Better that ten innocent persons suffer than that one guilty escape. Evidentiary standards for investigating and charging defendants (original) (raw)

BETTER THAT TEN INNOCENT PERSONS SUFFER THAN THAT ONE GUILTY ESCAPE

Evidentiary standards for investigating and charging defendants*

GABRIEL DOMÉNECH-PASCUAL
University of Valencia Law School
gabriel.domenech@uv.es
Miguel Puchades-Navarro
University of Valencia Law School
miguel.puchades@uv.es

Abstract

In most legal systems, a very high standard of evidence is established to reach a verdict of guilty. Accused individuals may not be convicted unless the prosecution proves their guilt beyond a reasonable doubt. This contrasts with the much lower standards of evidence required to make pre-trial decisions. This paper contributes to the law and economics literature on standards of proof in criminal law by extending the analysis to pretrial decisions on whether the criminal investigation (proceedings) against a person should go ahead or be terminated. The paper shows why optimal standards for these decisions should be lower than that for conviction: false negatives (e.g., wrongful non-indictments) have a more adverse impact on deterrence than false positives (e.g., wrongful indictments). The paper analyzes as well how those standards should vary depending on certain circumstances.

Keywords: criminal procedure, pre-trial decisions, charging standards, indictment, investigation, seizures, searches, arrests, judicial errors, optimal standard of evidence
JEL Classification: K14, K42

1. INTRODUCTION

The title of this paper obviously reverses the famous Blackstone quotation, “It is better that ten guilty persons escape than that one innocent suffer” (1769, p. 352). The idea formulated in this assertion, and by many other authors in different ways throughout history (Volokh 1997), is a cornerstone of criminal law in modern societies.

Criminal Courts inevitably make mistakes. They sometimes convict individuals who really did not commit the crimes charged, while other times they acquit defendants who were actually guilty. The Blackstonian formulation expresses that the former (type I) errors are worse than the latter (type II) errors. Courts should therefore try to especially avoid wrongful convictions rather than wrongful acquittals. Criminal law should hence be designed in a “biased” way in favor of defendants (Hylton and Kahnna 2007; Miceli 2009).

The most important manifestation of such asymmetric design is the requirement of a high standard of evidence in order to reach a verdict of guilty. This so-called principle of in dubio pro reo is well established in contemporary criminal systems. Under the case law of the Supreme Court of the United States, for instance, defendants

[1]

- The authors would like to thank participants in EALE, AEDE, and GLEA conferences (2013), and, in particular, Matteo Rizzolli, Fernando Gómez, and Naci Mocan for useful comments. All errors (both type I and II) remain ours.
  ↩︎

may not be convicted unless their guilt is proved “beyond a reasonable doubt.” 1{ }^{1} Similarly, the European Court of Human Rights has declared that the principle of the presumption of innocence requires that “any doubt should benefit the accused” 2{ }^{2}; it is a “basic requirement of criminal justice [that] the prosecution has to prove its case beyond reasonable doubt”; the in dubio pro reo is “one of the fundamental principles of criminal law” embedded in the right to a fair trial under Article 6§16 \S 1 of the European Convention of Human Rights 3{ }^{3}. The same principle is enshrined in Article 66 of the Rome Statute of the International Criminal Court (ICC), which states in its third paragraph that “in order to convict the accused, the Court must be convinced of the guilt of the accused beyond reasonable doubt.”

This strict standard sharply contrasts with that usually required in civil cases (e.g., “preponderance of the evidence”), according to which the plaintiff prevails if she or he manages to prove that her or his allegations are more likely to be true than not true. 4{ }^{4}

It should be pointed out, however, that the principle of in dubio pro reo is far from being universally applied even in criminal cases. On the contrary, it is only to be used in the very final stage of the criminal procedure, when the relevant court or jury has to assess the defendant’s guilt in order to either convict or acquit her or him. Before then, the general rule seems to be the opposite: in dubio contra civem.

Law enforcement and legal proceedings are usually structured as a multistage process 5{ }^{5}. State agents do not proceed in one fell swoop, but step by step. Before a final decision is made, several interim measures need to be taken. The relevant agents have to decide, for instance, whether or not to detain some people; carry out audits, searches or seizures of property; freeze certain assets; determine, after a preliminary hearing, whether or not there is enough incriminating evidence for a trial to be held, etc. And one can see that a lower standard of evidence is required for such measures to be adopted. Even when these decisions imply the deprivation of the liberty of the affected individuals, it is not necessary for the prosecution to prove that they are guilty beyond any reasonable doubt.

Under article 5§15 \S 1 © of the European Convention of Human Rights, for example, a mere “reasonable suspicion of having committed an offence” is needed to lawfully arrest or detain a person for the purpose of bringing him before the competent legal authority. As the European Court has declared, “Facts which raise a suspicion need not be of the same level as those necessary to justify a conviction or even the bringing of a charge, which comes at the next stage of the process of criminal investigation” 6{ }^{6}; it is only required that the elements of which the authorities have knowledge at the time when the order is issued are “reasonably sufficient to believe” that the detainee has committed an offence 7{ }^{7}.

[1]

In re Winship, 397 U.S. 358 (1970).
Judgment of the ECHR of 6 December 1988 (Barberà, Messegué and Jabardo v. Spain, 10590/83, § 77).
Judgment of the ECHR of 13 December 2011 (Adjarić v. Croatia, 20883/09, § 51).
See, for instance, Brooks (1982). For an economic analysis of this standard, Lando (2002).
See Kaplow (2013), who focuses on civil litigation.
Judgment of the ECHR of 28 October 1994 (Murray v. the United Kingdom, 14310/88).
Judgment of the ECHR of 11 January 2001 (N.C. v. Italy, 24952/94). ↩︎

Under U.S. law, there is a range of standards of evidence for pre-trial criminal proceedings. The arguably strictest one is that of “clear and convincing evidence” of guilt, which is established by many state statutes for detention on remand 8{ }^{8}.

The second standard is that of “probable cause,” which is explicitly provided for in the Fourth Amendment to the U.S. Constitution with respect to “searches” and “seizures” either of persons or things. The U.S. Supreme Court early observed that “the term ‘probable cause’, according to its usual acceptation, means less than evidence which would justify condemnation… It imports a seizure made under circumstances which warrant suspicion.” 9{ }^{9} This is also the standard for charging criminal defendants (i.e., for determining which cases go to trial). 10{ }^{10} And the U.S. Supreme Court has declared that probable cause, provided by the Federal Bail Reform Act of 1984, is a constitutionally permissible standard for pre-trial detention. 11{ }^{11}

The third standard is that of “reasonable suspicion.” The U.S. Supreme Court has stated that, “In some circumstances, an officer may detain a suspect briefly for questioning although he does not have ‘probable cause’ to believe that the suspect is involved in criminal activity, as is required for a traditional arrest… However, [it has] required the officers to have a reasonable suspicion, based on objective facts, that the individual is involved in criminal activity.” 12{ }^{12} This “is a less demanding standard than probable cause and requires a showing considerably less than preponderance of the evidence.” 13{ }^{13}

Lastly, the U.S. Supreme Court has declared that, in some cases (e.g., checkpoints permanently located on a highway in order to detect illegal aliens and sobriety checkpoint programs), state agents may carry out seizures even without any individualized suspicion against the affected individuals. 14{ }^{14} This represents a corner solution. Everyone who is investigated (e.g., who goes through the checkpoint) suffers detention, if only for a very brief moment. “Some quantum of individualized suspicion is usually a prerequisite to a constitutional search or seizure… But the Fourth Amendment imposes no irreducible requirement of such suspicion.” 15{ }^{15}

