Part I Theoretical Framework
4.3 Actor’s Interaction – Static Game
In the following sections, we analyze the optimal actions when the principal decides to perform the task herself, and then derive the optimal contract if the agent is hired and finally compare both options.
4.3.1 Optimal Effort Choice of Principal
If the principal does not hire the agent but has to decide whether to carry out the task herself or not to perform it at all, her optimal action is described by
Proposition 1: If the principal decides not to hire an agent, effort conditional on performing the task is
* ( P)
P
P
H L
e k
= − . However, performing the task by herself is optimal for the principal if and only if
( )2
2
P
P P P
P
H L
L F u
k
+ − − ≥ . (1.1)
Proof: In the present case, the principal can directly set the effort level. Thus, the problem to be solved equals
( )
max 1 ( )
P
P P P P P P
e u =e H + −e L −C e , (1.2)
s.t. UP ≥UP. The first-order condition characterizes equilibrium effort given performing the task (note that this condition is also sufficient, provided the assumptions concerning effort costs):
(H−LP) k− peP =0. (1.3)
Plugging this into the right hand side of (1.2) gives (1.1).
Q.E.D.
It follows from the foregoing that more effort is chosen if variable costs kP are lower and if the difference between good and bad outcome, H−LP, is larger. The same is true for the probability of performing the task at all, which is also affected by fixed costs, as well as the principal's utility in the bad outcome. A lower LP indicates that a failure is very detrimental to the principal's interest. For example, the political costs a state has to face in case of a bad outcome can outweigh any potential
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gains a military victory provides (see Section 4.2.1). In this case, it will be better for the principal to completely abstain from the task. We briefly illustrate this aspect with the US involvement in the Vietnam War between 1965 and 1975. Against the background of the Cold War, the US intervened in the war between the communist North Vietnam and anti-communist South Vietnam in order to halt the so-called communist expansion. Even though the US was never defeated on the battlefield, the military involvement in the Vietnam War proved to be a political disaster for the US administration.
The domestic public opinion turned against the US involvement and eventually caused the withdrawal of US troops from Vietnam particularly because of the brutal acts of war and the high civilian and troop causalities which became known to the US public during the war (Hess, 2009:154ff). In retrospective, one could therefore argue that it would have had been better for the US not to intervene in the Vietnam War in the first place.
4.3.2 Optimal Contract if a Specific Agent is Hired
Now, assume that the principal decides to hire a specific agent. Here, we assume that this is done competitively. All bargaining power is on the principal's side, who can therefore determine the terms of the contract. However, due to the exogenous bounds on compensation (which cannot be negative), the agent might still receive a rent.
Before exploring the agent's actual effort choice and associated incentives, we look at the first-best effort (from the principal's perspective) given the agent is hired and take it as a benchmark. This level would be FB ( A)
A
A
H L
e k
= − and might or might not be higher than the effort the principal would
choose.49 More precisely, eAFB≥e*P if ( )
( )
A A
P P
H L k
H L k
− ≥
− . On the one hand, lower marginal effort costs (kP ≥kA) rather make it optimal to have eAFB ≥e*P. On the other hand, a bad outcome is worse for the principal if it had performed the task itself (LA ≥LP), which reduces the need to induce effort.
If the agent's effort was verifiable, it would be optimal to induce eAFB. In this case, offering a bonus b e( )=c (A eAFB) would provide sufficient incentives, and the fixed wage would be set to cover the agent's fixed costs and outside option, i.e. wA =FA+uA =FA. However, the agent's effort is not contractible, and the contingent payment b can only be based on the realized outcome. Hence, it might be optimal to actually induce an effort level lower than eFBA . The reason is that due to the agent's limited liability constraint (i.e. the principal cannot punish the agent in case of a bad
49 FB
eA would be the agent's effort if the principal were able to directly choose it itself. In this case, the principal solves
max (1 ) ( )
A
A A A A A
e e H+ −e L −C e , where the first order condition gives eAFB.
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outcome), the agent must receive a rent absent fixed costs. If FA is sufficiently small, the agent receives a net rent, and the principal faces a trade-off between effort maximization and rent extraction.50 This induces him to implement an effort level that is below the first-best, even if the latter were enforceable.
Deriving these results, note that due to b being restricted to non-negative values, it will obviously be optimal to set b L( A)=0. If high output is realized, the agent receives the payment b H( )≡b. Given an arbitrary payment b, the agent chooses an effort level to maximize its expected utility level. This is captured by the incentive compatibility (IC) constraint
( )
* argmax 1 0 ( )
A A A A A A
e = u = +w e b+ −e ⋅ −C e .
