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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