## Abstract

This paper examines factors affecting the decision of whether or not to make certain harmful acts illegal. It considers factors related both to the cost of law enforcement and to the crime commission decision. On the enforcement side, illegality is limited by the existence of fixed notice and response costs, which are unrelated to the harm from the act, and also by costs of imposing punishment. In addition, illegality is limited by a finite marginal productivity of detection, one cause of which is legal error. On the commission side, illegality is limited if offenders have strictly positive benefits from committing the act. The paper concludes by examining how the optimal scope of law is affected by its “expressive function”, the idea that some people are deterred by the mere fact that an act is illegal. We specifically ask how the scope of law changes if more people behave in this way. The answer depends on whether “efficient” violations of the law are possible, which in turn depends on whether offenders’ gains are counted in welfare.

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## Notes

- 1.
Advances in GPS technology, however, have greatly reduced this cost, thus making pure Pigovian taxation a more viable enforcement option.

- 2.
- 3.
- 4.
Following this literature, throughout the article we refer to the government’s choice as criminalization, although the model is equally applicable to the illegalization of acts which are regulated through public and non-criminal enforcement.

- 5.
Thus, for example, if

*s*is a fine, \(\sigma =-1\), so the overall social cost of imposing s, including the cost to the offender, would be \((1+\sigma )s=0\), because the cost to the offender is exactly offset by the revenue received by the government. However, if \(\sigma >-1\), there is a net social cost of imposing*s*. - 6.
In truth, notification costs will not be completely fixed. For example, the government will choose how much effort to devote to notifying the public about a law based on the perceived harm. Think, for example, of public service announcements and billboards about drunk driving laws, or the decision of how far apart to space speed limit signs. We can safely abstract from this consideration because some notice costs are definitely fixed, and any variable costs can be captured by our other cost categories.

- 7.
For purposes of isolating factors limiting optimal criminalization, we focus on the most studied case of risk-neutrality. When this assumption is relaxed, gains can emerge from the deliberate random enforcement of criminal laws.

- 8.
There are models in which the optimal sanction is not maximal, even if it is a fine; for example, when offenders can invest in efforts to avoid capture (Malik 1990), when juries resist convicting offenders if punishment appears too harsh (Andreoni 1991), when type 1 errors (false convictions) are accounted for in welfare (Miceli 1991), and when criminal benefits are state-dependent (Mungan 2010). However, none of these factors is relevant here.

- 9.
- 10.
We assume throughout our analysis that \(x^{*}(h)\) is a proper maximizer of welfare.

- 11.
This follows the conventional view of the effects of legal error (see, e.g., Png 1986). For a recent debate on the impact of type 1 error on deterrence, see Lando (2006), Garoupa and Rizzolli (2012), Lando and Mungan (2018), and Mungan (2017). If type 1 errors do not reduce deterrence, as considered in Lando (2006), the results presented in proposition 1 may become more relevant.

- 12.
See the references in note 2 above.

- 13.
- 14.
This formulation treats reflexive obedience to the law as an all or nothing decision—either people have internalized the sentiment or not. An alternative approach would be to assume that people have varying “moral costs” of breaking the law (apart from any threatened legal sanction), which range from zero to infinite. The implications of this alternative approach are pointed out in note 16 below.

- 15.
Stigler (1970) first raised this issue.

- 16.
In the current formulation, we examine this question by doing a comparative static on \({\hat{h}}\) with respect to \(\beta\). In the more general formulation suggested in note 14 above, where people have varying moral costs of breaking the law, an increase in overall morality would be reflected by a rightward shift (in the sense of first-order stochastic dominance) in the distribution of this cost. The qualitative conclusions of such an approach would mirror the conclusions obtained from the simpler formulation in the text.

- 17.
See Miceli (2019, Chapter 8) for a preliminary version of this result.

