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(1)City University of New York (CUNY). CUNY Academic Works Dissertations, Theses, and Capstone Projects. CUNY Graduate Center. 1978. The Effects of Apprehension, Conviction and Incarceration on Crime in New York State Hope Corman The Graduate Center, City University of New York. How does access to this work benefit you? Let us know! More information about this work at: https://academicworks.cuny.edu/gc_etds/2199 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected].

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(3) 78-8670 CORMAN, Hope, 1948THE EFFECTS OF APPREHENSION, CONVICTION AND INCARCERATION ON CRIME IN NEW YORK STATE. City University of New York, Ph.D., 1978 Economics, general. University Microfilms International,. ©. C O P Y R IG H T B Y HOPE. CORMAN 1978. Ann Arbor, M ichigan 48106.

(4) T H E E F F E C T S O F A P P R E H E N S I O N , C O N V IC T IO N A N D IN C A R C E R A T IO N O N C R I M E IN N E W Y O R K S T A T E by HOPE. CORMAN. A d is s e r ta tio n s u b m itte d to the g ra d u a te F a c u lt y in E c o n o m ic s in p a r t i a l f u l f i l l ­ m e n t o f the r e q u ir e m e n ts f o r the d e g re e o f D o c to r o f P h ilo s o p h y , the C ity U n iv e r s it y o f N e w Y o r k .. 1978.

(5) T h is m a n u s c r ip t has been re a d and a c c e p te d f o r the G ra d u a te F a c u lty in E c o n o m ic s in s a tis fa c tio n o f the d is s e r ta tio n r e q u ir e m e n t f o r the d e g re e o f D o c to r o f P h ilo s o p h y .. [signature] l ) ) l. /. ■. '. h f r d a te. • d a te. C h a ir m a n o f E x a m in in g C o m m itte e. E x e c u tiv e O f f ic e r. H e rb e rt. G eyer. E lli o t t Z u p n ic k. S u p e r v is o r y C o m m itte e. T h e C ity U n iv e r s ity o f N ew Y o r k.

(6) ACKNOWLEDGEMENTS. T h e d a ta used in th is d is s e r ta tio n w a s c o lle c te d u n d e r tw o g ra n ts : g r a n t n u m o e r N I-S 9 -Q 1 1 5 a w a rd e d by the L a w E n fo r c e m e n t A s s is ta n c e A d m in is t r a t io n and one a w a r d e d by the D ru g A b u s e C o u n c il, In c . .. In. c o n ju n c tio n w ith the d a ta c o lle c tio n e f f o r t , a o le r e s e a r c h a s s is ta n c e w a s p r o v id e d by B a r b ie K a i s e r , A n d r e a P e d o ls k y , P h ilip R ic h a rd s o n and M a rg o t W illia m s .. R ic h a r d J u rg e s w a s th e le g a l c o n s u lta n t.. H e h e lp e d. to d e c ip h e r che N e w Y o r k S ta te c r im i n a l la w s and e x p la in e d the o p e r a tio n s o f the c o u r ts in N e w Y o r k S t a t e .. S p e c ia l th a n k s go to P a t r i c i a C ly n e. and John L y o n s o f th e N e w Y o r k S ta te D iv is io n o f C r i m i n a l J u s tic e S e r v ic e s .. T h e y h e lp e d a g r e a t d e a l in m a k in g the d a ta a c c e s s ib le and. m e a n in g f u l. M y g r e a t e s t d e b t is to M ic h a e l G r o s s m a n .. H e s p o n s o re d th is. d is s e r t a t io n and p ro v id e d u s e fu l g u id a n c e th ro u g h o u t the s tu d y .. A ls o ,. H e r b e r t G e y e r and E l l i o t t Z u p n ic k p ro v id e d u s e fu l c o m m e n ts . I a ls o w is h to e x p r e s s m y s in c e r e th a n k s to B e n ja m in S a c k m a r y and K . K . W o n g f o r t h e i r e n c o u ra g e m e n t and s u p p o r t.. F i n a l l y , m a n y th a n k s. go to N a n c y H u jik w ho s k i l l f u l l y ty p e d th is m a n u s c r ip t and w a s a lw a y s p a tie n t and h e lp fu l..

(7) TABLE OF CONTENTS. I N T R O D U C T I O N .......................................................................................................................... 1 I.. C R I M E A N D T H E I N D I V I D U A L ........................................................................................ 5 T h r e e S ta te s o f the W o r ld A . A C h a n g e in the P r o b a b ilit y o f A r r e s t B . A C h a n g e in the P r o b a b ilit y o f C o n v ic tio n and P u n is h m e n t C . A C h a n g e in the C o s ts o f A p p r e h e n s io n , C o n v ic tio n and P u n is h m e n t D . A n In c re a s e in the L e g a l o r Ille g a l W a g e E . S u m m a r y o f E ffe c ts o f E xo g en o u s S h ifts in th e L a w E n fo r c e m e n t V a r ia b le s and in the W age V a r ia b le s F o u r S ta te s o f the W o r ld. II.. 23 25. 27 29. 32 35 36 38 39. M E A S U R IN G L A W E N F O R C E M E N T V A R I A B L E S A N D T H E C R I M E R A T E IN N E W Y O R K S T A T E ........................................................................ 4 6 F e lo n y C a s e P r o c e s s in g M e a s u r in g C o u r t - R e la t e d P r o b a b ilit ie s and C o s ts D a ta on C o u r t O u tc o m e s C a lc u la tin g the C o u r t - R e la t e d P r o b a b ilit ie s C a lc u la tin g the C o u r t - R e la t e d C o s ts to the O ffe n d e r M e a s u r in g th e C r i m e R a te M e a s u r in g the P r o b a b ilit y o f A r r e s t. IV .. 20. T H E S U P P L Y O F O F F E N S E S .......................................................................................32 T h e B e h a v io r a l F u n c tio n R e v e r s e C a u s a lity T h e P r o b a b ilit y o f A r r e s t O th e r L a w E n fo r c e m e n t V a r i a b l e s In c a p a c ita tio n. III.. 5 13. 46 52 52 58 60 63 65. E C O N O M E T R IC S P E C I F I C A T IO N O F T H E M O D E L A N D R E G R E S S IO N R E S U L T S ................................................................................................... 70 E c o n o m e tr ic S p e c ific a t io n R e g r e s s io n R e s u lts R e v e r s e C a u s a lity In c a p a c ita tio n. 70 77 8 6 8 8.

(8) . C O N C L U S IO N S 3 IB L IO G R A P H Y.

(9) LIST. 1. 2. 3. 4. 5. 6 . 7.. OF. TABLES. C o n s u m p tio n P r o s p e c ts and P r o b a b ilit ie s : T h r e e S ta te s o f th e W o r l d .................................................................................. 6 C o n s u m p tio n P r o s p e c ts and P r o b a b ilit ie s : F o u r S ta te s o f th e W o r l d ........................................................................................3 0 C r i m e C a te g o r ie s in N e w Y o r k S t a t e ...........................................................4 7 R e p o r tin g R a te s and C r i m e W e i g h t s ..............................................................6 4 V a r ia b le s in th e R e g r e s s io n E q u a t i o n .......................................................... 71 R e g r e s s io n R e s u l t s ....................................................................................................7 9 E la s t i c it ie s o f th e In d e p e n d e n t V a r i a b l e s W ith R e s p e c t to th e C r i m e R a t e ........................................................................ 83.

(10) LIST. OF. ILLUSTRATIONS. F ig u r e 1. 2. 3. 4. 5.. M a r g in a l R e v e n u e and M a r g in a l C o s t C u r v e s ................................................ 12 E f f e c t o f a C h a n g e in (J) on /M a rg in a l C o s t ......................................................... 15 T h e E ffe c ts o f a C h a n g e in ($> on M C and M R C u r v e s ............................... 19 T h e E ffe c ts o f a C h a n g e in (|) on M C and M R C u r v e s ...............................22 C r i m i n a l J u s tic e P r o c e s s in g in N e w Y o r k S t a t e .....................................4 9.

(11) INTRODUCTION. S in c e the p a th -b r e a k in g w o r k on the e c o n o m ic s o f c r im e d e v e lo p e d by G a r y B e c k e r 1 , m a n y e c o n o m is ts h a v e fo c u s e d on the p r o b le m o f c r im e and c r im e c o n t r o l . ^. T h e s e a n a ly s e s u se a m o d e l w h ic h t r e a t s. c r im i n a l a c t iv it y d e c is io n s as la b o r s u p p ly d e c is io n s .. A n in d iv id u a l. r a t io n a lly d e c id e s w h e th e r o r not to c o m m it c r im e s d e p e n d in g on h is e x p e c te d g a in s fr o m c o m m it tin g c r i m e , h is e x p e c te d c o s ts and h is o p p o rtu n ity c o s t r e s u ltin g fr o m n o t w o r k in g in the le g a l s e c t o r .. C o s ts. a r e in c u r r e d b e c a u s e the o ffe n d e r m a y be a p p re h e n d e d and im p r is o n e d . It is e x p e c te d th a t an in c r e a s e in the p r o b a b ilit y o f p ris o n o r the le n g th o f p r is o n s e n te n c e im p o s e d w i l l d e t e r c r im e b e c a u s e the e x p e c te d g a in s f r o m c o m m it tin g c r im e s a r e r e d u c e d . T h e p r e s e n t a n a ly s is u s e s the s a m e b a s ic m o d e l o f the in d iv id u a l's c r im i n a l b e h a v io r d e c is io n s as the one d e v e lo p e d by B e c k e r and r e fin e d and a p p lie d by Is a a c E h r l i c h . ^ e xp a n d e d in tw o w a y s . m odel a re chang ed.. H o w e v e r , the B e c k e r / E h r l i c h m o d e l is. F i r s t , s o m e o f the b a s ic a s s u m p tio n s o f t h e i r. In t h e i r m o d e l, the p r o b a b ilit y o f a p p re h e n s io n. is a s s u m e d to be in d e p e n d e n t o f th e n u m b e r o f c r im e s c o m m it te d .. 4. A ls o , th e y a s s u m e th a t the s e n te n c e im p o s e d is a fu n c tio n o f the n u m b e r o f c r im e s c o m m it te d .. H e r e , th e s e a s s u m p tio n s a r e r e v e r s e d .. The.