In the Statute of the ICC, we could find three standards of evidence concerning the investigation and prosecution stages. They can be ranked, from lowest to highest, as follows. 16{ }^{16} First, the prosecutor may, and eventually has to, initiate an investigation if there is “reasonable basis to proceed” under this Statute (articles 15 and 53.1). Second, the Pre-Trial Chamber shall issue a warrant of arrest of a person if, inter alia, “there are reasonable grounds to believe that the person has committed a crime within the jurisdiction of the Court” (article 58.1.a). Third, the Pre-Trial Chamber shall confirm the

[1]

8 See Goldkamp (1985, pp. 33-38), and Williams (1994, p. 380).
Locke v. United States, 11 U.S. 7 Cranch 339, 348 (1813). See also Bell v. Wolfish, 441 U.S. 520, 533 (1979).
See, for instance, Ortman (2016).
United States v. Salerno, 481 U.S. 739, 755 (1987). Some authors, such as Alschuler (1986, p. 566) and Williams (1994, p. 366), argue that this standard of proof should be increased so that the guilt of the suspect is proved by “clear and convincing evidence” for him to be detained.
Brown v. Texas, 443 U.S. 47, 51 (1979).
Illinois v. Wardlow, 528 U.S. 119 (2000).
See United States v. Martinez-Fuerte, 428 U.S. 543 (1976); Michigan Department of State Police v. Sitz, 496 U.S. 444 (1990).
United States v. Martinez-Fuerte, 428 U.S. 543, 560-561 (1976).
See, for instance, Miraglia (2008); Ventura (2013); Ramsden and Chung (2015); Mariniello (2015), and Cross (2018). ↩︎

charges and commit the person for trial if “there is sufficient evidence to establish substantial grounds to believe that the person committed each of the crimes charged” (article 61.1).

In these and other legal systems, the amount of supporting evidence required to take these pre-trial measures could even be much lower than that needed under the preponderance of the evidence standard that is used as a general rule in civil trials. It seems that, when deciding on these measures, type II errors are worse than type I errors: it is better that one innocent person-or eventually hundreds of them-suffer than that one guilty escape. 17{ }^{17}

It must be pointed out that such measures can have two ultimate goals. The first one is deterrence. They can be necessary to investigate the facts and eventually apprehend and punish individuals who have already committed a crime in order to deter future offenses. The second goal is harm prevention. Some of those measures aim to stop ongoing or imminent crimes in order to hinder that they end up causing (more) harm. They thwart the “successful” commission of offenses (see Friehe and Tabbach 2013; Mungan 2018).

Several law enforcement pre-trial decisions might produce both deterrence and preventive effects. For instance, the arrest of a person could be necessary not only to ensure both the “person’s appearance at trial” and that she or he “does not obstruct or endanger the investigation or the court proceedings,” but also to “prevent the person from continuing with the commission of [the crime at issue] or a related crime” (article 58.1.b of Statute of the ICC).

Other decisions, by contrast, produce only deterrent effects. This is the case, for instance, of (i) the initiation of an investigation, (ii) the confirmation of charges, and (iii) the arrest of a person aimed at ensuring his appearance at trial if there is no risk of him continuing the commission of any crime.

Within the subset of pre-trial decisions, we could distinguish between those made on whether the criminal proceedings against a person should go ahead or be terminated, and those made on other alternatives. Instances (i) and (ii) belong to the former class, while instance (iii) belongs to the latter one, unless the ex ante probability of the suspect escaping criminal liability if not arrested equals 1 , as in such case, the no-arrest decision would inexorably lead to the termination, whereas the other one would enable the proceedings to be continued.

There is a considerable law and economics literature analyzing the evidentiary standards used in criminal law. The overwhelming majority of this literature has focused on those applied in the very final stage of the criminal proceedings, namely when deciding either to convict or acquit defendants (see, for instance, Mungan 2011; Rizzolli and Stanca 2012; Rizzolli and Saraceno 2013; Nicita and Rizzolli 2014; Garoupa 2017; Mungan 2017; Obidzinsky and Oytana 2019). There is hardly any paper on the standards established for previous stages.

If we are not wrong, only Mungan (2018) has analyzed the optimal standards for “stops” (i.e., for pre-trial decisions aimed at harm prevention) and shown that they are weaker than their analogs in trials, which require preponderance of the evidence in civil cases (as a general rule) and proof beyond a reasonable doubt in criminal ones. He also has demonstrated that suspicion-less stops can be optimal under certain circumstances

[1]

17 The U.S. Supreme Court has held permissible detention programs with success rates of 1.5%1.5 \% [Michigan Department of State Police v. Sitz, 496 U.S. 444, 455 (1990)] and even 0.12% [United States v. Martinez-Fuerte, 428 U.S. 543, 554 (1976)]. ↩︎

and, in particular, more likely optimal when: the inconvenience costs derived from stopping people are small; the percentage of the population attempting criminal offenses is high; law enforcers perform poorly in forming suspicions in that regard; and individuals are relatively unresponsive to expected sanctions.

This paper contributes to the law and economics literature on the standards of proof by extending the analysis to some criminal law enforcement decisions that (i) are made before trial (ii) on whether the criminal investigation (proceedings) against a person should go ahead or be terminated, and (iii) that produce only deterrent effects.

For the sake of illustration, we specifically consider the case of charging standards, although both the model and the results presented here can be smoothly applied to the standards of proof established to make other pre-trial decisions aimed at enforcing criminal law (e.g., the decision to initiate an investigation against a suspect) insofar as the following conditions are met: (i) after collecting some preliminary evidence, the relevant authority has to decide either to terminate the procedure or to continue it and move to the next procedural stage; (ii) choosing the latter alternative implies direct costs for certain individuals who are suspected of committing a crime and, eventually, for the rest of society; and (iii) in the next procedural stage(-s), the relevant authority can elicit additional information and distinguish between innocent and guilty individuals better than in the prior stage(-s), in order either to acquit or to convict them (or, eventually, to make another pretrial decision).

The paper proceeds as follows. Section II presents the basic model. Section III shows why the standard of evidence required to convict accused individuals is so high (“beyond any reasonable doubt”). The paper builds on Rizzolli and Saraceno (2013), who have shown that the costs of punishment can well explain the high standard of evidence required to reach a verdict of conviction. The rationale is that there is an asymmetry between the costs of wrongful conviction and wrongful acquittal. Both types of errors have the same negative impact on deterrence, but the former are costlier because of the direct costs of punishment. Section IV shows why the optimal standards of evidence required to investigate or charge suspects are lower than those established for convicting them. We will see that, when deciding on whether or not to adopt such measures against those suspected of committing criminal offenses, type II errors have a more intense impact on deterrence than do type I errors. We will also analyze how those optimal standards change depending on several factors. Section V concludes.

2. BASIC MODEL

Let us suppose that individuals choose between committing an offense and complying with the law by comparing the private benefits and costs they expect to derive from crime (Becker, 1968; Shavell and Polinsky, 2000).

