be replaced by

e = b (aERu - *ERd), (A12)

which states the irst part of Lemma 1.

Equivalently, let us show that when a = 0, the entrepreneur never chooses aA and aA such that

1 (aARu - aARd)<0. (A13)

By the same reasoning as before, when a = 0, the program solved by the entrepreneur is

max(Ru - Rd)\(auERu - adERd)- ± +(auERu - adRd)2 + Rd - I (A14)

s.t. aARd - aARu > 0, (A15)

aERu + aARu < Ru, (A16)

aERd + aARd < Rd (A17)

Because of the presence of the pure financier, the same solution can be attained whether equation (A15) is binding or not and is characterized by

auERu - adERd = Ru - Rd. (A18)

Given that a = 0, effort e is set at its first-best value, that is, e = (Ru - Rd).The value of the objective function is then

- [Ru - Rd]2 + Rd - I. (A19)

As a consequence, with no loss of generality, equation (A2) can be replaced by

a = 1 (aARu - aARd). (A20)

Proof of Proposition 1: The first step of the proof is to show that Lemma 1 still holds when one imposes AVC = 0 in the general program.The main difference with the case where AVC can be optimally chosen is that (PC) VC may not be binding.

Suppose first that e = 0. Effort a is given by equation (A20) and (PC)Fis binding. The program solved by the entrepreneur is written

maxi[Ru - Rd -(a\Ru - adARd)] + (a\Ru - adARd)+Rd - I - adARd (A21)

s.t. aERd - aERu > 0 (A22)

+-y(auARu - aARd)2 + aARd > 0 aERu + a\Ru < Ru, adRd + aAARd < Rd

(A23) (A24) (A25)

The optimal solution is to set aA = 0and aARu = 2(Ru - Rd). For the reasons mentioned in the proof of Lemma 1, this solution is feasible whether equation (A22) is binding or not.

Suppose next that a = 0. Effort e is given by equation (A12) and (PC)fis binding. (PC)vC is written

(aERu - aE,Rd)(auARu - aARd)+ aARd > 0.

(A26)

If aARu = aARd, the optimal solution of the program is given by equation (A18). Efort e is set at its irst-best level, given that a = 0.

If aARu < aARd (for instance, aARu = aARd - e, e >0) it is not possible anymore to induce the first-best level of effort e. Indeed, at the optimum, we have

auERu - adERd = Ru - Rd +

(A27)

which induces too large a level ofefort e compared to the irst best. The value of the objective function is strictly lower than in the case where aARu = aARd. Hence, Lemma 1 still holds when there is no inancial participation by the advisor.

The second step of the proof consists of solving the general program after replacing (IC)vC and (IC)e using the expressions in Lemma 1. Note that (PC)f is still binding and can also be replaced. After manipulations, the program to solve is the following:

max[b(aERu-aERd)+ i(aARu - a.ARd)][Ru - Rd -(auARu - a.ARd)]

- 2b(aERu - aERd)2 + Rd - I - aARd s.t. 2y(aARu - a.ARd)+b(aERu - adRd)(auARu - aAARd) + aARd > 0

auERu + auARu < Ru,

adERd + adARd < Rd,

(A28)

(A29) (A30)

(A3 )

where У e {u, d}. Note that equation (A29) representing (PC)vC cannot be binding if e > 0and a > 0. The constraint (PC) vC can only be binding if a = 0and aA = aA = 0, = 0, which corresponds to the case where the entrepreneur does not hire a con-

sultant. To establish Proposition 1, it will be demonstrated that the entrepreneur is strictly better off if (pc) VC is binding.

Setting adA = 0 is optimal since it lowers the expected outcome of the advisor, and increases the entrepreneurs proit without afecting the latters incentives to exert effort. Define x = aEru - aERd and y = aA ru. Equation (A29) states

- y2 +1xy > 0. (A32)

As X>0, it is automatically satisfied when y > 0, which implies that it is redon-dant compared to the feasibility constraint. The program solved by the entrepreneur is:

max -4;X2 +(\x + -y) (ru - rd - y)

s t y > 0

The objective function is concave if 2b > g and convex otherwise. The Lagrangian of the program is

l = -±-X2 + (\x +1 y] (ru - rd - y) + 1y. (A33)

2b Vb g /

The solutions must verify Вт

- = 0 x + (ru - rd - y) = 0 (A34)

BY = 0 bX + - (Ru - Rd - 2Y)=0 (A35)

l > 0, Y > 0, 1Y = 0

If l = 0, equations (A34) and (A35) imply Y = [(g - b)/(g - 2b)](Ru - Rd). Note, however, that this solution is not feasible if 2b> g (since Y must be positive). In that case, we must have Y = 0and X = Ru - Rd.If2b < g, Y = [(g - b)/(g - 2b)](Ru - Rd) is feasible, but recall that in that case, the objective function is convex, which means that Y deined above is a minimum. The maximum is then also deined by Y = 0and X = Ru - Rd.To conclude, it is optimal for the entrepreneur to set Y = 0, that is, not to hire a consultant. The optimal level of effort of the entrepreneur is then e = b(Ru - Rd) = eFB. Note that if e = eFB, the expected income of the pure inancier is at most equal to Rd, which means that this solution holds forAF< Rd. In case the entrepreneur needs to borrow more than Rd (say, if he is wealth constrained), it can be shown (using the same methodology) that the result of the