Bertrand Chopard

Edwige Marion

Ludivine Roussey

Does the Appeals Process Lower the Occurrence of Legal Errors?

Keywords: appeals, legal errors, crime deterrence. J.E.L. Classi…cation: K14, K4

This paper challenges the commonly held idea that the appeals process lowers the occurrence of legal errors. We show that, even if incorporating the right to bring an appeal in criminal adjudication directly o¤ers the opportunity to correct mistakes made at trial, the …nal impact on the appeals process on the accuracy of judicial decisions is ambiguous because of its e¤ect through trial court decision-makers'e¤ort and crime deterrence. We …nd out that according to (i) whether trial court decision-makers are rather reputation-concerned or socially-motivated, (ii) the distribution of the population of potential o¤enders and (iii) the marginal e¤ect of decision-makers'e¤ort on the probability of wrongful conviction compared to wrongful acquittal, implementing an appeals process may spur decision-makers to reduce their e¤ort and thus be detrimental to the accuracy of legal decisions. We show that this result is true whether an appeal can be brought both after an acquittal or a conviction at trial, or the right to bring an appeal is limited by double jeopardy.

Introduction

Appeals allow for parties in lawsuits to seek reconsideration of their arguments. There is a commonly held idea that appeals mechanically improve the accuracy of judicial decisions by o¤ering the opportunity to correct legal errors at trial. [START_REF] Posner | Economic Analysis of Law[END_REF] …rst highlighted this argument in his Economic Analysis of Law and [START_REF] Shavell | The Appeals Process as a Means of Error Correction[END_REF] formalized it. According to the latter, the appeals process may be understood as a means of error correction from the moment that litigants possess private information about whether an error has been made at trial and that appeal courts can verify these errors with higher precision than trial courts. Under these quite reasonable assumptions, a litigant will not bring an appeal if there is no mistake at …rst instance.

Consequently, new errors cannot be made on appeal. On the contrary, appeals can (imperfectly) correct errors made at …rst instance. Shavell goes further: not only does the appeals process lower the occurence of legal errors but it does so at a lower cost than the alternative of enhancing the quality of the trial process. These arguments justify the existence of appeals process in most adjudication systems. For example the right to bring an appeal has been introduced in France in 2001 for serious crimes in order to correct errors potentially EconomiX, CNRS and University Paris Ouest Nanterre La Défense y Paris Center for Law and Economics, University Paris II Panthéon-Assas z LIRAES, University Paris Descartes, Sorbonne Paris Cité x We are deeply indebted to Tim Friehe, Andrew Hanssen, and Matteo Rizzolli, for their accurate suggestions on some crucial issues. We also wish to thank the participants in the First International Meeting on Law and Economics in Nanterre, and one anonymous referee, who provided very helpful comments. This article was previously circulated as Appeals Process, Judicial Errors, and Crime Deterrence. 1 made by Cours d'Assises. Since then, an appeal can be brought either after convictions or acquittals, by the defendant or the prosecutor. It is worthwhile noticing that, in some countries, the right to bring an appeal is more limited. In the United States, double jeopardy thus forbids the prosecution to appeal from an acquittal verdict.

The purpose of this paper is to reconsider the commonly held idea that appeals necessarily lower the occurence of legal errors. This idea is probably true if the appeals process does not impact the behavior of trial court decision-makers, but it is likely to be challenged otherwise. We thus explore the possible changes in the behavior of trial court decision-makers (judges, juries...) after the introduction of an appeals process.

We show that if the appeals process can directly reduce the occurrence of legal errors, it may also indirectly either increase or decrease the occurrence of legal errors, according to the incentives given to the trial court decision-maker. Intuitively, the appeals process may change the behavior of trial court decision-makers in two opposite ways. On the one hand, it plays as a sanction mechanism for trial court decision-makers who fear reversal (reputation-concerned decision-makers). As such, it may spur them to devote more e¤ort to obtain accurate decisions than they would in the absence of an appeals process. On the other hand, since the appeals process mechanically reduces the occurence of legal errors, it may substitute to the e¤ort of trial court decisionmakers who are concerned with the quality of …nal verdicts (socially-motivated decision-makers). In this case, it dissuades them from making e¤orts because the same …nal quality of judicial decisions can be attained with less e¤ort thanks to the appeals process. Finally, we point out that this behavioral trade-o¤ is not neutral to the …nal quality of court decisions and that it may qualify the widespread idea that appeals always improve the quality of judicial systems. The originality of our paper is not only to consider a representative trial court decision-maker that can be both career-concerned and socially-motivated but also to take into account the interaction between the trial court decision-maker's behavior and crime deterrence. To the best of our knowledge, how these aspects in ‡uence the occurence of legal errors when the right to bring an appeal is introduced in a criminal procedure has never been investigated before. And yet, this is of great importance because this leads to reconsider the idea that the appeals process is always a good way of improving the accuracy of judicial decisions. This paper suggests that whether this idea is true or not especially depends on the interaction between the characteristics of trial court decision-makers and potential o¤enders.

Our paper is closely related to the literature on the appeals process. We already mentioned Shavell's seminal paper on the appeals process as a means of error correction (1995). In this paper, the basic model supposes that the likelyhood of error at trial depends only on the resources devoted by the State to the trial process. In an extension to the basic model, Shavell interestingly envisages the case in which trial court judges' e¤orts may in ‡uence the chance of error at trial. However, he only considers that trial court judges are reputation-concerned, so that the appeals process necessarily spurs them to expend e¤ort in order to make more accurate decisions and to avoid reversal. [START_REF] Shavell | The Appeals Process and Adjudicator Incentives[END_REF] more precisely develops the idea that the threat of appeal leads judges to make decisions that more closely resemble the socially optimal decision. [START_REF] Levy | Careerist Judges[END_REF] also studies the e¤ect of the appeals process on judges'incentives. He shows that in the presence of an appeals process, careerist judges may seek to provoke appeal, so as to signal their abilities and improve their reputation. [START_REF] Iossa | Information Provision and Monitoring of the Decision-Maker in the Presence of an Appeal Process[END_REF] analyze the degree of opportunism of the decision-maker when his decision may be subject to an appellate review. They …nd out di¤erent results depending on whether information on the dispute is provided by the parties themselves or by an independent investigator. More precisely, they show that information provision by the parties generates more e¢ cient monitoring through appeals and less opportunism by the deicison-maker than information provision by the investigator. [START_REF] Daughety | Appealing Judgments[END_REF] do not focus on the …rst-instance decision-maker's behavior but on parties incentives to bring an appeal when the appeals court can draw inference from the fact that an appeal is brought. Finally, a forthcoming paper by Oytana studies the raison d'être of the appeals process when judges make their decisions on the basis of information gathered by biased experts. In this context, Oytana shows that the appeals process limits the adverse impact of biased experts in terms of legal errors.