(12) 2. p r o b a b ilit y o f a r r e s t is a s s u m e d to v a r y w ith the n u m b e r o f c r im e s c o m m itte d and the c o s t is a s s u m e d to be in d e p e n d e n t o f the n u m b e r o f c r im e s c o m m itte d in the c u r r e n t p e r io d .. T h u s , th is a n a ly s is e x a m in e s. the e x te n t to w h ic h the c o n c lu s io n s o f the m o d e l o f c r im e c o n tro l c u r r e n t ly u s ed by a n u m b e r o f e c o n o m is ts a r e chang ed w h e n the a s s u m p ­ tio n s a r e c h a n g e d . T h e second w a y in w h ic h th is a n a ly s is e x p a n d s the b a s ic m o d e l is by a llo w in g a n u m b e r o f u n d e s ir a b le o u tc o m e s to a ffe c t the d e c is io n ­ m a k in g o f the p o te n tia l o ff e n d e r . is one d e s ir a b le o u tc o m e :. In th e B e c k e r / E h r l i c h a n a ly s is , th e r e. no a r r e s t and th e r e is one u n d e s ir a b le o u tc o m e. a r r e s t , c o n v ic tio n and im p r is o n m e n t. in c lu d e :. 5. H e r e , the u n d e s ir a b le o u tc o m e s. a r r e s t b u t no c o n v ic tio n ; a r r e s t and c o n v ic tio n b u t no in c a r c e -. r a tio n ; a r r e s t , c o n v ic tio n and a j a i l. 6. s e n te n c e ; and a r r e s t , c o n v ic tio n. and a p r is o n s e n te n c e . A n a g g re g a te c r im e s u p p ly fu n c tio n is d e r iv e d f r o m the m o d e l o f in d iv id u a l c r im i n a l b e h a v io r .. T h e in d iv id u a l's s u p p ly o f c r im e is a. fu n c tio n o f h is g a in s f r o m c o m m ittin g o ffe n s e s h is o p p o rtu n ity c o s ts o f n o t e n g a g in g in le g a l w o r k , and h is p r e fe r e n c e to w a rd r i s k .. In the. a g g r e g a te , c r im e is a fu n c tio n o f the p r o b a b ilit ie s and c o s ts o f a r r e s t , c o n v ic tio n and in c a r c e r a t i o n , a s w e ll a s o th e r v a r ia b le s n o t d i r e c t l y r e la t e d to the c r i m i n a l ju s t ic e s y s te m . T h e a g g r e g a te fu n c tio n is te s te d , u s in g 1970 d a ta f o r the 6 2 c o u n tie s in N e w Y o r k S t a t e .. T h e c o u n ty le v e l is the s m a lle s t u n it in w h ic h th e r e.

(13) 3. Is s e p a r a te a d m in is tr a tio n o f la w e n fo r c e m e n t a g e n c ie s .. T h e c o u n ty is. th u s th e s m a lle s t u n it o f o b s e rv a tio n in w h ic h th e re is s e p a r a te d e c is io n ­ m a k in g a b o u t the la w e n fo r c e m e n t v a r i a b l e s .. /Most o th e r s tu d ie s h a ve. te s te d the d e te r r e n c e m o d e l a t the S ta te w id e l e v e l , o r f o r m u n ic ip a li­ tie s o r s m a l l e r d iv is io n s . C h a p te r. 1. p r e s e n ts a d e s c r ip tio n o f th e t h e o r e tic a l m o d e l o f c r im e. and the in d iv id u a l.. In C h a p te r 2 , a d e s c r ip tio n o f the in d iv id u a l and. a g g re g a te s u p p ly - o f-o f fe n s e s fu n c tio n s a r e p r e s e n te d , as w e ll a s the h y p o th e s e s to be te s te d .. In C h a p te r 3 , th e r e is a d e s c r ip tio n o f the. c r im i n a l ju s t ic e s y s te m in N e w Y o r k S ta te and o f the d a ta u s ed to te s t the h y p o th e s e s .. C h a p te r 4 p r e s e n ts the e c o n o m e tr ic s e p c ific a tio n o f. the m o d e l, and r e g r e s s io n r e s u lt s . th e a n a ly s is a r e p r e s e n te d .. In C h a p te r 5 , the c o n c lu s io n s to.

(14) 4. FO O TNO TES. G a r y S . B e c k e r , " C r i m e and P u n is h m e n t: A n E c o n o m ic A p p r o a c h ," in G a r y B e c k e r and W i l l i a m L a n d e s , ( e d s . ) , E s s a y s in the E c o n o m ic s o f C r i m e and P u n is h m e n t (N e w Y o r k : C o lu m b ia U n iv e r s it y P r e s s 1 9 7 4 ). (T h e a r t i c l e w a s o r i g n i a l l y p u b lis h e d in 1968 in th e M a r c h / A p r i l is s u e o f T h e J o u r n a l o f P o l it ic a l E c o n o m y 7 8 . ) 1. 2. S e e , f o r e x a m p le , D a v id L . S jo q u is t , " P r o p e r t y C r i m e and E c o n o m ic B e h a v io r : S o m e E m p i r i c a l R e s u lt s ," A m e r ic a n E c o n o m ic R e v ie w 83 (J u n e 1 9 7 3 ); R . A . C a r r - H i l l and N . H . S t e r n , "A n E c o n o m e tr ic M o d e l o f the S u p p ly and C o n tr o l o f R e c o rd e d O ffe n s e s in E n g la n d and W a l e s , " J o u r n a l o f P u b lic E c o n o m ic s 2 (N o v e m b e r 1 9 7 3 ); E u g e n e S w im m e r , " M e a s u r e m e n t o f the E ffe c tiv e n e s s o f U rb a n L a w E n f o r c e m e n t - A S im u lt a ­ neous A p p r o a c h ," S o u th e rn E c o n o m ic J o u r n a l 4 0 ( A p r i l 1 9 7 4 ); L la d P h i ll i p s and H a r o ld V o t e y , J r . , "A n E c o n o m ic A n a ly s is o f th e D e f e r r e n t E ffe c t o f L a w E n fo r c e m e n t on C r i m i n a l A c t i v i t i e s , " J o u r n a l o f C r i m i n a l L a w , C r i m i n o l o g y , and P o lic y S c ie n c e 6 3 ( S e p te m b e r 1 9 7 2 ); Is a a c E h r l ic h , " P a r t ic ip a t io n in I lle g it im a t e A c t iv it ie s : A n E c o n o m ic A n a l y s i s , " in G a r y B e c k e r and W i l l i a m L a n d e s ( e d s . ) , E s s a y s in the E c o n o m ic s o f C r i m e and P u n is h m e n t, (N e w Y o r k : C o lu m b ia U n iv e r s it y P r e s s 1 9 7 4 ) ( A r t i c l e w a s o r i g i n a l l y p u b lis h e d in 1 97 3 in the M a y /J u n e is s u e o f T h e J o u r n a l o f P o l it ic a l E c o n o m y 8 1 ); D o n a ld M a th ie s o n and P e t e r P a s s e l l , " H o m ic id e and R o b b e ry in N e w Y o r k C ity : A n E c o n o m ic M o d e l, " J o u r n a l o f L e g a l S tu d ie s 5 ( J a n u a r y , 1976.) 3. E h r l ic h , " P a r t ic ip a t io n in Ille g it im a t e A c t i v i t i e s . ". 4. T h is a s s u m p tio n is r e la x e d in the a p p e n d ix to E h r l ic h 's a n a ly s is . H o w e v e r , E h r lic h o n ly a n a ly z e s one s p e c ia l c a s e : W h e r e an exo g e n o u s s h ift in the a v e r a g e p r o b a b ilit y c a u s e s the m a r g in a l p r o b a b ilit y to r is e by the s a m e p e r c e n ta g e a s the chang e in the a v e r a g e p r o b a b ilit y . 5. A lth o u g h th e s e a u th o rs do a llo w , t h e o r e t i c a l l y , f o r m a n y s ta te s o f th e w o r l d , th e y do n o t s p e c if ic a lly e x a m in e m o r e th an one u n fa v o ra b le o u tc o m e f o r the c r i m i n a l . T o be s p e c if ic , j a i l r e f e r s o r c o u n ty ) f a c i l i t y w ith a m a x im u m r e f e r s to in c a r c e r a t io n o f one y e a r f a c i l i t y . In c a r c e r a tio n r e f e r s to a. to in c a r c e r a t io n in a lo c a l (m u n ic ip a l s e n te n c e o f up to one y e a r . P r is o n o r m o r e in a s ta te c o r r e c t io n a l j a i l o r a p r is o n s e n te n c e ..

(15) 5. CHAPTER. C R IM E. AND. THE. I. IN D IV ID U A L. T h e m o d e l used to d e r iv e the s u p p ly o f o ffe n s e s fu n c tio n is one o f e x p e c te d u t i l i t y m a x im iz a t io n .. In d iv id u a ls w i l l m a x im iz e the e x p e c te d. u t i l i t y g iv e n by the fo llo w in g fu n c tio n : s (1 .1 ). E U (C s ,t cs) =. ^ 3=. TTs. U (C s , t c s ) .. 1. U t i l i t y is a fu n c tio n o f the a m o u n t o f m o n e y a v a ila b le to spend on c o n s u m p tio n goods (C ) a s w e ll as the tim e a v a ila b le f o r c o n s u m p tio n (tc ).. E x p e c te d u t i l i t y is the s u m o f the u t i l i t y in e a c h p o s s ib le s ta te o f. the w o r ld ( i . e . , o u tc o m e ) w e ig h te d by the p r o b a b ilit y o f th a t o u tc o m e . TheTT. v a r ia b le r e p r e s e n ts the p r o b a b ilit y o f s ta te s .. TTs ' s m u s t e q u a l o n e .). (T h e s u m o f the. In d iv id u a ls w i l l a llo c a te t h e i r tim e to le g a l. a c t i v i t i e s , ille g a l a c t iv i t i e s and c o n s u m p tio n a c t iv it ie s in a m a n n e r w h ic h w i l l r e s u lt in m a x im u m p o s s ib le e x p e c te d u t i l i t y .. T h r e e S ta te s o f the W o r ld A s u m e th a t th e re a r e th r e e p o s s ib le o u tc o m e s r e s u ltin g f r o m c o m m is s io n o f an o ffe n s e : no a r r e s t , a r r e s t b u t no c o n v ic tio n ; and a r r e s t , c o n v ic tio n and p u n is h m e n t..

(16) A s in d ic a te d in T a b le 1 , e a c h s ta te o f the w o r ld has a c e r t a in p ro o a b U ity a tta c h e d to i t . is ( 1 - P ) .. T h e p r o b a b ilit y o f s ta te o n e , w h e re th e r e a r e no a r r e s t s ,. P r e p r e s e n ts the p r o b a b ility o f a t le a s t one a r r e s t f o r c o m ­. m ittin g N n u m b e r o f o ffe n s e s in a g iv e n tim e s p a n . c o n v ic tio n and p u n is h m e n t g iv e n a r r e s t is Q .. T h e p r o b a b ility o f. T h u s , the p r o b a b ilit y o f. s ta te tw o , w h e r e th e r e is a r r e s t bu t no c o n v ic tio n , is P ( 1 - Q ) .. The. p r o b a b ilit y o f the m o s t u n fa v o r a o le o u tc o m e , s ta te t h r e e , is P Q .. TABLE 1 C O N S U M P T IO N P R O S P E C T S A N D P R O B A B I L I T I E S : T H R E E S T A T E S OF T H E W ORLD P r o b a b ilit y. S ta te o f the W o r ld 1. .. (1 -P ). No a r r e s t. 2.. A r r e s t b u t no C o n v ic tio n. 3.. A r r e s t , C o n v ic tio n and P u n is h m e n t. C o n s u m p tio n P r o s p e c t C 1 =W 1 + W j(N }fW jt u. P (1 - Q ). PQ. C 3 =W ' + W ; ^ f W fe't j - F a - F b. T h e c o n s u m p tio n p ro s p e c t in s ta te one is e q u a l to th e to ta l in c o m e : fix e d in c o m e ( W ’ ) p lu s ille g a l in c o m e p lu s le g a l e a r n in g s . W . is the (c o n s ta n t) w ag e f r o m c o m m ittin g an o ffe n s e and N is the n u m b e r o f o ffe n s e s c o m m it te d .. It is a s s u m e d , f o r s im p l i c it y , th a t one o ffe n s e is. c o m m itte d p e r h o u r .. Thus,. t im e s N is the ille g a l in c o m e .. The. (c o n s ta n t) h o u r ly w a g e in the le g a l s e c to r is r e p r e s e n te d by W i_an d ti_ r e p r e s e n ts the a m o u n t o f tim e a llo c a te d to g e n e r a tin g le g a l e a r n in g s . L e g a l e a r n in g s is. tim e s t. ..

