A1 Maximising expected utility

For given α and p, the criminal chooses x in order to maximise his expected utility, E, of expression [2], and hence x=x∗ is chosen such that ∂E∂x=0 (and ∂2E∂x2<0). Equation [3] follows: p(α−1)U′(w+(1−α)x)=(1−p)U′(w+x), and the solution is x∗=x∗(α,p;w), a function of α and p, given w; if E>U(w) and the following second order condition holds:

A2 Optimal expected utility elasticities and proposition 1

Substituting x=x∗(α,p) into eq. [2] gives maximal expected utility, E*, as a function of α and p, and hence an indifference map in (α, p) space. Differentiating with respect to p gives

which is zero, with (α, p) on an indifference curve, when

The latter condition reduces to:

which gives the slope of an indifference curve: dE∗dp≡∂E∗∂p+∂E∗∂α⋅dαdp=0. Equation [4] follows and Proposition 1 merely considers when this expression is greater than or less than 1.

A3 Implications of eq. [5] for crime deterrence

Based on eq. [5], the indifference surface in (α0,α1,p) space satisfies

In this case, the limiting condition to prevent the criminal from engaging in at least small-scale crime would be: U′(w+)U′(w−α0)<p(1−p)(α1−1)+U(w)−U(w−α0)U′(w−α0).

If he is loss-averse over the range (w−α0,w), it follows that U(w)−U(w−α0)U′(w−α0)>α0, but if he is loss-prone in that range then this inequality will be reversed. However, as expected, larger values of both α0 and α1 appear to be more of a deterrent.

A4 Comparative statics for the optimal levels of crime

Comparative statics derivatives for x* with regard to p,α0,α1 are as follows:

To ensure that eq. [5] produces a maximal value of E, the second order condition is

and hence the denominators of the expressions for ∂x∗∂p,∂x∗∂α0 and ∂x∗∂α1 will all be negative. It thus follows that at least one of U′′(w−α0+(1−α1)x∗) and U′′(w+x∗) must be negative.

Taking α1>1, if U′′(w−α0+(1−α1)x∗)<0 then ∂x∗∂α0<0 and ∂x∗∂α1<0, but if U′′(w−α0+(1−α1)x∗)>0 and hence U′′(w+x∗)<0, then ∂x∗∂α0>0 and the sign of ∂x∗∂α1 depends on whether the Arrow-Pratt measure of risk-aversion, −U′′(w−α0+(1−α1)x∗)U′(w−α0+(1−α1)x∗), is greater than or less than −1(α1−1)x∗; ∂x∗∂α1<0 if −U′′(w−α0+(1−α1)x∗)U′(w−α0+(1−α1)x∗)>−1(α1−1)x∗. Thus, if the criminal is risk-averse (in other words, loss-prone) at w−α0+(1−α1)x∗, then crime can be reduced by increasing α0 or α1. But, if he is risk-seeking (loss-averse) at w−α0+(1−α1)x∗, and therefore necessarily risk-averse at w+x∗, crime can be reduced by (i) decreasing α0 or (ii) increasing α1 if he is not too risk-seeking (loss-averse) but decreasing α1 if he is sufficiently risk-seeking (loss-averse) at w−α0+(1−α1)x∗.

A5 Propositions 3 and 4

Proposition 3 is a special case of the comparative statics results above and Proposition 4 follows since the ratio of elasticities in eq. [7] will be greater than 1 if and only if the Arrow-Pratt measure of risk aversion at w+(1−α)x∗,−U′′(w+(1−α)x∗)U′(w+(1−α)x∗), is between −1(α−1)x∗ and (αp−1)α(α−1)(1−p)x∗; the criminal is not so risk-averse and not so risk-seeking. If αp≤1, the criminal must be risk-seeking (i.e. loss-averse) at w+(1−α)x∗ but if αp>1 he could be risk-averse (loss-prone); and in either case (αp−1)α(1−p)>−1 since α>1.

A6 Optimal levels of crime for a variety of utility functions

Kahneman and Tversky:

where a,b,k,h are positive and U‾=U(w); and U′′(x)<0 for x>w but U′′(x)>0forx<w. In this case, the criminal’s expected utility is

It is clear that for x>0,∂E∂p<0 and ∂E∂α<0, both as expected, but whilst ∂E∂x>0 if α≤1, ∂E∂x<0 if α is sufficiently large and >1. The criminal’s optimal choice of x when α>1 is

x∗=1b−a(α−1)ln(1−p)kbpha(α−1), the choice being maximal if a(α−1)<b. Furthermore, the limiting condition for the criminal to engage in small-scale crime is kbha≡U′(w+)U′(w−)>p(α−1)(1−p), or equivalently (1−p)kbpha(α−1)>1, and hence to ensure x∗=0, α(>1) and p should be set such that the odds against detection, 1−pp are at most ha(α−1)kb.

Risk averse-Risk averse:

where a, b, k, h are positive; and U′(x)>0, U′′(x)<0 (x≠w). In this case,

and for x>0,∂E∂p<0 and ∂E∂α<0, as expected. Also,

and whilst the criminal would prefer α≤1, when α>1 his optimal (utility maximising) choice of x is: x∗=1b+a(α−1)ln(1−p)kbpha(α−1).

To ensure that the criminal gain, x∗=0,α(>1) and p should be chosen so that the odds against detection are at most ha(α−1)kb and hence kbha≤p(α−1)(1−p), the same condition as the limiting one to ensure that the criminal will not engage in small-scale crime.

