Crime  and  Punishment:  An  Economic  Approach  Gary  Becker  

 Thursday,  January  26,  2012  Economics  206-­‐  Team  Lewis  

Tevy  Chawwa,  Igor  Hernandez,  Nan  Li,  and  Laura  Paul    

I. Introduction  o How  many  resources  should  be  used  to  enforce  the  laws?  Is  there  an  optimal  

level  of  crime?  o The  optimal  level  of  enforcement  depends  of  costs  of  apprehension,  

conviction,  punishment  and  the  elasticity  of  the  supply  of  crime  with  respect  to  changes  in  enforcement    

• Economics  allows  us  to  take  these  parameters  and  to  minimize  social  loss.  

• Some  people  commit  crimes  when  others  do  not  because  their  costs  and  benefits  differ.    

II. Basic  Analysis    o The  Cost  of  Crime  

• The  US  Bureau  of  Justice  Statistics  reports     In  2010,  a  total  of  7.1  million  persons  under  “supervision  of  

adult  correctional  authorities”  (in  jail  or  on  probation).     Between  2008  and  2009,  there  were  a  total  of  86,975  federal  

sentences  imposed,  only  2,747  of  which  were  fines.   During  2007,  the  cost  of  the  nation's  police  protection,  

corrections,  and  judicial  and  legal  services  was  $228  billion,  an  increase  of  171%  since  1982,  after  adjusting  for  inflation.  

In  2007,  a  total  of  2.5  million  persons  were  employed  in  the  nation's  justice  system,  an  increase  of  93%  from  1982,  when  1.3  million  persons  were  employed.  

o The  Model  • Damages  

Harm,  H,  is  a  function  of  activity  level,  O.    - 𝐻! > 0,𝐻!! > 0  

The  gain  to  offenders,  G,  is  also  a  function  of  O.    - 𝐺! > 0,𝐺!! > 0  

Net  damage  to  society,  D,  is  a  function  of  the  two:  - 𝐷 𝑂 = 𝐻 𝑂 − 𝐺 𝑂  (3)  - 𝐷! > 0      ∀  𝑂 >  𝑂!    if    𝐷! 𝑂! > 0  

• The  Cost  of  Apprehension  and  Conviction   Police  activity,  A,  is  a  function  of  manpower,  materials  and  

capital.   Therefore,  the  cost  of  law  enforcement,  𝐶 = 𝐶(𝐴)  (6),  is  a  

function  of  this  activity.  

  2  

An  empirical  measure  of  law  enforcement  activity  could  be  the  number  of  offenses  cleared  by  conviction:  𝐴 ≅ 𝑝𝑂  (7)  

• Where  p  is  the  ratio  of  offenses  convicted  to  all  offenses  committed  (probability  of  conviction).    Substituting  (7)  into  (6)  and  differentiating  we  have   𝐶! =

!"(!")!"

= 𝐶!𝑂 > 0    and      𝐶! =!"(!")!"

= 𝐶!𝑝 > 0  (6,  8)  • The  Supply  of  Offenses    

The  number  of  crimes  that  an  individual,  j,  will  commit  is  a  function  of  the  probability  that  they  will  be  caught,  the  punishment  and  other  influences  on  criminal  activity:  - 𝑂! = 𝑂! 𝑝! , 𝑓! ,𝑢! ,

!!!!!!

< 0, !!!!!!

< 0      (12)  

- The  other  influences,  uj,  could  be  educational  attainment,  income  level,  or  cultural  influences.  

An  increase  in  𝑝!  compensated  by  an  equal  percentage  reduction  in  𝑓!  would  not  change  the  expected  income  from  an  offense,  but  could  change  the  expected  utility  through  risk.  - 𝐸𝑌! = 𝑝! 𝑌! − 𝑓! + 1− 𝑝! 𝑌! = 𝑌! − 𝑝!𝑓!  (fn  17)  

From  the  Expected  Utility  Maximization  Problem  - !!"!

!!!= 𝑈! 𝑌! − 𝑓! − 𝑈! 𝑌! < 0  and  

!!"!!!!

= −𝑝!𝑈!′ 𝑌! − 𝑓! < 0.  

- And  if  𝜀! > 𝜀! ,  it  can  be  shown  that  

- !! !!!!! !!! !!!!

> 𝑈!! 𝑌! − 𝑓!  

- Graphically,  if  the  criminal  is  risk  lover,      

  The  stronger  effect  of  p  could  reflect  other  forces.    

- Limited  government  budgets  result  in  “cheaper”  punishments.  

- Monetary  punishments  might  impact  rich  and  poor  differently.    

