A STRUCTURAL MODEL OF CRIME AND INEQUALITY IN COLOMBIA
Franc¸ois Bourguignon
DELTA, The World Bank, and Ecole Normale Supe´ rieure
Fabio Sanchez
CEDE, Universidad de los Audes
Jairo Nun˜ez
CEDE, Universidad de los Audes
Abstract Economic theory suggests that inequality should in uence crime positively. Yet, the evidence in favor of that hypothesis is weak. Pure cross-sectional analyses show signi cant positive effects but cannot control for xed effects. Time series and panel data point to a variety of results, but few turn out being signi cant. The hypothesis maintained in this paper is that it is a speci c part of the distribution, rather than the overall distribution as summarized by conventional inequality measures, that is most likely to in uence the rate of (property) crime in a given society. Using a simple theoretical model and panel data in seven Colombian cities over a fteen-year period, a structural model is proposed that permits identifying the precise segment of the population whose relative income best explains time changes in crime. (JEL: K42, D63, O15)
1. Introduction Inequality always ranked high among the potential economic determinants of property crime. The argument justifying this hypothesis is simple. Other things being equal, and in particular the probability of detection, the expected gain from crime may be taken as proportional to the mean income of the society under consideration, whereas the cost depends, through the opportunity cost of time and punishment in case of detection, on the potential income of the would-be criminal. Criminals in a society are thus likely to have a potential income at some distance below the mean income of society. It then follows that there will be more criminals the more people below this relative income threshold, or equivalently the more inequality there will be in society. 1 Acknowledgments: Paper presented at the Congress of the European Economic Association, Venice, August 2002. Views expressed in this paper are those of the authors and should not be attributed to the World Bank or any af liated organization. E-mail addresses: Bourguignon: Bourg@java.ens.fr; Nun˜ez: jnunez@minprotecciousocia l.gov.co; Sanchez: Fasunche@uniandes.edu 1. For a statement and a critique of this argument see Deutsch et al. (1992).
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The empirical evidence on such a relationship between crime and inequality is mixed. In what probably was the rst empirical paper on the economics of crime, Ehrlich (1973) found a signi cant relationship between the crime rate and the share of the population below half the median income across the United States. That cross-sectional relationship was more or less systematically con rmed by further work using different measures of inequality—see for instance Freeman (1996). Time series evidence is much weaker, however. Allen (1996) reports on several largely inconclusive studies of the aggregate crime rate in the United States, whereas Freeman (1996) mentions that no signi cant effect was found in a cross-section of time series for various metropolitan areas in the United States when controlling for xed effects. Entorf and Spiegler (2000) found some evidence of inequality across German laender in a panel data analysis of crime in Germany, but they do not consider inequality within laender. Cross-country evidence leads to analogous conclusions. Pure crosscountry data reveal a signi cant positive relationship between crime rates and various inequality measures, whether one uses of cial crime rates collected by the UN—Fajnzylber et al. (2002)— or data coming from victimization surveys—Soares (2000). 2 International panel data are dif cult to put together because of comparability problems across countries and over time. Fajnzylber et al. (2002) found the Gini coef cient not signi cant when using GMM on differenced time series, but they obtained a signi cant result when using an Arellano-Bower system estimator. An issue that should arise when trying to nd some empirical support for the basic hypothesis of the crime-inequality relationship is that of the representation of inequality one should use. Suppose that some redistribution takes place at the top of the distribution so that inequality in a given country diminishes. On the basis of the argument in the introduction of this paper, should it be expected that the crime rate will come down? Probably not if the redistribution is concerned with people high enough on the income scale not to be tempted by illegal activities. Thus, not any change in the distribution may cause a change in the crime rate and it is quite likely that using different inequality measures in regressions that try to explain the crime rate will lead to different results. The Gini coef cient might lead to a given result, but the Theil measure or the logarithmic deviation might lead to another one. The present paper proposes a structural model that precisely permits identifying what part of the distribution is the most relevant to explain crime over a given time period. 3 The empirical relevance of that model is tested by studying the evolution of criminality in the seven largest cities of Colombia over the last fteen years. In the perspective of the relationship between crime and inequality, Colombia indeed appears as an interesting case study. It is a high criminality 2. A short survey of this evidence is given in Gartner (2000). See also Eiden (1997). 3. A longer version of this paper with results of more standard econometric analysis of crime rates is Bourguignon, Nunez, and Sanchez (2002).
