Malice Aforethought

Philip A. Curry

doi:10.1515/rle-2015-0032

Article

Malice Aforethought

Philip A. Curry

Published/Copyright: July 12, 2016

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From the journal Review of Law & Economics Volume 13 Issue 1

Abstract

This paper examines why criminal intent matters in sentencing. In particular, it considers two types of crimes, opportunistic and premeditated. Opportunistic crimes are ones that present themselves to a criminal and can be deterred if the victim makes it too costly for the criminal through private property protection. Premeditated crimes are ones sought out by the criminal, and the effect of private property protection is simply to displace crime. This difference between deterrence and displacement leads to the result that it is optimal to punish premeditated crimes more. The extent to which this is true, however, lies in the response by potential victims. If victims protect themselves from premeditated crimes in ways that also protect them from opportunistic ones, then the difference in penalties is relatively less.

Keywords: crime; private property protection; criminal intent

JEL Classification: K14; D60

Acknowledgements

I would like to thank Martin Bouchard, Steve Easton, Steeve Mongrain, and David Scoones for comments on this paper. I also thank Marty Uhl for excellent research assistance.

Appendix

Proof to Lemma 1

Since variables aO and sO only appear in eq. (1), and aI and sI only appear in eq. (2), it is clear that aˆO is independent of sI and aˆI is independent of sO. Further, ∂aˆO∂sO and ∂aˆI∂sI can be found using the Implicit Function Theorem.

∂aˆO∂sO=pπaOgpsOπaaO1−GpsO<0

∂aˆI∂sI=qaMIpgpsIqaa<0

■

Proof to Proposition 1

Consider the derivative of total costs with respect to sO, which can be rewritten as

πO[−phg(psO)+c′(⋅)[p[1−G(psO)]−pg(psO)psO]]+c′(⋅)[πaO[1−G(psO)]psO∂a^O∂sO]

When this expression is evaluated at sˆI, the first term is equal to −πO∂a^I∂sI. We thus have

−πO∂aˆI∂sI+c′⋅πaO1−GpsOpsO∂aˆO∂sO>0

Thus the cost minimizing level of aI is greater than the cost minimizing level of aO. ■

Proof to Lemma 2

Since the equilibrium is symmetric, each agent’s avoidance expenditure is characterized by the equation

[∂πO(a∗)∂a[1−G(psO)]+∂q(a∗;a−i∗,MI)∂a]h+1=0

Applying the Implicit Function Theorem yields

∂a∗∂sO=pπaOgpsOπaaO1−GpsO+qaa1−GpsI<0

∂a∗∂sI=pqaMIgpsIπaaO1−GpsO+qaa1−GpsI<0

■

Proof to Proposition 2

As in the proof to Proposition 1, the derivative of total costs with respect to sO can be rewritten as

πO−phgpsO+c′⋅p1−GpsO−pgpsOpsO+c′⋅πaO1−GpsOpsO∂a∗∂sO

When this expression is evaluated at s˜I, the first term is equal to −πO∂a∗∂sI. We thus have

−πO∂a∗∂sI+c′⋅πaO1−GpsOpsO∂a∗∂sO>0

Thus the cost minimizing level of aI is greater than the cost minimizing level of aO. ■

Proof to Lemma 3

From Lemma 1 and 2 above, we have that

∂a^O∂sO=pπaOg(psO)πaaO[1−G(psO)]

∂a^I∂sI=qaMIpg(psI)qaa

∂a∗∂sO=pπaOgpsOπaaO1−GpsO+qaa1−GpsI

∂a∗∂sI=pqaMIg(psI)πaaO[1−G(psO)]+qaa[1−G(psI)]

First consider the effect of sI on aˆI and a∗. Holding sI the same, we can see that ∂aˆI∂sI<∂a∗∂sI if and only if

qaMIpg(psI)qaa<pqaMIg(psI)πaaO[1−G(psO)]+qaa

At a given sI, the numerator is the same, however the denominator is greater in the term on the right (and positive). Since both terms are negative, it must be that ∂a^I∂sI<∂a∗∂sI.

Now consider the effect of sO on aˆO and a∗. Holding sO the same, we can see that ∂a^O∂sO<∂a∗∂sO if and only if

pπaOg(psO)πaaO[1−G(psO)]<pπaOg(psO)πaaO[1−G(psO)]+qaa[1−G(psI)]

At a given sO, the numerator is the same, however the denominator is greater in the term on the right (and positive). Since both terms are negative, it must be that ∂a^O∂sO<∂a∗∂sO. ■

Proof to Theorem 1

Consider the first order conditions for the cost minimization problem:

−phπOgpsO+c′⋅πa1−GpsOpsO∂a∂sO−pgpsOπOpsO+pπO1−GpsO=0

−phgpsI+∂a∂sI+c′⋅1−GpsIp−pgpsIpsI=0

where ∂a∂sO and ∂a∂sI refer to either ∂aˆj∂sj or ∂a∗∂sj, j=O,I. Let us first consider the first order condition pertaining to sO. Suppose that avoidance activities are distinct and consider evaluating it s˜O. This yields

−phπOgps˜O+c′⋅πa1−Gps˜Ops˜O∂aˆO∂sO−pgpsOπOps˜O+pπO1−Gps˜O

Since ∂a^O∂sO<∂a∗∂sO, it therefore must be the case that this expression is positive, meaning that s˜O>sˆO.

Now consider the first order condition with respect to sI. Evaluating it at s˜I when avoidance activities are distinct yields

−phgps˜I+∂aˆI∂sI+c′⋅1−Gps˜Ip−pgps˜Ips˜I

In this case, however, since ∂a^I∂sI<∂a∗∂sI, it therefore must be the case that this expression is negative, meaning that s˜I<sˆI. Since s˜O<s˜I, we therefore have that sˆO<s˜O<s˜I<sˆI. ■

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Published Online: 2016-7-12

Published in Print: 2017-3-1

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Articles in the same Issue

https://doi.org/10.1515/rle-2015-0032

Keywords for this article

crime; private property protection; criminal intent