The present paper is organized as follows. Section 2 presents the model of legal errors correction. In section 3, we extend the model to study the direct and the indirect e¤ects of the appeals process on the occurrence of legal errors. In section 4, we discuss the double jeopardy clause. Section 5 concludes.

The model

We consider the case for a risk neutral decision-maker (judge, juries, prosecutors) who has enough in ‡uence to reduce the probability of judicial errors at the …rst order trial. He chooses some e¤ort level, e > 0, at convex cost (e) in terms of disutility of e¤ort, time, or any other resources. By increasing his e¤ort, e, he may reduce the occurrence of legal errors. We denote p(e) the likelihood of relaxing a guilty defendant, with p 0 (e) < 0 and p 00 (e) >, and (e), the likelihood of wrongly convicting an innocent individual, with 0 (e) < 0 and 00 (e) > 0. We assumed that p(e) > (e). This assumption re ‡ects the fact that a high standard of evidence is usually required in criminal procedure, thus limiting wrongful conviction (RiZzolli and Saraceno, 2013). The decision-maker is paid a gross wage w. In addition, he may su¤er a private loss associated with his own mistakes h, where h represents the social harm due to judicial errors and refers to the decisionmaker's sensitivity to social harm. Finally, we consider a continuum of risk neutral individuals each de…ned by a private bene…t from harmful act, as in Polinsky and Shavell (1984). We denote the latter b and take it to be distributed on a …nite support V (0; B), with cumulative distribution function F (:). The corresponding density f (:) is continuously di¤erentiable and positive in the interval (0; B), and equal to zero outside this interval. Given e, an individual will commit an harmful act if and only if his bene…t, b, exceeds the expected sanction, t[1 p(e) (e)]S bNA (e), where the value bNA (e) will be referred to as the borderline type when it is impossible to bring an appeal (NA means "no appeal"). Thus, there is a given probability (the product of the probability of detection, t, with the probability of conviction, 1 p(e) (e)) that an act will be subject to a sanction, S. A higher probability of relaxing a guilty individual increases the payo¤ of committing an act whereas a higher probability of wrongly convicting an innocent person will decrease the payo¤ of abiding by the law (Garoupa and Rizzolli, 2012). Following [START_REF] Png | Optimal Subsidies and Damages in the Presence of Judicial Error[END_REF], we also assume that the e¤ects of judicial errors on crime deterrence are symmetric, meaning that the two probabilities of legal errors have the same impact on the potential o¤ender, that is on the returns from honesty and from harmful act [START_REF] Lando | Does Wrongful Conviction Lower Deterrence?[END_REF], Rizzolli and Stanca (2012)). 1In this setting, the decision-maker chooses e to solve: Di¤erentiating u N A (e) with respect to e leads to lemma 1 below (all proofs are reported in appendix).

Lemma 1 There exists a unique e¤ ort level at equilibrium e N A de…ned by:

0 (e N A ) = h p 0 (e N A )[1 F ( bNA (e N A ))] 0 (e N A )F ( bNA (e N A )) f ( bNA (e N A ))tS[p 0 (e N A ) + 0 (e N A )][p(e N A ) (e N A )] ! . (1)
The …rst order condition can be rewritten: We …rst explore the term B N A p . It represents the marginal bene…t due to the fact that, by increasing his e¤ort by one unit, the decision-maker reduces the probability of relaxing a guilty defendant by p 0 (e) units, and the probability of wrongly convicting an innocent individual by 0 (e) units. Thus, the decision-maker reduces his expected private loss in case of a legal error by p 0 (e)(1 F ( bNA (e))) + 0 (e)F ( bNA (e)) h units because losses associated to both types of legal errors are weighted by the probabilities that the defendant is either guilty or innocent.

Second, term B N A d captures the deterrence e¤ect. By increasing his e¤ort by one unit, the decisionmaker deters crime. Then, it simultaneously reduces the likelihood to be faced with a guilty individual by

@(1 F ( bNA (e))) @e = @F ( bNA (e))
@e units, and increases the probability to be faced with an innocent individual by @(F ( bNA (e))) @e units. As it is more likely to wrongly relax a guilty individual than to wrongly convict an innocent individual (p(e) > (e)), such an increase of e by one unit is pro…table. More precisely, it reduces the decision-maker's expected loss associated with legal errors by

@ (1 F ( bNA (e))) @e
p(e) h + @F ( bNA (e)) @e (e) h units.

The second order condition for a maximum is: 2 (p(e) (e)) h+2f ( bNA (e))tS(p 0 (e)+ 0 (e))(p 0 (e) 0 (e)) h 0:

d 2 u N A (e) de 2 = f 0 ( bNA (e))( t 2 S 2 )(p 0 (e)+ 0 (e))
(

Following [START_REF] Kaplow | A Note on the Optimal Use of Nonmonetary Sanctions[END_REF], it appears that the …rst term in (3) is the product of the decision-maker's net marginal bene…t per individual deterred and the rate at which the population density changes, which is of indeterminate sign. The word "net" means that an increase in e has a deterrence e¤ect. It leads to a reduction in the risk to be faced with a guilty individual, and thus to wrongly relax him with probability p(e). But it leads also to an increase in the probability to be faced with an innocent individual, and thus to wrongly convict him with probability (e) < p(e) by assumption. The second term in (3) re ‡ects that for each additional individual deterred, the decision-maker simultaneously looses the private bene…t of a one-unit increase in his e¤ort to reduce the risk of wrongly relaxing a guilty individual ( p 0 (e) h) and wins the marginal bene…t of a one-unit increase in his e¤ort to reduce the risk of wrongly convicting an innocent one ( 0 (e) h). As long as the decision-maker's e¤ort is deterrent (p 0 (e) + 0 (e) < 0), fewer individuals then remain undeterred, and the second term in (3) is either positive or negative according to the sign of p 0 (e) 0 (e) which is indeterminate.

To sum up, whether the SOC will hold depends on the sign and the magnitude of the term p 0 (e) 0 (e), and the shape of distribution f (:) weighted by the gap between the two probabilities of legal errors, p(e) (e).

