Transcript Lecture 3. Equality of opportunity

Lecture 3. Equality of
opportunity
Erik Schokkaert (KULeuven,
Department of Economics)
Structure
1.
2.
3.
4.
Roemer's model of "equality of opportunity"
An application to optimal income taxation
An alternative: Van de gaer's approach
Comparing different approaches
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
Relation between effort and output for
various types
instruments
"Effort" dependent on type
high SES
5
8
low SES
cigarettes smoked
Equality of opportunity-criterion

"equalize" outcomes at a given level of π
(remember EWEP or EIER!)

"sum" over all the possible π-levels
Special cases

if everybody has the same π:
MAXIMIN

if there is only one type:
UTILITARIANISM
this is very different from the responsibility
axioms in Fleurbaey!
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
=>
OUTCOME AS A FUNCTION OF π
"Final" objective function:
(in the monotonic case) maximize the
average income of the worst-off type
Modelling behavioural reactions

individuals have utility function

hence,
Government budget constraint
B
Objective function: "maximize the average postfisc income of the worst-off type":
post-tax income = (1-a)x +c
The optimal tax rate


interpretation 1: η
interpretation 2: (B – A)
value of the objective function at
the observed policy
value of the objective
function at the EOPpolicy
value of the objective function
at the (proportional) benchmark
Refining the definition of "type"
3. An alternative: Van de gaer-approach
Comparing the rules

Roemer:

Van de gaer:

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
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)
Illustration
4. Comparing both approaches
Schokkaert, Van de gaer, Vandenbroucke, Luttens (Mathematical
Social Sciences, 2004)

Individuals differ in two dimensions
0  eL  e  1


Independently distributed with density functions fw(w) and
fe(e)
Quasi-linear utility function (cfr Roemer et al., 2003)
u (Y , L)  Y 


0  wL  w  1
1 
( L0 )
e 1 

Budget constraint Y=B+(1-t)wL
Labor supply L=(e(1-t)w)εL0
1
1
 ( L) 
GOVERNMENT REVENUE
CONSTRAINT
B (t )  LO t (1  t ) e1L f e (e)e dew1 L f w ( w) w1 dw
or:
B (t )  LO t (1  t )   (e) 1 ( w)
For later reference:
t BI
1

1  t BI 
SUBJECTIVE OUTCOME
EGALITARIANISM
OBJECTIVE OUTCOME
EGALITARIANISM
SUBJECTIVE OPPORTUNITY
EGALITARIANISM
OBJECTIVE OPPORTUNITY
EGALITARIANISM
Optimal subjective outcome egalitarian tax rate
t E (W )
1  t E (W )
1 


1
wL eL
 1 

   (e) 1 ( w) 
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
SUBJECTIVE OUTCOME
EGALITARIANISM
OBJECTIVE OUTCOME
EGALITARIANISM
SUBJECTIVE OPPORTUNITY
EGALITARIANISM
OBJECTIVE OPPORTUNITY
EGALITARIANISM
Optimal subjective opportunity egalitarian tax
rate
t I (W )

t S (W )
1  t I (W ) 1  t S (W )


1 

1
wL
 1 

  1 ( w) 
Smaller than optimal subjective outcome egalitarian
tax rate
Independent of the distribution of e
SUBJECTIVE OUTCOME
EGALITARIANISM
OBJECTIVE OUTCOME
EGALITARIANISM
SUBJECTIVE OPPORTUNITY
EGALITARIANISM
OBJECTIVE OPPORTUNITY
EGALITARIANISM
ADVANTAGE FUNCTION:
1 1

1 
A(Y , L )  Y 
( LO )  L 
g 1 
compare with utility function:
1

1 
u (Y , L)  Y 
( LO )  L 
e 1 
1
 as g increases, the burden of market
work, as perceived by the social planner
decreases
 if g goes to infinity, only income matters
(cf Roemer et al.)
t
tE(A)
tBI
tE(W)
eL  1 (e) 
  ( e)
(1,1)
e1L  1
 
eL  1
(1,wL)
g
(eL, wL)
Objective egalitarianism and subjective Paretoefficiency 1

1 
w
Individuals with larger values of i ei
(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.
Objective egalitarianism and subjective Paretoefficiency 2

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?)
SUBJECTIVE OUTCOME
EGALITARIANISM
OBJECTIVE OUTCOME
EGALITARIANISM
SUBJECTIVE OPPORTUNITY
EGALITARIANISM
OBJECTIVE OPPORTUNITY
EGALITARIANISM
t
tS(A)
tI(A)
tBI
tE(W)
e L
 ( e)

 1
  ( e)
e1L  1
 
eL  1
g
PROPOSITION: for a given value of g, t E ( A)  t I ( A)  t S ( A)
t
tE(A)
tS(A)
tI(A)
tBI
tE(W)
e L
 ( e)

 1
  ( e)
e1L  1
 
eL  1
g
Application: description of the sample
Optimal tax rates (subjective cases)
introducing opportunity considerations has a minor influence
important effects of ε
Results for ε=0.30
1
0,95
t
0,9
0,85
0,8
0,75
0,7
0
0,2
0,4
0,6
0,8
1
g
tE(A)
tS(A)
tBI
tE(W)
tS(W)
introducing “advantage” matters for low values of g
Introducing opportunity considerations has a minor influence
1
1
0,98
0,9
0,96
0,8
0,94
0,7
t
t
ε = 0.06 versus ε=1
0,92
0,6
0,9
0,5
0,88
0,4
0
0,2
0,4
0,6
0,8
1
0
0,2
0,4
tS(A)
tBI
0,8
1
g
g
tE(A)
0,6
tE(W)
tS(W)
tE(A)
tS(A)
tBI
Effects of ε: (a) level of optimal tax; (b) breakpoint
tE(W)
tS(W)
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?