Crime and Punishment: An Economic Approach

Gary Becker (JPE, 1968)

Notes by Team Grossman (Spring 2015)

Meagan Madden, Mavzuna Turaeva, Ke Xu

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Contents 1. INTRODUCTION .................................................................................................................................... 3

2. THE MODEL ........................................................................................................................................... 3

The Cost of Crime ...................................................................................................................................... 3

Key Notation for Reference ...................................................................................................................... 4

Model Components .................................................................................................................................. 5

a. Damages ........................................................................................................................................ 5

b. Apprehension and Conviction ....................................................................................................... 6

c. Supply of Offenses ........................................................................................................................ 7

d. Punishment ................................................................................................................................... 9

3. OPTIMALITY CONDITIONS .................................................................................................................. 10

4. SHIFTS IN THE BEHAVIORAL RELATIONS ........................................................................................... 11

a. Effect of 𝐷′ ...................................................................................................................................... 12

b. Effect of 𝐶𝑝 ..................................................................................................................................... 13

c. Effect of 𝜀𝑓 and 𝜀𝑓 .......................................................................................................................... 15

5. FINES ................................................................................................................................................... 16

Welfare Theorems and Transferable Pricing .......................................................................................... 16

Optimality Conditions ............................................................................................................................. 16

The Case for Fines ................................................................................................................................... 17

Compensation and the Criminal Law ...................................................................................................... 17

6. SOME APPLICATIONS (BECKER) ......................................................................................................... 18

The Effectiveness of the Public Policy ..................................................................................................... 18

A Theory of Collusion .............................................................................................................................. 19

7. EXTENSIONS/CRITICISM ..................................................................................................................... 19

“Prison”- Community .............................................................................................................................. 19

“Expensive” Prison Time ......................................................................................................................... 20

“Cost” of Judicial Error ............................................................................................................................ 20

Why does murder have a higher probability of conviction? .................................................................. 20

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1. INTRODUCTION There is a large economic literature related to crime which traces back to Gary Becker’s seminal

work “Crime and Punishment.” The article currently has 11,404 citations in Google Scholar.

Becker acknowledges the founding works of Bentham (An Introduction to the Principles of

Morals and Legislation, 1780) and Beccaria (On Crime and Punishments, 1764) who applied an

economic framework to decisions of crime and deterrence, and Becker suggests that his 1968

paper is “a resurrection, modernization, and […] improvement on these much earlier pioneering

studies.” Here, Becker acknowledges that zero crime is not optimal for society because

significant resources are devoted to fighting crime. Instead, he approaches crime from the

perspective of the losses that it inflicts on society. The concept of social loss was not novel in

Becker’s time (see The Social Loss of Dying Parents, Glaster and Strauss, 1964), but its

application to crime was a significant contribution.

The model is a function of social loss (borrowed from traditional welfare economics) which

incorporates the various costs associated with crime: (1) costs from criminal damages, (2) costs

to apprehend & convict criminals, and (3) social costs associated with punishment. The optimal

level of crime is that which minimizes the social loss function. The associated level expenditures

to fight crime and the associated penalties to deter crime can be derived from this optimum.

Becker then uses this theoretical framework to discuss public policy implications for crime and

punishment.

2. THE MODEL

The Cost of Crime Here, “crime” is considered any violation of the law, ranging from murder to tax-evasion to

traffic violations. As Becker states, crime is an economically important “industry” with

significant resources going to determent, enforcement, prosecution, and punishment of offenses,

as well as the damages lost (e.g. stolen property) from offenses. Table 1 gives a snapshot from

1965 of public expenditures on the crime “industry”.

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In 1965 the total cost of crime equaled 21 billion, which was about 4% of national income.

Becker argues that the numbers are significantly understated and it would be much higher if

other unmeasured social losses were accounted for. For example, the cost of murder is measured

by the loss of earnings of victims and excludes the value placed by society on life itself, thus

represents an unmeasured loss/cost to the society; or the cost of gambling excludes both the

utility to the gambler and “external” disutility to some clergy. Also, private expenditures are

significantly understated, as many crime preventions activities cannot be easily measured, for

examples, choosing to live in the suburbs, taking taxi instead walking, and etc.