In particular, if p 0 (e) 0 (e) is relatively low and negative, equation (3) fails unless f (:) is increasing at a su¢ cient rate in the relevant range. To illustrate consider the case of an uniform distribution: the …rst term in ( 3) is zero, and the e¤ort satisfying the FOC will be a maximum if and only if p 0 (e) 0 (e). In words, the decision-maker's e¤ort should rather reduce at the margin the risk of wrongly convicting an innocent individual insteand of reducing the risk of wrongly relaxing a guilty one for the SOC to hold. Finally, to ensure the existence of a maximum, we can rewrite equation (3) as follows:

Assumption 1:

"(b) < 2(p 0 (e) 0 (
An increase in either the social harm due to judicial errors, h; or the decision-maker's sensitivity to social harm of errors, ; implies an increase in decision-maker's e¤ ort.

An increase in the decision-maker's private loss associated with a legal error, h, contributes to rise both First, an increase in the penalty or the probability of detection reduces the probability of crime (by increasing bNA (e)). Since the probability of crime becomes lower, B N A p may either increase or decrease according to the sign and the magnitude of the di¤erence p 0 (e) 0 (e). On the one side, the decision-maker has a lower incentive to reduce the probability to wrongly relax a guilty individual simply because there is a lower probability to be faced with such an individual. On the other side, the opposite e¤ect is at stake when we consider the risk of wrongly convicting an innocent person. In this sense, the penalty and the probability of detection can be considered as either substitutes or complements for the decision-maker's e¤ort.

B N A
Second, raising the penalty or the probability of detection contributes also to either increase or decrease

B N A d .
Indeed, notice that:

B N A d = @F ( bNA (e)) @e (p(e) (e)) h = (p(e) (e)) hf ( bNA (e)) d bNA (e) de > 0.
It appears that the penalty and the probability of detection modify the magnitude of the positive e¤ect of the decision-maker's e¤ort on the decrease in the probability of crime through the two terms d bNA (e) de and f ( bNA (e)). First, d bNA (e) de is increasing with the penalty and the probability of detection. Thus, an increase in one of these two parameters strengthens the positive e¤ect of the decision-maker's e¤ort on crime deterrence.

Second, f ( bNA (e)) may either increase or decrease with the penalty or the probability of detection according to the rate at which the population density changes. Finally, remark that both these e¤ects are weighted by the gap between the two probabilities of legal errors, p(e) (e). Therefore, the lower is this gap, the lower is the e¤ect of the penalty and the probability of detection on the extent to which the decision-maker's e¤ort is pro…table through criminal decision making.

Put together, condition (4) resumes all these e¤ects and indicates whether the penalty and the probability of detection can be viewed as complements or substitutes for the decision-maker's e¤ort to reduce the occurrence of legal errors. Below, we study the interaction between the decision-maker's choice of e¤ort in reducing the risk of legal errors and the correction of legal errors by appeals courts.

3 Appeals process as a means of legal errors correction

Following [START_REF] Shavell | The Appeals Process as a Means of Error Correction[END_REF], it is assumed that an appeal is brought if and only if an error has been made at trial.

It supposes that litigants know whether an error has been made at trial and that the appeals court makes judicial decisions with higher precision than the trial court 2 . As Shavell, we abstract from the fact that appeals courts could draw inferences from the fact that litigants bring appeals and consequently systematically reverse trial court decisions. Indeed, using such inferences is …rst not what happens in practice: appeals courts cannot use this knowledge to make their decisions ; they fully reconsider cases. Second it is not socially desirable that appeals courts use inferences from the fact that appeals are brought. As explained by [START_REF] Shavell | The Appeals Process as a Means of Error Correction[END_REF] "if appeals courts were to reverse all decisions, on the basis of their inference that all appellants are the victims of error, then disappointed litigants who are not the victims of error clearly would have an incentive to bring appeals, for they could obtain sure reversal ". Consequently, the utility of the appeals process in error correction would be diminished. For these reasons, we assume that the probability of reversal by the appeals court, q 2 (0; 1) is exogenous and common knowledge. For sketch of simplicity, this probability does not depend on the type of legal error.

Under this new setting, the threshold value of the bene…t of crime equals:

bA (e) = tS[1 (1 q)( (e) + p(e))]:

Notice that bA (e) > bNA (e), which means that the probability of crime will decrease if an appeals court corrects legal errors if any.

We now consider that the decision-maker su¤ers a reputational cost or a disutility z if he is reversed by the appeals court. The decision-maker is thus now said to be socially-motivated when he is more sensitive to social harm than to his reputation ( h > z). Otherwise, we will refer to him as a reputation-concerned decision-maker ( h < z).

The decision-maker chooses e to solve:

max e n u A (e) = w (e) (1 F ( bA (e)))p(e)(( 1
q) h + qz) F ( bA (e)) (e)(( 1
q) h + qz)
o under the condition below in order to control for the problem's concavity Assumption 2:

"(b) < 2(p 0 (e) 0 (e))
(p 0 (e) + 0 (e))

1 (p(e) + (e))(1 q) (p(e) (e))(1 q) " A .

Di¤erentiating u A (e) with respect to e leads to lemma 2:

Lemma 2 There exists a unique e¤ ort level at equilibrium e A de…ned by:

0 (e A ) = f ( bA (e A )))( tS)(1 q)( 0 (e A ) + p 0 (e A )) (p(e A ) (e A )) ((1 q) h + qz) (1 F ( bA (e A )))p 0 (e A ) + F ( bA (e A )) 0 (e A ) ((1 q) h + qz) (5)
The …rst-order condition may be rewritten as follows:

0 (e) | {z } C = p 0 (e)(1 F ( bA (e))) + 0 (e)F ( bA (e)) ((1 q) h + qz) | {z } + B A p @F ( bA (e)) @e (p(e) (e)) ((1 q) h + qz) | {z } B A d . ( 6
)
where B A p and B A d are the decision-maker's marginal bene…ts of raising e by one unit.

We now explore the conditions under which the appeals process may encourage or discourage the decisionmaker's to increase his e¤ort according to the parameters of the model. To begin with, notice that the appeals process leads to a decrease (resp. an increase) in the decision-maker's e¤ort if

B p = B A p B N A p < 0 (resp. B p > 0) and B d = B A d B N A d < 0 (resp. B d > 0)
, that is if the appeals process reduces (resp. increases)

the marginal bene…ts associated with the reduction of the probability of a legal error and crime deterrence.

Below, we explore separately these two conditions.

First of all, we rewrite B d as follows:

B d = (p(e) (e)
) @F ( bA (e)) @e ((1 q) h + qz) (p(e) (e)) @F ( bNA (e)) @e h.

For given probabilities of legal errors, appeals process has an in ‡uence on the decision-maker's choice through two channels: (i) his private loss associated with a legal error (which moves from h to (1 q) h + qz), and

(ii) his capacity to deter crime at the margin (captured by respectively @F ( bA (e)) @e and @F ( bNA (e))

@e

).

