Oscillatory vs. stationary enforcement of law

Oscillatory vs. Stationary Enforcement of Law

Defx~rtment ofEconomics, Nutional Tuiwan University und tnstitute Arademia Sinica, Nankang, ‘Ihipei, Taiwan 11529

ofEconomics,

Introduction Despite the fact that the impact of decision-interdependency among individuals has been thoroughly analyzed by Jones (I 976), Schelling (1978), and Schotter (198 1) in various fields of economic research, the policy implication of illegal decision interdcpendency among potential criminals has not caught much attention of law and economics scholars. Lui (1986) and Chu (1990) argued that if most officials take bribes, in general it is more difficult to collect evidence from the colleagues of a suspect. As a result, the probability of conviction and the deterrence perception will be low. Benjamini and Maital (1985) and Schlicht (1985) noted that if most of one’s neighbors are evading taxes, the expected peer-group condemnation will be low if one does the same, which facilitates the decision of evading.’ In the above-mentioned research, one question left unanswered is how would the optimal enforcement structure be affected if potential criminals do have such decision interactions? The following two observations may help us answer the above question: First, it is not difticult to see that the reinforcing peer-group decision-inter-dependency among potential criminals tends to make a high crime rate higher, and similarly a low crime rate lower. Second, such a reinforcing momentum will not last forever because when the crime rate is very high, there usually exists public pressure asking for tightening the law enforcement in order to ensure social order, as described in Cooter anti Ulen (1988, p. 534). Similarly, when the crime rate is very low, people may also propose to relax the law enforcement in order to reduce the seemingly unnecessary enforcement cost. Thus, when the crime rate reaches its peak (trough), there is a natural tendency for it to go down (up), and therefore we may observe two coexistent oscillations in crime rates and in enforcement intensity. The above-mentioned coexistent oscillations of illegal activity and enforcement intensity, however, do not seem to be consistent with the conventional “optimal enforcement” argument.’ Indeed, when enforcement is tough, although the social order cost is low, the enforcement cost is high; whereas when enforcement is lax, although the cost of social order is high, the enforcement cost is low. This situation is not dissimilar to the well-known negative relationship between enforcement cost

304

Oscillatory 71s.stationary enforcement (flaw

and social order cost associated with most illegal or externality-creating activities. Given such a negative relationship, many law and economics scholars have suggested that there should exist an “optimal” enforcement level (somewhere in between the tough-lax extremes), which minimizes total social costs. If that is the case, oscillatory changes in the intensity of enforcement are clearly suboptimal. We notice that the traditional optimal enforcement argument was presented without taking into account the decision-interdependency among potential criminals; but the question is would the existence of such interdependency create any difference in the optimal enforcement structure: + rThe purpose of this paper is thus threefold: First, we set up a model to characterize one broad class of illegal decisioninterdependency and show that the macro index of illegal activities has a natural nonconvex dynamic pattern. Second, we explain why such a nonconvex pattern of crime rates will make the problem of optimal enforcement different from the conventional one. Third, we provide a possible efticiency justification for oscillatory law enforcement when there are illegal decision interactions among potential criminals. Beside this introductory part, this paper is arranged into three major sections. First we will present a “critical mass” model, which characterizes how a potential criminal affects and, at the same time, is affected by the attitude of others. It will be shown explicitly that the dynamic macro crime rate naturally has a nonconvex shape. Then we will discuss the impact of various law enforcement policies on the macro crime rate. Under some parametric specifications, it will be shown that every stationary enforcement policy can be dominated by .sorneoscillatory ones, and this result provides a possible justihcation for oscillatory law enforcement on the part of the government. Finally, the previous discussion is elaborated and possible future research directions are presented. A Simple Model of Traffic Order Evolution Person-to-person interactions are often modelled as multiple-player games. Even in the simplest case of two-person interactions, as Schelting (1976, p. 36) pointed out, there is no single mode of behavior that covers all cases. Different individual preferences correspond to different payoffs in a game matrix, which lead to distinct equilibrium patterns. In this paper we concentrate on a special class of interactions characterized by the model of critical muss. What the critical mass model involves is some activity that is self-sustaining once the measure of that activity passes a certain minimum level. For instance, one double-parks if many others do, one evades taxes if many of one’s friends do, one surges to the ticket window if most people do, and one crosses the lawn if plenty of others do? To make the following analysis more practical, it may be helpful to think of the macro “crime” rate as the number of trafhc violations, and individual drivers interact with each other in their attitude toward driving. The analysis can be easily applied to other models of critical mass. Individual Decisionmaking and the Mucro Crime Kate We shall consider a population with very large size N. Each member of the population randomly meets (one by one, say, at the crossroads) n others in each period. Because no one is in a position to know in advance whom he is going to come across,