Key Notation for Reference

Notation Variable Description Functional Relationships

𝑶𝒊𝒋 Activity level of i

th crime committed by j

th individual;

measured as # offenses

Supply function for j:

𝑂𝑗 = 𝑂𝑗(𝑝𝑗 , 𝑓𝑗 , 𝑢𝑗)

𝒑 Probability of conviction (measured as ratio of

convictions cleared to total offenses) Market supply function:

𝑂 = 𝑂(𝑝, 𝑓, 𝑢) 𝒇 Cost of punishment

𝒖 Additional determinants of criminal activity

𝑯𝒊 Harm from i th

crime (to society) 𝐻𝑖(𝑂𝑖)

𝑮𝒊 Gain from i th

crime (to offenders) 𝐺𝑖(𝑂𝑖)

𝑫 Net damage to society 𝐷(𝑂) = 𝐻(𝑂) − 𝐺(𝑂)

𝑨 Police & judicial “activity” to apprehend & convict

criminals

𝐴 = 𝑓(𝑚, 𝑟, 𝑐);

𝐴 ≅ 𝑝𝑂

𝐴 = ℎ(𝑝, 𝑂, 𝑎)

𝒎 Manpower devoted to apprehend & convict offenders

𝐴 = 𝑓(𝑚, 𝑟, 𝑐); 𝒓 Resources devoted to apprehend & convict offenders

𝒄 Capital devoted to apprehend & convict offenders

𝑪 Cost to apprehend & convict criminals 𝐶 = 𝐶(𝐴)

𝒇′ Social cost of punishment

𝑓′ ≡ 𝑏𝑓 𝒃

Multiplier that transforms individual cost of

punishment to social cost of punishment

𝑬𝑼 Expected utility of criminal activity 𝐸𝑈 = 𝑝 ∙ 𝑈(𝑌 − 𝑓) +

(1 − 𝑝) ∙ 𝑈(𝑌)

𝑳 Social loss from crime 𝐿 = 𝐿(𝐷, 𝐶, 𝑏𝑓, 𝑂)

𝐿 = 𝐷(𝑂) + 𝐶(𝑝, 𝑂) + 𝑏𝑝𝑓𝑂

�̂� Optimal number of offenses 𝑉 = 𝐺′(�̂�)

When 𝑏 = 0, 𝑉 = 𝐻′(�̂�) 𝑽 Marginal monetary value of punishments

𝑮 Private gain coming from criminal offenses

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Model Components

a. Damages The motivation behind outlawing or restricting an activity is a belief that other members of

society are harmed by the activity, resulting in diseconomies. Here, the level of harm from

criminal activities is a function of the number of criminal offenses. However, the individuals that

commit crimes receive some benefit, which is also a function of the number of offenses. It

follows that the net damage is the difference between the harm to society and gain to the

individual offender.

Variable Notation Description

𝑂𝑖 Criminal activity level of i

th crime; measured as #

offenses

𝐻𝑖 Harm from i th

crime (to society)

𝐺𝑖 Gain from i th

crime (to offenders)

𝐷 Net damage to society

Functional

Relationship Description

First- and Second-

Order Conditions Description

𝐻𝑖 = 𝐻𝑖(𝑂𝑖)

Harm from crime is

function of #

offenses

𝐻𝑖′ > 0 Harm increases with # offenses

𝐻𝑖" > 0 Increasing marginal harm

𝐺𝑖 = 𝐺𝑖(𝑂𝑖)

Gain from crime is

function of #

offenses

𝐺𝑖′ > 0

Gain to offenders increases with #

offenses

𝐺𝑖" < 0

Diminishing marginal gains to

offenders

Cost of Crime

a. Damages

b. Apprehension & Conviction

(Public Expenditures)

c. Supply of Offenses

d. Punishment

Protection & Apprehension

(Private Expenditures)

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𝐷(𝑂)= 𝐻(𝑂) − 𝐺(𝑂)

Net damage is

difference between

harm and gain

𝐷′ = 𝐻′ − 𝐺′ ? 0

Both 𝐻𝑖′ and 𝐺𝑖

′ > 0, so sign of 𝐷′

depends on relative magnitudes of

𝐻𝑖′ and 𝐺𝑖

′; Becker assumes 𝐷′ > 0

except in section V

𝐷" = 𝐻" − 𝐺" > 0

Increasing marginal harm for

additional offenses, assuming

𝐻𝑖" > 0 and 𝐺𝑖

" < 0

b. Apprehension and Conviction Public resources for police forces and courts are necessary to enforce laws and prosecute

criminals. In 1965 public expenditure on police and court “activity” totaled to over $3 billion

dollars. According to the estimates prepared for seven major felonies, the average cost per

offense was $500 and per person arrested was $2000.