The intuition for point (i) runs as follows. If the decision-maker is reputation-concerned, then implementing an appeals process creates an incentive for him to increase his e¤ort due to his disutility of being overturned.

Indeed, if h < z, the appeals process increases the decision-maker's private loss associated with legal errors because (1 q) h + qz > h. At the opposite, if the decision-maker is socially-motivated ( h > z), then implementing an appeals process spurs him to decrease his e¤ort because he now relies on the appeals court to correct his mistakes. The appeals process thus diminishes the decision-maker's private loss associated with legal errors ((1 q) h + qz < h).

To better understand point (ii), we use lemma 3 below:

Lemma 3 @F ( bA (e))
@e > @F ( bNA (e)) @e , 0 < 1 (1 q)[p(e)+ (e)] (1 q)(p(e)+ (e)) < "( bA (e)) < " A .

Here, the rate at which the population density changes (captured by the term "( bA (e))) determines how a one-unit increase in the decison-maker's e¤ort will either increase or decrease the marginal impact of his e¤ort on the probability to be faced with either a guilty or an innocent defendant (and by the way, on the risk to commit one speci…c type of legal error). To illustrate, consider anew the case of an uniform distribution.

Thus, "( bA (e)) = 0 and @F ( bA (e)) @e < @F ( bNA (e)) @e : the appeals process discourages the decision-maker to exert more e¤ort, everything else equal (in particular, the decision-maker's type, see point (i) above) because one more e¤ort is less deterrent with the appeals process. Now, we study B p or the way that the appeals process a¤ects the decision-maker's marginal bene…t resulting from the fact that, by increasing his e¤ort, he also reduces the probabilities of legal errors. By de…nition:

B p = p 0 (e)(1 F ( bA (e))) + 0 (e)F ( bA (e)) ((1 q) h+qz) p 0 (e)(1 F ( bNA (e))) + 0 (e)F ( bNA (e)) h .

Thus, B p depends on the way that the appeals process a¤ects (i) the decision-maker's private loss associated with legal errors, and (ii) the expected marginal e¤ect of the decision-maker's e¤ort on the probabilities of legal errors, that is the expression p 0 (e)(1 F ( bi (e))) + 0 (e)F ( bi (e)) for i = fA; N Ag.

The explanation of point (i) is identical to the one above and depends on the decision-maker's type:

either socially-motivated or reputation-concerned. Further, notice for point (ii) that the following is true. As bA (e) > bNA (e), we have B p < 0 , p 0 (e) > 0 (e) > 0. In words, recall that a one-unit increase in the decision-maker's e¤ort allows for a reduction of legal errors on both innocent individuals (with probability F ( bi (e)) for i = fA; N Ag) and guilty ones (with probability 1 F ( bi (e)) for i = fA; N Ag). As the appeals process directly deters crime (that is for a given e), fewers individuals remain undeterred. Thus, due to the appeals process, the decision-maker's expected bene…t of a reduction in the probability of wrongly relaxing a guilty individual becomes lower whereas the expected bene…t of a reduction in the probability of wrongly convicting an innocent individual becomes higher. As a consequence, we show that the appeals process will encourage the decision-maker to exert less e¤ort ( B p < 0) if and only if his e¤ort reduces more the probability of wrongly relaxing a guilty individual than the probability of wrongly convicting an innocent one, that is if

p 0 (e) > 0 (e) > 0.
Put together, we show that the appeals process has an indirect impact on the probabilities of legal errors through three channels: the type of the decision-maker, either socially-motivated or reputation concerned

( h > < z
), the magnitude of the marginal impact of the decision-maker's e¤ort on the probabilities of legal errors ( p 0 (e) > < 0 (e)), the sign and the magnitude of the rate at which the population density changes ("( bA (e)) > < 1 (1 q)[p(e)+ (e)]

(1 q)(p(e)+ (e)) , see Lemma 3). We summarize these e¤ects in Table 1 below: Table 1: appeals process and variations in the decision-maker's marginal bene…ts.

"( bA (e)) > 1 (1 q)[p(e)+ (e)] (1 q)(p(e)+ (e)) h > z h < z p 0 (e) > 0 (e) B d > < 0 B p < 0 B d > 0 B p > < 0 p 0 (e) < 0 (e) B d > < 0 B p > < 0 B d > 0 B p > 0 "( bA (e)) < 1 (1 q)[p(e)+ (e)]
(1 q)(p(e)+ (e)) h > z h < z p 0 (e) > 0 (e) B d < 0

B p < 0 B d > < 0 B p > < 0 p 0 (e) < 0 (e) B d < 0 B p > < 0 B d > < 0 B p > 0
We show in Table 1 that the e¤ect of the appeals process on the decision-maker's incentives is largely unclear. Intuitively, if the decision-maker is reputation-concerned ( h < z), concerned only by the risk of wrongly relaxing a guilty individual ( (e) = 0), and unsensitive to the way judicial errors may have a deterrence e¤ect, then we can conclude that the appeals process spurs him to exert more e¤ort to obtain more accurate decisions and avoid a reversal on appeal [START_REF] Shavell | The Appeals Process as a Means of Error Correction[END_REF][START_REF] Shavell | The Appeals Process and Adjudicator Incentives[END_REF]. We show here that adding on the fact that decision-makers may be rather socially-motivated, that there is a room for two types of legal errors, and that decision-makers can also internalize the impact of both their own e¤ort and the appeals process on the deterrence, quali…es the previous result.

More precisely, we show that a reputation-concerned decision-maker will still have unambiguously an incentive to increase his e¤ort only if "( bA (e)) > 1 (1 q)[p(e)+ (e)]

(1 q)(p(e)+ (e)) and p 0 (e) < 0 (e): see row 3 column 3 in Table 1. A socially-motivated decision-maker will have unambiguously an incentive to decrease his e¤ort if "( bA (e)) < 1 (1 q)[p(e)+ (e)] (1 q)(p(e)+ (e)) and p 0 (e) > 0 (e): see row 6 column 3 in Table 1. Thus, neither the decisionmaker's type by itself, nor the sensitivity of the probabilities of legal errors to the decision-maker's e¤ort by itself are su¢ cient to predict the way that the appeals process may have an indirect in ‡uence on the emergence of legal errors. Finally, remark that an appeals process will all the more encourage the decision-maker to reduce the risk of a legal error (i ) that the decision-maker is rather more reputation-concerned ( h < z), (ii ) that the elasticity of the density function of the private bene…t of crime for the threshold individual, "( bA (e)), is high, and (iii ) that the decision-maker is rather concerned by the reduction of the risk of wrongly convicting an innocent individual (in comparison to the risk of wrongly relaxing a guilty individual, p 0 (e) < 0 (e)).

