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Equality of Opportunity - Stanford Universityjuanfrr/eop.pdf · 1 Equality of Opportunity:...
Transcript of Equality of Opportunity - Stanford Universityjuanfrr/eop.pdf · 1 Equality of Opportunity:...
Equality of Opportunityand School Financing Structure
Juan Rios
January 4, 2017
Juan Rios Equality of Opportunity January 4, 2017 1 / 20
Motivation (I)
1 Normative Principles:
Compensation Principle:“Allocation is intended to compensateindividuals for their disadvantageous circumstances” (Roemer, 2012)Liberal Reward Principle: “Particular Inequalities due toresponsibilities should be left untouched” (Fleurbaey, 2008)
2 Crucial distinction
Circumstances:Responsibility:
Juan Rios Equality of Opportunity January 4, 2017 2 / 20
Motivation (I)
1 Normative Principles:
Compensation Principle:“Allocation is intended to compensateindividuals for their disadvantageous circumstances” (Roemer, 2012)Liberal Reward Principle: “Particular Inequalities due toresponsibilities should be left untouched” (Fleurbaey, 2008)
2 Crucial distinction
Circumstances:Responsibility:
Juan Rios Equality of Opportunity January 4, 2017 2 / 20
Motivation (I)
1 Normative Principles:
Compensation Principle:“Allocation is intended to compensateindividuals for their disadvantageous circumstances” (Roemer, 2012)Liberal Reward Principle: “Particular Inequalities due toresponsibilities should be left untouched” (Fleurbaey, 2008)
2 Crucial distinction
Circumstances: Affect opportunity setsResponsibility: Affect preferences
Juan Rios Equality of Opportunity January 4, 2017 2 / 20
Motivation (II)
“It is, of course, a deep philosophical question, with psychological andneurophysiological components, to determine exactly what constitutesthe complete set of circumstances for any given social problem. Inpractice, we choose some circumstances for the purpose of thecomputation, and define the partition of types with respect to those.We then arbitrarily attribute the variation in the acquisition of theobjective among those within a type entirely to differential effort.”(Roemer, 2003)
Juan Rios Equality of Opportunity January 4, 2017 3 / 20
Motivation (III)
Equality of Opportunity: force to equalize school investment
Transfers from Central Government
Property Taxes are efficient to finance local public schools (Tiebout)
Efficiency vs Equality of Opportunity
Property Taxes vs Central Transfers
Juan Rios Equality of Opportunity January 4, 2017 4 / 20
Motivation (III)
Equality of Opportunity: force to equalize school investment
Transfers from Central Government
Property Taxes are efficient to finance local public schools (Tiebout)
Efficiency vs Equality of Opportunity
Property Taxes vs Central Transfers
Juan Rios Equality of Opportunity January 4, 2017 4 / 20
The Research Question
What is the optimal public school financing structure?
Equality of Opportunity vs EfficiencyCentral Government Transfers vs Property Taxes
Optimal Policy depends on
Elasticities of house prices wrt to scholl investmentsHouse price distributionElasticity of children wages wrt public sch. inv.
Juan Rios Equality of Opportunity January 4, 2017 5 / 20
The Research Question
What is the optimal public school financing structure?
Equality of Opportunity vs EfficiencyCentral Government Transfers vs Property Taxes
Optimal Policy depends on
Elasticities of house prices wrt to scholl investmentsHouse price distributionElasticity of children wages wrt public sch. inv.
Juan Rios Equality of Opportunity January 4, 2017 5 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Related Literature
1 Equality of Opportunity: Fleurbaey (2008), Roemer (2012), Saezand Stancheva (2016), Jacquet et al (2015), Lockwood and Weinzierl(2015), Bergstrom and Dodds (2016).
