The role of audit thresholds in the misreporting of private information

Abstract

The accounting profession has faced considerable criticism in recent years for failing to effectively combat reporting manipulation. A particular point of contention is the use of audit thresholds. The tendency for auditors to suppress inconsistencies that are deemed immaterial has been viewed as an open invitation for abuse. This paper revisits the effectiveness of audits and the misreporting of private information in light of audit thresholds. The paper demonstrates that while audit thresholds may create incentives for misstatements, the predictability of such misstatements may actually serve to promote efficiency. In effect, an environment in which parties are expected to systematically bias their reports can bring the threat of audit consequences for further exaggeration to the forefront. Such a consideration also suggests that more relaxed audit thresholds (and the ensuing increase in equilibrium misstatements) may be condoned by report recipients and can actually lessen inefficiencies wrought by adverse selection.

Appendix

1.1 Proof of Proposition 1

The optimality of $\hat{c}(c) = c + \tau$ can be confirmed as follows. First, the outcome under any feasible proposed solution to (P) which entails another reporting function, denoted $(\tilde{c}(c),\tilde{t}(\hat{c}),\tilde{I}(\hat{c})),$ can be (at least weakly) improved upon by another contract such that $\hat{c}(c) = c + \tau.$ Further, this contract satisfies the constraints for a relaxed program, (P′), wherein the only (IC) constraints are for $\tilde{c} > c + \tau.$ The modified contract is $t(\hat{c}) = \tilde{t}(\tilde{c}(\hat{c} - \tau )) - \Upphi (\tilde{c}(\hat{c} - \tau ),\hat{c} - \tau )L$ and $I(\hat{c}) = \tilde{I}(\tilde{c}(\hat{c} - \tau ))$ for $\hat{c} \in [0 + \tau ,\bar{c} + \tau ]$ and $t(\hat{c}) = I(\hat{c}) = 0$ otherwise. In this case, with $\hat{c}(c) = c + \tau,$ $t(\hat{c}(c)) - I(\hat{c}(c))c - \Upphi (\hat{c}(c),c)L$ $= \tilde{t}(\tilde{c}(c)) - \tilde{I}(\tilde{c}(c))c - \Upphi (\tilde{c}(c),c)L.$ Since the right-hand side represents the (IR) under the original contract, (IR) is also satisfied under the modified contract. In this case, the relaxed program (IC) constraints require:

$$ \tilde{t}(\tilde{c}(c)) - \tilde{I}(\tilde{c}(c))c - \Upphi (\tilde{c}(c),c)L \ge t(\tilde{c}) - I(\tilde{c})c - \Upphi (\tilde{c},c)L,\quad\forall \tilde{c} > c + \tau . $$

(1)

Given the definition of $\Upphi (\tilde{c},c),$ the right-hand side of (1) is simply $t(\tilde{c}) - I(\tilde{c})c - \pi (\tilde{c} - c)L.$ By (IR), (1) is clearly satisfied for all $\tilde{c} > \bar{c} + \tau.$ And, by the definition of $t(\hat{c})$ under the modified contract, the right-hand side for the remaining constraints can be alternatively written as

$$ \tilde{t}(\tilde{c}(\tilde{c} - \tau )) - \tilde{I}(\tilde{c}(\tilde{c} - \tau ))c - \Upphi (\tilde{c}(\tilde{c} - \tau ),\tilde{c} - \tau )L - \pi (\tilde{c} - c)L . $$

(2)

By (IC) for the original contract,

$$ \tilde{t}(\tilde{c}(c)) - \tilde{I}(\tilde{c}(c))c - \Upphi (\tilde{c}(c),c)L \ge \tilde{t}(\tilde{c}(\tilde{c} - \tau )) - \tilde{I}(\tilde{c}(\tilde{c} - \tau ))c - \Upphi (\tilde{c}(\tilde{c} - \tau ),c)L . $$

(3)

