Mathematical Model of Crime and Unemployment

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Mathematical Model of Crime and Unemployment

Ruhul Amin

Associate Professor

Department of Mathematics, West Goalpara College, Assam, India.

Abstract:- In this paper, we are interested in possible contribution of mathematical modeling of crime. The concentration of criminal activities is not proportional in every area. Because criminal activities depend on socio-economic factors like population densities, unemployment, literacy rate, per capita income, schedule castes and schedule tribes etc. There is a correlation between the volume of crime and these socio-economic factors. The equation of the line of regression is established to interpret the nature of relationship between crimes and unemployment.

Keywords: Volume of crime, Correlation, Regression

INTRODUCTION

This is the simplest mathematical model which can be applied in many cases where relationship among variables actually exists. As for example, the relationship between the number of criminal convictions and the number of unemployed in a particular span of time in a society can be established.

Similarly, volume of crime and literacy, volume of crime and scheduled castes and tribes and volume of crime and per capita income in a certain area also exhibit some relationship.

The above relationship can be expressed in the form of an equation connecting the dependent variable Y and one independent variable X. More precisely, the equation takes the form

Y C BX

(1)

This is called the simplest regression equation, where C and B are said to be the regression coefficients.

Similarly, if more than one variable are considered then the regression equation can take another form. In particular, we already know that criminal activities are somewhat related with population density, per capita income, literacy rate, unemployment and proportion of scheduled castes and tribes etc.

Then the regression equation takes the following form:-

Y C B1X1 B2 X2 B3 X3 B4 X4 B5 X5

(2)

Where Y =Volume of crime per million population

X1 = Population density; X2 = Per capita income; X3 = Literacy rate;

X4 = Unemployment;

X5 = Percentage of scheduled castes and tribes; and C and Bi s are regression coefficients.

It is remarkable that the negative sign before the third and fourth terms in relation (2) indicates that the volume of crime reduces for increase of per capita income and literacy rate. Hence there is a negative relationship.

Per capita income and literacy have apparently an inverse relation (negative relation) with crime which suggests that as income levels and literacy rise, crime tends to decrease. The hypothesis is supported to the extent that the bulk of reported crime can be traced to the economically deprived sections and the illiterate on whom the full impact of law-enforcement is felt. It does not necessarily absolve the affluent and the literate from criminality which may assume more subtle forms which do not form part of Penal Code and also have the capacity to defy conventional law-enforcement [7]. On the other hand, unemployment has significantly positive correlation with crime, followed by population density. Although not very significant, the percentage of scheduled castes and tribes appears to have some positive relationship. The marginal significance of this factor can be ascribed to the fairly uniform proportion of this segment in all states. The relationship between the major socio-economic variables hold good for nearly high percentage of crime under the Indian Penal Code and establishes unemployment as the most significant criminogenic factor.

The above equation no. (2) suggests that the relationship between two variables is such as a change in one variable results in a positive or negative change in the other, also greater change in one variable results in a corresponding greater change in the other, is known as correlation.

LITERATURE REVIEW

The modern mathematical model on crime was initiated by G.S.Beckers model of rational criminal activity[1]. Becker assumed a social loss function which includes costs and benefits of crime. Its minimization determines how many resources and how much punishment should be used to enforce the law.

Isaac Ehrlich developed a model where crime as considered as goods and individuals make rational decisions in the market of crime with a hypothesis- a person commits a crime if his expected utility exceeds the utility he could get with legal activities [2][3][4].

Cambel et al. offer a differential methodological approach to the process by which crime rates changes over time [5]. Their approaches is similar that used in mathematical biology to describe how potential epidemics are either spread or contained in a population [8] . Cambel at al. considered the criminal activity as an epidemic problem. They described the dynamic of the crime rate growth by some differential equations.

Another model describing the interaction of three sociological species, termed as Owners, Criminals and Security Guards [6].In this model [Juan C. Nuno et al.] Criminal is the predator for the species Owners and Security Guards is the predator for the species Criminals. On the basis of pre-predator model they propose a system of three ordinary differential equations to account for the dynamics of Owners, Criminals and Security Guards.

Some modeler tried to relate crime rates to possible explicative variables through linear regressions [9]. The models assume that crime rate = f(explicative variables), where f(.) is a linear function and the explicative variables considered as average income, gender inequality, age, education level, race etc.

Preliminaries

  1. If x and y are two random variables then the correlation coefficients between x and y is denoted by r or rxy and is defined by

    x y xi yi

    r

    i

    i

    x2

    i i

    i

    i

    x 2

    N

    N

    i

    i

    y2

    y 2

    i

    i

    N

    where, 1 r 1, r has not units and is a mere number

    If r = 1, then there exist a perfect and positive correlation between the variables x and y. if r 1, then there exist a perfect

    and negative correlation between the variables, x and y. The above relation is known as Karl Pearsons correlation coefficients.

  2. The equation of the line of regression of y over x is

y y r y (x x )

x

Similarly, the equation of the line of regression of x over y is

x x r x y y

y

where x and y are the means of the values of x and y respectively. These two relations are known as Equation of line of regressions.

We have already discussed that the criminal activities are related with several factors such as population density, per capita income, literacy rate, unemployment etc.

These factors can be correlated positively or negatively or partially with the help of the regression equation (2). We can apply these mathematical or statistical concepts for the analysis of crime pattern.

Relationship between crime and unemployment

The figure in the following Table- 1 gives the number of unemployed and volume of crime in the states of India for the year 1971. We have to find out the coefficient of correlation for the given data. Also we shall find the equation of the line of regression to interpret the nature of relationship between crime and unemployment.