Next, we summarized the results of the comparative statics analysis in the proposition below. The explanation is similar to the one of Proposition 1.

Proposition 2 An increase in either the sanction, S, or the probability of detection, t, implies a decrease in the decision-maker's e¤ ort if and only if "( bA (e)) < (p 0 (e) 0 (e)) + 2(1 q)( 0 (e) (e) p 0 (e)p(e))

(p 0 (e) + 0 (e))(p(e) (e))( 1 q) .

An increase in either the social harm due to a judicial errors, h, or the decision-maker's sensitivity to social harm of errors, , implies an increase in the decision-maker's e¤ ort at equilibrium.

To supplement the previous analysis, we compare the expected social cost of legal errors without any appeals process (namely ESCE N A (e N A ) at equilibrium) with the expected social cost of legal errors when an appeals process prevails (ESCE A (e A ) at equilibrium), where:

ESCE N A (e N A ) = [1 F ( e b N A (e N A ))]p(e N A ) + F ( e b N A (e N A )) (e N A ) h and ESCE A (e A ) = [1 F ( e b A (e A ))]p(e A ) + F ( e b A (e A )) (e A ) (1 q)h.
If q = 0 then e A = e N A and ESCE N A (e N A ) = ESCE A (e A ). If q > 0, we have shown that the appeals process may either encourage or discourage the decision-maker to increase his e¤ort, and thus reduce the occurence of a legal error. Thus, even if the appeals process directly reduces the risk to commit a egal error (see the term 1 q in ESCE A (e A )), the e¤ect of the appeals process on the expected social cost of errors remains unclear because of its indirect e¤et via e A .

The double jeopardy

We now take into account the doctrine known in the US as double jeopardy. This constitutional clause refers to the impossibility of appealing acquittals by public prosecutors in criminal cases 3 . In our setting, this means that the appeals court may now only …nally relax a previously wrongly convicted person. Thus, if double jeopardy exists, it modi…es both the de…nition of the threshold value of the bene…t of crime and the decision-maker's utility. The borderline type then equals:

bDJ A (e) = tS (1 p(e) (1 q) (e))
where bA (e) > bDJ A (e). The appeals process is less deterrent because the risk of …nally wrongly relaxing a guilty person is higher (in comparison to the case where appeals process corrects both types of legal errors, as in section 3).

The decision-maker chooses e to solve:

max e n u DJ A (e) = w (e) (1 F ( bDJ A (e)))p(e) h F ( bDJ A (e)) (e)(( 1
q) h + qz) o .
in (B A d ) DJ . In words, it means that the reputation-concerned decision-maker's marginal bene…ts, (B A p ) DJ and (B A d ) DJ , may be lower because the decision-maker cannot be reversed (and does not su¤er a disutility z) if he does wrongly relax a guilty person. At the opposite, the socially-motivated decision-maker's marginal bene…ts, (B A p ) DJ and (B A d ) DJ , may be greater because the decision-maker cannot rely on appeals courts to …nally convict guilty defendants.

Second, we show that fewer individuals remain undeterred with double jeopardy: bA (e) > bDJ

A (e). The comparison between (B A p ) DJ and B A p , and the one between (B A d ) DJ and B A d thus become much more complex. Not surprisingly, they depend on parameters p 0 (e), 0 (e) and the rate at which the population density changes.

Finally, we explore the conditions under which the appeals process may still encourage or discourage the decision-maker's to increase his e¤ort according to the parameters of the model. To begin with, notice that the appeals process anew leads to a decrease (resp. an increase) in the decision-maker's e¤ort if

B p = (B A p ) DJ B N A p < 0 (resp. B p > 0) and B d = (B A d ) DJ B N A d < 0 (resp. B d > 0).
After some manipulations, we have:

B d = (p(e) (e)) h @F ( bDJ A (e)
) @e @F ( bNA (e)) @e ! and B p = p 0 (e) 0 (e) F ( bDJ A (e)) F ( bNA (e)) h 0 (e)F ( bDJ A (e))q(z h)

where F ( bDJ A (e)) F ( bNA (e)) > 0 and Lemma 5 @F ( bDJ A (e)) @e > @F ( bNA (e)) @e , 0 < 0 (e) (e)

1 p(e) (1 q) (e) p 0 (e)+(1 q) 0 (e) < "( bDJ A (e)).

Thus, the appeals process still has an indirect impact on the probabilities of legal errors through the type of the decision-maker, either socially-motivated or reputation concerned ( h > < z), the magnitude of the marginal impact of the decision-maker's e¤ort on the probabilities of legal errors ( p 0 (e) > < 0 (e)), the sign and the magnitude of the rate at which the population density changes ("( bDJ

A (e)) > < 0 (e) (e)
1 p(e) (1 q) (e) p 0 (e)+(1 q) 0 (e) ). We summarize these e¤ects in Table 2 below: Table 2: variations in the decision-maker's marginal bene…ts.

"( bDJ

A (e)) > 0 (e) (e)

1 p(e) (1 q) (e) p 0 (e)+(1 q) 0 (e) h > z h < z p 0 (e) > 0 (e) B d > 0 B p < 0 B d > 0 B p > < 0 p 0 (e) < 0 (e) B d > 0 B p > < 0 B d > 0 B p > 0 "( bDJ A (e)) < 0 (e) (e)
1 p(e) (1 q) (e) p 0 (e)+(1 q) 0 (e)

h > z h < z p 0 (e) > 0 (e) B d < 0 B p < 0 B d < 0 B p > < 0 p 0 (e) < 0 (e) B d < 0 B p > < 0 B d < 0 B p > 0 Note: we derive from Lemma 5 that B d > 0 , "( bDJ A (e)) > 0 (e) (e) 1 p(e) (1 q) (e)
p 0 (e)+(1 q) 0 (e) .

By comparing Table 1 (without double jeopardy) and Table 2 (with double jeopardy), we …rst notice that the discussion on B p is similar. As a result, the double jeopardy does not signi…cantly matter here.

Second, it appears that when the double jeopardy exists, the sign of B d does not depend on the type of the decision-maker anymore (contrary to the case where appeals courts may correct both types of legal errors).

