Crime, Punishment And Social Norms Jörgen W. Weibull And Edgar ...
CRIME, PUNISHMENT AND SOCIAL NORMS
Jörgen W. Weibull and Edgar Villa
Stockholm School of Economics and Boston University
November 4, 2005
Abstract.
We analyze the interplay between economic incentives and
social norms when individuals decide whether or not to engage in criminal ac-
tivity. More speci…cally, we assume that there is a social norm against criminal
activity and that deviations from the norm result in feelings of guilt or shame.
The intensity of these feelings is here endogenous in the sense that they are
stronger when the population fraction obeying the norm is larger. As a con-
sequence, a gradual reduction of the sanctions against criminal activity, or of
the taxation of legal incomes, may weaken the social norm against crime. Due
to the potential multiplicity of equilibria in our model, such a gradual change
may even induce a discontinuous increase in the crime rate. We show that law
enforcement policies may have dramatic and permanent e¤ects on the crime
rate, and lead to hysteresis. We also de…ne political equilibrium under major-
ity rule and show how a majority of individuals, who feel no guilt or shame
from violating the law, in political equilibrium can exploit a minority who do
have such feelings.
Keywords: crime, punishment, social norm, political equilibrium.
1.
Introduction
Human behavior seems to be in‡uenced both by economic incentives and social norms.
While sociologists have emphasized social norms since the time of Emile Durkheim,
economists have focused almost exclusively on economic incentives since the time
of Adam Smith. Most likely, the di¤erent approaches re‡ect the di¤erent subject
matters of the two disciplines; while sociologist mostly study behavior in social groups,
economists focus mainly on behavior in the marketplace. However, for some decisions
that people make, both social norms and individual economic incentives appear to
be involved. This paper is an attempt to bring together social norms and individual
rational choice in an economic model of crime following the seminal model of Becker
(1968). More precisely, we extend the traditional economic approach, where pecuniary
motives are the driving force of the criminal o¤enses, to encompass social norms in
such a way as to enable an analysis of the interplay between economic incentives and
1
CRIME, PUNISHMENT AND SOCIAL NORMS
2
social norms. The treatment of social norms is similar to that in Lindbeck-Nyberg-
Weibull (1999) and Huck-Kübler-Weibull (2004) in the way social norms are modeled.
The …rst of those studies analyzes the decision whether to work or live o¤ transfers,
and the second analyzes individual e¤ort decisions in team work. By contrast, we
here analyze individual decisions of whether or not to take part in criminal activity.
On top of the legal sanctions taken against criminals, there seems to exist in most
societies an implicit social norm to live o¤ legal activities rather than illegal. This
normative pressure can be more or less strong, and can take the form of “shame”
or “guilt”. As Elster (1989) has pointed out “if punishment was merely the price
tag attached to crime, nobody would feel shame when caught” p. 105. By shame is
usually meant embarrassment in front of others who have observed or know about
the deviation, while guilt refers to internalized shame, or embarrassment in front of
oneself. We believe that both shame and guilt are at work in connection with criminal
behavior.
We also believe that the intensity of the social norm to stay out of crime depends
in part on how many others in one’s peer group stay out of crime. The more frequent
crime is, the lesser the guilt and shame attached to it. Bentham (1789) referred
to this e¤ect as the syndrome of robberies without shame: “where robberies are
frequent, and unpunished, robberies are committed without shame” p. 156. Living
o¤ criminal activity is less shameful the more criminal activity there is in society.
While the existence of a social norm against crime is taken as a given here, the
intensity of it, as perceived by the individual, is endogenous in the model: it depends
positively on the population fraction adhering to it. Hence, the intensity of the social
norm becomes an equilibrium phenomenon, which, a priori, allows for the possibility
of multiple equilibria under given taxes and sanctions against crime.
We develop a relatively simple model and study in detail two cases. In the …rst
case, the return to criminal activity is uncorrelated with the individual’s potential
wage from legal work. This may, for example, be the case with illegal dealing with
drugs or weapons, auto-theft and robbery. In the second case, the returns to criminal
activity are positively correlated with potential wages: the higher an individual’s
potential wage from legal work, the higher are the expected returns from criminal
activity. This may plausibly be the case with tax evasion, economic extortions and
theft within …rms and organizations. We assume, throughout, that every individual
has only a binary choice: either to work o¤ legal or illegal activity. The individual
obtains material and immaterial utility from his or her choice, where the material
utility depends on the disposable income from legal work as well as on the sanctions
CRIME, PUNISHMENT AND SOCIAL NORMS
3
against crime and the probability of being caught. The immaterial utility emanates
from norm adherence. If driven only by guilt, this is independent of the probability
of being caught and convicted, while if driven by shame it does depend on this
probability.
Within this modelling framework we …rst analyze individual decision-making and
equilibrium outcomes under given policy parameters concerning taxation of legal work
and law enforcement measures. Each individual in the economy then decides whether
or not to engage in criminal activity, and these decisions are based on knowledge of
relevant policy parameters and on expectations about the going crime rate. The latter
is important, because it may in‡uence the utility from norm adherence. A crime rate
is an equilibrium outcome if no individual wants to change his or her individual
choice at that crime rate. Because of the endogeneity of the intensity of the social
norm, there may be multiple equilibria for given policy parameters. We establish a
su¢ cient condition for the existence of equilibrium and another su¢ cient condition
for its uniqueness. We also illustrate the equilibrium outcome and its comparative
statics properties by means of examples. Having studied equilibrium crime rates,
we turn to the question of how policy is determined. We call a policy p, such as
a combination a tax rate and a penalty for criminals, an equilibrium policy if (i)
it is consistent with equilibrium behavior and a balanced government budget, and
(ii) there exists no other such policy, p0, that would gain a strict majority against
p if individuals would vote earnestly according to the expected utility they obtain
under the respective policy. We illustrate the nature of equilibrium policies within
the context of a tax evasion example.
In order to high-light the interplay between economic incentives and social norms
we make several simplifying assumptions. One such assumption is that we take the
probability of catching a given criminal as independent of the current crime rate.
This is clearly unrealistic if the capacity of the law enforcement system is …xed and
given. However, our assumption makes the analysis and exposition easier, and, once
this case has been treated, it is not di¢ cult to extend the present analysis to the
case of an endogenous such probability — another source of potential equilibrium
multiplicity.1
The rest of the paper is organized as follows. The model is developed in section 2,
1 Such a generalization can be readily obtained by letting the punishment probability
be a
continuous and non-increasing function of the crime rate x, see section 2. See also section 4.1 where
the cost of maintaining a given probability
is an increasing function of x, that is, we then allow
for endogeneous law enforcement capacity.