Volume of crime and number of unemployment of India, 1971 Table -1

Sl. No.

Nae of States

Number of Unemployed = x

(Per Thousand Population)

Volume of Crime

(Per One Lakh Population)

1

Andhra Pradesh

336

106

2

Orissa

135

138

3

Karnataka

270

124

4

Tamil Nadu

459

144

5

Bihar

420

147

6

Uttar Pradesh

531

166

7

Gujarat

171

121

8

Maharashtra

430

195

9

Assam

789

175

10

Kerala

357

139

11

West Bengal

868

176

12

Haryana

100

82

13

Punjab

118

84

14

Rajasthan

139

142

15

Madhya Pradesh

315

211

16

Himachal Pradesh

45

73

17

Jammu & Kashmir

25

119

18

Tripura

30

114

19

Manipur

38

180

Source:

  1. Crime in India,Ministry of Home Affairs, New Delhi

  2. Statistical Abstracts, Central Statistical Organization, Government of India, New Delhi.

  3. Labour Bureau, Government of India.

Table -2

Calculation for correlation coefficient

Sl. No.

No. of Unemployment = x

Volume of Crime = y

(Per One Lakh Population)

x

u

u 2

y

v

v 2

uv

1

336

-21

441

106

-33

1089

693

2

135

-222

49284

138

-1

1

222

3

270

-87

7569

124

-15

225

1305

4

459

102

10404

144

5

25

510

5

420

63

3969

147

8

64

504

6

531

174

30276

166

27

729

4698

7

171

-186

34596

121

-18

324

3348

8

430

73

5329

195

56

3136

4088

9

789

432

186624

175

36

1296

15552

10

357

0

0

139

0

0

0

11

868

511

261121

176

37

1369

18907

12

100

-257

66049

82

-57

3249

14649

13

118

-239

57121

84

-55

3025

13145

14

139

-218

47524

142

3

9

-654

15

315

-42

1764

211

72

5184

-3024

16

45

-312

97344

73

-66

4356

20592

17

25

-332

110224

119

-20

400

6640

18

30

-327

106929

114

-25

625

8175

19

38

-319

101761

180

41

1681

-13079

u

1207

u2

1078329

v

5

v2

26787

uv

96271

Correlation coefficients

uv uv

r n

u2

u 2

v2

v2

n n

96271 (1207) (5)

19

0.586

1078329

(1207)2

26787

(5)2

19 19

Now, Regression coefficient of x on y b r x

r u

xy

uv u v

n

v2

y v

3.58

v2

n

Similarly, Regression coefficient of y on x

b r y r v

yx

x u

uv u v

n

u 2

0.096

0.10

u2

n

Now, the equation to the line of regression of x over y is

x x r x ( y y )

y

x x 3.58( y y)

(3)

Arithmetic average of unemployment

  1. assumed average u

    n

    357 1207 293.47

    19

    and Arithmatic average of volume of crime

  2. assumed average v

n

139 5

19

138.74

So the equation (2.3) becomes

x 293.47 3.58( y 138.74)

x 203.22 3.58y

(4)

and the equation to the line of Regression of y over x is

y y r y (x x )

x

y 138.74 0.10(x 293.47)

y 109.39 0.10x

(5)

150

150

Volume of crime(y-axis)

Volume of crime(y-axis)

These two regression equations show that as the unemployment increases the volume of crime also increases.

Correlation between unemployment and crime

Correlation between unemployment and crime

250

250

200

200

50

50

0

0

0

100

200

300

400

500

600

700

800

900

1000

0

100

200

300

400

500

600

700

800

900

1000

Nos. of unemployed(x-axis)

Nos. of unemployed(x-axis)

100

100

Fig. -1 Correlation between unemployment and crime

Fig. -1 Shows r 0 for standard data given in the Table -1

The Fig.-1 exhibits that as unemployment increases the volume of crime also increases.

CONCLUSION

The positive correlation coefficient r 0 shows that, the volume o crime increases as the unemployment increases. The two

equations of regression (4) and (5) represent straight line which exhibit that as unemployment increases the volume of crime also increases. The Fig.-1 also exhibits the same interpretation.

ACKNOWLEDGMENTS

The author thanks Dr. Atowar Rahman,Associate Professor of the Department of mathematics, B.P. Chaliha College, Nagarbera for many suggestions.

REFERENCES

  1. Becker, G.S. (1968) Crime and Punishment: An Economic Approach, Journal of Political Economy 76: 169-217.

  2. Ehrlich, I. (1973) Participation in Illegitimate Activities: A Theoretical and Empirical Investigation. Journal of political Economy, Vol. 18, 521-565.

  3. Ehrlich. I. (1975) The Deterrent Effect of Capital Punishment: A Question of Life and Death. American Economic Review 65: 397-417.

  4. Ehrlich, I. (1996) Crime, Punishment and the Market for Offences, Journal of Economic Perspectives, Vol.10, No.1, pp. 43-67.

  5. Cambel, M. & Ormerod, P. (1998) Social interaction and the dynamic of crime.

  6. Nuno, J. C., Herrero, M. A. & Primcerrio, M. (2008) A triangle model of criminality. PACS: 387, 2926-2936.

  7. Rao, S. Venu Gopal (1981), Dynamics of Crime Spatial and Socio-Economic Aspects of Crime in India IIPA, New Delhi.

  8. Murray, J.D.(1990) Mathematical Biology: Springer Verlag, Barlin

  9. Gordon, M.B.(2010) Random walk in literature on criminality: A partial and critical view on some statistical analyses and modeling approaches Euro. Jnl. Of Applied Mathematics, Vol.21, pp.325-348

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