Journal of Economics

, Volume 111, Issue 2, pp 105–130

Regulating a manager whose empire-building preferences are private information

Authors

  • Ana Pinto Borges
    • Núcleo de Investigação do Instituto Superior de Administração e Gestão (NIDISAG)
    • Faculdade de Economia, CEF.UPUniversidade do Porto
  • Didier Laussel
    • Aix-Marseille Université (AMSE), GREQAM, CNRS & EHESS
Article

DOI: 10.1007/s00712-012-0325-1

Cite this article as:
Borges, A.P., Correia-da-Silva, J. & Laussel, D. J Econ (2014) 111: 105. doi:10.1007/s00712-012-0325-1

Abstract

We obtain the optimal contract for the government (principal) to regulate a manager (agent) who has a taste for empire-building that is his/her private information. This taste for empire-building is modeled as a utility premium that is proportional to the difference between the contracted output and a reference output. We find that output is distorted upward when the manager’s taste for running large firms is weak, downward when it is strong, and equals a reference output when it is intermediate (in this case, the participation constraint is binding). We also obtain an endogenous reference output (equal to the expected output, which depends on the reference output), and find that the response of output to cost is null in the short-run (in which the reference output is fixed), whenever the manager’s type is in the intermediate range, and negative in the long-run (after the adjustment of the reference output to equal expected output).

Keywords

ProcurementRegulationAdverse selectionEmpire-buildingReservation utility

JEL Classification

D82H42H51I11

1 Introduction

Behavior of managers within firms is more complex than profit maximization.1 Its immediate determinants include: salary, security, status, power, prestige, social service and professional excellence (Niskanen 1971; Brehm and Gates 1997; Prendergast 2007). An observed willingness to manage a firm without significant monetary incentives should be attributed to these non-monetary factors (Wilson 1989).

The motivations of managers were synthesized and formalized by Williamson (1974), who concluded that, in addition to a preference for profit, managers also display a preference for expenses in staff and emoluments. Such preference for expenditure may accentuate the business cycle, by inducing a systematic accumulation of staff and emoluments during prosperity, with divestment frequently becoming necessary during adversity.

The empire-building motive is well documented (Donaldson 1984), and has been emphasized by Jensen (1986); Jensen (1993) 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.

Nevertheless, this tendency of managers towards empire-building has not been accounted for in the standard theory of regulation under asymmetric information (Baron and Myerson 1982; Guesnerie and Laffont 1984; Maskin and Riley 1984; Laffont and Tirole 1986). There, it is assumed that the managers’ objective is the maximization of the monetary transfers that they receive, net of the monetary value of the effort that they exert. We believe that extending the theory of regulation to incorporate the empire-building bias of managers will have a positive effect on the design of procurement contracts. Recently, Borges and Correia-da-Silva (2011) contributed in this direction, by extending the model of Laffont and Tirole (1986) to allow the manager of the firm to have a tendency for empire-building, modeled as a preference for high output.2

The intrinsic difficulty in the observation of managers’ preferences motivates us to pursue this line of research by studying the case in which the empire-building preference of the manager is unobservable by the government. In this paper, we consider a model in which the government (principal) offers a contract to a manager (agent) that has an unobservable preference for high output.3 The model can be straightforwardly adapted to deal with the case of firm’ owners offering a contract to a manager.

We model the preference for empire-building as a utility premium that is proportional to the difference between the contracted output and a reference output (it will actually be a utility penalty whenever the contracted output is lower than the reference output). The immediate effect of this bias is that the manager becomes willing to accept a lower payment if the contracted output is relatively high, and will demand a greater payment if it is relatively low. The government has, therefore, reasons to prefer facing a manager with a strong tendency for empire-building (as long as the optimal contract stipulates an output level that is greater than the reference output).

In the case of complete information, we find that the contracted output is a strictly increasing function of the observed manager’s preference for empire-building. There are two mechanisms that generate this unsurprising result. One is the fact that when this preference is stronger, the social marginal utility of output is greater. The other is related to the compensation scheme. The government can reward the manager by increasing output or by increasing the monetary transfer. When the preference for empire-building is stronger, the increase in output that is needed to substitute a monetary reward becomes lower.

If the contracted output is lower (resp. higher) than the reference output, then a stronger preference for empire-building implies a greater utility penalty (resp. premium) and, therefore, an increase (resp. a decrease) of the net monetary reward that the manager requires to participate. In this case, the manager has incentives to overstate (resp. understate) his/her taste for empire-building. Therefore, it may happen that managers with a strong preference for empire-building (sufficiently strong for the contracted output to be higher than the reference output) tend to understate their preference, while managers with a weak preference for empire-building (sufficiently weak for the contracted output to be lower than the reference output) tend to overstate their preference. There are countervailing incentives (Lewis and Sappington 1989; Maggi and Rodriguéz-Clare 1995; Jullien 2000; Laffont and Martimort 2002).

