# Multidimensional Screening with Complementary Activities: Regulating a Monopolist with Unknown Cost and Unknown Preference for Empire Building

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Remark 1.**An empire builder produces more output than a money-seeker (with the same efficiency), and an efficient manager produces with a lower cost than an inefficient manager (with the same tendency for empire building):

## 3. The Relaxed Problem

**Remark 2.**In any solution of the relaxed problem, we have:

**Remark 3.**If a downward incentive constraint is binding, the corresponding upward incentive constraint is surely satisfied if the activity levels satisfy the monotonicity property (2).

## 4. Several Possible Scenarios

#### 4.1. Case A: Low Probabilities of Intermediate Types

**Remark 4.**When Case A is optimal in the relaxed problem, output and marginal cost levels are ranked in the natural way, with a bunching of the worst types in each activity:

**Proposition 1.**Case A is optimal in the relaxed problem and in the fully constrained problem if $\mathrm{min}\left(\right)open="\{"\; close="\}">{\alpha}_{EM},{\alpha}_{IB}{\alpha}^{*}$, where ${\alpha}^{*}$ is a strictly positive function of the remaining parameters.

#### 4.2. Cases B and C: Equal Variabilities of β and δ

#### 4.2.1. Case B: Natural Ranking of Activity Levels

**Remark 5.**When Case B is optimal in the relaxed problem, output and marginal cost levels are ranked in the natural way:

**Proposition 2.**Case B is optimal in the relaxed problem and in the fully constrained problem when $\omega =1$ if $\rho \ge -\phantom{\rule{0.166667em}{0ex}}\frac{1-\lambda}{1+\lambda}$ and ${\alpha}_{IM}\in \left(\right)open="["\; close="]">{\alpha}_{IM}^{*},{\alpha}_{IM}^{**}$, where ${\alpha}_{IM}^{*}$ and ${\alpha}_{IM}^{**}$ are functions of the remaining parameters. The interval, $\left(\right)open="["\; close="]">{\alpha}_{IM}^{*},{\alpha}_{IM}^{**}$, is non-empty if and only if $\lambda \le {\lambda}^{*}$, where ${\lambda}^{*}$ is a strictly positive function of ${\alpha}_{EB}$ and ${\alpha}_{EM}$.

#### 4.2.2. Case C: Bunching of Intermediate Output Levels

**Remark 6.**When Case C is optimal in the relaxed problem, output and marginal cost levels are ranked as follows:

**Proposition 3.**Case C is optimal in the relaxed problem and in the fully constrained problem when $\omega =1$ if and only if ${\alpha}_{IM}\le {\alpha}_{IM}^{***}$, where ${\alpha}_{IM}^{***}$ is a strictly positive function of the remaining parameters. ☐

#### 4.3. Case D: Empire Building Dominance ($\Delta \delta \gg \Delta \beta $)

**Remark 7.**When Case D is optimal in the fully constrained problem, we must have $\omega >1$ and the following ranking of activity levels:

**Proposition 4.**Case D is optimal in the relaxed problem and in the fully constrained problem if $\Delta \beta <{\Delta}^{*}$, where ${\Delta}^{*}$ is a strictly positive function of the remaining parameters.

#### 4.4. Case E: Efficiency Dominance ($\Delta \beta \gg \Delta \delta $)

**Remark 8.**When Case E is optimal in the relaxed problem, output and marginal cost levels are ranked as follows:

**Proposition 5.**Case E is optimal in the relaxed problem and in the fully constrained problem if $\Delta \delta <{\Delta}^{**}$, where ${\Delta}^{**}$ is a strictly positive function of the remaining parameters.

