Income Mixing via Lotteries in an Equilibrium Sorting Model

Income Mixing via Lotteries in an Equilibrium Sorting Model Muharrem Ye¸silırmak University of Iowa and Federal Reserve Bank of St. Louis

March 5, 2012

1 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Introduction

Motivation

I I I I I

Tiebout model has been the workhorse of local public finance. ? It implies perfect income sorting in equilibrium. ? Not consistent with data. To overcome this criticism, we introduce lottery into Tiebout model. Lottery has the advantage of: 1. implementing pareto efficient allocation. 2. being more consistent with empirical facts that previous extensions of Tiebout model failed at.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Introduction

Fact 1: Mixing at Municipality Level 1 0.9

Cumulative Distribution Function

0.8 0.7 Mun. I Mun. II Mun. III Mun. IV Mun. V

0.6 0.5 0.4 0.3 0.2 0.1

0−

All

Munclus

10

10

−1

5 15

−2

0

−2

20

0

5

−3

25

0 0 5 5 0 0 5 0 5 00 0+ 20 15 12 −7 −6 −5 −4 −4 −3 −1 20 0− 5− 0− 60 50 45 40 35 30 75 15 12 10 Income Groups (in thousand dollars) Source: Census 2000

States 3 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Epple & Platt (1998)

I

I

Relies on preference and income heterogeneity to explain income mixing at the municipality level. Rich and poor people live in the same municipality because: 1. rich want to pay lower taxes and consume more housing. 2. poor want to receive higher transfers and consume less housing.

I

Not consistent with: 1. Income mixing at the micro neighborhood level. 2. Income mixing conditional on rent share in income.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 2: Mixing at Micro Neighborhood Level

I

I

A micro neighborhood consists of a dwelling and up to ten nearest neighbors. Ioannides (2004) finds that: 1. the correlation of two neighbors’ income is around 0.3. 2. there is considerable amount of income mixing at the micro neighborhood level.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 3: Income Mixing Conditional on Rent Share in Income Less Than 14%

V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+ Source:Census 2000

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 3: Income Mixing Conditional on Rent Share in Income Between 14% and 19%

V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+ Source: Census 2000

7 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 3: Income Mixing Conditional on Rent Share in Income Between 19% and 24%

V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+ Source: Census 2000

8 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 3: Income Mixing Conditional on Rent Share in Income Between 24% and 29%

V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+ Source: Census 2000

9 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Fact 3: Income Mixing Conditional on Rent Share in Income More Than 29%

V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+ Source: Census 2000

10 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Nechyba (1999)

I I

I I

Each municipality has several different house quality types. Each household with a particular income is endowed with all possible house types. Therefore, households with same income has different wealth. There is income mixing because: 1. Low income household with high wealth lives in high income municipality. 2. High income household with low wealth lives in low income municipality.

I

This model implies perfect sorting w.r.t. wealth.

?

Data

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

I I I I I

Uses logit function to get mixing. Subject to ”duplicates effect”. Ex May lead to biases in policy experiments. Implies perfect mixing w.r.t. wealth also. Not consistent with data. Data

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Summary of Facts

I

Suggest a quantitative theory consistent with: 1. Income mixing at the micro neighborhood level. 2. Income mixing at the municipality level. 3. Conditional on rent share in income, there is not perfect sorting w.r.t. income. 4. Higher median income municipalities have higher median house values. (Corr= 0.72) 5. High median house value municipalities have higher public spending per household. (Corr= 0.55) 6. Low median income municipalities are more unequal. (Corr= -0.59)

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Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Intuition for Facts 4

Indirect Utility

2

1

I1

I2

I3

Income

14 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Intuition for Facts

Indirect Utility

I1

I2

I3

Income

15 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Previous Literature

Intuition for Facts Lottery line

Indirect Utility

V2

V1

I1

I2

I3

Income

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Model House Types

Household

Income =y

Heterogeneous w.r.t. value,  quality, public spending, tax rate   17 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Environment

I I

I

Static model. There are M municipalities, H house quality types with fixed supply in each municipality. Municipalities are heterogeneous w.r.t.: 1. Residential property tax rate. 2. Per household public spending.