The cost of public expenditures depends on the “activity” produced by police, judges, juries, and

the sophistication level of technologies (e.g. fingerprinting, lie-detector tests). Becker models the

cost of apprehension and conviction (A&C) as a function of “activity” (𝐴). He uses the number

of convictions as a convenient approximation of 𝐴, and also uses a more general function for

activity. He derives the first and second order conditions for cost that are necessary for an

interior solution to the loss minimization problem.

Variable Notation Description

𝑝 Probability of conviction (measured as ratio of

convictions cleared to total offenses)

𝐴

Police & judicial “activity” to apprehend & convict

criminals; activity involves inputs of manpower (𝑚),

resources (𝑟), and capital (𝑐)

𝐶 Cost to apprehend & convict offenders

Functional

Relationship Description

First- and Second-Order

Conditions Description

𝐴 ≅ 𝑝𝑂

Becker suggests #

convictions (𝑝𝑂)

as approximate

measure of 𝐴

𝐴 = ℎ(𝑝, 𝑂, 𝑎)

More general

function

incorporates other

determinants of

“activity” (𝑎), e.g.

arrests;

ℎ𝑝 > 0

ℎ𝑂 > 0

ℎ𝑎 > 0

Reasonable assumptions that

activity level increases with 𝑝, 𝑂, 𝑎

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𝐶 = 𝐶(𝐴)

Cost is a function

of the activity

level;

First, assumes

𝐶 = 𝐶(𝑝𝑂)

𝐶′ =𝑑𝐶

𝑑𝐴> 0

A&C cost increases with activity

level (e.g. adding more police force)

𝐶𝑝 =𝜕𝐶(𝑝𝑂)

𝜕𝑝= 𝐶′𝑂 > 0

A&C cost increases with respect to

increases in probability of

conviction (i.e. if more people

convicted via higher police &

judicial activity, then costs are

higher for higher activity level)

𝐶𝑂 =𝜕𝐶(𝑝𝑂)

𝜕𝑂= 𝐶′𝑝 > 0

A&C cost increase wrt increases in

# of offenses (i.e. if more crimes

committed then activity level must

increase to maintain same

probability of conviction, thus cost

is higher for higher activity level)

𝐶𝑝𝑝 = 𝐶"𝑂2 > 0 Assumes that 𝐶" > 0, which implies

that the Hessian matrix of second

derivatives for the cost function is

positive definite can find a

minimum

𝐶𝑂𝑂 = 𝐶"𝑝2 > 0

𝐶𝑂𝑝 = 𝐶𝑝𝑂 = 𝐶"𝑝𝑂 + 𝐶′

> 0

Then, applies more

general function

𝐶 = 𝐶(𝑝, 𝑂, 𝑎)

𝐶𝑝 > 0

𝐶𝑂 > 0

𝐶𝑎 > 0

Follows from assumptions that

ℎ𝑝, ℎ𝑂 , ℎ𝑎 > 0

𝐶𝑝𝑝 ≥ 0

𝐶𝑂𝑂 ≥ 0

𝐶𝑝𝑂 ≅ 0

Sufficient assumptions to get interior

solution; suggests that first two

conditions are reasonable but third

is less reasonable

c. Supply of Offenses Why do people commit crime? Diverse theories suggest that skull type, genetic predisposition,

family up-bringing, and disenchantment with society are determinants of crime. Among the

various explanations, there is consensus that an increase in the probability of conviction or

punishment would generally decrease the number of offenses committed. Thus, Becker models

the supply of offenses as a function of 𝑓 and 𝑝, as well as other determinants 𝑢.

Variable Notation Description

𝑂𝑗 Crime level committed by j

th offender; measured as

# offenses

𝑝𝑗 Probability of conviction for j th

offender

𝑓𝑗 Cost of punishment to the j th

offender

𝑢𝑗

Various other determinants of criminal activity for j th

offender (e.g. income available through legal or

illegal means, arrest history, willingness to break the

law)

𝑌𝑗 Income from a criminal offense for j th

offender

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Functional

Relationship Description

First- and

Second-Order

Conditions

Description

𝑂𝑗 = 𝑂𝑗(𝑝𝑗, 𝑓𝑗 , 𝑢𝑗)

Supply of offenses

committed by

individual j is function

of 𝑝𝑗 , 𝑓𝑗 , 𝑢𝑗

𝑂𝑝𝑗=

𝜕𝑂𝑗

𝜕𝑝𝑗< 0

𝑂𝑓𝑗=

𝜕𝑂𝑗

𝜕𝑓𝑗< 0

If probability of conviction (𝑝) or

punishment (𝑓) increase, then the

supply of offenses with decrease

𝑂𝑢𝑗=

𝜕𝑂𝑗

𝜕𝑢𝑗 ? 0

Can reasonably anticipate the effects of

𝑢𝑗 components (e.g. an increase in

legal income should decrease #

criminal offenses)

