Lecture 3. Equality of opportunity
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Transcript of Lecture 3. Equality of opportunity
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Lecture 3. Equality of opportunity
Erik Schokkaert (KULeuven, Department of Economics)
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Structure
1. Roemer's model of "equality of opportunity"
2. An application to optimal income taxation
3. An alternative: Van de gaer's approach
4. Comparing different approaches
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1. Roemer's model of "equality of opportunity"
Make a distinction between characteristics for which persons are responsible ("effort") and for which they are not ("circumstances")
Persons who are identical wrt the “compensation characteristics” are of the same “type”
Persons who are identical wrt the “responsibility characterics” have exerted the same “effort” level
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Relation between effort and output for various types
instruments
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"Effort" dependent on type
cigarettes smoked
low SEShigh SES
5 8
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Equality of opportunity-criterion
"equalize" outcomes at a given level of π (remember EWEP or EIER!)
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"sum" over all the possible π-levels
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Special cases
if everybody has the same π:
if there is only one type:
MAXIMIN
UTILITARIANISM
this is very different from the responsibilityaxioms in Fleurbaey!
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2. Application: optimal income taxation circumstance (type): level of education of
parents outcome function - instruments φ:
post-tax income = (1 – a) x + c
therefore: φ=(a,c)
effort is the residual: π in income distribution per type
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=>OUTCOME AS A FUNCTION OF π
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"Final" objective function:
(in the monotonic case) maximize the average income of the worst-off type
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Modelling behavioural reactions
individuals have utility function
hence,
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Government budget constraint
B
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Objective function: "maximize the average post-fisc income of the worst-off type":
post-tax income = (1-a)x +c
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The optimal tax rate
interpretation 1: η interpretation 2: (B – A)
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value of the objective functionat the (proportional) benchmark
value of the objective function at the observed policy
value of the objective function at the EOP-
policy
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Refining the definition of "type"
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3. An alternative: Van de gaer-approach
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Comparing the rules Roemer: Van de gaer:
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both rules coincide: in the extreme cases (one type OR everybody the
same effort) if there is a dominance relation between the
different outcome functions
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In general: different intuitions
Compensation of results (Roemer): try to equalize outcomes for different types at the same effort level
Compensation at the level of opportunity sets (Van de gaer): try to equalize the value of opportunity sets of different types axiomatic analysis in Ooghe, Schokkaert, Van de
gaer (Social Choice and Welfare, February 2007)
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Illustration
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4. Comparing both approachesSchokkaert, Van de gaer, Vandenbroucke, Luttens (Mathematical Social Sciences, 2004) Individuals differ in two dimensions
Independently distributed with density functions fw(w) and fe(e)
Quasi-linear utility function (cfr Roemer et al., 2003)
Budget constraint Y=B+(1-t)wL Labor supply L=(e(1-t)w)εL0
10 eeL 10 wwL
11
0 )()(1
1),( LL
eYLYu
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G O V E R N M E N T R E V E N U E C O N S T R A I N T
o r : )()()1()( 1 wettLtB O
F o r l a t e r r e f e r e n c e :
1
1
BI
BI
tt
1 1 1)()()1()(L Le w weO dwwwfdeeefttLtB
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SUBJECTIVE OUTCOME EGALITARIANISM
OBJECTIVE OUTCOME EGALITARIANISM
SUBJECTIVE OPPORTUNITY
EGALITARIANISM
OBJECTIVE OPPORTUNITY EGALITARIANISM
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Optimal subjective outcome egalitarian tax rate
NOTE: worst-off individual has characteristics (eL,,wL)
Smaller than tBI
If eL decreases (the laziest person in society gets lazier), the optimal marginal tax rate will increase
)()(1
11 1
1
)(
)(
weew
t
t LL
WE
WE
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SUBJECTIVE OUTCOME EGALITARIANISM
OBJECTIVE OUTCOME EGALITARIANISM
SUBJECTIVE OPPORTUNITY EGALITARIANISM
OBJECTIVE OPPORTUNITY EGALITARIANISM
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Optimal subjective opportunity egalitarian tax rate
Smaller than optimal subjective outcome egalitarian tax rate
Independent of the distribution of e
)(1
111 1
1
)(
)(
)(
)(
ww
t
t
t
t L
WS
WS
WI
WI
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SUBJECTIVE OUTCOME EGALITARIANISM
OBJECTIVE OUTCOME EGALITARIANISM
SUBJECTIVE OPPORTUNITY EGALITARIANISM
OBJECTIVE OPPORTUNITY EGALITARIANISM
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A D V A N T A G E F U N C T I O N :
11
)(1
1),( LL
gYLYA O
c o m p a r e w i t h u t i l i t y f u n c t i o n :
11
)(1
1),( LL
eYLYu O
a s g i n c r e a s e s , t h e b u r d e n o f m a r k e t w o r k , a s p e r c e i v e d b y t h e s o c i a l p l a n n e r d e c r e a s e s
i f g g o e s t o i n f i n i t y , o n l y i n c o m e m a t t e r s ( c f R o e m e r e t a l . )
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g
tE(A)
tBI
t
tE(W)
Le)(
)(1
e
e
1
11
L
L
e
e
(1,1) (1,wL) (eL, wL)
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Objective egalitarianism and subjective Pareto-efficiency 1 Individuals with larger values of
(larger labor income) prefer a lower tax rate
Tax rates are not Pareto-efficient if smaller than tax rate preferred by (1,1) - easily
possible for large values of g (e.g. income as advantage);
larger than tax rate preferred by (eL, wL) - definitely true for low values of g.
ii ew 1
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Political feasibility? (but then why not go for the option of the median voter?)
Ethical trade-offs: Pareto-efficiency as a side-constraint reject subjectivism altogether (extreme case of
laundering subjective preferences?)
Objective egalitarianism and subjective Pareto-efficiency 2
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SUBJECTIVE OUTCOME EGALITARIANISM
OBJECTIVE OUTCOME EGALITARIANISM
SUBJECTIVE OPPORTUNITY EGALITARIANISM
OBJECTIVE OPPORTUNITY EGALITARIANISM
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g
tS(A)tI(A)
tBI
t
tE(W)
Le)(
)(1
e
e
1
11
L
L
e
e
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g
tE(A)
tS(A)tI(A)
tBI
t
tE(W)
Le)(
)(1
e
e
1
11
L
L
e
e
PROPOSITION: for a given value of g, )()()( ASAIAE ttt
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Application: description of the sample
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Optimal tax rates (subjective cases)
introducing opportunity considerations has a minor influence
important effects of ε
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Results for ε=0.30
introducing “advantage” matters for low values of gIntroducing opportunity considerations has a minor influence
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ε = 0.06 versus ε=1
Effects of ε: (a) level of optimal tax; (b) breakpoint
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Conclusion
It is possible to derive operational tax rules from rather complex objective functions
Real debate is about the choice of the objective function How to interpret equality of opportunity? How to trade off compensation versus responsibility? Where do “reference preferences” come from? What about (subjective) Pareto-efficiency? How to correct
"happiness" measures?