I I I

I

Continuum of households heterogeneous w.r.t. income. All households are renters and face zero mobility cost. Local government collects housing tax and spends all on local public good. Perfectly competitive, risk neutral firms supply lotteries.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Household Preferences

I

Households have identical preferences defined over: 4×M×H M X = {((cmh , qmh , Emh , πmh )H : h=1 )m=1 ∈ <+

M X H X

πmh = 1}

m=1 h=1 I

Preferences are represented by an expected utility form as follows: M X H X

U(cmh , qmh , Emh )πmh

m=1 h=1

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Assumptions on Preferences

Utility

mh

m’h’

Consumption

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Endowments

I

I

Each household is exogenously endowed with annual income measured in terms of numeraire consumption good. Income is distributed according to c.d.f. F (·) with support <+ .

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Lotteries

I I

I

Lottery over M × H alternatives. M Each lottery is characterized by probabilities ((πmh )H h=1 )m=1 and H M prizes ((zmh )h=1 )m=1 . Each lottery has zero expected gain: M X H X

zmh πmh = 0

m=1 h=1 I I

Aggregate receipts equal aggregate value of prizes distributed. Can be thought of as a financial contract.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Housing Market

I I I I

I I

M × H different house types. Quality of house type mh is denoted by qmh . qmh is treated as a parameter. qmh captures both housing services and neighborhood externalities from alternative mh. The supply of each house type, µmh > 0, is exogenously given. pmh denotes value of house type mh and pinned down from market clearing condition.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Household’s Decision Problem H M M Given {τm }M m=1 , {Em }m=1 , {{pmh , qmh }h=1 }m=1 , the household’s problem with income y is:

max

M X H X

M {{cmh }H h=1 }m=1 M {{zmh }H h=1 }m=1 M {{πmh }H h=1 }m=1

U(cmh , qmh , Em )πmh

m=1 h=1

subject to cmh + rmh + τm pmh = y + zmh M X H X

zmh πmh = 0

m=1 h=1 M X H X

πmh = 1

m=1 h=1

πmh ∈ [0, 1] cmh ≥ 0 24 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Household’s Decision Problem

I I

rmh is the annual rent for house type mh It is determined by no arbitrage condition: pmh =

∞ X t=0

I

Equivalently: rmh =

rmh (1 + ρ)t

ρ pmh 1+ρ

ρ: Real annual interest rate, exogenous.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Household’s Decision Problem

I

I

Solution of this problem is an allocation of households to house types. Aggregating over house types gives us the distribution of households in a municipality, denoted fm (y ).

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Local Public Good Provision

I

Given gm (p): E m = τm

X

p · gm (p)dp + NSAm

gm (p): Pdf of house value in municipality m. NSAm : Net state aid per household in municipality m, exogenous.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Equilibrium ∗ M An equilibrium is a collection of {fm (y ), gm (p ∗ )}M m=1 , {Em }m=1 , H M M ∗ ∗ H M {{pmh , rmh }h=1 }m=1 , {τm }m=1 , {NSAm }m=1 , {{µmh }h=1 }M m=1 and ∗ ∗ ∗ H for each household such that: optimal decisions {{cmh , πmh , zmh }h=1 }M m=1 ∗ ∗ ∗ H solves the decision problem of the household i) {{cmh , πmh , zmh }h=1 }M m=1 M ∗ H M given {Em∗ }M m=1 ), {{pmh }h=1 }m=1 and {τm }m=1 . ii) No arbitrage in the housing market. ∗ M iii) The equilibrium distributions {fm (y ), gm (p ∗ )}M m=1 and {Em }m=1 are consistent with households’ optimal decisions. iv) Housing market clears for each alternative mh: Z ∗ µmh = πmh dF (y )

v) Local government budget balances in each municipality m: X Em∗ = τm p ∗ · gm (p ∗ )dp ∗ + NSAm 28 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Model

Income Mixing in Equilibrium

Definition There is perfect income sorting among a set of locations if the support of income distribution in any location has an empty intersection with any other location’s support of income distribution.