𝑂 = 𝑂(𝑝, 𝑓, 𝑢)

Market supply of

offenses by all

individuals j,

simplified by using

average values 𝑝, 𝑓, 𝑢

𝑂𝑝 =𝜕𝑂

𝜕𝑝< 0

𝑂𝑓 =𝜕𝑂

𝜕𝑓< 0

𝑂𝑢 =𝜕𝑂

𝜕𝑢 ? 0

Same FOCs and SOCs as individual

supply functions

Becker uses theories of choice under uncertainty to explain the conditions for a person to engage

in an illegal activity: an offender’s expected utility from committing an offense exceeds the

utility from using his/her resources at other activities. Therefore, Becker argues, some people

become “criminals” not because they have different determinants for criminal behavior, but

because of different utilities and choice sets.

Functional Relationship First- and Second-Order

Conditions Description

𝐸𝑈 = 𝑝𝑈(𝑌 − 𝑓) + (1 − 𝑝)𝑈(𝑌)

𝐸𝑈𝑝 = 𝑈(𝑌 − 𝑓) − 𝑈(𝑌) < 0 Expected utility of an

offense decreases with

either an increase in 𝑝

or 𝑓 𝐸𝑈𝑓 = −𝑝𝑈′(𝑌 − 𝑓) < 0

Implication: An increase in the probability of being convicted “compensated” by an equal

percentage reduction in the “price” for convicted offense would not change expected income, but

would change the expected utility as it would affect the risk associated with the crime.

𝜀𝑝 = −𝐸𝑈𝑝

𝑝

𝑈= [𝑈(𝑌) − 𝑈(𝑌 − 𝑓)]

𝑝

𝑈

𝜀𝑓 = −𝐸𝑈𝑓

𝑓

𝑈= 𝑝𝑈′(𝑌 − 𝑓)

𝑓

𝑈

𝑈(𝑌) − 𝑈(𝑌 − 𝑓)

𝑓

𝑈′(𝑌 − 𝑓)

The widespread generalization that offenders are more deterred by the probability of conviction

than by the punishment itself turns out to imply that in the expected-utility approach that

offenders are risk referrers, at least in the relevant region of punishments.

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Attitude towards risk Increase in 𝒇 Increase in 𝒑

Risk – averse (𝑈′′ < 0) Greater effect Lower effect

Risk – neutral (𝑈′′ = 0) Same effect Same effect

Risk – seekers (𝑈′′ > 0) Lower effect Greater effect

Implication: the level of crime depends not on the efficiency of police and the level of

expenditure on combatting crime, but on the offender’s attitude towards risk. Thus, if offenders

were risk takers at some level of probability and price, and risk avoiders at other, then policy

makers could minimize social cost by selecting 𝑝 and 𝑓 in the regions where risk is preferred, i.e.

where “crime does not pay”.

d. Punishment In the model, the cost of punishment to offenders (𝑓) is the monetary equivalent of the type and

level of punishment. Empirically, this can be done easily when fines are the method of

punishment, but it is much more complicated to measure the cost of imprisonment or other types

of punishment.

Except for fines, most punishments hurt not only the offenders but other members of society as

well. For example, money from taxpayers is spent on probation, parole, and imprisonment.

Therefore, Becker uses the total social cost of punishment:

Total social cost of punishment = Cost to offenders + Cost (−gains) to society

Variable Notation Description

𝑓𝑗 Cost of punishment to the offender j

𝑓′ Social cost of punishment

Non-Measureable:

- Value placed on freedom - Value placed on restricted consumption

Costs of Imprisonment

Measureable:

- Discounted sum of earnings forgone

- Expenditures on prison personnel, facilities, food, etc.