2.1. Private Gains from Crime

Offenders get some benefits from engaging in illegal activities. Let wjw_{j} be the expected private gain individual jj obtains from committing a crime. It must be noted that wjw_{j} is an assessment of a subjective nature, which varies across individuals as a result of the existing differences between them in preferences, moral constraints, reputation concerns, risk aversion, opportunity costs of criminal activity, etc. A significant share of the population would thus probably experience low utility from committing crime while some individuals place a higher value on it. Let us assume that wjw_{j} is randomly distributed

according to a probability density function z(w)z(w) and a cumulative distribution function Z(w)Z(w). Therefore, normalizing population to 1,Z(w0)1, Z\left(w_{0}\right) represents the share of the population that would obtain gains from crime that are lower than w0w_{0}, while 1−Z(w0)1-Z\left(w_{0}\right) denotes the share of the population that would derive benefits from it higher than w0w_{0}.

Let RR denote, for simplicity reasons, the crime rate for the entire population. That is,

R=1−Z(wˉ)R=1-Z(\bar{w})

2.2. Probability of Being Investigated, Charged and Convicted

Law enforcers investigate only a part of the total population. Both criminals and law-abiding citizens thus face some probability of being investigated as a suspects of having committed an offense. Let pgp_{g} denote such probability for guilty individuals, and pip_{i} the probability for innocent ones. It is realistic to assume that pg≥pip_{g} \geq p_{i}

Before trial, there is certain net incriminating evidence ebe_{b} against individuals who are being investigated, whom we call “suspects.” The collection of this evidence is a costless activity and its amount only depends on whether or not they committed the offense at issue. We thus consider ebe_{b} as an exogenous random variable distributed according to two probability density distributions conditional on whether suspects are guilty or innocent. Let gb(eb)g_{b}\left(e_{b}\right) and ib(eb)i_{b}\left(e_{b}\right) denote the density functions of incriminating evidence ebe_{b} for guilty and innocent suspects, respectively, and Gb(eb)G_{b}\left(e_{b}\right) and Ib(eb)I_{b}\left(e_{b}\right) the corresponding cumulative distribution functions. We may reasonably assume that Ib(eb)I_{b}\left(e_{b}\right) first-order stochastically dominates Gb(eb)G_{b}\left(e_{b}\right), that is, Ib(eb)>Gb(eb)∀eb>0I_{b}\left(e_{b}\right)>G_{b}\left(e_{b}\right) \quad \forall e_{b}>0, which implies that the amount of incriminating evidence existing against factually guilty suspects before the trial is, on average, larger than the amount against innocent ones.

Suspects are indicted or charged if such net incriminating evidence ebe_{b} exceeds certain legal standard e^b\hat{e}_{b}. Otherwise, they are not. The probability of a factually guilty suspect being charged is therefore pg[1−Gb(e^b)]p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right] and that of an innocent one pi[1−Ib(e^b)]p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right].

We define the term “trial” in a very broad, non-technical sense, as the process carried out in order to determine the innocence or guilt of the person charged with a crime (i.e., the defendant) and, correspondingly, either to acquit (or terminate the proceedings) or to convict (or, eventually, continue the proceedings). It must be noticed that, in this sense, every defendant is “tried,” and that some “trials” take place, not before a Court, but in front of a police officer or a prosecutor.

During trial, a certain amount of net incriminating evidence eae_{a} is collected against the defendant. This amount could well depend on several factors, such as the personal abilities of the defense attorney and the prosecutor, the resources invested by them in the production of evidence, etc. We suppose, nevertheless, that eae_{a} only depends on whether or not the defendant committed the offense as charged. We also model eae_{a} as an exogenous random variable distributed according to two unimodal probability density functions conditional on whether the defendant is guilty or innocent. Let ga(ea)g_{a}\left(e_{a}\right) and ia(ea)i_{a}\left(e_{a}\right) denote the probability density functions of incriminating evidence eae_{a} for guilty and innocent defendants, respectively, and Ga(ea)G_{a}\left(e_{a}\right) and Ia(ea)I_{a}\left(e_{a}\right) the cumulative distributions. As before trial, it is assumed that Ia(ea)I_{a}\left(e_{a}\right) first-order stochastically dominates Ga(ea)G_{a}\left(e_{a}\right), that is, Ia(ea)>Ga(ea)∀ea>0I_{a}\left(e_{a}\right)>G_{a}\left(e_{a}\right) \quad \forall e_{a}>0. This implies that the amount of

incriminating evidence generated against a factually guilty defendant is, on average, larger than the amount collected against an innocent one.

It must be underlined that ebe_{b} and eae_{a} are not the same variables, nor are gb(eb)g_{b}\left(e_{b}\right) and ib(eb)i_{b}\left(e_{b}\right) the same functions as ga(ea)g_{a}\left(e_{a}\right) and ia(ea)i_{a}\left(e_{a}\right) respectively, although they could be related. The same applies to Gb(eb),Ib(eb),Ga(ea)G_{b}\left(e_{b}\right), I_{b}\left(e_{b}\right), G_{a}\left(e_{a}\right), and Ia(ea)I_{a}\left(e_{a}\right).

Once the trial has concluded, defendants are convicted and punished only if incriminating evidence exceeds certain standard e~a\tilde{e}_{a}. Otherwise, they are acquitted. Given a standard e~a\tilde{e}_{a}, the probability of a guilty defendant being convicted is 1−Ga(e~a)1-G_{a}\left(\tilde{e}_{a}\right) and that of an innocent one is 1−Ia(e~a)1-I_{a}\left(\tilde{e}_{a}\right).

The probability of a guilty individual being investigated, charged, and convicted is therefore pg[1−Gb(e~b)][1−Ga(e~a)]p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]\left[1-G_{a}\left(\tilde{e}_{a}\right)\right], and that of her or him being investigated, charged, and acquitted is pg[1−Gb(e~b)]Ga(e~a)p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right] G_{a}\left(\tilde{e}_{a}\right). Analogously, the probability of an innocent individual being investigated, charged, and convicted is pi[1−Ib(e~b)][1−p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right][1- Ia(e~a)]\left.I_{a}\left(\tilde{e}_{a}\right)\right], and that of her or him being investigated, charged, and acquitted is pi[1−Ib(e~b)]Ia(e~a)p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right] I_{a}\left(\tilde{e}_{a}\right).

2.3. Direct Costs from Charging Suspects and Convicting Defendants

Charging a suspect with a criminal offense entails a direct social cost CbC_{b} that equals the costs borne by this person, cbpc_{b p}, plus those borne by the rest of society, cbsc_{b s}. The former arise from: loss of reputation and self-esteem; emotional distress; demoralization effects of being publicly humiliated; time and money spent on hiring lawyers, etc. The latter might include, for instance, negative psychological, stigmatizing and monetary impacts on family members of individuals charged with crimes, and financial costs incurred by the State in order to carry out further proceedings aimed at investigating the facts and punishing the guilty, etc.

Analogously, punishment implies a direct social cost, CaC_{a}, which equals the sum of the costs borne by the convicted defendant, capc_{a p}, and the rest of society, casc_{a s}. We assume that both costs are higher than those derived from charging suspects with the same crimes: cbp≪capc_{b p} \ll c_{a p} and cbs≪casc_{b s} \ll c_{a s}.