Given $\tilde{c} \in (c + \tau ,\bar{c} + \tau ]$ (2) is no more than the right-hand side of (3). Hence, the modified contract satisfies the constraints to (P′). Furthermore, by construction the contract provides the same expected profit to the buyer:

$$ \int\limits_{0}^{{\bar{c}}} {[I(\hat{c}(c))X - } t(\hat{c}(c)) + \Upphi (\hat{c}(c),c)L]f(c)dc = \int\limits_{0}^{{\bar{c}}} {[\tilde{I}(\tilde{c}(c))X - } \tilde{t}(\tilde{c}(c)) + \Upphi (\tilde{c}(c),c)L]f(c)dc . $$

(4)

Because the solution is feasible in the relaxed program (P′) and provides equivalent buyer surplus as confirmed in (4), the proof is completed by noting that the solution to (P′) with $\hat{c}(c) = c + \tau$ satisfies the constraints for (P) and thus represents the solution to (P). This aspect is verified in the proof of Proposition 2. From the optimal contract verified in the proof of Proposition 2, it is also apparent that $\hat{c}(c) = c + \tau$ is the unique solution for all c such that $I(\hat{c}(c)) = 1$ in that no other reporting function can replicate the equilibrium outcome and satisfy the relevant constraints.

1.2 Proof of Proposition 2

Clearly, the solution for L = 0 is as in the no audit case. The solution for any L > 0 is determined from the following two steps.

• (i) The investment cutoff characterization can be confirmed by contradiction. Say there exist reports c′ and c′′, where c′ < c′′ and I(c′′) = 1 while I(c′) = 0. In this case, (IR) in (P′) for c′′ stipulates t(c′′) ≥ c′′ − τ and (IC) requires that t(c) − I(c)(c − τ) ≥ t(c′′) − I(c′′)(c − τ) − $\Upphi (c^{\prime \prime} ,c - \tau )L$ for all $c < c^{\prime \prime },$ or alternatively, t(c) − I(c)(c − τ) ≥ t(c′′) − (c − τ) − $\pi (c^{\prime\prime } - c + \tau )L.$ Given this, a simple contract perturbation provides strictly higher expected profit to the buyer: I(c′) = 1 and t(c′) = πL(c′ − τ) + (1 − πL)t(c′′) and all other terms unchanged. Clearly, (IR) for c = c′ − τ is satisfied at this solution. The only (IC) requirement to check is t(c) − I(c)(c − τ) ≥ t(c′) − (c − τ) − $\Upphi (c^{\prime } ,c - \tau )L$ for all c < $c^{\prime },$ or alternatively, t(c) − I(c)(c − τ) ≥ t(c′) − (c − τ) − $\pi (c^{\prime } - c + \tau )L.$ By (IR) in the original contract, t(c′′) − (c − τ) − $\pi (c^{\prime \prime } - c + \tau )L$ ≥ πL(c′ − τ) + (1 − πL)t(c′′) − (c − τ) − $\pi (c^{\prime } - c + \tau )L.$ Because the left-hand side of this inequality is less than t(c) − I(c)(c − τ) for all c < $c'$ given (IC) for the original contract and the right-hand side term is t(c′) − (c − τ) − $\pi (c' - c + \tau )L,$ (IC) is also satisfied in the contract perturbation. The contradiction is completed by noting that the buyer’s expected profit increases by X − (πL(c′ − τ) + (1 − πL)t(c′′)) > 0 for c = c′ − τ. The strict inequality follows from (a) X ≥ t(c′′) was a necessary condition for I(c′′) = 1 to be optimal and thus X > (c′ − τ)) and (b) πL < 1, given the regularity conditions Λ ≤ X and π < 1/($\bar{c}$ + τ).