- Subjective - Increases with duration of imprisonment

- Varies with income potential - Increases with duration of imprisonment

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𝑏 Multiplier that transforms 𝑓 to 𝑓′

For mathematical convenience, total social costs are expressed as the transformation of costs to

offenders by constant 𝑏:

𝒇′ ≡ 𝒃𝒇

3. OPTIMALITY CONDITIONS

Variable Notation Description

𝐿 Social loss from criminal offenses

𝐷 Damages

𝐶 Costs of apprehension & conviction

𝑏𝑓 Social cost of punishment

𝑂 # offenses

Functional

Relationship Description First- and Second-Order Conditions Description

𝐿 = 𝐿(𝐷, 𝐶, 𝑏𝑓, 𝑂) Social loss is function

of 𝐷, 𝐶, 𝑏𝑓, 𝑂

𝜕𝐿

𝜕𝐷> 0

𝜕𝐿

𝜕𝐶> 0

𝜕𝐿

𝜕𝑏𝑓> 0

Reasonable

assumptions that

social loss

increases with

increased damages,

A&C costs, or

social costs from

punishment

𝐿 = 𝐷(𝑂) + 𝐶(𝑝, 𝑂) + 𝑏𝑝𝑓𝑂

Less general, more

convenient function

for social loss equal

to loss in real income

from 𝐷(𝑂), 𝐶(𝑝, 𝑂),

and 𝑏𝑝𝑓𝑂 associated

with crime level (𝑂)

𝜕𝐿

𝜕𝑓= 𝐷′𝑂𝑓 + 𝐶′𝑂𝑓 + 𝑏𝑝𝑓𝑂𝑓 + 𝑏𝑝𝑂 = 0

𝜕𝐿

𝜕𝑝= 𝐷′𝑂𝑝 + 𝐶′𝑂𝑝 + 𝐶𝑝 + 𝑏𝑝𝑓𝑂𝑝 + 𝑏𝑓𝑂 = 0

First-order

optimality

conditions

𝜕2𝐿

𝜕𝑓2 = (𝐷+C)𝑂𝑓2 + 𝑏𝑝(1 − 𝐸𝑓)𝑂𝑓 > 0,

where 𝐸𝑓 =−𝜕𝑓

𝜕𝑂∙

𝑂

𝑓

Sufficient second-

order optimality

conditions to

minimize social

loss

For each of the first-order partials of the loss function, you can rearrange the terms, divide by 𝑂𝑓

or 𝑂𝑝, and substitute 𝜀𝑓 = −𝑓

𝑂𝑂𝑓 and 𝜀𝑝 = −

𝑝

𝑂𝑂𝑝 to get the following two equations:

𝐷′ + 𝐶′ = −𝑏𝑝𝑓 (1 −1

𝜀𝑓) (21)

𝐷′ + 𝐶′ + 𝐶𝑝1

𝑂𝑝= −𝑏𝑝𝑓(1 −

1

𝜀𝑝) (22)

Equations 21 and 22 represent the optimality condition that 𝑀𝐶 = 𝑀𝑅. The left-hand side is

interpreted as the marginal cost of increasing the number of offenses through a reduction in

11

either𝑓 (equation 21) or 𝑝 (equation 22). Similarly, the right-hand side gives the marginal

revenue with respect to 𝑓 or 𝑝. These optimality conditions are displayed in Figure 1 below.

With respect to either 𝑓 or 𝑝, the social loss is minimized when marginal costs are equal to

marginal revenues (the point of intersection). Marginal cost of increasing number of offenses, O,

is less when the probability, 𝑝, rather than “price” of punishment, 𝑓, is reduced. Same holds for

marginal revenues, but on the condition that Ɛ𝑝 > Ɛ𝑓 , where Ɛ𝑝 and Ɛ𝑓 are elasticities of 𝑝

and 𝑓. This is precisely the condition indicating that offenders are risk seekers and that “crime

does not pay”. The job of the policy makers is to choose certain level of 𝑝 and 𝑓 to satisfy the

optimal level �̂�.

4. SHIFTS IN THE BEHAVIORAL RELATIONS This section shows the effect of shifts in damages, costs and supply of offenses on the optimal

value of p and f. This section aims to explain why more damaging offenses, such as murder, are

punished more severely and more impulsive offenders, such as parking in a wrong spot, less

severely?

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a. Effect of 𝐷′

Shift Effect Observation Optimal number

of offenses

D' (increase in

marginal

damages) (Fig.2)

Increases MC by

changing either p

or f.

p and f move in the same

direction - increase. The more

severe the offense, the more

resources we spend in both

prevention and punishment

decreases

C' (increase in

marginal costs)

Same as in D' Same as in D' decreases

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b. Effect of 𝐶𝑝

Shift Effect Observation Optimal number of

offenses

Cp

(increase in other

components of cost)

(Fig. 3)

No direct effect on

MCf of changing

offenses, but reduces

MCp

p and f move in

opposite directions.

Reduces p and only

partially

“compensates” with

increase in f

increase

C' + Cp Some effect on MC.