Risk averse-Risk seeking:

a,b,k,h>0;U′′(x)>0 for x>w, U′′(x)<0 for x<w. In this case, when α>1 the criminal’s utility maximising choice of x is

if b<a(α−1), and to ensure x∗=0,α(>1) and p should satisfy kbha≤p(α−1)(1−p), that the odds against detection of the crime, 1−pp≤ha(α−1)kb. If b>a(α−1), the above solution represents a minimum and crime will be unlimited. If b=a(α−1), crime will be zero or unlimited depending on whether p>kk+h or p<kk+h.

Markowitz:

Let the criminal’s utility function be:

where a, b, k, h > 0, w is current wealth and U‾=U(w).

and hence $U′′(x)>0 if x<w−1a, $U′′(x)<0 if x>w+1b,$U′′(x)<0 if w−1a<x<w and $U′′(x)>0 if w<x<w+1b.

In this case, the benefit to the criminal of no detection (p=0) or no fine (α=0) is k[1−e−bx−bxe−bx] and, for any α,E=U‾+p∂E∂p+k1−e−bx−bxe−bx.

The criminal’s optimal choice of x>0 is x∗=1b−a(α−1) ln(1−p)kb2pha2(α−1)2, if b>a(α−1), with x=0 giving minimal E, and to ensure that x*=0, α(>1) and p should be set such that the odds against detection, (1−p)/p, are at most ha2(α−1)2/kb2; the same condition as the limiting one to ensure that the criminal will not engage in small-scale crime, in this case that p(1−α)2U′′(w−)+(1−p)U′′(w+)≤0, since U′(w−)=0=U′(w+).

A7 Comparative statics for a variety of utility functions

The comparative statics solutions eq. [6] with α0=0,α1=α for the case z=αx(α>1), and the above four utility functions, are as follows:

Kahneman and Tversky:

and

Since pha(α−1)e−a(α−1)x∗=(1−p)kbe−bx∗, the sign of ∂x∗∂α is the same as that of

which is negative if either (1−p)k≤ph or, (1−p)k>ph and b/a(α−1) is not between the two solutions x > 0 of ln(1−p)kphx=x−1. If b/a(α−1) is smaller than the larger solution and (1−p)k>ph then ∂x∗∂α>0, in which case it would be desirable to reduce α in order to reduce crime. In this latter unusual case, p must be sufficiently small, necessarily less than k/(k+h), and it is of course always desirable to increase p since ∂x∗∂p<0. It has been observed previously that to ensure x*=0, α and p should be set such that 1−pp≤ha(α−1)kb, in which case ∂x∗∂α>0 could not occur since it is then not possible to simultaneously have (1−p)k>ph and a(α−1)<b.

Risk averse-Risk averse:

Risk averse-Risk seeking:

Markowitz:

The sign of ∂x∗∂α is the same as that of

which is negative if either (1−p)k≤ph or, (1−p)k>ph and b/a(α−1) is greater than the larger of the two solutions x>0 of:

If (1−p)k>ph and b/a(α−1) is smaller than the larger solution then ∂x∗∂α>0, an unusual situation akin to that of a Giffen good.

A8 Relative deterrent elasticities for a variety of utility functions

With fine z=αx, α>1, consider expressions [4] and [7] for each of the utility functions given in this Appendix, with a view to assessing whether the criminal is more responsive to changes in p than to changes in α.

Kahneman and Tversky:

which is greater than 1 if and only if k+h−ke−bx∗−h(1+aαx∗)e−a(α−1)x∗>0.

Since pha(α−1)e−a(α−1)x∗=(1−p)kbe−bx∗, the condition reduces to:

which is clearly satisfied if αp≥1 but also satisfied if x∗>2/a(α−1)2 even if αp<1; and more generally. Also,

which is greater than 1 if and only if (1−αp)α(1−p)<a(α−1)x∗<1 or, in other words,

Although it seems that ∈E∗,p∈E∗,α and ∈x∗,p∈x∗,α are not likely to be small, there is no absolute guarantee that either of these ratios will be greater than 1. However, if α=1+β(β>0) so that z=x+βx consists of restitution, x, and an additional fine, βx, then

without any qualification other than, in the latter case, the obvious one that ∂x∗∂β≡∂x∗∂α<0; that a.(α−1)x∗<1. The empirical evidence seems to be imprecise but perhaps it could be interpreted as that the criminal is more responsive to changes in the detection probability, p, than to changes in the ‘fine’, β; that the corresponding ratio of elasticities is greater than 1. But, even without this interpretation, the Kahneman and Tversky model gives a consistent explanation of criminal behaviour, subject to possible constraints on parameters.

Risk averse-Risk averse:

∈E∗,p∈E∗,α=k1−e−bx∗+hea(α−1)x∗−1αx∗haea(α−1)x∗ > 1 iff k−h−ke−bx∗+h(1−aαx∗)ea(α−1)x∗>0.

∈x∗,p∈x∗,α=(α−1)α(1−p)1+a(α−1)x∗ > 1 iff a(α−1)x∗<(αp−1)α(1−p) or, equivalently,

Risk averse-Risk seeking:

∈E∗,p∈E∗,α=kebx∗−1+hea(α−1)x∗−1αx∗haea(α−1)x∗ > 1 iff k+h−kebx∗+h(aαx∗−1)ea(α−1)x∗<0.

∈x∗,p∈x∗,α=(α−1)α(1−p)1+a(α−1)x∗>1 iff a(α−1)x∗<(αp−1)α(1−p) or, equivalently,

Markowitz:

∈x∗,p∈x∗,α=(α−1)α(1−p)2−a(α−1)x∗ >1 iff 1+(1−αp)α(1−p)<a(α−1)x∗<2 or, equivalently, 1+1−αpα1−pbaα−1−1<In1−pkb2pha2α−12<2baα−1−1.