(17) 7. T h e c o n s u m p tio n in s ta te tw o is e q u a l to in c o m e in s ta te one m in u s the c o s ts ( d i r e c t , in d ir e c t and p s y c h ic ) a s s o c ia te d w ith a p p r e h e n s io n , F. .. D i r e c t c o s ts in c lu d e le g a l f e e s , the c o s ts o f a b a il b o n d , e tc .. cl. In d ir e c t c o s ts in c lu d e the v a lu e o f the tim e no lo n g e r a v a ila b le f o r g e n e r a tin g in c o m e ( o r f o r c o n s u m p tio n ).. A ls o , the s tig m a o f a r r e s t. m ig h t re d u c e the fu tu r e in c o m e s t r e a m o f the in d iv id u a l.. w o u ld. then in c lu d e the p r e s e n t v a lu e o f the fu tu re in c o m e s t r e a m w h ic h is lo s t . A n d , th e r e m ig h t a ls o be p s y c h ic c o s ts a s s o c ia te d w ith a r r e s t . T h e c o n s u m p tio n p r o s p e c t in s ta te th r e e s ta te one m in u s the c o s t o f a p p re h e n s io n , F c o n v ic tio n and p u n is h m e n t, F ^ . in c lu d e d in F. a. Cl. is e q u a l to the in c o m e in , and m in u s the c o s t o f. T h e o ffe n d e r in c u r s a ll o f the c o s ts. p lu s th e a d d itio n a l d i r e c t , in d ir e c t and p s y c h ic c o s ts. o f c o n v ic tio n and p u n is h m e n t.. F o r e x a m p le , w ith c o n v ic tio n , the. o ffe n d e r o b ta in s o r le n g th e n s h is c r im i n a l r e c o r d . p r e s e n t and fu tu r e in c o m e .. T h is m ig h t a ffe c t. A n d the p u n is h m e n t, w h ic h w o u ld in c lu d e. e it h e r p r o b a tio n a r y s u p e r v is io n o r in c a r c e r a t io n , w o u ld s u r e ly re d u c e p r e s e n t w e a lt h , and p o s s ib ly w o u ld re d u c e fu tu r e w e a lt h , a ls o . I t is a s s u m e d th a t: (1 .2 ). Fa> O ;. F b> O. ^ 1^. C3 .. and th e r e f o r e (1. .3 ). E q u a tio n ( 2 . 1 ) c an now be r e w r i t t e n as: O '-4). E U ( C s , t c s ) < 1 - P ) U ( C - , , t c 1 ) + P (1 - Q ) U ( C g . t c 2 ) + P Q U ( c 3 , t c 3 ) ..

(18) 8. T h e p r o b le m is to m a x im iz e e q u a tio n ( i . 4 ) s u b je c t to c o n s u m p tio n c o n s tr a in ts o f .5 ). th ro u g h C g , and the tim e c o n s t r a in t g iv e n by: t= t i+ tc + N .. S in c e th is a n a ly s is does no t in v e s tig a te the iss u e o f tim e a llo c a tio n b e tw e e n w o r k and c o n s u m p tio n , i t is a s s u m e d th a t d e c is io n s o f w h e th e r to w o r k o r u se tim e f o r le is u r e a r e m a d e in d e p e n d e n tly o f d e c is io n s c o n c e rn in g a llo c a tin g tim e b e tw e e n le g a l and i lle g a l w o r k .. T h u s , fo r. o u r p u r p o s e s , e q u a tio n (1 . 5 ) is re d u c e d to : (1 .6 ). t= t i+ N = t -tc .. and (■ •7). tc 1 = tc 2 stc 3 =tc -. A l s o , the e x p e c te d u t i l i t y fu n c tio n c an be e x p r e s s e d in t e r m s o f C-) and the F ^ 's r a t h e r than the C ^ 's . 0 -3 ). E q u a tio n (1 . 4 ) is now r e w r it t e n a s .. E U ( C s<t c )= (1 -P )U (C -i , t c )+P (1 - Q ) U ( C - i - F a , t c ) + P Q U ( C - i- F a - F b , t c). E q u a tio n (1 . 8 ) w i l l be m a x im iz e d w h e n t im e is o p t im a lly d is tr ib u te d b e t­ w e e n c r im e and le g a l w o r k . B e fo r e p r e s e n tin g the f i r s t - o r d e r c o n d itio n s f o r e x p e c te d u t i l i t y m a x im iz a t io n , s o m e a s s u m p tio n s need to be m a d e a b o u t the r e la tio n s h ip s b e tw e e n the p r o b a b ilit ie s and c o s ts o f u n d e s ir a b le o u tc o m e s and the n u m b e r o f o ffe n s e s c o m m it te d .. It is a s s u m e d , f o r s im p l i c it y , th a t the. n u m b e r o f o ffe n s e s c o m m itte d in the c u r r e n t tim e p e r io d w i l l n o t a ffe c t the c o s ts o f a p p re h e n s io n , c o n v ic tio n and p u n is h m e n t.. S in c e the. o p e r a tio n a l tim e span used in th is a n a ly s is is one y e a r , th is a s s u m p tio n.

(19) 9. m e a n s th a t the n u m b e r o f o ffe n s e s p r e v io u s ly c o m m itte d d u r in g the c u r r e n t y e a r do not a ffe c t the F . ' s .. S u c h an a s s u m p tio n , a lth o u g h it is a depai—. tu r e f r o m m o s t o th e r a n a ly s e s , is n o t u n r e a s o n a b le . T h e r a tio n a le f o r the F ^ 's to be in d e p e n d e n t o f N is th a t p a s t c r im i n a l r e c o r d is a s s u m e d to be a d e te r m in a n t in th e h a rs h n e s s o f t r e a t m e n t the o ffe n d e r r e c e iv e s d u rin g the p ro c e s s o f r e s o lv in g the a r r e s t in the c r im in a l ju s t ic e s y s te m . to r a is e the F ^ 's .. A n d h a r s h e r t r e a t m e n t is l i k e l y. B u t, o ffe n s e s c o m m itte d d u rin g the c u r r e n t y e a r. p r o b a b ly do n o t c o n s titu te. the m a jo r p a r t o f th e c r i m i n a l 's r e c o r d. b e c a u s e : ( 1 ) the c u r r e n t y e a r is o n ly p a r t o f the o ff e n d e r 's c r im i n a l " c a r e e r " and ( 2 ) m o s t c r im e s c o m m itte d d u rin g th is y e a r h a v e p ro b a b ly n o t b e en d e te c te d , and a r e e v e n le s s l i k e l y to h ave a lr e a d y r e s u lte d in a c o n v ic tio n .. T h u s , c r im e s c o m m itte d in o th e r y e a r s a r e a s s u m e d to be. th e m a jo r p a r t o f the c r i m i n a l r e c o r d , and o n ly c r im e s c o m m itte d in p a s t y e a r s a r e a s s u m e d to a ffe c t the F ^ 's . T h e p r o b a b ilit y o f a r r e s t in one y e a r , P , is th o u g h t to be a p o s itiv e fu n c tio n o f the n u m b e r o f o ffe n s e s c o m m itte d th a t y e a r . i s , a g a in , a d e p a r t u r e f r o m p r i o r a n a ly s e s .. 1. T h is a s s u m p tio n. B u t, i t s e e m s m o re. r e a s o n a b le to a s s u m e th a t the m o r e c r im e s an in d iv id u a l c o m m it s , the g r e a t e r h is c h a n c e s o f b e in g c a u g h t.. T h e s p e c ific fu n c tio n a l fo r m o f th is. p r o b a b ilit y d is tr ib u tio n is d is c u s s e d m o r e f u lly l a t e r in the a n a ly s is . T h e p r o b a b ilit y o f c o n v ic tio n and p u n is h m e n t is a s s u m e d to be in d e -.

(20) 10. p e n d e n t o f the n u m b e r o f o ffe n s e s c o m m it te d .. T h is a s s u m p tio n is. c o n s is te n t w ith p r o p e r t r i a l p r a c t ic e s w h ic h d ic ta te th a t the a c c u s e d is o n ly on t r i a l f o r th e p a r t i c u l a r o ffe n s e f o r w h ic h he w a s a r r e s t e d . P a s t r e c o r d is no t supposed to e n te r in to the d e te r m in a tio n o f g u ilt o r in n o c e n c e . T h e r e la tio n s h ip s b e tw e e n the n u m b e r o f o ffe n s e s and the p r o b a b i­ l i t i e s and c o s ts can be s u m m a r iz e d by the fo llo w in g e q u a tio n s : (1. .9 ). (1 .1 0 ). (1 .1 1 ). Fa. = G; o F b = O Sn. P = PN> O <S N. 6. <5Q SN. = O. T h e f i r s t - o r d e r c o n d itio n f o r c o n s tra in e d u t i l i t y m a x im iz a t io n is now found by d if f e r e n t ia t in g e q u a tio n ( 1 . 3 ) w ith r e s p e c t to N , th e n u m b e r o f o ffe n s e s c o m m it t e d .. (1.12). T h e r e s u l t is:. ^SU = (1 - P ) d U '1+P (1-Q )d U 2+ P Q ciU '3+ f= N j:i-Q )U 2+ Q U 3-. up = O w h e re : (1 .1 3 ). U-| = U ( C 1 , tc ). >O. sc1 u2=. U (C 2 ,tc ). U '2 = 5 u _ >. o. $c 2 U 3 = U (C 3 >tc ). and. U 43 = | U. > O.

(21) T h e f i r s t th r e e t e r m s o f e q u a tio n ( 1 . 1 2 ) a r e a w e ig h te d s u m o f the m a r g in a l u t i l i t i e s o f in c o m e in e a c h s ta te , m u ltip lie d by d ,. the. m a r g in a l r e t u r n f r o m a llo c a tin g an a d d itio n a l h o u r to c r im e r a t h e r than le g a l w o r k . m in u s U-) .. T h e la s t t e r m is S in c e. is a s s u m e d to be p o s itiv e and U-)> U 2 ^ U g , th e la s t. t e r m w i l l be n e g a tiv e . m u s t be p o s it iv e .. tim e s a w e ig h te d a v e r a g e o f U 2 and U 3 ,. In o r d e r to f u l f i l l the f i r s t - o r d e r c o n d itio n , d. T h a t i s , the w ag e in ille g a l w o r k m u s t e x c e e d the. le g a l w a g e . T h e f i r s t - o r d e r c o n d itio n c an be r e w r it t e n as: (1 .1 5 ). P N = d ( ( 1 - P ) U , 1 + P ( 1 - Q ) U 12 + P Q U ,q) U t-(1 -Q )U 2 -Q U 3. T h e l e f t s id e o f th is e q u a tio n c o u ld be in te r p r e t e d a s th e m a r g in a l c o s t, to the o f f e n d e r , o f c o m m it tin g an a d d itio n a l o ffe n s e .. T h e r ig h t s id e. c o u ld be in te r p r e t e d a s the o ffe n d e r 's m a r g in a l re v e n u e .. S in c e the. o ffe n d e r is a m o n o p o lis t in s u p p ly in g h is own la b o r (to < ;r im e ), he w i l l be in e q u ilib r iu m w h e r e h is m a r g in a l re v e n u e e q u a ls m a r g in a l c o s t. F ig u r e 1 i l lu s t r a t e s tw o p o s s ib ilit ie s f o r s ta b le e q u ilib r iu m .. If. b oth th e m a r g in a l p r o b a b ilit y o f a r r e s t and m a r g in a l re v e n u e c u r v e s f a ll a s N in c r e a s e s c u rv e .. the M C c u r v e w i l l be le s s s te e p ly s lo p e d than th e M R. I f b o th fu n c tio n s a r e in c r e a s in g w ith N , the M C c u r v e w i l l be. m o r e s te e p ly s lo p e d th en M R . o c c u r s w h e re :. M a t h e m a t ic a lly , the s ta b le e q u ilib r iu m.

