A Proof of Lemma 1

Maximizing ue(ee,ej,η) with respect to ee raises the following first-order condition for an interior solution:

This condition implies that if μ>ηej, the utility maximizing effort of E is strictly positive (i.e. eeBej,η>0). The second-order condition is:

From the implicit function theorem, we have:

With:

Thus:

This condition means that E’s effort decreases in J’s effort (i.e. ∂eeBej,η/∂ej≤0). Moreover, we have:

And:

These conditions mean that the magnitude of the decrease of E’s effort with respect to J’s effort increases in λ and η (i.e. ∂2eeBej,η/(∂ej∂λ)≤0 and ∂2eeBej,η/(∂ej∂η)<0). Finally, we have:

Thus, from the implicit function theorem:

This condition means that E’s effort increases with the complexity of the case at hand (∂eeBej,η/∂η≤0).

B Proof of Lemma 2

Let us define:

The complexity threshold ηˆB(ej) is found by solving (6) for ηˆ. This is equivalent to solving for ηˆ:

The right hand side of this equation is strictly positive since k > 0, which implies that the left hand side should also be strictly positive. This is the case only if μ>ηˆej, which implies (8).

Applying the implicit function theorem on (37), we have:

With:

And:

This implies:

Which means that the complexity threshold chosen by J is decreasing with respect to her own effort (i.e. ∂ηˆB(ej)/∂ej<0).

Given that ∂ηˆB(ej)/∂ej<0, J appoints an expert with a strictly positive probability if her effort is not too high, i.e. if:

Moreover, J does not appoint an expert with a strictly positive probability if her effort is not too low, i.e. if:

The two conditions (42) and (43) imply that ηˆB(ej)∈(0,1) only if the following condition holds:

It follows that if ej∈A, then ηˆB(ej)∈(0,1), and thus F(ηˆB(ej))∈(0,1).

C Proof of Lemma 3

We have:

D Proof of Lemma 4

Maximizing ujej,eeB(ej,η),ηˆ with respect to ej raises the following first-order condition:

From Assumption 1, the left hand side of (48) is strictly positive, implying that the right hand side should also be strictly positive, wich is true only if J’s effort is strictly positive (ejBηˆ>0).

Applying the implicit function theorem on (48), we have:

With:

According to (14) and Lemma 1, this expression is negative, thus implying from (49) that ∂ejBηˆ/∂ηˆ<0.

E Proof of Lemma 5

We have:

F Proof of Proposition 1

The proof follows from Lemma 1 and Lemma 2.

G Proof of Proposition 2

The proof follows from (24).

H Proof of Proposition 3

The policymaker chooses ex ante the socially optimal complexity threshold ηˆS according to:

By taking the derivative of wejB(ηˆ),eeB(ejB(ηˆ),η),ηˆ with respect to ηˆ, we obtain the following expression:

The social welfare is decreasing with respect to the complexity threshold if (54) is negative when evaluated for ηˆ=ηˆ∗.

Consider the first term to the right hand side of (54). We have:

Assume the policymaker chooses ex ante a complexity threshold ηˆ=ηˆ∗. This level of the complexity threshold cancels out ∂uj(ej,eeB(ej,η),ηˆ)/∂ηˆ, because it is the first-order condition resulting from (16). Moreover, from the first-order condition (48), we know that ∂uj(ejB(ηˆ),eeB(ejB(ηˆ),η),ηˆ)/∂ejB(ηˆ)=0. Thus, for ηˆS=ηˆ∗, ∂uj(ejB(ηˆ),eeB(ejB(ηˆ),η),ηˆ)/∂ηˆ=0.

Consider the second term to the right hand side of (54). We have:

From Assumption 1, ∂Q(ejB(ηˆ),eeB(ejB(ηˆ),η),ηˆ)/∂ejB(ηˆ)>0 and, from Lemma 4, we know that ∂ejB(ηˆ)/∂ηˆ<0. The first term to the right hand side of (56) is thus negative. Moreover, from Lemma 3, we have ∂Q(ej,eeB(ej,η),ηˆ)/∂ηˆ>0. Thus, (56) is negative if condition (26) is satisfied, and the second term to the right hand side of (54) is negative if (26) is satisfied (not satisfied) and Gj<1 (Gj>1).

Consider the third term to the right hand side of (54). We have:

From Lemma 5, we know that ∂cˉe(eeB(ej,η),ηˆ)/∂ηˆ>0 (the first term to the right hand side of (57) is positive), and that ∂cˉe(eeB(ejB(ηˆ),η),ηˆ)/∂eeB(ejB(ηˆ),η)>0. Finally, we know from Lemma 1 that ∂eeB(ejB(ηˆ),η)/∂ejB(ηˆ)<0 and from Lemma 4 that ∂ejB(ηˆ)/∂ηˆ<0. Thus, expression (57) is positive.