The intuition of this result is the following. Recall that a one-unit increase in the decision-maker's e¤ort has a deterrent e¤ect ( @F ( bDJ A (e)) @e > 0). Thus, there is a lower probability to be faced with an undeterred individual, and a higher probability to be faced with an innocent individual. Further, the magnitude of each variation is identical. As the likelihood of relaxing a guilty defendant is strictly superior to the probability of wrongly convicting an innocent individual, p(e) > (e), then such an increase in the decision-maker's e¤ort is pro…table. Now, by preventing appeals courts from …nally convicting undeterred individuals, double jeopardy then partially clears the reputation-concerned decision-maker's incentive to increase his e¤ort (due to his disutility of being overturned), and simultaneously partially clears the socially-motivated decision-maker's incentive to reduce his e¤ort (due to the free riding e¤ect). Finally, notice that this e¤ect (and indirectly the type of the decision-maker) is at stake for both types of legal errors as explained above. As a result, the decision-maker's type, either socially-motivated or reputation-concerned, has not anymore an in ‡uence on the way that appeals process indirectly in ‡uences the occurrence of legal errors via the so called deterrence e¤ect.

To sum up, even if the e¤ect of the appeals process on the decision-maker's incentives is less ambiguous, the theoretical prediction detailed in section 3 still holds. Further, let us de…ne the expected social cost of legal errors when an appeals process with double jeopardy prevails as follows:

ESCE DJ A (e DJ A ) = (1 F ( e b A (e DJ A )))p(e DJ A )h + F ( e b A (e DJ A )) (e DJ A )(1 q)h.
In comparison to the case where appeals courts may correct both types of legal errors, their expected social cost tends to be automatically higher because the direct e¤ect of the appeals process on the correction of legal errors is lower. But the indirect e¤ect remains unclear, and its magnitude may be either superior or inferior to the one exposed in section 3.

Conclusion

In this paper we have focused on how the introduction of an appeals process may indirectly impact the occurrence of legal errors, via trial court decision-makers'e¤orts. So far, the dominant idea has been that the appeals process lowers the occurence of legal errors both directly, by o¤ering the opportunity to reconsider cases and indirectly, by spurring trial court decision-makers to exert more e¤ort to avoid reversal on appeal.

By considering the fact that trial court decision-makers may be more concerned by errors than by reputation and that their behavior has a deterrent e¤ect on potential o¤enders, we have pointed out that the appeals process does not necessarily improve the accuracy of judicial decisions. Although we have adopted a setting in which the appeals process indeed directly (mechanically) improves the accuracy of judicial decisions (because an appeals is brought if and only if an error has been made at trial), our paper shows that whether the appeals process will …nally allow reducing legal errors also depends on a positive or negative indirect e¤ect through the trial court decision-makers'e¤orts. We have highlighted that this indirect e¤ect depends on the type of the decision-maker (reputation-concerned or socially-motivated), the distribution of the population of potential o¤enders and the marginal e¤ect of the decision-maker's e¤ort on the probability of wrongful conviction compared to the probability of wrongful acquittal. Our results are qualitatively the same whether an appeal can be brought after conviction and acquittal verdicts at trial, or appeals are limited by double jeopardy. On a social point of view, these results question the opportunity of incorporing an appeals process compared to the alternative of enhancing the quality of the trial process, as regards the occurrence of legal errors. Indeed, our results suggest that, according to the characteristics of decision-makers and potential o¤enders, investing in more skilled trial court decision-makers or increasing the number of decision-makers who hear each case, could be preferable to using an appeals process (even if it is a more costly solution) since implementing an appeals process may even increase the occurence of legal errors. We know that 00 (e) > 0, p 0 (e) < 0, p 00 (e) > 0, 0 (e) < 0, 00 (e) > 0 and p(e) (e) > 0. This implies that a su¢ cient condition for having u 00

Proofs

@e @S = @u 0 N A @S @u 0 N A @e ; @e @ = @u 0 N A @ @u 0 N A @e ; @e @h = @u 0 N A @h @u 0 N A @e
and @e @t = @u 0

N A @t @u 0 N A @e
where the derivatives @u 0 .

We know that h > 0 and @u 0 N A @e < 0 (by the second order condition). This implies that the sign of @e @S equals the sign of the bracketed term in the numerator of the expression above. After some manipulations, we …nd that: By a very similar token we can show that, under the same condition regarding "( bNA (e)), we have @e @t > 0. Finally, we have: Proof of Lemma 2. Di¤erentiating u A (e) with respect to e shows that u 0 A (e) is equal to 0 (e) ((1 q) h+qz) p 0 (e)[1 F ( bA (e))] + 0 (e)F ( bA (e)) (1 q)f ( bA (e))tS[p 0 (e)+ 0 (e)][p(e) (e)]((1 q) h+qz).

Di¤erentiating twice u A (e) with respect to e gives u 00 A (e) = 00 (e) + f 0 ( b3 (e))( t 2 S 2 )(1 q) 2 (p 0 (e) + 0 (e)) 2 (p(e) (e))((1 q) h + qz) + f ( bA (e))( tS)(1 q)(p 00 (e) + 00 (e))(p(e) (e))((1 q) h + qz) + f ( bA (e))( tS)(1 q)(p 0 (e) + 0 (e))(p 0 (e) 0 (e))((1 q) h + qz)

((1 q) h + qz) " f (
bA (e))( tS)(1 q)(p 0 (e) + 0 (e))p 0 (e) + (1 F ( bA (e))p 00 (e)+ f ( bA (e))( tS)(1 q)(p 0 (e) + 0 (e)) 0 (e)) + F ( bA (e)) 00 (e) # .

We know that 00 (e) > 0, p 0 (e) < 0, p 00 (e) > 0, 0 (e) < 0, 00 (e) > 0 and p(e) (e) > 0. This implies that a su¢ cient condition for having u 00 A (e) < 0 is f 0 ( bA (e))( t 2 S 2 )(1 q) 2 (p 0 (e) + 0 (e)) 2 (p(e) (e)) + f ( bA (e))tS(1 q)(p 0 (e) + 0 (e))(p 0 (e) 0 (e))

f ( bA (e))( tS)(1 q)(p 0 (e) + 0 (e))(p 0 (e) 0 (e))

> 0.

The inequality above is equivalent to Proof of Lemma 3. Note that @(F ( bA (e))) @e = f ( bA (e))tS(1 q) p 0 (e) + 0 (e) > 0.

"( bA (e)) < 2
It follows that @ @F ( bA (e)) @e @q = tS p 0 (e) + 0 (e) h f 0 ( bA (e))tS(1 q) (p(e) + (e)) f ( bA (e))

i :

As tS p 0 (e) + 0 (e) > 0 we have @(F ( bA (e))) @e > @(F ( bNA (e))) @e if and only if f 0 ( bA (e))tS(1 q) (p(e) + (e)) f ( bA (e)) > 0 or "( bA (e)) > 1 (1 q)(p(e) + (e)) (1 q)(p(e) + (e)) .