Direction depends on

the relative

importance of the

changes.

increase OR

decrease

Note: 𝐶𝑝 and 𝐶′ differ significantly between different kinds of offenses. It is easier to solve

rape or armed robbery, than a burglary or auto theft because of the availability of the personal

identification, i.e. if you cannot identify your offender (auto theft), the chances of catching

him/her are slim (Table 2). For example, n increase in salary of policemen increase both C' and

Cp, while improvement in technology, say, fingerprinting will reduce both, but not by the same

extent. Implication: although improvement in technology may or may not reduce p and optimal

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number of offenses, it does reduce f. It possibly explains why we are moving away from capital

punishment.

Table 2

*Adapted from the original paper. Only prison term in federal civil institutions and probability of

conviction of those found guilty in known offenses are included.

Murder Rape RobberyAggravated Assualt

Bugary LarcenyAuthoTheft

Probability of apprehension andconviction (percent)

57.9 37.7 25.1 27.3 13 10.7 13.7

Average prison time (months) 111 63.6 56.1 27.1 26.2 16.2 20.6

0

20

40

60

80

100

120

0

10

20

30

40

50

60

70

Probability of Conviction and Prison Time* (1960)

15

c. Effect of 𝜀𝑓 and 𝜀𝑓

Shift Effect Observation Optimal number of

offenses

Ɛf'

(increase in elasticity

of 𝑓)

(Fig. 4a.)

Given 𝑏 > 0,

Increases 𝑀𝑅𝑓 of

changing offenses

Decrease in optimal

𝑓 partially

“compensated” by

increase in 𝑝

increase

Ɛp'

(increase in elasticity

of 𝑝)

(Fig.4b.)

Given 𝑏 > 0,

increases MR of

changing offenses by

changing p

Decrease in optimal 𝑝

partially

“compensated” by

increase in 𝑓.

Increase

if 𝑏 = 0, i.e. in cases of zero social cost (fines), then both 𝑀𝑅 functions lie along horizontal axis

and changes in elasticities have NO effect

Note: The income of the firm would be larger if it could separate its total market into submarkets

by the level of elasticity of demand: higher prices would be charged in markets with lower

elasticities. In case of the “offense” market, total social loss could be reduced by “charging”

lower “prices” (p’s and f’s) in markets with lower elasticities.

Example: It is possible to separate persons who have committed the same type of crime into

different into groups that have different responses to punishment. For instance, groups of

unpremeditated murders versus serial killers (sane ones). Unpremeditated murders are supposed

to act impulsively and to be relatively irresponsive to the size of the punishment (i.e. low

elasticity of f). Or, young adults or insane people have lower elasticity of both 𝑝 and f.

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5. FINES

Welfare Theorems and Transferable Pricing A well-known condition in welfare economics is 𝑀𝐶 = 𝑃, so welfare depends only on levels and

not on the slopes of marginal cost and revenue functions. For Becker’s theory of social loss from

crime, the slopes incorporated in the elasticity of supply do matter. Why the difference? The

reason is that the “price” paid by the offenders is not fully transferred to the society, so there is a

social loss incurred. With no social loss as in case of fine (𝑏 = 0), the elasticity of supply would

drop out of the optimality condition and 𝑀𝐶 and 𝑀𝑅 would lie on the x-axis.

When there is a social cost incurred (𝑏 > 0), as with imprisonment, it means some of the

“payment” by offender is not transferred to the rest of the society. In such a case the elasticity of

the supply (ex. Eq. 21) of the offenses becomes important in the optimality condition, because it

determines the change in the social cost caused by the change in punishment, 𝑓.

Optimality Conditions In the next section, Becker looks several cases where 𝑏 = 0 and varies the parameters 𝑓 and 𝑝.

He derives the optimality conditions for each case in order to show the optimal level and

mechanism for the penalty.

Variable Notation Description

�̂� Optimal number of offenses

𝑽 Marginal monetary value of penalties

𝑮 Private gain coming from criminal offenses

𝑯 Harm to society

𝑪 Cost of apprehension and conviction

Case Optimality Condition Description Implication

𝑏 = 0

𝐶′ = 0

𝐷′(�̂�) = 0 Marginal damage at the

optimal is 0 Because 𝐶 = 0, then can set

probability of conviction at

highest possible value 𝑝 = 1

without cost. Thus, monetary

value of punishment is simply

equal to the level of punishment

𝑓 = 𝐻′(�̂�).

𝑯′ completely compensated by

𝒇.