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any ex-ante cooperative negotiation is out of the question. The interactions between any two individuals will be characterized by the following symmetric 2 X 2 game. There are two choices in everyone’s strategy space: I and II. Let us define strategy I as driving rudely and strategy II as driving courteously. The magnitude of payoffs is assumed to be as follows:

d>b,d>a>c Row chooser’s payoff

fi

listed

first.

penalty of adopting strategy In the above payoff matrix, P - r . 7~is the expected I, where n is the size of penalty on all detected rude drivers, and r is the probability of detection. First, let us consider the simple situation without enforcement and penalty (P = 0) and explain the relative magnitude of a, b, c, and d. The ideal situation is for both sides to drive courteously (max {u, 6, c, d} = d). If one expects the other driver to drive rudely, he would incline to requite like for like (a > c). Conversely, if one drives rudely but the driver he comes across happens to be courteous, he would be mortified for what he has done (d > b). Let 8, (t3,) be the probability that the row (column) player takes strategy I. There are three obvious Nash equilibria in this 2 x 2 game: (0,, 0,) = (1, I), (e,, 0,) = (0, 0), and (0,. Et,) = (x, x), where”

a-P

d -

d-b+p

d - (b - P)

xs

-c+d-(b-p)=a-c+d-b=a-rfd-b’

b + mr

(1)

Since one does not know in advance what strategies the others are going to adopt, he can only draw on his experience and make an educated guess. This, as we will see, gives rise to the mutual interactions among potential violators. Suppose there are p, proportion of people taking strategy I (and hence 1 - p, taking strategy II) at period t. One may come across n << N people on the street randomly, where N denotes the population size of the city. Thus, the number of people taking strategy I (denoted k) in this n-person sample will follow a binomial distribution:

Let z,, be the sample

mean

of this n-sample,

PL&t> x I $4) = phz, >

then nxT

I p,)

0

=1-i

n

v=,,

we have

pr(l

(2) -

p,)“--l =f(p,;x),

y

where x* is the largest integer no greater than 72~. Clearly, when p, = 0, the event j,, > x never happens, and hencef(O;x) = 0. Similarly,f(l;x) = 1. For the time being let us suppose, as Schelling (1978) did, that people are myopic in the sense that they use the sample mean they observe at period t to predict the proportion of strategy-I choosers in period t + 1.” Given this assumption, we show in Appendix 1 that in a large city (N is large), the dynamic evolution of p, is characterized by

P,+I

=

.f@,;x)

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Oscillatory 715.stationary enforcement Endogenous

Nonconvexity

of the Macro

of law Crime Rate

The functionf(p,;x) may have several possible shapes. First, if 0 < nx < 1, then x* = 0, andf(p,;x) = 1 - (1 - p,)“. This implies that dfydp, = n( 1 - p,)“-’ > 0, and d“7 dp: = -n(n - l)(l - p,)“-’ < 0. The second possibility is when n - 1 < nx < n so that x* = n - 1, then f function becomes f(p,;x) = p;. In this case it is easy to see thatfis increasing and convex. Both these cases are depicted in Figure 1. As we shall see in the next section, the value of x will be determined by the level of the government’s traffic law enforcement. Therefore, the above-mentioned two cases will stand only if the enforcement is extremely lax or tough. In what follows, we shall set aside these two extremes and concentrate on the third possibility: 1 c nx c n - 1, where we can rewriteJ‘(p,;x) as f(P,;x)

=

1 -

(1 -

P,)” -

i 0; y=l

fl( 1 -

P,)“_‘, 1 5 x* 5 7L -

1.