Consider EOp in Education
2 Local Public Economics: Tiebout (1956), Benabou (1996).
Take EOp into account
3 Optimal School Financing Structure: Feldstein (1975), Fleuerbaeyet al (2002), De Fraja (2002), Gottlieb and Moreira (2002), Kranich(2016)
EOp vs Efficiency of Local TaxesOptimality as function of elasticities
Juan Rios Equality of Opportunity January 4, 2017 6 / 20
Outline
1 Normative Criteria
2 Model
3 Optimal Policy
4 Future Directions
Juan Rios Equality of Opportunity January 4, 2017 7 / 20
Outline
1 Normative Criteria
2 Model
3 Optimal Policy
4 Future Directions
Juan Rios Equality of Opportunity January 4, 2017 7 / 20
Outline
1 Normative Criteria
2 Model
3 Optimal Policy
4 Future Directions
Juan Rios Equality of Opportunity January 4, 2017 7 / 20
Outline
1 Normative Criteria
2 Model
3 Optimal Policy
4 Future Directions
Juan Rios Equality of Opportunity January 4, 2017 7 / 20
Normative Criteria
Equality of Opportunity Function
u(α, γ, x)
α: Circumstances
γ: Responsibility
x : Policy
EOp(x) =
∫minα
u(α, γ, x)dF (γ)
Link to General EOp
Juan Rios Equality of Opportunity January 4, 2017 8 / 20
Normative Criteria
Equality of Opportunity Function
u(α, γ, x)
α: Circumstances
γ: Responsibility
x : Policy
EOp(x) =
∫minα
u(α, γ, x)dF (γ)
Link to General EOp
Juan Rios Equality of Opportunity January 4, 2017 8 / 20
Normative Criteria
Example
Table : Economy with 4 Types
Responsibility γ EOpCircumstances α Low High
∫minα
u(α, γ)dF (γ)
Selfish Parents 0 4 -Altruistic Parents 6 10 -
minα
u(α, γ) 0 4 4 ∗ Prob(γ = High)
EOp reconciled with Welfarism
If all heterogeneity is α: EOp = min u(α, x)
If all heterogeneity is γ: EOp =∫u(γ, x)dF (γ)
Link to Optimal Tax Results
Juan Rios Equality of Opportunity January 4, 2017 9 / 20
Normative Criteria
Example
Table : Economy with 4 Types
Responsibility γ EOpCircumstances α Low High
∫minα
u(α, γ)dF (γ)
Selfish Parents 0 4 -Altruistic Parents 6 10 -
minα
u(α, γ) 0 4 4 ∗ Prob(γ = High)
EOp reconciled with Welfarism
If all heterogeneity is α: EOp = min u(α, x)
If all heterogeneity is γ: EOp =∫u(γ, x)dF (γ)
Link to Optimal Tax Results
Juan Rios Equality of Opportunity January 4, 2017 9 / 20
Normative Criteria
First Best
1 EOp: Planner observes parents altruism
u(α, γ) = u(γ),∀αNo equality of outcome even in the first best
2 Mirrlees: Planner observes ability
u(θ) = u ∀θEquality of outcome
Juan Rios Equality of Opportunity January 4, 2017 10 / 20
Normative Criteria
First Best
1 EOp: Planner observes parents altruism
u(α, γ) = u(γ),∀αNo equality of outcome even in the first best
2 Mirrlees: Planner observes ability
u(θ) = u ∀θEquality of outcome
Juan Rios Equality of Opportunity January 4, 2017 10 / 20
Model
Assumptions
1 2 generations
2 Parents are heterogeneous only on altruism
3 Planner observes house prices
4 Uniform property tax rate across districts
5 Uniform house quality within district
Tractability
1 Housing supply totally inelastic
2 Single crossing of parents’ consumption and school investments withrespect to their altruism parameter.
What is in the model
Focus on trade off between the efficiency of property taxes andredistribution of central transfers.
Juan Rios Equality of Opportunity January 4, 2017 11 / 20
Model
Assumptions
1 2 generations
2 Parents are heterogeneous only on altruism
3 Planner observes house prices
4 Uniform property tax rate across districts
5 Uniform house quality within district
Tractability
1 Housing supply totally inelastic
2 Single crossing of parents’ consumption and school investments withrespect to their altruism parameter.
What is in the model
Focus on trade off between the efficiency of property taxes andredistribution of central transfers.
Juan Rios Equality of Opportunity January 4, 2017 11 / 20
Model
Assumptions
1 2 generations
2 Parents are heterogeneous only on altruism
3 Planner observes house prices
4 Uniform property tax rate across districts
5 Uniform house quality within district
Tractability
1 Housing supply totally inelastic
2 Single crossing of parents’ consumption and school investments withrespect to their altruism parameter.
What is in the model
Focus on trade off between the efficiency of property taxes andredistribution of central transfers.
Juan Rios Equality of Opportunity January 4, 2017 11 / 20
Model
Children’s Problem
vk(α, γ, x) ≡ maxc,e
uk(c , e; γ) s.t.
c ≤ wk(e, α, x
)
uk(·): Kids’ utility
α: Parents’ Preferences
γ: Responsibility
x : School District investment
c : Consumption
e: Effort in education
wk(·): Endogenous children income
Juan Rios Equality of Opportunity January 4, 2017 12 / 20
Model
Parents’ Problem
vp(α) ≡ maxh,l
{up(xh, l ;α
)s.t.
