which can thus be written as Ui(a, ci) and G(a, c): this is precisely how they appear in DGH (1997). It is then possible to directly apply their Proposition 4, and to conclude that truthful equilibria lead to Pareto-efficient allocations in generalized common agency games with perfect commitment by the principals with respect to announced payment schedules. In fact, all that production does within a perfect commitment framework is to define the set of possible equilibrium contributions, Ci.

Let us now consider the case in which there is no such perfect commitment mechanism. In this case, proceeding by backward induction in Γ, a subgameperfect equilibrium necessarily involves c<sup>d</sup> i<sup>=0:oncethepolicychosenbythe</sup> agent has been implemented, principals have no incentive whatsoever to make any further payment, for it would decrease their utilities. Any promise of a strictly positive c<sup>d</sup> i<sup>wouldnotbecredible,andtheagentwilltakeaccountof</sup> that by demanding that payments be made simultaneously to policy implementation. This means that all payments must be made before production, and that affects the amount of resources available for productive investment in equilibrium, since contributions and investments must both come from initial wealth, wi. Formally, we have C i0(wi) ≡ wi, which defines a new set of possible equilibrium contributions Ci0, and the utility functions must be now written as Ui[Ψi(a, wi − c<sup>s</sup> i<sup>)]andG[a, cs; Ψ(a, w −cs)].</sup> If wi < Ψi(ai, wi), which amounts only to assuming that the production technology available in this economy is more efficient than a simple storage technology, then Ci0 is a proper subset of Ci. A careful analysis of the aforementioned heuristic argument behind the proof of Proposition 4 of DGH (1997) shows that this suffices to break the validity of the proposition as a general result: it could be the case that there existed a<sup>∗</sup> and c<sup>∗</sup> such that Ci<sup>T(a∗,u0</sup> i<sup>), c∗</sup> i<sup>∈Ciand</sup> Ci<sup>T(a∗,u0</sup> i<sup>), c∗</sup> i<sup>∈/Ci0,forsomei∈L,sothattherevealedpreferenceargument</sup> is no longer valid. To put it another way, the absence of perfect commitment reduces the set of payments that are possible in equilibrium: there may be an allocation which Pareto-dominates the constrained equilibrium one and which is feasible in Ci, but which is not feasible in the restricted set Ci0.

Moreover, the above discussion shows that it is possible to think of total contributions, ci = c<sup>s</sup> i<sup>+ cd</sup> i<sup>,asbeingtherelevantdecisionvariables,foroneof</sup> its components will always be zero in equilibrium, regardless of which of the two cases is being analyzed - that is precisely what allows the equilibrium to be characterized just as it is in DGH (1997). One can see then that going from a perfect commitment setup to one in which such commitment is absent will change the utility function Ui[Ψi(a, wi − c<sup>s</sup> i<sup>) −cd</sup> i<sup>],andthismay alsochange the</sup> result of individual optimization. Therefore truthful equilibria may not lead to Pareto-efficient allocations in generalized common agency games without perfect commitment. Once the time-production binomial is taken into account, the absence of perfect commitment (such as might be permitted through a perfect credit market) changes the set of payments that are possible in equilibrium, so that the new set is a strict subset of the other.

As a final comment, we address the question of how important the full account of the consequences of a productive activity is in obtaining those re25