Delegation, externalities and organizational design

DELEGATION, EXTERNALITIES AND ORGANIZATIONAL DESIGN Axel Gautier Dimitri Paolini

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Delegation, Externalities and Organizational Design

November 9, 2007

Abstract

In a repeated interaction between a principal and two agents with inter-agents external-ities and asymmetric information, we show that optimal decentralization within the organi-zation is limited to the …rst period and across agents.

1

Introduction

In multi unit organizations, with externalities (the choices made by one unit a¤ect the pro…t of the others), two key factors will drive the task allocation problem: externalities and asymmetric information. In the absence of externalities, there is no need to coordinate agents’choices and pro…t is maximized in a fully decentralized structure where the agents have all the power. In the absence of asymmetric information, there is no reason to delegate and pro…t is maximized in a centralized organization where the principal keeps all the power. In an organization where there are both externalities and asymmetric information, bene…ts and costs are associated with delegation.

In a single period interaction, delegation is bene…cial because the decider has superior in-formation (Dessein, 2002). With repeated interaction, delegation has an additional bene…t: the principal can improve her knowledge of the agents’information by observing their past decisions (Gautier and Paolini, 2007). Delegation is a learning process: when the agent uses his private information to make better decision, the principal revises her beliefs by observing the agent’s choice and she can improve her decisions.

We model a two-period interaction between one principal supervising two agents (units). At each period, a project must be implemented in each unit. At time zero, the principal chooses the process of decision-making for the following periods. Projects are transferable control actions (Aghion et al., 2002): projects cannot be contracted out but control is contractible. Each agent has a piece of private information and he exerts an externality on the other. The question, in this context, is which decisions should be decentralized and when?

In line with Gautier and Paolini (2007), we show that optimal decentralization within the organization is limited to the …rst period. Being a learning process, delegation is unnecessary when the non-informed party has acquired the information. Delegation is also limited across agents. With correlated information, the principal acquire information that is partially redun-dant when she delegates to both agents. Hence, symmetric agents can be treated asymmetrically

Corresponding author. Tel.: +39 079 2017338; fax +39 079 2017338. E-mail address: dpaolini@uniss.it (D. Paolini).

y_{CREPP, HEC-ULg and CORE, Université Catholique de Louvain.} z_{DEIR, Università degli Studi di Sassari, and CRENoS}

within the organization. An agent receives control over the …rst project while the other does not. Necessary conditions for this asymmetric treatment of agents are repeated interaction and correlation between information.

2

The Model

There are three players, the principal and two agents, and two periods. Each period sees a project implemented in both units. Before the …rst period, the principal decides who will have the power to choose the project (dl_t) in periods t = 1; 2 in unit l = A; B.

Each agent l has private information represented by a state parameter l drawn out of

a set _{= f} 1; 2g with 1 < 2 and = 2 1. Each agent knows the true realization

of while the principal only knows the prior distribution. The distribution of ( A; B) over

is represented by a joint probability distribution fv11; v12; v21; v22g where vij = prob( A= i; B = j). Correlation can be measured by = jv11v22 v12v21j. To simplify, we assume

that: v11 = v22 = vii and v12 = v21 = vij: Then, =jvii 1₄j and it takes a value between 0

(independence) and 1₄ (perfect correlation).

The projects di¤er in just one dimension and there is a continuum of possible decisions: dl_t2 (0; +1):

Each unit is a pro…t center. Depending on l, there is one project maximizing pro…t in unit l. There are externalities between the two units. The agents maximize their unit’s pro…t, while the principal’s aim is to maximize total pro…t. Pro…ts in units l is:

l₌ l_dl 1 ( l dl₁)2 2 + l_dl 2 ( l dl₂)2 2 (d k 1+ dk2);

with l 6= k. lis maximized for dl_t= l+ l. Total pro…t A+ Bis maximized for dl_t= l+ l .

3

Decisions of the principal and the agents

Each decision dl_t is made by the person to whom the principal has allocated the right to decide. When the principal must decide on a project dl_t, her decision depends on the information she

has. Let us represent her information by a distribution of beliefs _ij over . The decision

dl_tthat maximizes total pro…t is:

dl_t= l+ E( l_{j ij}) ;

where E( l_{j ij}) denotes the expected value of l conditional on beliefs _ij.

When agent l must decide on a project dl_t, he must take into account that the principal will revise her beliefs after observing dl_t. This changes the agents’pro…t if the principal can make a decision after the agent i.e. only if dl₁ is delegated and dA₂ or dB₂ is not. If the principal cannot use the agent’s information, the agent chooses his preferred project dl_t= ( l+ l): In all other cases, the agents and the principal play a signaling game and we must search for equilibria. Usually, signaling games have multiple equilibria. We use Cho and Kreps (1987) intuitive criterion (IC) to re…ne the set of equilibria. We can establish that:

Proposition 1 Under delegation of dl₁, the only equilibrium that survives IC is the least costly separating (LCS) equilibrium.1

1

Proof. See appendix.

Proposition 1 implies that the agents disclose their private information by taking a state contingent decision when they receive control at period 1. Hence, if the principal delegates dl₁,

she learns l and improve her knowledge of k. Second period decisions are then based on a

more accurate information.

Corollary 2 If 2 2, the LCS equilibrium is dl₁( ) = l+ l.

