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Costly Rewards and Punishments

  • Frances Z. Xu Lee EMAIL logo
Published/Copyright: September 17, 2019
The B.E. Journal of Theoretical Economics
From the journal The B.E. Journal of Theoretical Economics Volume 20 Issue 1

Abstract

To punish an agent, the principal often incurs costs. I study a principal’s least costly reward and punishment scheme for an agent whose effort the principal cannot observe. I find the principal’s cost is sometimes minimized by using both costly rewards and costly punishments because (1) the agent has an outside option, or (2) a principal without commitment ability repeatedly interacts with the agent. I also find that when an agent’s effort is better at increasing the probability of a good outcome for the principal, the agent’s payoff may decrease, because the principal replaces rewards with punishments.

Keywords: costly punishment; principal-agent; moral hazard; outside option; credibility
JEL Classification: D23; L20; D86

Acknowledgements

The author is grateful for comments from two anonymous referees, Dan Bernhardt, Jacques Cremer, Jin Li, Albert Ma, Wing Suen and the audience at International Industrial Organization Conference

Appendix

A Proof of Lemma 1

Proof.

Suppose the optimal scheme has x1 < 0 and x2 < 0. Consider another scheme with x1′=[(1−p−θ)x1+(p+θ)x2]/(1−p−θ)+ϵ and x'2 = 0. Since IR is satisfied with x1,x2 , IR must also be satisfied with x'1 and x'2 by construction for any ε > 0. Since IC is satisfied with x1,x2 , x2−x1>cθ . I have x2′−x1′=x2−x1+11−p−θ(−x2)−ϵ>cθ+11−p−θ(−x2)−ϵ . Therefore, for any ϵ∈(0,11−p−θ(−x2)) , the IC holds for x1′,x2′ . Because the new scheme x1′,x2′ involves lower punishment for both outcomes than scheme x1,x2 , the principal’s cost is lower under x1′,x2′ .

Suppose the optimal scheme has x1 > 0 and x2 > 0. Suppose IR is not binding. Consider another scheme with x'1 = 0 and x2′=[(1−p−θ)x1+(p+θ)x2]/(p+θ)−ϵ>0 . By construction, it satisfies IR for ε small enough. Since the original scheme satisfies IC, x2>x1+cθ . This implies x2′−x1′>x1p+θ+cθ−ϵ>cθ if ε is small enough, which satisfies IC. Then this new scheme x1′,x2′ reduces the cost to implement the effort, which is a contradiction. Therefore, IR has to binding.

Consider another scheme with x'1 = 0 and x2′=[(1−p−θ)x1+(p+θ)x2]/(p+θ)>0 . By construction, it also satisfies IR. Since the original scheme satisfies IC, x2>x1+cθ . This implies x2′−x1′>x1p+θ+cθ>cθ , so the new scheme strictly relaxes IC. Therefore, x1′,x2′ implement the effort and results in the same cost to the principal.

Proof of Corollary 2, It follows from rewriting Proposition 1 in the following way:

Lemma 4.

Let uA denote the agent’s expected payoff under the optimal scheme.

  1. Under low outside option t<−(1−p)c/λ1+λ−p0 ,

    1. if θ<λ1+λ−p , the scheme is pure reward and uA=pθc>t ;

    2. if θ>λ1+λ−p , the scheme is pure punishment and uA=−1−pθc>t .

  2. Under intermediate outside option −(1−p)c/λ1+λ−p<t<0 ,

    1. if θ<λ1+λ−p , the scheme is pure reward and uA=pθc>t ;

    2. if λ1+λ−p<θ<−1−ptc , the scheme uses both reward and punishment and uA = t;

    3. if θ>−1−ptc , the scheme is pure punishment and uA=−1−pθc>t .

  3. Under intermediate outside option 0<t<pc/λ1+λ−p ,

    1. if θ<λ1+λ−p , the scheme is pure reward and uA=pθc>t ;

    2. if λ1+λ−p<θ<ptc , the scheme uses both reward and punishment and uA = t;

    3. if θ>ptc , the scheme is pure reward and uA = t.

  4. Under high outside option t>pc/λ1+λ−p ,

    1. if θ<ptc , the scheme is pure reward and uA=pθc>t ;

    2. if θ>ptc , the scheme pure reward and uA = t.

Proof of Proposition 2.

Proof.

As a preparation, we solve for the kink point of CC in the ( – x1)–x2 space:

λ(1+α)(−x1)+βx2=γλα(−x1)+(1+β)x2=γ

This gives

−x1=1λγ1+α+β=δθvλ,x2=γ1+α+β=δθv.