Theorem In equilibrium, there is never perfect income sorting among micro neighborhoods or among municipalities.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Exogenously Given

I I

I

M = 5 and H = 6 Residential property tax rate and Municipality I II III IV V

NSA, exogenous data: τ NSA (\$) 2.9% 485 2.5% 1,674 2.2% 471 3.4% 2,515 2.6% 1,161

Real annual interest rate is set ρ = 5%.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Housing Supply

I

Housing supply µmh , exogenous data: XXX Mun. I II III IV V

Quality XXX XXX Q1

0.0045 0.0006 10−5 0.0070 0.0020

Q2

Q3

Q4

Q5

Q6

0.0041 0.0083 0.0085 0.0210 0.0148

0.0193 0.0201 0.0122 0.0497 0.0534

0.0725 0.1054 0.0737 0.1316 0.2461

0.0104 0.0085 0.0373 0.0263 0.0468

0.0016 0.0002 0.0117 0.0023 10−5

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Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Income Distribution F (·) 0.14

0.12

Frequency

0.1

0.08

0.06

0.04

0.02

0

0 0 5 5 0 0 5 0 5 0 5 0 5 00 10 20 15 12 −7 −6 −5 −4 −4 −3 −3 −2 −2 −1 −1 0− 0− 5− 0− 60 50 45 40 35 30 25 20 15 10 75 15 12 10 Income Brackets (in thousand dollars)

20

0+

Census 2000

32 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Utility Function

I

The functional form for Bernoulli utility is: α U(cmh , qmh , Emh ) = qmh cmh (ln Emh )γ

where α > 0, γ > 0 and α + γ ≤ 1.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Estimation I I

H=6 32 parameters to be estimated, θ ≡ (α, γ, {{qmh }M=5 m=1 }h=1 ). To estimate the parameters we solve:

Z min θ

y

y∗ cmh (θ) dF (y ) − T1data y

!2 +

PM=5

∗ m=1 Em (θ) − T2data max{Em∗ (θ)}M=5 m=1 1 M

+

M=5 X H=6 X

!2

2 ∗ (θ) − 1) (Ipmh

m=1 h=1 I I

T1data : Average ratio of consumption to income in data=0.75 Det T2data : Ratio of mean public spending per household to maximum public spending per household across municipalities in data=0.73

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Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Estimation  ∗ = Ipm1

I

∗ 1 if pm2 ∈ [25000, 50000] 0 o.w.



∗ ∈ [50000, 90000] 1 if pm3 0 o.w.

=

∗ 1 if pm4 ∈ [90000, 175000] 0 o.w.

 ∗ = Ipm4

 ∗ = Ipm5

 ∗ = Ipm6

∗ ∈ [0, 25000] if pm1 o.w.

 ∗ = Ipm2

∗ pm3

1 0

∗ 1 if pm5 ∈ [175000, 400000] 0 o.w.

1 0

∗ if pm6 ∈ [400000, 1000000] o.w.

35 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Fit of Estimation

Param. α γ Mun. I q11 q12 q13 q14 q15 q16

Value 0.6 0.2

Target Mean Cons. Exp. Sh.

4.2861 5.0311 5.4378 5.8580 6.3651 7.1512

House House House House House House

mean(E ) max(E )

Value Value Value Value Value Value

Int. Int. Int. Int. Int. Int.

Data 0.75 0.73

Model 0.79 0.76

0\$-25,000\$ 25,000\$-50,000\$ 50,000\$-90,000\$ 90,000\$-175,000\$ 175,000\$-400,000\$ 400,000\$-1,000,000\$

36,221\$ 39,767\$ 48,300\$ 75,374\$ 204,760\$ 658,280\$

36 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Fit of Estimation Param. Mun. II q21 q22 q23 q24 q25 q26 Mun. III q31 q32 q33 q34 q35 q36

Value

Target

4.2493 4.9905 5.3951 5.8131 6.3177 7.0997

House House House House House House

Value Value Value Value Value Value

4.3696 5.1025 5.5027 5.9161 6.4150 7.1883

House House House House House House

Value Value Value Value Value Value

Data

Model

Int. Int. Int. Int. Int. Int.