2.4. Decision to Commit Crime

Individuals who are supposed to be rational and risk neutral will commit a crime if the expected private gains from infringing the law exceed the private costs of obeying it. The critical threshold at which those gains become larger than these costs can be denoted by w~\widetilde{w}. Individuals choose to commit crime if their expected net payoff from infringing the law exceeds their net payoff from obeying it, i.e., if:

wj−pg[1−Gb(e~b)]cbp−pg[1−Gb(e~b)][1−Ga(e~t)]cap>−pi[1−Ib(e~b)]cbp−pi[1−Ib(e~b)][1−Ia(e~a)]cap\begin{aligned} & w_{j}-p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right] c_{b p}-p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]\left[1-G_{a}\left(\tilde{e}_{t}\right)\right] c_{a p}>-p_{i}[1- \\ & \left.I_{b}\left(\tilde{e}_{b}\right)\right] c_{b p}-p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] c_{a p} \end{aligned}

That is, individuals engage in crime if wj>w~w_{j}>\widetilde{w} where

w~={pg[1−Gb(e~b)]−pi[1−Ib(e~b)]}cbp+{pg[1−Gb(e~b)][1−Ga(e~a)]−pi[1−Ib(e~b)][1−Ia(e~a)]}cap\begin{aligned} & \widetilde{w}=\left\{p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]-p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]\right\} c_{b p}+\left\{p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]\left[1-G_{a}\left(\tilde{e}_{a}\right)\right]-\right. \\ & \left.p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]\left[1-I_{a}\left(\tilde{e}_{a}\right)\right]\right\} c_{a p} \end{aligned}

3. OPTIMAL STANDARDS OF EVIDENCE FOR CONVICTION

3.1. Standard for Conviction that Maximizes Deterrence

Raising the standard of evidence for conviction produces a double effect on deterrence and, subsequently, on the volume of criminal activity. On the one hand, it increases the likelihood of guilty individuals being acquitted and, therefore, the net payoff from committing offenses, thereby pushing up the crime rate. On the other hand, it also lowers the probability of innocent individuals being wrongfully convicted, which reduces the costs of abiding by the law, which in turn pushes the crime rate down.

Proposition 1. The standard of evidence for conviction that maximizes deterrence e~a∗\tilde{e}_{a}^{*} is such that the marginal probability of being acquitted is the same for both innocent and guilty defendants, that is,

pi[1−Ib(e~b)]ia(e~a∗)=pg[1−e~b)]ga(e~a∗)\begin{gathered} p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right] i_{a}\left(\tilde{e}_{a}^{*}\right)=p_{g}[1- \\ \left.\left.\tilde{e}_{b}\right)\right] g_{a}\left(\tilde{e}_{a}^{*}\right) \end{gathered}

First order condition to minimize the crime rate (i.e., maximize deterrence) with respect to the standard for conviction is:

dRde~a=−z(w~){pi[1−Ib(e~b)]ia(e~a)−pg[1−Gb(e~b)]ga(e~a)}(cap−cbp)=0\frac{d R}{d \tilde{e}_{a}}=-z(\widetilde{w})\left\{\begin{array}{c} p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right] i_{a}\left(\tilde{e}_{a}\right) \\ -p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right] g_{a}\left(\tilde{e}_{a}\right) \end{array}\right\}\left(c_{a p}-c_{b p}\right)=0

This obviously leads to expression [3]. This is also the result we obtain if we derive w~\widetilde{w}-i.e., expression [2]-with respect to e~a\tilde{e}_{a} and equate to zero.

We can assume that there is at least one standard of evidence for conviction e~a\tilde{e}_{a} that allows law enforcers to distinguish between innocent and guilty individuals more accurately after the trial (when deciding on their conviction) than before (when charging them with a crime). That is,

Ia(e~a∗)Ga(e~a∗)>pg[1−Gb(e~b)]pi[1−Ib(e~b)]\frac{I_{a}\left(\tilde{e}_{a}^{*}\right)}{G_{a}\left(\tilde{e}_{a}^{*}\right)}>\frac{p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]}{p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]}

This means that there is at least one standard of evidence for conviction e~a\tilde{e}_{a} such that the probability of being investigated, indicted, and acquitted is greater for innocent individuals than for guilty ones, i.e., pi[1−Ib(e~b)]Ia(e~a)>pg[1−Gb(e~b)]Ga(e~a)p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right] I_{a}\left(\tilde{e}_{a}\right)>p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right] G_{a}\left(\tilde{e}_{a}\right).

And it also implies that there is at least one standard of evidence for conviction e~a\tilde{e}_{a} such that

Ia(e^a)ga(e^a)∼pg[1−Gb(e^b)]pi[1−Ib(e^b)]\frac{I_{a}\left(\hat{e}_{a}\right)}{g_{a}\left(\hat{e}_{a}\right)} \sim \frac{p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right]}{p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right]}

If this were not the case, the trial would be entirely pointless, and every defendant would be convicted. Indeed, if pi[1−Ib(e^b)]Ia(e^a)−pg[1−Gb(e^b)]Ga(e^a)<0p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right] I_{a}\left(\hat{e}_{a}\right)-p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right] G_{a}\left(\hat{e}_{a}\right)<0 ∀e^a>0\forall \hat{e}_{a}>0, then the standard for conviction minimizing crime rate would be e^a∗=0\hat{e}_{a}^{*}=0.

Given that, as Ga(ea)G_{a}\left(e_{a}\right) and Ia(ea)I_{a}\left(e_{a}\right) are continuous and differentiable over the interval (0,∞)(0, \infty), and G(e)≤I(e)∀eG(e) \leq I(e) \forall e, there exists a standard of incriminating evidence for conviction that maximizes deterrence e^a∗\hat{e}_{a}^{*} such that [3] holds, i.e., the marginal probability of a guilty individual being acquitted equals the marginal probability of an innocent one being acquitted.

Proposition 2. The negative impact on deterrence is the same for both type I errors and type II errors. 18{ }^{18}

Although the trial discriminates guilty from innocent individuals better than previous procedural stages, courts still make mistakes. They sometimes commit type I errors (i.e., wrongful convictions) or type II errors (i.e., wrongful acquittals). The probabilities of courts committing such type I or type II errors are, respectively,

Ela=pi[1−Ib(e^b)][1−Ella=pg[1−Gb(e^b)]Ga(e^a)\begin{aligned} & E_{l}^{a}=p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right][1- \\ & E_{l l}^{a}=p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right] G_{a}\left(\hat{e}_{a}\right) \end{aligned}

One can see that the crime rate increases equally with equal increases in the probabilities of both types of errors. Indeed,

dRdEla=dRElla=z(wˉ)(cap−cbp)\frac{d R}{d E_{l}^{a}}=\frac{d R}{E_{l l}^{a}}=z(\bar{w})\left(c_{a p}-c_{b p}\right)

This means that deterrence is maximized by minimizing the sum of both types of errors.

As Rizzolli and Saraceno (2013) have shown, punishment costs can at least explain the high standard of evidence required to reach a verdict of guilty. The rationale is that

[1]

18 In the law and economics literature, this result was first pointed out by Png (1986). However, some authors have questioned that wrongful convictions lower deterrence. See Lando (2006), and Lando and Mungan (2018). In favor of the standard result, see Garoupa and Rizzolli (2012). See also Mungan (2017), and Obidzinski and Oytana (2019). ↩︎

there is an asymmetry between the costs of wrongful conviction and wrongful acquittal. Both types of errors have the same negative impact on deterrence, 19{ }^{19} but the former are socially costlier because of the costs of punishment. 20{ }^{20} Let us examine this.