CRIME, PUNISHMENT AND SOCIAL NORMS
4
equilibrium crime rates are analyzed in section 3 and section 4 is devoted to political
equilibrium considerations.
2.
The model
Consider a continuum population of individuals, where each individual faces a binary
choice: whether or not to engage in some criminal activity. Let x denote the popu-
lation share of individuals who choose crime. An individual who does not choose to
engage in crime will be said to choose work, but we do not exclude the possibility
that criminals also work.
We assume the labor market to be perfectly competitive: each worker is paid
his or her marginal productivity or potential wage, w, which is exogenous and …xed.
Labor incomes are taxed at a constant rate . The disposable (life time) income to
an individual with potential wage w who chooses work is (1
) w. The disposable
income to the same individual, when instead choosing crime, is 1w +
if not caught
1
and convicted and otherwise 0w +
. We assume
0
0w +
<
,
(1)
0
1w +
1
that is, the disposable income of a criminal who is caught and convicted is lower
than that of a criminal who is not. The -parameters are non-negative and account
for a positive “correlation” between a criminal’s earnings, or disposable income, and
his or her potential wage. This can be due to full-time or part-time legal work.
Moreover, higher paid employee’s within a …rm’s hierarchy usually have more control
of and knowledge about the …rm’s resources, thereby opening up the possibility for
higher criminal earnings, all of which suggests 1 > 0. On the other hand, convicted
criminals may have lower life-time wage earnings due to lost work-time (as a share
of an individual’s active work life) spent in prison, and/or by way of paying a …ne
(which may be increasing in the individual’s potential wage rate) and/or by way of
the labor market’s negative wage response to workers who are ex-convicts, suggesting
0 < 1.
The -parameters represent returns from crime that are “uncorrelated” with the
individual’s potential wage. For certain crimes, such as tax evasion, the “wage-
sensitivity”parameters
would typically be positive while the -parameters may be
close to zero, while for other crimes, such as full-time drug dealing, the -parameters
would typically be close to zero and the -parameters positive. For
< 0,
may
0
0
be thought of as a …xed …ne to be paid by convicted individuals. Unlike
and
1
1,
and
0
0 can be more or less directly in‡uenced by policy (the length of sentence
and size of penalty).
CRIME, PUNISHMENT AND SOCIAL NORMS
5
So far, we have focused on purely economic consequences of work and crime. We
now turn to attitudes towards criminal activity. We assume that there is a social
norm against engaging in criminal activity and that the intensity of this norm is
endogenous. Individuals who choose crime loose utility from adherence to this social
norm, and this utility is non-increasing in the expected crime rate. In other words,
when many others are expected to adhere to the norm, a deviation from the norm
results in a (weakly) larger utility loss that when only few others are expected to
adhere to the norm.
Moreover, workers and o¤enders alike receive a disutility from criminal activity in
society at large. This disutility thus represents the externality that certain criminal
activities give rise to. An example is the fear, violence, restrictions on personal free-
dom and demoralizing in‡uence on the young that illegal trade in drugs or weapons
usually give rise to.
In sum, and more precisely, the expected utility associated with each of the two
choices, work and crime, respectively, are
UW = u (w
w; 1) + av (xe)
(xe)
(2)
and
UC = (1
) u ( 1w +
; h
; h
1
1) +
u ( 0w + 0 0)
(xe) ;
(3)
where
2 [0;1] is the probability of being caught and convicted, which we take
to be the same for all criminals. We assume the consumption utility function u :
R2++ ! R to be continuous and strictly increasing in both arguments, the …rst being
consumption of goods, the second, h, consumption of leisure (or, more generally,
any relevant “material” attribute that distinguishes the three di¤erent “roles” each
individual may be in: as a worker, a criminal at large, or as a caught and convicted
criminal). We normalize leisure of a non-criminal worker to one unit. Moreover, we
assume the norm-adherence utility function v : [0; 1] ! R+ to be strictly positive,
continuous and non-increasing, and the utility weight a 2 R that the individual
places on norm adherence to be positive or negative, and the externality function
: [0; 1] ! R+ to be nonnegative, continuous and non-decreasing. For the sake of
analytical tractability, we assume all individuals to have the same subutility functions
u, v and , while they may di¤er with respect to their individual potential wage, w,
and degree of norm attachment, a. Moreover, we assume that a is a positive a¢ ne
function of the punishment probability :
a =
+
;
(4)
CRIME, PUNISHMENT AND SOCIAL NORMS
6
for
2 R and
0. This representation can be derived from a psychological
consideration, namely, the distinction between guilt and shame, that is, whether or
not the disutility from norm deviation depends on own or others’disapproval. The
parameter
represents the degree to which norm adherence is driven by guilt— as in
the case of an internalized norm, where one’s disutility from deviating is insensitive to
whether or not others observe the deviation. By contrast, the parameter
represents
the degree to which norm adherence is driven by shame— as in the case of an externally
sanctioned norm, where one’s disutility from deviating is sensitive to whether or not
others observe the deviation.
To see that this distinction indeed gives rise to (4), temporarily replace the current
assumption of a norm-adherence utility gain, av (x), when choosing “work", with an
equally large (and thus behaviorally equivalent) norm-adherence utility loss, from
choosing "crime". To a criminal who is not caught and convicted, let this loss be
v (x), while the loss to a criminal who is caught and convicted is ( + ) v (x).
Assuming that only the criminal knows about his or her crime when not caught and
convicted, the utility loss in the …rst case represents guilt while that in the second
case represents both guilt and shame— assuming that others in society learn about
the crime of a convicted criminal. This alternative utility representation results in
exactly the same utility di¤erence between “work” and “crime” as in the present
model, under equation (4), and is hence behaviorally equivalent with the present
formulation.
Finally, note that the expected crime rate, xe, is the same in equations (2) and (3).
It is thus independent of the individual’s own choice. This simplifying assumption is
reasonable in a large population, individuals then arguably neglect the e¤ect of their
own choice on the crime rate in society at large. More exactly, viewing the utility
from norm adherence as a function of the expected population share of others (in
society at large, or in one’s peer group) who choose crime, but the externality as
a function of the total population share of criminals, the argument xe in the term
av (xe) should be interpreted as the population share of others who choose crime. If
the total population, which we treat as a continuum, instead were …nite, say of size
N , then we should have (xe + 1=N ) instead of (xe) in equation (3). However,
by continuity of , the di¤erence between these two function values tend to zero as
N ! 0. Thus, our analysis also applies approximately to …nite but large populations.
