private provision of public good and endogenous income ... · private provision of public good and...

Private provision of public good andendogenous income inequality

Manash Ranjan Gupta1, Debasis Mondal2

7th Annual International Conference on Public Finance and Public

Policy, CTRPFP.

Jan 03, 2018

1Economic Research Unit, Indian Statistical Institute, Kolkata Centre, 203 B.T. Road, Kolkata-700108.

E-mail: manashranjan@isical.ac.in2Corresponding author: Department of Humanities and Social Sciences, Indian Institute of Technology Delhi,

Room no - MS 606 Hauz Khas, New Delhi - 110016; E-mail: debasis@hss.iitd.ac.in

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Motivation

I Three important results in existing literature:

1. Higher inequality between contributor and non-contributorraises aggregate provision.

2. (Neutrality Theorem) If income is redistributed over donors,total provision remains unchanged [Warr (1983), BBV (1986)]

3. (Group size) Any increase in (aggregate) income of the donorswould never decrease the aggregate level of provision.

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Motivation

I In reality, inequality is endogenous to the system. In existingliterature, this is exogenous (say, a transfer from poor to rich).

I This would certainly raise the aggregate level of provision.

I (BBV(1986, pp.43))

“Thus, if an economy evolves toward a more equaldistribution of income we can expect the amount of publicgoods that would be provided voluntarily to diminish.This means that the case for government provision in theinterest of efficiency would become stronger as theincome distribution becomes more equal and mighteventually overcome the advantages of private provision.”

I The role of non-contributor (poor people) is minimal in existingliterature. There is no production.

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Motivation

I In reality, inequality is endogenous to the system. In existingliterature, this is exogenous (say, a transfer from poor to rich).

I This would certainly raise the aggregate level of provision.

I (BBV(1986, pp.43))

“Thus, if an economy evolves toward a more equaldistribution of income we can expect the amount of publicgoods that would be provided voluntarily to diminish.This means that the case for government provision in theinterest of efficiency would become stronger as theincome distribution becomes more equal and mighteventually overcome the advantages of private provision.”

I The role of non-contributor (poor people) is minimal in existingliterature. There is no production.

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Motivation

I In reality, inequality is endogenous to the system. In existingliterature, this is exogenous (say, a transfer from poor to rich).

I This would certainly raise the aggregate level of provision.

I (BBV(1986, pp.43))

“Thus, if an economy evolves toward a more equaldistribution of income we can expect the amount of publicgoods that would be provided voluntarily to diminish.This means that the case for government provision in theinterest of efficiency would become stronger as theincome distribution becomes more equal and mighteventually overcome the advantages of private provision.”

I The role of non-contributor (poor people) is minimal in existingliterature. There is no production.

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Our work

I We introduce production

I Build a simple general equilibrium set up of public good with richand poor people.

I Income inequality is endogenous to our model.

I Our model show the joint evolution of inequality and aggregatedonation

I We also talk of optimal tax-transfer policy where rich are taxed andfraction of the revenue are transferred to the poor as direct cashtransfer.

I Cash versus kind (giving through public good), e.g., Coate(1995)

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Our work

I We introduce production

I Build a simple general equilibrium set up of public good with richand poor people.

I Income inequality is endogenous to our model.

I Our model show the joint evolution of inequality and aggregatedonation

I We also talk of optimal tax-transfer policy where rich are taxed andfraction of the revenue are transferred to the poor as direct cashtransfer.

I Cash versus kind (giving through public good), e.g., Coate(1995)

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Our work

I We introduce production

I Build a simple general equilibrium set up of public good with richand poor people.

I Income inequality is endogenous to our model.

I Our model show the joint evolution of inequality and aggregatedonation

I We also talk of optimal tax-transfer policy where rich are taxed andfraction of the revenue are transferred to the poor as direct cashtransfer.

I Cash versus kind (giving through public good), e.g., Coate(1995)

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Framework

I A group of rich people - H

I A group of poor people - L

I Rich holds capital and labour - capital k percapita and one unit oflabour.

I Poor holds only labour (one unit) and no capital

Capital Labour Income (per-capita)Rich k 1 rk+wPoor 0 1 w

I Aggregate Capital = K = k ∗ H, Labour = H + L

I Preferences are identical - Cobb Douglas in private and public

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Framework

I A group of rich people - H

I A group of poor people - L

I Rich holds capital and labour - capital k percapita and one unit oflabour.

I Poor holds only labour (one unit) and no capital

Capital Labour Income (per-capita)Rich k 1 rk+wPoor 0 1 w

I Aggregate Capital = K = k ∗ H, Labour = H + L

I Preferences are identical - Cobb Douglas in private and public

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Framework

I A group of rich people - H

I A group of poor people - L

I Rich holds capital and labour - capital k percapita and one unit oflabour.