The tendency of managers to annouce an intermediate level of their preference for output implies that: (i) the output of the extreme types is not distorted (because the other types do not tend to mimic them); (ii) output is distorted below the perfect information level for high types (this removes the manager’s incentives to understate their type, because the utility premium would decrease); (iii) output is distorted above the perfect information level for low types (this removes the manager’s incentives to overstate their type, because the utility penalty would increase); (iv) output is at its reference level for a bunch of intermediate types (this also removes the manager’s incentives to misrepresent their type, because the attainable utility would remain equal to the reservation utility).

The reference output can be seen as a typical output level, that the manager associates with neither a utility premium nor a utility penalty. It may be natural to keep the reference output fixed when we estimate the short-run effects of an exogenous shock. But it is equally natural to consider an endogenous reference output, which may adjust in response to exogenous shocks, especially if we are interested in estimating the long-run effects of these shocks. We also address this possibility, by making the reference output equal to the expected output with respect to the probability distribution of manager’s types (notice that this expected output depends on the reference output itself).

The consideration of an endogenous reference output is useful to improve our understanding of the response of output to a change in the production cost. For example, in the case of managers that have an intermediate taste for empire-building, the fact that the contracted output is equal to the reference output implies that the cost-sensitivity of output is driven by the adjustment of the reference output. Therefore: in the short-run (in which the reference output is fixed), output does not respond to small variations in cost (remains equal to the reference output); while in the long-run (after the adjustment of the reference output to equal the expected output), the sensitivity of output to cost becomes negative (being driven by the adjustment of the reference output).

We believe that we are pioneer in assuming the existence of private information about the preference for empire-building. But, of course, we are not the first to investigate the economic effects of deviations from profit-maximizing behavior. The early managerial discretion models (Baumol 1959; Marris 1963; Williamson 1974) aimed to investigate the implications of various hypotheses concerning the objective functions of managers, explicitly recognizing that these may not coincide with profit-maximization. Baumol (1959) argued that some firms maximize sales subject to a profit constraint, thus producing more than if they operated as profit-maximizers. These firms respond to external shocks in a very different way. For example, if there is an increase in fixed costs or if a lump-sum tax is imposed, a firm whose managers have a preference for staff will reduce its output, staff employment and other perquisites.

A recent strand of literature is taking into account the preference of managers for empire-building.4 The focus in this literature has been to develop models of financial structure based on the desire to curb management’s empire-building tendencies, assuming that firms are owned by shareholders who don’t observe cash-flows or investment decisions. The empire-building bias is typically incorporated by assuming managerial private benefits of control (Grossman and Hart 1988) that are either proportional to investment (Hart and Moore 1995; Zwiebel 1996) or to the gross output from investment (Stulz 1990).

It is worthwhile to compare our results with those of the early managerial discretion models and of the models of optimal financial structure. Our result that output is an increasing function of the manager’s preference for output is in line with managerial discretion theory, but results from a quite different mechanism. Instead of being the straightforward implication of managerial discretion over output, it follows from the manager’s participation constraint: a larger output yields a larger utility to the manager who, then, becomes willing to accept a lower monetary transfer. Our result differs, however, from those in the above-mentioned literature on optimal financial structure, where empire-building tendencies do not necessarily lead to an empirical prediction of overinvestment (or overproduction) on average.5 Relatively to the sensitivity of output to cost, we find that, unlike what is predicted by managerial discretion models, increases in marginal cost may have no effect on output (as long as the reference output is kept fixed).

The paper is organized as follows. Section 2 describes the model and Sect. 3 analyzes the benchmark case of complete information. In Sect. 4, we derive the optimal incentive scheme and provide the general results. Section 5 presents an illustration for quadratic social value functions and an uniform distribution over types (with an exogenous reference output and with an endogenous reference output). Finally, Sect. 6 offers some concluding remarks.

2 The model

We consider a model of procurement in which the government (principal) delegates to a firm (agent) the provision of a public good. The manager of the firm cares about profit, \(t\), but also about the output level, \(q\). The manager’s utility function is:
$$\begin{aligned} U = t + \delta \left(q - q_{ref}\right)\!, \end{aligned}$$
where \(\delta \), the marginal utility of output to the manager, is a private information parameter, drawn according to a probability distribution over an interval \(\left[\underline{\delta },\overline{\delta }\right] \in \mathbb R _+\), with strictly positive density, \(f(\delta )>0\), and monotone hazard rates, \(\frac{d}{d \delta }\left[\frac{f(\delta )}{1-F(\delta )}\right]>0\) and \(\frac{d}{d \delta }\left[\frac{f(\delta )}{F(\delta )}\right]<0\) (where \(F(\delta )\) denotes the cumulative distribution function). The reference output, \(q_{ref} \ge 0\), is independent of \(\delta \) and may be interpreted as a “normal” output, such that producing this output neither implies a utility increase nor a utility penalty.6

The manager accepts to participate if and only if \(U(\delta ) \ge 0, \ \forall \delta \in \left[\underline{\delta },\overline{\delta }\right]\).

The production cost is \(\beta q\), where the marginal cost, \(\beta > 0\), is observable by the government. The social value of output is \(S(q)\), with marginal social value being strictly positive and decreasing, \(S^{\prime }(q) > 0\) and \(S^{\prime \prime }(q) < 0\), for any \(q \in \left[0,\overline{q}\right)\). We also set \(S(0) = 0\) and \(S^{\prime }(\overline{q}) = 0\) (where \(\overline{q}\) can be interpreted as fully covering the needs of the population).