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country by international standards with rather large variations of the crime rate and inequality ever since the mid-1980s—see Sanchez and Nunez (1997). The paper is organized as follows. A theoretical model is developed in the next section which leads to a structural econometric speci cation of the way distributional characteristics, together with other crime determinants, should affect the crime rate. Section 3 is devoted to the estimation of that model in the Colombian case and to the identi cation of that part of the distribution of income that turns out to be important for property crime. According to the estimates found, would-be criminals in Colombia appear to be recruited in that part of the population whose standard of living lies below 80 percent of the mean. The proportion of people within that range and their mean income, relative to the overall population mean, thus are what matters for explaining variations in the crime rate. 2. A Structural Econometric Model of the Crime Rate Following the argument in Bourguignon (2000), criminals are assumed to be those people whose economic resources, y, are below a threshold that depends on the expected net pro t from crime. The latter corresponds to the expected loot, x, corrected by a term, a, that depends on the probability of being caught, p, the sanction, q, if caught— expressed as a proportion of economic resources—and a set of variables summarizing the attitude of the individual and/or the social group he/she belongs to with respect to crime. Without loss of generality, all these variables may be summarized by a single parameter, h, which will be referred to as honesty. The statistical distribution of that attribute is supposed to be de ned on the support [h1, h2] with density function g¼. It is also assumed to be independent of economic resources. Formalizing the previous argument, an individual with honesty h will get into crime if y # x z a( p, q, h). The proportion of criminals among people with honesty h is therefore given by F[ x z a( p, q, h)] where F¼ is the cumulative function of the distribution of economic resources. Aggregating over all levels of honesty leads then to the following overall crime rate in the population C5Nz
E
h2
F@ x z a ~ p, q, h!# g~h!dh,
(1)
h1
where N is a scale factor that stands for the number of crimes committed by a criminal, which is assumed here to be constant. 4 4. This may be a restrictive assumption. In theory, variations in the number of crimes by criminal may matter as much as the number of criminals to explain overall changes in the crime rate. See Deutsch et al. (1992).
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Let Ct be the crime rate observed at period t (51, 2, . . . T) and let Xt be a vector of explanatory variables that excludes inequality summary measures. The standard econometric models of crime essentially relates the crime rate, Ct, to these variables Xt through a linear speci cation that also includes a summary measure of inequality, G(Ft). The speci cation thus is: (2)
C t 5 X t z b 1 g z G~F t ! 1 u t
where b and g are coef cients to be estimated and ut is the usual error term.5 There is no rigorous justi cation behind speci cation (2). The point is simply to identify variables that tend to covary with the crime rate and which might be considered as causes of crime. Yet, if the microeconomic model (1) seems a good representation of aggregate criminal behavior, then the adequate speci cation should be of type (1) rather than (2). To see what this implies, consider that the explanatory variables, Xt, may now be taken as the direct or indirect observed determinants of the net pro t from crime, conditionally on h at time t. If the relationship is linear, this may be represented as: @ x z a~ p, q, h!# t 5 y t z @bX t 1 m t 1 h#
(3)
In that expression, it is reasonably assumed that the net expected pro t from crime is proportional to the mean income of the population, yt with a coef cient of proportionality depending on observed determinants of crime, Xt, the honesty parameter, h, and a set of unobserved determinants with total effect mt. b is a set of coef cients to be estimated. It is the analogous of b in the standard model.6 Substituting (3) into (1) leads to: Ct 5 N z
E
h2
F t ~bX t 1 m t 1 h! g~h!dh
(4)
h1
where Ft¼ now stands for the distribution of income normalized by the mean income of the population—i.e., the distribution of relative incomes. Under the assumption that the time variations of Xt b 1 mt are relatively small, a linear approximation may be used. It leads to: C t /N >
E
h2
F t ~h! g~h!dh 1 @X t b 1 m t # h1
E
h2
f t ~h! g~h!dh
(5)
h1
where ft¼ stands for the density of the relative income distribution at time t. It now remains to make an assumption on the distribution of the honesty 5. For estimation of models of type (3) see the rich list of references in Gartner (2000). 6. There is no need to affect a coef cient to the honesty parameter h, since its scale has been left unspeci ed. Note that like bX and m, (3) implies that h necessarily has the dimension of a relative income.
