In his seminal article on the allocation of time to crime, Isaac Ehrlich (1973) derives five interesting theoretical results. He uses a state-preference diagram to derive one result, retreating to mathematics for deriving the remaining four results. This note shows that all five results can easily be derived from an alternative and simpler diagrammatical exposition that involves intersection of curves rather than tangency between curves.

Gideon Yaniv

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 1 
  
The allocation of time to crime: A simple diagrammatical exposition 
  
 
 
Gideon Yaniv  
COM Academic Studies 
Abstract 
In his seminal article on the allocation of time to crime, Isaac Ehrlich (1973) derives five interesting theoretical results. 
He uses a state-preference diagram to derive one result, retreating to mathematics for deriving the remaining four 
results. This note shows that all five results can easily be derived from an alternative and simpler diagrammatical 
exposition that involves intersection of curves rather than tangency between curves.
I am indebted to an anonymous referee for helpful comments and suggestions 
Citation: Gideon Yaniv, (2009) ''The allocation of time to crime: A simple diagrammatical exposition'', Economics Bulletin, Vol. 29 no.1 pp. 
457-464. 
Submitted: Dec 11 2008.   Published: March 27, 2009. 
 
        1
1.  Introduction 
 
Most theoretical models of criminal behavior consider the decision to commit a crime as 
binary: the individual may either participate in criminal activity or in legitimate work [e.g., 
Becker (1968), Glaeser et al (1996), Polinsky and Shavell (2000)]. An exception is Isaac 
Ehrlich’s (1973) seminal article on the allocation of time to crime, which allows the individual 
to divide his time between criminal activities and legitimate work. Two decades later, Richard 
Freeman (1996) supported empirically Ehrlich’s approach, arguing that  
 
“Many youths combine crime and work or shift between them readily. 
Because most criminals are self-employed, and because the U.S. job market 
has considerable flux, crime and legitimate work are not dichotomous 
choices for most young men. Joe holds a job, robs someone he meets on a 
dark empty street and sells drugs on the weekend.”(p. 34).  
 
Freeman’s evidence renews interest in Ehrlich’s approach and provides motivation for 
revisiting his work. 
 
Ehrlich derives five theoretical results: (a) a sufficient condition for entry into crime – 
regardless of the attitude toward risk – is that the marginal differential return from crime 
exceeds the marginal expected penalty at the position where no time is devoted to crime; (b) a 
risk-neutral offender will spend more time in crime relative to a risk avoider, whereas a risk 
preferrer will spend more time in crime relative to both; (c) an increase in the probability of 
apprehension will reduce the time spent in crime - regardless of the offender’s attitude toward 
risk; (d) an increase in the marginal penalty will reduce the time spent in crime of a risk 
avoider and a risk-neutral offender, but might increase the time spent in crime of a risk 
preferrer; (e) a 1 percent increase in the probability of apprehension accompanied by a 1 
percent decrease in the marginal penalty will not affect the time spent in crime of a risk-
neutral offender, but will decrease the time spent in crime of a risk preferrer and will increase 
the time spent in crime of a risk avoider. The last three results relate to offenders who are only 
partially engaged in crime rather than fully specializing in it. Otherwise they may not respond 
at all.  
 
Ehrlich uses a state-preference diagram to demonstrate the optimal solution of a risk avoider 
as a point of tangency between a convex indifference curve and a concave opportunity 
boundary. He also uses the diagram to derive result (a) above. However, he retreats to 
mathematics for deriving all other results. The purpose of the present note is to show that all 
five results can easily be derived from an alternative diagrammatical exposition. The proposed 
exposition is also simpler than the state-preference diagram, as it involves intersection of 
curves rather than tangency between curves. Consequently, the optimal solution of a risk 
preferrer, which in a state-preference diagram necessitates a cumbersome tangency between 
two concave curves, becomes clearer and graphically comparable with the optimal solutions of 
the other risk-attitude types.          
  
2.  Ehrlich’s model 
 
Ehrlich considers an individual who, at the beginning of a given period, may allocate a given 
amount of time, T, between criminal activity, S, and legitimate work, T− S. The monetary    2
value of his returns from crime is m(S), where m′(S) > 0 and m″(S) < 0. The monetary value of 
his returns from legitimate work is w(T− S), where w′(T− S) > 0 and w″(T− S) < 0. With 
probability p the individual will be apprehended at the end of the period and punished. The 
monetary value of the punishment is assumed to be F(S), where F′(S) > 0 and F″(S) ≥ 0. 
Denoting initial assets by W0, the individual’s net assets at the end of the period will be  
 
                                                     0 () ( ) WWm S w T S
+ = ++ −                                              (1) 
 
if he is not apprehended, and   
 
                                               0 () ( ) () WWm S w T SF S
− =+ + −−                                      (2) 
                  
if he is apprehended. The individual seeks the value of S that maximizes his expected utility 
from the alternative net asset levels 
 