(22) 12. (1. &MC SN. . 16). N j> M R y «5N. M C = F n ;M R =. eJ ( ( 1 - P ) U , 1 + P ( 1 - Q ) U ' 2 ^ Q U ,3 u r (i-Q ) u2 -Q u 3. O r , s u b s titu tin g e q u a tio n (1 .1 5 ) into (1 . 16) w e o b ta in : (1 .1 7 ). PN ^. d 2 ( ( 1 - P ) U " 1 + P ( 1 - Q ) U " 2 + P Q U " 3 )+ (d M R -K iP n ) __________________________________ ((1 - Q ) U , q+ Q U ,3 - U >i ) U -|-(1 -Q )U 2 -Q U 3. ■’A i A c.. iAC. ,r. f»\ c --lAfc. ftt-- ‘A>\. I. N. N. N (B ). (A ) F iq .. 1. M a r g in a l R e v e n u e and M a r g in a l C o s t C u r v e s (A ) R e p r e s e n ts c a s e w h e re b oth M C and M R a r e f a llin g (B ) R e p r e s e n ts c a s e w h e re b oth M C and M R a r e r is in g. T h e s a m e r e s u lt is a c h ie v e d by a s s u m in g the e x p e c te d u t i l i t y fu n c tio n is c o n v e x to the o r i g i n . (1 .1 3 ). ■U e u \* N , -5 N. E x p re s s e d th is w ay:. ,2. d‘'( P ( 1 - Q ) U " 2 -H=QU" 3 + ( 1 - P ) U " 1). +. = 4 = 2 P N ( d X ( 1 - Q ) U ,2 + Q U ,3 - U ' 1) + P n n C (1 -Q )U 2 + Q U 3 - U i ). q. o .. No c o n c lu s io n a b o u t U " c a n be m a d e by e x a m in in g the s e c o n d -o r d e r c o n d itio n e x p re s s e d in e q u a tio n (1 . 1 8 ) .. I f the in d iv id u a l is r i s k a v e r s e. and t h e r e f o r e 'J" is le s s than z e r o , the f i r s t t e r m w i l l be n e g a tiv e bu t the sec o n d t e r m w i l l be p o s it iv e , s in c e U-)' is le s s than U 2 ' and U 3 ’ ..

(23) 13. i f the in d iv id u a l is r i s k n e u t r a l, and t h e r e f o r e U " e q u a ls z e r o , then m u s t be p o s itiv e to f u l f i l l e q u a tio n. (1. .1 8 ).. T h e la w e n fo r c e m e n t v a r ia b le s w h ic h e n te r in to th e c o n d itio n d e s ­ c r ib in g the o p tim a l n u m b e r o f o ffe n s e s a r e : P , the p r o b a b ilit y o f a r r e s t ; Q , th e p r o b a b ilit y o f c o n v ic tio n and p u n is h m e n t g iv e n a r r e s t ; F a , the c o s t o f a p p re h e n s io n ; and F ^ , the c o s t o f c o n v ic tio n and s e n te n c e g iv e n a p p r e h e n s io n .. A l s o , th e w a g e s in the le g a l and ille g a l s e c to r s a f f e c t. tim e a llo c a tio n to c r i m e .. W h a t w i l l be the e ffe c t on c r im e o f an in c re a s e. in e a c h o f th e s e v a r ia b le s ? A.. A C H A N G E IN T H E P R O B A B I L I T Y O F A R R E S T A s s ta te d e a r l i e r , the p r o b a b ilit y o f a r r e s t is a s s u m e d to b e , in. p a r t , a fu n c tio n o f th e n u m b e r o f o ffe n s e s c o m m it t e d .. T h e p r o b a b ilit y o f. a r r e s t is a ls o a fu n c tio n o f an exo g e n o u s v a r i a b l e , x .. T h e g e n e ra l fo rm. o f th is p r o b a b ilit y fu n c tio n is : (1 .1 9 ). P = P (N ,x ) A n e xo g e n o u s s h ift in P is c a u s e d by a c h an g e in the e x o g e n o u s. v a r i a b l e in the P fu n c tio n , x .. B y d if f e r e n t ia t in g e q u a tio n. (1. . 1 2 ) w ith. r e s p e c t to N , the e f fe c t on th e o p tim a l n u m b e r o f o ffe n s e s , N * , o f an in c r e a s e in x can be show n a s fo llo w s : (1 .2 0 ). dN* dx. =_1. (d P x ( U ' 1 - ( 1 - Q ) U ^ - Q U , 3 )+ P n x ( U 1 - ( 1 - Q ) U. 2. -Q U 3 ). A. w h e re : (1 .21). ; S *. p nx. = jP n S*. T h e s ig n o f th is fu n c tio n is n o t d e te r m in e d w ith o u t f u r t h e r a s s u m p tio n s.

(24) 14. a b o u t the P fu n c tio n .. I f it is a s s u m e d th a t b o th P and Pf\j r i s e by th e s a m e. p e r c e n ta g e w ith a c h an g e in x , then it c an be show n th a t the e ffe c t o f x on the n u m b e r o f o p tim a l o ffe n s e s c o m m itte d w i l l be n e g a tiv e r e g a r d le s s o f th e r i s k p r e fe r e n c e o f th e in d iv id u a l T h e a b o ve c o n c lu s io n w i l l n o t hold w hen one m a k e s o t h e r a s s u m p ­ tio n s a b o u t the b e h a v io r o f th e P fu n c tio n .. F o r e x a m p le , ta k e the c a s e. w h e r e x is r e p r e s e n te d by (J) , th e p r o b a b ilit y o f a r r e s t f o r one o ffe n s e , and P r e p r e s e n ts. the p r o b a b ilit y o f a t le a s t one a r r e s t f o r N o ffe n s e s .. I f it a s s u m e d th a t th e p r o b a b ilit y o f a r r e s t f o r one o ffe n s e. is in d e p e n d e n t. o f the to ta l n u m b e r o f o ffe n s e s c o m m it t e d , then one can u se th e b in o m ia l d is tr ib u tio n to fin d P a s fo llo w s : (1 .2 2 ). P. =. 1. - ( 1 -(J))N. F o r s m a ll v a lu e s o f Cj), the a b o v e f o r m u la c an be a p p ro x im a c e d u s in g th e 3. fo llo w in g : (1 .2 3 ). P=1-(e ^ N). F ir s t - a n d S e c o n d - p a r t ia l d e r iv i t i v e s o f th is fu n c tio n w ith r e s p e c t to (p and N a r e e x p re s s e d b e lo w : (1 .2 4 ). P N =(t)e. ~^N. >. °. (1 .2 5 ). P X= P<|)= N e-N<^>^ O. (1 .2 6 ). Pn. (1 .2 7 ). P NN. x. =. (1 -< p N ) ^ O a s < ( ) N f l. = -4>2 e - (t) N < O. I t is o b v io u s th a t th e a s s u m p tio n o f an e q u a l p e rc e n ta g e c h an g e in P and P]^ w h e n x d is t r ib u t io n .. ((j)). c h a n g e s is u n r e a s o n a b le , u s in g the a b o ve p r o b a b ilit y. A ls o , th e s ig n o f P [\ j x '■s a m b ig u o u s .. T h e m a r g in a l.

(25) 15. p r o b a b ilit y o f a r r e s t w i l l in c re a s e o n ly f o r o ffe n s e s le s s than 1/cj>.. For. a la r g e n u m b e r o f o ff e n s e s , che m a r g in a l p r o b a b ility o f a r r e s t w i l l f a l l . F ig u r e 2 i l lu s t r a t e s the s h ift in the P N ( m a r g in a l c o s t) fu n c tio n w ith a s h ift in the e x o g e n o u s v a r i a b l e , h e re r e p r e s e n te d by the p r o b a b ilit y o f a r r e s t f o r one o ffe n s e . s lo p e d .. It is a s s u m e d th a t the M C c u r v e is n e g a tiv e ly. T h e m a r g in a l re v e n u e is h e ld c o n s ta n t.. I f the o p tim a l n u m b e r o f o ffe n s e s is le s s th an 1/<|3, a s h ift in P ^ , c a u s e d by an e xo g e n o u s in c r e a s e in the p r o b a b ilit y o f a r r e s t f o r one o f ­ fe n s e , w i l l r e s u lt in a new e q u ilib r iu m to the l e f t o f th e o r ig in a l e q u ili­ b r iu m .. S u ch a s h ift in th e m a r g in a l c o s t fu n c tio n is illu s t r a t e d in F ig u r e. 2 , panel A .. I f N * (th e o p tim a l n u m b e r o f o ffe n s e s ) is g r e a t e r than 1 /4 ) ,. a. s h ift in the M C c u r v e c a u s e d by an in c r e a s e in the p r o b a b ilit y o f a r r e s t f o r one o ffe n s e w i l l r e s u lt in a n e w e q u ilib r iu m p o in t w h e r e m o r e o ffe n s e s a r e c o m m it te d .. F ig 2. T h is s h ift is in d ic a te d in p a n e l B o f F ig u r e 2 .. e f fe c t o f a c h an g e in ($ on m a r g in a l c o s t (A ) N * < 1/(t> (B ) N * > 1/<t> M C is m a r g in a l c o s t c u r v e b e fo r e th e s h ift M C ' is m a r g in a l c o s t c u r v e a f t e r th e s h ift.

(26) 16. It is not s u r p r is in g than an in c r e a s e in the m a r g in a l p r o b a b ilit y o f a p p r e h e n s io n , m a r g in a l re v e n u e h e ld c o n s ta n t, w i l l r e s u lt in f e w e r o ffe n s e s ; and a f a ll in the m a r g in a l p r o b a b ilit y o f a r r e s t f o r one o ffe n s e (an d th e r e fo r e f o r N o ffe n s e s ) w ill r e s u lt in m o r e o ffe n s e s .. W h a t is. s u r p r is in g is th a t an in c re a s e in th e p r o b a b ility o f a r r e s t f o r one o ffe n s e and th e r e f o r e f o r N o ffe n s e s w i l l lo w e r the m a r g in a l p r o b a b ilit y o f a r r e s t , P N , in s o m e c a s e s .. In the s a m p le d a ta , d e s c r ib e d l a t e r , the a v e r a g e. v a lu e f o r the p r o b a b ilit y o f a r r e s t f o r one o ffe n s e is found to be b e tw e e n .1 5 and . 2 0 .. T h u s , f o r fe w e r than fiv e to s e v e n o ffe n s e s c o m m itte d. a n n u a lly , a r is e in the in d iv id u a l p r o b a b ility o f a r r e s t w o u ld be e x p e c te d to in c r e a s e the m a r g in a l p r o b a b ility o f a r r e s t and d e t e r c r i m e .. For. fr e q u e n t o f f e n s e r s , an in c re a s e in the p r o b a b ilit y o f a r r e s t f o r an in d iv id u a l o ffe n s e , h o ld in g M R c o n s ta n t, is e x p e c te d to e n c o u ra g e c r im e by lo w e r in g the m a r g in a l p r o b a b ility o f a r r e s t . So f a r , the m a r g in a l re v e n u e o f c o m m ittin g an o ffe n s e has been h e ld c o n s ta n t.. O ne c an e x a m in e the to ta l e ffe c t o f a chan g e in (p on N. by ta k in g the d i f f e r e n t i a l o f e q u a tio n (1 . 1 5 ) . H e r e <4 ) one o f the enogeuous v a r ia b le s is a llo w e d to v a r y a s w e ll a s N , the endogenous v a r ia b le : (1. .2 3 ). PN<{> d(t)+P N N d N = M R < |)d ^ + M R N dN. w he r e : (1 .2 9 ). P NN ~ ^ PN <N. MRryj —& M R 5N.

