Proof of Proposition EC Sample Clauses
The 'Proof of Proposition EC' clause requires a party to provide evidence or documentation supporting a specific proposition, referred to as EC, within the agreement. In practice, this may involve submitting data, reports, or other forms of verification to demonstrate that the conditions or assertions related to Proposition EC have been met. This clause ensures that claims or obligations tied to Proposition EC are substantiated, thereby promoting transparency and reducing the risk of disputes over unverified assertions.
Proof of Proposition EC. 5 d u u u u d d u u d
Proof of Proposition EC. 1 We have shown that Jd(w) and Ju(w) summarized at the beginning of Section EC.1 are upper bounds of the societal utility of any incentive compatible contracts starting from states d and u, respectively, under different conditions. Or, equivalently, they satisfy that Jd′ (w) ≥ −1, Ju′ (w) ≥ −1, and boundary conditions Jd(0) = vd and Ju(0) = vu, and that Φt defined in (57) (or equivalently (EC.22) with νt = 1) is non-positive almost surely. Hence, to prove that they are upper bounds of any contracts, we need to further verify that if Φt defined in (EC.22) is non-positive almost surely when νt = 0. Hence, following (EC.22), the following conditions rJd(Wt−) ≥ −µdJd(Wt−) + rWt−Jd′ (Wt−) + µd[qtWt− − (1 − qt)Ht]Jd′ (Wt−) + µd(qtJu(0)+(1 − qt)Ju(Wt− + Ht)), Wt− ≥ 0, and rJu(Wt−) ≥ −µ¯uJu(Wt−) + rWt−Ju′ (Wt−) + µ¯u[qtWt− + (1 − qt)Ht]Ju′ (Wt−) + µ¯u(qtJd(0)+(1 − qt)Jd(Wt− − Ht)), Wt ≥ 0, for any −Ht ≤ Wt− and qt ∈ [0, 1] imply that it is optimal to induce effort from the agent before contract termination. They are further equivalent to
Proof of Proposition EC. 4 u d u d u u EC.5. Proofs in Section EC.3
Proof of Proposition EC. 2 First, we could rearrange (EC.4)-(EC.6) as the fol- lowing (µ + r)Jm(w) = µ Jm µd + rw! , w ∈ [0, w¯ − β ] . (EC.25) d d d u µ m u d −cu + (rw + µuβu)1w<w¯ Jm′ (w) = (µu + r)Jm(w) − R − µuJm(w − βu) + h 2µ + r βui 1w=w¯ , w ∈ [βu, w¯m] , Jm(βu) − Jm(0) (EC.26) Jm(w) = Jm(0) + u w , w ∈ [0, βu] . u u βu Then, the corresponding differential equations for V m(w) and V m(w) are (µ + r)V m(w) = µ V m µd + rw! , w ∈ [0, w¯ − β ], (EC.27) d d d u µ m u d (rw + µuβu)1w<w¯ V m′ (w) = (µu + r)V m(w) + cu − R − µuV m(w − βu), w ∈ [βu, w¯m] , (EC.28)