# Mathematical Model of Crime and Unemployment Text Only Version

#### 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 . 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. 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 .

Cambel et al. offer a differential methodological approach to the process by which crime rates changes over time . Their approaches is similar that used in mathematical biology to describe how potential epidemics are either spread or contained in a population  . 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 .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 . 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

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