(27) 17. T h e to ta l e ffe c t o f a chang e in (J) on N is found by n e a r r a n g in g the t e r m s in e q u a tio n (1 .3 0 ). (1. .2 8 ):. dN =. P N(p - M R ^. d(J). M R n -P ,\ jn. a c c o r d in g to e q u a tio n (1 . 1 6 ) , M R N ^ P n on the r ig h t s id e o f e q u a tio n (1. . 31). (1. n. * T h e r e f o r e , the d e n o m in a to r. . 3 0 ) m u s t be n e g a tiv e .. 2 . a s P[\|(p - MR<j) " y. dN ^ O d<t>. Thus:. O. T h u s , h o ld in g M R f ix e d , (1 .3 2 ). dN > dCf). C. as P n a 4 * r. C.. T h is e f f e c t , o f $ on the M C c u r v e , has a lr e a d y been illu s t r a t e d in F i g . 2 . H o w , h o ld in g M C fix e d , and ta k in g the p a r t i a l d e r iv i t i v e o f th e r ig h t s id e o f e q u a tio n ( 1 . 1 5 ) w ith r e s p e c t to (p: (1 .3 3 ). dN^_0 d(j>. as MR<|)= d(P <|)( 1 -Q )U 'o -iP (|)Q U ' 3 - P ( | ) U ' i ) ^ U 1 -(1 -Q )U. 2. -Q U 3. S in c e U i> U 2 > U g , the d e n o m in a to r m u s t be p o s it iv e .. E q u a tio n (1 .3 3 ). i l lu s t r a t e s th a t i f the in d iv id u a l is a r i s k p r e f e r e r , m a r g in a l re v e n u e w i l l f a ll w ith an in c r e a s e in th e in d iv id u a l p r o b a b ilit y o f a r r e s t ,( J ) . C o n v e r s e ly , i f the in d iv id u a l is a r i s k a v o id e r , the m a r g in a l re v e n u e c u r v e w ill s h ift to the r ig h t a s (|) in c r e a s e s .. T h a t i s , i f M C is h e ld. c o n s ta n t and <^) is in c r e a s e d , r i s k a v o id e r s w i l l tend to in c r e a s e t h e i r o p tim a l n u m b e r o f c r im e s and r i s k p r e f e r e r s w i l l re d u c e c r i m e .. The. M R c u r v e does noc s h ift f o r r i s k n e u tr a l in d iv id u a ls . T h e fa c t th a t an in c re a s e in the p r o b a b ilit y o f a r r e s t e n c o u ra g e s. O.

(28) 18. r i s k a v o id e r s to c o m m it c r im e s m a y s t r i k e the r e a d e r as c o u n t e r in t u it iv e . N o te , h o w e v e r , th a t w hen the in d iv id u a l is r i s k a v e r s e , the m a r g in a l u t i l i t y o f C a t the u n fa v o ra b le o u tc o m e s (C 2 and C 3 ) is g r e a t e r than the m a r g in a l u t i l i t y o f C a t the fa v o r a b le o u tc o m e (C -|).. B y in c re a s in g P ,. th e r e is a g r e a t e r p r o b a b ilit y o f an u n fa v o ra b le o u tc o m e and th e r e f o r e a g r e a t e r p r o b a b ilit y o f an o u tc o m e w ith g r e a t e r m a r g in a l u t i l i t y o f C . T h u s , it is r e a s o n a b le th a t an in c r e a s e in P w o u ld in c re a s e the m a r g in a l. u t i l i t y (r e v e n u e ) f r o m c r im e f o r r i s k a v o id e r s and d e c r e a s e. th e m a r g in a l u t i l i t y (r e v e n u e ) f r o m c r im e f o r r is k p r e f e r e r s . F ig u r e 3 illu s t r a t e s the s h ifts in b o th M C and M R c u r v e s . a s s u m e d , h e r e , th a t m o s t o ffe n d e r s c o m m it le s s th a n. 1. I t is. /(p c r im e s p e r. y e a r and th e r e f o r e th a t M C is in c re a s e d a t the e q u ilib r iu m n u m b e r o f o ffe n s e s , N * . F o r r i s k p r e f e r e r s (p a n e l A ) a r is e in th e exo g e n o u s p r o b a b ilit y o f a r r e s t f o r one o ffe n s e , 0 , w i l l s h ift the M R d o w n w a rd and w i ll t h e r e f o r e re d u c e th e e q u ilib r iu m n u m b e r o f o ffe n s e s . a v o id e r s , the M R c u r v e w i l l s h ift u p w a r d s .. In p a n e l. F o r r is k , th e s h ift in. th e M R c u r v e is n o t a s g r e a t a s the s h ift in the M C c u r v e , and the o p tim a l n u m b e r o f o ffe n s e s w i l l d e c lin e .. In p a n e l B 2 , the s h ift in the. M R c u r v e has a g r e a t e r e ffe c t on the n u m b e r o f o ffe n s e s than th e s h ift in th e M C c u r v e , and m o r e o ffe n s e s w i l l be c o m m it te d ..

(29) 19. * *. F ig . 3. N* (A). At. T h e e ffe c ts o f a c h an g e in cj> on M C and M R C u r v e s (A ) In d iv id u a l is a R is k P r e f e r e r ( B ) In d iv id u a l is a R is k A v o id e r 1 . N * 1 <• N * 2. N * 1 > N*.

(30) 20. T h e o v e r a ll e ffe c t on c r im e o f a change in th e e x o g e n o u s c o m p o n e n t o f the p r o b a b ilit y o f a r r e s t is now show n to be a m b ig u o u s .. U s in g the. p la u s ib le p r o b a b ilit y d is tr ib u tio n c ite d in e q u a tio n (1 . 2 3 ) , the e f f e c t o f an in c r e a s e in x (Cp) on N w i l l p r o b a b ly be n e g a tiv e , if the in d iv id u a l is a r i s k p r e f e r e r o r r i s k n e u t r a l.. T h e e f fe c t o f an in c re a s e in x on the. n u m b e r o f o ffe n s e s r i s k a v o id e r s c o m m it w i l l p o s s ib ly be n e g a tiv e a ls o . T h u s , in the e m p ir i c a l t e s t , p r e s e n te d l a t e r in th is a n a ly s is , w e w o u ld e x p e c t to fin d a n e g a tiv e r e la tio n s h ip b e tw e e n c r im e and the p r o b a b ilit y o f a r r e s t f o r one o ffe n s e .. H o w e v e r , no r e la tio n s h ip o r e v e n a p o s itiv e. r e la t io n s h ip , w o u ld n o t be in c o n s is te n t w ith the th e o r y . B.. A C H A N G E IN T H E P R O B A B IL I T Y O F C O N V IC T IO N A N D P U N I S H M E N T W h e n the p r o b a b ilit y o f c o n v ic tio n and p u n is h m e n t g iv e n a r r e s t (Q ). in c r e a s e s , th e o ffe n d e r fa c e s a g r e a t e r c h a n c e o f b e in g c o n v ic te d and p u n is h e d upon a r r e s t .. T h e p r o b a b ilit y o f c o n s u m p tio n p ro s p e c t C g. is h ig h e r , the p r o b a b ilit y o f c o n s u m p tio n p r o s p e c t Cq is l o w e r , and the p r o b a b ilit y o f c o n s u m p tio n p r o s p e c t C. 1. is u n c h a n g e d .. T h u s , an in c re a s e. in Q w i l l lo w e r the e x p e c te d in c o m e , b u t w i l l n o t a ffe c t the in c o m e in e a c h s ta te . T h e o v e r a ll e ffe c t o f a chan g e in the p r o b a b ilit y o f c o n v ic tio n and p u n is h m e n t on the n u m b e r o f o ffe n s e s c o m m itte d c a n be e x a m in e d in a s i m i l a r fa s h io n to the e f f e c t on N o f a chang e in (j), o r in a n y o f the e x o g e n o u s v a r ia b le s . A n e xo g e n o u s c h an g e in Q. r e s u lt s in th e fo llo w in g :.

(31) 21. (1 .3 4 ). dN _. =. P NQ. dQ. (1 . 3 5 ) and. -. MRq. ------------------------MRn - Pn n. dN ^ O dQ <-. as. 1. P , \jq ~ M R q. its O. S in c e P do es not v a r y w ith Q , P ^ jq is z e r o , and a c h an g e in Q w i ll no t a f fe c t the M C c u r v e .. T a k in g the p a r t i a l d e r iv i t i v e o f M R w ith r e s p e c t. to Q and s u b s titu tin g into e q u a tio n (1. .3 6 ). dN } dQ <. O as. M R q. (1. .3 5 ) w e fin d :. = M R (u 3 -U 2 ). + (d P U ^ -d P U ^ ) > O ----------------------------------- ^ U -i- ( 1 - Q ) U o - Q U 3. T h e d e n o m in a to r in the M R q fu n c tio n m u s t be p o s itiv e s in c e U-|^ U 2 > U 3 M R , d e fin e d in e q u a tio n. (1. . 1 6 ) , is a s s u m e d to be p o s itiv e .. If the. in d iv id u a l is r is k n e u t r a l, the M R fu n c tio n w i l l f a ll a s Q is r a is e d . T h is r e s u lt is o b ta in e d s in c e U g ) Ug; and U " = 0 . R is k p r e f e r e r s w i l l a ls o e x p e r ie n c e a d o w n w a rd s h if t in the M R c u r v e w hen Q in c r e a s e s .. T h is s h ift is illu s t r a t e d in fig u r e 4 , p a n e l A .. T h u s , r i s k p r e f e r e r s w i l l re d u c e c r im e w hen Q in c r e a s e s . T h e r e s u lta n t n u m b e r o f o ffe n s e s c o m m itte d w ith a c h an g e in Q is a m b ig u o u s in th e c a s e o f the r i s k a v e r s e in d iv id u a l. th e. T h e f i r s t t e r m in. n u m e r a t o r o f e q u a tio n ( 1 . 3 6 ) can be thou g h t o f a s the r i s k - n e u t r a l. e f f e c t , s in c e the s ig n o f th is t e r m d o e s not depend on the r i s k p r e fe r e n c e o f th e in d iv id u a l.. In the c a s e o f th e e ffe c t a chan g e in Q on the M R c u r v e ,. the r i s k - n e u t r a l e ffe c t is n e g a tiv e . T h e second t e r m in the n u m e r a t o r o f e q u a tio n ( I . 3 6 ) d ep en d s on the r i s k p r e fe r e n c e o f th e in d iv id u a l.. In the c a s e o f the r is k a v o id e r ,.