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parameter, h. Without any information available, this assumption is necessarily arbitrary. For the sake of simplicity, it is assumed in what follows that h is uniformly distributed over the interval [h1, h2].7 Under that assumption, it may is easily shown that (5) becomes: C t /N > f t~h 1 , h 2 ! 1 c t ~h 1 , h 2 !@X t b 1 m t #
(6)
f t ~h 1 , h 2 ! 5 F t ~h 2 ! 2 c t @Y t ~h 1 , h 2 ! 2 h 1 #
(7)
with:
ct ~h 1 , h 2 ! 5
F t ~h 2 ! 2 F t ~h 1 ! h2 2 h1
(8)
where Yt(u, v) stands for the mean relative income of all people whose income is between u and v times the overall population mean at time t. Expression (7) may be interpreted as a measure of inequality of the distribution of income that focuses exclusively on people whose income lies in the interval [h1, h2]. In effect, the function wt¼ combines various types of information on the shape of the Lorenz curve in that interval: how many people are there, and how poor are they. It increases when the proportion of people below the relative income limit, h2, increases or when the mean relative income of people in the interval [h1, h2] falls. It may therefore be interpreted as a measure of inequality. Speci cation (6) thus is similar to the standard mode where the crime rate is supposed to depend linearly on some measure of inequality and on the explanatory variables, Xt. There is a big difference between the two speci cations, though. It is that the effect of the explanatory variables, Xt, on the crime rate in (6) is ltered by a term that depends on the distribution of income at time t. This term, ct¼, is the average density of the distribution within the interval [h1, h2]. In other words, crime determinants are now assumed to affect the crime rate proportionally to the number of people who lie in an income range low enough to make them potential criminals. This structural speci cation essentially depends on the two parameters, h1 and h2, the role of which is essentially to de ne what part of the distribution of income really matters for explaining variations in the crime rate. The point is now to estimate econometrically these two parameters, as well as the coef cients, b, that represent the way the explanatory variables, X, affect criminal behavior. 7. Other distributional assumptions are possible but they would lead to much more complicated numerical calculations.
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3. Estimation Procedure and Results The preceding model was estimated on Colombian data. The dependent variable is the annual rate of property crime per 100,000 inhabitants reported to the police in each of the seven largest cities from 1986 to 1998. Distribution data are from the September survey of the “Encuestas de hogares.” Income distribution is de ned on household income per capita with households being weighted by size.8 Labor-market related explanatory variables are extracted from the same surveys, whereas crime environment variables were collected from various sources. A standard model of type (2) was rst estimated with alternative summary inequality measures, G(Ft)— e.g., Gini, Theil, and Atkinson measures with various inequality aversion parameters. The model also included several labormarket indicators, the homicide detection rate, guerilla and drug activity, year and city dummies and the lagged crime rate to account for crime hysteresis. 9 Results were disappointing. Few variables were signi cant, outside xed effects. Income inequality was barely signi cant, although more so for speci c inequality measures, like the Gini or Atkinson with low inequality aversion. Yet, this variety of results con rmed at the same time the basic intuition that the way inequality is represented matters very much for explaining crime.10 Coming back now to the structural model discussed in the preceding section, one may rst notice that, after adding the index i to account for the city dimension of the data, the following change of variable: Z it 5
C it /N 2 f it ~h 1 , h 2 ! cit ~h 1 , h 2 !
(9)
leads to a standard linear regression speci cation. From (6) it indeed comes that the ‘transformed crime rate,’ Zit, is given by: Z it 5 b z X it 1 m it
(10)
Estimates of coef cients b may thus be obtained by standard ordinary least squares. But the problem is to nd estimates of the coef cients N, h1, and h2 allowing for the computation of the ‘transformed crime rate’ for each city i and each time period t. This is done by iterating on a grid of values and selecting the combination that minimizes the sum of square residuals of equation (10). Under usual normality and i.i.d. assumptions of the random terms, mit, it may be shown that this is equivalent to standard maximum likelihood estimation. Of course, this procedure relies on the implicit assumption that the distribution of the honesty parameter is the same across cities and time periods. This iterative computing intensive technique—note that the variables fit 8. Various alternative de nitions were used but that one worked better. 9. For a theoretical justi cation of this hysteresis effect, see Sah (1991). 10. These results are discussed at length in Bourguignon, Nunez, and Sanchez (2002).