(1 + t)P(xh
)≤ l}
up(·): Parents’ utility
h: District index
t: Property tax rate
P(xh): Price of a house in h with xhl : Parents labor supply (wp ≡ 1)
upl (α) ≡ 1 ∀α
No need to redistribute across parents:
⇒ upx (α) = (1 + t)P ′(xh(α)
)Juan Rios Equality of Opportunity January 4, 2017 13 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)
αM
P1 P2
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
Decreasing -upux
(α)
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
Decreasing -upux
(α)
αL
αH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
Decreasing -upux
(α)
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Parents’ Problem: Reduced Form
u(x ,P;α
)= up
(x , (1 + t)P;α
)
P
x
x(P)αM
P1 P2
Decreasing -upux
(α) ⇒ Sorted Equilibrium
αLαH
Link to Competitive Equilibrium Definition
Juan Rios Equality of Opportunity January 4, 2017 14 / 20
Model
Planner’s Problem
maxx(·)
{∫αvp(α)dF (α) + β
∫minα
vk(α, γ, x)dF (γ) s.t.
∫ ∞0
x(p)− tpdH(p) ≤ R
}
β: Discount factor
P ∼ H(·)R: Exogenous Revenue
Juan Rios Equality of Opportunity January 4, 2017 15 / 20
Model
Elasticities
P(ξ, I ) = argmaxP
u(x(P) + ξP + I ,P;α
)
1 ε(P) = x ′(P)P
∂P(ξ,I )∂ξ
∣∣∣ξ=I=0
: Elasticity of house prices with respect to
the marginal public school resources
2 η(P) = x ′(P)∂P(ξ,I )∂I
∣∣∣ξ=I=0
: Level Effect Parameter
3 εc(P) = ε(P) + η(P): Slutsky Equation
Juan Rios Equality of Opportunity January 4, 2017 16 / 20
Model
Elasticities
P(ξ, I ) = argmaxP
u(x(P) + ξP + I ,P;α
)
1 ε(P) = x ′(P)P
∂P(ξ,I )∂ξ
∣∣∣ξ=I=0
: Elasticity of house prices with respect to
the marginal public school resources
2 η(P) = x ′(P)∂P(ξ,I )∂I
∣∣∣ξ=I=0
: Level Effect Parameter
3 εc(P) = ε(P) + η(P): Slutsky Equation
Juan Rios Equality of Opportunity January 4, 2017 16 / 20
Optimal Policy
Optimal Policy
x ′opt(P)
t − x ′opt(P)= εc(P)h(P)P
(∫ ∞P
1− g(p)− βeop(p)−t − x ′opt(p)
x ′opt(p)η(p)dH(p)
)−1
and
∫ ∞0
xopt(p)− tpdH(p) = R for P ≤ Pt
Observations
Larger x ′opt(P) implies less redistribution.
Juan Rios Equality of Opportunity January 4, 2017 17 / 20
Optimal Policy
P
x
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗P∗ + dP∗
Juan Rios Equality of Opportunity January 4, 2017 18 / 20
Optimal Policy
P
x tP
t
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗P∗ + dP∗
Juan Rios Equality of Opportunity January 4, 2017 18 / 20
Optimal Policy
P
x tP
t
x(P)
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗P∗ + dP∗
Juan Rios Equality of Opportunity January 4, 2017 18 / 20
Optimal Policy
P
x
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗P∗ + dP∗
Juan Rios Equality of Opportunity January 4, 2017 18 / 20
Optimal Policy
Intuition
Consider a perturbation dx ′(P∗) at P∗
P
x
t
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗
Juan Rios Equality of Opportunity January 4, 2017 19 / 20
Optimal Policy
Intuition
Consider a perturbation dx ′(P∗) at P∗
1 BE: [t − x ′(P∗)]εc (P∗) P∗
x′(P∗)h(P∗)dP∗dx ′(P∗)
P
x
t
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗
BE
Juan Rios Equality of Opportunity January 4, 2017 19 / 20
Optimal Policy
Intuition
Consider a perturbation dx ′(P∗) at P∗
1 BE: [t − x ′(P∗)]εc (P∗) P∗
x′(P∗)h(P∗)dP∗dx ′(P∗)
2 ME: −∫∞P∗ dH(p)dx ′(P∗)dP∗
3 EopE:∫∞P∗ g(p) + βeop(p)dH(p)dx ′(P∗)dP∗
4 LE:∫∞P∗
t−x′(p)x′(p)
η(p)dH(p)dx ′(P∗)dP∗
P
x
t
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗
ME+EopE+LE
Juan Rios Equality of Opportunity January 4, 2017 19 / 20
Optimal Policy
Intuition
Consider a perturbation dx ′(P∗) at P∗
1 BE: [t − x ′(P∗)]εc (P∗) P∗
x′(P∗)h(P∗)dP∗dx ′(P∗)
2 ME: −∫∞P∗ dH(p)dx ′(P∗)dP∗
3 EopE:∫∞P∗ g(p) + βeop(p)dH(p)dx ′(P∗)dP∗
4 LE:∫∞P∗
t−x′(p)x′(p)
η(p)dH(p)dx ′(P∗)dP∗
At the optimum the sum of these effects is zero.
