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Some mechanisms of terror cycles
Journal of Economic Behavior & Organization 67 (2008) 644–656
Some mechanisms of terror cycles夽 Satya P. Das ∗ Indian Statistical Institute, Delhi Centre, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India Received 7 February 2007; received in revised form 31 March 2008; accepted 31 March 2008 Available online 18 April 2008
Abstract There is empirical evidence of cycles in transnational terrorist activities over time. The existing informal and formal works on terror cycles suggest that some form of interaction between the actions of a terrorist organization and a defending state is necessary for cycles to occur. This paper attempts to initiate a systematic investigation into some possible mechanisms behind terror cycles by modeling fear as a stock and analyzing the dynamic behavior of a terrorist organization and a defending state in a variety of scenarios. © 2008 Published by Elsevier B.V. JEL classification: C7; D9 Keywords: Terrorism; Cycle; Oscialltions; Bifurcation; Security-deterrence measures; Predator–prey structure
1. Introduction While the 9/11 terrorist attack has spurred a general interest in the economics of terrorism, the literature on terrorism dates back to 1970s. Works prior to this unprecedented event include Halperin (1976), Landes (1978), Mickolus (1982), Sandler et al. (1983), Im et al. (1987), Enders et al. (1992), Brophey-Baermann and Conybeare (1994), O’Brien (1996), Pizam and Smith (2000), Enders and Sandler (2000) and Feichtinger et al. (2001), among others. Interestingly and in particular, on the empirical side, Enders et al. (1992) and Enders and Sandler (2000) have discovered cycles in the time-series data on various indices of terrorist activities around the globe. For example, the last study mentioned has examined quarterly data from 1970:1 to 1996:2 and considered indices of transnational terrorist activity in the form of (i) number of terrorist incidents, (ii) number of casualties including death, (iii) proportion of incidents ending in casualties, and (iv) proportion of events involving one or more individuals being killed. The first two indices contain a ‘long-term’ primary cycle (18.57 quarters) and a ‘medium-term’ secondary cycle (7.65 quarters), while (iv) has two very ‘long-term’ cycles, one primary (58.18 quarters) and the other secondary (23.98 quarters).1 There is however little theoretical work on how terror cycles may arise. Informal accounts have been forwarded however. For example, Enders and Sandler (2000) argue that there may be clustering in terrorist incidence (one incident 夽
The paper has benefited from an anonymous referee’s comments as well as a presentation of a previous version at the South and Southeast Asia Econometric Society Meetings held in Chennai, India, during December 2006. ∗ Tel.: +91 11 4149 3936; fax: +91 11 4149 3981. E-mail address: [email protected]. 1 Considering more recent data, Enders and Sandler (2005) have shown that 9/11 has not led to any structural break in the time series of various indices of terrorism. Hence the presumption of cycles remains even when post 9/11 data are taken into account. 0167-2681/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.jebo.2008.03.012
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encouraging other attempts in a copycat fashion) that would invite public pressure resulting in more safeguards and measures against terrorism. This would lead to a decline in terrorist activities, and as this happens, safety measures may be downgraded, increasing the likelihood of successful attacks in subsequent periods, and the cycle would repeat. These authors mention other scenarios, such as political election cycles triggering terror cycles.2 Feichtinger et al. have outlined a formal model of terrorism cycle in the context of the tourism industry. As a country invests in tourism, it attracts a large number of tourists. This invites terrorism, which, in turn, turns away tourists and then terrorists. As terrorist incidences decline, the government again starts to invest in tourism, and the cycle continues. More recently, Faria (2003) has formulated an intertemporal model of behavior by a terrorist organization and a defending state. The two entities respectively build terrorist capital and enforcement/security stock, and cycles result from the cross-effects. It is fair to say, however, that this nascent literature on terror cycle is rather ad hoc and unsystematic. For instance, Faria’s model directly postulates that (1) the terrorist organization invests more in terrorist capital if the stock of enforcement capital increases, and (2) as terrorist activities rise, a defending state invests more in enforcement capital. However, both these strategic relationships should be ideally “derived” from first principles via some notion of gametheoretical equilibrium concept. This paper analyzes the behavior of a terrorist organization and a defending state, in which interactive responses are derived rather than assumed. In the process it also models ‘fear’ as a stock variable over time. The models developed in this paper are quite simple, aimed at initiating a systematic theoretical investigation of possible mechanisms behind terror cycles. 2. General framework The ultimate goal of a terrorist organization is to obtain some political concession for its sympathizers from a state or group of states. From the state’s viewpoint, granting such concessions is costly. Unable to obtain these through political channels, organizations originate among those who demand the concessions. They decide to unleash terror against the state either out of grudge or to strengthen their bargaining position in negotiating their demand at some future date. However, as long as the political objectives are not attained to an acceptable degree, the problem can be seen in the forms of a conflict in which the ‘goal’ of a terrorist organization is to inflict damage on the state and that latter’s prime objective is to defend itself. This scenario is taken as the primitive of our analysis. In other words, we abstract from the ‘long-term’ issues or political processes involved, although, as discussed in the concluding section, our model can be narrowly interpreted as accommodating political concessions. It will be maintained throughout that there are only two parties or players involved: one terrorist organization (the Organization) and one defending state (the State). The Organization derives its utility from the damage it is able to inflict on the State. Let xt denote the overall scale of attack within a unit of time, meaning the product of number of attacks and size of each attack; xt can be measured in terms of people killed or injured and value of properties damaged. All attacks are assumed to be of the same size. There are two types of costs associated with an increase in xt : (a) an increased chance of the plan of action being detected ahead of time and (b) a rise in resource cost. The State chooses the extent of security-deterrence measures, st . The probability of detection depends on st . (If detection occurs, it is assumed to foil an attack.) There are costs of st , both pecuniary and non-pecuniary (such as loss of civil liberty). Let p(st , xt ) denote the probability of detection of an attack or activity. The expected damage is equal to Xt = A[1 − p(st , xt )]xt , where xt is the ‘potential damage’, meaning damage incurred or inflicted if the terrorist task is not detected. Assuming the law of large numbers, let Xt be interpreted as the actual damage incurred or inflicted. The distinguishing feature of terrorism as opposed to other forms of conflict is the sense of fear it is meant to generate among the common mass. We view ‘fear’ at a given point of time as a stock that has a hang-over effect into the subsequent period. Thus, fear at time t depends on fear experienced at t − 1. It also depends on damage occurring at time t. Let Ft+1 denote the stock of fear at the end of period t or at the beginning of period t + 1. We assume Ft+1 = Xt g(Ft ), with g(0) = 0, g > 0 > g , g (0) = ∞ and g (∞) = 0. This means that the more heightened the 2
Also see Frey (2004, p. 16) for a brief discussion on terror cycles.
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existing stock of fear, the greater is the panic generated from a given magnitude of current, successful terrorist acts. It implies that if the existing stock of fear is already high, a small incidence can cause a serious panic problem. It can also be argued that, over time, fear may depreciate as memory recedes or people get ‘used to’ it; this element will be introduced later. Our analysis considers various mechanisms of cycle over time of Ft , xt and Xt within the broad scenario just described. In the literature, cycles in dynamic models are recognized in two alternative ways. In their seminal paper on credit cycles, Kiyotaki and Moore (1997) identify ‘cycles’ with damped oscillations, whereas in the tradition of Grandmont (1985), Farmer (1986), Caballe et al. (2006) and many others, cycles are endogenous, meaning limit cycles or closed invariant curves. Both approaches have their merits and limitations. In the former, cycles would not last without a series of external shocks, but it accommodates the notion of persistence of cycles and how parametric variations affect such persistence (see Kiyotaki and Moore, 1997). Endogenous cycle models look for intervals of parameter configurations under which cycles continue forever. While this is theoretically more satisfying, the conditions for stability of such cycles in dimensions higher than one are not subject to straightforward or intuitive interpretations and finding these conditions in the first place is an analytically (and even computationally) daunting task. Further, there is no such notion of a parametric change affecting the persistence of cyclic behavior since endogenous cycles mean a ‘permanent’ cycle. However, the underlying mechanism behind why a cyclical dynamics arises is the same in either approach. In this paper, we look for and examine both. In what follows, specific models of behavior are explored that may lead to terror cycles. 3. A model of choice by the defending state We begin by considering a situation where the number and the size of attacks in each period are given, that is, xt is constant (normalized to one) over time. The purpose is to demonstrate that cycles may arise from the optimal behavior of the State with respect to choosing security-deterrence measures. Suppressing the argument xt in the p( ) function, we have Xt = A[1 − p(st )] and Ft+1 = A[1 − p(st )]g(Ft ).
(1)
We assume ps > 0 ≥ pss . Let the total cost of security-deterrence be denoted as c(st )/ with both c and c positive. We interpret A as a measure of disembodied (destructive) technology in the hands of the Organization and as a parameter representing security-deterrence technology. At each time period t, the State essentially faces a static problem, that is to minimize Ft+1 + c(st )/. The first-order condition is −Aps (st )g(Ft ) + c (st ) = 0.
(2)
The second-order condition is met since pss ≤ 0 and c > 0. Eq. (2) yields: st = s(Ft ). +
(3)
By substituting this function into the expression of Ft+1 , we have a dynamic equation in Ft . With F0 given, it has a well-defined dynamics: Ft+1 = A[1 − p(s(Ft ))]g(Ft ) ≡ f (Ft ).
