# Optimal Incentives in a Principal–Agent Model with Endogenous Technology

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## Abstract

**:**

## 1. Introduction

## 2. The Framework

## 3. The Equilibrium

**Proposition**

**1.**

## 4. Agent’s Risk Aversion and the Provision of Incentives

#### 4.1. The Effects of a Change of the Agent’s Risk Aversion on Incentives

#### 4.2. Agent’s Risk Aversion and Incentives: The Conditions for a Positive Link

**Condition**

**1.**

**Condition**

**2.**

**Proposition**

**2.**

- (i)
- When Condition 1 is satisfied, both the direct and indirect effects have the same sign and a lower agent’s risk aversion r unambiguously increase ${\beta}^{*}$ (i.e., $\partial {\beta}^{*}/\partial r<0)$ as in the standard principal–agent model.
- (ii)
- When Condition 1 does not hold, the total effect of r on ${\beta}^{*}$ can either be negative or positive, depending on the magnitude of the direct and of the indirect effects.
- (iii)
- When Condition 2 holds, the indirect effect have the opposite sign of the direct effect and larger size; therefore, a lower agent’s risk aversion r unambiguously decreases ${\beta}^{*}$ (i.e., $d{\beta}^{*}/dr>0)$, which is an opposite result to the one usually obtained in the standard principal–agent model.

#### 4.3. An Example

## 5. Extensions and Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Holmstrom, B.; Milgrom, P. Aggregation and linearity in the provision of intertemporal incentives. Econometrica
**1987**, 55, 303–328. [Google Scholar] [CrossRef] - Rao, C.; Hanumantha, H. Uncertainty, entrepreneurship, and sharecropping in India. J. Polit. Econ.
**1971**, 79, 578–595. [Google Scholar] - Allen, D.W.; Lueck, D. Risk preferences and the economics of contracts. Am. Econ. Rev.
**1995**, 85, 447–451. [Google Scholar] - Aggarwal, R.; Samwick, A. The other side of the trade-off: the impact of risk on executive compensation. J. Polit. Econ.
**1999**, 107, 65–105. [Google Scholar] [CrossRef] - Core, J.; Guay, W. Estimating the value of employee stock option portfolios and their sensitivities to price and volatility. J. Account. Res.
**2002**, 40, 613–630. [Google Scholar] [CrossRef] - Wulf, J. Authority, risk, and performance incentives: evidence from division manager positions inside firms. J. Ind. Econ.
**2007**, 55, 169–196. [Google Scholar] [CrossRef] - Prendergast, C. The tenuous trade-off between risk and incentives. J. Polit. Econ.
**2002**, 110, 1071–1102. [Google Scholar] [CrossRef] - Wright, D.J. The risk and incentives trade-off in the presence of heterogeneous managers. J. Econ.
**2004**, 83, 209–223. [Google Scholar] [CrossRef] - Legros, P.; Newman, A.F. Beauty is a beast, frog is a prince: assortative matching with nontransferabilities. Econometrica
**2007**, 75, 1073–1102. [Google Scholar] [CrossRef] - Serfes, K. Risk sharing vs. incentives: contract design under two-sided heterogeneity. Econ. Lett.
**2005**, 88, 343–349. [Google Scholar] [CrossRef] - Serfes, K. Endogenous matching in a market with heterogeneous principals and agents. Int. J. Game Theory
**2008**, 36, 587–619. [Google Scholar] [CrossRef] - Li, F.; Ueda, M. Why do reputable agents work for safer firms? Financ. Res. Lett.
**2009**, 6, 2–12. [Google Scholar] [CrossRef] - Ackerberg, D.; Botticini, M. Endogenous matching and the empirical determinants of contract form. J. Polit. Econ.
**2002**, 110, 564–591. [Google Scholar] [CrossRef] - Mookherjee, D.; Ray, D. Contractual structure and wealth accumulation. Am. Econ. Rev.
**2002**, 92, 818–849. [Google Scholar] [CrossRef] - Barros, F.; Macho-Stadler, I. Competition for managers and market efficiency. J. Econ. Manag. Strategy
**1998**, 7, 89–103. [Google Scholar] [CrossRef] - Dam, K.; Perez-Castrillo, D. The principal-agent matching market. Front. Theor. Econ.
**2006**, 2. [Google Scholar] [CrossRef]