We will treat the quadruple ( ; 0;
; ) as a vector of policy instruments, per-
0
ceived as …xed and given by individuals when they decide whether or not to engage
CRIME, PUNISHMENT AND SOCIAL NORMS
7
in criminal activity.2
3.
Equilibrium crime rates
If each individual chooses the alternative with the highest expected utility, then the
choice crime is optimal for an individual with norm attachment a and potential wage
w if and only if
u (w
w; 1) + av (xe)
(1
) u ( 1w +
; h
; h
1
1) +
u ( 0w + 0 0) .
(5)
For an individual with potential wage w who expects a crime rate xe there thus exists
a unique critical degree of norm attachment, ao (w; xe), such that it is optimal for the
individual to choose crime if and only if the individual’s norm attachment a does not
exceed this critical degree, which is determined from indi¤erence in (5):
ao (w; xe) =
u (w) =v (xe) ;
(6)
where
u (w) = (1
) u ( 1w +
; h
; h
1
1) +
u ( 0w + 0 0)
u (w
w; 1) :
(7)
In words,
u (w) is the individual’s expected consumption utility gain from crime, as
compared with work, given the tax rate , punishment probability , and punishment
( 0;
). In other words, ao (w; xe) is the ratio between the expected consumption
0
utility gain from crime— that depends on the individual’s potential wage w as well as
on the (here …xed) policy parameters
and — and the utility from norm adherence—
that depends on the expected crime rate xe.
We note that the critical degree of norm attachment, ao (w; xe), is non-decreasing
and continuous in the expected crime rate, xe. Moreover, this critical degree of norm
attachment is positive if and only if crime results in a higher expected consumption
utility than work. We also note that the externality, not surprisingly, is irrelevant
to individuals’ decisions; it depends only on the expected population share xe of
criminals (which we assumed to be independent of the individual’s own decision, see
above).
We assume that all individuals simultaneously choose between work and crime, for
a given tax rate
and punishment probability . By de…nition, a pro…le of such indi-
vidual choices constitutes a Nash equilibrium if and only if every individual’s choice
2 Arguably also h0, the amount of leisure that a caught and convicted criminal has, is in‡uenced
by policy. However, we will not analyze this aspect.
CRIME, PUNISHMENT AND SOCIAL NORMS
8
is optimal, given the others’ choices. A crime rate x will be called an equilibrium
crime rate if it is consistent with some Nash equilibrium pro…le. Let
be the cumu-
lative probability distribution function (c.d.f.) of “types” (a; w) in the population,
where we recall that a =
+
, and let x be the associated crime rate, that is, the
population share that chooses crime.3 We then have:
Proposition 1. A crime rate x 2 [0;1] is a Nash equilibrium crime rate if and only if
it satis…es the …xed-point equation (8), where F : [0; 1] ! [0;1] is de…ned in equation
(9). There exists at least one Nash equilibrium.
x = F (x)
(8)
F (x) = Z +1Z ao(w;x)d (a;w)
(9)
w=0
a= 1
Proof : For any expected crime rate xe 2 [0;1], the subset C
T = R
R+
of individual types (a; w) for whom crime is optimal is Borel measurable, by (6),
and their population share equals x = F (xe). Hence, a strategy pro…le is a Nash
equilibrium if and only if F (xe) = xe. Being continuous, the function F has at least
one …xed point (by the intermediate value theorem applied to f (x) = x
F (x), a
continuous function with f (0)
0 and f (1)
1. End of proof.
In words, F (xe) is the crime rate that results if all individuals expect the crime
rate to be xe. The …xed-point equation (8) expresses the requirement that individuals’
crime-rate expectations be ful…lled, xe = F (xe), an assumption of “perfect foresight”
or “rational expectations.”
A solution x to equation (8) will be called an equilibrium crime rate. In the
special cases when the subutility v (x) from norm adherence is independent of the
crime rate x, the right-hand side in (8) is a constant and the equilibrium crime
rate unique. However, the function F is in general non-decreasing, and therefore
multiple equilibrium crime rates is a possibility. Figure 3 is drawn for a logistic
norm-adherence utility function v and exponentially distributed norm attachments
a, under the further assumption that every individual has the same potential wage
w (the numerical speci…cations of all diagrams are provided in the appendix at the
end of the paper).
3 For any expected crime rate xe the subset of individuals who will choose crime is Borel measur-
able, by (6), and hence their population share x is well-de…ned. The function F is continuous and
hence has at least one …xed point.
CRIME, PUNISHMENT AND SOCIAL NORMS
9
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
Figure 1: The equilibrium crime rate equation.
In this example, there are three equilibrium crime rates, x
0:2, x
0:7 and
x
1. The intuition for the multiplicity is that if the crime rate is low (high) then
the utility from norm adherence — the guilt or shame from criminal activity — is
strong (weak). Consequently, few (many) individuals choose to live o¤ crime. Thus,
two societies with the same potential wage, norm attachment distribution, penal code
and police monitoring may di¤er signi…cantly in their crime rates. That the highest
equilibrium crime rate is close to 1 depends on
u (w) being positive, that is, that
“crime pays”in terms of expected utility from income, and that the utility from norm
attachment is close to zero when the crime rate is close to one (see Figure 3). We
also note that the intermediate equilibrium crime rate is unstable with respect to
perturbations of expectations about others’ behavior: if an individual expects the
crime rate to be slightly higher (lower), then it is optimal to choose crime (work).
Equation (8) also de…nes the Nash equilibrium correspondence that maps para-
meter combinations to the associated (non-empty) set of equilibrium crime rates. We
illustrate this correspondence by means of a diagram that we will discuss in two dis-
tinct cases: one when norm adherence is driven solely by guilt (
= 0), the other
when norm adherence is driven solely by shame (
= 0). Consider Figure 3 below.
Imagine …rst that norm adherence is driven by guilt, that is, that
= 0. The diagram
then illustrates the equilibrium correspondence that maps the expected consumption
utility gain from crime,
u (w), de…ned in equation (7), to equilibrium crime rates,
CRIME, PUNISHMENT AND SOCIAL NORMS
10
for
exponentially distributed. We see that the correspondence has a fold, and that
it rises from zero to one as
u (w) increases from zero to 0:4. For values of
u (w)
between approximately 0:1 and 0:37, there are three equilibrium crime rates, at each
end-point of this interval there are two equilibria, and outside the interval a unique
equilibrium. It is the endogeneity of the strength of the social norm, as represented
by the function v, that causes the dramatic fold in the equilibrium correspondence.