(4)
Notice that this is a generalized logistic equation, studied extensively in the mathematical literature on cycles and chaos. There is a positive effect of Ft on Ft+1 through the g( ) function. Furthermore and particularly, there is a negative effect of Ft on Ft+1 via the response function of the State, a necessary condition for the oscillatory behavior of Ft . Because the fear stock has a hangover effect, a higher stock of fear in one period triggers a greater security-deterrence measure in the next, which tends to reduce terror and hence fear in that period. This process generates cycles. Technically, as long as (a) the f ( ) function has a “hump” (say at F 0 ), (b) the hump occurs above the 45◦ line in the Ft -to- Ft+1 map, and (c) f 2 (F 0 ), the second iteration of f at F 0 , is positive, we have df/dFt < 0 at the steady state, and oscillatory dynamics is present. If −1 < df/dFt < 0, a temporary or a permanent perturbation of a parameter would
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lead to an oscillatory (but a stable) adjustment path of Ft to the steady state. This implies an oscillatory adjustment path of Xt , which, unlike fear, is observable. More about the oscillation pattern can be learned if we assume particular functional forms of p( ), g( ) and c( ) β β β β functions. Let pt = St , g(Ft ) = Ft , 0 < β < 1 and c(st ) = st2 /2. Then st = AFt and Ft+1 = A(1 − AFt )Ft . The steady state is defined by F ∗ 1−β = A(1 − AF ∗ β ). It is easy to show that a steady state exists and it is unique. Further, dFt+1 β(1 − 2AF ∗ β ) = . (5) β dFt F ∗ 1 − AF ∗ The solution of F ∗ is such that the denominator is unambiguously positive, but the numerator can be negative. Consider parametric configurations such that (5) is negative and its modulus less than unity. For instance, if β = 0.5, there is a closed-form solution of F ∗ equal to A/(1 + A2 ). Hence dFt+1 /dFt |F ∗ ∈ (−1, 0) as long as A2 ∈ (1, 3). We can then define the decay rate (in the neighborhood of the steady state) as3 ∗β dFt+1 = 1 − β(2AF − 1) . φ =1− dFt 1 − AF ∗ β From the steady-state condition it is easy to prove that d(AF ∗ β )/d > 0, implying dφ/d < 0. That is, as long as oscillations occur, an improvement in deterrence technology reduces the decay rate (or increases the persistence) of oscillations. This is because the higher the magnitude of , the greater is the negative effect of Ft on Ft+1 via the adjustment in the security-deterrence measures.4 Consider the implications of a change in the parameter A. The steady-state condition and the implied derivative dF ∗ /dA lead to F ∗ 1−β d(AF ∗ β ) dφ = < 0. >0⇒ 1−2β ∗ 2 dA dA (1 − β)F + A β Thus technology/efficiency improvement of the Organization reduces the decay rate and therefore increases the persistence of oscillation because a higher A implies a greater response of the security-deterrence measures to an increase in Ft . The model can exhibit endogenous cycles as well. If −df/dFt , evaluated at the steady state, is equal to one at some parameter value (say at A = A0 ) and d2 f/dFt dA|A=A0 = 0, then a flip bifurcation occurs. Higher values of A imply two-period cycles, four-period cycles, and even chaos. For example, √ if we√continue with the same functional forms and further assume β = 0.5 and = 1, we have Ft+1 = A(1 − A Ft ) Ft . If 1 < A√ < 2, the hump occurs to above the 45◦ line and f 2 (F 0√ ) > 0. It is easy to compute that a flip bifurcation occurs at A = 3. Thus endogenous cycles and chaos occur if A ∈ ( 3, 2). √ Fig. 1 illustrates convergence to a two-period cycle of the expected damage from terror, for the case of A = 3.2 and the initial value of F0 equal to 0.15. 3.1. Depreciation of fear We have assumed a positive feedback effect of fear in period t − 1 into period t. Along with this effect, it is plausible that, all else the same, fear depreciates over time. Accordingly, let us postulate that Ft+1 = A[1 − p( )]g(Ft ) − ΔFt , where Δ ∈ (0, 1) is the rate of depreciation. Given such a dynamics, there can be oscillations even if there is no adjustment in the scale of terrorist activity or security-deterrence. It is because depreciation of fear itself amounts to a negative effect of Ft on Ft−1 . The critical assumptions are g > 0 > g , lim g > Δ and lim g < Δ, which imply a hump-shaped map of Ft into Ft+1 . 3 4
F →0
This is analogous to Kiyotaki and Moore who define a decay rate of this kind in their study of credit cycle. This applies to the cyclicity of Xt as well since dst+1 Ag (F ∗ ) dFt+1 dFt+1 dXt+1 = = = . dXt dst Ag (F ∗ ) dFt dFt
F →∞
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Fig. 1. Two-period cycle of Xt .
For instance, if g(Ft ) = Ft , and p(·) = p˜ ∈ (0, 1), a constant, the steady state is at F ∗ 1−β = A(1 − p)/(1 ˜ + Δ) and 1−β = βA(1 − p)/Δ. ˜ Hence F˜ < F ∗ if and only if β < Δ/(1 + Δ). the hump occurs at F˜ Fear depreciation is thus an added element for the cyclical nature of the dynamics. Having noted this, we suppress it from now on to focus on the implications of choice behavior by the State and/or the Organization. β
4. A model of choice by the organization We now consider a scenario where the Organization optimally chooses the scale of attack. More specifically, let either the number of attacks be constant while the size varies over time or vice versa. The State remains passive in that it selects a constant st over time. Let the detection function be p = p(x) with px > 0 and px + xpxx ≥ 0. As mentioned in Section 1, apart from raising the probability of detection, there is a higher resource cost of enhancing xt . Resource use by a terrorist organization is essentially a dynamic problem. Without infusion of new resources in terms of manpower, funds and so on, more resource use in one period would leave less for the next. For our purpose, assume that resources are a composite input, so that xt itself represents the amount of resources expended at time t. Let Zt denote the stock of resources in the hands of the Organization in the beginning of period t (similar to ‘terrorist capital’ of Faria).5 Further, let Bt denote the infusion of new resources at time t by the sympathizers of the ‘cause’ held by the Organization. It can be argued that Bt depends on the success of the Organization in inflicting damage in the previous period (Xt−1 ). It can also be counter-argued that sympathizers are die-hards in their belief and do not condition their contribution on period-by-period ‘achievements’. It could depend on how far the sympathizers’ fundamental demands are met, or what their economic status is. The model abstracts from these considerations and assume that Bt = B, a constant. Thus the dynamic resource constraint facing the Organization is Zt+1 = Zt + B − xt .
(6)
Turning to its objective function, we take the position that the Organization is an entity that does not look far into the future. At time t its utility is derived from the fear it inflicts upon the State in that period and the amount of resources it leaves for the next period. In other words, the decision making of the Organization is “rolling.” We postulate that it maximizes Ft+1 + γV (Zt+1 ), where V > 0 > V and γ > 0.6 More explicitly, A[1 − p(xt )]xt g(Ft ) + γV (Zt + B − xt ) is the objective function, with xt as the decision variable. The first-order condition is the following: A[1 − p(xt ) − xt px (xt )]g(Ft ) − γV (Zt + B − xt ) = 0.
5 6
But Zt does not enter the production function of terror. This is akin to warm-glow bequest motive in macroeconomics.
(7)
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It is easy to check that the second-order condition holds. The above equation implicitly yields: xt = x(Ft , Zt ). +
+
(8)
An increase in Ft enhances the marginal benefit from attacks and thus encourages the Organization to engage in a higher scale of attacks. This implies ∂xt /∂Ft > 0. A higher Zt means more resources at the disposal of the Organization, a part of which goes to finance activities in the current period; thus ∂xt /∂Zt > 0. We now substitute (8) into the expressions of Ft+1 and Zt+1 and obtain: Ft+1 ≡ f (Ft , Zt ) = A[1 − p(x(Ft , Zt ))]x(Ft , Zt )g(Ft ) Zt+1 ≡ h(Ft , Zt ) = Zt + B − x(Ft , Zt ).
(9) (10)
In what follows it is shown that this 2 × 2 dynamic system can exhibit cycles, and the genesis of cyclicity lies in its “predator–prey” structure—the cross-effects fZ and zF having opposite signs. Thus it is unlike the one-variable case where cyclicity requires the own effect from one period to the next being negative. An increase in Zt leads to an increase in xt . This tends to enhance the initial stock of fear in the following period. On the other hand, a higher initial stock of fear in the current period implies a higher level of xt and, thus a lower initial stock of resources in the hands of the Organization in the following period. Hence Ft and Zt are respectively parallel to the “predator” and the “prey.” We first characterize the steady state. Denoting the steady-state values by an asterisk, Eq. (6) yields x∗ = B. The Ft+1 function yields F ∗ = A[1 − p(x∗ )]x∗ g(F ∗ ) = A[1 − p(B)]Bg(F ∗ ). The assumed properties of the g( ) function imply a unique positive solution of F ∗ . Given x∗ and F ∗ , the first-order condition (7) solves Z∗ .7 The Jacobian matrix of the system (9) and (10) has the following elements: ∂x F ∗ g (F ∗ ) ∗ ∂x ∗ fF = γV (Z ) + >0 > 0; fZ = γV (Z ) ∗ ∂Ft (F ∗ ,Z∗ ) g(F ) ∂Zt (F ∗ ,Z∗ ) ∂x ∂x ∂xt < 0; z = 1 − ∈ (0, 1) since ∈ (0, 1). zF = − Z ∂Ft (F ∗ ,Z∗ ) ∂Zt (F ∗ ,Z∗ ) ∂Zt (F ∗ ,Z∗ ) The cross-partials fZ and zF having opposite signs forms the predator–prey structure and the necessary condition for eigen roots to be complex conjugates. The necessary and sufficient condition is that −4fZ zF − (fF − zZ )2 > 0.
