Distributive Justice, Public Policies and the Comparison of Legal Rules: Quantify the “Price of Equity”

Marco Fabbri; Diogo GC Britto

doi:10.1515/rle-2016-0005

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Distributive Justice, Public Policies and the Comparison of Legal Rules: Quantify the “Price of Equity”

Marco Fabbri und Diogo GC Britto

Veröffentlicht/Copyright: 2. September 2017

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Aus der Zeitschrift Review of Law & Economics Band 14 Heft 3

Abstract

This paper proposes a quantitative approach to study two methodological problems arising when a costly redistribution of resources is implemented through public policies or legal rules: (a) aggregating individual into social preferences and (b) choosing the object of maximization. We consider a redistribution intervention that reduces inequality but diminishes total wealth and we specify a set of social welfare functions combining different preferences aggregation methods and maximands. For each social welfare function, we calculate its “price of equity”, defined as the maximum fraction of total wealth that a society is willing to sacrifice in order to implement the redistribution. Comparing the prices for equity across different social welfare function specifications, we identify systematic relationships and we rank them according to the efficiency-equity orientation. Results show that social welfare functions characterized by aggregation methods conventionally considered equity-oriented may reject redistribution interventions that are evaluated as welfare-improving by social welfare functions using efficiency-oriented aggregation methods. Similarly, social welfare functions considered equity-oriented because using utility as object of maximization may reject distributive policies that are evaluated as welfare-improving by social welfare functions using wealth as maximand. We argue that the quantitative approach proposed, by expounding the trade-off between equity and efficiency connected to different social welfare functions, may prove useful in areas of public law where policy-makers have to engage in the choice of a normative criterion for the evaluation of social welfare. Additionally, our results may inform rule-makers interested in comparing the distributive effects of alternative legal rules in special circumstances where private remedies can efficiently achieve redistribution goals.

Keywords: efficiency; law and economics; Nash; Rawls; social welfare function

JEL Classification: D04; D61; D63; K10

Appendix

Table 2:

Minimal level of K required to guarantee welfare improvement - homogeneous groups.

Utility Func. Form	Bentham	Nash	Rawls
Wealth	1	wAwB−ϵ	∅
Polynomial	(1/ϵ){[wAα+wBα−(wB−ϵ)α]1/α−w1}	wAwB−ϵ	∅
Logarithmic	wAwB−ϵ	(1/ϵ)[elnwAlnwBlnwB−ϵ−wA]	∅
Exponential	(1/ϵ){−(1/α)ln[e−αwA+e−αwB−e−α(wB−ϵ)]−wA}	1ϵ−1αln1−(1−e−αwA)(1−e−αwB)1−e−α(wB−ϵ)−wA	∅

Table 3:

Minimal level of K required to guarantee welfare improvement - non-homogeneous groups.

Utility	Bentham	Nash	Rawls
Wealth	N−jj	(1/ϵ)[(wAjwBN−j(wB−ϵ)N−j)(1/j)−wA]	∅
Poly.	(1/ϵ){[jwAα+(N−j)wBα−(N−j)(wB−ϵ)αj]1/α−wA}	(1/ϵ)[(wAjwBN−j(wB−ϵ)N−j)(1/j)−wA]	∅
Log.	(1/ϵ)[(wAjwBN−j(wB−ϵ)N−j)(1/j)−wA]	(1/ϵ){e[(lnwA)j(lnwB)N−j(lnwB−ϵ)N−j]1/j−wA}	∅
Exp.	(1/ϵ){−(1/α)lnje−αwA+(N−j)e−αwB−(N−j)e−α(wB−ϵ)j−w1}	(1/ϵ){(−1α)ln[1−((1−e−αwA)j(1−e−αwB)N−j(1−e−α(wB−ϵ))N−j)1/j]−wA}	∅

B.1 Prices of equity for homogeneous groups: Proofs of level of k reported in Table 2

For each case, we derive the minimum k such that SW(W′)≥SW(W), where W′=(w1+kϵ,w2−ϵ)W=(w1,w2).

Bentham Case:

Wealth: u(wi)=wi

(w1+kϵ)+(w2−ϵ)≥w1+w2

⟺k≥1

Polynomial: u(wi)=wiα

(w1+kϵ)α+(w2−ϵ)α≥w1α+w2α

⟺k≥1ϵ{[w1α+w2α−(w2−ϵ)]1/α−w1}

Logarithmic: u(wi)=ln(wi)

ln(w1+kϵ)+ln(w2−ϵ)≥ln(w1)+ln(w2)

⟺k≥w1w2−ϵ

Exponential: u(wi)=1−e−αwi

(1−e−α(w1+kϵ))+(1−e−α(w2−ϵ))≥(1−e−α(w1))+(1−e−α(w2))

⟺k≥(−1αϵ)ln[1+e−α(w2−w1)(1−e−αϵ)]

Nash Case:

Wealth: u(wi)=wi

(w1+kϵ)(w2−ϵ)≥w1w2

⟺k≥w1w2−ϵ

Polynomial: u(wi)=wiα

(w1+kϵ)α(w2−ϵ)α≥w1αw2α

⟺k≥w1w2−ϵ

Logarithmic: u(wi)=ln(wi)

ln(w1+kϵ)ln(w2−ϵ)≥ln(w1)ln(w2)

⟺k≥1ϵ{exp[ln(w1)ln(w2)ln(w2−ϵ)]−w1}

Exponential: u(wi)=1−e−αwi

(1−e−α(w1+kϵ))(1−e−α(w2−ϵ))≥(1−e−αw1)(1−e−αw2)

⟺k≥(−1αϵ)ln[1−e−αw2+e−α(w2−w1)1+e−α(w2−ϵ)]

B.2 Prices of equity for non-homogeneous groups: Proofs of level of k reported in Table 3

Let W=[jw1,(N−j)w2] be the status quo and W′=[jw1+kϵ,(N−j)w2−ϵ)], j∈}0, N}, be the new allocation in which agent A increases her wealth by kϵ, and agent B has her wealth reduced by ϵ.^[15]Table 3 in Appendix A shows the price of equity for each SWF combination.

Bentham Case:

Wealth: u(wi)=wi

∑i=1j(wi+kϵ)+∑i=j+1N(wi−ϵ)≥∑i=1Nwi

⟺k≥(N−j)j

Polynomial: u(wi)=wiα

∑i=1j(wi+kϵ)α+∑i=j+1N(wi−ϵ)α≥∑i=1N(wi)α

⟺k≥(1/ϵ){[jwAα+(N−j)wBα−(N−j)(wB−ϵ)αj]1/α−wA}

Logarithmic: u(wi)=ln(wi)

∑i=1jln(wi+kϵ)+∑i=j+1Nln(wi−ϵ)≥∑i=1Nln(wi)

⟺k≥1ϵwAjwB(N−j)(wB−ϵ)(N−j)1/j−wA

Exponential: u(wi)=1−e−αwi

∑i=1j(1−e−α(wi+kϵ))+∑i=j+1N(1−e−α(wi−ϵ))≥∑i=1N(1−e−α(wi))

⟺k≥−1αϵln1+(N−j)je−α(wB−wA)(1−eαϵ)

Nash Case:

Wealth: u(wi)=wi

Πi=1j(wi+kϵ)Πi=j+1N(wi−ϵ)≥Πi=1Nwi

⟺k≥1ϵwAjwBN−j(wB−ϵ)N−j(1/j)−wA

Polynomial: u(wi)=wiα

Πi=1j(wi+kϵ)αΠi=j+1N(wi−ϵ)α≥Πi=1Nwiα

⟺k≥1ϵwAjwBN−j(wB−ϵ)N−j(1/j)−wA

Logarithmic: u(wi)=ln(wi)

Πi=1jln(wi+kϵ)Πi=j+1Nln(wi−ϵ)≥Πi=1Nln(wi)

⟺k≥1ϵexp(lnwA)j(lnwB)N−j(ln(wB−ϵ))N−j1/j−wa

Exponential: u(wi)=1−e−αwi

Πi=1j(1−e−α(wi+kϵ))Πi=j+1N(1−e−α(wi−ϵ))≥Πi=1N(1−e−αwi)

⟺k≥1ϵ−1αln1−(1−e−αwA)j(1−e−αwB)N−j(1−e−α(wB−ϵ))(N−j)1/j−wA

B.3 Efficiency-improving transfers

The proofs of results for E-I transfers summarized in the table above follow analogously to those of the R-H case. There are only two differences, respectively that w2−w1<0 and that ϵ is now bounded by w2 (an individual cannot be left with negative wealth) instead of w2−w1.

Table 4:

Generality of results under EI transfers.

Maximand/SWF Form	Bentham	Nash	Rawls
Wealth	∅	{B−W; B−P; B−L; N−P}	EI¬R
Polynomial	}B−W}	}B−W; B−P; B−L; N−W}	EI¬R
Logarithmic	}B−W; B−P}	}B−W; B−P; B−L; N−W; N−P}	EI¬R
Exponential	}B−W}	}B−W; B−E}	EI¬R

Acknowledgements

The idea of this paper was inspired by a conversation of the authors with Francesco Parisi. We are grateful to professor Parisi for his suggestions and comments on older versions of the paper. We also thank Emanuela Carbonara, Robert Cooter, Giuseppe Dari-Mattiacci, Michael Faure, Jonathan Klick, Louis Visscher, conference and workshop participants at the 2013 European Association of Law and Economics Annual Meeting, the 2013 German Association of Law and Economics Annual Meeting, the Institute of Law and Economics at Erasmus University Rotterdam and two anonymous referees for helpful comments. The usual disclaimer applies.

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efficiency; law and economics; Nash; Rawls; social welfare function