In the classical case without a social norm (
=
= 0 for all individuals), the equi-
librium correspondence is just a step function, jumping up from zero to one as the
consumption utility gain from crime runs from negative to positive. Hence, besides
potentially causing a fold — multiple equilibria — the social norm against crime, as
modelled here, reduces the equilibrium crime rate when this utility gain is small and
positive.
x
1
0.75
0.5
0.25
0
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
co
c ns. utili
o
ty ga
ty g in
a
Figure 2: The equilibrium crime rate correspondence, mapping consumption utility
gains to crime rates.
This diagram provides some comparative statics insights. First, an increase in
the punishment probability
decreases the expected consumption utility to crime,
while an increase in the income tax rate
increases it. Hence, as the population
attaches less and less value to norm attachment, or as crime pays better and better,
the equilibrium crime rate will increase, from almost 0% to almost 100%, but not
continuously; there has to be at least one discrete upward jump as the consumption
CRIME, PUNISHMENT AND SOCIAL NORMS
11
utility gain from crime
u (w) increases gradually from below 0:1 to above 0:37,
approximately.
Let us make a heuristic thought experiment based on this diagram.4 Suppose
that the government gradually increases the punishment probability
, which de-
creases
u (w), ceteris paribus, starting from a value to the right of the fold, like
u (w) = 0:5, and ending at a value to the left of the fold, such as
u (w) = 0:05.
The equilibrium crime rate will then decrease from about 100% to about 7%. Ac-
cording to the diagram, the crime rate necessarily makes a downward “jump”at some
intermediate punishment probability. A gradual policy change thus results in a sud-
den and drastic fall in the value that people attach to the norm “not to engage in
criminal activities”and thus also to their aggregate behavior. Reversing the thought
experiment, that is, gradually reducing the punishment probability such that
u (w)
is increased from
u (w) = 0:05 to
u (w) = 0:5, will likewise necessarily result in
an upward jump at some point in time. If expectations-formation has inertia, in the
sense that the equilibrium crime rate is expected to change only gradually when the
punishment probability is changed marginally (whenever this is compatible with ag-
gregate behavior), then we will have hysteresis: the upward jump will take place at a
higher value of
u (w) (i.e. lower punishment probability) than the downward jump.
In other words, if we think of the norm not to engage in crime as a form of social
capital, then a gradual increase in the punishment probability leads to appreciation
of that capital, while a gradual decrease in the punishment leads to depreciation of
that capital, with hysteresis.
So far, we assumed that norm adherence was driven by pure guilt. Next, imagine
that norm adherence is driven by pure shame, that is,
= 0. Also then would the
equilibrium correspondence have a graph like that in Figure 3. However, now it would
be the equilibrium correspondence that maps
u (w) =
(on the horizontal axis) to
equilibrium crime rates (on the vertical axis).
3.1.
Crime earnings uncorrelated with potential wage.
We here study in
some more detail the case when the disposable income to criminals is independent of
the potential wage rate. This may be the case of full-time illegal trade in drugs or
weapons. More precisely, we now study cases when 1 = 0 = 0. For this particular
study, we assume that all individuals have the same degree of norm attachment, a =
+
, but that wages w are distributed according to some cumulative distribution
function G.
4 See Lindbeck, Nyberg and Weibull (1999, 2003) for discussions of social norm dynamics.
CRIME, PUNISHMENT AND SOCIAL NORMS
12
For any individual with norm attachment a =
+
in a society with crime
rate x there thus exists a unique critical wage value, wo (x), such that it is optimal
to choose crime if and only if the individual’s potential wage w does not exceed this
critical value. The critical wage is determined from the indi¤erence in the decision
equation (5) which yields
1
wo (x) =
u 1 [(1
) u (
; h
; h
1
1
1) +
u ( 0 0)
( +
) v (x)] ;
(10)
where u 1 is the inverse of the function u ( ; 1) : R++ ! R.
The equilibrium equation (8) can be written now as
1
x = G
u 1 [(1
) u (
; h
; h
1
1
1) +
u ( 0 0)
( +
) v (x)] .
(11)
This equation de…nes a correspondence that maps each policy triplet ( ;
; ), where
0
> 0 is a caught and convicted criminal’s consumption, to the associated set of
0
equilibrium crime rates x (recall that we …xed 0 at zero).
In order to facilitate the analysis of this correspondence, we assume equal leisure
in all three roles of an individual, logarithmic consumption utility function, and that
the wage density g (w) = G0 (w) is positive for all positive wages w > 0. The c.d.f. G
then has an inverse, and equation (11) can be re-written as
1
G 1 (x) exp [( +
) v (x)] =
exp [(1
) ln
+
ln
] :
(12)
1
1
0
The right-hand side of (12) is a continuous function of the tax rate , the consump-
tion
of convicted criminals, and the punishment probability , strictly increasing
0
in the …rst two and strictly decreasing in the third. Likewise, the left-hand side is
a continuous function of the crime rate x and the punishment probability
. It is
non-decreasing in the latter (strictly increasing i¤
> 0), and strictly increasing in x
if the utility from norm-adherence is relatively insensitive to the crime rate. In such
cases, the equilibrium crime rate x is uniquely determined by ,
and . Moreover,
0
the equilibrium crime rate is then increasing in the tax rate — since a higher income
tax rate makes work less attractive — and in the consumption level of convicted
criminals, but decreasing in the punishment probability, for obvious reasons. More
precisely, by way of di¤erentiation of the left-hand side in (12) with respect to x, one
obtains:
Proposition 2. The equilibrium crime rate is unique under condition [M] below. It
is increasing in
and
, and decreasing in .
0
CRIME, PUNISHMENT AND SOCIAL NORMS
13
1
[M ]
v0 (x) g G 1 (x) G 1 (x) >
+
8x
By contrast, if the left-hand side in equation (12) is not increasing in x, then
certain policies ( ;
; ) admit multiple equilibrium crime rates x. See Figure 3,
0
drawn for a case when the norm-adherence utility function is logistic in the crime rate
and the potential wage w is distributed according to a Weibull (0; 1 ; 2) distribution.
2
The wavy curve is the graph of the left-hand side in equation (12), viewed as a
function of x, and the straight line gives the value of the right-hand side of the same
equation.
LHS
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
Figure 3: The left-hand side in equation (12) as a function of the crime rate x.