The government finances public good provision using a distortionary mechanism (taxes, for example), so that the social cost of raising one unit of money is \(1+\lambda \), with \(\lambda >0\). The welfare of consumers is the difference between the social value of the public good and the cost of financing its provision, \(S(q)-(1+\lambda )(t+\beta q)\). In the model of Laffont and Tirole (1986, 1993) and, more generally, in the regulation literature, the government is assumed to maximize the sum of the consumers’ welfare with the utility of the firm. With the incorporation of an empire-building tendency, it becomes natural to consider two possibilities:
  1. (i)

    social welfare, \(W(\delta )\), is the sum of the consumer’s welfare with the utility of the firm (this case corresponds to setting \(k=0\), below);

     
  2. (ii)

    social welfare, \(W(\delta )\), is the sum of the consumer’s welfare with the profit of the firm, while the empire-building component of the manager’s utility is excluded (corresponds to setting \(k=1\), below).7

     
The problem of the government (maximization of expected social welfare) is:
$$\begin{aligned} \max _{q,t}\int _{\underline{\delta }}^{\overline{\delta }} \left[ S(q)-(1+\lambda )(t+\beta q)+U-k\delta (q - q_{ref}) \right] f(\delta ) \ d\delta \end{aligned}$$
subject to
$$\begin{aligned} U\ge 0. \end{aligned}$$

3 The case of complete information

As a benchmark, we study the case in which there is no asymmetry of information between the government and the firm (the government is able to observe \(\delta \)).

The problem of the government can be written as:
$$\begin{aligned} \max _{q,U}\left\{ S(q)-\left( 1+\lambda \right) \beta q-\lambda U + \left( 1+\lambda -k \right) \delta \left(q - q_{ref}\right)\right\} \end{aligned}$$
subject to
$$\begin{aligned} U \ge 0. \end{aligned}$$
We make the following assumptions, similar to those in Laffont and Tirole (1986), for the problem to be well-behaved.

 

Assumption 1

 
  1. (i)

    \(\lambda > 0\);

     
  1. (ii)

    \(\forall q \in \left[0,\overline{q}\right), \ S^{\prime \prime }(q) < 0\);

     
  2. (iii)

    \(S^{\prime }(0) > \left(1+\lambda \right)\left(\beta - \underline{\delta }\right)+ k\underline{\delta }\);

     
  3. (iv)

    \(\beta > \frac{1+\lambda -k}{1+\lambda } \overline{\delta } \).

     
 

Under Assumption 1: (i) guarantees that the participation constraint is binding, (ii) ensures that the second order condition is satisfied, (iii) ensures a positive output level, and (iv) is necessary for the equilibrium output to be lower than \(\overline{q}\).

With \(\lambda >0\), the objective function is strictly decreasing in \(U\), therefore the participation constraint is binding. We can replace \(U=0\) in the objective function and then solve for \(q\). The corresponding first-order condition is:
$$\begin{aligned} S^{\prime }(q)=(1+\lambda )\left( \beta -\delta \right) +k\delta . \end{aligned}$$
(1)
And the second order condition is: \(S^{\prime \prime }(q) < 0\) (always satisfied).
Since the participation constraint is binding: the utility level of the manager is \(U_c^*(\delta )=0\); and the net transfer is:
$$\begin{aligned} t^*_c(\delta ) = \delta \left[q_{ref} - q^*_{c}(\delta )\right]. \end{aligned}$$
Observe that the net monetary transfer from the government to the firm, \(t^*_c\), is positive if and only if the output, \(q_c^*\), is lower than the reference output, \(q_{ref}\) (because we have normalized the reservation utility to zero).8

The optimality condition (1) equates the marginal benefit of output to consumers, \(S^{\prime }(q)\), and the social marginal cost. Using this condition, the optimal output level, \(q_{c}^*\), can be obtained. It is clear that when the regulator cares about the non-monetary component of the manager’s utility (\(k=0\)), the level of output is higher and the net monetary transfer is lower than when it does not care (\(k=1\)). Observe also that the optimal output level is an increasing function of the preference for output parameter, even when the regulator does not care at all about the utility that the manager derives from output (\(k=1\)). This occurs because increasing the output implies an increase in the non-monetary component of the manager’s utility, decreasing, therefore, the monetary transfer which is necessary to induce participation.

We also find that the social welfare is increasing (resp. decreasing) with the intensity of the empire-building preference, \(\delta \), if and only if the optimal output, \(q_{c}^*\), is higher (resp. lower) than the reference output, \(q_{ref}\). Applying the Envelope Theorem:
$$\begin{aligned} \frac{dW^*_c}{d\delta } = \frac{\partial W^*_c}{\partial \delta } = (1+\lambda -k) \left(q_c^* - q_{ref}\right). \end{aligned}$$
Since \(q_c^*\) is increasing in \(\delta \), we conclude that social welfare, \(W^*_c\), is a convex function of \(\delta \):
$$\begin{aligned} \frac{d^{2} W^*_c}{d\delta ^2} > 0. \end{aligned}$$
In the following section, we determine the optimal procurement contract in the presence of an unobservable tendency for empire-building.