𝑉 = 𝐺′(�̂�)

Marginal $ value of

punishments should equal

marginal gain from

committing the offense

𝑉 = 𝐻′(�̂�)

Marginal $ value of

punishments should equal

marginal harm to society

𝑏 = 0

𝐶′ > 0

𝑝 = 1

𝐷′(�̂�) + 𝐶′(�̂�, 1) = 0

Optimality condition also

incorporates the marginal

cost of apprehension &

conviction

The optimal fine should equal

marginal harm plus marginal cost:

𝑓 = 𝐻′(�̂�) + 𝐶′(�̂�, 1).

Fines should also compensate

for costs to apprehend &

convict offenders.

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𝑏 = 0

𝐶′ > 0 𝑓 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐷′(�̂�) + 𝐶′(�̂�, 𝑝)

+ 𝐶𝑝(�̂�, �̂�)1

𝑂𝑝= 0

𝑝 allowed to vary so

optimal condition

incorporates partials wrt

probability of conviction

Sign of 𝐶𝑝 depends on whether

penalty is changed through fine 𝑓

or probability of conviction 𝑝

The Case for Fines

Proposition: Social Welfare is increased if fines are used (whenever feasible)

Pros Cons

Fines do not use up social resources (no

expenditure on guards, supervisory personnel,

etc) (Table 1).

Need to know the 𝐺′, 𝐻′ and 𝑀𝐶 – not easy to

come by

Determination of 𝑂and 𝑓 is simplified Immoral*

Use of other 𝑓 in addition requires the

knowledge of elasticities.

No amount of money could compensate for

murder.

Other 𝑓 fail to compensate. Moreover, they

require “victims” to spend additional

resources to carry out punishment.

Whenever 𝐻 exceeds the resources of the

offender, they have to “pay” in other way:

through prison time (“debtor prisons”) **.

No anger and fear towards ex-convicts who

have not “paid their debt to society”.

Fines will discriminate against those who

cannot pay them. Thus, rich people will be

able to “buy” their freedom.

* Counterargument: fine is price in monetary unit, imprisonment is measured in time units.

** That’s why more serious crimes are punished by prison time, parole, or sometimes, death

sentence.

“Fairness” of the imprisonment in lieu of the full fine for those who cannot pay it depends on the

value placed on the prison time, which varies with offender’s financial standing.

“Value” of prison time

Poor Rich

Low: spend more time in planning their

offenses, like court appearances, to reduce

𝑝. The cost of conviction (fine) is relatively

larger to the value of their time

High: spend more money on planning

offenses, like hire good lawyers, even bribing

someone to reduce 𝑝. The cost of conviction

is smaller than the value of the prison time.

For example, in New York State, Class A Misdemeanors can be punished by a prison term as

long as one year or a fine no larger than $1000 (by effectively setting the exchange rate between

dollar and value of the prison time as $1000 365 = $2.74 ⁄ for a day). Often the poor are forced

to choose the prison.

Compensation and the Criminal Law If punishment by optimal fines became the norm:

The aim of legal proceedings would be not punishment and deterrence but assessment of

“harm”.

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“Criminal” would be defined not by the nature of the activity, but by the offender’s

inability to compensate for the “harm”

Example: Monopolies are outlawed in the US because they pose constraints on trade that are

harmful to society. When accused in practices defendants often, become subject of damage suits

or are jailed. With optimal the gains to them would be less than “harm” they cause, thus, they

would cease their “harmful” activities.

However, it is not easy to correctly measure all the “harm” done and mistakes are inevitable. But

Becker argues that with experience the “margin of error would decline and rules of thumb would

develop”.

6. SOME APPLICATIONS (BECKER) Discussions of external economies are usually perfectly symmetrical to those of diseconomies,

however, there is no equivalent to police authorities to “apprehend” and “convict” benefactors.

There exist such awards as medals, titles, or Nobel Prize, to reward benefactors, but it’s a

relatively small number.

Similarities: the Model 𝐴 (𝐵) Net social advantage from

𝐵, benefits

𝐷(𝑂)

𝐾(𝐵, 𝑝1) Cost of apprehending and

rewarding, where 𝑝1 is the

probability of doing so

𝐶 = 𝐶(𝑝, 𝑂, 𝑎)

𝐵(𝑝1, 𝑎, 𝑣) Supply of benefits, where

𝑎 is reward per benefit and

𝑣 is other determinant

𝑂 = 𝑂(𝑝, 𝑓, 𝑢)