Proof of Proposition 2. Since u 0 A (e; x) = 0 (e) ((1 q) h + qz) p 0 (e)[1 F ( bA (e))] + 0 (e)F ( bA (e))

(1 q)f ( bA (e))tS p 0 (e) + 0 (e) (p(e) (e)) ((1 q) h + qz) = 0 for x = fS; ; h; tg, then by applying the implicit function theorem, we have @e @S = @u 0 A @S @u 0 A @e ; @e @ = @u 0 A @ @u 0 A @e ; @e @h = @u 0 A @h @u 0

A @e and @e @t = @u 0 A @t @u 0 A @e where the derivatives @u 0 A @x for x = fS; ; h; tg and the derivative @u 0 A @e are calculated respectively at the points (e; x) such that u 0 A (e; x) = 0 for x = fS; ; h; tg. Thus @e @S = ((1 q) h + qz) .

We know that (1 q) h + qz > 0 and @u 0 A @e < 0 (by the second order condition). This implies that the sign of @e @S equals the sign of the bracketed term in the numerator of the expression above. After some manipulations, we …nd that: By a very similar token, we can show that under the same condition, we have @e @t > 0. Finally, we also have: d bDJ A (e) de 0 (e)((1 q) h + qz))

@e @ = ( 1
(1 F ( bDJ A (e))p 00 (e) h F ( bDJ A (e)) 00 (e)((1 q) h + qz).

1 F

 1 ( bNA (e)))p(e) h F ( bNA (e)) (e) h o where the last two terms of u N A (e) represent respectively the expected cost of wrongful acquittal weighted by the probability to be faced with a guilty individual (or the probability of crime), and the expected cost of wrongful conviction weighted by the probability to be faced with an innocent indivudual. Note that neither the social harm of legal errors (h) nor the decision-maker's sensitivity to social harm of errors ( ) di¤er from one type of legal error to the other one.

1 F

 1 ( bNA (e))) + 0 (e)F ( bNA (e)) The term C represents the decision-maker's disutility of raising e by one unit. Together, terms B N A p and B N A d are the decision-maker's total marginal bene…t of raising e by one unit under two e¤ects: the reduction of the probability of legal errors and crime deterrence.

Proposition 1

 1 where "(b) = f 0 (b) f (b) b is the elasticity of the density. The static comparative results are summarized in Proposition 1 below: An increase in either the sanction, S, or the probability of detection, t, implies a decrease in the decision-maker's e¤ ort if and only if "( bNA (e)) < (p 0 (e) 0 (e)) + 2( 0 (e) (e) p 0 (e)p(e)) (p 0 (e) + 0 (e))(p(e) (e)) .

p

  and B N A d , thus encouraging him to increase his e¤ort because legal errors are more costly. The e¤ect of the penalty, S, and the probability of detection, t, on the optimal level of e¤ort is ambiguous. A variation of these parameters may a¤ect B N A p and B N A d in a di¤erent way.

Proof of Lemma 1 .

 1 Di¤erentiating u N A (e) with respect to e shows that u 0 N A (e) is equal to 0 (e) hp 0 (e)[1 F ( bNA (e))] h 0 (e)F ( bNA (e)) hf ( bNA (e))tS[p 0 (e) + 0 (e)][p(e) (e)]. Di¤erentiating twice u N A (e) with respect to e gives u 00 N A (e) = 00 (e) + f 0 ( bNA (e))( t 2 S 2 )(p 0 (e) + 0 (e)) 2 (p(e) (e)) h +f ( bNA (e))( tS)(p 00 (e) + 00 (e))(p(e) (e)) h + f ( bNA (e))( tS)(p 0 (e) + 0 (e))(p 0 (e) 0 (e)) h h " f ( bNA (e))( tS)(p 0 (e) + 0 (e))p 0 (e) + (1 F ( bNA (e))p 00 (e)+ f ( bNA (e))( tS)(p 0 (e) + 0 (e)) 0 (e) + F ( bNA (e)) 00 (e) # .

1 F

 1 N A (e) < 0 is f 0 ( bNA (e))( t 2 S 2 )(p 0 (e) + 0 (e)) 2 (p(e) (e)) + f ( bNA (e))tS(p 0 (e) + 0 (e))(p 0 (e) 0 (e)) f ( bNA (e))( tS)(p 0 (e) + 0 (e))(p 0 (e) 0 (e)) is satis…ed under Assumption 1. Finally, notice for the existence of an interior solution (the function u N A being continuously di¤erentiable twice) that lim ( bNA (0))) hcF ( bNA (0)) hf ( bNA (0))tS[a + c][p(0) (0)] > 0 with a = lim e!0 p 0 (e) < 0 and c = lim Since u 0 N A (e; x) = 0 (e) ( h) p 0 (e)[1 F ( bNA (e))] + 0 (e)F ( bNA (e)) f ( bNA (e))tS p 0 (e) + 0 (e) (p(e) (e)) h = 0 for x = fS; ; h; tg, then by applying the implicit function theorem, we have

  x = fS; ; h; tg and the derivative at the points (e; x) such that u 0 N A (e; x) = 0 for x = fS; ; h; tg. Thus @e @S = h f 0 ( e b N A (e))t 2 S(1 (p(e) + (e)))(p 0 (e) + 0 (e))(p(e) (e)) f ( e b N A (e))t(p 0 (e) + 0 (e))(p(e) (e)) + f ( e b N A (e))t(1 (p(e) + (e)))(p 0 (e)

f 0 (

 0 e b N A (e))t 2 S(1 (p(e) + (e)))(p 0 (e) + 0 (e))(p(e) (e)) f ( e b N A (e))t(p 0 (e) + 0 (e))(p(e) (e))+ f ( e b N A (e))t(1 (p(e) + (e)))(p 0 (e) 0 (e)) ! > 0 is equivalent to: "( bNA (e)) > (p 0 (e) 0 (e)) + 2( 0 (e) (e) p 0 (e)p(e)) (p 0 (e) + 0 (e))(p(e) (e)) .