It is shown in Appendix 2 that thefcurve in (4) is uniformly increasing in p,, and has a unique inflexion point at x*/(n - l), as shown in Figure 1. It is also clear that

P <+,

I

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thefcurve is neither concave nor convex. Given thatf(O;.) = 0 andf(l;.) = 1, there are clearly three steady states for P,. Although there have been several models explaining the interdependency of individuals’ choices, our analysis is nevertheless unique. By assuming that people make decisions on the basis of their previous binomial sampling, we were able to derive the exact formula, and the exact injkxion points of the state transition rule. Thus, instead of proposing vaguely that there may be multiple equilibria, as in Arthur ( 1989) and Gordon (1989), we can conclude that as long as 1 G nx G n - 1 holds, there are exactly thw equilibria, as drawn in Figure 1. These properties are very helpful to developing our understanding about the shape of the crime-rate transition path and our later discussion of government policies. Figure 1 shows that the dynamic evolution of p, satisfies the four properties of a “complex system” characterized in Arthur (1989), namely, possible multiple equilibria, possible inefficiency, lock-in, and path dependence. These properties can be explained from Figure 1: (1) In the case of 1 s YU c n - 1, the three equilibria are A, B, and C, and only 8 and C are stable. (2) As long as R and C involve different social costs, certainly one of them is inefficient. (3) The system is path-dependent because different initial points [starting from (A&) or (A,C)] 1ead to different steady states. (4) A government policy can change the value of x, and hence the shape off, but a stable equilibrium will not be responsive to any marginal change of x. This implies that a stable equilibrium is “locked in” and the government has to make some significant change in order to extricate itself from a stable but inefficient situation.” In the next section, we shall compare the impact of stationary and oscillatory policies on equilibrium outcomes. Welfare Analysis of Different Enforcement Policies ‘To make the model more realistic for policy analysis, let us assume that there are q, proportion of (well-mannered) people who never drive rudely and qn proportion of (rash) people who always drive rudely. With these modifications, the dynamic transition rule of the state variable P, becomes

PIt, = q2 + (1 -91 - 4n)f(P,;4 =

g(P,;x),

(5)

which is shown in Figure 2, with the number of steady states being possibly one, two, or three. Let us assume that violation fines collected by the government are paid back to the public in lump sum so that they are private but not social cost. Thus, if we normalize the population size to be one, the expected social return of traffic interactions in period t will be up: + (b + c)p,( 1 - p,) + d( 1 - p,)’ = B,. Enforcement cost C,‘,is assumed to be a linear function of detection probability: C, = (Y . T,, and hence the net social welfare would be IV, = B, -

C, = al-): + (6 + c)P,(l

- p,) + d( 1 -

P,)’ -

CxYt-

W(P,,?-,),

(6)

where the relationship between r and p,, , is governed by (1) and (5). With 6 the social discount rate, the present value of total social welfare is X7=,, VW,. In the traffic violation case, at least, the penalty size cannot be frequently altered, but the detection probability can be changed by adjusting police forces. So in our later analysis, we shall study the impact caused by changing r. It is easy to see from (1) that tightened enforcement (increasing r) would increase x. If such an increase is

P ,+,

I

P,

significant enough to cause the threshold integer x * to go upward, then individuals would be more likely to adopt strategy II, the curve g&x) would shift down, and the corresponding steady state would also reduce. Let W(r$*(r)) denote the total social welfare associated with T and its corresponding steady-state traffic order p*(r). Clearly, whether a change in r is worth doing depends upon the sign of dW/dr.