I Poor holds only labour (one unit) and no capital

Capital Labour Income (per-capita)Rich k 1 rk+wPoor 0 1 w

I Aggregate Capital = K = k ∗ H, Labour = H + L

I Preferences are identical - Cobb Douglas in private and public

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Preferences

I A representative Rich agent maximizes:

uH(c1H , c2H , ..., cnH ,G ) = log

(n∑

i=1

cθiH

) 1θ

+ log(G ); θ ∈ (0, 1).

subject to

n∑i=1

picHi = w + (rk − gj)(1− t) +δt∑H

j=1(rk − gj)

H. (1)

For t = 0,

n∑i=1

piciH + gH = w + rκ; G =∑H

gH > 0.

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Preferences

I A representative Rich agent maximizes:

uH(c1H , c2H , ..., cnH ,G ) = log

(n∑

i=1

cθiH

) 1θ

+ log(G ); θ ∈ (0, 1).

subject to

n∑i=1

picHi = w + (rk − gj)(1− t) +δt∑H

j=1(rk − gj)

H. (1)

For t = 0,

n∑i=1

piciH + gH = w + rκ; G =∑H

gH > 0.

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Preferences

I A representative Poor agent maximises:

uL(c1L, c2L, ..., cnL,G ) = log

(n∑

i=1

cθiL

) 1θ

+ log(G ); θ ∈ (0, 1).

subject to

n∑i=1

picLi = w +(1− δ)t

∑Hj=1(rk − gj)

L. (2)

For t = 0,n∑

i=1

piciL = w

I Poor do not contribute as long as G > w .

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Preferences

I A representative Poor agent maximises:

uL(c1L, c2L, ..., cnL,G ) = log

(n∑

i=1

cθiL

) 1θ

+ log(G ); θ ∈ (0, 1).

subject to

n∑i=1

picLi = w +(1− δ)t

∑Hj=1(rk − gj)

L. (2)

For t = 0,n∑

i=1

piciL = w

I Poor do not contribute as long as G > w .

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Production

I Public good: one-to-one in labour: LG = G

I Private goods:(i) Fixed cost: α units of capital for any variety.(ii) Variable cost: β units of labour per unit of output.

I Monopolistic competition in private good’s variety.

I Price per unit p = βwθ :

Profit = π = px − βwx = 1−θθ βwx

I public good price is pG = w = 1, [public good is the numeraire].

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Inquality measure

I Income distribution net of voluntary contribution.

I In a two class society where the richest x% of population equallyshare y% of all income and others equally share the remainder,

I Gini = y − x

I Fraction of rich = HH+L

I Rich’s share in income (net of donations) = H+(rK−G)(1−t+δt)H+L+rK−G .

Gini =H + (rK − G )(1− t + δt)

H + L + rK − G− H

H + L;

⇒ Gini =1− t(1− δ)− H

H+L

1 + H+LrK−G

. (3)

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Inquality measure

I Income distribution net of voluntary contribution.

I In a two class society where the richest x% of population equallyshare y% of all income and others equally share the remainder,

I Gini = y − x

I Fraction of rich = HH+L

I Rich’s share in income (net of donations) = H+(rK−G)(1−t+δt)H+L+rK−G .

Gini =H + (rK − G )(1− t + δt)

H + L + rK − G− H

H + L;

⇒ Gini =1− t(1− δ)− H

H+L

1 + H+LrK−G

. (3)

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Inquality measure

I Income distribution net of voluntary contribution.

I In a two class society where the richest x% of population equallyshare y% of all income and others equally share the remainder,

I Gini = y − x

I Fraction of rich = HH+L

I Rich’s share in income (net of donations) = H+(rK−G)(1−t+δt)H+L+rK−G .

Gini =H + (rK − G )(1− t + δt)

H + L + rK − G− H

H + L;

⇒ Gini =1− t(1− δ)− H

H+L

1 + H+LrK−G

. (3)

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Inquality measure

I Income distribution net of voluntary contribution.

I In a two class society where the richest x% of population equallyshare y% of all income and others equally share the remainder,

I Gini = y − x

I Fraction of rich = HH+L

I Rich’s share in income (net of donations) = H+(rK−G)(1−t+δt)H+L+rK−G .

Gini =H + (rK − G )(1− t + δt)

H + L + rK − G− H

H + L;

⇒ Gini =1− t(1− δ)− H

H+L

1 + H+LrK−G

. (3)

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Equilibrium

I Free entry: π = rα⇒ x = αθ(1−θ)β r

I Capital market clears: n = Kα .

I Labour market clears:

L + H = G + nβx ⇒ L + H = G +Kθ

1− θr

I From the FOC of utility maximization:

ncHp = G ⇒ G (1 + H)− rK = H

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Equilibrium

I Solutions:

G =H + L− θL

1 + θH

rK = (1− θ)H2 + LH + L

1 + θH

I Assumption: H + L > 11−θ for G > 1 [poor do not contribute]

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Results

I As non-contributor’s size increases, aggregate provision of publicgoods goes up

I Income inequality between a contributor and non-contributor goesup

I Aggregate wealth of the contributor rises- due to larger supply of non-contributors.

I This is a case where rich might welcome poor in their community.

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Results

I As non-contributor’s size increases, aggregate provision of publicgoods goes up

I Income inequality between a contributor and non-contributor goesup

I Aggregate wealth of the contributor rises- due to larger supply of non-contributors.