Figure 4 below shows the equilibrium correspondence that maps punishment prob-
abilities
to equilibrium crime rates x when the income tax rate
and consumption
of convicted criminals are kept …xed (at the levels
= 0:25 and
= 0:1) as in the
0
0
speci…cation of Figure 3). The thicker and folded curve is the graph of the equilibrium
correspondence when all individuals have norm attachment parameters
= 1 (guilt)
and
= 1 (shame), while the thinner curve corresponds to the case of zero norm
attachment — the standard economics model of crime and punishment (initiated by
Becker (1968)). We see that the presence of a social norm against committing crimes
suppresses the equilibrium crime rate, less at low punishment probabilities and more
at medium to high punishment probabilities, and that the endogeneity of the social
CRIME, PUNISHMENT AND SOCIAL NORMS
14
norm causes again a dramatic fold in the equilibrium correspondence. In particular,
condition [M] is violated in this example, since, by proposition 2 the condition is
su¢ cient for the non-existence of folds in this correspondence.
x
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
pi
Figure 4: The equilibrium correspondence that maps punishment probabilites to
crime rates.
Let us make again a heuristic thought experiment based on this diagram. Suppose
that the government gradually increases the punishment probability , starting from
a value to the left of the fold, such as
= 0:25, and ending at a value
= 0:5 to the
right of the fold. The crime rate will then decrease from about 95% to about 10%.
According to the diagram, the crime rate necessarily makes a downward “jump” at
some intermediate punishment probability. A gradual policy change thus results in a
sudden and drastic fall in the value that people attach to the norm “not to engage in
drug dealings" and thus also to their aggregate behavior. If expectations-formation
shows inertia then we will have again hysteresis: the upward jump will take place at
a lower punishment probability than the downward jump.5
3.2.
Crime earnings correlated with potential wage.
We here study the case
when criminals’ disposable income is positively related to their potential wages, as
may be the case with tax evasion. Indeed, Allingham and Sandmo (1972) emphasize
5 The two probabilities being approximately
= 0:26 and
= 0:35
CRIME, PUNISHMENT AND SOCIAL NORMS
15
that tax evasion may be subject to shame or stigma. Tax evaders may be quite
embarrassed when they are caught under-reporting their income. We here outline
how our model can be applied to such forms of illegal behavior.
Individuals face a binary choice when reporting their income to the tax authorities:
to either report the full income, w, or only the fraction w, where
2 (0;1) is …xed
and given. Reported incomes are taxed at a constant rate
2 (0;1). Thus, an
individual with income w who reports truthfully pays w in taxes while an under-
reporting individual pays only
w. If an individual is caught under-reporting, then
she must pay the amount withheld, (1
) w, and a fraction
2 (0;1) of her
remaining disposable income (or, equivalently, spend time in prison and thereby lose
work time). We will call
the penalty rate. Hence, in the notation of section 2:
1 = 1
;
0 = (1
) (1
) and
=
= 0:
1
0
We note that
0 < 1
<
1; the disposable income from under-reporting
and being caught and convicted is lower than the disposable income from reporting
truthfully, which in its turn is lower than the disposable income from under-reporting
and not being caught and convicted.6 The crime rate x 2 [0;1] is now the population
fraction of tax evaders. We assume leisure to be the same in all three roles of an
individual, h0 = h1 = 1, and suppress this constant argument in the consumption
utility function.
The expected utility associated with each of the two choices, report truthfully and
underreport, respectively, are thus
UW = u [(1
) w] + av (x)
(x)
(13)
and
UC = (1
) u [(1
) w] + u [(1
) (1
) w]
(x)
(14)
where a 2 R is the utility weight that the individual places on adherence to the social
norm to report truthfully.
In the special case of logarithmic subutility from consumption, the potential wage
cancels out when comparing UW with UC, and we obtain
1
u (w)
(1
) ln
+
ln (1
) :
1
6 Parameters are assumed to be such that 0 > 0, that is, caught and convicted tax evaders are
left with a positive disposable (life time) income.
CRIME, PUNISHMENT AND SOCIAL NORMS
16
from (7). In other words, individuals’ choices are independent of their potential
wages, and the consumption utility gain from crime is increasing in the tax rate
and decreasing in the punishment probability .
We brie‡y consider this case. First, suppose that all individuals have the same
positive norm attachment a. Then the crime rate is unique, x = 0, if
u (w)
av (1):
even if all others were to defect from the norm, truthful reporting is preferable.
Likewise, x = 1 is the unique equilibrium crime rate if
u (w)
av (0): even if no
one else were to defect from the norm, under-reporting is preferable. However, in the
remaining case, that is, when av (1) <
u (w) < av (0), there are three equilibria,
namely, x = 0, x = 1, and x = v 1 [ u (w) =a]. The reason for this multiplicity is
simple. If the crime rate is expected to be x = v 1 [ u (w) =a], then all individuals
are indi¤erent between reporting truthfully and under-reporting. Hence, this crime
rate is an equilibrium. If the crime rate is expected to be lower (higher), then the
unique optimal choice is truthful reporting (under-reporting).7
Secondly, suppose individuals di¤er in their norm attachment, and suppose this is
driven solely by shame. Indeed, it seems reasonable to conjecture that shame is more
prominent than guilt in the present case of tax evasion, since tax evasion convictions
easily become public and may cause considerable social stigma. Hence, we now have
= 0, while
is distributed according to some c.d.f. B. Equation (8) then becomes
1
1
1
x = B
ln
+ ln (1
)
,
(15)
v (x)
1
a …xed-point equation with a graph identical to that in Figure 3, under suitable
parameter speci…cations.8
The equilibrium equation (15) also de…nes a correspondence that maps policies,
( ; ; ), to the associated set of equilibrium crime rates, x. Figure 5 below shows
the the graph of this equilibrium correspondence, for a …xed tax rate and punishment
probability— that is, the correspondence from
to x, given
and
(for the same
numerical speci…cation as in the preceding footnote). The thinner curve corresponds
to the case of no attachment to the social norm, that is when B (z) = 0 for all z < 0
and B (z) = 1 for all z
0. In this case, all individuals have the same preferences
and thus make identical choices. Therefore, the equilibrium crime rate drops from 1
7 This also shows that the equilibrium crime rate x = v 1 [ u (w) =a] is unstable with respect
to perturbations of expectations about others’behavior.
8 More precisely, if v is logistic,
exponentially distributed with mean value 1,
= 0:2,
=
0:2,
= 0:4 and
= 0:5, then
u (w) = 1
ln 1
+ ln (1
)
1
' 0:2, just as in Figure 3 (see
appendix).
CRIME, PUNISHMENT AND SOCIAL NORMS
17
to 0 as the penalty rate
increases from below to above a certain critical value
o
(here approximately 0:35, see appendix).
x
1
0.75
0.5
0.25
0
0
0.125
0.25
0.375
0.5
0.625
0.75
0.875
1
gamma
ga
Figure 5: The equilibrium correspondence that maps penalty rates to equilibriium
crime rates.