Suppose we are now at point A of Figure 2, which is sustained by very strict trafhc law enforcement with cost too high to afford persistently. Suppose the government is trying to relax the enf-orcement in order to cut costs. When r reduces, x reduces, as we can see from equation (1). If the critical integer x* also reduces, the curve g(p,;x) will shift up, and the steady-state value p* will increase to B. As the g(.,.) curve shifts upward, a larger percentage of people is adopting strategy I, and the traffic order is gradually worsening. However, the nonconvexity of g implies that as r reduces there may be a.jump in p*(r), and hence a,jump in social costs. This could be seen in Figure 2, where a slight reduction of r at the point C would make the steady-state p value

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change from p*, to p*,]. Similarly, there would also be a jump if one tries to increase r at the point E. In fact, no p value brtween p*, and p*, cc&d ever be sustained as a steady .state. Thus, for stationary policies, the government a very -faces only two choices: good traffic order sustained by strict - enforcement (AC area) or a very bad traffic order as a result of lax enforcement (Et; area). Now we calculate the social welfare associated with a stationary enforcement policy (detection probability). When r = 1’1, let p*(i) be the steady state(s) achieved. The total discounted social welfare under stationary enforcement policy r’ will be

2 6’w,

= ,;, 8’w(p*(;),rc)

=

w;pyif)

s

TW:,

,

,=,I

where

sp stands

for stationary

policy.

Instead of restricting our attention to stationary policies, we could also consider the following oscillatory policy that shuttles between tough and lax enforcement. ‘Ii) match our notations with later numerical discussion, let us suppose that the lax (tough) enforcement probability is respectively r” (t”). Suppose at period zero p,, = p*(o), which may be very large because there is essentially no enforcement. Suppose the government decides that starting from period one, 1’will be increased to T-‘,a very strict enforcement. Then p, will follow the path p,, , = g(p,$), the traffic order is gradually improving, and the social welfare in each period will be W, = W(p,,r”). ‘l‘he state variable p, eventually converges to p*(5) (Figure 3). After staying at p*(5) for a while, suppose at period T, the government decides to relax enforcement to r,, from a period T, + 1 onward. The 1_‘,will then follow the path g(p,;x”), the traffic order is gradually deteriorating, and the social welfare in each period is W, = W&r”‘). The government can repeat the above-mentioned cycles once p, converges to p*(O). ‘I‘his cyclical enforcement policy would generate a sequence of W,‘s, and the discounted total social welfare will be denoted as TWj:,,,, where the superscript 0 specifies the cycle’s starting point p*(O). Similarly, if at period zero pl, = p*(5), which is very small, we assume that the government starts the cyclical policy by first reducing r to 1.”and then, asp, converges to p*(O), increasing r to r’, and so on. Discounted total social welfare so obtained will be denoted as TW,‘,,,<. For starting points p,, = p*(i) for any other i, the government will first adopt r”(r’) if p*(i) is small (large) and will begin to switch policies (between P and 1”) afterwards. WC shall demonstrate below that under some parametric specifications, TW:,, < TW:,,,,, Vi, that is, the government will have no incentive to stay at arly stationary state. Numrriccll Exavnp1r.t We assume a = 4, h = 8, c = 0, d = IO, T = 5, cx E (0, .5, l.O}, and 6 E 1.95, .90, 235, .80}. For each of the 12 cases, we calculate p*(i), TW’,,,, and TW’,,,,,, respectively, and the results are summarized in Tables l-3. In the case of stationary policies, we consider six enforcement probability values: T- = (1.“, . , r;) = (0, .I, 2, .Y, .4, .5}, and the corresponding x’s will be x8 = (d - b + +rr)l(a - c + d - b).’ It can be shown that when i = 0, 3, 4, 5, p*(i) has only one value, whereas when i = I, 2, p*(i) has two possible values. In the latter case we shall list both of them in the tables. When (Y =

0, traffic law enforcement is costless, and the government will have no incentive to leave a steady state with good traffic order [e.g., p*(5)]. U n d er these circumstances, oscillatory enforcement cannot be justified. But when (Y = 1 (Table 3), p*(5) starts to lose its attractiveness because the enforcement cost to sustain p*(5) is too high. At any time t with p, = p*(5), the government can always realize higher total social welfare by making oscillatory enforceme,nt swing between r’ and r”. The same also holds true forp, = p*(i), i = 0, 1, 2, 3, 4. Th’ IS implies that all these stationary policies can be dominated by the r” - P shuttling policy.’ In Table 2, where c1 = .5, the situation is more interesting. It can be seen that when 6 = .95, staying at the second and the fourth stationary state will result in higher social welfare, and hence the government will have no reason to make any policy change. But when 6 reduces to 30, every stationary policy will be dominated by the corresponding oscillatory one. It is interesting to note that the relationship between the desirability of cycles and the size of the discount rate is consistent with the findings in the literature of complex dynamics of Brock (1988).