4 The optimal incentive scheme

In the asymmetric information case, the government does not know the manager’s marginal utility of output, \(\delta \). At the moment of contracting, the government only knows the prior probability distribution of \(\delta \).

Thanks to the Revelation Principle, we can restrict (without loss of generality) our attention to incentive compatible direct revelation mechanisms.9

The timing of the game is the following:
  1. (1)

    nature chooses the preference for output of the manager, \(\delta \in \left[\underline{\delta },\overline{\delta }\right]\);

     
  2. (2)

    the government offers a contract to the manager, \([q(\tilde{\delta }),t(\tilde{\delta })]\), specifying a level of output and a net payment that depend on the preference for output that is announced by the manager, \(\tilde{\delta }\);

     
  3. (3)

    the manager either accepts the contract, announces a type, \(\tilde{\delta }\), produces \(q(\tilde{\delta })\) and receives the net transfer \(t(\tilde{\delta })\), or rejects the contract and receives a null payoff.

     
In this section, we will consider that if the manager rejects the contract, the government receives a large negative payoff. The government is, therefore, forced to participate.10

4.1 The firms’s optimization problem

The government offers a contract to the manager such that he/she receives a net transfer \(t(\tilde{\delta })\) for producing the contracted output level \(q(\tilde{\delta })\), and has to pay an extreme penalty, \(t(\tilde{\delta }) = - \infty \), if the output level turns out to be different from \(q(\tilde{\delta })\).

For any \(\delta \in \left[ \underline{\delta },\overline{\delta } \right]\), truthful revelation must maximize the utility of the manager:
$$\begin{aligned} \delta \in arg\max _{\tilde{\delta } \in \left[\underline{\delta },\overline{\delta }\right]}\left\{ t(\tilde{\delta })+\delta \left[q(\tilde{\delta }) - q_{ref}\right]\right\} . \end{aligned}$$
(2)
Let \(V(\delta )\) be the manager’s value function (attainable utility as a function of \(\delta \)):
$$\begin{aligned} V(\delta ) = \max _{\tilde{\delta }\in \left[ \underline{\delta },\overline{\delta } \right]}\left\{ t(\tilde{\delta })+\delta \left[q(\tilde{\delta }) - q_{ref}\right]\right\} = t(\delta )+\delta \left[q(\delta ) - q_{ref}\right]. \end{aligned}$$
If announcing \(\tilde{\delta } \ne \delta \) is not optimal (2), then the output, transfer and value functions are almost everywhere differentiable (see Appendix A.1.1).
The first-order condition of (2) implies that the output and transfer functions vary in opposite directions:
$$\begin{aligned} t^{\prime }(\delta )+\delta q^{\prime }(\delta ) = 0. \end{aligned}$$
(3)
From the Envelope Theorem, we obtain the first-order incentive compatibility constraint:
$$\begin{aligned} V^{\prime }(\delta ) = q(\delta )- q_{ref}, \end{aligned}$$
(4)
which tells us that the derivative of the value function with respect to \(\delta \) is equal to the difference between the output level and the reference output.
The second-order incentive compatibility constraint is derived from the second-order condition of utility maximization (see Appendix A.1.2):11
$$\begin{aligned} q^{\prime }(\delta ) \ge 0. \end{aligned}$$
(5)
It implies that the output level is non-decreasing with the manager’s preference for output. Thus, from (3), the net transfer is non-increasing.

4.2 The government’s optimization problem

The objective of the government is to maximize expected social welfare, \(E_\delta \left[W(\delta )\right]\):
$$\begin{aligned}&\displaystyle {\max _{q(\delta ),V(\delta )}} \int _{\underline{\delta }}^{\overline{\delta }} \left\{ S\left[q(\delta )\right] -(1+\lambda ) \beta q(\delta )-\lambda V(\delta )\right.\nonumber \\&\qquad \qquad \left. + \left(1+\lambda -k \right)\delta \left[q(\delta ) - q_{ref}\right] \right\} f(\delta ) \ d\delta , \end{aligned}$$
(6)
subject to, \(\forall \delta \in \left[ \underline{\delta } , \overline{\delta } \right]\), the incentive conditions (4) and (5), and the participation constraint
$$\begin{aligned} V(\delta )\ge 0. \end{aligned}$$
Observe that the incentive compatibility conditions, (4) and (5), imply that \(V(\delta )\) is a convex function. Therefore, the participation constraint is binding over an interval, that we denote by \(\left[ \delta _0 , \delta _1 \right]\), with \(\underline{\delta } \le \delta _0 \le \delta _1 \le \overline{\delta }\).12

The following characterization is a strightforward consequence of the incentive compatibility conditions.

