• Aucun résultat trouvé

Theoretical model

Dans le document Heterogeneity and international economics (Page 77-84)

terrorism and counter-terrorism

2.2 Theoretical model

Consider an economy where there is a large number of identical households whose pref-erences are symmetric across space. Each member of the household supplies one unit of labor inelastically, such that the labor input is L = 1. We assume that population does not grow. In this Ramsey Cass Koopmans setting, the household’s utility function is given by R0u(c(t))e−ρtdt where c(t) is the rate of consumption at time t and ρ > 0 is the rate of time preference. The functional form of the utility function is CRRA;

u(c(t)) = [c(t)1−θ −1]/1−θ with θ the measure of risk aversion and θ > 0. The

inter-9For an account of counter-terrorism effectiveness, see Drakos and Giannakopoulos (2009) for transna-tional counter-terrorism and Enders and Sandler (1993) on more general considerations.

2.2. Theoretical model 65 temporal utility function is assumed to be additively separable and discounted:

U(C(0)) =

Z 0

u(c(t))e−ρtdt (2.1)

The household divides its per capita income from capital, r(t)a(t), and labor, w(t), into consumption and saving - in the form of accumulating assets, a(t) - so as to maximize lifetime utility. Its dynamic budget constraint is given by

˙

a(t) =r(t)a(t) +w(t)c(t) (2.2) where the dot notation denotes a time derivative. The no-Ponzi game condition holds such that

is the discount factor based on continuous discounting with changing in-terest rate r(t) and s > t. Such condition prevents the representative household’s debt to grow faster asymptotically than the real interest rate. The household optimization problem is to maximize utility (2.1), subject to its budget constraint (2.2), limitation on debt (2.3), initial stock of assets a(0) and inequality restrictionc(t)≥0:

maxc(t) U =R0u(c(t))e−ρtdt The Hamiltonian of the optimization problem is given by

H(c, a, µ) =u(c(t)e−ρt) +µ(t)[r(t)a(t) +w(t)c(t)] (2.4)

where µ are the undertemined Lagrange multipliers. Each µ can be interpreted as the shadow value of a unit of capital in period t, i.e. the discounted marginal utility from having more Kt. The solution of equation (2.4) results in the following optimal growth rate equation:

˙ c c = 1

θ[r(t)−ρ]

whereθ = u00(c(t))c(t)u0c(t) is the Euler equation, in other words the elasticity of marginal utility of consumption. Consumption will grow over time when the discount rate ρ is smaller than the rate of return on assets r(t).

Production uses capital and labor with a neoclassical technology. The per capita produc-tion funcproduc-tion is given by y(t) = f(k(t)) wherey(t) is the per capita output at timet, f(.) is increasing, concave and Inada and k(t) is the physical capital stock per worker at time t. In the absence of adjustment costs and uncertainty, the return to capital, r(t), equates the marginal product of capital, f0(k(t)). Thus the market rate for consumer is given by r(t) =f0(k(t)).

Without population growth, technological progress and capital depreciation, the model steady-states are typically given by

k = α ρ

!1−α1

and c =f(k)−k

where α is the capital intensity from the Cobb-Douglas production function10.

In prevention of terrorist attacks the state provides security through counter-terrorism expenditures G(t). These expenditures are financed through a flat rate of income tax τ which is determined by the amount of collected tax from the households,g(t) =τ[r(t)k(t)].

Intuitively, τ is best set when it ensures a status quo security level without inducing fear among households which would further depress investment or consumption11. The per capita production function given counter-terrorism policies now takes the form:

y(t) =k(t)αg(t)1−α

where 0 < α <1. It is assumed that investments in counter-terrorism is the only public good provided by the state such that no fraction of g(t) is allocated to either public consumption or investment other than security. Independently of terrorist attacks, the capital accumulation equation now is diminished by the amount g:

k˙ =f(k(t))−c(t)g(t)

10IfY(t) =F(K(t), AL(t)) andy(t) =f(k) =kα, thenkis given by the above expression when ˙c= 0.

