Rent-seeking and capital accumulation.

Barelli, Paulo ; De Abreu Pessoa, Samuel

I. INTRODUCTION

Rent-seeking activities are likely to generate a negative impact on the long-run performance of an economy by diverting resources away from growth-promoting activities. In addition, it is widely documented that economies are not free of rent-seeking activities, especially less-developed economies. It is therefore important that issues in development economics are analyzed taking into account the presence of rent-seeking activities. We propose a way of incorporating rent-seeking activities in a standard model of economic growth, the neoclassical model of capital accumulation. We solve the model and present its main properties, and use it to analyze some topics in development economics.

In order to add a rent-seeking sector to the model, we introduce an "aggregate rent-seeking technology" that determines the fraction of the productive sector's output that is captured by the rent-seeking sector. With this reduced-form approach, there are two relevant economic decisions: static and dynamic. The static is the factor allocation problem between the two sectors (for a given level of productive factors), and the dynamic is the consumption-investment allocation problem. We define a static equilibrium and a steady-state equilibrium and provide the conditions that the aggregate rent-seeking technology ought to satisfy in order for these equilibria to exist, and also further conditions ensuring uniqueness.

In terms of the dynamics of the model, we show that one can have an economy whose capital stock increases and per capita gross domestic product (GDP) decreases, This means that, for instance, the poor performance of sub-Saharan countries can be explained on the grounds that, with a capital-intensive rent-seeking sector, the increased capital stock leads to relatively more rent-seeking activities and to poor performance in GDP growth.

The factor allocation decision is determined by the relative returns of the two sectors. In turn, returns are determined by the structure of the economy: an extreme case of a rent-seeking activity is robbery, and the efficiency of the police in preventing robbery determines the returns to productive factors employed in robbery. Accordingly, we capture "institutional efficiency" as a measure of the rent-seeking technology relative to the productive sector's technology.

The main result of the model is that there is a monotone relation between institutional efficiency and welfare: the more efficient are the institutions in preventing rent-seeking activities, the greater is the welfare of the representative agent. We show that it is possible that long-run per capita GDP increases as institutions become less efficient in preventing rent-seeking activities. So long-run per capita GDP is not a good measure of the negative impact of institutional efficiency. But the effect on welfare is unambiguous: during the transition following an improvement in the rent-seeking technology, consumption decreases so much that it overcomes the (possible) long-run increase in income.

The logic behind the result above is the following. A change in institutional efficiency generates two effects in welfare: (1) A static effect, because it changes what is captured by the rent-seeking sector and (2) a dynamic effect, because it generates a distortion in capital accumulation. We name these two effects as Tullock and Harberger effects, respectively. (1) The unambiguous result means that the Tullock effect dominates the Harberger effect (while the Tullock effect is unambiguous, as worse institutions means that more is captured by the rent-seeking sector, the Harberger effect can go in the opposite direction when rent-seeking is capital intensive). Hence, welfare is reduced by inefficient institutions because productive factors are employed in rent-seeking activities. When both sectors operate under the same technology, the Harberger effect is zero; given that it is always dominated by the Tullock effect and it is only non-zero when technologies differ, we may say that it is of second order of importance in welfare terms. Hence, the driving force is the distortion in factor allocation (the Tullock effect).

We then provide remarks on the insights that the model provides to some topics in development economics: (1) A monopolist rent-seeker is less harmful to the economy than a competitive rent-seeking sector. This follows from the fact that a monopolist produces less output than a competitive industry (see Murphy, Shleifer, and Vishny 1993). Other things equal, we expect a relatively poor performance of economies moving out of a centralized system into a market-oriented system, providing a rationale for the recent experience of economies that used to be colonies or belong to the Soviet Union. (2) The failure in foreign aid in helping the receiving country can be attributed to the fact that the aid benefits relatively more the rent-seeking sector. (3) And there is an "endogenous total factor productivity (TFP)" perspective built in the model: as productive factors move between sectors in the economy, the measured TFP changes. That is, because TFP is measured as a residual after considering that all productive factors are allocated in productive activities, the fact that some factors are allocated in rentseeking activities shows up in TFP estimates. During transitional dynamics, reallocations of productive factors generate a transition path for TFP as well. This fact provides some rationale for the observed paths of measured TFPs in some countries.

The paper is organized as follows. Section II relates our work with the previous literature. The model is presented in Section III. In addition to presenting the assumptions behind the aggregate rent-seeking technology, we define the static and dynamic equilibria and also show their existence and uniqueness. Section IV presents the comparative statics results, and also the properties of the transition dynamics. Sections V and VI present the additional results mentioned above and some concluding remarks.