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Journal of the European Economic Association TABLE 1. REGRESSION RESULTS
Explanatory variables Youth unemployment rate Participation rate Log real wage Log population Log mean schooling Log homicide detection rate Guerilla activity (dummy) Log drug income per inhabitant Lagged Log (transformed) crime rate R2 Mean square error Number of observations
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PANEL CRIME RATE: TRANSFORMED CRIME RATESa Coef cients
Standard error
T-statistic
7.959 25.688 1.064 22.215 2.985 0.257 0.008 1.224 0.313
2.65 3.53 0.67 3.15 1.59 0.15 0.00 0.40 0.15 0.831 0.134 84
3.00 21.61 1.58 20.70 1.88 1.77 2.97 3.02 2.15
a
See estimation method in text. Other variables included are dummies for cities and years, coef cients are not reported. Dependent variable is the log of transformed crime rate (h1 5 .0; h2 5 .8).
and cit must be evaluated on the basis of micro distribution data for each city and each time period at each iteration—was applied to the set of available data. The lagged transformed crime rate, Zit21, was introduced in (10) to account for hysteresis whereas xed city and year effects were controlled for. Finally, the model was estimated in logarithms rather than in absolute values. Iterating on N, it turned out that an order of magnitude of ten crimes per year and per criminal gave the best results. This value was kept in all the subsequent estimation work. Iterating on (h1, h2) then led to the best estimates h1 5 0, h2 5 .8, although there was practically no difference in the sum of the squared residuals of (10) between h1 5 0 and h1 5 .2 or .3. The latter property clearly re ects the very low density of the income distribution near zero income per capita. According to these results, that part of the population which most matters for time uctuations in the crime rate thus are those individuals whose welfare lies below 80 percent of the mean of the whole population. It is the proportion of those people in the population, their mean relative income and the average density of the distribution in that relative income range that better explains time variations in the crime rate within cities. On average over all observations, approximately 60 percent of the population is in that relative income range. Table 1 shows the OLS estimates of the coef cients in equation (10) when the transformation of the crime rate is made through (9) on the basis of h1 5 0, h2 5 0.8. The reported variance of the estimates is that given by OLS, which tends to under-estimate the actual variance because the imprecision coming from the estimation of (h1, h2) is not taken into account. 11 Unlike the results found with the standard models, the regression run on the (log) transformed crime rates appear rather satisfactory, even though not always 11. A more rigorous estimation of this covariance matrix could be obtained by using standard maximum likelihood techniques.
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signi cant or with the expected sign. These results look especially attractive when one takes into account that the estimation procedure controls for xed city and year effects, which means that all effects are actually estimated on the basis of within city time variations of the crime rate and the distribution of income. The coef cients that appear in Table 1 shed some interesting light on the determinants of uctuations in the crime rate in Colombia, besides of course those uctuations that may be explained by distributional changes and are already included in the transformation (9) of the crime rate. The signi cance of this transformation may be evaluated by comparing the sum of squared residuals of equation (10) when the left-hand side variable is the crime rate and when it is the ‘transformed’ crime rate. In effect the former is the same as the latter with h1 5 0, h2 5 `. The difference in favor of the transformed model and therefore the role of inequality into crime appears to be strongly signi cant. Some other variables are signi cant, too, while others are ambiguous or have surprising effects. The youth unemployment rate has the expected positive and signi cant effect on the crime rate. Note that the coef cient reported in Table 1 overestimates the true quasi-elasticity of the crime rate, because the dependent variable is not the logarithm of the crime rate itself but of the crime rate minus the variable wit¼. As the latter is positive, the quasi-elasticity of the crime rate with respect to youth unemployment is actually smaller than the reported coef cient. The effect of the participation rate is more or less the opposite of the effect of unemployment, although just below the limit of statistical signi cance. To the extent that variations in this participation rate may account for changes in disguised unemployment, this could be expected. Finally, the (log) real wage is estimated to have a positive effect on crime. Although not statistically signi cant, this effect is somewhat surprising since one would have expected a priori that any evolution favorable to labor in general would reduce crime. However, it must be taken into account that the change in the real wage is to be interpreted at constant unemployment and participation rates, as well as at constant distribution of income among the poorest 60 percent. It is not clear what its effect should be under these conditions. Another possible interpretation is that a change in the real wage increases the potential loot of property crime by more than the average income, because wage workers are predominantly in the top 40 percent of the distribution. As far as general population variables are concerned, it turns out that autonomous population changes in cities have no signi cant impact on crime whereas those in the mean schooling of the working age population is almost signi cant and positive—recall that the effects of the long-run common trend in urban population and its mean schooling are accounted for by the year dummies included in the regression. Unlike what might have been expected, these results suggest that the migration movements which may be responsible for deviations of cities’ population growth from a common trend have no impact on crime. In theory, this should be the case only if crime prevention and police expenditures
