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proposition goes through: The entrepreneur never hires a consultant. However, because outside financing is too large, he is induced to exert a level of effort strictly lower than the first best. More formal proof is available upon request. □ Optimal Contract when the Revenue of the Pure Financier Is Not Constrained to Be Nondecreasing Using Lemma 1, the program of the entrepreneur becomes: max UuuERu - adERd)2 + \vuARu - adARd){auERu - adERd) + adERd 4 ,cceA,Avc Af - -(I -(Ave + Af)) (A36) s.t. 2-y(aARu - aARd)2 + 1(aARu - aARd)(aERu - adRd) + aAA)Rd > Avc (A37) (j((uERu - aERd)+ -(kARu - aARd)) (Ru - Rd -(aERu - aERd) -(aARu - adARd)) + Rd -(aARd + aERd)> Af (A38) (add, add, aA, aA)> 0 (A39) 1 - (ad + aA)> 0 (A40) 1 - (ad + aA)> 0 (A41) where У e {u, d}. The participation constraints are binding. If they were not, increasing AF and Avc would increase the entrepreneurs expected income without affecting the advisors incentives. Replace then AF and Avc in the objective function. The program is written miax -2-(aARu - aARd)2 + (Ru - Rd- adRd)+ 1(aARu - aARd)} ad ,aA, - 2b (addRu - adRd)2 + Rd - I (A42) s.t. aA > 0; ad > 0, 1 - (aA + ad )> 0; 1 - (aA + )>0 (A43) where У e {u,d}. Consider first not taking into account the feasibility constraints, and define X = aFRu - adRd and Y = aARu - aARd. The objective function is concave since the Hessian is negative semidefinitive. First-order conditions of the maximization of the objective function give X = Y = Ru - Rd. It is straightforward to see that if feasible, this solution corresponds to the irst-best levels ofefort being exerted. Replacing XandY by their value, and using the fact that aE + aA < 1, it follows that this solution is feasible if and only if 2(Ru - Rd) + aERd + aARd < Ru. (A44) Since the smallest possible value for aEd and aAd is 0, it follows that irst-best levels of effort can be implemented if and only if Ru < 2Rd. If Ru> 2Rd, one must write down the Lagrangian L of the program, including all the feasibility constraints described above: L = - 2g (aARu - aARd) + (Ru - Rd) b (aERu - aERd)+1 (aARu - aARd) - 2b (aERu - aERd)2 + l1aERu + 2aERd + l3aARu + haARd +l5(Ru -(aERu + aARu)) + l6(Rd -(aERd + aARd)) (A45) Straight application of the theorem of Kuhn-Tucker and tedious algebra give the following solution: aE* Rd = aA* Rd = 0 aE* Ru = gRd +b+Rbu-Rd), (A46) u * ru = bRd+g(Ru-Rd) A g+b . To conclude, note that a*>0 in both cases. Also, (PCVC) binding implies that AVC>0 under the optimal contract: The results of Proposition 2 still hold. The maximal amount of outside financing is given by A*C + A% Replacing the parameters of the contract by their optimal value gives the following: if Ru < 2Rd, A*C + A* < Rd - 2L(Ru - Rd)2; (A47) C F 2g , d ** ** (Ru - 2Rd)[3gbRu + 2gRd(g - 2b)l if Ru>2Rd, A*C + A* <(--+v----- VC 2b(g + b)2 + Ri1 - f). (A48) If the entrepreneur has to raise an amount ofoutside capital larger than the values deined above, the previously deined optimal contract cannot hold anymore and the value of the project decreases, which corresponds to the results ofPropo-sition 3. The main diferences with the case where the revenue of the inancier is nondecreasing are that (1) eforts are higher and (2) the inancier needs to invest a strictly positive amount of capital (AF*> 0) while her contribution is neutral when her revenue is nondecreasing. □ Proof of Proposition 2: The program to be solved is the same as in the previous section, except that equation (17) must be added to the program. Note that the limited liability constraint represented in equation (A40) becomes redundant, as it is automatically satisied when equation (17) holds. The new Lagrangian is the following: L = - 2- (aARu - aARd)2 + (Ru - Rd) (adR - adR) + - (aAR - aAARa) (adRu - adRd)2 + XR + + haARu + /UaARd + X5(Ru - Rd -(adRu - adRd)-(aARu - aARd)) + X6(Rd -(adRd + aARd)) (A49) Again, straight application of the theorem of Kuhn-Tucker gives (A50) Replace ad, aA, ad and aA in (PCF) and (PCvc) to obtain R- adRd adARd u n (Ru - Rd)2 - + )2 A51) A52) Note that the solutions presented in equation (A50) imply that a*>0 and that the minimum value of A\/C is strictly positive, which concludes the proof of Proposition 2. □ Proof of Proposition 3: Define -m+Rd- (a53) and use equations (A51) and (A52) to state that under the optimal contract Af +avc< i n. If I< In, the project can entirely be financed by outside capital and the entrepreneurs participation is useless. In that case, the value of the project is v/ = (-2 + b + f-) (Ru - Rd)2 + Rd - I. (A54) 2-b(- + -) 1 2 3 [ 4 ] 5 6 |