C.Y. CYRUSCHU ‘FABLE 1. Net social

welfare

under

stationary

6 = .95 i

P*(i)

0 1

.800 ,217 .797 ,203 ,782 .200 .200 .200

84.72 153.51 84.92 156.32 85.67 156.73 156.79 156.80

vr~~~.~ 2. Net social

125.47 124.93 125.51 125.96 125.78 126.09 126.11 126. I 1

welfare

under

TWn,,*

42.41 76.76 42.46 78.16 42.83 78.37 78.40 78.40

63.10 63.26 63.14 64.14 63.39 64.26 64.28 64.28

stationary

and

P*(i)

TW:,

TWL

,799 ,217 ,797 .202 ,782 ,200 ,200 ,200

84.81 148.51 79.92 146.3 1 75.66 141.73 136.79 131.79

145.35 144.81 145.39 145.83 145.66 145.98 146.00 146.00

,799 ,217 .797 ,202 ,782 ,200 ,200 .200

welfare

84.8 1 143.51 74.92 136.31 65.66 126.73 116.79 106.79

cyclical

TW:, 28.27 51.17 28.3 1 52.1 I 28.56 52.24 52.26 52.27

under

TW:, 42.40 74.25 39.96 73.15 37.83 70.86 68.39 65.89

stationary

165.24 164.70 165.28 165.72 165.55 165.86 165.88 165.89

(CX= 0)

TW:,,, 73.09 73.26 73.13 74.13 73.39 74.26 74.27 74.28

and cyclical

83.09 83.26 83.13 84.13 83.39 84.26 84.27 84.28

S = .80

TW:,,,,

Tw:,

41.84 42.72 4 I .88 43.47 42.12 43.59 43.59 43.59

21.20 38.38 21.23 39.08 21.42 39.18 39.20 99.20

policies

TW$ 28.27 49.50 26.64 48.77 25.22 47.24 45.59 43.93

TW:,<,e 48.5 1 49.38 48.54 50.13 48.78 50.24 50.25 50.25

enforcement

policies

6 = .85

28.27 47.83 24.97 45.43 21.88 42.24 38.93 35.59

TW,,, 31.05 32.59 3 1.08 33.23 31.31 33.32 3 3 .33 33.33

((Y = .5) 6 = .HO

6 = .85

6 = .90

42.40 71.75 37.46 68.15 32.83 63.36 58.39 53.39

policies

enfbrcernent

8 = .90

6 = .9.5

0 1 1 2 2 3 4 5

enforcement 6 = .x5

TW:,

6 = .95

TABLE 3. Net social

cyclical

6 = .90 TY,,,e

TM/’‘P

and

311

55.17 56.05 55.21 56.80 55.45 56.90 56.92 56.92

TW:,, 21.20 37.12 19.98 36.57 1X.91 35.43 34.19 32.94

TW:,,,, 36.04 37.58 36.07 38.22 36.30 38.31 38.33 3 8 .‘3.‘3

(CI = 1 .O) 6 = .x0

21.20 35.87 18.73 34.07 16.41 3 1.68 29.19 26.69

41.04 42.58 4 1.07 43.22 41.30 43.31 43.33 43.33

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Oscillatory vs. .ttationay er@wccmmt 01 luw The Efficient Enforcement Policy