Proposition 1

(Level of output and sign of net transfer) The participation constraint, \(V(\delta ) \ge 0\), binds at \([\delta _0 , \delta _1]\), where \(\underline{\delta } \le \delta _0 \le \delta _1 \le \overline{\delta }\).
  1. (i)

    For \(\delta < \delta _0\), the output is lower than the reference output, \(q(\delta ) < q_{ref}\), and the monetary transfer is positive, \(t(\delta ) > 0\);

     
  2. (ii)

    For \(\delta _0 \le \delta \le \delta _1\), the output is equal to the reference output, \(q(\delta ) = q_{ref}\), and the monetary transfer is null, \(t(\delta ) = 0\);

     
  3. (iii)

    For \(\delta > \delta _1\), the output is higher than the reference output, \(q(\delta ) > q_{ref}\), and the monetary transfer is negative, \(t(\delta ) < 0\).

     
 

 

Proof

See Appendix A.2. \(\square \)

To further characterize the solution, we start by solving a relaxed problem in which condition (5) is ignored (Appendix A.3.1). Then, we check that the solution of this relaxed problem is the solution of the general problem (Appendix A.3.2).

Proposition 2

(Behavior of output and net transfer) The behavior of output and net transfer is such that:
  1. (i)

    For \(\delta < \delta _0\), the output is distorted upwards, \(q(\delta ) > q^*_c(\delta )\), and strictly increasing, \(q^{\prime }(\delta ) > 0\), and the monetary transfer is positive, \(t(\delta ) > 0\), and strictly decreasing, \(t^{\prime }(\delta ) < 0\);

     
  2. (ii)

    For \(\delta _0 \le \delta \le \delta _1\), the output is constant and equal to the reference output, \(q(\delta ) = q_{ref}\), and the monetary transfer is null, \(t(\delta ) = 0\);

     
  3. (iii)

    For \(\delta > \delta _1\), the output is distorted downwards, \(q(\delta ) < q^*_c(\delta )\), and strictly decreasing, \(q^{\prime }(\delta ) < 0\), and the monetary transfer is negative, \(t(\delta ) < 0\), and strictly increasing, \(t^{\prime }(\delta ) > 0\).

     
 

 

Proof

See Appendix A.3. \(\square \)

The distortion of output results, obviously, from the need to induce truth-telling. If the manager happens to have a relatively strong preference for high output, contracted output is below the perfect information level. This means that it would be preferable to “pay” more in output and less in money. But in this range the principal must avoid that the higher types mimic the lower. Paying more in output to type \(\delta \) would make it even more costly to prevent type \(\delta +\epsilon \) from mimicking type \(\delta \). Of course that this argument no longer holds at \(\delta =\overline{\delta }\), and this is why distortion disappears for the highest type (see Figs. 1, 2 and 3 in the next section).
https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0325-1/MediaObjects/712_2012_325_Fig1_HTML.gif
Fig. 1

Output and utility of the manager with \(k=0\), \(\alpha =3\), \(\lambda =0.5\), \(\beta =1.2\) and \(q_{ref}=1.5\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0325-1/MediaObjects/712_2012_325_Fig2_HTML.gif
Fig. 2

Output and utility of the manager with \(k=0\), \(\alpha =3\), \(\lambda =0.5\), \(\beta =1.2\) and \(q_{ref}=2.8\)

https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0325-1/MediaObjects/712_2012_325_Fig3_HTML.gif
Fig. 3

Output and utility of the manager with \(k=0\), \(\alpha =3\), \(\lambda =0.5\), \(\beta =1.2\) and \(q_{ref}=1.1\)

5 Characterizing the optimal contract

In this section, we characterize the optimal contract in the case of a quadratic social value function of output and a uniform distribution over types.

5.1 Exogenous reference output

In order to illustrate our general results, we now assume that the social value of output is \(S(q)=\alpha q-\frac{1}{2}q^{2}\) and that \(\delta \) is uniformly distributed on \(\left[0,1\right]\).

With complete information, the solution is:
$$\begin{aligned} q_c^*(\delta )&= \alpha -(1+\lambda )\beta + (1+\lambda -k) \delta , \\ t_c^*(\delta )&= \delta \left[ q_{ref}-q_c^*(\delta ) \right]. \end{aligned}$$
Assumption 1 (iii), which in this case is \(\alpha >(1+\lambda )\beta \), ensures that the complete information output is always positive.
We have bunching if and only if the reference output is not too large nor too small. Precisely, assuming that \(q_{ref} \in \left[ \alpha - (1+\lambda ) \beta - \lambda , \alpha - (1+\lambda )\beta + 1+2\lambda -k \right]\), the participation constraint binds in the interval \(\delta \in [\delta _0,\delta _1]\), where (see Appendix A.4.1):
$$\begin{aligned} \delta _0\! =\! \max \left\{ 0,\frac{q_{ref}\! -\! \alpha \!+\!(1\!+\!\lambda )\beta }{1\!+\!2\lambda \!-\!k}\right\} \ \text{ and} \ \delta _1 \!=\! \min \left\{ \frac{q_{ref}- \alpha + (1+\lambda )\beta + \lambda }{1\!+\!2\lambda -k},1\right\} .\nonumber \\ \end{aligned}$$
(7)
Without the above restriction on the value of the reference output, we obtain: \(\delta _0 > 1\) (if \(q_{ref} > \alpha - (1+\lambda )\beta + 1+2\lambda -k\)), meaning that the participation constraint only binds at \(\delta =1\); or \(\delta _1 < 0\) (if \(q_{ref} < \alpha - (1+\lambda ) \beta - \lambda \)), meaning that it only binds at \(\delta =0\).