(11) √ We now show the existence of cycles by considering a specific example. Let B = 1, pt = αxt , g(Ft ) = Ft and V (·) = ln(·). This leads to the following steady-state expressions: x∗ = 1, F ∗ = [A(1 − α)]2 and Z∗ = 1/[ξ(1 − 2α)(1 − α)], where ξ ≡ A2 /γ. We assume α < 0.5. Using these, the elements of the Jacobian matrix reduce to 1 (1 − 2α)2 γξ 2 (1 − α)2 (1 − 2α)3 fF = ; fZ = 1+ 2 2 (1 − α)[2α + ξ(1 − α)(1 − 2α) ] 2α + ξ(1 − α)(1 − 2α)2 1 − 2α 2α ; zZ = zF = − . 2 2 2γξ(1 − α) [2α + ξ(1 − α)(1 − 2α) ] 2α + ξ(1 − α)(1 − 2α)2 The eigen roots depend on two parameters: α and ξ.8 For any given value of α, (11) is equivalent to a quadratic expression in ξ being positive. As an example, if α = 0.1, the expression (11) reduces to (0.672)ξ − (0.082944)ξ 2 − √ 0.065308642 > 0. If we choose ξ = 1/ 2, this condition is met and the modulus of the complex conjugates is less than one. Therefore, assuming that initially the system is in the steady state, a temporary or a permanent shock would lead to an oscillatory transition path converging to the steady state. Fig. 2 depicts the dynamics of the attempted and realized damage originating from a temporary (one-period) decline in B by 10 percent. Finally, endogenous cycles can arise via the Hopf bifurcation theorem and existence of an invariant closed curve in Ft and Zt . It requires that (a) the roots of the Jacobian matrix be complex conjugates (of the form u ± v), (b) (u ± v)n = 0 7 8
As a simple comparative statics, an increase in B implies more attacks and more successful attacks in the long run. Although the parameter γ appears in fZ and zF , the product fZ zF and therefore the roots are independent of γ.
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Fig. 2. Effects of a temporary negative shock to B.
for n = 1, 2, 3, 4, (c) there exists a parameter, say λ, such that for λ = λ0 , ||u ± v|| = 1, and (d) du ± v/dλ|λ0 = 0.9 In our example, if ξ ≈ 0.443672 ≡ ξ0 , the roots are 0.85975674 ± (0.51070481)i, with modulus ≈ 1, and, as ξ is varied the modulus takes values higher and lower than one. Thus conditions (a), (c) and (d) are satisfied. Because the roots have both real and imaginary parts, the condition (b) is also met. Thus a closed invariant curve exists for a range of values close to ξ0 .10 4.1. Number and size of attacks both endogenous Thus far the terrorist activity of the Organization has been summarized through a single variable xt , denoting the ‘overall’ scale of attack. Implicitly, either the number of attacks (frequency) or the size of each attack was endogenous, but not both. We now consider the case where both are endogenous and strategically chosen. Let yt denote the size of each attack and nt the number of such attacks (i.e., xt = yt nt ). It is reasonable to postulate that the detection probability increases with both yt and nt , but if the probability of detection is only a function of yt nt , then yt and nt are analytically indistinguishable. However, this is only a very special, razor-edge case. In what follows, we assume that (∂p/∂yt )/nt = (∂p/∂nt )/yt , implying that p cannot be written as a function of yt nt and thus yt and nt are analytically distinguished.11 We show below that cyclicity can also arise in this case. The expected damage from terror has the expression A[1 − p(yt , nt )]xt g(Ft ), which can be rewritten as A[1 − p(xt /nt , nt )]xt g(Ft ), where xt = yt nt . The Organization maximizes A[1 − p(xt /nt , nt )]xt g(Ft ) + γV (Zt + B − xt ), having two choice variables, xt and nt . The first-order conditions are respectively xt A 1 − p(·) − py (·) − γV (·) = 0 (12) nt py (·)xt − pn (·)n2t = 0.
(13)
In particular, the last equation implicitly expresses nt as a function of xt independent of Ft or Zt . Substituting this function into (12), it follows that as long as the second-order conditions are met, we have xt increasing in both Ft and Zt , as in the simpler model. This implies that the predatory–prey structure of the dynamic system in Ft and Zt continues to hold, which is capable of generating cycles. What is the relationship between the size and the number of attacks? Eq. (13) yields: dnt py + yt pyy − nt pny = . dyt pn + nt pnn − yt pny 9
(14)
See, for example, Azardiadis (1993, p. 100). Like most works on two or higher dimensional dynamic systems, in this paper we are limited to considering only the existence of limit cycles, not their stability (e.g., Torre, 1977; Azardiadis, 1993; Faria, 2003; Reichlin, 1986 is an exception). In any event, stability conditions generally depend on higher-order derivatives of dynamic equations that do not possess economic or intuitive interpretations. 11 Das and Lahiri (2006) consider a game model where the damage function satisfies increasing returns to scale, capturing the contagious effect of fear. Increasing returns to scale also allow yt and nt as separate variables irrespective of the nature of the p function. 10
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There seems no a priori reason to argue that the cross-partial pny should be of any particular sign. If we assume that pny 0 and pj + jt pjj > 0, j = y, n, then more can be said.12 From Eq. (14), dnt /dyt > 0 (i.e., the size and the number of attacks are complementary to each other). This would imply that dxt /dyt > 0. Thus, xt , yt and nt would all move together over time.13 4.2. Choice of security-deterrence by the state It is natural to introduce the behavior of the State into the current framework. For simplicity however, let us now revert back to the situation where either nt or yt is endogenous. Let pt = p(st , xt ), satisfying ps > 0 ≥ pss , px > 0, px + xpxx > 0 and psx ≥ 0. At each time t, both the State and the Organization essentially face static objective functions. These are min A[1 − p(st , xt )]xt g(Ft ) + st
c(st ) ;
maxxt A[1 − p(st , xt )]xt g(Ft ) + γV (Zt + B − xt ).
Assuming non-cooperative Nash behavior, the following are the respective first-order conditions: −Aps (st , xt )xt g(Ft ) + c (st ) = 0
(15)
A[1 − p(st , xt ) − xt px (st , xt )]g(Ft ) − γV (Zt + B − xt ) = 0.
(16)
These equations yield: ∂st > 0, ∂Ft
∂xt ≷ 0, ∂Ft
∂st > 0, ∂Zt
∂xt > 0. ∂Zt
As the stock of fear increases, the State steps up st . This has a negative effect on xt . Coupled with the positive effect of Ft on xt , the net effect of Ft on xt is ambiguous. An increase in Zt leads to an increase in xt for the same reason as in the earlier model, which implies a higher st via the State’s best response function (15); thus ∂st /∂Zt > 0. Both st and xt being flow variables, the dynamics of the system is expressed, as before, in terms of the state variables, Ft and Zt . At constant st , the cross-effect of Zt on Ft+1 and that of Ft on Zt+1 were respectively positive and negative, and these were the basis of cyclicity. How does the choice behavior of the State affect the cross-effects? As st adjusts: fZ = γV
∂xt ∂st − Aps (st , xt ) ≷ 0; ∂Zt ∂Zt
zF = −
∂xt ≷ 0. ∂Ft
Thus the predatory–prey effect is somewhat weakened. That is, while individually the choices of xt and xt are agents of cyclicity, they neutralize each other’s effect to some extent. However, Appendix 1 shows that in the case of psx = 0, fZ and zF do have opposite signs; that is, there are asymmetric cross-effects, even though, individually, their signs are still ambiguous. Thus cycles can arise.14 Consider an example where B = 1, p = μst + αxt ,
β
g(Ft ) = Ft ,
c(st ) =
st2 2
and
V (·) = ln(·).
(17)
Appendix 1 works out the elements of the Jacobian matrix. More specifically, if α = 0.15, β = 0.5, γ = 1, = 2, μ = 0.1 and A = 0.6, the roots are complex conjugates with modulus less than one. Fig. 3 draws the corresponding dynamics due to a temporary decline in B by 10 percent. 12 13
A linear p( ) function is obviously a special case of this. If pny = 0, the second-order conditions require that 2py + ypyy > 0, y(py + ypyy ) + n(pn + npnn ) > 0 and −Ag(Ft )
py [y(py + ypyy ) + n(pn + npnn )] + n(py + ypyy )(pn + npnn ) − γV ( ) < 0. n[y(py + ypyy ) + n(pn + npnn )]
These conditions are met under the specified restrictions. 14 By virtue of continuity, there are two other cases in which cycles may well be present: (a) p sufficiently close to zero and (b) a small enough sx variation in st .
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Fig. 3. A temporary decrease in B.
Endogenous cycles may emerge also. In the specific example we are looking at, we can treat A as the bifurcating parameter. If A ≈ 0.55975, the roots are complex conjugates with modulus ≈1. Therefore, a closed invariant cycle exists. 5. Convex adjustment cost of scale of attack and security-deterrence measures One main point emerging from the foregoing analysis is that fear dynamics together with the change of strategy over time either by the State with regard to security-deterrence measures or by the Organization with respect to the scale of attacks may be sufficient to generate cyclical dynamics. In this section we demonstrate that fear dynamics is not necessary, while the interaction between the behavior of the two players can generate cycles—if there are dynamic convex adjustment costs of changing security-deterrence levels and the scale of attacks. Let such costs be denoted by L(st − st−1 ) and N(xt − xt−1 ), where L , L , N and N are all positive. The rationale behind these costs is similar to that behind investment or output adjustment cost in the dynamic theory of firm behavior. Varying the scale of deterrence measures over time may entail the cost of changing the very nature of security systems, displacement cost of security personnel and so on, in addition to the cost associated with changing the intensity of measures at a given point of time. Similarly, varying the scale of attack over time may involve changing the form of attacks, involving new planning and hence new set-up costs. Let L(st − st−1 ) = λ(st − st−1 )2 /2 and N(xt − xt−1 ) = η(xt − xt−1 )2 /2. We abstract from the dynamic resource management problem facing the Organization and assume, instead, that the marginal cost of resources used in terrorist activity is constant (denoted by w). The State minimizes A[1 − p(st , xt )]xt + c(st )/ + λ(st − st−1 )2 /2, and the Organization’s objective is to maximize A[1 − p(st , xt )]xt − wxt − η(xt − xt−1 )2 /2. Assuming Nash competition, the respective first-order conditions are −Aps (st , xt )xt +
c (st ) + λ(st − st−1 ) = 0
A[1 − p(st , xt ) − xt px (st , xt )] − w − η(xt − xt−1 ) = 0.