## 5. Concluding remarks

## 6. Appendix

#### 6.1. Relaxed Problem

**Proof of Remark 3**

#### 6.2. Case A

**Proof of Remark 4**

**Proof of Proposition 1**

#### 6.3. Case B

**Proof of Remark 5**

**Proof of Proposition 2**

#### 6.4. Case C

**Proof of Remark 6**

**Proof of Proposition 3**

#### 6.5. Case D

**Proof of Remark 7**

**Proof of Proposition 4**

#### 6.6. Case E

**Proof of Remark 8**

**Proof of Proposition 5**

## Acknowledgements

## Conflict of Interest

## References

- Armstrong, M.; Rochet, J.-C. Multidimensional screening: A user’s guide. Eur. Econ. Rev.
**1999**, 43, 959–979. [Google Scholar] - Basov, S. Multidimensional screening. In Studies in Economic Theory; Volume 22, Springer: Berlin, Germany, 2005. [Google Scholar]
- Laffont, J.J.; Tirole, J. Using cost observation to regulate firms. J. Polit. Econ.
**1986**, 94, 614–641. [Google Scholar] [CrossRef] - Borges, A.P.; Correia-da-Silva, J. Using cost observation to regulate a manager who has a preference for empire building. Manchester School
**2011**, 79, 29–44. [Google Scholar] [CrossRef] - Niskanen, W.A. Bureaucracy and the representative government; Aldine Press: Chicago, IL, USA, 1971. [Google Scholar]
- Donaldson, G. Managing corporate wealth: The operation of a comprehensive financial goals system; Praeger: New York, NY, USA, 1984. [Google Scholar]
- Jensen, M.C. Agency costs of free cash flow, corporate finance and take-overs. Am. Econ. Rev.
**1986**, 76, 323–329. [Google Scholar] - Jensen, M.C. The modern industrial revolution, exit, and the failure of internal control systems. J. Finance
**1993**, 48, 831–880. [Google Scholar] [CrossRef] - Borges, A.P.; Correia-da-Silva, J.; Laussel, D. Regulating a manager whose empire building preferences are private information. J. Econ. forthcoming. [CrossRef]
- Brighi, L.; D’Amato, M. Two-dimensional screening: a case of monopoly regulation. Research in Econ.
**2002**, 56, 251–264. [Google Scholar] [CrossRef] - Rochet, J.-C.; Stole, L.A. The economics of multidimensional screening. Econometric Society Monographs
**2003**, 35, 150–197. [Google Scholar]

^{1.}The book by Basov [2] is also an important reference for the treatment of multidimensional screening problems.^{2.}The tendency of managers for empire building has been studied, among others, by Niskanen [5] and documented by Donaldson [6]. Jensen [7,8] has emphasized it as an origin of excess investment and output: “Managers have incentives to cause their firms to grow beyond the optimal size. Growth increases managers’ power by increasing the resources under their control. It is also associated with increases in managers’ compensation, because changes in compensation are positively related to the growth in sales.”^{3.}In the contributions of Brighi and D’Amato [10] and Rochet and Stole [11], the case of four possible types characterized by the realization of two binary variables is not explored. While Brighi and D’Amato [10] focus exclusively on the case of two types that differ in two dimensions (and, therefore, cannot be ranked a priori), Rochet and Stole [11] provide a general study of multidimensional screening, but in the discrete case, only explore in detail an example that also has two types that differ in two dimensions.^{4.}The downward (resp. upward) constraints are those that require that a worse (resp. better) type should not benefit from mimicking a better (resp. worse) type. One speaks of a diagonal constraint when the two types cannot be ranked: each type is better in one dimension and worse in the other.^{5.}It is usual to assume that $-{S}^{\u2033}\left(q\right){\psi}^{\u2033}\left(e\right)>(1+\lambda )$. See, for example, Laffont and Tirole [3]. For the specific functions on which we focus, this is equivalent to $\lambda <1$.^{6.}See Appendix 6.1 for further details.

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Borges, A.P.; Laussel, D.; Correia-da-Silva, J.
Multidimensional Screening with Complementary Activities: Regulating a Monopolist with Unknown Cost and Unknown Preference for Empire Building. *Games* **2013**, *4*, 532-560.
https://doi.org/10.3390/g4030532

**AMA Style**

Borges AP, Laussel D, Correia-da-Silva J.
Multidimensional Screening with Complementary Activities: Regulating a Monopolist with Unknown Cost and Unknown Preference for Empire Building. *Games*. 2013; 4(3):532-560.
https://doi.org/10.3390/g4030532

**Chicago/Turabian Style**

Borges, Ana Pinto, Didier Laussel, and João Correia-da-Silva.
2013. "Multidimensional Screening with Complementary Activities: Regulating a Monopolist with Unknown Cost and Unknown Preference for Empire Building" *Games* 4, no. 3: 532-560.
https://doi.org/10.3390/g4030532