𝑏1 Faction of 𝑎 that is the net

loss to society

𝑏

Π = 𝐴(𝐵) − 𝐾(𝐵, 𝑝1) − 𝑏1𝑝1𝑎𝐵 Profit function showing

increase in income to

society

𝐿 = 𝐷(𝑂) + 𝐶(𝑝, 𝑂) + 𝑏𝑝𝑓𝑂

…………

Differences: Implications 𝜀𝑝 > 𝜀𝑓 – at the margin, benefactors are risk

avoiders

𝜀𝑝 > 𝜀𝑓 – at the margin, offenders are risk

seekers

Optimal 𝑝1 and 𝑎 are in the regions where

“benefits do pay”: the marginal income

available to benefactor is greater than the one

available to them in a less risky activity

Optimal 𝑝 and 𝑓 are in the regions where

“crime does not pay”: the marginal income

of criminals is less than that available to

them in a less risky legal activity

The Effectiveness of the Public Policy What determines the “effectiveness” of public efforts to discourage offenses? The model shown

earlier can be used to answer this question if social welfare is measured by income and if

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E = 𝑀

𝐵

Where E means “effectiveness”, 𝑀 – maximum feasible increase in income and 𝐵 – increase in

income if all offenses causing net damage were abolished.

𝑀 is achieved by choosing optimal values of 𝑝 and 𝑓. E essentially depends on 𝐶, and elasticities

of 𝑝 and 𝑓. The smaller the cost and the greater the elasticities, the smaller the cost of reducing

offenses and thus the greater the E.

E differs among different offenses more because of differences in 𝐶, than in elasticities of

response. The cost 𝐴&𝐶 is higher for those crimes where there is a longer time period between

commission and detection of an offense. It means that effectiveness is greater for robbery than

for a related felony like burglary than for, say, antitrust regulation.

A Theory of Collusion “Collusion” – efforts by competing firms to collude in order to obtain monopoly profits.

𝐺(𝑂) - positively related to the elasticity 𝑀𝐶 curve and is inversely related to elasticity of their

collective demand curve.

𝑂- is setting price below or producing more than is agreed by colluding firms.

𝐻(𝑂)- harm to collusion depends on the number of violation and the elasticities of the 𝑀𝐶 and

Demand curves.

𝐶 = 𝐶(𝑝, 𝑂, 𝑎) − cost of discovering and “apprehending” violators. Similarly, it is increasing in

𝑝, 𝑂, 𝑎, where 𝑎 is the cost of punishing violators. Since fines are rules out, methods like

predatory price-cutting or violence has to be used, which are less costly than legal recourse. Thus

“syndicates” in US have advantage.

Elasticities are crucial: the prices and output levels would be set close to competitive ones, if

demand is elastic, there are many buyers and sellers, if punishments are less transferrable, and if

the government is hostile towards collusion.

7. EXTENSIONS/CRITICISM

“Prison”- Community This model primarily deals with the society as a whole: how to minimize the social loss to

society by reaching an optimum level of crimes. An interesting extension to the model would be

to consider communities that have federal prisons vis-à-vis those that don’t:

For “prison” - communities’ is 𝐺(𝑂) higher or lower?

Are such communities inclined to favor prison time over fines as a method of

punishment?

Are offenders in “prison”-communities more risk seeking?

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“Expensive” Prison Time Supplementing fines with prison time is common practice, and in considering the “value” of the

prison time it seems that it is taken from the perspective of the poor person. An interesting study

would be to determine an “optimum exchange rate” - $/prison time, that would minimize social

cost to society. Consider the following:

“Harm” inflicted on offender who serves the jail time instead of paying fine: stigma,

being away from work, family… The length of prison time matters here

How significant is the length of time on recidivism

“Cost” of Judicial Error 𝑝 – the probability of conviction, as calculated as a ratio of the numbers of conviction over the

total number of offenses, is an important variable in Becker’s constrain minimization problem.

Nowhere in the model, however, is the probability of judicial error, 𝑝𝑒 , incorporated. In solving

the constrained minimization problem it would also be interesting to consider the following:

The social cost of futile legal proceedings

The Social loss of convicting innocent

𝜀𝑝 of convictions as , 𝑝𝑒 increases, i.e. are judges more inclined to leniency if they

know 𝑝𝑒 is high

Why does murder have a higher probability of conviction? Evidence suggests (Table 2) that there is a positive correlation between the severity of the crime

and the probability of conviction. Becker explains the correlation by the fact that “personal”

crimes are easier to solve because the traces of personal identification are present. While we

agree with the author, we also believe that there must be stronger reasons for that. Such heinous

crimes as murder and rape generate much more indignation, i.e. the emotional damage to society

is much higher. Thus, the society is much more driven and “unconsciously” allocates more

resources (monetary and physic) into solving them.