For all penalty rates
>
o, the expected consumption utility from crime falls
short of that from work, so no one then chooses tax evasion, even in the absence
of a social norm against such behavior. Hence, the unique equilibrium tax evasion
rate is zero, both with and without attachment to the social norm. For su¢ ciently
low penalty rates (below 0:07, approximately), the expected consumption utility from
crime is so high that all individuals choose crime, even with norm attachment. For
intermediate penalty rates, 0:07 <
< 0:3, there are three equilibrium crime rates,
in the case with a positive norm attachment. The intermediate equilibrium crime
rate is unstable under expectation perturbations, and thus does not seem plausible
as a prediction. The highest equilibrium crime rate would be close to one while the
lowest equilibrium crime rate lies in the interval between 0 and 0:45, and decreases
smoothly as the penalty rate increases.
Suppose that initially the penalty rate is high (above 0:35), and then gradually
reduced, under the social norm. With inertia in expectations formation, we would
then see how the tax evasion rate would gradually rise and then suddenly jump up,
from about 45% to 100% , as the penalty rate gradually falls below 0:07. Compared
CRIME, PUNISHMENT AND SOCIAL NORMS
18
with the case with no attachment to the social norm, this jump thus takes place at
a much lower penalty rate. The social norm keeps the crime rate down as long as
the economy is locked in on the lower equilibrium. Had the penalty rate thereafter
been gradually increased, then hysteresis would be observed: the evasion rate would
remain very high until the penalty had reached the value 0:3.
A society that is locked in at the lowest of the three equilibria thus has a lower rate
of tax evasion than can be explained by economic incentives alone. Frey and Feld (7)
point out that under purely economic incentives, as in the original Allingham-Sandmo
model, tax evasion should be much higher than what is empirically observed, given the
low level of deterrence in most countries.9 The present model can potentially explain
such empirical observations. The model provides an explanation of why tax evasion
empirically seems to be less pervasive than can be explained by purely economic
incentives.
4.
Crime-deterrence policy as political equilibrium
The model can be used for policy analysis, in particular for analyses of alternative
crime deterrence measures. This can be done normatively— to see what policies maxi-
mize a given welfare function— or positively— in order to predict policy as an outcome
of a democratic voting process.
As noted above, the present model contains three potential instruments for crime
deterrence: the punishment probability , that is, the probability that a given crimi-
nal be caught and convicted, and, indirectly 0 and
, where we recall that
,
0
0w +
0
is the income to an individual with potential wage rate w who has been caught and
convicted. Hence,
0 in part depends on the extent to which criminals also work
and pay taxes, but also in part on the duration of imprisonment (in terms of lost
lifetime wage earnings), and
may be interpreted as a …xed …ne to be paid by an
0
individual who is caught and convicted (if the …ne in part depends on the individual’s
potential wage rate, then this will also in‡uence 0).
We illustrate these possible uses of the model by way of discussing an example.
But …rst some general points.
4.1.
Public budget balance.
Let C ( ; 0;
; x) be the cost to government of
0
keeping the punishment probability at the level
and the duration of imprisonment
9 Frey and Feld (7) argue that to achieve the high compliance rates of 80% observed (respectively
low tax evasion rates) the Arrow-Pratt risk aversion measure would have to be unreasonably high
(around 30) in the original Allingham-Sandmo model to match what is currently observed for the
United States and Switzerland. See also Sandmo (13) and Pommerehne and Weck-Hannemann (11).
CRIME, PUNISHMENT AND SOCIAL NORMS
19
and the …nes such that they result in 0 and
, when the crime rate is x. Plausibly,
0
this cost is increasing in all arguments.
Suppose that the income tax
is the sole source of revenue for the government.
Given the government’s commitments to other public expenditures, E
0, the gov-
ernment budget is then balanced if and only if
R ( ; 0;
; x) = C ( ;
; x) + E;
(16)
0
0;
0
where the left-hand side, R ( ; 0;
; x), is the total tax revenue collected from work-
0
ers (those who choose work and those who choose crime but also work part time),
when the crime rate is x. While tax revenues are plausibly decreasing and contin-
uous in the crime rate, at …xed policy parameters, expenditures are increasing and
continuous. Therefore, if other public spending, E, is not excessive in relation to the
income tax rate, there will exist a unique crime rate that balances the government
budget (where the downward-sloping revenue curve intersects the upward-sloping ex-
penditure curve). A quintuple s = ( ; 0;
; ; x) that satis…es the budget equations
0
(16) and the equilibrium equation (8) will be called a balanced equilibrium state of
the economy, and we denote this set S .
A normative approach to crime deterrence policy can thus be developed by way
of maximization of a welfare function over the set of balanced equilibrium states,
de…ned by these two equations.
4.2.
Political equilibrium.
By a political equilibrium we mean a balanced equi-
librium state such that no other balanced equilibrium state is preferred by a majority
of voters. More exactly, the voting situation for each individual may be thought
of as a vote between a current policy p = ( ; 0;
; ), and some “opposing” pol-
0
icy p0 = ( 0; 0 ; 0 ; 0), where the current crime rate x is uniquely determined by
0
0
the condition that s = ( ; 0;
; ; x) be a balanced equilibrium state, and where
0
( 0; 0 ; 0 ; 0) is such that s0 = ( 0; 0 ; 0 ; 0; x0) is a balanced equilibrium state (where
0
0
0
0
again x0 is uniquely determined by p0). All voters expect the current crime rate x
under the incumbent policy p, and they all expect some crime rate x0 under the
alternative policy p0.
The expected utility to an individual with potential wage rate w > 0 and norm
attachment parameters
and
in any state s = ( ; 0;
; ; x) is
0
U (s; w; ; ) = max fUW;UCg,
where
UW = u (w
w; 1) + ( +
) v (x)
(x)
(17)
CRIME, PUNISHMENT AND SOCIAL NORMS
20
and
UC = (1
) u ( 1w +
; h
; h
1
1) +
u ( 0w + 0 0)
(x) .
(18)
In other words, each individual anticipates to make an optimal individual choice were
the state s to materialize.
We de…ne a policy p = ( ; 0;
; ) to be a political equilibrium policy, or an
0
unbeatable policy under majority rule, if the “current” state s = ( ; 0;
; ; x) is a
0
balanced equilibrium state and is preferred by a (weak) majority over any alternative
balanced equilibrium state s0 = ( 0; 0 ; 0 ; 0; x0), that is, if
0
0
Z +1Z +1 H[U(s;w; ; ) U(s0;w; ; )]d (a;w) 1=2 8s02S , (19)
w=0
a= 1
where H is the indicator function de…ned by H (x) = 1 for x
0 and H (x) = 0 for
x < 0.10
A positive approach to crime deterrence policy can thus be developed by way
of identifying the (potentially empty) subset S0
S of balanced equilibrium states
that satis…es (19). Given this subset, the associated subset P 0 of political equilibrium
policies is de…ned as those policies ( ; 0;
; ) for which s = ( ;
; ; x)
0
0;
0
2 S0.
We illustrate this abstract machinery by way of an example.