Having demonstrated the possible inefficiency of stationary enforcement policies, we are about to explain the reason. Intuitively, oscillatory policies such as the ones presented above can never be efficient if the law enforcement agency is typically assumed to maximize a concave objective function over a convex feasible set.” As the convexity refers to a set of variables indexed by time, cycles could not be optimal because averaging over two points of a cycle will exploit the convexity of the function and therefore increase the objective value achieved. However, as we explained in the previous section, the mutual interaction of individual decisions has created a natural nonconvexity in the macro rate of traffic violation. This, in turn, makes an oscillatory policy possibly better because now averaging over two points of a cycle may end up at a point that cannot he .swtained by any stationary policy but can be attained by an oscillatory one. One implication of the possible inefficiency of stationary enforcement is that if the government intends to search for an “optimal” enforcement policy, the traditional time-invariant setup of Becker (1968) is intrinsically misleading. With a nonconvex dynamic state transition rule (5), the government’s problem would become max 2 ‘I

s.t.

x, =

d a -

S’W(p,,r,)

,= (1

b +

r,Tr

c + (1 -

0

(7)

P ,+ I = ,9(P,;x,) The problem in (7) will generate an optimal sequence & T-Y, . . , T-Y, , which may or may not converge to a stationary value. It may be worth studying what exactly the necessary or sufficient conditions are for (7) to have “complex” enforcement dynamics, but so far analytical results along this line are few and have only restrictive applications.“’ A Final Remark on Crime Cycle and Enforcement Cycle As Cooter and Ulen (1988) and Wilson (1983) pointed out, U.S. crime statistics over the last 50 years showed that the amount of a wide variety of crimes declined from a peak in the mid- 1930s to a trough in the early 196Os, then began a rapid rise from the early 1960s until the mid or late 197Os, and began to fall slowly in the 1980s. There are two often-mentioned explanations for these crime cycles: .l‘he first relates the peak of the crime rate to the possibly unfair distribution of income in the period of rapid economic growth; the second hypothesis suggests that crime cycles are related to the cyclical age structure brought by fertility cycles.” What was rarely stressed in previous discussions of this topic was the interaction between crime waves and enforcement policies. As we mentioned in the first section, when the crime rate is high, there usually exists public pressure asking for tightening of law enforcement in order to improve social order. Thus, as the crime rate reaches its peak, there is a natural tendency for it to go down. Furthermore, if the dynamic time path of crime rates is nonconvex, a tightened law enforcement may cause a perdent fall of crime rates, as shown by the zigzag curve from F to A in Figure 3. Similarly, when the crime rate is low, people

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may propose to relax the law enforcement in order to reduce the seemingly unnecessary enforcement costs, and this proposal could also cause a persistent increase in crime rates, as shown from A to F. As such, aside from the exogenous demographic or economic shocks, the above-mentioned public pressure, together with the nonconvexity of dynamic path, form an cndo~~nou~ force of prolonging crime cycles. This mutual interaction between crime rates and enforcement policy seems to be an important factor in interpreting empirical data, and perhaps one can set up a model to refine the existing empirical literature in the future. Appendix

I: Deriving Equation (3)

Given p,, it is shown in (2) that a particular individual i will have probabilityf(p,;x) observe a sample mean, denoted as z;,, larger than x. Let

to

Suppose we pick a sample of size M and calculate the corresponding average of w: w,,, = C;l1, w’lM, then clearly o,,, will have a sampling distribution with mean f(~,;x). By Khinchine’s theorem (see Theil, 197 1, p. 3X), as M + 2, G,,, converges to J(&;x) in probability. Thus, in a city with a very large number of drivers, there will beJ(p,;.r) proportion of them that has w’ = 1 (or observes z:, > x). Under Schelling’s myopic prediction assumption, person i with w’ = 1 would expect to meet more than x proportion of strategy-l takers in period t + 1, and therefore, according to the specification of the 2 X 2 traffic interaction game, i’s best response is to adopt strategy 1 in period t + 1. As such, in a large city the dynamic evolution of p, will be as described in (3). Appendix II Differentiating

the R.H.S.

of (4) with respect

to p, yields

Since

by expanding

the terms in the square bracket of the above equation, one finds that most of them cancel with each other, and df/dp, can be further simplified as

df

( 1 n -

i&g=”

x*

1

I:* (1 - pi)“-‘* -’ > 0.