Equilibrium effort is independent of the fixed payment w, which is set at the beginning of the relationship and thus cannot be used to give incentives. Since the conditions to use the first order approach are satisfied, we have equilibrium effort
* A
A
e b
=k .
It follows from the foregoing that the agent’s equilibrium effort depends on b and kA. More effort is chosen if variable costs kA are lower and b is large. Conditional on hiring the agent, the principal sets b and w to
( )
maxuP =e HA( − + −b) 1 eA LA−w, subject to
(IC)
* A
A
e b
=k (IR)
2
2 0
A
A A A A
u =e b k− e −F + ≥w (LL)
50 In cases in which the agent is risk-neutral but cannot be punished, the optimal contract must satisfy a so-called limited liability constraint additional to incentive and participation constraints (Laffont and Martimort, 2001:149).
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Proposition 2: Assume the agent is hired.51 Then, equilibrium effort depends on FA which determines whether the agent has to get a rent. More precisely, there exist values FA and FA, with FA <FA, ii) Effort is at half its efficient level, * ( )
2
Thus, for each level of effort that the principal would like to induce exists a corresponding contract which ensures participation and incentive compatibility (Laffont and Martimort, 2001:151). For relatively high fixed costs, the principal does not have to give the agent an extra rent and thus implements first-best effort. However, this case can only be optimal if doing nothing is less worth than a failure, i.e., if uP <LA. This aspect is further explored in Proposition 5. For low fixed costs, the binding limited liability constraint allows the agent to extract some rent. Then, the principal faces a trade-off between surplus maximization and rent extraction, making it optimal to only implement half of first-best effort and paying the agent a rent. For intermediate levels of fixed costs, this trade-off is also present. Then, however, it is optimal to only reduce effort and keep the agent at its outside utility. Thus, effort falls with lower fixed costs, until it reaches half the efficient level, where it becomes optimal to grant the agent a rent. Furthermore, the principal's payoff decreases with FA as long as
In the following, we compare the principal's decision to perform the task herself with her decision to hire the agent. Generally, we can make the following statement:
Lemma: Performing the task herself rather than hiring the agent becomes relatively better for the principal for higher levels of L FP, A and kA, and for lower levels ofL FA, P and kP. The impact of a higher H is ambiguous and depends on the size of k kA, P,LA and LP.52
51 See Appendix for the proof to Proposition 2.
52 See Appendix for the proof to Lemma.
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Not surprisingly, it becomes less attractive to hire the agent if the latter ceteris paribus faces higher operating costs that it must be compensated for, and if a bad outcome after performing the task itself, LP, is less problematic. The next proposition captures the importance of the difference between LA and LP, i.e., the principal's payoff after a failure, dependent on whether the agent was hired or not. In this case, a larger difference always makes it relatively better to hire the agent.
Proposition 3: A larger difference between LA and LP reduces uPP−uPA.53
The fact that the domestic public opinion might be rather indifferent towards the death of PMSCs personnel compared to the death of regular soldiers leads to a problematic outcome: It not only reduces the implemented effort if the agent is hired, but also makes it more likely that the agent is actually hired. In case of international military interventions, this result connotes a problem for the country in which the intervention takes place (i.e. the host state). Even though the host state mainly cares about how good the job is done - about the level of the implemented effort - it is possible that the wrong player (from the host state's perspective), namely the PMSC instead of the principal's own forces, carries out the task. We further explore this issue in Section 4.5.
Before, we present a further proposition that describes another case of when hiring the agent is always better than performing the task oneself, namely when we assume FP ≥FA and the agent's fixed costs are sufficiently high.
Proposition 4: Assume FP ≥FA and (H L )2 2
A A
A
F k
≥ − . Then, it is always better for the principal to
hire the agent compared to performing the task herself.
Two aspects drive this result. First of all, high fixed costs alleviate the necessity to give the agent a rent, also making it optimal to induce first-best effort if the agent is hired. Since the principal's fixed costs are even higher, the need to compensate the agent for his fixed costs does drive the principal to perform the task herself. Furthermore, LA ≥LP reduces the principal's loss if the task fulfilment is not successful.
However, if the conditions of Proposition 4 are satisfied, it will be very likely that the task it not performed at all. Above, we described that it seems sensible to assume LA ≤uP, i.e. that a failure of the task gives the principal a lower payoff than if it had abstained from performing it from the beginning. Then, we have
53 See Appendix for the proof to Proposition 3.
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In all other cases, namely if
( )2
= = < − Then, hiring the agent is always associated with substantially less
effort than if the principal performs the task herself. However, the latter can be optimal, namely if
2 2