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Note that the principal cannot hope to do better than the solution to (C), as the constraints in (C) must be satisfied by the final allocation. But if the principal offers the contract defined by the solution to (C), there is no state of nature in which the total wage bill net of the disutility of effort can be increased by changing the report or the effort level. Furthermore, by construction, (C) embodies the optimal insurance scheme (subject to the AIC constraint) between the supervisor and the agent. Thus, no side contract between the supervisor and the agent forms, and the principal can indeed guarantee himself the solution to (C). We call this fact the equivalence principle: the principal can restrict himself to contracts that do not induce the agent and the supervisor to collude, once the relevant coalition incentive constraints are introduced.21 We have thus obtained Proposition 2: When the supervisor and the agent can collude, the final allocation satisfies conditions (a) through (e) of lemma 1.

Let us now comment on the outcome under collusion. Lemma 1 (d) says that a distortion in effort is imposed only when the state of productivity is low and is not observed by the supervisor; (c) stems from (CIC 2) and the fact that the effort is the same in states 3 and 4. Thus, the total wage bill is the same in states 3 and 4. However, the supervisor'sand the agent's wages vary between these two states, in spite of risk aversion. The point is that in state 3, the agent

20. Let me check that the ignored constraints are also satisfied by the solution to (C).

From (e), we know that g(e2 - AO).

W- g(e3) W2Together with (d) and the convexity of g, this equality implies W3 - g(e3 + AO) W2 - g(e), so that the agent's incentive compatibility constraint in state 2 is satisfied. Furthermore, from (a), we have S3 + W3 - g(e3 + AO) S2 + W2 - g(2), so that the coalition incentive compatibility constraint in state 2 is also satisfied.

21. The coalition then does not form. Note that the allocation between the supervisor and the agent that results from (C) is optimal given the (conditional) wage bill and the agent's IC constraint; thus the solution to (C) could also be obtained by the principal by letting the supervisor and the agent collude. An extreme example occurs when the principal gives the supervisor the total (conditional) wage bill and lets the supervisor subcontract with an agent.

This content downloaded from 59.65.123.66 on Mon, 28 Oct 2013 01:51:38 AM All use subject to JSTOR Terms and Conditions 196 / JOURNAL OF LAW, ECONOMICS, AND ORGANIZATION 11:2, 1986 can claim that the state of productivity is low and the supervisor cannot provide evidence to the contrary. The agent must then be paid a high wage in order not to shirk. In state 4, optimal insurance calls for a lower wage for the agent than in state 3. But the supervisor must then obtain a higher wage in state 4 than in state 3, in order for the agent not to bribe the supervisor to conceal the state of productivity. This increase in the supervisor's wage represents a cost of obtaining the information.

The coalition incentive compatibility constraint in state 1-which induces the supervisor to reveal that the state of productivity is low-is not binding.

This is very naturalbecause in the low state of productivity, the agent prefers to have an excuse for generating a low profit. We interpret the result that (CIC

1) is not binding, while (CIC 2) is, as the idea that the supervisor naturallyacts as an advocate for the agent.

To make it less costly to induce the supervisor to reveal that productivity is high (state of nature 4), the principalwould want to give him a low salary(S3) is he claims he has observed nothing and the profit is high. However, the supervisor'swage in state 3 cannot be lower than that in state 2 (from[CIC 3']).

Thus S3 S2. This constraint in turn leads to a lower S2. This explains why the supervisor's wage in state 1 is higher than in state 2, despite the fact that the supervisor is quite willing to reveal the low state of productivity.

The two extreme cases of risk aversion for the supervisor lead to particularly simple results (see the appendix for a derivation). The supervisor is risk-neutralif V is linear; he is infinitely risk-averse if he cares only about his lowest possible wage.

Proposition3: If the supervisor is risk-neutral,the principalrealizes the same profit as in the collusion-free case. Up to a fixed cost So, everything is as if the principal monitored the agent directly and had the information structure {sl = O, S2 = 53 0, s4 = 0} = (that is, the supervisor's information structure).

Proposition 4: If the supervisor is infinitely risk-averse, the principal pays a fixed wage So to the supervisor; he then has the information structure {si = f, s2 = S3 = s4 = 0} to monitor the agent.

The interpretation of propositions 3 and 4 is as follows.

A risk-neutral supervisor can own (be a residual claimant for) the vertical structure without any loss in terms of insurance. Thus, the principal can sell the vertical structure to the supervisor at a price equal to the expected profit minus the supervisor's reservation wage. The hierarchy then boils down to a two-tier structure between the supervisor and the agent. But we know that there is no room for collusion in a two-tier structure. Thus, the outcome is the collusion-free one.

In the examples mentioned in part 2, the supervisor is farfrom being made the residual claimant for the vertical structure. This suggests that proposition 3 is of limited interest in many cases.

The case of infinite risk aversion is clearly extreme. The motivation for studying it is that it very starklyillustrates the nature of the supervisor-agent coalition. The supervisor receives a constant wage like in the collusion-free case; however, he deliberately ignores the informationhe receives about the good state of productivity. He reveals only the informationhe receives about the bad state of productivity. Again, this behavior amounts to acting as an advocate for the agent.