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## Appendix

### Appendix

### Proof of Proposition 1

Parts (i)–(iv): First, consider the case where \({\underline{b}}=0\). Note that when \(c=0\), \(\sigma =-1\), and

*L*(*x*) has a strictly positive lower bound, \({\bar{h}}\), when \(p^{\prime }(0)\) is finite. Thus, \(\frac{\partial W_{1}}{\partial x}<0\) for all *x* and all \(h<{\bar{h}}\). Moreover, \(W_{0}(h)-W_{1}(0,h,\sigma ,k,c)=k\ge 0\), and, thus, \(W_{0}(h)\ge W_{1}(0,h,\sigma ,k,c)\) for all *x* and all \(h<{\bar{h}}\) when \(c=0\), \(\sigma =-1\), and \(p^{\prime }(0)\) is finite. Additionally, \(\frac{ \partial W_{1}}{\partial c},\frac{\partial W_{1}}{\partial \sigma }<0\) and \(\frac{\partial W_{0}}{\partial c},\frac{\partial W_{0}}{\partial \sigma }=0\) for all *x*, and, therefore \(W_{0}(h)\ge W_{1}(0,h,\sigma ,k,c)\) for all *x* and all \(h<{\bar{h}}\) whenever \(p^{\prime }(0)\) is finite.

When \(\underset{x\rightarrow 0}{\lim }p^{\prime }(0)=\infty\), it follows that \(\underset{x\rightarrow 0}{\lim }L(x)=0\) and \(L(x)>0\) for all \(x>0\). Thus, \(\underset{x\rightarrow 0}{\lim }[W_{0}(0)-W_{1}(0,0,\sigma ,k,c)]=k\ge 0\), and, therefore, \(W_{0}(0)>W_{1}(x,0,\sigma ,k,c)\) for all \(x>0\). But, \(\frac{\partial W_{1}}{\partial c},\frac{\partial W_{1}}{\partial \sigma },\frac{\partial W_{1}}{\partial k}<0\), and, thus, \(\underset{ x\rightarrow 0}{\lim }[W_{0}(0)-W_{1}(x,0,\sigma ,k,c)]>0\) and \(W_{0}(0)-W_{1}(x,0,\sigma ,k,c)>0\) for all *x* whenever \(\sigma >-1\), \(c>0\) or \(k>0\). This implies that whenever one of these conditions are met, there exists \({\bar{h}}>0\) such that \(W_{0}(0)-W_{1}(x,h,\sigma ,k,c)>0\) for all *x* and all \(h<{\bar{h}}\).

The above observations together imply that if \({\underline{b}}=0\) it follows that whenever \(\sigma >-1\), \(c>0\), \(k>0\) or \(p^{\prime }(0)\) is finite, there exists some \({\hat{h}}>0\) such that \(h<{\hat{h}}\) implies that any policy with \(\psi =1\) and \(x>0\) is inefficient.

Part (v): The derivative of welfare can be expressed as

We previously noted that if \({\underline{b}}=0\), it is optimal to criminalize all harmful acts, i.e. \({\hat{x}}(h)>0\) for \(h>0\) where \({\hat{x}}\) denotes the optimal solution when \({\underline{b}}=0\), and that \({\hat{x}}\) is increasing in *h*. Since, \(p(0)=0\) and \(p^{\prime }>0\), then if \({\underline{b}}>0\) there exists \({\underline{x}}>0\) such that \({\hat{b}}={\underline{b}}\) for \(x\le {\underline{x}}\). Thus, \(\frac{\partial {\hat{b}}}{\partial x}=0\) for \(x\le {\underline{x}}\), which implies that (13) negative in this range. Now define \({\hat{h}}\) by \({\underline{x}}={\hat{x}}(\hat{h })\). It follows that \(h<{\hat{h}}\) implies \(x^{*}=0\). \(\square\)

### Proof of Proposition 2:

Note that when \(c=0\) and \(\sigma =-1\),

When \(p^{\prime }(0)-q^{\prime }(0)\) is finite, \({\hat{L}}(x)\) has a strictly positive lower bound \({\bar{h}}\). Thus, the proof follows from arguments very similar to those made in the proof of proposition 1. \(\square\)

### Proof of Proposition 3

(i) Totally differentiating (10) gives:

From this we compute:

The sign of this expression depends on the numerator, which is (11).

(ii) We also get

which is positive. Thus, as a larger fraction of offender gains are counted, the scope of law contracts. \(\square\)

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Miceli, T.J., Mungan, M.C. The limit of law: factors influencing the decision to make harmful acts illegal.
*Econ Gov* (2021). https://doi.org/10.1007/s10101-021-00255-w

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### Keywords

- Crime
- Illegality
- Law enforcement
- Expressive function of law

### JEL Classification

- K14
- K42