0\$-25,000\$ 25,000\$-50,000\$ 50,000\$-90,000\$ 90,000\$-175,000\$ 175,000\$-400,000\$ 400,000\$-1,000,000\$

37,982\$ 41,446\$ 49,797\$ 75,256\$ 208,170\$ 665,420\$

Int. Int. Int. Int. Int. Int.

0\$-25,000\$ 25,000\$-50,000\$ 50,000\$-90,000\$ 90,000\$-175,000\$ 175,000\$-400,000\$ 400,000\$-1,000,000\$

40,698\$ 46,712\$ 60,941\$ 117,800\$ 279,200\$ 987,880\$ 37 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Estimation

Fit of Estimation Param. Mun. IV q21 q22 q23 q24 q25 q26 Mun. V q31 q32 q33 q34 q35 q36

Value

Target

4.2077 4.9377 5.3389 5.7534 6.2537 7.0291

House House House House House House

Value Value Value Value Value Value

4.2263 4.9665 5.3707 5.7882 6.2920 7.0730

House House House House House House

Value Value Value Value Value Value

Data

Model

Int. Int. Int. Int. Int. Int.

0\$-25,000\$ 25,000\$-50,000\$ 50,000\$-90,000\$ 90,000\$-175,000\$ 175,000\$-400,000\$ 400,000\$-1,000,000\$

33,725\$ 36,658\$ 43,702\$ 64,809\$ 180,610\$ 574,760\$

Int. Int. Int. Int. Int. Int.

0\$-25,000\$ 25,000\$-50,000\$ 50,000\$-90,000\$ 90,000\$-175,000\$ 175,000\$-400,000\$ 400,000\$-1,000,000\$

37,463\$ 40,543\$ 47,752\$ 68,901\$ 192,450\$ 635,750\$ 38 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Distribution in Municipality I: Data vs. Model (Not Targeted) 1 0.9

Cumulative Distribution Function

0.8 0.7 Data Model

0.6 0.5 0.4 0.3 0.2 0.1 0 10

0−

−2

15

0

5

0

5

−1

10

−2

20

−3

25

0 0 5 5 0 0 5 0 5 00 20 15 12 −7 −6 −5 −4 −4 −3 −1 0− 5− 0− 60 50 45 40 35 30 75 15 12 10 Income Groups (in thousand dollars)

0+

20

39 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Distribution in Municipality II: Data vs. Model (Not Targeted) 1 0.9 Data Model

Cumulative Distribution Function

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10

0−

−2

15

0

5

0

5

−1

10

−2

20

−3

25

0 0 5 5 0 0 5 0 5 00 20 15 12 −7 −6 −5 −4 −4 −3 −1 0− 5− 0− 60 50 45 40 35 30 75 15 12 10 Income Groups (in thousand dollars)

0+

20

40 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Distribution in Municipality III: Data vs. Model (Not Targeted) 1 0.9

Cumulative Distribution Function

0.8 0.7 Data Model

0.6 0.5 0.4 0.3 0.2 0.1 0 10

0−

−2

15

0

5

0

5

−1

10

−2

20

−3

25

0 0 5 5 0 0 5 0 5 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

−3

30

0+

20

41 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Distribution in Municipality IV: Data vs. Model (Not Targeted) 1 0.9

Cumulative Distribution Function

0.8 0.7 Data Model

0.6 0.5 0.4 0.3 0.2 0.1 0 10

0−

−2

15

0

5

0

5

−1

10

−2

20

−3

25

0 0 5 5 0 0 5 0 5 00 20 15 12 −7 −6 −5 −4 −4 −3 −1 0− 5− 0− 60 50 45 40 35 30 75 15 12 10 Income Groups (in thousand dollars)

0+

20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Distribution in Municipality V: Data vs. Model (Not Targeted) 1 0.9