(18) (19)
The intuition behind cyclical dynamics lies in these two equations. The best response function of the State implies that an increase in xt , the scale of attacks, induces an increase in security-deterrence measures st , while that of the Organization implies that an increase in st forces the Organization to reduce its scale of attack. Hence the contemporaneous crosseffects (i.e. the signs of the slopes of best response functions) are opposite. In the presence of adjustment costs, st and st−1 are positively associated, as are xt and xt−1 . Thus the contemporaneous cross-effects between st and xt translate into an asymmetric lagged crossed effects, which is same as a predatory–prey structure. The State is the ‘predator’ and the Organization the ‘prey’. Formally, in Fig. 4, SS and OO denote the best function of the State and that of the Organization, respectively. The left and right panels show the effects of increases in st−1 and xt−1 , respectively. We see that an increase in st−1 reduces xt and an increase in xt−1 leads to an increase in st . Algebraically, Eqs. (18) and (19) implicitly yield a difference-equation system of the following form: st+1 = s(st , xt );
xt+1 = x(st , xt ),
(20)
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Fig. 4. Effects of increases in st−1 and xt−1 .
with (∂st+1 /∂xt ) > 0 > (∂xt+1 /∂st ). Appendix 2 shows that the determinant |Δ1 |, where Δ1 ≡ ((∂st+1 /∂xt )(∂xt+1 /∂xt )) − ((∂st+1 /∂xt )(∂xt+1 /∂st )), has modulus less than unity. Thus, if the roots are complex conjugates, their modulus equals |Δ1 | < 1. Hence endogenous cycles cannot arise. However, damped oscillations can. In what follows, a specific example is worked out and the necessary and sufficient conditions for damped oscillations are derived. Choose p(st , xt ) = 1 − μst − αxt and c(st ) = st2 /2. Then (18) and (19) reduce to a system of linear difference equations: 1 −Aμxt + A(1 − μst − 2αxt ) − w − η(xt − xt−1 ) = 0, + λ st − λst−1 = 0; which yield: λ(2Aα + η) ∂st+1 = ; ∂st D
Aμη ∂st+1 = ; ∂xt D
Aμλ ∂xt+1 =− ; ∂st D
η(1/ + λ) ∂xt+1 = ∂xt D
where D ≡ (1/ + λ)(2Aα + η) + A2 μ2 > 0. 15 It is straightforward to derive that roots of the system (20) are complex conjugates if and only if λ λ 1 2 2α − < μ2 (21) η A η that is, the necessary and sufficient condition for damped oscillation is that the adjustment–cost–coefficients ratio is neither too high nor too low. This is intuitive, because the lag effects imply asymmetric cross-effects. The condition (21) is illustrated in Fig. 5; it is met as long as λ/η ∈ (a, b). The decay rate has the expression: λη φ =1− . (2Aα + η)((1/) + λ) + A2 μ2
15
The steady-state conditions are
2Aα + η
+ A2 μ2 s∗ − Aμηx∗ = Aμ(A − w);
We assume A > w, which ensures positive solutions of
s∗
Aμλs∗ + 2Aα and
x∗ .
1
+ λ + A2 μ2 x∗ = (A − w)
1
+λ .
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Fig. 5. The necessary and sufficient condition for oscillations.
Fig. 6. A permanent security-deterrence technology shock.
We see that a higher (i.e. an improvement in deterrence technology) implies a lower decay rate or greater persistence, which is same as what was obtained in the first model. However, unlike in the first model, an increase in A makes the cycle less persistent. This is because the cycle does not depend on the dynamics of fear.16 Fig. 6 illustrates the dynamic effects of a permanent security technology shock of 5 percent. Suppose that initially = 2, w = 0.64, α = 0.15, μ = 1, λ = η = 5, and A = 1.6. The steady-state values are: s∗ = 0.90909091 and x∗ = 0.3030303. Consider a permanent increase in from 2 to 2.1. The new steady state is given by: s∗ = 0.91304348 and x∗ = 0.28985507. Fig. 6 depicts the dynamics of st and xt in terms of their percent deviations from the new steady state. A couple of remarks are in order. First, the linear component of the marginal adjustment cost function (equal to the quadratic component of the total adjustment cost function) is necessary for oscillations to occur locally. It is because, otherwise, the marginal effects of a change in st−1 or xt−1 on both st and xt are zero in the neighborhood of the steady state.17 Oscillations arise in the presence of adjustment cost functions of higher order as long as the linear component is present, for example, when L (st − st−1 ) = κ(st − st−1 ) + π(st − st−1 )m where m > 1. 16
The linearity of the system lends itself to further characterization of the nature of oscillation. The cycle length equals 2π/θ, where
θ = tan
−1
(4μ2 (λ/η)) − ((1/A) − 2α(λ/η))2 (1/A) + (2λ/A) + (2αλ/η)
.
We observe that a proportionate increase in λ and η implies a decline in θ and hence an increase in the cycle length. Increase in and A have similar qualitative effects, although individual effects are ambiguous. 3 2 17 For instance, suppose that L(s − s t t−1 ) = (st − st−1 ) /3. The first-order condition (18) is now −Aps (st , xt )xt + c / + π(st − st−1 ) /2 = 0. The partial derivative of the left-hand-side expression with respect to st−1 is zero when st = st−1 .
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Second, the above model yields only damped oscillations. Endogenous cycles arise once fear dynamics is introduced. Appendix 3 presents a specific case where the State chooses st , there are adjustment cost associated with st and there is fear dynamics, while the Organization is passive. The model exhibits dynamics with two state variables, Ft and st . For some parameter values, flip bifurcation and endogenous cycles emerge, while for some other values, the model features damped oscillation. Needless to say, the same characterization would extend to cases where the Organization is not passive. 6. Concluding remarks The paper has presented some dynamic models of behavior by a terrorist organization and a defending state that illustrate various ways in which the interaction between the two may exhibit terror cycles in terms of damped oscillations or limit cycles. Fear is viewed as a stock with a ‘memory’ or a hangover effect, while it is also influenced by current damage from terror. Our very first model shows that such an evolution of fear together with an optimal choice of security-deterrence by the state leads to a reduced-form dynamic equation of fear, which is in the form of a generalized logistic equation. Thus even and odd period stable cycles (and chaos) can occur. The next two models incorporate variations of choice behavior of the terrorist organization over time with respect to the scale of attacks. Coupled with fear dynamics but with or without variations in security-deterrence measures by the defending state, the optimal behavior of the organization can lead to cyclical dynamics. The last section demonstrates that fear dynamics is not necessary. As long as variations over time of security-deterrence measures undertaken by the state and scale of attacks planned by the organization are subject to convex adjustment costs, their interaction can lead to cycles. The models features are quite simple. Several extensions suggest themselves. For example, security-deterrence is the only instrument of counter-terrorism policy considered in our model. A more complete analysis should include pre-emptive measures as an endogenous variable that would affect the overall capability of the terrorist organization as well as the flow of resources into its hands.18 Our model assumes one organization and is therefore unable to accommodate copycat effects. It also assumes one defending state. How strong is the presumption of cycles in the presence of multiple terrorist groups and multiple defending states? Sandler and Siqueira (2006) consider the problem of externalities stemming from counterterrorism policies when there are multiple defending states, but theirs is a static model, not meant to address dynamic issues. We have assumed perfect information. Obviously, terrorist strikes are uncertain, and there are serious problems of informational asymmetry; the size and location of planned attacks are known to a terrorist organization but not to a defending state. It is also important to consider learning by either player about the behavioral or technology parameters pertaining to the other and analyze its implications towards dynamics of terror. Notice that security-deterrence as a choice variable of the State can be alternatively interpreted as political concessions. The costs of such concessions by the State are analogous to the cost of security-deterrence measures. Its benefit lies in terms of the Organization going ‘softer’ on the State, which is similar to the negative effect of security-deterrence on expected damage. Hence political concessions can interact with ‘fear’ to generate cycles, as in, for example, the very first model introduced in Section 3.19 However, a fuller analysis of terrorism and political process should incorporate bargaining, negotiations, and so on. It is hoped that the analysis of this paper serves as a useful bench-mark for investigating some of these extensions and further understanding the nature of terror cycles. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jebo. 2008.03.012.
18 19
The variables Zt and Bt would be decreasing functions of the scale of pre-emptive measures, and this will affect the dynamics of terror. I am grateful to a reviewer for suggesting this mechanism of cycle in terms of political concessions and fear.
