and Whinston (1986) and generalized by Proposition 4 in DGH (1997, p. 761), establishes the Pareto efficiency of truthful Nash equilibria, i.e. equilibria in which every principal offers a truthful payment schedule relative to some utility level (essentially meaning that principals will give the compensating variation to the agent, provided that such paymentis feasible). It will be helpful to go through the argument that underlies its proof<sup>30</sup> : Assume there were a policy vector a<sup>∗</sup> and a payment vector c<sup>∗</sup> that Pareto-dominated the truthful equilibrium pair of {a<sup>0</sup> , C<sup>0</sup> }(with respect to utility levels u<sup>0</sup> i<sup>).As principal i must be at least</sup> as well off as in equilibrium, and once payments reduce its utility, it must be the case that c<sup>∗</sup> i<sup>≤C</sup> i<sup>T(a∗,u0</sup> i<sup>),forthisisbyassumptionatruthfulschedule.Hence</sup> the agent cannot strictly prefer a<sup>∗</sup> and c<sup>∗</sup> to the equilibrium values, following a revealed preference logic: once a<sup>∗</sup> and {Ci<sup>T(a∗,u0</sup> i<sup>)}i∈Lwereavailable,yethe</sup> chose {a<sup>0</sup> , C<sup>0</sup> }, then he must not prefer the former to the latter. Given that c<sup>∗</sup> i<sup>≤C</sup> i<sup>T(a∗,u0</sup> i<sup>),itfollowsthathealsodoesnotprefera∗andc∗totheformer,</sup> for his utility is increasing in each principal’s payment. It must therefore be true that the strict inequality that is required to characterize Pareto-dominance is valid for some principal i: some of the principals must strictly prefer a<sup>∗</sup> and c<sup>∗</sup> i to the equilibrium values. This would mean, however, that such principal would not be optimizing in equilibrium: he could have offered c<sup>∗</sup> i<sup>inexchangefora∗,</sup> and the agent would have accepted, for he would still be receiving the truthful contributions Cj<sup>T(a∗,u0</sup> j<sup>)fromeveryotherprincipalj(andC</sup> j<sup>T(a∗,u0</sup> j<sup>)≥c∗</sup> j<sup>,as</sup> was seen above). This means that {a<sup>0</sup> , C<sup>0</sup> } was not an equilibrium, and this contradiction establishes the Pareto efficiency of truthful equilibria.

Let us now assume that the common agency game takes place in a production economy: the agent’s choice affects the principals’ production function, Ψi, and individual utilities depend on this function’s (scalar) output. The production technology also uses as an input the resources directly invested by each principal, which will be denoted ki. Therefore our setup may be summed up as follows: each principal has continuous preferences Ui[Ψi(a, ki), ci], which is increasing in Ψi and decreasing in ci. Ψi is increasing in ki and satisfies the Inada conditions; the agent has continuous preferences G[a, c; Ψ(a, k)], where Ψ is the vector of production outputs and k is the vector of ki. The main point to bear in mind is that the existence of a productive activity stresses the importance of time in the common agency game: resource availability after production is not the same as before it takes place. Under imperfect commitment, it is the availability of resources before production which determines the feasibility constraints for individual principals.

In order to capture this point, it is convenient to rewrite the common agency problem in a generalized framework, as a three-stage game, rather than the two stages in which it is usually modeled. More specifically, we shall consider that: at stage one, principals announce payment schedules Ci(a) ∈ Ci, just as usual. At the second stage, the agent chooses a policy vector a ∈ A, and the principals decide how much they will pay simultaneously with the implementation of the

> 30 The heuristic argument is provided by DGH (1997). In order to find a formal proof, however, one must refer to Dixit, Grossman and Helpman (1999).

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