Common use of Proof of Proposition EC Clause in Contracts

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)