(32) 22. LJ*3 > U. ancl the r is k - p r e f e r e n c e e ffe c t is to r a is e M R w hen Q in c r e a s e s .. T h e o v e r a ll e f f e c t , th e n , is u n c e r t a in .. I f the r i s k - n e u t r a l e ffe c t is. s tr o n g e r tnan the r is k - p r e f e r e n c e e f f e c t , c r im e s w i l l be re d u c e d ( F ig u r e 4 , p a n e l B -|).. If the r is k - p r e f e r e n c e e ffe c t is s tr o n g e r than the r i s k - n e u t r a l. e f f e c t , r i s k a v o id e r s w i l l in c r e a s e t h e i r c r im i n a l a c t iv it y ( F ig u r e 4 , p a n e l E ^ ).. N *‘ N* (A) \. H * 'N * (B ,). F ig . 4. N* 1. T h e e ffe c ts o f a chan g e in Q on M C and M R C u r v e s (A ) In d iv id u a l is a R is k P r e f e r e r (B ) In d iv id u a l is a R is k A v o id e r 1. R is k N e u tr a l e ffe c t > R is k P r e f e r e n c e e ffe c t 2 . R is k P r e f e r e n c e e ffe c t > R is k N e u tr a l e ffe c t.

(33) 23. F o r p u rp o s e s o f th is a n a ly s ts , It Is a s s u m e d th a t m o s t r i s k a v o id e r s w i l l be d e t e r r e d by an in c re a s e in Q and th a t , in the a g g r e g a te , the e ffe c t o f an in c re a s e in Q on c r im e w i l l be n e g a tiv e . C.. A C H A N G E IN T H E C O S T S O F A P P R E H E N S I O N , O O N V IC T IO N A N D P U N IS H M E N T A n in c re a s e in F a , the c o s t o f a p p r e h e n s io n , w i l l lo w e r the. c o n s u m p tio n p ro s p e c ts in both o f the u n fa v o ra b le s ta te s o f the w o r ld . A n in c re a s e in F. , th e c o s t o f c o n v ic tio n and p u n is h m e n t, n e t o f the. a r r e s t c o s t, w i l l lo w e r the c o n s u m p tio n p r o s p e c t o n ly i f the p e rs o n is a r r e s t e d and c o n v ic te d . C 0 and C g .. T h a t i s , an in c r e a s e in F a w i l l re d u c e both. A n in c r e a s e in F b w i l l o n ly lo w e r C 3 .. T h e e ffe c ts o f F a and F b on the n u m b e r o f o ffe n s e s c o m m itte d can be found by ta k in g the d if f e r e n t i a l o f e q u a tio n ( 1 . 1 5 ) w ith r e s p e c t to N and F a ; and w ith r e s p e c t to N and F b , r e s p e c tiv e ly : (1 .3 7 ). dN _ =. PM Fb - M R F a. dFa. M R n ~ P NN. and (1 .3 8 ). dN _ dFK. =. PNFb - M R F b --------------------------M R n - P NN. S in c e P does n o t v a r y w ith F a n o r w ith F b , it is e a s ily show n th a t: (1 .3 9 ). _dN. >. F a <*. C. as. MRp. = M R ( -(1 - Q ) U ' 2 - Q U ' 3 )+[;dP (1 - Q ) U -d P Q U " 3 ]. 3. "2. > O.. U 1 -(1 -Q )U and. 2. -Q U 3.

(34) 24. n.4G >. dN y dFb C. 0 as M R F. = b. M R ( - Q U Tg )+ (S P Q U " 3 ) ? ---------------------------------------U 1 - (1 -Q )U 2 -Q U 3. O.. T h e f i r s t t e r m on the r ig h t-h a n d s id e o f both e q u a tio n s (1 .3 9 ) and (1. . 4 0 ) r e p r e s e n ts th e r i s k - n e u t r a l e f f e c t .. I f U" w e r e O , th e n f o r an. in c r e a s e in e it h e r F a o r F p , the M R c u rv e w o u ld s h if t d o w n w a rd and c r im e w o u ld be r e d u c e d .. F a w i l l h ave a g r e a t e r e f fe c t than F p in re d u c in g. c r im e c o m m itte d by r i s k a v o id e r s . T h e r i s k p r e fe r e n c e e ffe c t is r e p r e s e n te d b y th e second t e r m in the n u m e r a t o r o f e q u a tio n s (1 .3 9 ). and (1 .4 0 ). F o r r i s k p r e f e r e r s , the. r i s k p r e fe r e n c e e ffe c t is the s a m e sig n as the r i s k - n e u t r a l e f f e c t . S in c e U " is p o s it iv e , th e r i s k - p r e f e r e n c e e ffe c t is n e g a tiv e , and the r i s k - p r e f e r e r s w i l l re d u c e t h e i r c r im i n a l a c t iv it y e ven m o r e than r is h - n e u t r a l in d iv id u a ls .. A g a in a chan g e in F a w i l l h ave a g r e a t e r. im p a c t on c r im e th an F p . F o r the r i s k a v o id e r , the r is k p r e fe r e n c e e ffe c t is p o s itiv e in the c a s e o f an in c r e a s e in F a o r F p . t h e i r c r i m i n a l a c t iv it y a s m u c h to r i s k .. T h u s , r i s k a v o id e r s w i l l n o t re d u c e a s th e y w o u ld i f th e y w e r e in d iffe r e n t. In f a c t , it is p o s s ib le f o r r i s k a v o id e r s to in c re a s e t h e i r. c r im i n a l a c t iv it y w h e n F a o r F p in c r e a s e s . F ig u r e 4 , w h ic h illu s t r a t e s the e ffe c ts on M R , M C and c r im e o f an in c r e a s e i n Q , c an a ls o be used to i l lu s t r a t e the e ffe c t o f an in c re a s e in F.

(35) 25. or Fb.. T h e M R c u r v e s h ifts d o w n w a rd f o r the r i s k p r e f e r e r , and m a y. s h ift d o w n w a rd o r u p w a rd f o r the r i s k a v e r s e in d iv id u a l.. F o r p u rp o s e. o f th is a n a ly s is , i t is a s s u m e d th a t f o r m o s t in d iv id u a ls , the M R c u r v e w i l l s h ift d o w n w a rd and th a t an in c re a s e in F & o r F ^ w i ll d e t e r c r i m e . D.. A N IN C R E A S E IN T H E L E G A L O R I L L E G A L W A G E T h e le g a l and ille g a l w a g e s a f fe c t the to ta l in c o m e and the m a r g in a l. in c o m e in e ac h s ta te o f the w o r l d , f o r in d iv id u a ls w ho a r e e n g ag ed in b oth c r im e and le g a l w e a lth - g e n e r a tin g a c t iv i t i e s .. T h e e f f e c t on c r im e. o f a chan g e in ille g a l w a g e s is found by ta k in g the d i f f e r e n t i a l o f e q u a tio n (1. . 1 5 ) , a llo w in g N and W j to v a r y :. (1 .4 1 ). dN. =. PnW[. ~ MR W.. ---------------------------. d /vi. MRN. " P NN. S in c e P does n o t v a r y w ith W , i t is e a s ily show n th a t: C1 - 4 2 ). [(1 - P U i + P (1 - Q ) U ’ 2 + P Q U ’ 3 ] + [d C l-P )U " 1 N + d P ( 1 - Q ) U ,,2 N + d P Q U " 3 N >. ‘ dN y. dW j <. O as. M R yy 1. = + [ M R ( ( 1 - Q ) U ,p N -fQ U ' 3 N - U U 1 -(1 -Q )U 2 -Q U 3. , 1. N )] 'lO. T h e f i r s t t e r m o f the n u m e r a t o r in e q u a tio n (1 . 4 2 ) r e p r e s e n ts the r i s k n e u tr a l e f f e c t . o f e q u a tio n. I f U " w e r e z e r o , then the f i r s t t e r m o f the n u m e r a t o r. (1 .4 2 ) w o u ld be p o s itiv e and a ll o th e r t e r m s w o u ld be z e r o .. T h u s , th e r i s k n e u tr a l e ffe c t o f an in c r e a s e in W j is to in c r e a s e the n u m b e r o f o ffe n s e s c o m m it te d . T h e r i s k - p r e f e r e n c e e ffe c t o f an in c r e a s e in W j on c r im e c o m m itte d.

(36) 26. b y r i s k p r e f e r e r s is a m b ig u o u s s in c e the second t e r m in the n u m e r a t o r o f e q u a tio n (1 . 4 2 ) w i l l be p o s it iv e , b u t the th ir d t e r m w i ll be n e g a tiv e . A n d th e o v e r a ll e f f e c t o f an in c re a s e in W j on th e n u m b e r o f c r im e s the r i s k p r e f e r e r w i l l c o m m it is a ls o a m b ig u o u s . T h e e ffe c t o f W j on the n u m b e r o f c r im e s c o m m itte d by r i s k a v o id e r s is a ls o a m b ig u o u s .. F o r r i s k a v e r s e in d iv id u a ls , th e second t e r m in the. n u m e r a t o r o f e q u a tio n (1 .4 2 ) w i l l be n e g a tiv e b u t the t h ir d t e r m w i l l be p o s itiv e .. In the a g g r e g a te , the e ffe c t o f an in c re a s e in W j on c r im e is. a s s u m e d f o r th is a n a ly s is , to be p o s itiv e . T h e e f f e c t , on c r i m e , o f a chan g e in le g a l w a g e s is found by ta k in g the d if f e r e n t ia l o f e q u a tio n (1 . 1 5 ) , a llo w in g N and W L to v a r y : (1 .4 3 ). — ” dW L. pn w l. . “. m r n. m r w l p nn. S in c e P d o e s n o t v a r y w ith W ^ , it is e a s ily show n th a t: (1 .4 4 ). [ - ( 1 - P ) U ' t - P ( 1 - Q ) U ' 2 - P Q U ’ 3 ]+ dN \ O a s M R v v l =' [d (1 -P )U " 1 t L + d P (1 -Q )U " 2 t L + HW,i_^ dW dPQ Ug + jJ + M R T ( 1 - Q ) U ' 2 t + Q U ,3 t L - U ' 1 + i J. U 1 - ( 1 -Q )U 2 -Q U 3 T h e f i r s t t e r m o f th e r i s k n e u tr a l e f f e c t .. n u m e r a t o r o f e q u a tio n. (1. . 4 4 ) r e p r e s e n t s th e. I f U " w e r e z e r o , th e f i r s t t e r m o f the n u m e r a t o r o f. e q u a tio n (1 .-44) w o u ld be n e g a tiv e and a ll o th e r t e r m s w o u ld be z e r o . T h u s , the r i s k n e u tr a l e f fe c t o f an in c r e a s e in n u m b e r o f o ffe n s e s c o m m itte d .. is to d e c r e a s e the.