Cumulative Distribution Function

0.8 0.7 Data Model

0.6 0.5 0.4 0.3 0.2 0.1 0 10

0−

−2

15

0

5

0

5

−1

10

−2

20

−3

25

0 0 5 5 0 0 5 0 5 00 20 15 12 −7 −6 −5 −4 −4 −3 −1 0− 5− 0− 60 50 45 40 35 30 75 15 12 10 Income Groups (in thousand dollars)

0+

20

43 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing in Representative Municipalities (Not Targeted) Data Model

V

Municipality

IV

III

II

I

10

0−

5

−1

10

0

−2

15

5

20

−2

0

−3

25

0 0 5 0 5 5 00 125 150 200 −5 0−6 0−7 −3 5−4 0−4 −1 − − − 45 3 30 5 4 6 75 100 125 150 Income Groups (in thousand dollars)

0+ 20

44 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing Conditional on Rent Share in Income Less Than 14%: Data vs. Model (Not Targeted)

Data V

Model

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+

45 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing Conditional on Rent Share in Income Between 14% and 19%: Data vs. Model (Not Targeted) data model V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+

46 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing Conditional on Rent Share in Income Between 19% and 24%: Data vs. Model (Not Targeted) data model V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+

47 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing Conditional on Rent Share in Income Between 24% and 29%: Data vs. Model (Not Targeted) data model V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+

48 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Income Mixing Conditional on Rent Share in Income More Than 29%: Data vs. Model (Not Targeted) data model V

Municipality

IV

III

II

I

0−10

10−20

20−35 35−50 50−75 Income Groups (in thousand dollars)

75−100

100+

49 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Results

Facts 4 & 5 & 6

Correlations (Not Targeted in Estimation) Median Income & Median House Value Median House Value & Per Household Public Spending Median Income & Gini Index of Income

Data 0.7250 0.5571 -0.5993

Model 0.7611 0.5687 -0.3754

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Income Mixing via Lotteries in an Equilibrium Sorting Model Computational Experiment

What is the experiment?

I

I I

I

Try to understand the effect of variation in residential property tax rates on income sorting. Quantify the effect of voting with feet on income sorting. To this end, set the mean tax rate (2.7%) as the tax rate in each municipality. Under benchmark parameters solve the equilibrium once more.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Computational Experiment

Benchmark vs. Experiment

Mun. I II III IV V

Med. Inc.(\$) Bench. Expe. 62,394 50,277 51,658 59,931 78,215 79,637 39,185 20,552 25,596 35,510

Pub. Exp. Per HH.(\$) Bench. Expe. 3,084 2,783 3,693 4,191 5,351 7,225 5,131 4,526 3,305 3,425

Med. House Val.(\$) Bench. Expe. 74,983 73,773 72,893 80,272 157,000 176,130 63,734 64,939 72,686 74,364

Other

52 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Computational Experiment

A Measure of Income Sorting

I

We use the following measure of sorting: PM

S=

ym BG m=1 λm ln y R = y y M WG Σm=1 λm y Nm y ln y fm (y )dy m

I I I I I

I

m

λm is the income share of municipality m y m is the mean income in municipality m y is the mean income in the society Nm is the measure of households living in municipality m BG + WG = Theil Index

Higher S implies higher income sorting in the society.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Computational Experiment

Sorting: Benchmark vs. Experiment

I

I

Benchmark Experiment

WG 0.5324 0.5033

BG 0.2658 0.2952

S 0.4992 0.5865

S increases by 17% under experiment.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Conclusion

Conclusion

I

We introduced a different mixing mechanism, namely lottery that: 1. can implement Pareto efficient allocation. 2. is more consistent with empirical facts compared to previous papers. 3. is not prone ”duplicates effect”.

I I

We carried out a computational experiment. Without any variation in residential property tax rates, there is 17% more sorting in the society.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Conclusion

Future Research

I

Study effects of some public policies like: 1. mixed income housing policy. 2. decentralized vs. centralized provision of public good.