4.3.
Tax-evasion: an example.
Suppose that all individuals have logarithmic
consumption utility and that their norm attachment is driven solely by shame ( = 0).
Suppose, moreover, that the shame parameter
is distributed according to some con-
tinuous cumulative probability distribution function B. We focus on the special case
of tax evasion when
= 0, that is, when all tax evaders report zero income. More-
over, assume that the penalty
paid by convicted tax evaders is not a source of
revenues to the government (more realistically, part of these penalties would consti-
tute government income). The punishment probability
is assumed to be …xed, and,
at that punishment probability, the cost of maintaining and enforcing the legal code
is assumed to be an a¢ ne increasing function of the crime rate: C = C0 + cx, where
C0 > 0 is the …xed cost and c > 0 the constant marginal cost. This is the case if the
capacity of the law enforcement system can be adjusted, at a cost, so as to keep the
punishment probability at a given level.
Hence, for any crime rate x, government revenues emanate from only one source,
namely from those who work and pay taxes, and these revenues have to …nance its
10 In this de…nition, the preference may be required to be strict or weak and likewise with the
majority. We here take both to be weak.
CRIME, PUNISHMENT AND SOCIAL NORMS
21
expenses:
(1
x) w = C0 + cx + E;
(20)
where C0+E > 0 is …xed, and w is the average wage. (Recall that individual decisions
are independent of one’s own wage.) The only remaining policy variables are , the
income tax rate, and , the penalty rate paid by tax evaders. More precisely, a tax
evaded with wage w pays his or her tax debt, w, and from the remaining income,
(1
) w, the share .
Combining the public budget balance equation with the equilibrium equation (15),
we obtain the following equation in
and :
w
C
1
0
E
w
C
1
= B v
0
E
ln (1
)
ln (1
) !: (21)
w + c
w + c
Note that if the penalty rate is maximal,
= 1, then the consumption utility of a
caught and convicted criminal is minus in…nity. Hence, the crime rate is then zero
and, by (20), the tax rate is
C0 + E
0 =
;
(22)
w
and where we recall that w is national income.11 The policy ( ; ) = ( 0; 1) is ideal
for every worker, since the tax rate is minimal and the crime rate is zero, hence
minimizing the cost of law enforcement and also maximizing the utility to a worker
from norm adherence.
Under what conditions is the policy ( ; ) = ( 0; 1) a political equilibrium? The
answer is there should exist no alternative balanced-budget equilibrium policy under
which a majority would be tax evaders and obtain a higher expected utility than
when working and paying income taxes according to the tax rate 0. More precisely,
let ~ be the median value of the norm-attachment parameter
, and assume = 0,
that is, that there is no externality from tax evasion. Then
Proposition 3. ( 0; 1) is not a political equilibrium policy if and only if (23) and
(24) hold for some policy ( ; ) that satis…es (21).
~
(1
) (1
)
(1
v(0)
0)1=
e
(23)
2 0 + c=w
(24)
11 To see how this follows from equations (20) and (21), note that for
= 1, the right-hand side
of the latter equation becomes zero, and hence w = C0 + E. Inserting this expression for C0 + E
in the …rst equation, we obtain
x w = cx, which has x = 0 as its unique solution.
CRIME, PUNISHMENT AND SOCIAL NORMS
22
Proof: The policy ( 0; 1) cannot be beaten under majority vote by a policy that
results in a majority of workers, since these would be better o¤ under ( 0; 1). Hence,
only a policy ( ; ) satisfying (21) and such that equation (20) gives x
1=2 can
beat ( 0; 1). Note that x
1=2 holds in a budget-balanced equilibrium if and only if
(24) holds. (The latter inequality is obtained by solving for x in (20) and requiring
x
1=2.) Hence, a competing policy ( ; ) results in x
1=2 i¤ (24) holds, and such
a policy beats ( 0; 1) if and only if the expected utility to a tax evader is higher than
when working and paying income tax 0 under policy ( 0; 1). Using (17) and (18),
the latter condition boils down to
(1
) ln w +
ln ((1
) (1
) w) > ln ((1
0) w) +
v (0)
or, equivalently,
ln ((1
) (1
)) > ln (1
0) +
v (0) :
This inequality needs to hold for all
~ in order for x to be at least 1=2. This is
condition (23). End of proof.
The proposition has certain comparative statics implications, which can be sum-
marized as follows. Let
P0 = ( ; ) 2 [0;1]2 : eqs. (23-24) hold .
We say that ( 0; 1) is more easily a political equilibrium the smaller P0 is. It follows
from the above proposition that the policy ( 0; 1) is more easily a political equilibrium
the stronger the social norm is against tax evasion, the higher the probability is for
a tax evader to be caught and convicted, and the higher is the marginal cost (as a
share of national income) of law enforcement:
Corollary 1. ( 0; 1) is more easily a political equilibrium, the higher ~v (0) is, the
higher
is, and the higher c=w is.
Can any other policy be a political equilibrium? Since ( 0; 1) is the ideal policy
for workers, political equilibria with a majority of tax evaders are the only possible
alternatives. In order to address this question, …rst note that (21) de…nes the penalty
rate
as a function of the tax rate . More exactly, equation (21) is satis…ed if and
only if
= g ( ), where
1
g ( ) = max 0; 1
(1
)
exp B 1
0
v
0
.
(25)
+ c=w
+ c=w
CRIME, PUNISHMENT AND SOCIAL NORMS
23
Let ^ be the lowest tax rate that is compatible with a (weak) majority of tax evaders.
By (20), x
1=2 requires
^, where
^ = 2 0 + c=w:
(26)
Hence, a necessary condition for the existence of political equilibrium with a majority
of tax evaders is ^ < 1. Let
1
= min arg max (1
) exp B 1
0
v
0
.
