inequality aversion and risk aversion · inequality aversion and risk aversion christopher p....

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



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



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?



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



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



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)



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



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))



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



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



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).



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



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



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+)



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



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 ′

.



Introduction

Basic setup

An example

Moredefinitions

Conclusion

c U (x)

W ′

W

Figure: Proof of main result



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



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 .



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)

inequality aversion and risk aversion · inequality aversion and risk aversion christopher p....

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