I

The efficient lottery mechanism can be used in city size distribution literature to model households’ city choice.

56 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Tiebout Model

Household

Municipalities

Income =y

Heterogeneous w.r.t. house price,  quality, public spending, tax rate

Back

57 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Indirect Utility

Tiebout Model

2

1

Back

I1

I2

Income 58 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Fact 1: All Municipalities 1

0.9

Cumulative Distribution Function

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0 5 5 0 0 5 0 5 0 5 0 5 00 10 0+ 20 15 12 −7 −6 −5 −4 −4 −3 −3 −2 −2 −1 −1 20 0− 0− 5− 0− 60 50 45 40 35 30 25 20 15 10 75 15 12 10 Income Groups (in thousand dollars) Source: Census 2000

Back 59 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Clustering Municipalities

I

I I

Municipality is defined as the smallest possible geographical unit that has its own government. We concentrate on Rhode Island municipalities. ? We group the original 14 municipalities into 5 representative municipalities. ?

Back

60 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Why Rhode Island?

I

I

I

I

Local property tax makes up on average 60% − 70% of total municipal revenue. School choice is constrained by the boundary of municipality of residence. Net migration to Rhode Island is around 1.3% of total population between 2000 − 2004. Rhode Island has the smallest land area among all states.

Back

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Clustering Municipalities

I I

We use Hierarchical Clustering method. Cluster municipalities with respect to: 1. Residential property tax rate 2. Housing supply 3. Net state aid per household

I

This method optimally gives us 5 different clusters.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Clustering Municipalities 1 0.9

Normalized Sum of Within Group Variances

0.8 0.7

0.6 0.5 0.4

0.3 0.2 0.1

0

1

2

3

4

5

6 7 8 Number of Groups

9

10

11

12

13

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Clustering Municipalities Municipality Barrington Bristol Central Falls Cranston East Providence Narragansett New Port North Providence Pawtucket Providence Tiverton Warwick West Warwick Woonsocket

Cluster III III III V II III III I V IV III V I II 64 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Characteristics of Representative Municipalities

I II III IV V

Frac. of HH’s 0.1 0.15 0.15 0.23 0.37

Med. Inc.(\$) 39,613 34,963 44,529 26,867 40,788

Med. House Val.(\$) 113,250 116,300 158,766 101,700 111,866

Per HH Pub. Exp.(\$) 3,296 4,071 6,559 5,831 4,296

Back

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Fact 1: All States

I

Define S as percent of statewide income variation explained by within municipality variance of income. State S State S Arizona 0.91 Missouri 0.84 California 0.88 New York 0.88 Florida 0.89 North Carolina 0.92 Georgia 0.88 Pennsylvania 0.88 Illinois 0.85 Rhode Island 0.94 Massachusetts 0.88 Texas 0.88 Michigan 0.85 Virginia 0.81 Minnesota 0.86 Wisconsin 0.90

Back

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Perfect Wealth Sorting in Nechyba (1999) 4

Indirect Utility

2

1

w1

w2

w3

Wealth

Back 67 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Wealth Mixing in Data 100 90 80

% of Municipalities

70 60 50 40 30 20 10 0

0

100

200

300 400 Wealth (in thousand dollars)

500

600

700

Source: Census 2000

Back 68 / 79

Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Duplicates Effect

I I I I

Consider there are two alternatives: car and bus. The indirect utility from car is twice that of bus⇒ Vc = 2Vb The logit framework implies pc = 2pb . So pc = 2/3 and pb = 1/3.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Duplicates Effect

I I I I I

Let’s add midibus to the alternative set. Assume bus and midibus are close substitutes and Vc = 2Vb = 2Vm Now pc = 1/2 and pb = pm = 1/4. pc drops from 2/3 to 1/2 unexpectedly. This is called ”duplicates effect”. Back

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Deducing T1data

I

I

Consider the household with income y who randomizes between alternatives mh and m0 h0 . The budget constraint normalized by income for this household in both states are: y∗ ∗ cmh zy∗ r∗ τm pmh + mh + = 1 + mh y y y y y∗ ∗ z y ∗0 0 cm r∗0 0 τm0 pm 0 h0 0 h0 + mh + =1+ mh y y y y

