lecture 3. equality of opportunity

Lecture 3. Equality of opportunity

Erik Schokkaert (KULeuven, Department of Economics)

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

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

cigarettes smoked

low SEShigh SES

5 8

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:

if there is only one type:

MAXIMIN

UTILITARIANISM

this is very different from the responsibilityaxioms 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 post-fisc 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 functionat the (proportional) benchmark

value of the objective function at the observed policy

value of the objective function at the EOP-

policy

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 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

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

SUBJECTIVE OUTCOME EGALITARIANISM

OBJECTIVE OUTCOME EGALITARIANISM

  SUBJECTIVE OPPORTUNITY

EGALITARIANISM  

OBJECTIVE OPPORTUNITY EGALITARIANISM

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

SUBJECTIVE OUTCOME EGALITARIANISM

OBJECTIVE OUTCOME EGALITARIANISM

SUBJECTIVE OPPORTUNITY EGALITARIANISM

OBJECTIVE OPPORTUNITY EGALITARIANISM

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

SUBJECTIVE OUTCOME EGALITARIANISM

OBJECTIVE OUTCOME EGALITARIANISM

SUBJECTIVE OPPORTUNITY EGALITARIANISM

OBJECTIVE OPPORTUNITY EGALITARIANISM

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 . )

g

tE(A)

tBI

t

tE(W)

Le)(

)(1

e

e

1

11

L

L

e

e

(1,1) (1,wL) (eL, wL)

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

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

SUBJECTIVE OUTCOME EGALITARIANISM

OBJECTIVE OUTCOME EGALITARIANISM

SUBJECTIVE OPPORTUNITY EGALITARIANISM

OBJECTIVE OPPORTUNITY EGALITARIANISM

g

tS(A)tI(A)

tBI

t

tE(W)

Le)(

)(1

e

e

1

11

L

L

e

e

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

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

introducing “advantage” matters for low values of gIntroducing opportunity considerations has a minor influence

ε = 0.06 versus ε=1

Effects of ε: (a) level of optimal tax; (b) breakpoint

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?