11As noted by Ocal and Yildirim (2010) acts of terror are intended to incite fear, insecurity, and intimidation, which can significantly undermine consumer and investor confidence in the medium run.

2.2. Theoretical model 67 The effects of counter-terrorism expenditures on the growth rate can take two channels.

First, an increase in the tax rate τ induces the representative household to adjust the accumulation of assets such as to keep consumption constant. Investment in capital k(t) diminishes which lowers the total output as well as consumption. Second, an increase in the share of counter-terrorism expenditures over output, g(t)y(t), can raise ∂y(t)∂k(t) which has a positive impact on γ. The derivative of γ with respect to g(t)/y(t) is given by

∂γ

∂(g(t)/y(t)) = 1

θα0 g(t) k(t)

!

The growth rate increases with g(t)/y(t) if g(t)/k(t) is small enough such that α0 > 1 and declines with α0 < 1. The optimal size of government that maximizes the growth rate γ corresponds to the condition for productive efficiency, namely α = 1. It follows that α =g(t)/y(t) = τ which gives rise to an inverse hump-shaped relationship between counter-terrorism spending g(t) and the growth rate γ.

Terrorist groups are rational12 agents devoting their resources to terrorism in order to maximize their long-term objectives which can be summarized as a redistribution of power, influence and wealth (Frey and Luechinger 2004). To achieve those goals terrorist groups develop short-term strategies that are to destabilize the polity and the economy with maximal media attention. Accordingly, terrorists set the optimal amount of incidents when the marginal benefits from undertaking an additional attack equates the incurred marginal costs. In other words, a given terrorist group maximizes its optimal number of incidents and targets such that it gains maximal public attention while eroding both the state stability and economic resources, subject to its budget constraint. The immediate impact of terrorism on the economy is to destroy a fraction δA of capital k(t) which imposes material costs on the population in order to force the government to comply with their demands. Such material damages impacts the rate of return13 r(t) =f0(k(t))−δA. To take into account the mitigating effect of counter-terrorism on terrorist attacks, a measure of government efficiency, ϕ, is introduced which intervenes at the following level:

δT =δAϕg(t) k(t)

12See Becker (1968).

13The destruction effect of terrorist attacks is modelled, as in Collier (1999) for the case of wars, through the rate of capital depreciation.

In other words, the effective rate of destruction δT is composed of the destruction rate of capital following a terrorist attack, δA, minus the amount of counter-terrorism expendi-tures to capital unitsg(t)/k(t), subject to the efficiency of the state destruction prevention, ϕ. Since the actual level of terrorist activities is dependent upon the costs, benefits and opportunity costs of terrorism, for a counter-terrorism measure to be effective in limit-ing the impact of terrorism on growth, it has to reduce either one of those parameters.

This means that an efficient counter-terrorist measure increases the costs of performing an attack; reduces the benefits associated with it; or makes non-violent alternatives to terrorists more attractive. Deterrence policies, i.e. increased level of security, commonly aim at increasing the costs of an attack by making it more difficult to undertake or by punishing its actors more severely (Frey and Luechinger 2003)14.

Frey and Luechinger (2004) suggest decentralization of power and changes in media reporting practices to reduce the benefits of terror. Decentralized systems are those in which central entities play a lesser role in any or all of these dimensions. In such systems, central governments possess a smaller share of fiscal resources, grant more administrative autonomy, and/or cede a higher degree of responsibility for political functions (Schneider 2003). Intuitively a polity and society with many different centres - such as a federal-ist state - is more difficult to destabilise than a system where all political and economic powers are centralized. In general, decentralization decreases the level of immediate inter-dependency and consequently the degree of potential damage. The attraction of violent actions on the part of terrorists is diminished as they prove to have less effect on the political stability and aggregate economic activity (Dreher and Fischer 2011). Making alternative options more attractive to terrorists is to raise the opportunity costs of terror (Frey and Luechinger 2003). The opportunity costs faced by terrorist groups consist in the utility they could gain by engaging in non-violent activities. For instance introduc-ing separatist claims or other political motives into the political debate could potentially decrease the number of terrorist incidents15. Similarly and as noted by Glaeser (2005),