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Deducing T1data

I

These imply that: y∗ y ∗ cmh πmh

y

+

y∗ y ∗ cm 0 h 0 πm 0 h0

y

=

y∗ πm 0 h0

I I

cy∗

∗ zy∗ r∗ τm pmh 1 + mh − mh − y y y

y∗ πmh

y∗ y ∗ mh Define ζ y ≡ πmh y + πm 0 h 0 R Notice T1data = y ζ y dF (y ).

!

∗ z y ∗0 0 r∗0 0 τm0 pm 0 h0 1+ mh − mh − y y y

y∗ cm 0 h0 y

+ !

.

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Deducing T1data I

y

ζ =1− I

∗ y ∗ rmh πmh

y

+

∗ y ∗ rm0 h0 πm 0 h0

y

! − 21

∗ y ∗ τm rmh πmh

y

+

∗ y ∗ τm0 rm0 h0 πm 0 h0

!

y

Find the average of ζ across households as: Z

y

ζ dF (y ) = 1 − y

r y

! − 21τ

r y

!

! r y

I

Following Davis and Ortalo-Magne (2010), set

I

Set τ = 0.016 which is the average residential property tax rate in Rhode Island in 2000. These imply T1data = 0.75 Back

I

= 0.18

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Income Distribution in Municipality I: Benchmark vs. Experiment 1 0.9

Cumulative Distribution Function

0.8 0.7

0.6

Benchmark Experiment

0.5 0.4

0.3 0.2 0.1

0 10

0−

5

−1

10

− 15

20

0

5

−2

20

−3

25

0 0 5 5 0 0 5 5 0 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

−3

30

0+ 20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Income Distribution in Municipality II: Benchmark vs. Experiment 1 0.9

Cumulative Distribution Function

0.8 0.7 Benchmark Experiment

0.6 0.5 0.4

0.3 0.2 0.1

0 10

0−

5

−1

10

− 15

20

0

5

−2

20

−3

25

0 0 5 5 0 0 5 5 0 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

−3

30

0+ 20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Income Distribution in Municipality III: Benchmark vs. Experiment 1 0.9

Cumulative Distribution Function

0.8 0.7

0.6 0.5

Benchmark Experiment

0.4

0.3 0.2 0.1

0 10

0−

5

−1

10

− 15

20

−3

25

0 0 5 5 0 0 5 0 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

5

0

5

−2

20

−3

30

0+ 20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Income Distribution in Municipality IV: Benchmark vs. Experiment 1 0.9

Cumulative Distribution Function

0.8 0.7 Benchmark Experiment

0.6 0.5 0.4

0.3 0.2 0.1

0 10

0−

5

−1

10

− 15

20

0

5

−2

20

−3

25

0 0 5 5 0 0 5 5 0 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

−3

30

0+ 20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Income Distribution in Municipality V: Benchmark vs. Experiment 1 0.9

Cumulative Distribution Function

0.8 0.7 Benchmark Experiment

0.6 0.5 0.4

0.3 0.2 0.1

0 10

0−

Back

5

−1

10

− 15

20

0

5

−2

20

−3

25

0 0 5 5 0 0 5 5 0 00 20 15 12 −7 −6 −5 −4 −4 −1 0− 5− 0− 60 50 45 40 35 75 15 12 10 Income Groups (in thousand dollars)

−3

30

0+ 20

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Income Mixing via Lotteries in an Equilibrium Sorting Model Appendix

Computational Algorithm Guess E for each municipality    Guess p for each house type

Solve household’s problem and find excess demand for each  house type

ED=0 for all house types?

No

Yes

Update Prices  Update E

Done

Yes

Convergence for E?

No

79 / 79

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Mar 5, 2012 - Fit of Estimation. Param. Value. Target. Data ..... School choice is constrained by the boundary of municipality of residence. â» Net migration to ...

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