- Risk  aversion  level  and  time  preference  will  change  the  impact  of  punishments  in  the  future.    

• In  the  US,  it  can  take  years  for  criminal  cases  to  

Yj#$ fj Yj#

Utility

  3  

be  processed.  This  also  allows  for  discounting.  • Conviction  lag  in  other  countries  could  instead  

increase  in  f  substantially.   Extension:  entrepreneurs,  like  criminals,  are  risk  lovers  and  

generate  externalities.  - Call  p  probability  of  recognition  and  f  reward.  - Would  p  have  a  larger  effect  on  entrepreneurs?  - How  would  f  incentivize  entrepreneurs?  - Could  we  turn  criminals  into  (legal)  entrepreneurs  with  

positive  externalities?  • Punishments  

The  cost  of  punishment  varies  between  individual  and  type  of  punishment.  

Punishments  are  translated  in  to  a  monetary  cost  by  the  coefficient  b.    - For  example,  fines  create  little  to  no  cost  to  society,  so  

𝑏 = 0.      - Whereas  imprisonment  represents  a  high  cost  to  

society,  so  𝑏 > 0.  This  is  because  society  must  pay  the  costs  of  the  prison  operation.  

- Is  b  and  effective  way  to  monetize  punishments?  By  how  much  does  it  vary?  Is  it  correlated  with  the  number  of  offenses?  

Would  remorse  on  the  part  of  the  criminal  remove  the  need  for  punishment?    

 III. Optimality  Conditions  

o Societies  could  reduce  the  number  of  crimes  to  near  zero,  however,  the  cost  to  society  of  enforcement  would  be  more  than  that  of  allowing  some  crimes.    

o Social  loss  is  made  up  of  cost  to  offenders  and  cost  to  society  and  is  a  function  of  the  damages  to  society,  cost  of  law  enforcement,  costs  of  punishment,  and  the  number  of  offenses.  

• 𝐿 = 𝐿 𝐷,𝐶, 𝑏𝑓,𝑂  and  𝐿 = 𝐷 𝑂 + 𝐶 𝑝,𝑂 + 𝑏𝑝𝑓𝑂  (18).   Where  !"

!"> 0, !"

!"> 0, !"

!"#> 0,  and  bpfO  is  the  total  social  loss  

from  punishments.  • What  about  other  influences  on  crime  and  interaction  between  the  

variables?  o Our  objective  is  to  minimize  social  loss,  L.  Marginal  costs  should  be  equal  to  

the  marginal  revenue  in  the  optimal  condition.  • The  marginal  cost  of  changing  the  number  of  offenses  through  a  

change  of  punishment,  f:  𝐷! + 𝐶! = −𝑏𝑝𝑓(1− !!!)    (21).  

• The  marginal  cost  of  changing  the  number  of  offenses  through  a  change  of  the  probability  of  being  caught,  p:  

  4  

 𝐷! + 𝐶! + 𝐶!!!!= −𝑏𝑝𝑓(1− !

!!)  (22).  

• Where  the  elasticities  of  punishment  and  probability  of  being  caught  are:  𝜀! = − !

!𝑂!  and  𝜀! = − !

!𝑂!  (23).  

The  death  penalty  (highest  form  of  punishment)  upholds  the  assumption  that  elasticity  of  punishment  is  relatively  inelastic  ( 𝜀! < 1).      

o Only  risk  preference  and  marginal  utility  of  crime  can  determine  the  marginal  benefit  of  crime.  Policy  (determining  punishment  level  and  probability  of  being  caught)  is  what  defines  the  marginal  cost.    

o Extensions  • If  offenders  were  all  risk  neutral,  what  would  be  the  socially  optimal  

p  and  f?  • What  if  we  turned  this  into  a  principal-­‐agent  model?  • In  the  case  where  f  is  too  high,  and  a  conviction  would  mean  a  big  

social  loss,  the  judges  and  juries  would  not  be  so  inclined  to  convict  people.  

o We  can  introduce  a  measure  of  price  discrimination  into  social  loss:    

• 𝑣 = !!= !!!

!  (fn  30).  

• Additional  social  loss  is  created  when  criminals  are  not  caught  and  punished  as  compared  to  when  they  are.    

 IV. Shifts  in  Behavioral  Relations    

  Variable   Example   Effects   O*  

1  Marginal  Damages,  𝐷! = !"

!"  increases  

Corresponds  to  different  kinds  of  offenses  

• !"!", !"!"  increase,  

• p*    and  f*  increase  Decreases  

2  

Marginal  cost  of  apprehension  and  conviction,  𝐶! = !"

