Complement or Substitute? Punishment and Self-Interested Enforcement

Ken Yahagi

doi:10.1515/bejeap-2024-0467

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Complement or Substitute? Punishment and Self-Interested Enforcement

Ken Yahagi

Veröffentlicht/Copyright: 11. August 2025

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Aus der Zeitschrift The B.E. Journal of Economic Analysis & Policy

Abstract

This study examines the problems that law enforcement agencies face with self-interested agents who (1) prefer to punish wrongdoers and reduce crime rates and (2) would suffer a loss of reputation resulting from law enforcement investigations that do not lead to convictions. Next, we examine how agents’ enforcement responses change with sanction size. This is fundamental to how higher-ranking authorities (e.g., the judiciary) can implement sanction policies to mitigate law enforcement agency problems. Our analysis shows that enforcement and sanctions can function as either complements or substitutes. When agents have strong preferences for reducing crime, stricter sanctions complement enforcement and increase deterrence. Conversely, when concerns about reputation loss dominate, stricter sanctions substitute for enforcement and may decrease deterrence. Consequently, a non-maximal sanction may be optimal when sanctions substitute for enforcement. This yields a novel justification for imposing non-maximal sanctions based on the behavior of the enforcer.

Keywords: legal error; non-maximal sanction; principal-agent problem; intrinsic motivations

JEL Classification: K42

Corresponding author: Ken Yahagi, Faculty of Economics, Seikei University, 3-3-1 Kichijoji-kitamachi, Musashino-shi, Tokyo 180-8633, Japan, E-mail: ken-yahagi@econ.seikei.ac.jp

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP22K13413

Acknowledgments

This work was supported by JSPS KAKENHI (grant number JP22K13413). I thank the editors-in-chief and two anonymous referees, Yoshinobu Zasu and Kohei Kamaga, as well as the participants of the Japanese Law and Economics Association and the Economics Workshop at Sophia University, for their helpful comments and suggestions.

Appendix

Proof of Lemma 1

The first-order condition for the agent is

(14) L e = m q ∂ q ∂ b ̄ b ̄ e − r E E e − c e = 0 .

According to the implicit function theorem, we have

(15) ∂ e * ∂ f = − L e f L e e ,

where L_ee < 0 and

(16) L e f = m q q b ̄ e b ̄ f − m q b ̄ e f − r E E E e E f − r E E e f .

Since we have

(17) b ̄ f = ( p 1 − p 2 x ) = b ̄ f > 0

(18) b ̄ e f = p e 1 − p e 2 x = b ̄ e f > 0

(19) E f = b ̄ f p 2 ( 1 − x ) = E f > 0

(20) E e f = b ̄ e f p 2 ( 1 − x ) + b ̄ f p e 2 ( 1 − x ) = E e f > 0 ,

incorporating the first-order condition (8) leads to

(21) L e f = m q q ︸ ? b ̄ e b ̄ f ︸ + − r E E ︸ ? E e E f ︸ + + c e f ︸ + .

Then, if the agent has a greater preference for lowering the crime rate and is less concerned with reputation costs (i.e., m q q ( b ̄ e b ̄ f ) + c e / f > r E E ( E e E f ) ), we have ∂e*/∂f > 0. However, if the agent cares about reputation loss and does not have a private preference for lowering the crime rate (i.e., m q q ( b ̄ e b ̄ f ) + c e / f < r E E ( E e E f ) ), we have ∂e*/∂f < 0.

Proof of Lemma 2

The simple calculation indicates that the deterrence threshold is b ̄ = p 1 ( e * ( f ) ) − p 2 ( e * ( f ) ) x f and the marginal effect of f on b ̄ is

(22) ∂ b ̄ ∂ f = b ̄ e ∂ e * ∂ f | e = e * + b ̄ f | e = e * = p e 1 − p e 2 x f ︸ + ∂ e * ∂ f ︸ ? + p 1 − p 2 x ︸ +

Then, if ∂e*/∂f > 0 or ∂ e * / ∂ f > − ( p 1 − p 2 x ) / p e 1 − p e 2 x f , we have ∂ b ̄ / ∂ f > 0 . On the other hand, if ∂ e * / ∂ f < − ( p 1 − p 2 x ) / p e 1 − p e 2 x f , we have ∂ b ̄ / ∂ f < 0 .

Proof of Proposition 1

By incorporating the first-order condition (8), the marginal effect of f on the utilitarian social welfare function is

(23) S W f = ( h − b ̄ ) ∂ b ̄ ∂ f ︸ ? − m q ( p 1 − p 2 x ) ︸ + − r E ( p 1 − p 2 x ) p 2 ( 1 − x ) ︸ − .

Then, if SW_f > 0, we have f* = F. However, if SW_f = 0, we have f* < F. If f* is an interior solution, we can have ∂f*/∂h > 0 or ∂f*/∂h < 0. Assuming the second-order condition is satisfied SW_ff < 0, the implicit function theorem with respect to SW_f = 0 indicates ∂ f * / ∂ h = − S W f h / S W f f = − [ ∂ b ̄ / ∂ f ] / S W f f . Then, ∂ b ̄ / ∂ f > 0 leads to ∂f*/∂h > 0, and ∂ b ̄ / ∂ f < 0 leads to ∂f*/∂h < 0.

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Received: 2024-12-24

Accepted: 2025-07-31

Published Online: 2025-08-11

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https://doi.org/10.1515/bejeap-2024-0467

Schlagwörter für diesen Artikel

legal error; non-maximal sanction; principal-agent problem; intrinsic motivations