Criminal offenses directly cause harm to society. Let us suppose that such harm is always higher than the private gains derived from infringing the law, and that crime always implies a direct, net social cost, which we denote by hh. Let us also assume that the total net harm crime directly causes society is proportional to the number of offenses committed. For simplicity reasons, we normalize the population to 1 , so the number of offenders equals the crime rate RR and the number of innocent individuals is 1−R1-R.

Besides the harm directly resulting from offenses, crime makes it necessary to incur the costs of enforcing the law, that is of charging suspects, Cb=cbp+cbsC_{b}=c_{b p}+c_{b s}, and punishing them, Ca=cap+casC_{a}=c_{a p}+c_{a s}. Let us denote TT as the sum of all the social costs caused by crime and deterrence of crime, which include:

Direct costs of crime
RhR h
Costs of charging suspects with committing a crime

{Rpg[1−Gb(eˉb)]+(1−R)pi[1−Ib(eˉb)]}Cb\left\{R p_{g}\left[1-G_{b}\left(\bar{e}_{b}\right)\right]+(1-R) p_{i}\left[1-I_{b}\left(\bar{e}_{b}\right)\right]\right\} C_{b}

And costs of conviction

{Rpg[1−Gb(eˉb)][1−Ga(eˉa)]+(1−R)pi[1−Ib(eˉb)][1−Ia(eˉa)]}Ca\left\{R p_{g}\left[1-G_{b}\left(\bar{e}_{b}\right)\right]\left[1-G_{a}\left(\bar{e}_{a}\right)\right]+(1-R) p_{i}\left[1-I_{b}\left(\bar{e}_{b}\right)\right]\left[1-I_{a}\left(\bar{e}_{a}\right)\right]\right\} C_{a}

Therefore, total social cost derived from crime can be represented as follows:

T=Rh+{Rpg[1−Gb(eˉb)]+(1−R)pi[1−Ib(eˉb)]}Cb+{Rpg[1−Gb(eˉb)][1−Ga(eˉa)]+(1−R)pi[1−Ib(eˉb)][1−Ia(eˉa)]}}\left.\begin{array}{l} T=R h+\left\{\begin{array}{c} R p_{g}\left[1-G_{b}\left(\bar{e}_{b}\right)\right]+ \\ (1-R) p_{i}\left[1-I_{b}\left(\bar{e}_{b}\right)\right] \end{array}\right\} C_{b}+ \\ \left\{\begin{array}{c} R p_{g}\left[1-G_{b}\left(\bar{e}_{b}\right)\right]\left[1-G_{a}\left(\bar{e}_{a}\right)\right]+ \\ (1-R) p_{i}\left[1-I_{b}\left(\bar{e}_{b}\right)\right]\left[1-I_{a}\left(\bar{e}_{a}\right)\right] \end{array}\right\} \end{array}\right\}

This expression can also be written as

ParseError: KaTeX parse error: Expected & or \\ or \cr or \end at position 267: …t)\right] C_{a}\̲r̲i̲g̲h̲t̲]\}+p_{i}\left[…

Proposition 3. The standard of evidence for conviction that minimizes social costs, eˉax∗\bar{e}_{a}^{x *}, is higher than the standard for conviction that maximizes crime deterrence, eˉax\bar{e}_{a}^{x}. That is,

[1]

19 Nonetheless, if individuals are risk averse, wrongful convictions have a stronger negative impact on deterrence than wrongful acquittals. See Rizzolli and Stanca (2012).
We are not going to consider other factors which can also create an asymmetry between both types of errors and, therefore, justify such a high standard of evidence. See, for instance: Mungan (2011) with respect to the costs incurred by non-criminals in order to avoid false convictions; Nicita and Rizzolli (2014) for behavioral explanations; and Garoupa (2017), who argues that “a high standard of proof might be needed because law enforcers have stronger than average taste for punishment.” ↩︎

e^a++>e^a+\hat{e}_{a}^{++}>\hat{e}_{a}^{+}

Indeed, first-order condition is:

∂T∂e^a=∂R∂e^a{h+[pg[1−Gb(e^b)]−pi[1−Ib(e^b)]]Cb+[pg[1−Gb(e^b)][1−Ga(e^a)]−pi[1−Ib(e^b)][1−Ia(e^a)]]Ca}+R{pi[1−Ib(e^b)]ia(e^a)−pg[1−Gb(e^b)]ga(e^a)}Ca−pi[1−Ib(e^b)]ia(e^a)Ca=0\begin{aligned} \frac{\partial T}{\partial \hat{e}_{a}}=\frac{\partial R}{\partial \hat{e}_{a}}\{h & +\left[p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right]-p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right]\right] C_{b} \\ & +\left[p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right]\left[1-G_{a}\left(\hat{e}_{a}\right)\right]-p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right]\left[1-I_{a}\left(\hat{e}_{a}\right)\right]\right] C_{a}\} \\ & +R\left\{p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right] i_{a}\left(\hat{e}_{a}\right)-p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right] g_{a}\left(\hat{e}_{a}\right)\right\} C_{a} \\ & -p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right] i_{a}\left(\hat{e}_{a}\right) C_{a}=0 \end{aligned}

It must be noticed that the standard of evidence for conviction which maximizes deterrence e^a∗\hat{e}_{a}^{*} does not satisfy this equation. Indeed, if e^a=e^a∗\hat{e}_{a}=\hat{e}_{a}^{*}, then ∂R∂e^a=0\frac{\partial R}{\partial \hat{e}_{a}}=0, and, from [3], pi[1−Ib(e^b)]ia(e^a)−pg[1−Gb(e^b)]ga(e^a)=0p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right] i_{a}\left(\hat{e}_{a}\right)-p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right] g_{a}\left(\hat{e}_{a}\right)=0. In that case

∂T∂e^a∣e^a=e^a∗=−pi[1−Ib(e^b)]ia(e^a)Ca<\left.\frac{\partial T}{\partial \hat{e}_{a}}\right|_{\hat{e}_{a}=\hat{e}_{a}^{*}}=-p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right] i_{a}\left(\hat{e}_{a}\right) C_{a}<

This means that TT is still decreasing at the point where the standard of evidence for conviction minimizes crime rate. Consequently, TT can still be reduced if a standard higher than e^a∗\hat{e}_{a}^{*} is established.

Raising the standard of conviction within the interval [0,e^a∗]\left[0, \hat{e}_{a}^{*}\right] obviously reduces total social cost. As criminal activity decreases, both direct costs of crime and direct costs of charging and punishing people diminish. Above e^a∗\hat{e}_{a}^{*}, costs of punishment continue to converge to zero, but social harm resulting from the growth of the crime rate and costs of charging suspects start to rise at an increasing rate, which will finally lead to an increase of the total social costs.