(27)
2[^;1]
+ c=w
+ c=w
This is the minimal tax rate among those that maximize tax evaders’expected utility
in balanced-budget equilibrium where these constitute a weak majority. For continu-
ous functions B 1 and v, the maximand is continuous and hence the set of maximzers
is non-empty and compact (by Weierstrass’Maximum Theorem), so
is then well-
de…ned, granted ^ < 1. Let
= g (
). This is the unique penalty rate that makes
(
;
) a balanced equilibrium policy. If there is a political equilibrium with a ma-
jority of tax evaders, then the tax rate in that equilibrium needs to be
and the
penalty needs to be
since otherwise policy (
;
) would defeat that policy under
majority rule. Also the converse holds:
Proposition 4. Suppose that ^ < 1 and B 1 is continuous. Then (
;
) is a
political equilibrium if and only if
~
(1
) (1
)
(1
v(0)
0)1=
e
.
(28)
Proof : Suppose that ^ < 1, B 1 is continuous and (28) holds. Then (
;
)
results in a balanced equilibrium state with a (weak) majority of tax evaders, and, by
(28) their expected utility is at least as high as under the optimal policy for workers.
Hence, no policy ( ; ) with
< ^ can obtain a majority against (
;
). But nor
can any policy with
> ^ since
maximizes tax evaders’expected utility across all
balanced-budget equilibria where these constitute a weak majority. End of proof.
It is harder to discuss the comparative statics of this proposition than of the
previous, since the candidate tax rate,
, is more indirectly de…ned in terms of the
primitives than is 0. However, in some special cases, the conditions of the proposition
are fairly transparent.
We here focus on the special case when a majority attaches no weight to the social
norm, that is, when ~ = 0, while the remaining minority attaches some positive weight
CRIME, PUNISHMENT AND SOCIAL NORMS
24
> 0 on the norm. Let thus x
1
1 = B (0) > 1=2. The minority of individuals who
care about the norm would be willing to report their incomes honestly even if the
others don’t and even if there were no penalty for tax evaders ( = 0), granted that
1
v (x
ln (1
) ,
(29)
1
1)
1
where
is the going tax rate. Suppose this is chosen such that the tax revenue from
the “honest”minority is su¢ cient to …nance all public expenditures, that is,
= 1,
where
C0 + cx1 + E
1 =
< 1.
(1
x1) w
In the absence of a penalty for tax evaders, the “dishonest” majority will choose
tax evasion, and thus ( 1; 0) is a political equilibrium. We see in (29) that, not
surprisingly, this is easier to obtain the larger is the utility
v (x
1
1) that the honest
derive from adhering to the norm of truthful income declaration when the population
fraction x1 = B (0) violate the norm. The larger that utility is, the more vulnerable
are the honest to exploitation by the dishonest.
u
2
1.5
1
0.5
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
x1
Figure 6: The political equilibrium correspondence that maps the share of tax
evaders to the norm attachment of the honest.
We end by illustrating this graphically. Figure 6 plots the minimal utility value
of norm adherence, u =
v (x), needed for a given population share x = B (0)
1
CRIME, PUNISHMENT AND SOCIAL NORMS
25
1=2 of individuals with no norm attachment for the policy ( 1; 0) to be a political
equilibrium, see Appendix for the numerical speci…cation. The higher the utility of
norm adherence for the honest is, the larger is the share of tax evaders that can
exploit them in political equilibrium.
5.
Appendix
We here provide the numerical speci…cations used in the diagrams.
5.1.
Figures 1 and 2.
Suppose that all individuals have the same potential wage
w, that their norm attachments, a, are exponentially distributed with mean E (a) = 1,
and that the norm-adherence utility function v is logistic. Figure A1 shows the graph
of the logistic norm-adherence utility function
1
v (x) =
:
(30)
1 + exp (8x
4)
v
1
0.75
0.5
0.25
0
0
0.25
0.5
0.75
1
x
Figure A1
With this norm-adherence utility function and exponentially distributed norm at-
tachment, the …xed-point equation (8) takes the (doubly exponential) form12
u (w)
x = 1
exp
[1 + exp (8x
4)] ,
(31)
E (a)
12 The right-hand side is formally equivalent with a Gumbel, or doubly exponential, probability
distribution function.
CRIME, PUNISHMENT AND SOCIAL NORMS
26
where
u (w) is given by (7). Figure 1 is drawn for
u (w) = 0:2. Figure 2 uses (31)
allowing
u (w) to vary implicitly with the crime rate x.
5.2.
Figure 3 and 4.
A Weibull ( ; ; ) c.d.f. G (x) is zero for all x < , while
for x
:
x
G (x) = 1
exp
:
The inverse of this c.d.f., when
= 0, is
1
1
G 1 (x) =
ln
.
1
x
Let the potential wage w be Weibull (0; 1 ; 2) distributed, see its density in Figure A2,
2
and that the norm-adherence utility function v is logistic.
g(w)
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
0.5
1
1.5
2
w
Figure A2
The wavy curve of Figure 3 is drawn for w distributed Weibull (0; 1 ; 2),
= 0:65,
2
= 1 and
= 0:35. The other parameters were set such that
= 0:25, ln
= 0:22
1
(which corresponds to the consumption utility for a worker earning a potential wage
in the upper tail of the wage distribution of 1:25 after tax) and ln
=
2:3 (the
0
approximate consumption utility for a worker earning a potential wage in the lower
tail of the wage distribution of 0:1 after tax). We have therefore that the straight line
CRIME, PUNISHMENT AND SOCIAL NORMS
27
in Figure 3 is given by
1
1
exp [(1
) ln
+
ln
] =
exp [0:65 ln (1:25) + 0:35 ln (0:1)]
1
1
0
1
0:25
' 0:69:
Figure 4 is based on this same speci…cation maintaining …xed the tax rate
= 0:25
while letting the punishment probability
vary implicitly with x such that (8) is
satis…ed.
5.3.
Figure 5.
The thick curve of this diagram represents the case with a social
norm that takes the form of pure shame ( = 0 and
distributed exponentially with
mean E ( ) = 1) and uses the following set up: v is a logistic norm-adherence utility
function as in (30),
= 0:3,
= 0:2 and
= 0:25. The …xed-point equation (??) is
then
1
x = 1
exp
(1 + exp (8x
4)) 0:23
0:35
which is the equation that implicitly relates x to
and generates the thick curve in
Figure 5. The thin curve, on the other hand, represents the case in which
=
= 0.
The critical punishment probability that makes every individual indi¤erent between
work and evading taxes is then given by
ln (1
)
ln (1
)
~ = ln (1
)
ln (1
)
ln (1
) ' 0:4:
(32)
Nobody evades taxes if
> ~ while everyone does so if
< ~.
5.4.
Figure 6.
The numerical speci…cation behind this diagram is w = 1, C0 +
E = c = 0:1, and
= 0:5. The resulting political equilibrium condition (29) then
becomes
1 + x
u
1 ln 1
0:1 1 x
or, equivalently,
1
x
u
ln
.
0:9
1:1x
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