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References Azardiadis, C., 1993. Intertemporal Macroeconomics. Blackwell Publishers, Cambridge. Brophey-Baermann, B., Conybeare, J.A.C., 1994. Retaliating against terrorism: rational expectations and the optimality of rules versus discretion. American Journal of Political Science 38, 196–210. Caballe, J., Jarque, X., Michetti, E., 2006. Chaotic dynamics in credit constrained emerging economies. Journal of Economic Dynamics and Control 30, 1261–1275. Das, S.P., Lahiri, S., 2006. A strategic analysis of terrorist activity and counter-terrorism policies. Contributions to Theoretical Economics: The Berkeley Electronic Press Journals in Theoretical Economics 6, Article 6. Enders, W., Sandler, T., 2000. Is transnational terrorism becoming more threatening? A time-series investigation. Journal of Conflict Resolution 44, 307–332. Enders, W., Sandler, T., 2005. After 9/11: is it all different now? Journal of Conflict Resolution 49, 259–277. Enders, W., Parise, G., Sandler, T., 1992. A time-series analysis of transnational terrorism: trends and cycles. Defense Economics 3, 305–320. Faria, J.R., 2003. Terror cycles. In: Studies in Nonlinear Dynamics and Econometrics. The Berkeley Electronic Press 7, Article 3. Farmer, R., 1986. Deficits and cycle. Journal of Economic Theory 40, 77–88. Feichtinger, Hartl, G., Kort, P.M., Novak, A.J., 2001. Terrorism control in the tourism industry. Journal of Optimization Theory and Applications 108, 283–296. Frey, B.S., 2004. Dealing with Terrorism—Stick or Carrot? Edward Elgar, Cheltenham, England. Grandmont, J-M., 1985. On endogenous competitive business cycles. Econometrica 53, 995–1045. Halperin, E., 1976. Terrorism in Latin America. Sage Publications, Beverly Hills, California. Im, E.I., Couley, J., Sandler, T., 1987. Cycles and substitutions in terrorist activities: a spectral approach. Kyklos 45, 238–255. Kiyotaki, N., Moore, J., 1997. Credit cycles. Journal of Political Economy 105, 211–248. Landes, W.M., 1978. An economic study of US aircraft hijacking, 1961–1976. Journal of Law and Economics 21, 1–31. Mickolus, E.F., 1982. International Terrorism: Attributes of Terrorist Events, 1968–1977 (ITERATE 2). Inter-University Consortium for Political and Social Research, Ann Arbor. O’Brien, S.P., 1996. Foreign policy crises and the resort to terrorism: a time-series analysis conflict linkages. Journal of Conflict Resolution 40, 320–335. Pizam, A., Smith, G., 2000. Tourism and terrorism: a quantitative analysis of major terrorist acts and their impact on tourism destinations. Tourism Economics 6, 123–138. Reichlin, P., 1986. Equilibrium cycles in an overlapping generations economy with production. Journal of Economic Theory 40, 89–102. Sandler, T., Tschirhart, J., Cauley, J., 1983. A theoretical analysis of transnational terrorism. American Political Science Review 77, 36–54. Sandler, T., Siqueira, K., 2006. Global terrorism: deterrence versus pre-emption. Canadian Journal of Economics 50, 1370–1387. Torre, J., 1977. Existence of limit cycles and control in complete Keynesian system by theory of bifurcations. Econometrica 45, 1457–1466.
Some mechanisms of terror cycles夽 Satya P. Das ∗ Indian Statistical Institute, Delhi Centre, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India Received 7 February 2007; received in revised form 31 March 2008; accepted 31 March 2008 Available online 18 April 2008
Abstract There is empirical evidence of cycles in transnational terrorist activities over time. The existing informal and formal works on terror cycles suggest that some form of interaction between the actions of a terrorist organization and a defending state is necessary for cycles to occur. This paper attempts to initiate a systematic investigation into some possible mechanisms behind terror cycles by modeling fear as a stock and analyzing the dynamic behavior of a terrorist organization and a defending state in a variety of scenarios. © 2008 Published by Elsevier B.V. JEL classification: C7; D9 Keywords: Terrorism; Cycle; Oscialltions; Bifurcation; Security-deterrence measures; Predator–prey structure
1. Introduction While the 9/11 terrorist attack has spurred a general interest in the economics of terrorism, the literature on terrorism dates back to 1970s. Works prior to this unprecedented event include Halperin (1976), Landes (1978), Mickolus (1982), Sandler et al. (1983), Im et al. (1987), Enders et al. (1992), Brophey-Baermann and Conybeare (1994), O’Brien (1996), Pizam and Smith (2000), Enders and Sandler (2000) and Feichtinger et al. (2001), among others. Interestingly and in particular, on the empirical side, Enders et al. (1992) and Enders and Sandler (2000) have discovered cycles in the time-series data on various indices of terrorist activities around the globe. For example, the last study mentioned has examined quarterly data from 1970:1 to 1996:2 and considered indices of transnational terrorist activity in the form of (i) number of terrorist incidents, (ii) number of casualties including death, (iii) proportion of incidents ending in casualties, and (iv) proportion of events involving one or more individuals being killed. The first two indices contain a ‘long-term’ primary cycle (18.57 quarters) and a ‘medium-term’ secondary cycle (7.65 quarters), while (iv) has two very ‘long-term’ cycles, one primary (58.18 quarters) and the other secondary (23.98 quarters).1 There is however little theoretical work on how terror cycles may arise. Informal accounts have been forwarded however. For example, Enders and Sandler (2000) argue that there may be clustering in terrorist incidence (one incident 夽
The paper has benefited from an anonymous referee’s comments as well as a presentation of a previous version at the South and Southeast Asia Econometric Society Meetings held in Chennai, India, during December 2006. ∗ Tel.: +91 11 4149 3936; fax: +91 11 4149 3981. E-mail address: [email protected]. 1 Considering more recent data, Enders and Sandler (2005) have shown that 9/11 has not led to any structural break in the time series of various indices of terrorism. Hence the presumption of cycles remains even when post 9/11 data are taken into account. 0167-2681/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.jebo.2008.03.012
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encouraging other attempts in a copycat fashion) that would invite public pressure resulting in more safeguards and measures against terrorism. This would lead to a decline in terrorist activities, and as this happens, safety measures may be downgraded, increasing the likelihood of successful attacks in subsequent periods, and the cycle would repeat. These authors mention other scenarios, such as political election cycles triggering terror cycles.2 Feichtinger et al. have outlined a formal model of terrorism cycle in the context of the tourism industry. As a country invests in tourism, it attracts a large number of tourists. This invites terrorism, which, in turn, turns away tourists and then terrorists. As terrorist incidences decline, the government again starts to invest in tourism, and the cycle continues. More recently, Faria (2003) has formulated an intertemporal model of behavior by a terrorist organization and a defending state. The two entities respectively build terrorist capital and enforcement/security stock, and cycles result from the cross-effects. It is fair to say, however, that this nascent literature on terror cycle is rather ad hoc and unsystematic. For instance, Faria’s model directly postulates that (1) the terrorist organization invests more in terrorist capital if the stock of enforcement capital increases, and (2) as terrorist activities rise, a defending state invests more in enforcement capital. However, both these strategic relationships should be ideally “derived” from first principles via some notion of gametheoretical equilibrium concept. This paper analyzes the behavior of a terrorist organization and a defending state, in which interactive responses are derived rather than assumed. In the process it also models ‘fear’ as a stock variable over time. The models developed in this paper are quite simple, aimed at initiating a systematic theoretical investigation of possible mechanisms behind terror cycles. 2. General framework The ultimate goal of a terrorist organization is to obtain some political concession for its sympathizers from a state or group of states. From the state’s viewpoint, granting such concessions is costly. Unable to obtain these through political channels, organizations originate among those who demand the concessions. They decide to unleash terror against the state either out of grudge or to strengthen their bargaining position in negotiating their demand at some future date. However, as long as the political objectives are not attained to an acceptable degree, the problem can be seen in the forms of a conflict in which the ‘goal’ of a terrorist organization is to inflict damage on the state and that latter’s prime objective is to defend itself. This scenario is taken as the primitive of our analysis. In other words, we abstract from the ‘long-term’ issues or political processes involved, although, as discussed in the concluding section, our model can be narrowly interpreted as accommodating political concessions. It will be maintained throughout that there are only two parties or players involved: one terrorist organization (the Organization) and one defending state (the State). The Organization derives its utility from the damage it is able to inflict on the State. Let xt denote the overall scale of attack within a unit of time, meaning the product of number of attacks and size of each attack; xt can be measured in terms of people killed or injured and value of properties damaged. All attacks are assumed to be of the same size. There are two types of costs associated with an increase in xt : (a) an increased chance of the plan of action being detected ahead of time and (b) a rise in resource cost. The State chooses the extent of security-deterrence measures, st . The probability of detection depends on st . (If detection occurs, it is assumed to foil an attack.) There are costs of st , both pecuniary and non-pecuniary (such as loss of civil liberty). Let p(st , xt ) denote the probability of detection of an attack or activity. The expected damage is equal to Xt = A[1 − p(st , xt )]xt , where xt is the ‘potential damage’, meaning damage incurred or inflicted if the terrorist task is not detected. Assuming the law of large numbers, let Xt be interpreted as the actual damage incurred or inflicted. The distinguishing feature of terrorism as opposed to other forms of conflict is the sense of fear it is meant to generate among the common mass. We view ‘fear’ at a given point of time as a stock that has a hang-over effect into the subsequent period. Thus, fear at time t depends on fear experienced at t − 1. It also depends on damage occurring at time t. Let Ft+1 denote the stock of fear at the end of period t or at the beginning of period t + 1. We assume Ft+1 = Xt g(Ft ), with g(0) = 0, g > 0 > g , g (0) = ∞ and g (∞) = 0. This means that the more heightened the 2
Also see Frey (2004, p. 16) for a brief discussion on terror cycles.
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existing stock of fear, the greater is the panic generated from a given magnitude of current, successful terrorist acts. It implies that if the existing stock of fear is already high, a small incidence can cause a serious panic problem. It can also be argued that, over time, fear may depreciate as memory recedes or people get ‘used to’ it; this element will be introduced later. Our analysis considers various mechanisms of cycle over time of Ft , xt and Xt within the broad scenario just described. In the literature, cycles in dynamic models are recognized in two alternative ways. In their seminal paper on credit cycles, Kiyotaki and Moore (1997) identify ‘cycles’ with damped oscillations, whereas in the tradition of Grandmont (1985), Farmer (1986), Caballe et al. (2006) and many others, cycles are endogenous, meaning limit cycles or closed invariant curves. Both approaches have their merits and limitations. In the former, cycles would not last without a series of external shocks, but it accommodates the notion of persistence of cycles and how parametric variations affect such persistence (see Kiyotaki and Moore, 1997). Endogenous cycle models look for intervals of parameter configurations under which cycles continue forever. While this is theoretically more satisfying, the conditions for stability of such cycles in dimensions higher than one are not subject to straightforward or intuitive interpretations and finding these conditions in the first place is an analytically (and even computationally) daunting task. Further, there is no such notion of a parametric change affecting the persistence of cyclic behavior since endogenous cycles mean a ‘permanent’ cycle. However, the underlying mechanism behind why a cyclical dynamics arises is the same in either approach. In this paper, we look for and examine both. In what follows, specific models of behavior are explored that may lead to terror cycles. 3. A model of choice by the defending state We begin by considering a situation where the number and the size of attacks in each period are given, that is, xt is constant (normalized to one) over time. The purpose is to demonstrate that cycles may arise from the optimal behavior of the State with respect to choosing security-deterrence measures. Suppressing the argument xt in the p( ) function, we have Xt = A[1 − p(st )] and Ft+1 = A[1 − p(st )]g(Ft ).