                                                ( ) (1 ) ( ) ( ) EU W p U W pU W
+ − =− + .                                     (3) 
 
Substituting (1) and (2) in (3) and differentiating with respect to S, the first-order condition for 
expected-utility maximization is given by  
 
                                                 
'( ) '( )
'( ) '( ) (1 ) '( )
dS p UW
FS dS p UW
−
+ =
−−
,                                      (4) 
 
where  '( ) '( ) '( ) dS mS wT S ≡− − is the marginal differential return from crime. An interior 
solution requires that the left-hand side of (4) is positive, or that  '( ) 0 dS>  and 
'( ) '( ) 0 FS dS −>  for some 0 < S < T.
1 Ehrlich uses a state-preference diagram to 
demonstrate the solution of condition (4) for a risk avoider, interpreting the left-hand side of 
(4) as the slope of an opportunity boundary and its right-hand side as the slope of an 
indifference curve which holds expected utility constant. He also uses the diagram to derive 
result (a) above. All other results are derived mathematically from the first-order condition 
with no reference to the diagram. 
 
 
3.  An alternative diagrammatical exposition 
 
Let the first-order condition be rewritten as 
 
                                                  
(1 ) '( ) '( )
'( ) '( ) '( )
pd S U W
pF S pd S U W
−
+
−
=
−
.                                          (4’) 
 
Denote  the  left- and  right-hand  side  of  (4’) by α(S) and β(S), respectively. Figure 1 depicts 
 
_____________________________ 
 
1 A prerequisite for this is that d′(0) > 0 (otherwise the individual will never engage in crime) and F ′(T) − d′(T) > 
0 (otherwise the individual will always specialize in crime).    3
 
 
 
 
  
  
●  
●  
●  
() S α   
() S
risk avoider
β
 
() S
risk neutral
β
 
() S
risk preferrer
β
  
S  
1  
T   0  
(0) α  
() S β  
() S α  
Figure 1: The allocation of time to crime  
c  
b  
a     4
α(S) and β(S) as a function of S. The assumptions on the return and penalty functions imply 
that α′(S) < 0. Hence, the α(S) curve declines from left to right. For any S > 0, the value of 
β(S) and the sign of β′(S) depend on the individual’s attitude toward risk: for a risk-neutral 
individual, β(S) = 1 and β′(S) = 0; for a risk avoider, β(S) > 1 and β′(S) > 0; for a risk 
preferrer, β(S) < 1 and β′(S) < 0. Hence, the β(S) curve is a horizontal line at 1 for a risk 
neutral, rises from left to right above 1 for a risk avoider, and declines from left to right below 
1 for a risk preferrer.
2 When S = 0, β(S) = 1 for all individuals, regardless of the attitude 
towards risk. Equilibrium is obtained at the point of intersection between the α(S) and β(S) 
curves. 
 
A glance at Figure 1 immediately confirms result (b): the equilibrium level of S for a risk 
avoider (point a) is lower than that of a risk neutral (point b), which is lower than that of a risk 
preferrer (point c). A sufficient condition for obtaining equilibrium at some S > 0 (rather than 
at S = 0) is that the α(0) point lies above 1. This implies that d ′(0) > pF ′(0), which confirms 
result (a). An increase in p will shift the entire α(S) curve downward, generating lower 
equilibrium levels for all risk-attitude types (points a’, b’, and c’ in Figure 2). This confirms 
result (c). An increase in F′(S) (for a given level of S) will  generate a similar shift in the α(S) 
curve, but it may affect the β(S) curve as well, because it raises F(S) and reduces W
−. For a 
risk avoider, the β(S) curve will shift upward, strengthening the former effect in reducing S 
(point a” in Figure 2). For a risk preferrer the α(S) curve will shift downward, acting to 
increase S. The combined effects of the shift in the α(S) and β(S) curves might lead to a higher 
equilibrium level for S (point c” in Figure 2), which confirms result (d).
3 Finally, a 1 percent 
increase in p accompanied by a 1 percent decrease in F′(S) will leave the product pF′(S) intact. 
Differentiating α(S) with respect to p, holding pF ′(S) constant, yields 
 
                                                 2
[ ( )] '( )[ '( ) '( )]
[' ( ) ' ( ) ]
Sd S d S p F S
pp F S p d S
α ∂−
=
∂−
,                                        (5) 
 
the sign of which is determined by the sign of  '( ) '( ) dS p FS − . The latter expression is 
positive as long as α(S) > 1, zero when α(S) = 1, and negative if α(S) < 1. It thus follows that 
the α(S) curve will shift upward in its segment above 1 and downward in its segment below 1 
(dotted curve in Figure 3). This will act to raise the equilibrium level of S for a risk avoider 
(point a’), but to lower it for a risk preferrer (point c’). In addition, the fall in F(S) resulting 
from  the  decrease  in  F′(S)  will shift the β(S) curve downward for a risk avoider and upward 
 
 
__________________________________________ 
2 For convenience, the β(S) curve is drawn as if β″(S) < 0 for a risk avoider and β″(S) > 0 for a risk preferrer. 
This, however, must not be the case. The results would not be affected if the sign of β″(S) was the opposite to the 
one assumed in the diagram. 
  