Moreover, the possibility of a corner solution is to be ruled out. The total cost function cannot continuously decrease. Indeed, if the standard of evidence for conviction were so high that every indicted suspect were acquitted, e^amax⁡\hat{e}_{a}^{\max }, then 1−Ia(e^amax⁡)=01-I_{a}\left(\hat{e}_{a}^{\max }\right)=0, 1−Ga(e^amax⁡)=0,ia(e^amax⁡)=01-G_{a}\left(\hat{e}_{a}^{\max }\right)=0, i_{a}\left(\hat{e}_{a}^{\max }\right)=0, and ga(e^amax⁡)=0g_{a}\left(\hat{e}_{a}^{\max }\right)=0, in which case the first derivative of the total cost function would be positive:

∂T∂e^a∣e^a=e^amax⁡=∂R∂e^amax⁡{h+{pg[1−Gb(e^b)]−pi[1−Ib(e^b)]}Cb}>0\left.\frac{\partial T}{\partial \hat{e}_{a}}\right|_{\hat{e}_{a}=\hat{e}_{a}^{\max }}=\frac{\partial R}{\partial \hat{e}_{a}^{\max }}\left\{h+\left\{p_{g}\left[1-G_{b}\left(\hat{e}_{b}\right)\right]-p_{i}\left[1-I_{b}\left(\hat{e}_{b}\right)\right]\right\} C_{b}\right\}>0

OPTIMAL EVIDENTIARY STANDARDS FOR CHARGING INDIVIDUALS

4.1. Standard for Charging Individuals that Maximizes Deterrence

Raising the standard for charging suspects also has two effects on deterrence. First, it increases the likelihood of guilty individuals not being charged and, subsequently, not being punished, which in turn increases the net payoff from committing offenses, thereby pushing the crime rate up. Second, it also lowers the probability of innocent individuals being wrongfully charged and convicted, which reduces the costs of abiding by the law, which in turn pushes the crime rate down.

Proposition 4. If there is a positive evidentiary standard for charging suspects that maximizes deterrence, e~b∗\tilde{e}_{b}^{*}, i.e., there is an interior solution, this standard is such that the marginal expected cost derived from indictment is the same for both innocent and guilty individuals.

Indeed, first-order condition to minimize crime rate RR leads to the following equation:

piib(e~b∗){cbp+[1−Ia(e~a)]cpp}=pggb(e~b∗){cbp+[1−Ga(e~a)]cpp}p_{i} i_{b}\left(\tilde{e}_{b}^{*}\right)\left\{c_{b p}+\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] c_{p p}\right\}=p_{g} g_{b}\left(\tilde{e}_{b}^{*}\right)\left\{c_{b p}+\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] c_{p p}\right\}

Such marginal expected cost equals the marginal probability of being charged, which is piib(e~b∗)p_{i} i_{b}\left(\tilde{e}_{b}^{*}\right) for innocent suspects and pggb(e~b∗)p_{g} g_{b}\left(\tilde{e}_{b}^{*}\right) for guilty ones, times the personal cost from indictment, cbpc_{b p}, plus the expected cost of being convicted, which is [1−Ia(e~a)]cpp\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] c_{p p} for innocent defendants and [1−Ga(e~a)]cpp\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] c_{p p} for guilty ones.

Proposition 5. The evidentiary standard for indictment that maximizes deterrence might equal zero, which means that every investigated individual should be indicted.

Indeed, if equation [15] never holds, there is such a corner solution. Thus, every suspect is to be indicted if and only if:
pggb(e~b){cbp+[1−Ga(e~a)]cap}>piib(e~b){cbp+[1−Ia(e~a)]cap}∀e~b>0p_{g} g_{b}\left(\tilde{e}_{b}\right)\left\{c_{b p}+\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] c_{a p}\right\}>p_{i} i_{b}\left(\tilde{e}_{b}\right)\left\{c_{b p}+\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] c_{a p}\right\} \forall \tilde{e}_{b}>0

Proposition 6. Wrongful non-indictments produce a stronger negative impact on deterrence than wrongful indictments.

When charging suspects, crime rate increases more with an increase in the probability of type II errors than with an equal increase in the probability of type I errors. Let the probability of a law enforcer committing such a type I error be

Elb=pi[1−Ib(e~b)]E_{l}^{b}=p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]

And the probability of her committing an analogous type II error be

Elib=pgGb(e~b)[18]E_{l i}^{b}=p_{g} G_{b}\left(\tilde{e}_{b}\right)[18]

It is easy to see that the marginal effects on crime rate of an increase in the probability of type I errors and type II errors are, respectively,

dRdEIR=z(wˉ){cbp+[1−Ia(e^a)]cap}[19]dRdEIIR=z(wˉ){cbp+[1−Ga(e^a)]cap}[20]\begin{aligned} & \frac{d R}{d E_{I}^{R}}=z(\bar{w})\left\{c_{b p}+\left[1-I_{a}\left(\hat{e}_{a}\right)\right] c_{a p}\right\}[19] \\ & \frac{d R}{d E_{I I}^{R}}=z(\bar{w})\left\{c_{b p}+\left[1-G_{a}\left(\hat{e}_{a}\right)\right] c_{a p}\right\}[20] \end{aligned}

Given that Ga(e^a)<Ia(e^a)G_{a}\left(\hat{e}_{a}\right)<I_{a}\left(\hat{e}_{a}\right), then

dRdEIIR>dRdEIR\frac{d R}{d E_{I I}^{R}}>\frac{d R}{d E_{I}^{R}}

The intuition behind this result is straightforward. Type I errors certainly reduce the benefits of obeying the law, cbp+[1−Ia(e^a)]capc_{b p}+\left[1-I_{a}\left(\hat{e}_{a}\right)\right] c_{a p} being the reduction for each mistake, but not so much as type II errors increase the benefits of engaging in crime, cbp+[1−Ga(e^a)]capc_{b p}+\left[1-G_{a}\left(\hat{e}_{a}\right)\right] c_{a p} being the marginal increase. When charged, individuals bear two types of costs: those directly derived from being charged and the expected costs of being eventually convicted. The former are equal for both guilty and innocent defendants, cbpc_{b p}. The latter, on the contrary, are higher for the former, insofar as their probability of being convicted after the trial is also higher for them, 1−Ga(e^a)>1−Ia(e^a)1-G_{a}\left(\hat{e}_{a}\right)>1-I_{a}\left(\hat{e}_{a}\right).

The corollary is clear: If lawmakers and enforcers want to maximize deterrence, they should try to specially avoid letting guilty individuals escape, rather than to prevent innocent ones from being charged. Even more, as we have seen, under some circumstances there is a corner solution, so that every investigated individual should be charged, which implies not committing any type II errors but only type I errors.

4.2. Comparative Statics

In view of equation [15], it is not very difficult to predict how the standard for charging suspects that maximizes deterrence will change if other variables change. Factors increasing expected costs for guilty individuals will lower such standard, whereas factors increasing those costs for innocent ones will raise it.

Thus, e^b∗\hat{e}_{b}^{*} will be lower: (1) the lower (higher) the probability of innocent (guilty) individuals being investigated, pi(pg^)p_{i}\left(p_{\hat{g}}\right); (2) the higher (lower) the probability of innocent (guilty) individuals being acquitted, Ia(e^a)(Ga(e^a))I_{a}\left(\hat{e}_{a}\right)\left(G_{a}\left(\hat{e}_{a}\right)\right); (3) the lover the direct private costs of charging suspects, cbpc_{b p}, and the higher the direct private costs of conviction cppc_{p p}. Our comparative statics analysis provides, indeed, the following results:
(i) If the probability of factually guilty individuals being investigated increases (as a consequence, for instance, of an improvement of the criteria applied to determine the target population), then the charging standard that minimizes crime rate will be lower. As more criminals now enter into the investigation pool, the marginal benefits of reducing that standard in order to catch them are higher than before. Conversely, if the probability of innocent individuals being investigated increases, then the charging standard that maximizes deterrence will be higher.