II. RELATION TO OTHER WORKS

Although there has been some interest in the literature on rent-seeking and unproductive activities, there does not exist a systematic literature on the subject. There are contributions on this issue coming from macroeconomics, political economy, international trade, common pool problems, and economic history. In this short section, without being comprehensive, we survey these contributions and show how our work relates to them.

Murphy, Shleifer, and Vishny (1993) and Acemoglu (1995) build simple static models of allocation of a single productive factor in the presence of competitive rent-seeking. Tullock (1980), Skaperdas (1992), Hirsbleifer (1995), and Grossman and Kim (1995) also present models without capital accumulation, and in which a fixed number of individuals or bands or groups fight for a slice of a pie, which in some cases is endogenously determined by the production decision of the contenders. Grossman and Kim (1995) considered that in addition to the two usual activities in this literature--production and predation--there is another activity, protection. The lack of free entry in the rent-seeking activity means that in equilibrium rents are not fully dissipated.

Krueger (1974) and Bhagwati and Srinivasan (1980) present models of rent-seeking in an international trade economic environment (some form of the HOV model). In this literature, many of the results rest on the specific interaction among the unproductive activity and other distortions related to international trade. In particular, the unproductive activity might be welfare improving, which is not the case in our setup.

Tornell and Velasco (1992), Benhabib and Rustichini (1996), and Grossman and Kim (1996) present models in which capital accumulation takes place in a dynamic game framework. Agents face the strategic choice of how much production to appropriate, which might generate less incentives for production and capital accumulation. The explicit game-theoretic formulation used in those papers is to be contrasted with our assumption of perfect competition, where a large number of rent-seeking firms compete for the appropriation of Sector 1's output. Strategic considerations are summarized by the aggregate rent-seeking technology, the contest success function (CSF), and by the free entry condition. As a result, our formulation is simpler, and, because those models use a single factor technology with constant returns to scale (basically a variant of the AK model), it is our understanding that our model is the first attempt to incorporate rent-seeking in the neoclassical model of capital accumulation. Moreover, the first two models do not take into consideration the resources employed the opportunity cost of having resources allocated in rent-seeking activities (Tullock effect), which turns out to be the most important effect of the model.

The original contribution of Tullock (1967) is taken as the background of our formalization. Jones (1988) provided a very illuminating account of the world economic history in terms of a struggle between rent-seeking and productive activities. (2) North (1990, 1994), Baumol (1990), Olson (1992), and Murphy, Shleifer, and Vishny (1993) among others were also instrumental in shaping the hypothesis that institutions form the fundamental structure of incentives that eventually drives all results in a market economy. One of the purposes of setting up the present model was to provide a formalization of these ideas under a general and standard macroeconomic framework.

III. THE MODEL

The model is an extension of the neoclassical model of capital accumulation. In that model there is just one good produced by a large number of firms under a constant returns to scale technology that employs capital and labor, renting these services from households. The representative household makes her intertemporal decision optimally taking into account the income stream she will receive from her renting of capital and labor. Institutions are usually introduced, in a macroeconomic setup, as a wedge between what firms produce and the income they earn. That is, output of the firms is summarized by an aggregate production function, F(K, L), and firms' income is given by a fraction of that output, (1 - [tau]) F(K, L). The "tax rate" [tau] represents any sort of distortion that might characterize the economy, which could be a tax itself. In general, it can be identified with the efficiency of the institutional background of the economy. The extension considered here is to give a specific formulation for the "tax rate" [tau].

In particular, we assume that there exists another sector in the economy, called the rent-seeking sector (also Sector 2). Like the productive sector (Sector 1), the rent-seeking sector combines capital and labor to produce an output, but this output is a service, and not another good. That is, it is an effort to capture goods produced in Sector 1. The more service is produced, the larger the amount of goods that have transferred toward Sector 2. Calling [Y.sub.1] and [Y.sub.2] the output levels of Sectors 1 and 2, respectively, the idea above can be stated as follows: Sector 1 keeps (1 - [tau]([Y.sub.2]))[Y.sub.1] and Sector 2 is able to confiscate [tau]([Y.sub.2])[Y.sub.1] goods from Sector 1, where [tau] is an increasing function of the transfer services, [Y.sub.2]. Formally, the burden imposed by the rent-seeking sector on the productive sector is a negative externality, which would not emerge if property rights were fully enforced. The function [tau] is characterized below (we denote it by g to reserve the symbol [tau] for a bona fide tax rate).