(1)
We assume ps > 0 ≥ pss . Let the total cost of security-deterrence be denoted as c(st )/ with both c and c positive. We interpret A as a measure of disembodied (destructive) technology in the hands of the Organization and as a parameter representing security-deterrence technology. At each time period t, the State essentially faces a static problem, that is to minimize Ft+1 + c(st )/. The first-order condition is −Aps (st )g(Ft ) + c (st ) = 0.
(2)
The second-order condition is met since pss ≤ 0 and c > 0. Eq. (2) yields: st = s(Ft ). +
(3)
By substituting this function into the expression of Ft+1 , we have a dynamic equation in Ft . With F0 given, it has a well-defined dynamics: Ft+1 = A[1 − p(s(Ft ))]g(Ft ) ≡ f (Ft ).
(4)
Notice that this is a generalized logistic equation, studied extensively in the mathematical literature on cycles and chaos. There is a positive effect of Ft on Ft+1 through the g( ) function. Furthermore and particularly, there is a negative effect of Ft on Ft+1 via the response function of the State, a necessary condition for the oscillatory behavior of Ft . Because the fear stock has a hangover effect, a higher stock of fear in one period triggers a greater security-deterrence measure in the next, which tends to reduce terror and hence fear in that period. This process generates cycles. Technically, as long as (a) the f ( ) function has a “hump” (say at F 0 ), (b) the hump occurs above the 45◦ line in the Ft -to- Ft+1 map, and (c) f 2 (F 0 ), the second iteration of f at F 0 , is positive, we have df/dFt < 0 at the steady state, and oscillatory dynamics is present. If −1 < df/dFt < 0, a temporary or a permanent perturbation of a parameter would
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lead to an oscillatory (but a stable) adjustment path of Ft to the steady state. This implies an oscillatory adjustment path of Xt , which, unlike fear, is observable. More about the oscillation pattern can be learned if we assume particular functional forms of p( ), g( ) and c( ) β β β β functions. Let pt = St , g(Ft ) = Ft , 0 < β < 1 and c(st ) = st2 /2. Then st = AFt and Ft+1 = A(1 − AFt )Ft . The steady state is defined by F ∗ 1−β = A(1 − AF ∗ β ). It is easy to show that a steady state exists and it is unique. Further, dFt+1 β(1 − 2AF ∗ β ) = . (5) β dFt F ∗ 1 − AF ∗ The solution of F ∗ is such that the denominator is unambiguously positive, but the numerator can be negative. Consider parametric configurations such that (5) is negative and its modulus less than unity. For instance, if β = 0.5, there is a closed-form solution of F ∗ equal to A/(1 + A2 ). Hence dFt+1 /dFt |F ∗ ∈ (−1, 0) as long as A2 ∈ (1, 3). We can then define the decay rate (in the neighborhood of the steady state) as3 ∗β dFt+1 = 1 − β(2AF − 1) . φ =1− dFt 1 − AF ∗ β From the steady-state condition it is easy to prove that d(AF ∗ β )/d > 0, implying dφ/d < 0. That is, as long as oscillations occur, an improvement in deterrence technology reduces the decay rate (or increases the persistence) of oscillations. This is because the higher the magnitude of , the greater is the negative effect of Ft on Ft+1 via the adjustment in the security-deterrence measures.4 Consider the implications of a change in the parameter A. The steady-state condition and the implied derivative dF ∗ /dA lead to F ∗ 1−β d(AF ∗ β ) dφ = < 0. >0⇒ 1−2β ∗ 2 dA dA (1 − β)F + A β Thus technology/efficiency improvement of the Organization reduces the decay rate and therefore increases the persistence of oscillation because a higher A implies a greater response of the security-deterrence measures to an increase in Ft . The model can exhibit endogenous cycles as well. If −df/dFt , evaluated at the steady state, is equal to one at some parameter value (say at A = A0 ) and d2 f/dFt dA|A=A0 = 0, then a flip bifurcation occurs. Higher values of A imply two-period cycles, four-period cycles, and even chaos. For example, √ if we√continue with the same functional forms and further assume β = 0.5 and = 1, we have Ft+1 = A(1 − A Ft ) Ft . If 1 < A√ < 2, the hump occurs to above the 45◦ line and f 2 (F 0√ ) > 0. It is easy to compute that a flip bifurcation occurs at A = 3. Thus endogenous cycles and chaos occur if A ∈ ( 3, 2). √ Fig. 1 illustrates convergence to a two-period cycle of the expected damage from terror, for the case of A = 3.2 and the initial value of F0 equal to 0.15. 3.1. Depreciation of fear We have assumed a positive feedback effect of fear in period t − 1 into period t. Along with this effect, it is plausible that, all else the same, fear depreciates over time. Accordingly, let us postulate that Ft+1 = A[1 − p( )]g(Ft ) − ΔFt , where Δ ∈ (0, 1) is the rate of depreciation. Given such a dynamics, there can be oscillations even if there is no adjustment in the scale of terrorist activity or security-deterrence. It is because depreciation of fear itself amounts to a negative effect of Ft on Ft−1 . The critical assumptions are g > 0 > g , lim g > Δ and lim g < Δ, which imply a hump-shaped map of Ft into Ft+1 . 3 4
F →0
This is analogous to Kiyotaki and Moore who define a decay rate of this kind in their study of credit cycle. This applies to the cyclicity of Xt as well since dst+1 Ag (F ∗ ) dFt+1 dFt+1 dXt+1 = = = . dXt dst Ag (F ∗ ) dFt dFt
F →∞
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Fig. 1. Two-period cycle of Xt .
For instance, if g(Ft ) = Ft , and p(·) = p˜ ∈ (0, 1), a constant, the steady state is at F ∗ 1−β = A(1 − p)/(1 ˜ + Δ) and 1−β = βA(1 − p)/Δ. ˜ Hence F˜ < F ∗ if and only if β < Δ/(1 + Δ). the hump occurs at F˜ Fear depreciation is thus an added element for the cyclical nature of the dynamics. Having noted this, we suppress it from now on to focus on the implications of choice behavior by the State and/or the Organization. β
4. A model of choice by the organization We now consider a scenario where the Organization optimally chooses the scale of attack. More specifically, let either the number of attacks be constant while the size varies over time or vice versa. The State remains passive in that it selects a constant st over time. Let the detection function be p = p(x) with px > 0 and px + xpxx ≥ 0. As mentioned in Section 1, apart from raising the probability of detection, there is a higher resource cost of enhancing xt . Resource use by a terrorist organization is essentially a dynamic problem. Without infusion of new resources in terms of manpower, funds and so on, more resource use in one period would leave less for the next. For our purpose, assume that resources are a composite input, so that xt itself represents the amount of resources expended at time t. Let Zt denote the stock of resources in the hands of the Organization in the beginning of period t (similar to ‘terrorist capital’ of Faria).5 Further, let Bt denote the infusion of new resources at time t by the sympathizers of the ‘cause’ held by the Organization. It can be argued that Bt depends on the success of the Organization in inflicting damage in the previous period (Xt−1 ). It can also be counter-argued that sympathizers are die-hards in their belief and do not condition their contribution on period-by-period ‘achievements’. It could depend on how far the sympathizers’ fundamental demands are met, or what their economic status is. The model abstracts from these considerations and assume that Bt = B, a constant. Thus the dynamic resource constraint facing the Organization is Zt+1 = Zt + B − xt .
(6)
Turning to its objective function, we take the position that the Organization is an entity that does not look far into the future. At time t its utility is derived from the fear it inflicts upon the State in that period and the amount of resources it leaves for the next period. In other words, the decision making of the Organization is “rolling.” We postulate that it maximizes Ft+1 + γV (Zt+1 ), where V > 0 > V and γ > 0.6 More explicitly, A[1 − p(xt )]xt g(Ft ) + γV (Zt + B − xt ) is the objective function, with xt as the decision variable. The first-order condition is the following: A[1 − p(xt ) − xt px (xt )]g(Ft ) − γV (Zt + B − xt ) = 0.
5 6
But Zt does not enter the production function of terror. This is akin to warm-glow bequest motive in macroeconomics.
(7)
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It is easy to check that the second-order condition holds. The above equation implicitly yields: xt = x(Ft , Zt ). +
+
(8)
An increase in Ft enhances the marginal benefit from attacks and thus encourages the Organization to engage in a higher scale of attacks. This implies ∂xt /∂Ft > 0. A higher Zt means more resources at the disposal of the Organization, a part of which goes to finance activities in the current period; thus ∂xt /∂Zt > 0. We now substitute (8) into the expressions of Ft+1 and Zt+1 and obtain: Ft+1 ≡ f (Ft , Zt ) = A[1 − p(x(Ft , Zt ))]x(Ft , Zt )g(Ft ) Zt+1 ≡ h(Ft , Zt ) = Zt + B − x(Ft , Zt ).
