Does the Appeals Process Reduce the Occurrence of Legal Errors?

Proof A.

The second-order condition for a maximum is:

As c′′(e)>0, ((1−q)σh+qz)>0, and (1−F(b˜(e)))α′′(e)+F(b˜(e))β′′(e)>0, a sufficient condition for the second-order condition to hold is:

with d2(1−F(b˜(e)))de2=−f′(b˜(e))p2s2(1−q)2(α′(e)+β′(e))2+f(b˜(e))ps(1−q)(α"(e)+β"(e)) (which is positive when f′(b˜(e)) is negative or zero). ■

Proof B.

Consider α(e)=β(e)=ae, 0<a<e, and a cost of effort c(e)=ce.\ The private benefit of crime is uniformly distributed on [0,B] ,\ and σ=z=0. Using eq.(1), e∗=2sDa(1−q)Bc. Next, by replacing e∗ in eq. (4), we derive eq. (4) with respect to q.\ This derivative is positive if asD>8(1−q)Bc, or B≪D.

Proof C.

The second-order condition for a maximum is:

A sufficient condition for the second-order condition to hold is:

The sufficient condition eq. (9) is more likely to hold if d2(1−F(b˜(e)))de2 is positive, that is, if f′(b˜(e)) is negative or zero and if α′(e)>β′(e). However, even if these two conditions hold, the sufficient condition may fail if hβ is sufficiently large compared to hα. Thus, whether the second-order condition eq. (8) holds now depends not only on the sign and magnitude of the term α′(e)−β′(e) and the shape of distribution f(.) but also on the magnitude of the cost difference for the two types of errors (hβ−hα).

Proof D.

According to the implicit function theorem, deCdq=−d2uC(e,q)dedqd2uC(e,q)de2, where d2uC(e,q)dedq and d2uC(e,q)de2 are calculated such that duC(e,q)de=0. According to the analysis of the second-order condition (8), d2uC(e,q)de2 is more likely to be negative if f′(b˜(e)) is negative or zero, α′(e) is higher than β′(e) and hβ is not too large compared to hα. Moreover, we have:

If σhα>z and σhβ>z, the first and second terms on the right-hand side of eq. (10) are negative. The third term is negative unless hβ is sufficiently large compared to hα. The fourth term is negative if α′(e)>β′(e), unless hβ is sufficiently large compared to hα. The last term is negative if f′(b˜(e)) is negative or zero, unless hβ is sufficiently large compared to hα.

Proof E.

[E] The second-order condition is:

with d2(1−F(b˜I(e)))de2=−f′(b˜I(e))p2s2(1−q)2(α′(e))2+f(b˜I(e))ps(1−q)α"(e) (which is positive when f′(b˜I(e)) is negative or zero). Thus, a sufficient condition for equation eq. (11) to hold is that f′(b˜I(e)) is negative or zero.

Proof F.

According to the implicit function theorem, deIdq=−d2uI(e,q)dedqd2uI(e,q)de2, where d2uI(e,q)dedq and d2uI(e,q)de2 are calculated such that duI(e,q)de=0. According to the analysis of the second-order condition (11), d2uI(e,q)de2 is negative. Moreover, we have:

which is negative when σh>z and f′(b˜I(e))≤0 (recall that f′(b˜I(e)) is more likely to be negative when the second-order condition holds). Consequently, deIdq is negative (the trial court decision-maker free rides on the quality control function of the appeals process) when she is more concerned with legal errors than with her reputation and few individuals attribute high benefits to committing illegal acts.

Proof G.

The second-order condition is:

and a sufficient condition for it to hold is:

with d2(1−F(b˜DJ(e)))de2=−f′(b˜DJ(e))p2s2(α′(e)+(1−q)β′(e))2+f(b˜DJ(e))ps(α"(e)+(1−q)β"(e)) (which is positive when f′(b˜DJ(e)) is negative or zero). The first term on the left-hand side of sufficient condition (13) is positive if f′(b˜DJ(e)) is negative or zero, unless z is sufficiently large compared to σh (i.e., unless the reputation cost associated with a reversal is sufficiently high). The second term is positive if α′(e)>β′(e) , unless z is sufficiently large compared to σh. If f′(b˜DJ(e)) is negative or zero, α′(e) is higher than β′(e) and σh>z, then sufficient condition (13) holds.

Proof H.

We use the implicit function theorem to analyze when deDJdq is likely to be negative, that is, when the trial court decision-maker may free ride on the quality control function of an appeals process. According to the implicit function theorem, deDJdq=−d2uDJ(e,q)dedqd2uDJ(e,q)de2, where d2uDJ(e,q)dedq and d2uDJ(e,q)de2 are also calculated such that duDJ(e,q)de=0. According to the analysis of second-order condition (12), d2uDJ(e,q)de2 is more likely to be negative if f′(b˜DJ(e)) is negative or zero, α′(e) is higher than β′(e) and σh>z. Moreover, we have:

with d1−Fb˜DJ(e)dq<0 and d2(1−F(b˜DJ(e)))dedq>0 if f′(b˜DJ(e)) is negative of zero. Thus, d2uDJ(e,q)dedq is more likely to be negative when f′(b˜DJ(e)) is negative or zero and when α′(e)>β′(e), which is consistent with the second-order condition. Moreover, when σh>z, the second-order condition is also more likely to hold, and the first three terms on the right-hand side of eq. (14) are more likely to be negative. However, the last term on the right-hand side of eq. (14) is more likely to be positive.