1 | This could also explain the counterintuitive tendency of wealthier peasants to tend safer crops than poor peasants in medieval sharecropping (Ackerberg and Botticini [13]). A few other authors have contributed to this important relationship in principal–agent models. Among them, Mookherjee and Ray [14] who model an infinitely repeated interaction among principals and agents randomly matched at each period, Barros and Macho-Stadler [15] who look into a situation where several principals compete for an agent and Dam and Castrillo [16] who propose a model to analyze an economy with several principals and agents in order to characterize the set of stable outcomes. |

2 | As it will be clear next, if the technological relationship between efficiency and riskiness would be reversed, i.e., ${k}^{\prime}>0$, then the equilibrium outcome would be trivial as the optimal choice of the principal is always the more efficient and safer technology. |

3 | We here omit some details of the analysis as the complete description of the solution can be found in Holmstrom and Milgrom [1]. |

4 | It is worth remarking that in this framework the principal own the technology and, therefore, she will always offer a payment scheme giving the agent an expected utility equal to his reservation utility (i.e., the agent’s certainty equivalent) $\delta $. |

5 | The first order condition of the problem in Equation (6) is $d\pi /de=1-ke-r{k}^{2}e{\sigma}^{2}=0$ and the second order condition is always satisfied as ${d}^{2}\pi /d{e}^{2}=-k-r{k}^{2}{\sigma}^{2}<0$. |

6 | |

7 | Note that the first two components of Equation (12) are positive, while the third one is negative. |

8 | For example, this is always the case if k is bounded and ${lim}_{{\sigma}^{2}\to 0}(-{k}^{\prime})=+\infty $. |

9 | A straightforward comparison shows that, differently from our model, the endogenous matching models only consider the direct and the indirect riskiness effect (e.g., Serfes [10]), but not the indirect efficiency effect. Thus, whereas under positive assortative matching the riskiness effect is negative (since riskier principals attract more risk-averse agents), under negative assortative matching the indirect riskiness effect is positive (since now riskier principals are matched with less risk-averse agents) and the final effect of risk on incentives may, in this case, be ambiguous. |

10 | Note that the threshold ${\widehat{E}}_{k\sigma}$ is not necessarily always positive. Clearly, when ${\widehat{E}}_{k\sigma}<0$, Condition 2 can never be satisfied and there is always a negative relationship between the agent’s degree of risk aversion and the incentives provided by the principal as in the standard principal–agent framework. |

11 | As will be clear below, $\eta <1/2$ is necessary to obtain an interior solution. |

12 | We thank an anonymous reviewer for pointing this quite natural and interesting extension of the model to our attention. |

13 |

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**MDPI and ACS Style**

Marini, M.A.; Polidori, P.; Teobaldelli, D.; Ticchi, D.
Optimal Incentives in a Principal–Agent Model with Endogenous Technology. *Games* **2018**, *9*, 6.
https://doi.org/10.3390/g9010006

**AMA Style**

Marini MA, Polidori P, Teobaldelli D, Ticchi D.
Optimal Incentives in a Principal–Agent Model with Endogenous Technology. *Games*. 2018; 9(1):6.
https://doi.org/10.3390/g9010006

**Chicago/Turabian Style**

Marini, Marco A., Paolo Polidori, Désirée Teobaldelli, and Davide Ticchi.
2018. "Optimal Incentives in a Principal–Agent Model with Endogenous Technology" *Games* 9, no. 1: 6.
https://doi.org/10.3390/g9010006