Note that the description above using a tax rate [tau] may suggest another interpretation for what we call rent-seeking activities. One could think of Sector 2 as a productive sector that is subsidized by the government, who collects revenue by taxing Sector 1. The difference is that the output of Sector 2 would then be [tau]([Y.sub.2])[Y.sub.1] + [Y.sub.2], and the analysis below would have to be changed accordingly.

A. Aggregate Rent-Seeking

From a technological perspective, the major distinction between productive and rent-seeking activities is that production of rent-seeking services requires the use of output as an input, in addition to other productive factors. One can only "seek rents" that have been produced. The productive activity, on the other hand, requires only capital and labor services. Let G be the total amount of output which is extracted from the productive sector by the rent-seeking sector. Assume that G = G([theta][Y.sub.1], [Y.sub.1]), where the function G is homogeneous of the first degree, [Y.sub.2] is the total output of transfer services, and [theta] describes the quality of the institutional set. A high (low) [theta] represents a bad (good) institutional background. Using the homogeneity of G, we write

(1) G = g ([theta] [[Y.sub.2]/[Y.sub.1]]) [Y.sub.1] = g ([theta]y) [Y.sub.1],

where [y.sub.2] [equivalent to] [[Y.sub.2]/L], y [equivalent to] [[y.sub.2]/[y.sub.1]], and g ([theta][Y.sub.2]/[Y.sub.1]) [equivalent to] G([theta][Y.sub.2]/[Y.sub.1], 1). The function g is the share of the output of the productive sector that is extracted by the rent-seeking sector. (3) As anticipated above, the share g is the tax rate [tau] that firms in Sector 1 take which is given below:

g ([theta]y) = -[tau].

The function g is called aggregate rent-seeking technology. It is a function of the relative output, y = [y.sub.2]/[y.sub.1], multiplied by an institutional variable [theta]. We assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These are natural assumptions. We will impose additional assumptions on g as needed. For now, the maintained assumption is Assumption 1.

For any [theta]y [greater than or equal to] 0, let [[alpha].sub.g] ([theta]y) [equivalent to] [theta]y(g'([theta]y)/ g ([theta]y)) (1/1 - g ([theta]y)). In order to interpret [[alpha].sub.g] ([theta]y), note that it is the normalized elasticity of g([theta]y), where the normalization is given by the term 1/(1-g([theta]y)). More precisely, [[alpha].sub.g]([theta]y) is the relation between the percent increase in what is captured by Sector 2, [theta]yg'([theta]y)/g([theta]y), and the absolute decrease in what is not captured by Sector 2, 1-g([theta]y). Hence, [[alpha].sub.g]([theta]y)<1 can be interpreted as a situation of decreasing returns to the aggregate rent-seeking activity, and [[alpha].sub.g]([theta]y) > 1 can be interpreted as a situation of increasing returns. We can now state:

ASSUMPTION 1. (No Reversal) Either 0 < [[alpha].sub.g]([theta]y) < 1 for all [theta]y [member of][0, [infinity]], or [[alpha].sub.g] ([theta]y) > 1 for all [theta]y [member of] [0, [infinity]].

No Reversal means that either there are always decreasing returns or there are always increasing returns to the aggregate rent-seeking activity. Note that we used the compactified interval [0, [infinity]] in the statement, which ensures that [[alpha].sub.g]([theta]y) is bounded away from 1 in both cases. Below we show that No Reversal ensures existence and uniqueness of a static equilibrium.

Observe that if [[alpha].sub.g]([theta]y) is constant and equal to [[alpha].sub.g], then

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is one candidate for a functional form for g.

In Section III.F, we will see that [[alpha].sub.g]([theta]y) > 1 generates unstable dynamics, and that [[alpha].sub.g] ([theta]y) < I seems to be the more plausible case. In fact, we have:

PROPOSITION 1. If [[alpha].sub.g] ([theta]y) < 1 for all [theta]y [greater than or equal to] 0, then [lim.sub.[theta]y [right arrow]0] g'([theta]y) = [infinity], and if [[alpha].sub.g] ([theta]y) > 1 for all [theta]y [greater than or equal to] O, then [lim.sub.[theta]y [right arrow]0] g'([theta]y) = 0.

Proof. See Appendix C.2.

That is, [[alpha].sub.g]([theta]y) < 1 for all [theta]y implies that an "Inada" condition ([lim.sub.[theta]y[right arrow]0] g'([theta]y) = [infinity]) is satisfied. The very first unit of rent-seeking effort generates large returns. On the other hand, if [[alpha].sub.g]([theta]y) > 1 for all [theta]y, then the opposite is true: the very first unit of rent-seeking effort does not generate any return.