I This is a case where rich might welcome poor in their community.

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Results

I As non-contributor’s size increases, aggregate provision of publicgoods goes up

I Income inequality between a contributor and non-contributor goesup

I Aggregate wealth of the contributor rises- due to larger supply of non-contributors.

I This is a case where rich might welcome poor in their community.

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Results

I As non-contributor’s size increases, aggregate provision of publicgoods goes up

I Income inequality between a contributor and non-contributor goesup

I Aggregate wealth of the contributor rises- due to larger supply of non-contributors.

I This is a case where rich might welcome poor in their community.

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Results: Non-Contributor ↑

12 12 12

1 1 1 1 1 1

12 12 12

1 1 1 1 1 1 1

I G ↑ , r ↑, per capita income (contributor) ↑, inequality ↑

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Results - Contributor ↑, k constant

12 12 12

1 1 1 1 1 1

12 12 12 12

1 1 1 1 1 1 1

I G uncertain, r ↓, per capita income (contributor) ↓,inequality ↓

I If G ↓ ⇒ Group size paradox! For G ↑, welfare ↑ even when(contributor’s) income↓.

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Results - Contributor ↑, Agg Cap constant

12 12 12

1 1 1 1 1 1

9 9 9 9

1 1 1 1 1 1 1

I G uncertain, r ↑, per capita income (contributor) ↓,inequality ↓

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Results - Capital Redistribution

12 12 12

1 1 1 1 1 1

18 18

1 1 1

1 1 1 1 1 1

I G ↑, r ↓, per capita income (contributor) ↑, inequality ↑

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Results - Increase in Capital

12 12 12

1 1 1 1 1 1

18 18 18

1 1 1

1 1 1 1 1 1

I G unchanged, r unchanged, per capita income (contributor)unchanged, inequality unchanged- only welfare of all goes up (as private variety goes up)

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Government Taxes and Transfer

n∑i=1

picHi = w + (rk − gj)(1− t) +δt∑H

j=1(rk − gj)

H. (4)

n∑i=1

picLi = w +(1− δ)t

∑Hj=1(rk − gj)

L. (5)

I More transfer to poor (δ ↓), given tax rate, lowers inequality, but Gis ambiguous.

I BBV(1986) writes -

“Equalizing income redistributions that involve anytransfers from contributors to non-contributors willdecrease the equilibrium supply of the public good.”

I Higher tax rate (t ↑) ⇒ G ↑ and inequality ↓.

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Government Taxes and Transfer

n∑i=1

picHi = w + (rk − gj)(1− t) +δt∑H

j=1(rk − gj)

H. (4)

n∑i=1

picLi = w +(1− δ)t

∑Hj=1(rk − gj)

L. (5)

I More transfer to poor (δ ↓), given tax rate, lowers inequality, but Gis ambiguous.

I BBV(1986) writes -

“Equalizing income redistributions that involve anytransfers from contributors to non-contributors willdecrease the equilibrium supply of the public good.”

I Higher tax rate (t ↑) ⇒ G ↑ and inequality ↓.

33 / 40

Government Taxes and Transfer

n∑i=1

picHi = w + (rk − gj)(1− t) +δt∑H

j=1(rk − gj)

H. (4)

n∑i=1

picLi = w +(1− δ)t

∑Hj=1(rk − gj)

L. (5)

I More transfer to poor (δ ↓), given tax rate, lowers inequality, but Gis ambiguous.

I BBV(1986) writes -

“Equalizing income redistributions that involve anytransfers from contributors to non-contributors willdecrease the equilibrium supply of the public good.”

I Higher tax rate (t ↑) ⇒ G ↑ and inequality ↓.

34 / 40

Welfare and Optimal tax

I We assume Govt represents the interest of the rich.

I Market provides too little of the public good.

I Market provides too high a return to capital.

I Socially desirable inequality is positive (but less than themarket level)

I It is optimal to tax the entire capital income of the rich for δ = 1.Otherwise, optimum tax is interior.

35 / 40

Welfare and Optimal tax

I We assume Govt represents the interest of the rich.

I Market provides too little of the public good.

I Market provides too high a return to capital.

I Socially desirable inequality is positive (but less than themarket level)

I It is optimal to tax the entire capital income of the rich for δ = 1.Otherwise, optimum tax is interior.

36 / 40

Welfare and Optimal tax

I We assume Govt represents the interest of the rich.

I Market provides too little of the public good.

I Market provides too high a return to capital.

I Socially desirable inequality is positive (but less than themarket level)

I It is optimal to tax the entire capital income of the rich for δ = 1.Otherwise, optimum tax is interior.

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Welfare and Optimal tax

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Welfare and Optimal tax

topt =[H + L(1− θ)]θ(H − 1)

(H + L)θ(H + θ)− Lθ(Hθ + 1)− δ(1 + θ)Hθ≤ 1.

When δ = 1,topt = 1.

Alternatively, for δ = 0,

topt =Hθ(H − 1) + L(1− θ)θ(H − 1)

Hθ(H + θ) + L(1− θ)θ(H − 1)< 1.

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Thank you!Comments and suggestions?

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