As mentioned above, we may wonder what would happen if the agent were able to produce verifiable reports himself. Let us assume away the supervisory function, and let us endow the agent with full information in all.states of nature (as earlier) and with verifiable informationabout the state of productivity in states 1 and 4 (thus, we transfer the supervisor's technology to the agent). Do we obtain the same outcome as with a supervisor (the outcome with a supervisor is the solution to [C], whether or not the agent can produce verifiable information in states 1 and 4)? The answer is provided in Proposition 5: Assume the agent can produce verifiable informationhimself.

Except in the case of supervisor's infinite risk aversion, there is still scope for a supervisory function.

The idea behind proposition 5 (the proof of which is straightforwardand therefore not provided) is simple. In the absence of a supervisor, the agent will release only information that is favorable to him, that is, only evidence about the bad state of nature. In particular, we have W3 = W4 (and e3 = e4).

Thus, the solution differs from (and is dominated by) the solution with a supervisor. This point is particularlyclear in the case of the supervisor's risk neutrality. The supervisor, who is then the owner of the vertical structure, prefers to be informed about the good state of productivity, informationhe can obtain only if he collects verifiable information himself.

## STRUCTURES

3.3. GENERAL

## COALITIONAL

In the previous section, we assumed that only the supervisor and the agent can form a coalition. There is no a priori reason to impose such a restriction.Consider first the outcome obtained in part 3.1, when no coalition is feasible, and introduce the possibility of a supervisor-principalcoalition. This coalition could induce the supervisor not to release the evidence in state 1 or in state 4. Clearly, there is no point in doing so in state 4 (W3 W4 and e3 = e4). It can also be shown that the main contract can be designed so that the supervisor reveals his signal in state 1.22 Thus, the collusion-free

outcome is immune to a coalition between the supervisor and the principal.

Similarly, it is easily seen that it is also immune to a coalition between the agent and the principal.

We now investigate what kind of allocation can be implemented by the principal when all types of bilateral coalitions are allowed. By allocation, we mean the final allocation that results from the parties' optimizing behavior given the main contract and the side contracts.

A final allocation is said to be coalition-proof if there exists no state of nature in which a coalition can increase its aggregate payoff by changing a variable (effort, report) that is controlled by a member of the coalition.

Proposition 6: The solution to (C) is coalition-proof.

Proposition 6 says that the main contract defined by program(C), in which a potential coalition between the supervisor and the agent is accounted for, is more generally coalition-proof. Thus, if the principal offers this contract, it is an equilibrium for the other parties to accept the contract and for all parties not to expect or suggest any side contract.23 The proof of proposition 6 (supplied in the appendix) starts by describing the mechanism more completely (in particular, it defines what happens if the observed {profit, report} pair is not one of the four equilibrium ones), and shows that the solution to (C) can indeed be implemented when all coalitions are allowed.

Proposition 6 shows that the principal need not worry about the effect that his potential coalitions with the agent and the supervisor have on the optimal contract for the supervisor-agent coalition. The corresponding coalition incentive compatibility constraints are not binding. In this sense, the relevant coalition is that between the supervisor and the agent. Thus, collusion naturally arises at the organization'snexus of informed parties, that is, within a group that can manipulate the informationobtained by the rest of the organization (here, by the principal).24 I have not showed that the equivalence principle holds (while I did so when only the supervisor-agent coalition is feasible). Hence, we may wonder whether, given an extensive form for the formationof coalitions, the principal can do better when he can form coalitions than when he cannot (given, or course, that the other two parties correctly anticipate these coalitions if the main

contract gives scope for them). To answer this question, one must posit an extensive form for the game of coalition formation. For instance,suppose that in the coalition formation game, the supervisor and the agent form their coalition last. Then the constraints (CIC 1) through (CIC 3) must be satisfied by the final allocation. The final allocation must also satisfy (SIR), (AIR), and (AIC) (this last property holds for any game of coalition formation). Thus, the principal cannot do better than the solution to (C). Together with proposition 6, this implies that the outcome of the game with general coalitions is the same as the one with only the supervisor-agent coalition.

**4. COALITIONS AND ORGANIZATIONAL BEHAVIOR**

4.1. WHAT DO SUPERVISORS DO?

Before deriving some implications for hierarchicalorganizations, it is useful to discuss the role of supervisors in the light of the previous model. I again assume away productive activities by the supervisor to focus on the supervisory function. Also, I assume that the supervisor and the agent do collude (the factors of collusion are discussed in the next two sections).

We saw that the supervisor's information is more costly to obtain under collusion. For example, in the extreme case in which the supervisor is not willing to bear any income risk, everything is as if the principal hired a collusion-free (honest) supervisor who could observe that the agent's environment is unfavorable, but would never observe that this environment is favorable:the supervisor acts as an advocate (see proposition 4). But even in this extreme case, the supervisor is useful in producing verifiable evidence in the unfavorable state of productivity.