A Note on the Optimality of Domain-specific Liability

Tim Friehe; Eric Langlais; Elisabeth Schulte

doi:10.1515/rle-2022-0002

Article

A Note on the Optimality of Domain-specific Liability

Tim Friehe , Eric Langlais and Elisabeth Schulte

Published/Copyright: June 21, 2022

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From the journal Review of Law & Economics Volume 18 Issue 2

Abstract

This note analyzes the socially optimal allocation of liability when both consumers and the environment incur harm from the activity of a monopolistic firm. We show that the marginal welfare effect from a greater extent of loss shifting depends on the domain of harm (consumer vs. environment) and the relationship between the harm level and the level of output (proportional vs. non-proportional). Starting from the relevant benchmark of full compensation in both domains, reducing the firm’s liability for environmental harm is welfare-improving whereas reducing the firm’s liability for consumer harm is welfare-decreasing when harm increases more than proportionally with the quantity produced.

Keywords: liability; cumulative harm; product liability; environmental harm

JEL Classification: K13; Q50

Corresponding author: Tim Friehe, Marburg Centre for Institutional Economics (MACIE), University of Marburg, Marburg, Germany, Email: tim.friehe@uni-marburg.de

Acknowledgments

We are grateful for the helpful comments received from an anonymous reviewer on a former version of the manuscript.

Appendix

A.1 Welfare Maximization

We specify welfare as

(A.1) W = a − b 2 q − c ( x ) q − γ h ( x ) q θ

and obtain the first-order conditions

(A.2) W q ( q , x ) = a − b q − c ( x ) − γ θ h ( x ) q θ − 1 = 0

(A.3) W x ( q , x ) = − c ′ ( x ) q − γ h ′ ( x ) q θ = 0 .

From these conditions, we obtain

(A.4) W q q ( q , x ) = − b − γ ( θ 2 − θ ) h ( x ) q θ − 2 < 0

(A.5) W q x ( q , x ) = − c ′ ( x ) − γ θ h ′ ( x ) q θ − 1

(A.6) W x x ( q , x ) = − c ″ ( x ) q − γ h ″ ( x ) q θ < 0 .

At the welfare-maximizing combination of safety and output, the social planner fulfills condition (A.3). Using a reformulation of this expression to restate the cross-partial derivative as

(A.7) W q x ( q , x ) = − γ ( θ − 1 ) h ′ ( x ) q θ − 1 > 0 ,

we find that

(A.8) D = W q q W x x − ( W q x ) 2 > 0

because

γ 2 q 2 ( θ − 1 ) [ ( θ 2 − θ ) h h ″ − ( θ − 1 ) 2 ( h ′ ) 2 ] > 0

by θ > 1 and the assumption that H is strictly convex.

A.2 Comparative Statics Results

Starting from the profit equation specified in Eq. (5) in the main document,

Π ( q , x ) = ( a − b q − c ( x ) ) q − Λ M h ( x ) q θ ,

we obtain the first-order conditions,

Π q ( q , x ) = a − 2 b q − c ( x ) − θ Λ M h ( x ) q θ − 1 = 0 Π x ( q , x ) = − c ′ ( x ) q − Λ M h ′ ( x ) q θ = 0 .

From these conditions, we obtain

(A.9) Π q q ( q , x ) = − 2 b − Λ M ( θ 2 − θ ) h ( x ) q θ − 2 < 0

(A.10) Π q x ( q , x ) = − c ′ ( x ) − Λ M θ h ′ ( x ) q θ − 1

(A.11) Π x x ( q , x ) = − c ″ ( x ) q − Λ M h ″ ( x ) q θ < 0 .

At the profit-maximizing combination of safety and output, the firm fulfills condition (7). Using a reformulation of this expression to restate the cross-partial derivative as

(A.12) Π q x ( q , x ) = − Λ M ( θ − 1 ) h ′ ( x ) q θ − 1 > 0 ,

we find that

(A.13) G = Π q q Π x x − ( Π q x ) 2 > 0

because

Λ M 2 q 2 ( θ − 1 ) [ ( θ 2 − θ ) h h ″ − ( θ − 1 ) 2 ( h ′ ) 2 ] > 0

by θ > 1 and the assumption that H is strictly convex.

The comparative-static properties of the firm’s problem follow from

(A.14) Π q q Π q x Π x q Π x x d q d x = − Π q Λ M − Π x Λ M d Λ M

where

(A.15) Π q Λ M = − θ h ( x ) q θ − 1 < 0

(A.16) Π x Λ M = − h ′ ( x ) q θ > 0 .

This implies that the output and safety levels change as follows with a change in Λ_M:

(A.17) d q d Λ M = Π q x Π x Λ M − Π q Λ M Π x x G = q 2 θ − 1 Λ M θ [ ( h ′ ) 2 − h h ″ ] − Λ M ( h ′ ) 2 q 2 θ − 1 − c ″ q θ θ h G < 0

(A.18) d x d Λ M = Π q x Π q Λ M − Π x Λ M Π q q G = − h ′ q θ 2 b G > 0

where the sign in (A.17) stems from the convexity of H.

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Published Online: 2022-06-21

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https://doi.org/10.1515/rle-2022-0002

Keywords for this article

liability; cumulative harm; product liability; environmental harm