The analytical expressions for output, net transfer and manager’s utility are presented in Appendices A.4.2 and A.4.3.

Qualitatively, the solution is characterized by the following propositions.

Proposition 3

(The intermediate reference output case) When \(q_{ref} \!\in \! [\alpha -(1\!+\!\lambda )\beta ,\alpha -(1+\lambda )\beta +1+\lambda -k]\), we find that (see Fig. 1):
  1. (i)

    for \(\delta \in [\delta _0,\delta _1]\), there is bunching, with \(q(\delta )=q_{ref}\), \(t(\delta ) = 0\) and \(V(\delta )=0\) (binding participation constraint);

     
  2. (ii)

    for \(\delta \in [0,\delta _0)\), output is distorted upwards, \(q(\delta )>q_{c}^*\), and the monetary transfer is positive, \(t(\delta )> 0\);

     
  3. (iii)

    for \(\delta \in (\delta _1,1]\), output is distorted downwards (\(q(\delta )<q_{c}^*\)) and the monetary transfer is negative, \(t(\delta )< 0\).

     
 

 

Proof

See Appendices A.4.2. and A.4.3. \(\square \)

When \(\delta \) increases from 0 to 1, the output level first increases (until \(\delta _0\)), being larger than the complete information output, then it is constant (between \(\delta _0\) and \(\delta _1\)), and then increases again, being lower than the complete information optimal output.

The following propositions describe what happens when the reference output \(q_{ref}\) is outside the interval defined in Proposition 3.

Proposition 4

(The large reference output case) When \(q_{ref} \in \left( \alpha - (1+\lambda )\beta + 1 + \lambda \right.\)\(\left.-k , \alpha - (1+\lambda )\beta + 1+2\lambda -k \right)\), there is bunching at the top (see Fig. 2):
  1. (i)

    for \(\delta \in [\delta _0,1]\), there is bunching, with \(q(\delta )=q_{ref}\), \(t(\delta ) = 0\) and \(V(\delta )=0\) (binding participation constraint);

     
  2. (ii)

    for \(\delta \in [0,\delta _0)\), output is distorted upwards, \(q(\delta )>q_{c}^*\), and the monetary transfer is positive, \(t(\delta )> 0\).

     
 

 

Proof

See Appendix A.4.2. \(\square \)

Proposition 5

(The small reference output case) When \(q_{ref} \in \left( \alpha - (1+\lambda ) \beta - \lambda , \alpha - (1+\lambda )\beta \right)\), there is bunching at the bottom (see Fig. 3):
  1. (i)

    for \(\delta \in \left[0,\delta _1\right]\), there is bunching, with \(q(\delta )=q_{ref}\), \(t(\delta ) = 0\) and \(V(\delta )=0\) (binding participation constraint);

     
  2. (ii)

    for \(\delta \in \left( \delta _1,1\right]\), output is distorted downwards, \(q(\delta )<q_{c}^*\), and the monetary transfer is negative, \(t(\delta )<0\).

     
 

 

Proof

See Appendix A.4.3. \(\square \)

 

The figures illustrate the case in which the government incorporates the non-monetary component of the manager’s utility in the welfare function (\(k=0\)). If the government does not care about the preference for empire-building (\(k=1\)), the interval over which the output is constant and the participation constraint of the manager is binding moves to the right and, outside this interval, the output and the manager’s utility are lower.

We can use our results to study how the output responds to changes in cost, for a given reference output level. An increase in marginal cost, \(\beta \), entails a reduction of the output level both when the manager’s marginal utility is low \(\left( \delta \in \left[ 0,\delta _0\right] \right)\) and when it is high (\(\delta \in \left[\delta _1,1\right]\)). However, for intermediate values (\(\delta \in \left[ \delta _0,\delta _1\right]\)), output does not change in response to (small) changes in cost.

Finally, we show that a sufficient condition for social welfare, \(W(\delta )\), to be positive for all \(\delta \in \left[ 0,1\right] \) (which implies that the regulator is happy to offer a contract to all possible types) is that the reference output is not larger than twice the optimal output in the absence of empire-building tendencies.

Proposition 6

(The government’s participation constraint) If \(q_{ref}\le 2\left[\alpha \!-\!(1\!+\!\lambda )\beta \right]\), then the social welfare is positive, \(W(\delta ) > 0\), for all \(\delta \in \left[0,1\right]\).