!"  

increases  

Increase  in  benefits  to  police  or  judges  

• !"!", !"!"  increases  

• p*    and  f*  increase  Decreases  

3  

Marginal  cost  of  changing  the  probability  that  an  offense  is  convicted,  𝐶! =

!"!"  increases  

Quantity  of  patrolling  police  officers  increases    

• No  direct  effect  to  𝑀𝑅!  

• Reduces  𝑀𝑅!  • Reduces  p*  and  

only  partially  compensates  with  increase  in  f  

Increases  

4  

Marginal  costs  of  changing  the  probability  and  conviction  C’  and  𝐶!  increase  

Increase  the  wages  of  police  officers  

• f*  increase    • Can  increase  or  

decrease  p*  

Could  increase  or  decrease  

  5  

5  

Marginal  costs  of  changing  the  probability  and  conviction  C’  and  𝐶!  decrease  

Improved  police  technology  

• f*    decrease  • Can  increase  or  

decrease  p*  

Could  increase  or  decrease  

6   If  𝑏 > 0,  !"!"  decreases  

If  the  elasticity  of  f  is  low,  people  commit  crimes  no  matter  the  punishment  (like  teenagers)  

• f*    decrease  • p*  increase  • 𝑀𝑅!  Increases  

Increases  

7   If  𝑏 > 0,  !"!"  decreases  

If  the  elasticity  of  p  is  low,  people  commit  crimes  no  matter  the  probability  

• f*  increase    • p*  decrease  • 𝑀𝑅!Increases  

Increases  

8   If  𝑏 = 0,  !"!"  has  no  

effect     • No  effect   No  effect  

9   If  b  increases  The  cost  of  the  punishment  to  society  increases  

• f*    decrease  • p*  increase  • 𝑀𝑅!and  𝑀𝑅!  

increase  

Increases  

 V. Fines  

o Optimality  Conditions  • Assumption:  b=0,  p=1,  C’=0,  D’  can  be  0  or  negative  • Optimal  level  of  fines  is  equal  to  full  compensation  for  the  damage.  

𝐿 = 𝐷! 𝑂 = 0  (24)   𝑉 = 𝐺! 𝑂 = 𝐻! 𝑂  (25,  26)   𝑓 = 𝑉 = 𝐻! 𝑂    (27)  

• This  means  fine  should  be  set  at  the  level  that  just  compensate  the  harm  of  the  offense  to  the  society     If  𝐶! > 0,  then  𝐷! 𝑂 + 𝐶! 𝑂, 1 = 0  (28)      𝑓 = 𝐻! 𝑂 +  𝐶! 𝑂, 1  (29)  

• In  this  case,  fine  should  be  set  to  compensate  both  the  harm  of  offense  and  the  cost  of  catching  and  convicting  offender.     Every  offense  can  fully  be  compensated  by  a  fine,  𝑏 = 0.     You  have  to  assume  that  𝐷’   = 𝐻’− 𝐺’ < 0.  In  other  words,  the  

private  gain  can  be  higher  than  the  compensation  to  society.     Under  those  two  conditions,  you  can  create  a  fine  that  

compensates  for  the  damage.    o The  Case  for  Fines  

• Arguments  for  fines   Fines  do  not  use  up  social  resources.  

  6  

Easy  to  use  in  setting  optimal  level  of  punishment  and  does  not  require  an  understanding  of  the  elasticity  of  supply.    

Comply  with  the  status  quo  principle—they  redistribute  wealth  to  put  the  harmed  back  to  how  things  were  before  the  crime.  Fines  do  not  easily  cause  anger  or  fear.  

• Arguments  against  fines   Moral  issue:  offenses  can  be  bought  at  a  price.  However,  

imprisonment  can  also  be  measured  in  monetary  terms.   When  𝑏 > 1,  the  richer  the  offender,  the  larger  his  elasticity  

with  respect  to  fines.   When  𝑏 = 0,  the  poorer  the  offender,  the  larger  his  elasticity  

with  respect  to  fines.  • Certain  crimes  are  so  bad  that  no  amount  of  money  can  compensate.    

Those  convicted  are  thus  debtors  to  the  society.   The  optimal  prison  terms  are  actually  in  favor  of  offenders  

because  the  punishment  will  never  be  equal  to  the  cost  inflicted.  

• Compare  social  cost  to  the  cost  for  the  offender.  Do  fines  compensate  social  costs  of  crime?   Imprisonment  increases  social  cost.    

