In the model of this
Assume process (D, P ) and parameters (θ→,θ←) as given. Let the set S5= (y,w)∈?+3:for ∀y∈?+2,w≤wˆ , in which wˆ is taken as defined in (25). Given an initial state (y1,y2,w)∈?+3, there is a process C * that E-64d leads to a trajectory (Y1*,Y2*,W*) such that:(i)(Y1*,Y2*,W*)∈S5, almost surely with respect to t;(ii)C* is nondecreasing;(iii)one has∫0∞1(Y1*,Y2*,W*)∈int(S5)dCs*=0.Then, V(y,w)=Ey,w∫0τe−rs[(1−α)dCs*+dD1s+dD1s+θ→dP→s+θ←dP←s]D,P,θ.Remark 3.
Given V1 = V2 = 1, the principal will be indifferent to the possible specifications that the processes D1 and D2 may assume. This result has two implications. Firstly, this indifference allows assuming that these processes are Markovian, as proceeded in the previous subsection (from Assumption 1). In fact, by assuming a cost associated with the verification and enforceability related to the establishment of dDt from a history of realizations of Yt, and being the principal indifferent to that process, Gaia is natural to imagine that she accepts to minimize that cost through a Markovian structure (in which she needs to take account of only the current value of Yt). Secondly, the optimal contract will not be necessarily unique, i.e., the one derived in this paper, as resulting from that assumption, can be just one among others possible from different specifications (which not necessarily Markovian) for the processes D1 and D2.