(9) (10)
In what follows it is shown that this 2 × 2 dynamic system can exhibit cycles, and the genesis of cyclicity lies in its “predator–prey” structure—the cross-effects fZ and zF having opposite signs. Thus it is unlike the one-variable case where cyclicity requires the own effect from one period to the next being negative. An increase in Zt leads to an increase in xt . This tends to enhance the initial stock of fear in the following period. On the other hand, a higher initial stock of fear in the current period implies a higher level of xt and, thus a lower initial stock of resources in the hands of the Organization in the following period. Hence Ft and Zt are respectively parallel to the “predator” and the “prey.” We first characterize the steady state. Denoting the steady-state values by an asterisk, Eq. (6) yields x∗ = B. The Ft+1 function yields F ∗ = A[1 − p(x∗ )]x∗ g(F ∗ ) = A[1 − p(B)]Bg(F ∗ ). The assumed properties of the g( ) function imply a unique positive solution of F ∗ . Given x∗ and F ∗ , the first-order condition (7) solves Z∗ .7 The Jacobian matrix of the system (9) and (10) has the following elements: ∂x F ∗ g (F ∗ ) ∗ ∂x ∗ fF = γV (Z ) + >0 > 0; fZ = γV (Z ) ∗ ∂Ft (F ∗ ,Z∗ ) g(F ) ∂Zt (F ∗ ,Z∗ ) ∂x ∂x ∂xt < 0; z = 1 − ∈ (0, 1) since ∈ (0, 1). zF = − Z ∂Ft (F ∗ ,Z∗ ) ∂Zt (F ∗ ,Z∗ ) ∂Zt (F ∗ ,Z∗ ) The cross-partials fZ and zF having opposite signs forms the predator–prey structure and the necessary condition for eigen roots to be complex conjugates. The necessary and sufficient condition is that −4fZ zF − (fF − zZ )2 > 0.
(11) √ We now show the existence of cycles by considering a specific example. Let B = 1, pt = αxt , g(Ft ) = Ft and V (·) = ln(·). This leads to the following steady-state expressions: x∗ = 1, F ∗ = [A(1 − α)]2 and Z∗ = 1/[ξ(1 − 2α)(1 − α)], where ξ ≡ A2 /γ. We assume α < 0.5. Using these, the elements of the Jacobian matrix reduce to 1 (1 − 2α)2 γξ 2 (1 − α)2 (1 − 2α)3 fF = ; fZ = 1+ 2 2 (1 − α)[2α + ξ(1 − α)(1 − 2α) ] 2α + ξ(1 − α)(1 − 2α)2 1 − 2α 2α ; zZ = zF = − . 2 2 2γξ(1 − α) [2α + ξ(1 − α)(1 − 2α) ] 2α + ξ(1 − α)(1 − 2α)2 The eigen roots depend on two parameters: α and ξ.8 For any given value of α, (11) is equivalent to a quadratic expression in ξ being positive. As an example, if α = 0.1, the expression (11) reduces to (0.672)ξ − (0.082944)ξ 2 − √ 0.065308642 > 0. If we choose ξ = 1/ 2, this condition is met and the modulus of the complex conjugates is less than one. Therefore, assuming that initially the system is in the steady state, a temporary or a permanent shock would lead to an oscillatory transition path converging to the steady state. Fig. 2 depicts the dynamics of the attempted and realized damage originating from a temporary (one-period) decline in B by 10 percent. Finally, endogenous cycles can arise via the Hopf bifurcation theorem and existence of an invariant closed curve in Ft and Zt . It requires that (a) the roots of the Jacobian matrix be complex conjugates (of the form u ± v), (b) (u ± v)n = 0 7 8
As a simple comparative statics, an increase in B implies more attacks and more successful attacks in the long run. Although the parameter γ appears in fZ and zF , the product fZ zF and therefore the roots are independent of γ.
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Fig. 2. Effects of a temporary negative shock to B.
for n = 1, 2, 3, 4, (c) there exists a parameter, say λ, such that for λ = λ0 , ||u ± v|| = 1, and (d) du ± v/dλ|λ0 = 0.9 In our example, if ξ ≈ 0.443672 ≡ ξ0 , the roots are 0.85975674 ± (0.51070481)i, with modulus ≈ 1, and, as ξ is varied the modulus takes values higher and lower than one. Thus conditions (a), (c) and (d) are satisfied. Because the roots have both real and imaginary parts, the condition (b) is also met. Thus a closed invariant curve exists for a range of values close to ξ0 .10 4.1. Number and size of attacks both endogenous Thus far the terrorist activity of the Organization has been summarized through a single variable xt , denoting the ‘overall’ scale of attack. Implicitly, either the number of attacks (frequency) or the size of each attack was endogenous, but not both. We now consider the case where both are endogenous and strategically chosen. Let yt denote the size of each attack and nt the number of such attacks (i.e., xt = yt nt ). It is reasonable to postulate that the detection probability increases with both yt and nt , but if the probability of detection is only a function of yt nt , then yt and nt are analytically indistinguishable. However, this is only a very special, razor-edge case. In what follows, we assume that (∂p/∂yt )/nt = (∂p/∂nt )/yt , implying that p cannot be written as a function of yt nt and thus yt and nt are analytically distinguished.11 We show below that cyclicity can also arise in this case. The expected damage from terror has the expression A[1 − p(yt , nt )]xt g(Ft ), which can be rewritten as A[1 − p(xt /nt , nt )]xt g(Ft ), where xt = yt nt . The Organization maximizes A[1 − p(xt /nt , nt )]xt g(Ft ) + γV (Zt + B − xt ), having two choice variables, xt and nt . The first-order conditions are respectively xt A 1 − p(·) − py (·) − γV (·) = 0 (12) nt py (·)xt − pn (·)n2t = 0.
(13)
In particular, the last equation implicitly expresses nt as a function of xt independent of Ft or Zt . Substituting this function into (12), it follows that as long as the second-order conditions are met, we have xt increasing in both Ft and Zt , as in the simpler model. This implies that the predatory–prey structure of the dynamic system in Ft and Zt continues to hold, which is capable of generating cycles. What is the relationship between the size and the number of attacks? Eq. (13) yields: dnt py + yt pyy − nt pny = . dyt pn + nt pnn − yt pny 9
(14)
See, for example, Azardiadis (1993, p. 100). Like most works on two or higher dimensional dynamic systems, in this paper we are limited to considering only the existence of limit cycles, not their stability (e.g., Torre, 1977; Azardiadis, 1993; Faria, 2003; Reichlin, 1986 is an exception). In any event, stability conditions generally depend on higher-order derivatives of dynamic equations that do not possess economic or intuitive interpretations. 11 Das and Lahiri (2006) consider a game model where the damage function satisfies increasing returns to scale, capturing the contagious effect of fear. Increasing returns to scale also allow yt and nt as separate variables irrespective of the nature of the p function. 10
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There seems no a priori reason to argue that the cross-partial pny should be of any particular sign. If we assume that pny 0 and pj + jt pjj > 0, j = y, n, then more can be said.12 From Eq. (14), dnt /dyt > 0 (i.e., the size and the number of attacks are complementary to each other). This would imply that dxt /dyt > 0. Thus, xt , yt and nt would all move together over time.13 4.2. Choice of security-deterrence by the state It is natural to introduce the behavior of the State into the current framework. For simplicity however, let us now revert back to the situation where either nt or yt is endogenous. Let pt = p(st , xt ), satisfying ps > 0 ≥ pss , px > 0, px + xpxx > 0 and psx ≥ 0. At each time t, both the State and the Organization essentially face static objective functions. These are min A[1 − p(st , xt )]xt g(Ft ) + st
c(st ) ;
maxxt A[1 − p(st , xt )]xt g(Ft ) + γV (Zt + B − xt ).
Assuming non-cooperative Nash behavior, the following are the respective first-order conditions: −Aps (st , xt )xt g(Ft ) + c (st ) = 0
(15)
A[1 − p(st , xt ) − xt px (st , xt )]g(Ft ) − γV (Zt + B − xt ) = 0.
(16)
These equations yield: ∂st > 0, ∂Ft
∂xt ≷ 0, ∂Ft
∂st > 0, ∂Zt
∂xt > 0. ∂Zt
As the stock of fear increases, the State steps up st . This has a negative effect on xt . Coupled with the positive effect of Ft on xt , the net effect of Ft on xt is ambiguous. An increase in Zt leads to an increase in xt for the same reason as in the earlier model, which implies a higher st via the State’s best response function (15); thus ∂st /∂Zt > 0. Both st and xt being flow variables, the dynamics of the system is expressed, as before, in terms of the state variables, Ft and Zt . At constant st , the cross-effect of Zt on Ft+1 and that of Ft on Zt+1 were respectively positive and negative, and these were the basis of cyclicity. How does the choice behavior of the State affect the cross-effects? As st adjusts: fZ = γV
∂xt ∂st − Aps (st , xt ) ≷ 0; ∂Zt ∂Zt
zF = −
∂xt ≷ 0. ∂Ft
Thus the predatory–prey effect is somewhat weakened. That is, while individually the choices of xt and xt are agents of cyclicity, they neutralize each other’s effect to some extent. However, Appendix 1 shows that in the case of psx = 0, fZ and zF do have opposite signs; that is, there are asymmetric cross-effects, even though, individually, their signs are still ambiguous. Thus cycles can arise.14 Consider an example where B = 1, p = μst + αxt ,
β
g(Ft ) = Ft ,
c(st ) =
st2 2
and
V (·) = ln(·).
(17)
Appendix 1 works out the elements of the Jacobian matrix. More specifically, if α = 0.15, β = 0.5, γ = 1, = 2, μ = 0.1 and A = 0.6, the roots are complex conjugates with modulus less than one. Fig. 3 draws the corresponding dynamics due to a temporary decline in B by 10 percent. 12 13
A linear p( ) function is obviously a special case of this. If pny = 0, the second-order conditions require that 2py + ypyy > 0, y(py + ypyy ) + n(pn + npnn ) > 0 and −Ag(Ft )
py [y(py + ypyy ) + n(pn + npnn )] + n(py + ypyy )(pn + npnn ) − γV ( ) < 0. n[y(py + ypyy ) + n(pn + npnn )]
These conditions are met under the specified restrictions. 14 By virtue of continuity, there are two other cases in which cycles may well be present: (a) p sufficiently close to zero and (b) a small enough sx variation in st .
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Fig. 3. A temporary decrease in B.
