L  ⎧⎪ f (e)l 

if q q*,

⎩⎪ 0 otherwise。

Being benevolent, the regulator chooses the auditing standard q*, the enforcement level e and the penalty l so as to maximize the social surplus from auditing quality net of the associated enforcement cost e, subject to the incentive-compatibility constraint of auditors。 Formally, the problem is to maximize the second-best welfare level W sb :

max

l,q*,e

W sb (q*) q*(1p)(I VL ) C(q*) e

(8)

subject to the incentive compatibility constraint:

F (q*) C(q*) F (q) C(q) f (e)l

for any q q* , (9)

where the auditor’s fee F on both sides of the inequality corresponds to the prescribed audit quality expected by investors, while the cost C depends on the quality level actually chosen by the auditor。

As in Becker (1968), for any positive enforcement level it is optimal to set the penalty at the

maximum  feasible level:14

l l*。 To obtain the optimal enforcement level, we use the incentive

compatibility constraint (9) with equality, since the optimal policy requires this constraint to be binding。 If not, the regulator could increase welfare by lowering enforcement e, for any given l*。 Next, notice that, in case of non-compliance, the auditor would optimally deviate to a zero quality level, since this would minimize his cost。 Finally, since the detection probability f(e) is monotonically increasing, it can be inverted to yield the optimal enforcement:

14 To see why, notice that if the penalty were set at a lower level, increasing it would enable the regulator to decrease enforcement e while keeping L constant。 The social surplus in the objective function would be unchanged but the enforcement cost would be lower, so that welfare would be higher。

e(q*) f 1(C(q*) / l*) 。 (10)

From the properties of the enforcement and audit technologies, it is immediate that the optimal enforcement e* is an increasing and convex function of the audit standard q*, and a decreasing function of the maximum penalty l*。15 The positive relationship between enforcement and audit standards highlights their complementarity: a more demanding audit standard invites non- compliance by auditors, so that it must be assisted by more intensive monitoring by the regulator。

Replacing the optimal enforcement (10) into the objective function, the problem of   maximizing

(8) can be rewritten as:

max

q*

Y q*(1p)(I VL ) C( p*) e(q*) , (11)

whose first-order condition is

(1p)(I VL ) C '(q*) e '(q*)

(12)

Under our hypotheses on the limiting behavior of the

C(q)

and

f (e)

functions, this optimality

condition identifies an interior solution q*  0 。 More importantly, it implies that:

Proposition 1 (Second-best audit standard)。 The optimal audit standard q* is smaller than the first-best standard q fb 。

The proof of this proposition (and subsequent ones) is in the Appendix。 The intuition for why the optimal standard is lower than the first-best level is simple: the regulator must take into account the

resource  cost  of  enforcing it。 It  is  interesting to  explore  how  the  optimal standard depending on the parameters of the economy:

q*  varies

Proposition 2 (Comparative statics)。 The optimal standard is decreasing in the fraction of successful companies p and increasing in the required investment I, in the efficiency of the auditing and in the efficiency of the enforcement technology。