3 The effect of a change in the marginal penalty on the time spent in crime is particularly difficult to demonstrate 
in a state-preference diagram as it involves a change in slopes of both the opportunity boundary and the 
indifference curves. In a review of Ehrlich’s model, Pyle (1982) attempts to depict it, holding, however, the 
individual’s utility level constant. Furthermore, he considers the response of a risk avoider only, for whom the 
slope of the indifference curve increases with S while the slope of the boundary opportunity decreases. For a risk 
preferrer, the slope of the indifference curve decreases as well, making it practically impossible to demonstrate 
his response diagrammatically. Similarly, it is practically impossible to demonstrate result (e) in a state-
preference diagram.        5
 
 
 
 
 
 
      
   
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
a'  
Figure 2 : The individual's response to an increase in the probability         
                 of apprehension or the marginal penalty                                               
             
b'   b  
c  
c'  
c''  
S   T   
() S β  
0  
() S α
a''  
a  
(0) α  
1     6
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
  
  
  
  
  
  
  
             Figure 3: The individual’s response to a 1 percent increase in the probability  
                             of apprehension accompanied by a 1 percent decrease in the  
                             marginal penalty        
 
 
●  
●  
●   ●   
●  
●   
●  
a'  
() S α  
T    0  
1    b   
c'
c'   
c   
a'
a  
() S β  
(0) α  
S     7
for a risk preferrer, strengthening the effect on S of the clockwise shift in the α(S) curve 
(points  a’ and c’, respectively). Result (e) is thus confirmed as well. 
 
 
4.  Concluding remarks 
 
In his seminal article on participation in illegitimate activities, Ehrlich (1973) derives five 
theoretical results which link the allocation of time to crime to the individual’s attitude toward 
risk. He uses a state-preference diagram, which involves indifference curves defined over the 
individual’s final wealth when apprehension takes place and when it does not, to demonstrate 
one result and mathematics for deriving the rest. This diagrammatical exposition, probably 
more conventional for models at that time, is interesting but complicates the graphical analysis 
and hides the intuition behind the results. The present paper transforms Ehrlich’s first-order 
condition into equality with two terms, graphically presented as a function of the time devoted 
to crime. One term consists of the ratio between the marginal utilities when apprehension takes 
place and when it does not, which is fully determined by the individual’s attitude toward risk. 
The other term consists of the ratio between the (expected) marginal differential return from 
crime when apprehension does not take place and when it does, which is determined by the 
assumptions on the returns from legitimate work and criminal activity as well as by the 
monetary value of the punishment. This exposition is considerably helpful for understanding 
the driving force behind the results, because they derive from intersections of curves rather 
than from tangency between curves. There are also opportunities for further research based on 
this approach, such as inquiring how the results would change if individuals are risk loving in 
term of losses but risk abiding in terms of gains, as argued by prospect theory. 
 
 
References 
 
Becker, G.S. (1968) “Crime and Punishment: An Economic Approach”, Journal of Political 
Economy 76, 169-217. 
       
Ehrlich, I. (1973) “Participation in Illegitimate Activities: A Theoretical and Empirical 
Investigation”, Journal of Political Economy 81, 521-565.  
 
Freeman, R. B. (1996) “Why Do So Many Young American Men Commit Crimes and What 
Might We Do About It?,” Journal of Economic Perspectives 10, 25-42. 
 
Glaeser, E.L., Sacerdote, B., and J.A. Scheinkman (1996) “Crime and Social Interactions”, 
Quarterly Journal of Economics 111, 507-548. 
 
Polinsky, A.M., and S. Shavell (2000) “The Economic Theory of Public Enforcement of 
Law”,  Journal of Economic Literature 83, 45-76. 
 
Pyle, D.J. (1984) The Economics of Crime and Law Enforcement, Palgrave Macmillan: New 
York.  

Crime and Punishment: An Economic Approach”,

Crime and Social Interactions”,

Participation in Illegitimate Activities: A Theoretical and Empirical Investigation”,

The Economic Theory of Public Enforcement of Law”,

The Economics of Crime and Law Enforcement, Palgrave Macmillan:

Why Do So Many Young American Men Commit Crimes and What Might We Do About It?,”

The allocation of time to crime: A simple diagrammatical exposition

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