 

Proof

See Appendix A.4.4. \(\square \)

In Fig. 4, we observe that if the government cares about the managerial empire-building utility (\(k=0\)), the social welfare increases with \(\delta \). If the government disregards the managerial empire-building utility in the welfare function (i.e., when \(k\) increases), the social welfare increases in \(\delta \in \left[ 0,\delta _{0}\right)\), is constant in \(\delta \in \left[ \delta _{0},\delta _{1}\right]\), and increases again in \(\delta \in \left( \delta _{1},1\right]\).
https://static-content.springer.com/image/art%3A10.1007%2Fs00712-012-0325-1/MediaObjects/712_2012_325_Fig4_HTML.gif
Fig. 4

Social welfare for \(k=0\) versus \(k=1\), with \(\alpha =3\), \(\lambda =0.5\), \(\beta =1.2\), \(q_{ref}=1.5\)

5.2 Endogenous reference output

Until now, we considered that the reference output was exogenous. Suppose now that there is a large population of symmetric principals that are randomly matched with a large population of managers that differ in their tendency for empire-building. If we consider that the utility premium or penalty that is associated with the output level results from the comparison between the manager’s output and the average output across the population of managers (“keeping up with the Joneses”), then it is natural to assume that the reference output is the expected output of the population of firms, with respect to the managers’ types.13 Since this expected value depends on the reference output itself, the equilibrium value of the reference output is to be determined by stipulating that the reference output should equal the expected output, \(q_{ref}=E_{\delta }\left[ q(\delta )\right]\).

With an endogenous reference output, we cannot have a “large reference output” nor a “small reference output”. An “intermediate reference output” must emerge because, for the expected output to coincide with the reference output, if some managers produce less than the reference output, other managers must produce more than the reference output.

Proposition 7

(Endogenous reference output) The endogenous reference output level is such that only the “intermediate reference output case” occurs at equilibrium.

 

Proof

Suppose that we have an endogenously “large reference output”. Then, for \(\delta \in \left[ \underline{\delta }, \delta _0 \right]\), we have \(q(\delta ) < q_{ref}\), while, for \(\delta \in \left[ \delta _0, \overline{\delta } \right]\), we have \(q(\delta ) = q_{ref}\). Therefore, \(E_{\delta }\left[ q(\delta )\right] < q_{ref}\), which contradicts the hypothesis that \(q_{ref}\) is an endogenous reference output.

A similar argument applies to contradict the possibility of a “small reference output”. \(\square \)

With quadratic social value of output and a uniform distribution over types, whenever the reference output is assumed to take an intermediate value, the expected output is given by:
$$\begin{aligned} E_\delta \left[ q(\delta )\right]&= \int _0^{\delta _0} \left[\alpha -(1+\lambda )\beta +(1+2\lambda -k)\delta \right] d\delta +\int _{\delta _0}^{\delta _1} q_{ref} d\delta \nonumber \\&+\int _{\delta _1}^1 \left[\alpha - (1+\lambda )\beta +(1+2\lambda -k)\delta -\lambda \right] d\delta . \end{aligned}$$
(8)
Equating (8) to \(q_{ref}\), we obtain the unique equilibrium value of \(q_{ref}\) as:
$$\begin{aligned} q_{ref}^*= \alpha - (1+\lambda )\beta + \frac{1}{2}(1+\lambda -k). \end{aligned}$$
(9)
Observing that \(q_{ref}^* \in \left[\alpha -(1+\lambda )\beta ,\alpha +(1+\lambda )(1-\beta )-k\right]\), we validate, using Proposition 3, that the endogenous reference output takes an intermediate value.

The corresponding endogenous threshold values \(\delta _0\) and \(\delta _1\) are given by:

 
$$\begin{aligned} \delta _0= \max \left\{ 0,\frac{1}{2} - \frac{\lambda }{2 (1+2\lambda -k)}\right\} \quad \text{ and}\quad \delta _1 = \min \left\{ \frac{1}{2} + \frac{\lambda }{2 (1+2\lambda -k)} ,1\right\} . \nonumber \\ \end{aligned}$$
(10)
Making the reference output endogenous in this way leads us to reconsider what has been said at the end of the previous section. It is true that, for a given reference output, the output may not respond to “small” changes in cost when the managers’ types are intermediate. However, the (endogenous) reference output itself will adjust in response to changes in cost. Then, it makes sense to distinguish between short-run and long-run cost-elasticity of output. In the short-run, the reference output is given, for instance because the contracts have already been signed between the principals and the managers, and output can be insensitive to changes in costs or taxes for a group of projects. However, in the long run, as the old managers leave their jobs, new contracts are passed referring to the new reference output, and output converges to its new equilibrium value. It follows that the elasticity of output with respect to cost is going to be larger (in absolute value) in the long-run than in the short-run.

Proposition 8

(Cost-elasticity of output) After an increase in the production cost (\(\beta \)), the output of the managers with low and high preference for output (\(\delta \le \delta _0\) and \(\delta \ge \delta _1\)) decreases in the short-run and is not affected by the subsequent adjustment of the reference output. On the other hand, the output of the managers with an intermediate preference for output (\(\delta _0 \le \delta \le \delta _1\)) remains unchanged in the short-run and only decreases in the long-run.