• (ii) For a given investment cutoff, c _P , consider a relaxed program (P′′) where $c \ge c_{P} - {\tau \mathord{\left/ {\vphantom {\tau {[1 - \pi L]}}} \right. \kern-\nulldelimiterspace} {[1 - \pi L]}}$ are subject only to (IR) and $c < c_{P} - {\tau \mathord{\left/ {\vphantom {\tau {[1 - \pi L]}}} \right. \kern-\nulldelimiterspace} {[1 - \pi L]}}$ are subject only to (IC) for $\tilde{c} = c_{P}.$ In this case, each constraint is binding (otherwise, the buyer’s expected profit could be improved by simply lowering the relevant payment): $t(\hat{c}) = 0$ for $\hat{c} > c_{P},$ $t(\hat{c}) = \hat{c} - \tau$ for $\hat{c} \in [c_{P} - {{[\tau \pi L]} \mathord{\left/ {\vphantom {{[\tau \pi L]} {[1 - \pi L]}}} \right. \kern-\nulldelimiterspace} {[1 - \pi L]}},c_{P} ],$ and $t(\hat{c}) = c_{P} - \tau - \pi (c_{P} - \hat{c} + \tau )L$ for $\hat{c} < c_{P} - \,{{[\tau \pi L]} \mathord{\left/ {\vphantom {{[\tau \pi L]} {[1 - \pi L]}}} \right. \kern-\nulldelimiterspace} {[1 - \pi L]}}.$ This represents the solution presented in the proposition. The optimality of these payments then follows from the fact that the contract also satisfies the constraints of (P′) and (P) and thus, for any L, also represents the solution to (P).

Finally, note that for any c _P , expected equilibrium payments are decreasing in L, while the equilibrium investment decision is unchanged. Thus, at the solution to (P), the buyer sets L = Λ. The solution outlined in the proposition follows.

1.3 Proof of Proposition 3

For any c _P , expected payments are decreasing in π, while the investment decision is unchanged. Thus, the owner’s surplus at the solution must be increasing in π. As for τ, for any c _P the buyer invests if and only if c ≤ c _P  − τ. In this case, for any τ′′ > τ′, denote a c _P under τ′ by c _P ′. Setting a c _P of c _P ′′ = c _P ′ + (τ′′ − τ′) under τ′′ ensures the equilibrium investment decision is unchanged. Furthermore, for this cutoff choice, expected equilibrium payments are lower under τ′′ than under τ′. Thus, the owner’s surplus at the solution must also be increasing in τ.

1.4 Proof of Proposition 4

The buyer’s surplus for a given c _P is as in $B(c_{P} ).$ Taking the derivative of $B(c_{P} )$ with respect to c _P yields the first order condition in the proposition. Finally, taking the second order condition yields:

$$ \frac{{d^{2} B(c_{P} )}}{{dc_{P}^{2} }} = - f(c_{P} - \tau ) - (1 - \pi \Uplambda )f(c_{P} - \tau /(1 - \pi \Uplambda )) + (X - (c_{P} - \tau ))f'(c_{P} - \tau ) $$

(5)

Given the first order condition, (5) is equivalent to:

$$ - f(c_{P} - \tau ) - (1 - \pi \Uplambda )f(c_{P} - \tau /(1 - \pi \Uplambda )) + \frac{{(1 - \pi \Uplambda )F\left( {c_{P} - \tau /(1 - \pi \Uplambda )} \right)}}{{f\left( {c_{P} - \tau } \right)}}f'(c_{P} - \tau ) $$

(6)

Clearly, if $f'\left( {c_{P} - \tau } \right) \le 0$ at the solution, the second order condition is satisfied. If $f'(c_{P} - \tau ) > 0,$ dropping the negative second term in (6) and simplifying yields:

$$ \begin{array} {ll}\frac{{d^{2} B(c_{P} )}}{{dc_{P}^{2} }} &< - f(c_{P} - \tau )\left[ {1 - \frac{{(1 - \pi \Uplambda )F\left( {c_{P} - \tau /(1 - \pi \Uplambda )} \right)f'(c_{P} - \tau )}}{{[f\left( {c_{P} - \tau } \right)]^{2} }}} \right] \\ &< - f(c_{P} - \tau )\left[ {1 - \frac{{F\left( {c_{P} - \tau } \right)f'(c_{P} - \tau )}}{{[f\left( {c_{P} - \tau } \right)]^{2} }}} \right]. \end{array} $$

(7)

By H′(c) > 0, the last term in (7) negative and, hence, the second order condition is satisfied.

