Proof of Proposition EC Sample Clauses

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. 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)

Proof of Proposition EC. 4 u d u d u u EC.5. Proofs in Section EC.3

Proof of Proposition EC. 5 d u u u u d d u u d