Endogenous cycles may emerge also. In the specific example we are looking at, we can treat A as the bifurcating parameter. If A ≈ 0.55975, the roots are complex conjugates with modulus ≈1. Therefore, a closed invariant cycle exists. 5. Convex adjustment cost of scale of attack and security-deterrence measures One main point emerging from the foregoing analysis is that fear dynamics together with the change of strategy over time either by the State with regard to security-deterrence measures or by the Organization with respect to the scale of attacks may be sufficient to generate cyclical dynamics. In this section we demonstrate that fear dynamics is not necessary, while the interaction between the behavior of the two players can generate cycles—if there are dynamic convex adjustment costs of changing security-deterrence levels and the scale of attacks. Let such costs be denoted by L(st − st−1 ) and N(xt − xt−1 ), where L , L , N and N are all positive. The rationale behind these costs is similar to that behind investment or output adjustment cost in the dynamic theory of firm behavior. Varying the scale of deterrence measures over time may entail the cost of changing the very nature of security systems, displacement cost of security personnel and so on, in addition to the cost associated with changing the intensity of measures at a given point of time. Similarly, varying the scale of attack over time may involve changing the form of attacks, involving new planning and hence new set-up costs. Let L(st − st−1 ) = λ(st − st−1 )2 /2 and N(xt − xt−1 ) = η(xt − xt−1 )2 /2. We abstract from the dynamic resource management problem facing the Organization and assume, instead, that the marginal cost of resources used in terrorist activity is constant (denoted by w). The State minimizes A[1 − p(st , xt )]xt + c(st )/ + λ(st − st−1 )2 /2, and the Organization’s objective is to maximize A[1 − p(st , xt )]xt − wxt − η(xt − xt−1 )2 /2. Assuming Nash competition, the respective first-order conditions are −Aps (st , xt )xt +
c (st ) + λ(st − st−1 ) = 0
A[1 − p(st , xt ) − xt px (st , xt )] − w − η(xt − xt−1 ) = 0.
(18) (19)
The intuition behind cyclical dynamics lies in these two equations. The best response function of the State implies that an increase in xt , the scale of attacks, induces an increase in security-deterrence measures st , while that of the Organization implies that an increase in st forces the Organization to reduce its scale of attack. Hence the contemporaneous crosseffects (i.e. the signs of the slopes of best response functions) are opposite. In the presence of adjustment costs, st and st−1 are positively associated, as are xt and xt−1 . Thus the contemporaneous cross-effects between st and xt translate into an asymmetric lagged crossed effects, which is same as a predatory–prey structure. The State is the ‘predator’ and the Organization the ‘prey’. Formally, in Fig. 4, SS and OO denote the best function of the State and that of the Organization, respectively. The left and right panels show the effects of increases in st−1 and xt−1 , respectively. We see that an increase in st−1 reduces xt and an increase in xt−1 leads to an increase in st . Algebraically, Eqs. (18) and (19) implicitly yield a difference-equation system of the following form: st+1 = s(st , xt );
xt+1 = x(st , xt ),
(20)
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Fig. 4. Effects of increases in st−1 and xt−1 .
with (∂st+1 /∂xt ) > 0 > (∂xt+1 /∂st ). Appendix 2 shows that the determinant |Δ1 |, where Δ1 ≡ ((∂st+1 /∂xt )(∂xt+1 /∂xt )) − ((∂st+1 /∂xt )(∂xt+1 /∂st )), has modulus less than unity. Thus, if the roots are complex conjugates, their modulus equals |Δ1 | < 1. Hence endogenous cycles cannot arise. However, damped oscillations can. In what follows, a specific example is worked out and the necessary and sufficient conditions for damped oscillations are derived. Choose p(st , xt ) = 1 − μst − αxt and c(st ) = st2 /2. Then (18) and (19) reduce to a system of linear difference equations: 1 −Aμxt + A(1 − μst − 2αxt ) − w − η(xt − xt−1 ) = 0, + λ st − λst−1 = 0; which yield: λ(2Aα + η) ∂st+1 = ; ∂st D
Aμη ∂st+1 = ; ∂xt D
Aμλ ∂xt+1 =− ; ∂st D
η(1/ + λ) ∂xt+1 = ∂xt D
where D ≡ (1/ + λ)(2Aα + η) + A2 μ2 > 0. 15 It is straightforward to derive that roots of the system (20) are complex conjugates if and only if λ λ 1 2 2α − < μ2 (21) η A η that is, the necessary and sufficient condition for damped oscillation is that the adjustment–cost–coefficients ratio is neither too high nor too low. This is intuitive, because the lag effects imply asymmetric cross-effects. The condition (21) is illustrated in Fig. 5; it is met as long as λ/η ∈ (a, b). The decay rate has the expression: λη φ =1− . (2Aα + η)((1/) + λ) + A2 μ2
15
The steady-state conditions are
2Aα + η
+ A2 μ2 s∗ − Aμηx∗ = Aμ(A − w);
We assume A > w, which ensures positive solutions of
s∗
Aμλs∗ + 2Aα and
x∗ .
1
+ λ + A2 μ2 x∗ = (A − w)
1
+λ .
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Fig. 5. The necessary and sufficient condition for oscillations.
Fig. 6. A permanent security-deterrence technology shock.
We see that a higher (i.e. an improvement in deterrence technology) implies a lower decay rate or greater persistence, which is same as what was obtained in the first model. However, unlike in the first model, an increase in A makes the cycle less persistent. This is because the cycle does not depend on the dynamics of fear.16 Fig. 6 illustrates the dynamic effects of a permanent security technology shock of 5 percent. Suppose that initially = 2, w = 0.64, α = 0.15, μ = 1, λ = η = 5, and A = 1.6. The steady-state values are: s∗ = 0.90909091 and x∗ = 0.3030303. Consider a permanent increase in from 2 to 2.1. The new steady state is given by: s∗ = 0.91304348 and x∗ = 0.28985507. Fig. 6 depicts the dynamics of st and xt in terms of their percent deviations from the new steady state. A couple of remarks are in order. First, the linear component of the marginal adjustment cost function (equal to the quadratic component of the total adjustment cost function) is necessary for oscillations to occur locally. It is because, otherwise, the marginal effects of a change in st−1 or xt−1 on both st and xt are zero in the neighborhood of the steady state.17 Oscillations arise in the presence of adjustment cost functions of higher order as long as the linear component is present, for example, when L (st − st−1 ) = κ(st − st−1 ) + π(st − st−1 )m where m > 1. 16
The linearity of the system lends itself to further characterization of the nature of oscillation. The cycle length equals 2π/θ, where
θ = tan
−1
(4μ2 (λ/η)) − ((1/A) − 2α(λ/η))2 (1/A) + (2λ/A) + (2αλ/η)
.
We observe that a proportionate increase in λ and η implies a decline in θ and hence an increase in the cycle length. Increase in and A have similar qualitative effects, although individual effects are ambiguous. 3 2 17 For instance, suppose that L(s − s t t−1 ) = (st − st−1 ) /3. The first-order condition (18) is now −Aps (st , xt )xt + c / + π(st − st−1 ) /2 = 0. The partial derivative of the left-hand-side expression with respect to st−1 is zero when st = st−1 .
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Second, the above model yields only damped oscillations. Endogenous cycles arise once fear dynamics is introduced. Appendix 3 presents a specific case where the State chooses st , there are adjustment cost associated with st and there is fear dynamics, while the Organization is passive. The model exhibits dynamics with two state variables, Ft and st . For some parameter values, flip bifurcation and endogenous cycles emerge, while for some other values, the model features damped oscillation. Needless to say, the same characterization would extend to cases where the Organization is not passive. 6. Concluding remarks The paper has presented some dynamic models of behavior by a terrorist organization and a defending state that illustrate various ways in which the interaction between the two may exhibit terror cycles in terms of damped oscillations or limit cycles. Fear is viewed as a stock with a ‘memory’ or a hangover effect, while it is also influenced by current damage from terror. Our very first model shows that such an evolution of fear together with an optimal choice of security-deterrence by the state leads to a reduced-form dynamic equation of fear, which is in the form of a generalized logistic equation. Thus even and odd period stable cycles (and chaos) can occur. The next two models incorporate variations of choice behavior of the terrorist organization over time with respect to the scale of attacks. Coupled with fear dynamics but with or without variations in security-deterrence measures by the defending state, the optimal behavior of the organization can lead to cyclical dynamics. The last section demonstrates that fear dynamics is not necessary. As long as variations over time of security-deterrence measures undertaken by the state and scale of attacks planned by the organization are subject to convex adjustment costs, their interaction can lead to cycles. The models features are quite simple. Several extensions suggest themselves. For example, security-deterrence is the only instrument of counter-terrorism policy considered in our model. A more complete analysis should include pre-emptive measures as an endogenous variable that would affect the overall capability of the terrorist organization as well as the flow of resources into its hands.18 Our model assumes one organization and is therefore unable to accommodate copycat effects. It also assumes one defending state. How strong is the presumption of cycles in the presence of multiple terrorist groups and multiple defending states? Sandler and Siqueira (2006) consider the problem of externalities stemming from counterterrorism policies when there are multiple defending states, but theirs is a static model, not meant to address dynamic issues. We have assumed perfect information. Obviously, terrorist strikes are uncertain, and there are serious problems of informational asymmetry; the size and location of planned attacks are known to a terrorist organization but not to a defending state. It is also important to consider learning by either player about the behavioral or technology parameters pertaining to the other and analyze its implications towards dynamics of terror. Notice that security-deterrence as a choice variable of the State can be alternatively interpreted as political concessions. The costs of such concessions by the State are analogous to the cost of security-deterrence measures. Its benefit lies in terms of the Organization going ‘softer’ on the State, which is similar to the negative effect of security-deterrence on expected damage. Hence political concessions can interact with ‘fear’ to generate cycles, as in, for example, the very first model introduced in Section 3.19 However, a fuller analysis of terrorism and political process should incorporate bargaining, negotiations, and so on. It is hoped that the analysis of this paper serves as a useful bench-mark for investigating some of these extensions and further understanding the nature of terror cycles. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.jebo. 2008.03.012.
18 19
The variables Zt and Bt would be decreasing functions of the scale of pre-emptive measures, and this will affect the dynamics of terror. I am grateful to a reviewer for suggesting this mechanism of cycle in terms of political concessions and fear.
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