P
x
t
dx ′(P∗)
x(P)
dx ′(P∗)dP∗
P∗
Juan Rios Equality of Opportunity January 4, 2017 19 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ ux(α(p))
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dF γ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ ux(α(p))
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dF γ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ
∫upv ukc
∂wk
∂x dFγ|α(γ|α(p))− upc∂p∂x
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dF γ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ
∫upv ukc
∂wk
∂x dFγ|α(γ|α(p))− upc∂p∂x
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dF γ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ
∫upv ukc
∂wk
∂x dFγ|α(γ|α(p))− upc∂p∂x
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dγ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Optimal Policy
Planner’s weights
1 g(p) = 1λ
∫upv ukc
∂wk
∂x dFγ|α(γ|α(p))− upc∂p∂x
Lump-sum transfers if this is the only concern (Tiebout) Link to Details
2 eop(p) =
{1λ
∫ukc
∂wk
∂x dFγ|α(γ|0) if p = P(α = 0)
0 otherwise
All weight on α = 0 under equality of opportunity
Observations
∂wk
∂x (p) is relevant to compute parents’ weights
∂wk
∂x (P(α = 0)) is relevant to compute children’s weights
Overlapping Generations + income tax ⇒ fiscal externality on wages
If γ is partially circumstances ⇒ F becomes F
Juan Rios Equality of Opportunity January 4, 2017 20 / 20
Future Directions
Where to go?
Endogenous human capital formation
Bequests and access to credit
Endogenous local taxes
Overlapping generations + Endogenous R
Heterogeneity of preferences for district quality
Private schools and externalities
Juan Rios Equality of Opportunity January 4, 2017 21 / 20
Generalized EOp
m(γ, x) = minα
u(α, γ, x) ∈ M
EOp : M → R+: Increasing Operator.
EOp(m(γ, x)
): Equality of Opportunity Function
Table : Generalized EOp
Responsibility γ Social Criteria EOPCircumstances α Low High min
γm(γ)
∫m(γ)dF (γ)
Selfish Parents 0 4 - -Altruistic Parents 6 10 - -
m(γ) 0 4 0 4 ∗ Prob(γ = High)
Back to EOp
Juan Rios Equality of Opportunity January 4, 2017 22 / 20
Income Taxation
All weight on lower circumstances and equal weights overresponsibilities justifies larger tax rates (Saez and Stancheva)
All weight on circumstances with enough weight on high responsibilityjustifies EITC (Jacquet et al)
Arbitrary weights on circumstances and equal weights onresponsibilities optimal tax less progressive (Lockwood and Weinzierl)
If planner can observe circumstances he could use “tagging”
Back to Example
Juan Rios Equality of Opportunity January 4, 2017 23 / 20
Competitive Equilibrium
A CE is an allocation for the parents {c(α), h(α)}, for the children{c(h, γ), e(h, γ)} a price schedule P(h, x), a wage production functionw(e, x , h) and a policy schedule x(P) such that.
1 {c(α), h(α)} solves the parents problem given P() and x()
2 {c(h, γ), e(h, γ)} solves the children problem given w(·, x , h).
3 Good markets clear∫c(α) + (1 + t)P(h(α), x)dF (α) = W
4 Housing markets clear F (α) = H(h(α))
Back to Reduced Form
Juan Rios Equality of Opportunity January 4, 2017 24 / 20
Case without EOp concerns
If eop(p) = 0⇒ ux(α) = ux ≡ λ ∀αProperty taxes x ′(P) = t ∀P ensure that LHS = 0 = RHS
“Tiebout Result”
Back to Planner’s Weights
Juan Rios Equality of Opportunity January 4, 2017 25 / 20