We consider two cases in turn.

Case 1. θ<λ1+λ−p . This case corresponds to Figure 1. The principal prefers rewards to punishments.

When cθ≤γ1+β=δθv1−δ(1−p−θ) , the left most point on IC (hitting the vertical axis) satisfies CC. Therefore, the optimal scheme is pure reward: x1 = 0 and x2=cθ .

When cθ>γ1+β=δθv1−δ(1−p−θ) , IC intersects the upper part of CC as long as the kink point of CC is above or on IC. That is,

δθv≥cθ−1λδθv⇒cθ≤1+λλγ1+α+β=1+λλδθv.

Otherwise, the solution does not exist and the effort cannot be motivated.

Solving for the intersection of IC and the upper part of CC:

(−x1)+x2=cθλα(−x1)+(1+β)x2=γ

gives solution: x1=−(1+β)cθ−γ1+β−λα=−(1−δ(1−p−θ))cθ−δθv1−(1+λ)δ(1−p−θ) and x2=γ−λαcθ1+β−λα=δθv−λδ(1−p−θ)cθ1−(1+λ)δ(1−p−θ) .

Case 2. θ>λ1+λ−p . The principal prefers punishments to rewards.

When cθ≤γλ(1+α)=δθvλ(1−δ(p+θ)) , the right most point of IC (hitting the horizontal axis) satisfies CC. Therefore, the optimal scheme is pure punishment: x1=−cθ and x2 = 0.

When cθ>γλ(1+α)=δθvλ(1−δ(p+θ)) , IC intersects the lower part of CC as long as the kink point of CC is above or on IC. As in Case 1, this condition is:

cθ≤1+λλδθv.

Otherwise, the solution does not exist and the effort cannot be motivated.

Solving for the intersection of IC and the lower part of CC:

(−x1)+x2=cθλ(1+α)(−x1)+βx2=γ

gives solution: x1=−γ−βcθλ(1+α)−β=−δθv−δ(p+θ)cθλ−δ(1+λ)(p+θ) and x2=λ(1+α)cθ−γλ(1+α)−β=λ(1−δ)(p+θ)cθ−δθvλ−δ(1+λ)(p+θ) .

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Published Online: 2019-09-17

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/bejte-2018-0131
Keywords for this article
costly punishment; principal-agent; moral hazard; outside option; credibility

Articles in the same Issue

  1. Research Articles
  2. Optimal Forestry Contract with Interdependent Costs
  3. Bi and Branching Strict Nash Networks in Two-way Flow Models: A Generalized Sufficient Condition
  4. Pay-What-You-Want in Competition
  5. Two Rationales for Insufficient Entry
  6. Students’ Social Origins and Targeted Grading
  7. Pricing, Signalling, and Sorting with Frictions
  8. On the Economic Value of Signals
  9. The Core in Bertrand Oligopoly TU-Games with Transferable Technologies
  10. Reasoning About ‘When’ Instead of ‘What’: Collusive Equilibria with Stochastic Timing in Repeated Oligopoly
  11. Timing Games with Irrational Types: Leverage-Driven Bubbles and Crash-Contingent Claims
  12. Costly Rewards and Punishments
  13. Blocking Coalitions and Fairness in Asset Markets and Asymmetric Information Economies
  14. Strategic Activism in an Uncertain World
  15. On Equilibrium Existence in a Finite-Agent, Multi-Asset Noisy Rational Expectations Economy
  16. Optimal Incentives Under Gift Exchange
  17. Public Good Indices for Games with Several Levels of Approval
  18. Vagueness of Language: Indeterminacy under Two-Dimensional State-Uncertainty
  19. Winners and Losers of Universal Health Insurance: A Macroeconomic Analysis
  20. Behavioral Theory of Repeated Prisoner’s Dilemma: Generous Tit-For-Tat Strategy
  21. Flourishing as Productive Tension: Theory and Model
  22. Notes
  23. A Note on Reference-Dependent Choice with Threshold Representation
  24. Regular Equilibria and Negative Welfare Implications in Delegation Games
  25. Unbundling Production with Decreasing Average Costs
  26. A Simple and Procedurally Fair Game Form for Nash Implementation of the No-Envy Solution
  27. Decision Making and Games with Vector Outcomes
  28. Capital Concentration and Wage Inequality
  29. Annuity Markets and Capital Accumulation
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