VI. Some  Applications    o Use  a  profit  function  to  show  the  increase  in  income  from  optimal  benefits:  

• Π = 𝐴 𝐵 − 𝐾 𝐵,𝑝! − 𝑏!𝑝!𝑎𝐵   (33)   At  the  margin,  benefactors  are  risk  avoiders,  if  the  following  

holds:    - !"#

!!!

!!!> !"#

!"!!  (fn  61)  

- where  𝐸𝑈 = 𝑝!𝑈 𝑌 + 𝑎 + 1− 𝑝! 𝑈 𝑌  - by  differentiating,  ! !!! !!(!)

!> 𝑈′(𝑌 + 𝑎).  

o The  Effectiveness  of  Public  Policy  • 𝐸 = ! !! ![! ! !! !,! !!!!!

! !! !! !!  (fn  65)  

where  𝑝, 𝑓, and  𝑂  denote  optimized  values.  𝑂!  is  the  number  of  offenses  that  would  occur  if  𝑝 = 𝑓 = 0,  and  𝑂!  is  the  value  of    O  that  minimizes  D.  

o A  Theory  of  Collusion  • Collusion  is  a  good  fit  for  industries  where  supply  elasticity  is  high  

and  demand  elasticity  is  low.  • Regarding  punishment,  the  most  desirable  form  is  fine  (no  collective  

loss).      

VII. Criticism  o On  the  Criminals’  risk  attitudes  

  7  

• Becker  argues  that  because  𝜀! > 𝜀! ,  then  the  criminals  should  be  risk  lovers  (fn  19)    

• However,  Brown  and  Reynolds  (1973)  observe  that  this  result  only  holds  because  of  the  definition  of  loss  used,  which  is  the  difference  between  what  happened  (being  caught)  and  what  might  have  happened   A  more  common  approach  is  if  the  criminal  is  successful  with  

the  offense,  he  wins  G,  and  if  he  is  caught,  he  loses  L,  and  therefore  the  expected  utility  can  be  written  as  - 𝐸𝑈 = 𝑝𝑈(𝑊 − 𝐿)+ (1− 𝑝)𝑈(𝑊 + 𝐺),  where  𝑊  is  the  

initial  wealth  and  𝑈(∙)  is  a  von  Neumann-­‐Morgernstern  utility  function.  

- Taking  derivatives  with  respect  to  𝑝, 𝐿  and  𝐺,  they  show    • !"#

!"= 𝑈 𝑊 − 𝐿 − 𝑈(𝑊 + 𝐺),  

• !"#!"

= −𝑝𝑈′ 𝑊 − 𝐿 < 0    

• !"#!"

= (1− 𝑝)𝑈′ 𝑊 + 𝐺 < 0    

- Then  𝜀! = − !"#!"

!!"= − 𝑈 𝑊 − 𝐿 − 𝑈 𝑊 + 𝐺 !

!"  and  

𝜀! = − !"#!"

!!"= 𝑝𝑈! 𝑊 − 𝐿 !

!"  .  

- Whether    𝜀! ⋛ 𝜀!  depends  on    ! !!! !! !!!

!⋛ 𝑈′ 𝑊 − 𝐿  or    

! !!! !! ! ![! ! !! !!! ]!

⋛ 𝑈′ 𝑊 − 𝐿 .    

• If  𝑈’’ > 0  (risk  lovers),  we  know  that  [! ! !! !!! ]!

> 𝑈′ 𝑊 − 𝐿  and  therefore  𝜀! > 𝜀! .     We  want  to  show  that  if  𝑈’’ < 0,  𝜀! > 𝜀!  can  also  hold.    

 • In  the  graph,  𝑈′ 𝑊 − 𝐿  can  be  measured  as   !!!

!  and  [! ! !! !!! ]

!  

can  be  measured  as !!!!,  and   ! !!! !! !

!  can  be  measured  as   !!!

!  

(assuming  𝐺 = 𝐿)  .    

  8  

• Whether    𝜀! ⋛ 𝜀!  depends  on  !!!!

+ !!!!

⋛ !!!!  or    

𝐷 − 𝐸 + 𝐵 − 𝐶 ⋛ (𝐴 − 𝐶).    • We  know  that  𝐴 − 𝐶 >  𝐵 − 𝐶,  but  if  (𝐷 − 𝐸)  is  large  enough,  

that  is,  if  [𝑈(𝑊 + 𝐺)  –  𝑈(𝑊)]  is  large  enough,  then  𝜀! > 𝜀!  even  for  risk  avoiders.  

