Inequality (aversion), Risk (aversion), and Social Welfare

Bianca Mulaney EC426

October 3, 2016

Questions 1.  What is the argument for basing social

welfare function on individual expected-utility functions?

2.  Can one therefore associate inequality aversion with risk aversion?

3.  What are the problems highlighted by Carlsson et al. (2005) and by Cowell and Schokkaert (2001) in making this association?

Measuring Inequality •  Objective: measure of inequality – Why would we want to measure inequality?

•  Example: assess whether or not a taxation scheme affects inequality (compare pre-tax to post-tax income distributions) •  Example: compare income distribution in the past

to today

•  How do we measure inequality?

Historical Context •  Inequality measurements: historically measures

of dispersion (variations about average) –  quartile measure (Bowley) –  mean difference (Gini)

•  Dalton (1920): we are not just interested in the distribution of income, but in the effects of the distribution of income on the distribution of welfare

•  How do we define this relation between income and welfare?

Social Welfare •  Dalton: we need to consider which social

welfare function we would employ •  Assumptions: –  Increase in welfare corresponds to increases in income –  Welfare has a finite limit –  Welfare can go below zero

•  Thus: •  SWF would be “additively separable and

symmetric function of individual incomes”

Connecting Social Welfare to Inequality •  Atkinson (1970) uses Dalton’s assumption of

SWF to define relationship between inequality and risk

•  SWF: (assume U(y) is increasing and concave) •  Measure of inequality = ratio between

current level of SW to what level would be if income were equally distributed

•  .

Measure of inequality from SWF •  Need to make invariant w.r.t. linear

transformations: introduce concept of yEDE = equally distributed equivalent level of income

•  =

•  New measure of inequality:

•  This allows us to borrow from the theory of decision-making under uncertainty.

Translating inequality to risk

•  Borrowing terminology from the theory of decision-making under uncertainty:

•  yEDE analogous to “certainty equivalent” –  Where “certainty equivalent” is the guaranteed amount of

money that would be as desirable as a risky asset

•  I analogous to “proportional risk premium” –  Where “proportional risk premium” is the extra money the

expected return on a risky asset needs to exceed for someone to hold the risky asset

Harsanyi (1953) •  Also wrote about inequality and risk •  Cardinal utility: used in both welfare

economics and risk •  SWF: we need value judgments (unbiased) •  Impartial observer – Because you are impartial (and you have no

inkling of what your outcome might be), SW choice is now analogous to a choice involving risk!

– E.g. ‘how much more would you be willing to risk being at the ends of the income distribution in society, given you had as good a chance of becoming poor as becoming rich?’

Based on the theory… •  Decreased risk aversion (more risk-loving)

is associated with: – Higher variability/spread of distribution – Higher inequality – Decreased inequality aversion

•  Conversely, increased risk aversion is associated with: –  Smaller spread – Lower inequality – Higher inequality aversion

Amiel et al. (1999) •  Assume a SWF exists from which risk and

inequality ratings can be inferred •  Intuit iso-elastic constant relative

inequality aversion from SWF:

•  The more concave the utility function, the larger the relative inequality aversion

ßεisthedegreeofinequalityaversion

BUT this is not necessarily true in practice

•  Why? •  A. Self-interest –  The future matters (perception of income risk) –  People might not be risk averse

•  B. Altruism –  People might be affected by others’ welfare –  “Individual” inequality aversion: even if we knew we

were high on the distribution (relatively wealthy) ourselves, we have WTP for a more equal society

•  C. Other issues –  People might not have a well-defined preference order –  People are biased (we don’t have veils of ignorance)

•  Overall, the problem is that we assume a welfare function exists from which we can infer inequality or risk preferences.

•  How do we separate the effects of risk aversion or inequality aversion from welfare (utility function)?

•  See examples from Cowell & Schokkaert (2001):

Examples from Cowell & Schokkaert (2001)

•  1. Kroll and Davidovitz (1999): inequality is independent from risk

•  2. Schokkaert et al. (1997): UE insurance •  3. Cowell & Schokkaert (2001): insurance

for an unborn child (‘veil of ignorance’) •  (explained on next slides…)

Example #1: Kroll & Davidovitz (1999) •  Ran an experiment with 211 children, faced with a

choice between two scenarios: –  “common gamble”: everyone draws the same income xi from the

distribution of income F(x1,x2,…,xn) – will have complete equality (everyone has same income)

–  “individual gamble”: everyone independently draws an xi from F(x1,x2,…,xn) – potentially an unequal situation (people have different incomes)

–  the “common gamble” might be made if someone is inequality averse OR if they want to reduce personal income risk •  Risk aversion was measured as willingness to pay to reduce uncertainty of

income; it was separated from inequality aversion by randomly varying the amount of the ‘bonus’ given for choosing common vs. individual gamble

•  Results: independent of risk aversion, all children were inequality averse –  Inequality is independent from risk

Example #2: Schokkaert et al. (1997) •  Analyzed Flemish workers’ unemployment (UE) insurance

preferences •  Asked two questions to separate inequality aversion from risk

aversion: –  1 (measures inequality aversion) do you want the present system of unemployment insurance

to be more/less generous (or unchanged)? –  2 (measures risk aversion) what is your willingness to pay for additional insurance coverage?