14Applying rational choice theory to terrorism, Anderton and Carter (2005) point out that deterrence strategies reduce terrorist activities if the opportunities cost of terrorism increases. As for benevolence strategies the authors show that these strategies can indeed reduce terrorism if the non-violent alternatives are price elastic. Should the demand for non-violent alternatives be inelastic, then benevolence strategies might even increase terrorist activities.

15This effect aims in particular at domestic terrorism. It is not probable that the parliament addresses certain claims, i.e. anti-capitalist, antidemocratic or anti-Western, of international terrorists. Nonethe-less, targeted societies have opened political and societal debates on a number of issues (integration of religious communities, wearing of religious or political symbols in public places, freedom of speech and

2.2. Theoretical model 69 individual hatred toward specific groups or nations can emerge from misinformation and manipulation by political leaders. Such misinformation is less likely in democracies where the media can prevent the spreading of alternative facts. Mueller (2004) also argues for strengthening democratic institutions to counter terrorism.

Subsequently, ϕis an index of democracy (raising the opportunity cost)16, federalism and market economy (decreasing the benefits of terror). Deterrence policy (raising the cost of performing an attack) is captured through the counter-terrorism expendituresG(t).

Equilibria with terrorism and counter-terrorism17

Given terrorism and counter-terrorism policies, the interest rate r(t) is impacted in the following fashion:

r(t) = (1τ)f0[k(t)−δT] = (1−τ) f0(k(t))−δAϕ(g(t) k(t))

!

such that the optimization problem for the representative household becomes:

max

The Hamiltonian of the optimization problem (2.4) is now defined by H(c, k, µ) =u(c(t)e−ρt) +µ(t)

"

(1−τ)r(t)k(t)+w(t)c(t)

#

(2.5) This optimization problem leads to the following set of differential equations:

k˙ =f(k(t))−δTk(t)c(t)g(t) (2.6)

of association, etc.) following the emergence of a new global threat which has highlighted a number of failures from past integration policies and the perceived lack of opportunities for some communities.

16The literature is unclear whether democracies are likelier targets than autocratic regimes (Blomberg, Hess, and Orphanides 2004; Tavares2004). In his paper Tavares (2004) finds that the cost of terror is smaller in democracies.

17See Appendix for the full derivations.

˙ The steady state equilibrium is defined as an equilibrium path in which capital-labor ratio, consumption and output are constant, thus f0(k) = 1−τ1 (ρ+δT) implying that higher taxes to finance counter-terrorism operations would decrease k since f0(·) is decreasing.

The resulting equilibrium path of consumption is defined by c =f(k)−δTkg

and is decreasing inδT and in per capita counter-terrorism expendituresg(t). The steady-state value of capital is given by

k = α(1τ) ρ+δT

!1/(1−α)

(2.8) Thus, the steady-state capital will be higher if capital is more productive (the capital share α is higher), counter-terrorism measures are more effective, and will be lower if consumers are more impatient and destruction rates of physical capital after a terrorist attacks are higher. To get a per capita income equation we substitute the solution k of equation (2.8) into the production function y(t) = k(t)αg(t)1−α at steady-state. Log-linearizing the resulting expression as in Mankiw, Romer, and Weil (1992) yields the per capita income growth equation18:

where γ is the growth rate of output per efficient worker, i.e. the difference between the log of output per efficient worker at the present period and its initial value logy0, β = 1−α, and A0 is the technological progress. Replacing logA0 with its valuea+that represents different country-specific influences on growth (technological progress, climate, institutions, etc.) yields the per capita income growth regression expressed in the next section.

18Derivations are shown in the Appendix.

Dans le document Heterogeneity and international economics (Page 77-84)

Documents relatifs