• Using  this  new  definition  of  loss  means  that  if  authorities  make  off-­‐setting  changes  in  p  and  L  rather  than  p  and  f,  the  expected  income  would  change   For  criminals  that  are  risk  avoiders  (something  theoretically  

possible),  the  scenario  in  which  p  decreases  and  L  increases  could,  in  some  cases,  increase  the  number  of  offenses.    

• Graphically,  if  p  reduces  and  L  moves  to  L’,  we  could  end  up  moving  from  A*  to  B**,  which,  for  a  criminal  that  is  risk  averse  (as  shown  previously,  this  case  cannot  be  ruled  out),  and  the  number  of  offenses  could  increase.  

   

VIII. Conclusion  o This  paper  shows  that  crime  can  be  seen  as  a  problem  of  allocation  of  

resources,  where  criminals  are  trying  to  maximize  their  utility,  and  authorities  are  trying  to  minimize  the  social  losses  of  crime.      

o Benefits  from  an  offense  come  from  the  preferences  of  criminals.  However,  the  number  of  offenses  is  determined  once  authorities  choose  the  optimal  values  of  conviction  and  punishment.    

o If  criminals  are  considered  to  be  risk  lovers,  then  the  effect  of  changes  in  the  probability  of  conviction  could  be  higher  than  the  effect  of  changes  in  the  level  of  punishment.  On  the  other  hand,  increasing  the  probability  of  conviction  is  associated  with  higher  levels  of  enforcement  (and  higher  costs  of  activity).    

o Fines  are  more  suitable  than  other  punishments  because  they  create  a  higher  reduction  in  social  loss,  however,  there  are  some  other  considerations  that  allow  the  use  of  other  punishments.    

  9  

o This  model  can  be  applied  to  interpret  other  types  of  problems  such  as  optimal  benefit,  effectiveness  of  public  policy  and  theory  of  collusion.  

o There  are  some  other  considerations  that  the  model  does  not  address,  such  as  the  effect  of  enforcement  error  and  the  implications  of  elasticity  in  the  risk  preferences  among  criminals.  A  dynamic  model  of  crime  could  be  also  an  important  extension  of  this  paper.    

IX. References  o Becker,  Gary,  1968,  “Crime  and  punishment:  an  economic  approach.”  Journal  

of  Political  Economy  76:  169-­‐217.  o Brown,  W.  and  Reynolds,  M.  1973,  “Notes,  Comments,  and  Letters  to  the  

Editor,  Crime  and  “Punishment”:  Risk  Implications.”  Journal  of  Economic  Theory  6,  508-­‐514.  

o Cassidy,  T.,  Koumpias,  A.,  Liang,  X.  and  Zhou,  Y.  2011,  “Notes  for  Gary  Becker’s  “Crime  and  Punishment:  An  Economic  Approach.”    

X. Variables  Variable   Description  

𝐻!   Harm  (to  society)  from  the  ith  criminal  activity  𝑂!   Criminal  activity  level  (number  of  offenses)  𝐺   Gain  to  offenders  (criminals)  𝐴   Law  enforcement  activity,  𝐴 = 𝐴 𝑚, 𝑟, 𝑐  a  function  of  manpower,  materials  

and  capital.    𝐶   Cost  of  police  and  court  activity  𝑃   Probability  that  an  offense  (crime)  is  cleared  by  conviction  

ℎ 𝑝,𝑂,𝑎   A  function  of  activity,  A,  where,  𝑖𝑓  ℎ!, ℎ! ,𝑎𝑛𝑑  ℎ! > 0  𝑡ℎ𝑒𝑛  𝐶!,𝐶! ,𝑎𝑛𝑑  𝐶! > 0  𝑂!   The  number  of  offenses  that  one  would  commit  during  a  particular  period  𝑝!   The  probability  of  conviction  per  offense  𝑓!   The  punishment  per  offense  (e.g.  fines,  torture,  death)  𝑢!   All  other  influences  on  criminal  activity  (e.g.  education  level,  additional  income  

for  law-­‐abidingness)  𝑌!   Income  (monetary  and  psychic)  from  committing  an  offense  𝑓!   Social  cost  of  crime  𝑏   Transforms  punishment  into  monetary  cost  (to  society).  For  example,  

𝑏 ≅ 0  for  fines  and  𝑏 > 1  for  torture  𝑂   The  optimal  number  of  offenses  

𝐺 𝑂   The  marginal  private  gain  of  crime  𝑉   Monetary  value  of  marginal  penalties  𝜇   The  expected  punishment,  𝜇 = 𝑝𝑓  𝜎   The  variance,  𝜎! = 𝑝(1− 𝑝)𝑓!  𝑣   The  coefficient  of  variance  (price  discrimination  of  punishment)