(37) 27. A g a m , the r is k - p r e f e r e n c e e ffe c t f o r r i s k a v o id e r s and r i s k p r e f e r e r s is a m b ig u o u s .. T h e o v e r a ll e f fe c t o f a chan g e in W. on the c r im i n a l. a c t iv i t i e s o f r i s k a v o id e r s and r i s k p r e f e r e r s is n o t c l e a r .. H o w e v e r,. w e a s s u m e chat the r i s k n e u tr a l (n e g a tiv e ) e ffe c t is s tr o n g e r than the a m b ig u o u s r i s k p r e fe r e n c e e ffe c t f o r m o s t in d iv id u a ls .. T h a t is , w e. a s s u m e th a t an in c re a s e in W L w i l l d e t e r m o s t in d iv id u a ls . E . S U M M A R Y O F E F F E C T S O F E X O G E N O U S S H I F T S IN T H E L A W E N F O R C E M E N T V A R I A B L E S A N D IN T H E W A G E V A R I A B L E S A n in c r e a s e in a ll o f the la w e n fo r c e m e n t v a r ia b le s c ite d in s e c tio n s A th ro u g h C a r e e x p e c te d to re d u c e c r i m e , e ve n thoug h f o r s o m e o ffe n d e r s an in c r e a s e in the la w e n fo r c e m e n t v a r ia b le s m ig h t in c re a s e c r i m e . S i m i l a r l y , f o r the w a g e v a r i a b l e s , le g a l and ille g a l w a g e s , an exo g e n o u s in c r e a s e is e x p e c te d to h a v e a s p e c ific e ffe c t (n e g a tiv e f o r le g a l w a g e s and p o s itiv e f o r ille g a l w a g e s ) a lth o u g h f o r s o m e o ffe n d e r s it is p o s s ib le f o r the e ffe c ts to be r e v e r s e d .. T h u s , an in c re a s e in the fo llo w in g. v a r ia b le s w o u ld o ro b a b ly d e t e r c r im e : x , the e xo g e n o u s c o m p o n e n t o f th e p r o b a b ilit y o f a r r e s t v a r ia b le ; F a , the c o s t o f a r r e s t ; F ^ , the c o s t o f c o n v ic tio n and p u n is h m e n t n e t o f a r r e s t c o s t; Q , th e p r o b a b ilit y o f c o n v ic tio n and p u n is h m e n t g iv e n a r r e s t ; and W j_ , th e w a g e f o r le g a l w o rk .. A n in c r e a s e in W ^, th e w a g e f o r c o m m ittin g c r i m e s , w i l l u s u a lly. have a p o s itiv e e ffe c t on th e n u m b e r o f c r i m e s c o m m it te d . T h e h y p o th e s e s s ta te d a b o ve a r e q u ite a b it w e a k e r th a n th o s e c ite d by E h r lic h and th o s e used by m o s t s u b s e q u e n t a u th o rs in the f i e l d .. The. d if f e r e n t h y p o th e s e s a b o u t the e ffe c t on c r im e o f la w e n fo r c e m e n t and.

(38) 28. w ag e v a r ia b le s r e f l e c t the c h an g es m a d e in the a s s u m p tio n s o f the E h r lic h m o d e l.. E h r l i c h , in h is tw o s t a t e s - o f - t h e - w o r l d m o d e l,. a s s u m e s F to be a fu n c tio n o f the n u m b e r o f c r im e s and P (h is c o m p o s ite p r o b a b ilit y v a r i a b l e ) to be in d e p e n d e n t o f the n u m b e r o f c r im e s c o m m it te d . He c o n c lu d e s th an an in c re a s e in P w i l l u n a m b ig u o u s ly d e t e r c r i m e .. An. in c r e a s e in the a v e ra g e F w i l l u n a m b ig u o u s ly d e t e r r i s k a v o id e r s and m ig h t d e t e r r i s k p r e f e r e r s .. W h e n he in c re a s e s. the n u m b e r o f s ta te s. o f th e w o r ld , E h r lic h r e t a in s h is b a s ic h y p o th e s e s . W e fin d th a t a n y s u p p ly o f c r im e m o d e l w h ic h h y p o th e s iz e s th a t F w i l l v a r y a c c o rd in g to the n u m b e r o f o ffe n s e s c o m m itte d and th a t P is e x o g e n o u s ly d e te r m in e d w i l l r e s u lt in s tr o n g e r d e te r r e n c e o f r i s k a v o id e r s , w h e th e r P o r F is in c r e a s e d .. I f one b e lie v e s th a t m o s t in d iv id u a ls in th is. w o r ld a r e r i s k a v e r s e , then a m o d e l w h ic h h y p o th e s iz e s s tro n g e ffe c ts on th e a c t iv it ie s o f r i s k a v e r s e in d iv id u a ls w i l l h y p o th e s iz e s tro n g e ffe c ts o v e r the e n t ir e p o p u la tio n .. A nd a m o d e l o f c r im i n a l a c t iv it y. w h ic h h y p o th e s iz e s F to v a r y a c c o rd in g to the c r im i n a l in te n s ity o f an in d iv id u a l and P. to be in d e p e n d e n t o f the c r im i n a l a c t iv it ie s o f the. in d iv id u a l is a m o d e l w h ic h r e s u lt s in s tr o n g d e t e r r e n c e o f r i s k a v o id e r s in t h e i r c r im i n a l a c t iv i t i e s . B y c h o o s in g a d if f e r e n t s e t o f a s s u m p tio n s in th is a n a ly s is , a s s u m p tio n s w h ic h a r e b e lie v e d to be re a s o n a b le o n e s , the d e te r r e n c e a r g u m e n t is h e re found to be w e a k e r th a n i t has p r e v io u s ly been s ta te d . T h e n u m b e r o f s ta te s o f the w o r ld do es n o t a f fe c t the s tr e n g th o f the.

(39) 29. h y p o th e s e s . F o u r S ta te s o f the W o r ld T h e m o d e l is now e x te n d e d to in c lu d e fo u r s ta te s o f the w o r ld . T h e s e fo u r o u tc o m e s a r e d e s c r ib e d in T a b le 2 .. T h e d iffe r e n c e b e tw e e n. the th r e e -o u tc o m e m o d e l, d e s c r ib e d a b o v e , and th is m o d e l is th a t i t is no/vpossible f o r an in d iv id u a l to be c o n v ic te d b u t to a v o id in c a r c e r a t io n . T h e r e a r e now th r e e u n fa v o ra b le o u tc o m e s : the o ffe n d e r m ig h t be a r r e s t e d but n o t c o n v ic te d ; the o ffe n d e r m ig h t be a r r e s t e d and c o n v ic te d b u t n o t in c a r c e r a te d ; and the o ffe n d e r m a y be a r r e s t e d and c o n v ic te d and in c a r c e r a t e d .. T h e r e a r e c o s ts a s s o c ia te d w ith a r r e s t , c o n v ic tio n. and in c a r c e r a t io n . T h e r e s u lt s o f the th r e e -o u tc o m e m o d e l a r e e a s ily e x te n d e d to the fo u r o u tc o m e c a s e .. It is a s s u m e d th a t R , lik e Q , is n o t a fu n c tio n o f the. n u m b e r o ffe n s e s c o m m it te d .. A ls o , F. , the c o s t o f in c a r c e r a t io n n e t. o f a r r e s t and c o n v ic tio n c o s ts , is r o t a fu n c tio n o f th e n u m b e r o f o ffe n s e s c o m m it t e d .. T h e e f fe c t o f an e xo g e n o u s chan g e in R is e x p e c te d to be. s i m i l a r to th e e ffe c t o f a c h an g e in Q . u n a m b ig u o u s ly d e t e r c r i m e .. O n ly f o r r i s k p r e f e r e r s w i l l R. It is a ls o e x p e c te d th a t an in c r e a s e in F c. w o u ld d e t e r c r im e f o r a ll r i s k p r e f e r e r s and f o r s o m e r i s k a v o id e r s . F o r r i s k p r e f e r e r s , a r is e in F c w i l l h ave a s m a l l e r d e t e r r e n t e ffe c t than a c o m p a r a b le r i s e in F ^ o r F a ..

(40) 30. TABLE. 2. C o n s u m p tio n P r o s p e c ts and P r o b a b ilit ie s :. P r o b a b ilit y. S ta te o f the W o r ld. -P. F o u r S ta te s o f the W o r ld. C o n s u m p tio n P r o s p e c t C - ,= W '+ W LN + w L t L. 1.. No a r r e s t. 1. 2.. A r r e s t b u t no C o n v ic tio n. P(1 - Q ). C 2 = W , + W iN + W L tL - F a. 3.. A r r e s t , C o n v ic tio n , b u t no In c a r c e r a tio n. P Q (1 - R ). C 3 = W + W .N + W i_ ti_ -F a - F. 4.. A r r e s c , C o n v ic tio n and In c a r c e r a tio n. PQR. C =W '+ W LN + W L t L - F a - F b- F c. 5. S u m m a r iz in g the e f fe c t o f exo g e n o u s c h a n g e s in the la w e n fo r c e m e n t v a r ia b le s in the fo u r s t a t e - o f - t h e - w o r l d c a s e , an in c re a s e in a n y o f the fo llo w in g v a r ia b le s is e x p e c te d to d e t e r c r im e : x , the e xo g e n o u s c o m p o n e n t o f P ; Q , the p r o b a b ilit y o f c o n v ic tio n g iv e n a r r e s t ; R , the p r o b a b ilit y o f in c a r c e r a t io n g iv e n c o n v ic tio n ; F a , the c o s t o f a r r e s t ; F ^ , th e c o s t o f c o n v ic tio n n e t o f a r r e s t c o s ts ;F c , th e c o s t o f in c a r c e r a t io n n e t o f a r r e s t and c o n v ic tio n c o s ts .. A ls o , an in c r e a s e in W - , the w ag e f o r ille g a l. a c t iv i t i e s , is e x p e c te d to e n c o u ra g e c r im e and an in c r e a s e in W |_ , the w a g e f o r le g a l in c o m e -g e n e r a tin g a c t i v i t i e s , is e x p e c te d to d e t e r c r i m e . I f a n y o f th e s e v a r ia b le s w e r e not found to h ave t h e i r p r e d ic te d e ffe c t on c r i m e , h o w e v e r , no in c o n s is te n c y w ith the m o d e l c o u ld be c o n c lu d e d ..

(41) 31. FOOTNOTES. ^T h e B e c k e r /E h r lL c h a n a ly s e s use a c o m b in e d p r o b a b ilit y o f a r r e s t , c o n v ic tio n and im p r is o n m e n t. T h e y a s s u m e t h a t, in a g iv e n tim e p e r io d , th is p r o b a b ilit y is in d e p e n d e n t o f the n u m b e r o f c r im e s c o m m it te d . T h e le n g th o f th e tim e p e r io d is one y e a r in the e m p ir ic a l a n a ly s is . In the a p p e n d ix to h is a r t i c l e , E h r lic h d o e s c o n s id e r th e c a s e w h e r e P v a r i e s w ith th e a m o u n t o f tim e s p e n t in c r i m e . H o w e v e r , he o n ly c o n s id e r s the c a s e w h e r e the m a r g in a l and a v e r a g e p r o b a b ilit ie s chan g e b y the s a m e p e rc e n ta g e w ith an in c r e a s e in th e a m o u n t o f t im e s p e n t c o m m ittin g c r i m e s . p. I f P and Pfsj r i s e by the s a m e p e rc e n ta g e w hen x in c r e a s e s then:. sp /. sp n /. p. p n. S u b s titu tin g into e q u a tio n (1 .2 0 ): dN* =. 1. raPPCU^ -(1 - Q ) U g -Q U g ) + P P N(U 1 -( 1 -P )U 0. 2. -Q U 3 )j.. *. [d P (u ’ - ( 1 -Q )U 2 - Q U dN dx. -^T O. as. )+ P N ( U. 1 - ( 1. - Q ) U 2 - Q U 3 )J. d P (U ^ -(1 -Q )U ^ -Q U g ) +. p ^ -o -Q ^ -Q U g ). B u t, f r o m e q u a tio n. 3. 4,. o. (1 .1 2 ):. <JuJ = d P (U ^ -(1 - Q ) U 2 - P Q U ^ ) + P n ( U 1 -(1 - Q ) U 2 - Q U 3 ). ° o o. 3P =. 1. dN*. ^. ’ k O s in c e d U 1 ^ O. _(1_(^))N Let then but. and. Z = (1 -(J ))N In Z = N In (1-<t)) In ( 1 - $ ) # - (J). P = 1 - e - ^*^.