From (A I), one can easily derive

d2f dp: = 71

n-l

( ) .p

P;*-’ (I - p,y-I*--

x* - p,(n _ [

1)

1 (

314

Oscilluto~

71s.stntionayy rnforrrmrnt

of

1~7~

which will be positive (negative) if /I, is less (larger) than x*/(n - 1). Equations (Al) and (A2) allow us to draw the picture off‘ for the case 1 5 nx 5 n - 1. ‘The f‘curve is uniformly increasing in p, and has a unique inflexion point at x*/(n - I).

Notes I

2.

3.

4.

5.

6.

7.

8. 9. 10. 1 I.

Benjamini and Maital emphasimd the connection between one’s subjective ~~ohah&ly of being detected and one’s perception of other- people’s behavior. Schlicht argued that the moral condcrnnation COJLof law-breaking will lx low if most others a,-e breaking the law. ‘I’hc cspetted cost of violation is the product of perceived detection /~ohahi/i/~ and the co\f (penalty) after being detertrd ‘l‘he rescarch on optimal criminal policy was initiated by the seminal paper of Bee ker (1968). I.ater extensions in various directions can bc found in. for example. Stern (1978) and I’ylt (1983). Other multiperson interaction models include the famous px-isoncr’s dilemma. musical chairs, etc. Alternative model5 COT-respond to val-iant social outcome5 and ncetl “tipping,” dif.ferent policy interventions (if ;n~y) to improve ef’firiency. WC contentx~ate on the critical mass model because it can generate interesting equilibrium dynarnirs and contains rich policy implications. See Schelling (1978, <:h. Z3)for- morr elaborated discussion. ‘I‘he peer-group condemrlatioll cost can hc ~mhodied in the above analysis as follows. It is often believed that a condemnation pressux-e is pl-esent only when the rude driver ih caught and that the sile of corlclern~~atior~ cost is negatively related to the proportion of rude drivers in the society. Specifically, we suppose 7~ = y - ppI, with y > 6 > 0 (y > p implies that 71 is still positive when p! * 1). ‘Thus, Iquation (1) above beromcs x = rl ~ h + y~i[fd ~ h) + (U ~ c) + pr]. lt is easy to see that no11e of our following analysis M,ill he affected hy this change. The other approach is to assume that the moral pressure cost of r-ude dl-iving is incurred with certainty. I-egar-dless of whcthetone is caught. Intercstcd readers can I-ead (;or-don ( 1989) for more deailed anal+. Schelling (197X, p. X4) noted that it is the time lag of people’s cxpcctatiorr~ that gencl-ates interesting cyclical equilibrium. Even if pcoplc adopt more complex expectatioll-llpdatirlg rules, as long as thel-e are time lags our analysis still applirs. For the case of’ Kaycsian expcctation updating, WC also derive the f’ol-mula of macro crime I-ate transition and repeat the numerical analysis of the next section. I‘hc result is qualitatively inditterent and is thcl-efol-e omitted. Details are available from 1he author on request. The only exception is in Arthur (1989), where he briclly discussed the possibility 01 “shaking” the system into new configurations so AS to lcavc an inferior lot al steady state. See also Schlicht (1985). It is shown in the previous section that a change in r could effectively shift the curve s(P,;r) only when this change is significant enough to altcl- the threshold integer x*. ‘l‘hus, although I may have infinite values, under a given TI only a finite numhcr of them could make X(T) = d ~ b + r~/(a - r + d - h) integers. As such, ti)r welf’are analysis we only have to consider finite r’s with their corI-esponding x’s being integers bctause any I with x(r) between two neighboring integers (m < x(r) < n) will have the same enforcement impact as r,,,, but have why we can, without any higher enforcement cost than I,,,, where K,,,= x ‘(ra). This explains loss OF genel-ality, consider only finite cases. Policy options should also include cases where T > 5. But readers should be able to verify that our conclusions still hold in these cases. See Boldrin (1988) for more dctailcd explanations. See Brock (1988) and the references therein for detailed analysis. But a closer examination shows that these two explanations do not seem sufficient to account for the crime waves we ohservc. See Cooter and Ulen (1988, Ch. 12), for explanations.

3 15

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