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were growing at the same rate as population. Unfortunately, available information in this respect is extremely partial and, as will be seen next, not necessarily consistent. With respect to the positive effect of mean schooling, one possible interpretation is that, after controlling for the size of the population, changes in that variable correspond to changes in the demographic structure of the population. As age is probably the characteristic most directly linked to schooling, this variable might show the effect on crime of the population becoming younger or older in a way that departs from the trend common to all cities.12 Two out of the three crime environment variables are signi cant and have the expected sign. Other things being the same, drug activity in the city and guerilla activity in the region may be supposed to divert police attention from other criminals, which should increase the propensity to commit property crime. This is what is observed. More problematic is the positive and almost signi cant sign obtained for the homicide detection rate. This variable was supposed to be a proxy for the level of police activity relatively to the number of homicides. 13 The positive sign obtained in the regression seems to imply some kind of crowding-out of property crime detection by arrests for homicides—i.e., the same diversion phenomenon that is behind the guerilla and drug coef cients. Of course, it would be much better to use directly the size of the police force or expenditures on police as an explanatory variable. Unfortunately, this information is not available for the complete period under analysis. The last variable in the list is the lagged (log) transformed crime rate. The coef cient is positive and signi cant. It suggests some sizable hysteresis of changes in the crime rate. According to that coef cient, 30 percent of a shock in the crime rate at a point of time is carried over to the next year. This also implies that long-run changes in the crime rate associated with permanent changes in the explanatory variables are higher than the coef cients reported in Table 1 by approximately 40 percent. 4. Conclusion This paper investigated an original structural speci cation of an econometric model of the crime rate in the seven largest cities of Colombia that takes into account the simple fact that only a speci c part of the income distribution should matter in determining the aggregate crime rate in a society. This speci cation led to the conclusion that would be criminals in Colombia were to be found among those people living in households where income per capita was below 80 12. Of course, this effect is to be taken conditionally on the distribution, so that the effect of a higher mean level of schooling on crime cannot be that effect that goes through the distribution. 13. Assuming that the number of arrests for homicides, A, depends on the number, H, of homicides and the size, P, of the police force through the following simple ‘production ’ function: A 5 HmP12 m. Then the detection rate A/H is given by (P/H)12 m.
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percent of the mean. A corollary of this is that distributional changes among those people who are above that limit are likely to have no signi cant in uence on the crime rate. Interestingly enough, it turns out that this structural speci cation leads to an overall explanation of crime that appears better than that of the standard model where inequality enters additively and is described by conventional inequality measures. References Allen, R. (1996). “Socioeconomic Conditions and Property Crime: A Comprehensive Review and Test of the Professional Literature.” American Journal of Economics and Sociology, 55(3), pp. 293–308. Bourguignon, F. (2000). “Crime, Violence and Inequitable Development.” In Annual Bank Conference in Development Economics: 1999, edited by B. Pleskovic and J. Stiglitz. Washington, D.C.: The World Bank. Bourguignon, F., J. Nunez, and F. Sanchez (2002). “Crime Determinants in Colombia: Reduced Form vs Structural Form Estimates.” Mimeo. Cede, Universidad de los Andes. Deutsch, J., U. Spiegel, and J. Templeman (1992). “Crime and Income Inequality: An Economic Approach.” Atlantic Economic Journal, 20, pp. 46 –54. Ehrlich, I. (1973). “Participation in Illegitimate Activities: A Theoretical and Empirical Investigation.” Journal of Political Economy, 81, pp. 521–565. Eiden, G. (1997). “The Economics of Crime: Survey and Bibliography,” Working paper in Law and Economics, University of Oslo. Entorf, H. and H. Spengler (2000). “Socioeconomic and Demographic Factors of Crime in Germany: Evidence from Panel Data of the German States.” International Review of Law and Economics, 20(1), pp. 75–106. Fajnzylber, P., D. Lederman, and N. Loayza (2002). “What Causes Violent Crime?” European Economic Review, 46(7), pp. 1323–1357. Freeman, R. (1996). “Why Do So Many Young American Men Commit Crimes and What Might We Do About It?” Journal of Economic Perspectives, 10(1), pp. 25– 42. Gartner, R. (2000). “Cross-Cultural Aspects of Interpersonal Violence: A Review of International Empirical Evidence,” mimeo. University of Toronto, Canada. Sah, R. (1991). “Social Osmosis and Patterns of Crime.” Journal of Political Economy, 99, pp. 1272–1295. Sanchez, F. and J. Nunez (1997). “Desigualdad y crimen en Colombia,” Mimeo, Cede, Universidad de los Andes, Colombia. Soares, R. Reis, (2000). “Development Crime and Punishment: Accounting for International Differences in Crime,” mimeo. Department of Economics, University of Chicago.