(42) 32. CHAPTER. II. THE SUPPLY OF O FFENSES The. B e h a v io r a l. F u n c tio n. G iv e n the v a lid it y o f the g e n e r a l a p p ro a c h and o f the im p lic a tio n s o f the m o d e l d e v e lo p e d in the p r e c e d in g c h a p t e r , a b e h a v io r a l fu n c tio n can now be s p e c ifie d .. T h is fu n c tio n r e la t e s the n u m b e r o f o ffe n s e s an. in d iv id u a l w i l l c o m m it in a g iv e n tim e p e r io d to the v a r ia b le s w h ic h a r e a s s u m e d to a f f e c t h is c r im i n a l a c t i v i t y .. T h e g e n e r a l f o r m o f th is. b e h a v io r a l fu n c tio n is: (2 * 1 ). c ij = fjC p ij > ^ a ij t ^ ^ ij >. [j > ^ ij » ^ ij » ^ i j. V a r ia b le s < |)y ,Q y and R y a r e , r e s p e c t iv e ly , the p r o b a b ilit ie s of: a r r e s t f o r one o ffe n s e , c o n v ic tio n g iv e n a r r e s t and in c a r c e r a t io n g iv e n c o n v ic tio n .. T h e " i ” s u b s c r ip t r e f e r s to the ith c r im e ty p e and the " j". s u b s c r ip t r e f e r s to the jt h in d iv id u a l.. T h e c o s ts o f a p p re h e n s io n a r e. r e p r e s e n te d by F a ^ , the c o s ts o f c o n v ic tio n , n e t o f a p p re h e n s io n c o s ts , a r e r e p r e s e n te d by F 13^ and the c o s ts o f i n c a r c e r a t i o n , n e t o f a r r e s t and c o n v ic tio n c o s t s , a r e r e p r e s e n te d by f=rCy •. w a g e f o r the j t h in d iv i­. d u a l f o r the ith c r im e ty p e is s y m b o liz e d by W y . the le g a l w a g e r a t e f o r the j t h in d iv id u a l.. A n d W j_j r e p r e s e n ts. T h e n u m b e r o f c r im e s o f the. ith type by the j t h in d iv id u a l is in d ic a te d by C y ..

(43) 33. A n a d d itio n a l v a r i a b l e , efj, is added to d en o te o th e r v a r ia b le s w h ic h m a y a f fe c t the in d iv id u a l's p r o p e n s ity to c o m m i t a c r i m e o f ty p e i . a re a num ber. T h e re. o f fa c to r s w h ic h h a ve no t y e t b e en in c lu d e d in the. a n a ly s is , and w h ic h m ig h t a ffe c t the s u p p ly o f c r i m e .. F o r e x a m p le ,. e m ig h t in c lu d e : in c o m e o th e r than w a g e s (th e W ' v a r ia b le ) ; f a c to r s such a s a g e o r m a r it a l s ta tu s w h ic h m ig h t a f fe c t a ttitu d e s to w a rd r is k ; p r i o r c r i m i n a l r e c o r d , w h ic h m ig h t a f f e c t the c o s t o f a p p r e h e n s io n , c o n v ic tio n and p u n is h m e n t; o r c o s ts and g a in s f r o m a lt e r n a t iv e c r i m i n a l a c t iv i t i e s . T h e a n a ly s is o f the b e h a v io r a l im p lic a tio n s o f the m o d e l d e v e lo p e d in th e la s t c h a p te r r e v e a ls the fo llo w in g s e t o f h y p o th e s e s r e g a r d in g the e f f e c t o f the v a r ia b le s e n te r in g e q u a tio n ( 2 . (2 .2 ). $ c Lj. 5. o. Soij. $. 1. ) on to ta l o ffe n s e s :. O. ^ i j. $c ij. i R ij. *. O. JW y. <_ -. O. S^ij. L. ^. -. O. *®ij Scij. J c ... ^ cij 1. /. _ -. O. ^Lj '. °. * Cij ^. -. °. Cu. A s noted in the la s t c h a p t e r , a d e p a r t u r e f r o m a n y o f th e s e h y p o th e s e s d oes n o t in d ic a te an in c o n s is te n c y w ith the m o d e l. I f a ll in d iv id u a ls w e r e a l i k e , the b e h a v io r a l fu n c tio n ( 2 . 1 ) c o u ld a ls o s e r v e as an a g g re g a te s u p p ly fu n c tio n in a g iv e n p e r io d o f t i m e ,.

(44) 34. I f e q u a tio n ( 2 . 1 ) w e r e o f a l i n e a r f o r m , i t c o u ld be r e w r it t e n a s:. (2 . 2 ). If th e c o e f fic ie n ts ( i . e . , th e a 's and the b 's ) w e r e the s a m e f o r a ll in d iv i­ d u a ls , the in d iv id u a l e q u a tio n s c o u ld e a s ily be s u m m e d o v e r a ll in d iv id u a ls in a c o u n ty .. D iv id in g the t e r m s in the s u m m a tio n e q u a tio n by the p o p u la ­. tio n o f the c o u n ty , th e a g g re g a te s u p p ly o f c r im e e q u a tio n f o r the k th c o u n ty is: (2 .3 ). w h e re n. is th e s iz e o f the p o p u la tio n in the k th c o u n ty .. T h e b e h a v io r a l fu n c tio n c an now be w r it t e n f o r a ll c o u n tie s a s : (2 .4). n tK is th e n u m b e r o f c r im e s p e r c a p ita o f the ith ty p e in the k th c o u n ty . T h e " k " s u b s c r ip t r e f e r s to th e kth c o u n ty . T h e r e a r e tw o c o n s id e r a tio n s w h ic h m u s t be e x a m in e d m o r e f u lly b e fo r e te s tin g the h y p o th e s e s ( 2 . 2 ) o f the a g g re g a te b e h a v io r a l fu n c tio n (2 .4 ). and in t e r p r e t in g the r e s u l t s .. c a u s a lit y .. F i r s t , th e r e is th e p r o b le m o f r e v e r s e. T h a t i s , the le v e l o f c r im e in the c o m m u n ity m ig h t a ffe c t. d e c is io n s a b o u t th e le v e l o f la w e n fo r c e m e n t in the c o m m u n ity .. The. seco nd c o n s id e r a tio n is the s e p a ra tio n o f tw o fo r m s o f d e te r r e n c e :.

(45) 35. 1. ) by d is c o u r a g in g in d iv id u a ls f r o m c o m m ittin g o ffe n s e s b e c a u s e o f lo w. e x p e c te d g a in s ( d e t e r r e n c e ) and. 2. ) by p r e v e n tin g o ffe n d e r s fr o m. c o m m ittin g a d d itio n a l o ffe n s e s th ro u g h in c a r c e r a tio n (in c a p a c ita tio n ). R e v e r s e C a u s a lity T h e le v e l o f c r im e in a c o m m u n ity m ig h t a ffe c t both the s u p p ly and d e m a n d f o r la w e n fo r c e m e n t.. O n the d e m a n d s id e , i f a c o m m u n ity had. a h ig h c r im e l e v e l , and p e r c e iv e d a " c r im e p r o b l e m ," the c it i z e n r y m ig h t d e m a n d m o r e p r o t e c t io n .. C o n v e r s e ly , c o m m u n itie s w ith l i t t l e. c r im e w o u ld p ro b a b ly n o t a llo c a te m a n y r e s o u r c e s f o r la w e n fo r c e m e n t. B y o n ly c o n s id e r in g the d e m a n d f o r la w e n fo r c e m e n t, one w o u ld e x p e c t to o b s e rv e a p o s itiv e r e la tio n s h ip b e tw e e n c r im e and la w e n fo r c e m e n t. C o n s id e rin g o n ly the e ffe c ts on c r im e o f the d e m a n d fo r la w e n fo r c e m e n t, an in c r e a s e in c r im e w i l l r a is e the d e m a n d f o r la w e n fo r c e m e n t.. A n d , c o n s id e r in g o n ly th e s u p p ly - o f-o f fe n s e s fu n c tio n , the. r e la tio n s h ip b e tw e e n c r im e and la w e n fo r c e m e n t is n e g a tiv e .. B o th o f. th e s e e ffe c ts a r e o b s e r v e d to g e th e r w h e n a n a ly z in g the r e la tio n s h ip b e tw e e n c r im e and la w e n fo r c e m e n t. A t h ir d f a c t o r w h ic h a ffe c ts the o b s e rv e d r e la tio n s h ip b e tw e e n c r im e and la w e n fo r c e m e n t is the s u p p ly o f la w e n fo r c e m e n t e f f e c t .. H o ld in g. c r im i n a l ju s t ic e in p u ts c o n s ta n t, an in c r e a s e in th e n u m b e r o f c r im e s c o m m itte d is e x p e c te d to re d u c e th e le v e l o f la w e n fo r c e m e n t p r o v id e d . F o r e x a m p le , in th e c a s e o f a r r e s t s , h o ld in g the s iz e and q u a lit y o f the p o lic e fo r c e c o n s ta n t, an in c r e a s e in the n u m b e r o f c r im e s c o m m itte d.

(46) 36. w i l l re d u c e the r a t e o f a r r e s t s p e r c r im e c o m m it te d . T h e s im u lta n e o u s n a tu r e o f the r e la tio n s h ip b e tw e e n the s u p p ly o f la w e n fo r c e m e n t, the d e m a n d f o r la w e n fo r c e m e n t and the s u p p ly o f o ffe n s e s m u s t be ta k e n into a c c o u n t b e fo re in te r p r e t in g the r e s u lts o f the te s ts o f h y p o th e s e s in e q u a tio n ( 2 . 2 ) . T h e la w e n fo r c e m e n t v a r ia b le s in e q u a tio n ( 2 . 4 ) w h ic h c o u ld be r e la t e d to the c r im e r a te in a s im u lta n e o u s fa s h io n a r e :. 4. >, the a v e r a g e. p r o b a b ilit y o f a r r e s t f o r one o ffe n s e ; Q , the a v e r a g e p r o b a b ilit y o f c o n v ic tio n g iv e n a r r e s t ; R , the a v e r a g e p r o b a b ility o f in c a r c e r a tio n g iv e n c o n v ic tio n ; F a , the c o s t, to the o f f e n d e r , o f b e in g a r r e s t e d ; F 13, th e c o s t to the o ffe n d e r o f b e in g in c a r c e r a t e d , n e t o f a r r e s t and c o n v ic ­ tio n c o s ts . B e lo w is a b r i e f d e s c r ip tio n o f s u p p ly and d e m a n d c o n s id e r a tio n s f o r the la w e n fo r c e m e n t v a r ia b le s . T H E P R O B A B IL IT Y O F A R R E S T T h e s u p p ly o f o ffe n s e s fu n c tio n can be s im p ly r e p r e s e n te d by: (2 .5 ) w h e re. (C /N ) = f ( $ ,x ) 4. ) is the p r o b a b ilit y o f a r r e s t f o r one o f f e n s e ; x r e f e r s to a ll o th e r. v a r i a b l e s , and w e h y p o th e s iz e th a t <$(C/N)/§C$> < O .. T h e dem and fo r. p o lic e in p u ts c an be s im p ly r e p r e s e n te d by: (2 . 6 ). r = h (C /N ,y ). w h e r e r is s o m e m e a s u r e o f p o lic e in p u ts , C / N is th e c r i m e m e a s u r e and y is a c o m p o s ite o f a ll o th e r v a r ia b le s w h ic h a ffe c t the d e m a n d fo r.

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