•  Results: varied depending on personality and whether or not the workers currently faced high UE risk themselves –  Altruists (those who said they would be willing to pay not only for their own UE risk, but

also for someone less off than themselves) were more willing to make the system more generous and pay for additional coverage

–  Those with greater UE risk wanted a more generous system but did not want to pay more for additional insurance coverage

–  Those with higher incomes were willing to pay more for additional insurance coverage, but wanted to make the system LESS generous

•  Bottom line: the current design of the insurance component impacted how people evaluated it –  e.g. because the current UE insurance scheme has a ceiling (people with incomes above a

certain amount are not eligible), those with higher incomes would not gain anything from a more generous system of UE insurance, and were thus opposed to i

Example #3: Cowell & Schokkaert (2001) •  Thought experiment: pretend you are an unborn child (or the

parent of an unborn child and you don’t know you or your child’s (expected) wealth/income)

•  Catch 22: rationally, you’d want to insure against unlucky situations from age 0, but as an infant you cannot do this…and if you wait until adulthood to decide whether or not you want insurance you are no longer behind the ‘veil of ignorance’ that being ‘unborn’ confers

•  Thus, the welfare state intervenes and redistributes between lucky and unlucky children (income tax and other forms of social insurance) –  This is not a Pareto-improvement from the point of view of

adults, but from the point of view of an unborn child (before the veil of ignorance has been lifted, or “ex ante”) it is a Pareto-improvement. => There is a link between evaluating risk ex ante and evaluating income inequality ex post.

–  Should we adopt an ex ante approach to social policy?

Example from Carlsson et al. (2005) •  Experiment: 324 undergrad students choosing hypothetical

societies for their grandchildren •  Choose between various income distributions in two settings:

–  (1) test for risk aversion: involves uncertainty (you don’t know what your grandchild’s income will be, just that it will be somewhere within the range of the income distribution you choose)

–  (2) test for inequality aversion: no uncertainty (given that you know your grandchild will always receive mean income, choose an income distribution)

•  Results: people were both inequality-averse and risk-averse BUT had a lower relative risk aversion –  In scenario (2) (respondents know their grandchild will receive mean income),

people had a tendency to choose distributions that were relatively less risk-averse than what they had chosen in scenario (1)

–  Strong correlation between risk & inequality aversion

(FromCarlssonetal.)Hypothe;calincomedistribu;onsfromwhichtherespondentschose:

Example from Carlsson et al. cont’d

•  Econometric analysis: – Females (vs. males) more risk- and

inequality-averse – Left-wing voters more risk-averse – Business and technology students (vs. other

students) less risk- and inequality-averse •  Although studying economics specifically had no significant

effect on risk or inequality aversion – Positive correlation between risk

aversion and inequality aversion •  (possible genetic/cultural explanations for this?)

Example from Carlsson et al. cont’d •  Implication for welfare: SMRS (social marginal rate of

substitution = social welfare ratio of giving $1 to person i instead of j)

•  Violation of monotonicity! (where the line crosses the x-axis: beyond a certain level of income, any income increase results in a decrease in social welfare, ceteris paribus)

InequalityaversionRiskaversion

Overall insight •  The theory is too simplistic •  Empirical evidence can provide a more

nuanced view of the relationship between inequality and risk

References (by date) •  Dalton (1920). “The measurement of the inequality of

incomes.” [link] •  Harsanyi (1953). “Cardinal utility in welfare economics and in the

theory of risk-taking.” [link] •  Aigner and Heins (1967). “A social welfare view of the measurement

of income inequality.” [link] •  Atkinson (1970). “On the measurement of inequality.” [link] •  Schokkaert et al. (1997). “Individual preferences concerning

unemployment compensation: Insurance and solidarity.” [link] •  Kroll and Davidovitz (1999). “Choices in egalitarian distribution:

Inequality aversion versus risk aversion.” [link] •  Cowell and Schokkaert (2001). “Risk perceptions and distributional

judgements.” [link] •  Carlsson et al. (2005). “Are people inequality-averse, or just risk-

averse?” [link]