Ω = 1, ..., ω finite set of states of the world
N = 1, ..., n finite set of agents
Consumption space is RΩ+
Prior π = (π1, ..., πω) over states of the world (say fullsupport)
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Preferences R on RΩ+ are increasing and continuous; they
are also risk averse
Risk aversion requires that for all x ∈ RΩ+, (π · x) R x
Every preference can be represented by its certaintyequivalent function U i
U i : RΩ+ → R+ satisfies
(U i (x), ..., U i (x)
)I x
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Fixed-profile aggregation exercise
To keep analysis simple, assume W : RN+ → R operates
directly on utils (in terms of certainty equivalents)
Thus group utility of allocation (x1, ..., xn) isW (U1(x1), ..., Un(xn))
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Household utility becomes UW : RΩ+ → R defined by
UW (x) = max∑i x
i=xW (U1(x1), ..., Un(xn))
Household maximizes social utility across all allocations ofx
Note: UW not necessarily a certainty equivalentrepresentation
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
A basic example
Let’s consider three social welfare functions:
1 Maxmin: Wmin(u1, ..., un) = mini ui
2 Utilitarian: WU(u1, ..., un) = ∑i ui
3 Maxmax: Wmax (u1, ..., un) = maxi ui
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
A basic example
Let’s consider three social welfare functions:
1 Maxmin: Wmin(u1, ..., un) = mini ui
2 Utilitarian: WU(u1, ..., un) = ∑i ui
3 Maxmax: Wmax (u1, ..., un) = maxi ui
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
A basic example
Let’s consider three social welfare functions:
1 Maxmin: Wmin(u1, ..., un) = mini ui
2 Utilitarian: WU(u1, ..., un) = ∑i ui
3 Maxmax: Wmax (u1, ..., un) = maxi ui
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
A basic example
Let’s consider three social welfare functions:
1 Maxmin: Wmin(u1, ..., un) = mini ui
2 Utilitarian: WU(u1, ..., un) = ∑i ui
3 Maxmax: Wmax (u1, ..., un) = maxi ui
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Let’s assume for all i , U i is homogeneous:U i (αx) = αU i (x) for α > 0 and quasiconcave
Fix a riskless aggregate bundle (c , c , ..., c) ∈ RΩ+
Can compare risk aversion of different induced householdpreferences by studying at least as good as sets for(c , c, ..., c).
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
For each agent i , denote by
Ui (c) = x ∈ RΩ
+ : U i (x) ≥ U i (c , c , ..., c)The at least as good as set
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Now compute at least as good as set for UW at (c , c , ..., c) foreach of the three social welfare functions
1 Maxmin: ∑i (1/n)U i (c)
2 Utilitarian: conv⋃
i Ui (c)
3 Maxmax:⋃
i Ui (c)
Maxmin, Maxmax, both subset of utilitarian
No relation between maxmin and maxmax (in general)
The “inequality neutral” social welfare function results inleast risk averse household preference
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Now compute at least as good as set for UW at (c , c , ..., c) foreach of the three social welfare functions
1 Maxmin: ∑i (1/n)U i (c)
2 Utilitarian: conv⋃
i Ui (c)
3 Maxmax:⋃
i Ui (c)
Maxmin, Maxmax, both subset of utilitarian
No relation between maxmin and maxmax (in general)
The “inequality neutral” social welfare function results inleast risk averse household preference
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Now compute at least as good as set for UW at (c , c , ..., c) foreach of the three social welfare functions
1 Maxmin: ∑i (1/n)U i (c)
2 Utilitarian: conv⋃
i Ui (c)
3 Maxmax:⋃
i Ui (c)
Maxmin, Maxmax, both subset of utilitarian
No relation between maxmin and maxmax (in general)
The “inequality neutral” social welfare function results inleast risk averse household preference
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Now compute at least as good as set for UW at (c , c , ..., c) foreach of the three social welfare functions
1 Maxmin: ∑i (1/n)U i (c)
2 Utilitarian: conv⋃
i Ui (c)
3 Maxmax:⋃
i Ui (c)
Maxmin, Maxmax, both subset of utilitarian
No relation between maxmin and maxmax (in general)
The “inequality neutral” social welfare function results inleast risk averse household preference
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
A few formal definitions
Standard Arrow-Pratt comparative notions of inequalityaversion and risk aversion
Utility U more risk averse than U ′ if for all(c , c, ..., c) ∈ RΩ
+
U(x) ≥ U(c , ..., c)⇒ U ′(x) ≥ U ′(c , c, ..., c)
Social welfare W more inequality averse than W ′ if for all(u, u, ..., u) ∈ RN
+
W (y) ≥ W (u, ..., u)⇒ W ′(y) ≥ W ′(u, ..., u)
Definitions appear similar, but relate to different spaces(RΩ
+ vs. RN+)
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Further, social welfare function W is inequality averse (inan absolute sense) if for all (u1, ..., un)
W (∑i
1
nui , ..., ∑
i
1
nui ) ≥ W (u1, ..., un)
In environments of certainty, we should always allocateequitably (follows from certainty equivalent representation)
Dividing a (certain) dollar equally is always at least asgood as any other allocation
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Theorem
If W and W ′ are inequality averse, and W is more inequalityaverse than W ′, then for any (U1, ..., Un), UW is more riskaverse than UW ′
.
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
c U (x)
W ′
W
Figure: Proof of main result
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Does result extend to non inequality averse social welfarefunctions?
No (by the example in the beginning)
But we do have one result
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Theorem
For any social welfare function W , the household preferenceinduced by the utilitarian welfare function WU is less risk aversethan that induced by W .
Inequalityaversion andrisk aversion
Christopher P.Chambers
Introduction
Basic setup
An example
Moredefinitions
Conclusion
Conclusion
We conclude that inequality neutral social welfarefunctions induce least risk averse societies/households
An inequality neutral social welfare function is a functionof total amount of money held by society (in risklessenvironments)
Thus, a society which tends to maximize national income,etc, will tend to be as risk neutral as possible (it will berisk averse whenever individuals are all risk averse)