∂e^b∗∂pi>0, and ∂e^b∗∂pg<0\frac{\partial \hat{e}_{b}^{*}}{\partial p_{i}}>0, \text { and } \frac{\partial \hat{e}_{b}^{*}}{\partial p_{g}}<0

Therefore, the higher the ratio pgpi\frac{p_{g}}{p_{i}}, i.e., the better law enforcers perform when targeting the subset of the population being investigated (i.e., the higher the proportion of guilty individuals included in this subset), the lower the standard e^b∗\hat{e}_{b}^{*}.
(ii) If the proportion of factually innocent defendants being acquitted increases (for instance, because of an improvement in the evidentiary technology used at trial), the standard for charging suspects that minimizes the crime rate will be lower. On the contrary, if the probability of acquitting guilty individuals increases, that standard will rise.

∂e^b∗∂ta<0, and ∂e^b∗∂σa>0\frac{\partial \hat{e}_{b}^{*}}{\partial t_{a}}<0, \text { and } \frac{\partial \hat{e}_{b}^{*}}{\partial \sigma_{a}}>0

Therefore, the higher the ratio Ga(e^a)Ia(e^a)\frac{G_{a}\left(\hat{e}_{a}\right)}{I_{a}\left(\hat{e}_{a}\right)}, i.e., the less accurately the trial allows distinguishing between guilty and innocent individuals, the higher the standard e^b∗\hat{e}_{b}^{*}.
(iii) If private direct costs of being charged decrease, the asymmetry between wrongful non-charging decisions and wrongful charging decisions becomes more acute, as the expected costs of being convicted (which are higher for guilty defendants than for innocent ones) intensify their positive impact on deterrence, thereby reducing the standard for charging suspects that minimizes the crime rate. And when private costs of punishment increase, deterrence is also improved, reducing the standard for indictment that maximizes deterrence.

∂e^b∗∂cbp>0, and ∂e^b∗∂cap<0\frac{\partial \hat{e}_{b}^{*}}{\partial c_{b p}}>0, \text { and } \frac{\partial \hat{e}_{b}^{*}}{\partial c_{a p}}<0

Therefore, the higher the ratio capcbp\frac{c_{a p}}{c_{b p}}, i.e., the ratio between private direct costs of conviction and private direct costs of being charged, the lower the standard e^b∗\hat{e}_{b}^{*}.

Analogously, a corner solution, where e^b∗=0\hat{e}_{b}^{*}=0 and, therefore, every individual being investigated is charged, will exist when the ratios pgpi:[1−Ga(e^a)][1−ta(e^a)]\frac{p_{g}}{p_{i}}: \frac{\left[1-G_{a}\left(\hat{e}_{a}\right)\right]}{\left[1-t_{a}\left(\hat{e}_{a}\right)\right]} and capcbp\frac{c_{a p}}{c_{b p}} are sufficiently high.

Proposition 7. If there exists a positive standard of evidence for charging suspects that minimizes the crime rate (i.e., an interior solution), this standard is lower than that which maximizes social cost.

Indeed, first-order condition for the evidentiary standard for indictment that minimizes social costs e~b∗∗\tilde{e}_{b}^{* *} is:

∂T∂e~b=∂R∂e~b{h+pg[1−Gb(e~b)][CB+[1−Ga(e~a)]CA]}−pi[1−Ib(e~b)][CB+[1−Ia(e~a)]CA]}+R{piib(e~b)[CB+[1−Ia(e~a)]CA]−pggb(e~b)[CB+[1−Ga(e~a)]CA]}−piib(e~b)[CB+[1−Ia(e~a)]CA]\begin{aligned} \frac{\partial T}{\partial \tilde{e}_{b}}=\frac{\partial R}{\partial \tilde{e}_{b}}\left\{h+\right. & \left.p_{g}\left[1-G_{b}\left(\tilde{e}_{b}\right)\right]\left[C_{B}+\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\right\} \\ & -p_{i}\left[1-I_{b}\left(\tilde{e}_{b}\right)\right]\left[C_{B}+\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\} \\ & +R\left\{p_{i} i_{b}\left(\tilde{e}_{b}\right)\left[C_{B}+\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\right. \\ & \left.-p_{g} g_{b}\left(\tilde{e}_{b}\right)\left[C_{B}+\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\right\}-p_{i} i_{b}\left(\tilde{e}_{b}\right)\left[C_{B}+\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right] \end{aligned}

Let us note that the standard for charging suspects which maximizes deterrence e~b∗\tilde{e}_{b}^{*} does not satisfy this equation. Indeed, if e~b=e~b∗\tilde{e}_{b}=\tilde{e}_{b}^{*}, then

∂T∂eb∣e~b=e~b∗=−R{pggb(e~b∗)[CB+[1−Ga(e~a)]CA]}−(1−R){piib(e~b∗)[CB+[1−Ia(e~a)]CA]}<\begin{aligned} & \left.\frac{\partial T}{\partial e_{b}}\right|_{\tilde{e}_{b}=\tilde{e}_{b}^{*}}=-R\left\{p_{g} g_{b}\left(\tilde{e}_{b}^{*}\right)\left[C_{B}+\left[1-G_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\right\}-(1-R)\left\{p_{i} i_{b}\left(\tilde{e}_{b}^{*}\right)\left[C_{B}+\right.\right. \\ & \left.\left.\left[1-I_{a}\left(\tilde{e}_{a}\right)\right] C_{A}\right]\right\}< \end{aligned}

Therefore, the total social costs function is still decreasing at the point where the standard of evidence of charging suspects minimizes crime rate. Hence, we can conclude that, if there is an interior solution, the evidentiary standard for charging suspects that minimizes social costs is higher than that which minimizes the crime rate. That is,

e~b∗∗>e~b∗\tilde{e}_{b}^{* *}>\tilde{e}_{b}^{*}

Raising the standard for indictment within the interval [0,e~b∗[\left[0, \tilde{e}_{b}^{*}\right[ obviously reduces both the crime rate and the total social cost. There are less offenses, less indictments, and less convictions. If, by contrast, it is raised above e~b∗\tilde{e}_{b}^{*}, costs of indictment and punishment will continue to (either asymptotically or monotonically) converge to zero, but social harm resulting from the growth of the crime rate will rise at an increasing rate, which will finally lead to an increase of the total social costs.

Proposition 8. The evidentiary standard for indictment that minimizes total cost cannot be so high that no one is indicted (nor convicted).

Indeed, if such standard is that high, e~bmax⁡\tilde{e}_{b}^{\max }, then 1−Ib(e~bmax⁡)=0,1−1-I_{b}\left(\tilde{e}_{b}^{\max }\right)=0,1- Gb(e~bmax⁡)=0,ib(e~bmax⁡)G_{b}\left(\tilde{e}_{b}^{\max }\right)=0, i_{b}\left(\tilde{e}_{b}^{\max }\right), and ib(e~bmax⁡)=0i_{b}\left(\tilde{e}_{b}^{\max }\right)=0, and then:

∂T∂e~b∣e~b=e~bmax⁡=∂R∂e~bh>0\left.\frac{\partial T}{\partial \tilde{e}_{b}}\right|_{\tilde{e}_{b}=\tilde{e}_{b}^{\max }}=\frac{\partial R}{\partial \tilde{e}_{b}} h>0

Total cost is thus increasing at e~bmax⁡\tilde{e}_{b}^{\max }, which means that it might decrease if a lower evidentiary standard for charging suspects is established.