1.5 Proof of Proposition 5

At the solution, investment is undertaken in equilibrium if and only if c ≤ c _P  − τ, where c _P solves $c_{P} - \tau + \frac{{(1 - \pi \Uplambda )F\left( {c_{P} - \tau /(1 - \pi \Uplambda )} \right)}}{{f\left( {c_{P} - \tau } \right)}} = X.$ Alternatively, investment is undertaken if and only if $c \le c',$ where $c'$ solves $c' + \frac{{(1 - \pi \Uplambda )F\left( {c' - [\tau \pi \Uplambda ]/[1 - \pi \Uplambda ]} \right)}}{{f\left( {c'} \right)}} = X.$ By the implicit function theorem,

$$ \frac{dc'}{d\pi } = \frac{{ - \Uplambda (1 - \pi \Uplambda )F\left( {c' - [\tau \pi \Uplambda ]/[1 - \pi \Uplambda ]} \right) - \tau \Uplambda f\left( {c' - [\tau \pi \Uplambda ]/[1 - \pi \Uplambda ]} \right)}}{{(1 - \pi \Uplambda )\left. {[{{d^{2} B(c_{P} )} \mathord{\left/ {\vphantom {{d^{2} B(c_{P} )} {dc_{P}^{2} ]}}} \right. \kern-\nulldelimiterspace} {dc_{P}^{2} ]}}} \right|_{{c_{P} = c' + \tau }} }} $$

(8)

In (8), the numerator is negative, and by the second order condition, the denominator too is negative. Thus, ${{dc'} \mathord{\left/ {\vphantom {{dc'} {d\pi }}} \right. \kern-\nulldelimiterspace} {d\pi }}$ is positive. Similarly, by the implicit function theorem,

$$ \frac{dc'}{d\tau } = \frac{{ - \pi \Uplambda f\left( {c' - [\tau \pi \Uplambda ]/[1 - \pi \Uplambda ]} \right)}}{{\left. {[{{d^{2} B(c_{P} )} \mathord{\left/ {\vphantom {{d^{2} B(c_{P} )} {dc_{P}^{2} ]}}} \right. \kern-\nulldelimiterspace} {dc_{P}^{2} ]}}} \right|_{{c_{P} = c' + \tau }} }} $$

(9)

Following the previous logic, (9) too is positive. The proof is complete by noting that $c' < X$ and, hence, higher c′ equates to greater efficiency.

1.6 Proof of Proposition 6

Using Proposition 4, the effective hurdle as a function of the threshold is:

$$ K(\tau ) = \frac{X + \Uplambda \pi (\tau )\tau }{2 - \Uplambda \pi (\tau )} $$

(10)

Given the regularity conditions $\Uplambda \le X$ and $\pi (\tau ) < {1 \mathord{\left/ {\vphantom {1 {[X + }}} \right. \kern-\nulldelimiterspace} {[X + }}\tau ],\,K(\tau ) < X$ for all τ. Thus, if $K^{\prime}(0) > 0,\,\tau^{*} > 0.$ This condition is $\frac{{\Uplambda [X\pi^{\prime}(0) + \pi (0)[2 - \Uplambda \pi (0)]]}}{{[2 - \Uplambda \pi (0)]^{2} }} > 0$ which is equivalent to that in the proposition. Since $\mathop {Lim}\limits_{\tau \to \infty } K(\tau ) = X/2 < K(0),$ when the condition for $\tau^{*} > 0$ is satisfied $\tau^{*}$ is sure to be interior. Because maximizing efficiency entails maximizing $K(\tau ),\,\tau^{*}$ satisfies $K^{\prime}(\tau^{*} ) = 0.$ Using (10), this is $\frac{{\Uplambda [\pi^{\prime}(\tau )[X + 2\tau ] + \pi (\tau )[2 - \Uplambda \pi (\tau )]]}}{{[2 - \Uplambda \pi (\tau )]^{2} }} = 0;$ simplifying yields the condition $\pi (\tau )[2 - \Uplambda \pi (\tau )] = - \pi^{\prime}(\tau )[X + 2\tau ]$ as in the proposition.