4

Optimal organization

Given that there is an externality between the two units, any form of delegation has a cost for the principal because there is no coordination in the project choices. Nevertheless, delegation bene…ts the principal because (i) the decisions of the agents are based on better information than those of the principal and (ii) the principal improves her knowledge of the state parameter after observing delegated decisions. In line with that, delegation of second period decisions should never occur.2

Starting from a fully centralized organization, let us de…ne the marginal bene…t of delegating …rst period projects to the agents. Delegating one project amounts to a change of total pro…t equals to M b(1) = 2+1₂(1 + 8 2) 2: The marginal bene…t of delegating a second decision is M b(2) = 2+1₂(1 8 2) 2: Hence, if M b(1) > 0, the total pro…t increases if dl₁ is delegated to agent l. Likewise, if M b(2) > 0, both …rst period projects must be delegated to the agents. Clearly, M b(1) M b(2), that is the marginal bene…t of delegating a …rst project is higher than the marginal bene…t of delegating a second one because the information contained in a second signal is partially redundant (at least for > 0). Hence, if M b(1) > 0 > M b(2), the principal optimally delegates to only one agent.

Proposition 3 The optimal organization is: centralization for 2 ₁₊₈2 22, delegation of dl₁

to agent l for 2_{2 [1+8}2 22;

2 2

1 8 2] and delegation of dA₁ and dB₁ to A and B for 2

2 2

1 8 2.

The striking result of this paper is the optimality of limited delegation in a repeated context. Delegation is limited to the …rst period (Gautier and Paolini, 2007) and across agents. Agents can be treated asymmetrically within the organization. The principal selects one delegate that will be responsible for the production of the signal and keeps control over the decisions concerning the other agent.

The quality of the signal produced by the agent depends on the degree of correlation. M b(1) is increasing in : when the information are correlated the value of a unique signal is higher. Conversely, M b(2) decreases in : the informational content of the second signal decreases with the degree of correlation. Figure 1 illustrates the optimal organizational structure as a function of the correlation parameter. Asymmetric treatment of the agents is more likely when the agents have correlated information. Note also that in a one period model, there is no learning associated with delegation and agents will be treated symmetrically.

In this model, we have analyzed a dynamic task allocation problem with externalities and asymmetric information. An alternative mechanism could be that the agents communicate to

2_{It is obvious if} 2 2_{. If} 2_> 2_{, the LCS is such that d}l

1( ) al+ l and this makes the principal

weakly worse-o¤. In this case, delegation of a second period decision should be envisaged. As we shall see, if

2_> 2_{, the principal prefers to centralize all the decisions even if the LCS is d}l

1( ) = al+ l. Hence, delegation

the principal and the principal chooses the actions. Crawford and Sobel (1982) show that communication is not perfectly informative. In a message game both agents reveal their private

information to the principal who then takes all decisions if 4 2 2. Clearly there exists

a parameter set where communication fails and delegation is optimal. This con…rms Dessein (2002) who recognizes that "Delegation is typically a better instrument to use the local knowledge of the agent than communication."

-6 1/4 0 2 2 2 4 2 4 2 3 2 2 1 8 2 2 2 1+8 2 M b(1) < 0 M b(1) > M b(2) > 0 M b(1) > 0 > M b(2)

Figure 1: Optimal organization

A

Proof of proposition 1

In a two-players game, IC selects the Riley outcome if there are only two states and if the payo¤ function satis…es the single crossing condition. To prove proposition 1, we replicate the argument in this three-players game.

Suppose that agent k truly reveals his private information if he controls dk₁. Then, whatever the choice of dl₁ by agent l, he will not be able to change the principal’s belief about k. Hence, the game played by agent l and the principal is similar to a two-players game. In this game, the standard sorting condition is satis…ed; This is su¢ cient to kill all the pooling and the separating equilibria but the Riley outcome.

Suppose that agent k does not disclose his private information at period one either because he does not control dk₁ or because he plays a pooling equilibrium. In this case, disclosing the value of l changes the principal’s beliefs on both l and k. We must then identify the state i

in which the agent has no incentive to hide his private information (if it exists).

If agent l plays a pooling equilibrium, the second period decisions are: dk₂ = k+ 1+ 2

2 ,

k = A; B. If, in state i, the agent manages to signal his type, the second period decisions

change to: dl₂ = l+ i and dk2 = k+ vii i+ vij j , i; j = 1; 2. We must show that there

exists a state i 2 in which the agent has would prefer to inform the principal. Replacing

the decisions in the pro…t functions, we can show that such a state always exists. Consequently pooling equilibria are eliminated by IC.

If agent l plays a separating equilibrium, the second period decisions are: dl₂( i) = l+ i

and dk₂ = k+ vii i+ vij j . If, in state i, the agent deviates, he changes the second period

decisions to: dl₂= l+ j and dk2 = k+ vij i+ vii j , i; j = 1; 2. We must show that there

in the pro…t functions, we can show that indeed, deviating cannot be pro…table in both states. Hence, we have our standard sorting conditions and only the LCS survives IC.

Acknowledgement

We would like to thank Francis Bloch, Bernard Caillaud, Jacques Crémer, Tito Pietra, Jacques Thisse, and seminar participants in CRENoS, IRES and Liège for their comments. The second author thanks Italian MIUR for …nancial support.

References

[1] Aghion, P., Dewatripont, M., Rey, P. (2002) On Partial Contracting. European Economic Review 46(4-5): 745-753

[2] Cho, I. K., Kreps, D. (1987) Signaling Games and Stable Equilibria. Quarterly Journal of Economics 12(2): 179-221

[3] Crawford, V., Sobel, J. (1982) Strategic Information Transmission. Econometrica 50(6): 1431-1452

[4] Dessein, W. (2002) Authority and Communication in Organizations. Review of Economic Studies 69(4): 811-838

[5] Gautier, A., Paolini, D. (2007) Delegation and Information Revelation, Journal of Institu-tional and Theoretical Economics, forthcoming.

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