Proposition 9. The evidentiary standard for indictment that maximizes social cost might equal zero, so that every investigated individual is to be indicted.

A necessary, but not sufficient, condition for this result is that the standard for indictment that maximizes deterrence equals zero. If this standard is positive, the former one will be, too. Let us recall that if e~b∗>0\tilde{e}_{b}^{*}>0, then e~b∗∗>e~b∗\tilde{e}_{b}^{* *}>\tilde{e}_{b}^{*}.

For that corner solution to exist, the marginal harm directly caused by crime must be sufficiently high to compensate the reduction in indictment and conviction costs that an increase in the standard for charging suspects would imply. In particular, e~b∗∗=0\tilde{e}_{b}^{* *}=0 if and only if

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4.4. Comparative Statics

As total cost always decreases when the crime rate falls, it is easy to conclude that factors increasing deterrence reduce total cost. We can thus expect that changes in variables that reduce the standard for charging people that maximizes deterrence also reduce the standard for charging people that minimizes total costs, and it is quite intuitive to guess the effect of other factors not affecting deterrence.

The social cost of charging and convicting individuals obviously makes the cost of law enforcement rise without affecting the crime rate, so we can expect that the higher the standard for charging suspects that minimizes total cost, the higher the social cost of both charging and convicting individuals. And it seems quite obvious that the higher the social harm caused by crime, the higher should be the efforts to reduce the crime rate and the lower should be the optimal standard of evidence for charging suspects.

Indeed, let us see the effect of each variable using comparative static analysis.
(i) An increase in the social net harm caused by each offense obviously increases direct costs of crime without affecting either deterrence or the costs of charging and convicting people. Such increase thus makes it more costly to raise the charging standard above the point where deterrence is maximized, e~b∗\tilde{e}_{b}^{*}, thereby pushing down the charging standard that minimizes total social costs, e~b∗∗\tilde{e}_{b}^{* *}.

∂eb∗∗∂h<0[30]\frac{\partial e_{b}^{* *}}{\partial h}<0[30]

(ii) Increases in the social costs of either charging or punishing individuals obviously makes the costs of law enforcement rise, but it does not affect crime levels. If indicting or punishing people becomes costlier, without deterring additional offenses, the respective optimal standards will obviously go up.

∂eb∗∗∂cbs>0, and ∂eb∗∗∂cas>0[31]\frac{\partial e_{b}^{* *}}{\partial c_{b s}}>0, \text { and } \frac{\partial e_{b}^{* *}}{\partial c_{a s}}>0[31]

(iii) Increases in the private costs of individuals being charged will have two effects. On the one hand, as already noted, deterrence will diminish, thereby increasing the direct costs of crime. On the other hand, the costs of law enforcement will increase, as more people will be charged and convicted. Both effects will push up the socially optimal standard for indictment. By contrast, increases in the private cost of punishment reduce the crime rate, but their impact on law enforcement cost is not so evident. Given our assumption that the probability of being investigated, indicted, and acquitted is greater for innocent persons than for guilty ones, we might conclude that increasing the private cost of conviction reduces the standard for charging suspects that minimizes total social cost.

∂eb∗∗∂cbp>0, and ∂eb∗∗∂cap<0\frac{\partial e_{b}^{* *}}{\partial c_{b p}}>0, \text { and } \frac{\partial e_{b}^{* *}}{\partial c_{a p}}<0

(iv) Increases in the probability of innocent defendants being acquitted IaI_{a}, improve deterrence and unambiguously reduce the number of individuals being indicted and punished, thereby pushing down the standard for indictment that minimizes social cost. By contrast, an increase in the probability of guilty defendants being acquitted GaG_{a} produces two opposite effects: first, it reduces the number of convictions and, subsequently, punishment costs; second, it undermines deterrence, which increases both the direct cost of crime and the number of indictments and convictions. It is not obvious which effect is stronger, but it can be demonstrated that the second one is. Therefore,

∂eb∗∗∂Ia<0, and ∂eb∗∗∂Ga>0[33]\frac{\partial e_{b}^{* *}}{\partial I_{a}}<0, \text { and } \frac{\partial e_{b}^{* *}}{\partial G_{a}}>0[33]

(v) Similarly, increases in the probability of innocent citizens being investigated lower deterrence and clearly augment the number of individuals being indicted and convicted, which raises the optimal standard for indictment. By contrast, an increase in the probability of guilty individuals being investigated produces two effects: first, it increases the number of indictments and convictions, thereby increasing law enforcement costs; second, it boosts deterrence, which reduces both direct cost of crime and indictment and punishment costs. It can also be demonstrated that the latter effect is stronger.

∂eb∗∗∂pi>0, and ∂eb∗∗∂pg<0\frac{\partial e_{b}^{* *}}{\partial p_{i}}>0, \text { and } \frac{\partial e_{b}^{* *}}{\partial p_{g}}<0

5. CONCLUSIONS

It is difficult to directly compare the standards for pre-trial decisions with those for conviction, since they are related to different kinds of evidence. Nevertheless, it is reasonable to assert that the former is “lower” than the latter.

Countless legal provisions and opinions suggest so. It has been said that a very high standard of evidence is established in order to reach a verdict of guilty. Defendants may not be convicted unless proven guilty “beyond a reasonable doubt.” By contrast, a lower standard of evidence is required when public agents adopt measures aimed at investigating crimes or preventing the risk of suspects fleeing and avoiding prosecution or punishment, destroying evidence, influencing witnesses, or committing any additional offense. Then it is not necessary for the prosecution to prove that they are guilty beyond any reasonable doubt.

This paper contributes to the law and economics literature on the standards of proof by extending the analysis to pre-trial decisions on whether the criminal investigation against a person should go ahead or be terminated.

Taking decisions to charge as an illustrative example, the present paper tries to explain why the evidentiary standard for making such pre-trial decisions is relatively low. In particular, we show that false negatives (e.g., wrongful non-indictments) have a more intense impact on deterrence than false positives (e.g., wrongful indictments). The decision to charge (i.e., investigate, prosecute, charge, etc.) entails two types of costs for the affected individuals: those immediately derived from it and those of an eventual conviction. The former costs are the same for both guilty and innocent individuals. The latter costs, on the contrary, are higher for guilty individuals, since their probability of conviction is, on average, higher as well. False negatives have, consequently, a more negative impact on deterrence than false positives. Type I errors certainly reduce the benefits of obeying the law, but not so much as type II errors increase the benefits of engaging in crime. If lawmakers and enforcers want to maximize deterrence, they should therefore try to specially avoid letting guilty individuals escape, rather than to prevent innocent ones from being charged.

The paper shows that the evidentiary standard for indicting suspects that maximizes deterrence will be lower (or a corner solution where every suspect is indicted will be more likely) as: private costs of being indicted (punished) decrease (increase); the probability of innocent (guilty) defendants being acquitted increases (decrease); the probability of innocent (guilty) individuals being investigated decreases (increases); and acquitted defendants are compensated for the harm suffered as a consequence of being charged.

Changes in those variables produce the same effect on the standard for indictments that minimize total social costs. Moreover, this standard will be lower as: the harm directly caused by crime increases; and the social costs of indictment (punishment) decrease (increase).

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Better that ten innocent persons suffer than that one guilty escape. Evidentiary standards for investigating and charging defendants (original) (raw)