 

Proof

See Appendix A.4.5. \(\square \)

Two remarks need to be made regarding the way the endogenous reference output was determined.14 First, the endogenous reference output is related to the output decisions of the principals. In a large world (with an infinite number of principals), the behaviour of an individual principal has no effect on the average output level, and thus it is consistent to assume that each principal takes \(q_{ref}\) as a given. However, this would not be the case in a small world (with a finite number of principals). Second, the endogenous reference output may be seen as being related to the reservation utility of the manager.15 If we assume that the outside option of the manager is to remain unemployed, then it is reasonable to assume that \(q_{ref}\) is independent of the manager’s tendency for empire-building. However, if the outside option corresponded to managing an alternative project, possibly producing a type-dependent output level, then \(q_{ref}\) should be considered to be type-dependent. We believe that these two lines of investigation are of interest and leave them for future work.

6 Concluding remarks and extensions

In this paper, we analyzed the optimal regulation of a firm when the manager’s empire-building preference (manager’s utility of output) is private information and the reservation utility is type-dependent. We showed that the optimal output function is such that output is distorted upward when the manager has a low preference for output, downward when he/she has a high preference for output and equals his/her reference output when he/she has an intermediate preference for output (in this case, the participation constraint is binding).

These results have implications for the cost-sensitivity of output. In the short-run, when the reference output can be considered as exogenous, the output of firms whose managers’ types are intermediate are not going to respond to small variations in their unit cost of production, while the output of firms whose managers’ types are either small or large will respond in the expected direction.16 The overall sensitivity of output to cost variations depends on the relative numerical importance of these three groups of firms, or, using the terminology of this model, on the length of the interval \(\left[\delta _0,\delta _1\right]\). In the long-run, the reference output adjusts progressively to its new equilibrium level, and so the cost-elasticity of output is larger.

For the sake of simplicity, we assumed throughout this paper that the government can observe the unit cost of production, and we simultaneously ignored the issue of moral hazard. Analyzing a model where both the marginal cost and the manager’s utility for output are private information would be fruitful. This will be the subject of future research.

Footnotes
1

Empirical evidence of deviations from profit-maximization was provided by Chetty and Saez (2005) and by Brown et al. (2007), who studied the response of corporations to the 2003 dividend tax cut in the USA.

 
2

The preference for high output is related to the preference for staff considered by Williamson (1974).

 
3

More precisely, the government offers an incentive compatible set of contracts, one for each possible type of manager.

 
4

See, for example, the works of Stulz (1990), Hart and Moore (1995), Harris and Raviv (1966), Li and Li (1996), Zwiebel (1996), Arya et al. (1999) and Kanniainen (2000).

 
5

Stulz (1990), Hart and Moore (1995) and Li and Li (1996) concluded that there is overinvestment when the free cash-flow from previous investments is higher than expected, and underinvestment when it is lower. Assuming that managers have private information about productivity, Harris and Raviv (1966) concluded that there is underinvestment when productivity is higher than expected and overinvestment when productivity it is lower.

 
6

In Sect. 5.2, we propose an endogenous determination of this reference level of output.

 
7

We also allow for intermediate cases, i.e., any \(k \in (0,1)\).

 
8

In some circumstances, managers may be willing to pay a fraction of the production costs, as in some volunteering activities or internships.

 
9

By the revelation principle (Myerson 1979), given a Bayesian Nash equilibrium of a game of incomplete information, there exists a direct mechanism that has an equivalent equilibrium where the players truthfully report their types. A direct-revelation mechanism is said to be incentive compatible if, when each individual is expecting the others to be truthful, then he/she has interest in being truthful.

 
10

In the light of Proposition , it will be clear that the government obtains a positive payoff for all types \(\delta \in \left[\underline{\delta },\overline{\delta }\right]\) if and only if the reference output, \(q_{ref}\), is below a certain threshold.

 
11

In Appendix A.1.3 we show that the first-order and second-order incentive compatibility constraints, (4) and (5), are equivalent to truth-telling (2).

 
12

This includes, for example, the case in which the participation constraint is only binding at one of the extremes of the interval (\(\delta _0 = \delta _1 = \underline{\delta }\) or \(\delta _0 = \delta _1 = \overline{\delta }\)).

 
13

This is in the spirit of Kőszegi and Rabin (2006), where the reference point of an agent corresponds to his/her rational expectation about a certain outcome.

 
14

We thank two anonymous referees for pointing out these two issues.

 
15

If we interpret the choice behavior of the managers as resulting from the objective function \(U = t + \delta q\), with participation requiring \(U \ge \delta q_{ref}\).

 
16

This looks like the result obtained by Sweezy (1939) in the very different framework of “the kinked oligopoly demand curve”.

 
17

See Basov (2005), pp. 124–126.

 
18

The participation constraint must be binding for some \(\delta \), otherwise the government could increase expected social welfare by reducing the transfers to all types of managers without violating the participation and the incentive constraints.

 

Acknowledgments

We are grateful to Inés Macho-Stadler, David Pérez-Castrillo and two anonymous referees for very useful comments and suggestions, and we thank participants in the 2009 SAET Conference and the 3rd Economic Theory Workshop in Vigo. Ana Pinto Borges and João Correia-da-Silva thank Fundação para a Ciência e Tecnologia and FEDER for financial support (research grants PTDC/EGE-ECO/114881/2009 and PTDC/EGE-ECO/111811/2009).

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© Springer-Verlag Wien 2012