1 F

 1 ( bNA (e))] + 0 (e)F ( bNA (e)) hf ( bNA (e))tS p 0 (e) + 0 (e) (p(e) (e)) p 0 (e)[1 F ( bNA (e))] + 0 (e)F ( bNA (e)) f ( bNA (e))tS p 0 (e) + 0 (e) (p(e) (e))

  (p 0 (e) 0 (e)) (p 0 (e) + 0 (e)) 1 (p(e) + (e))(1 q) (p(e) (e))(1 q) which holds under Assumption 2. Finally, notice for the existence of an interior solution (the function u A being continuously di¤erentiable twice) that lim e!0 u 0 A (e) = ((1 q) h+qz)a(1 F ( bA (0))) ((1 q) h+qz)cF ( bA (0)) ((1 q) h+qz)f ( bA (0))tS[a+c][p(0) (0)] > 0 because a = lim e!0 p 0 (e) < 0, c = lim e!0 0 (e) < 0. Finally, we have lim

  e b A (e))t 2 S(1 (1 q)(p(e) + (e)))(1 q)(p 0 (e) + 0 (e))(p(e) (e))f ( e b A (e))t(1 q)(p 0 (e) + 0 (e))(p(e) (e))+f ( e b A (e))t(1 (1 q)(p(e) + (e)))(p 0 (e)

f 0 (

 0 e b A (e))t 2 S(1 (1 q)(p(e) + (e)))(1 q)(p 0 (e) + 0 (e))(p(e) (e)) f ( e b A (e))t(1 q)(p 0 (e) + 0 (e))(p(e) (e)) + f ( e b A (e))t(1 (1 q)(p(e) + (e)))(p 0 (e) 0 (e)) ! > 0 is equivalent to: "( bA (e)) > (p 0 (e) 0 (e)) + 2(1 q)( 0 (e) (e) p 0 (e)p(e)) (p 0 (e) + 0 (e))(p(e) (e))(1 q) .

Proof of Lemma 4 .( 1 F

 41 q)h p 0 (e)[1 F ( bA (e))] + 0 (e)F ( bA (e)) (1 q) 2 hf ( bA (e))tS[p 0 (e) + 0 (e)][p(e) (e)] ) p 0 (e)[1 F ( bA (e))] + 0 (e)F ( bA (e)) (1 q) 2 f ( bA (e))tS[p 0 (e) + 0 (e)][p(e) (e)] Di¤erentiating u DJ A (e) with respect to e shows that the derivative of u DJ A (e) is equal to0 (e) + f ( bDJ A (e))( tS)((1 q) 0 (e) + p 0 (e))(p(e) h (e) ((1q) h + qz)) ( bDJ A (e)))p 0 (e) h + F ( bDJ A (e)) 0 (e) ((1q) h + qz) = 0Di¤erentiating twice u DJA (e) with respect to e gives u DJ 00 A (e) = 00 (e) f 0 ( bA (e)) d bDJ A (e) de tS(p 0 (e) + (1 q) 0 (e))(p(e) h (e)((1 q) h + qz)) f ( bDJ A (e))tS(p 00 (e) + (1 q) 00 (e))(p(e) h (e)((1 q) h + qz)) f ( bDJ A (e))tS(p 0 (e) + (1 q) 0 (e))(p 0 (e) h 0 (e))((1 q) h + qz)) +f ( bDJ A (e)) d bDJ A (e) de p 0 (e) h f ( bDJ A (e))

For instance,[START_REF] Rizzolli | Judicial Errors and Crime Deterrence : Theory and Experimental Evidence[END_REF] explain how wrongful conviction may become more detrimental to deterrence than wrongful acquittal if we introduce risk-aversion and loss aversion in the standard model of public enforcement.

Regarding the relevance of the second hypothesis,[START_REF] Shavell | The Appeals Process and Adjudicator Incentives[END_REF] argues that the society invests special e¤orts in selecting appeals court decision-makers to ensure that their preferences are more aligned with society's than trial court decision-makers'.For example, appeals courts decide in larger panels. One can also think that appeals court judges are more experienced.

See Rizzolli (2010) for a discussion on the history and spirit of double jeopardy.

under the condition below in order to control for the problem's concavity.

Assumption 3:

(p(e) h (e)((1 q) h + qz)) "(b) p 0 (e) + (1 q) 0 (e) 2 p 00 (e) + (1 q) 00 (e) (1 p(e) (1 q) (e))

< 2 p 0 (e) + (1 q) 0 (e) (1 p(e) (1 q) (e)) p 0 (e) h 0 (e)((1 q) h + qz) .

Di¤erentiating u DJ A (e) with respect to e leads to lemma 4 below:

Lemma 4 There exists a unique e¤ ort level at equilibrium e DJ A de…ned by:

The …rst order condition can be rewritten:

The term C represents the decision-maker's disutility of raising e by one unit. Terms (B A p ) DJ and (B A d ) DJ are the decision-maker's total marginal bene…ts of raising e by one unit, associated with the reduction of the probability of legal errors and crime deterrence. In comparison to the benchmark detailed in section 3, double jeopardy changes the decision-maker's marginal bene…ts at two levels.

By comparing the two FOC ( 6) and ( 8), we …rst show that the double jeopardy adds the term q(z h)p 0 (e)(1 F ( bDJ A (e))) in (B A p ) DJ and the term q(z h) (p(e) (e))

@F ( bDJ A (e)) @e

We know that 00 (e) > 0, p 0 (e) < 0, p 00 (e) > 0, 0 (e) < 0, 00 (e) > 0, p(e) (e) > 0 and

tS(p 0 (e) + (1 q) 0 (e) > 0. This implies that a su¢ cient condition for having u DJ 00 A (e) < 0 is f 0 ( bDJ A (e))t 2 S 2 (p 0 (e) + (1 q) 0 (e)) 2 (p(e) h (e)((1 q) h + qz)) f ( bDJ A (e))tS(p 00 (e) + (1 q) 00 (e)) (p(e) h (e)((1 q) h + qz))

2f ( bDJ A (e))tS(p 0 (e) + (1 q) 0 (e)) p 0 (e) h 0 (e)((1 q) h + qz) < 0.

The inequality above is equivalent to (p(e) h (e)((1 q) h + qz)) "(b) p 0 (e) + (1 q) 0 (e)

p 00 (e) + (1 q) 00 (e) (1 p(e) (1 q) (e))

< 2 p 0 (e) + (1 q) 0 (e) (1 p(e) (1 q) (e)) p 0 (e) h 0 (e)((1 q) h + qz) which holds under Assumption 3. Finally, notice for the existence of an interior solution that we have

@(F ( bDJ

A (e))) @e = f ( bDJ A (e))tS p 0 (e) + (1 q) 0 (e) > 0.

It follows that @ @F ( bDJ A (e)) @e @q = tS f 0 ( bDJ A (e)) (e)tS(p 0 (e) + (1 q) 0 (e)) f ( bDJ A (e)) 0 (e) :

Thus, we have @(F ( bDJ A (e))) @e > @(F ( bDJ N A (e))) @e if and only if "( bDJ A (e)) > 0 (e) (e)

1 p(e) (1 q) (e) p 0 (e) + (1 q) 0 (e) > 0.