B. Firms

Sector j = 1,2 consists of [N.sub.j] identical firms operating under the same technology. Sector 1's output is the homogeneous good in the economy, and Sector 2's output is a transfer service (an effort to capture Sector 1's output). Firm i in Sector j combines capital, [K.sub.ji], and labor, [L.sub.ji], according to a constant returns to scale technology [F.sub.j] to produce output [Y.sub.ji]. That is, [Y.sub.ji] [equivalent to] [F.sub.j]([K.sub.ji], [L.sub.ji]).

Part of what a firm in Sector 1 produces is captured by the firms operating in Sector 2. That is, the rent-seeking activity acts like a tax rate [tau] on the output of each firm in Sector 1: firm i in that sector keeps only (1 - [tau]) [Y.sub.1i] of its output. Under perfect competition, firm i's program is to

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [r.sub.1] and [w.sub.1] are the rental and wage rates prevailing in Sector 1.

For firms in Sector 2, instead of a tax on their output we have a variable q that determines the effectiveness of the service produced by the firms in Sector 2 to capture Sector 1's output. That is, q = [tau][Y.sub.1]/[Y.sub.2]. Hence q is endogenously determined in equilibrium. But each film in Sector 2 takes q as given. It follows that each firm in Sector 2 solves

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sub.2] and [w.sub.2] are the rental and wage rates prevailing in Sector 2.

The first-order conditions are given by

(5) [r.sub.1] = (1 - [tau])[f'.sub.1]([k.sub.1])

(6) [w.sub.1] = (1 - tau])[[f.sub.1]([k.sub.1]) - [k.sub.1] [f'.sub.1]([k.sub.1])

(7) [r.sub.2] = q[f'.sub.2] ([k.sub.2])

(8) [w.sub.2] = q [[f.sub.2]([k.sub.2]) - [k.sub.2][f.sub.2]([k.sub.2])],

where [f.sub.j] [equivalent to] [F.sub.j]([k.sub.j], 1) and [k.sub.j] [equivalent to] [K.sub.ji]/[L.sub.ji], j = 1,2.

Sector j's total output is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In per capita terms, [y.sub.j] [equivalent to] [Y.sub.j]/L = [l.sub.j] [f.sub.j]([k.sub.j]) where L is the population and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Sector j's labor share.

C. Static Equilibrium

The static equilibrium is an equilibrium in the allocation of productive factors between the two sectors, for given levels of productive factors and institutional efficiency (k and [theta]). The definition of equilibrium uses the underlying two-sector structure of the model. The idea is that each combination of output levels of both sectors determines a marginal rate of transformation and is in turn determined by the latter. The equilibrium is a fixed point of this mutual determination.

We, therefore, take an aggregate approach to the short-run supply side of the economy. Because the model is meant to capture the influence of rent-seeking activities in a macroeconomic framework, the aggregate approach can be justified. Nevertheless, in Appendix B, we provide some microstructure to the rent-seeking sector to demonstrate that the static equilibrium defined below is compatible with a richer microstructure.

Definition and Existence. With perfect factor mobility and an interior solution, it must be the case that [r.sub.1] = [r.sub.2] and [w.sub.1] = [w.sub.2], otherwise all productive factors would be allocated in just one of the industries. From the two-sector general equilibrium model, we know that

(9) [r.sub.j] = [p.sub.j] [f'.sub.j] ([k.sub.i])

(10) [w.sub.j] = [p.sub.j] [[f.sub.j]([k.sub.j]) - [k.sub.j] [f'.sub.j] ([k.sub.j])],

where [p.sub.j] is the price of Sector j's output, j = 1,2. Comparing Equations (5)-(8) with (9) and (10), we have that the situation where [p.sub.1] = 1 - [tau] and [p.sub.2] = q, or q/1 - [tau] = p, is an equilibrium, where p = [p.sub.2]/[p.sub.1] is the marginal rate of transformation (MRT).

In words, as factors are reallocated between industries, p shows how y changes and q/1 - [tau] shows how the relative output in terms of goods changes. In equilibrium, those changes must be equal.

Because q = [tau][Y.sub.1]/[Y.sub.2], we have [tau] = qy. Using g = [tau] and [y.sub.i](p, k) as Sector i's static supply function, (4) we have:

DEFINITION 1. The symmetric static equilibrium with perfect factor mobility is defined by a MRT p such that

(11)

p = [g([theta]y(p, k))/y(p,k)] [1/1-g ([theta]y(p, k))] [equivalent to] H(p,k, [theta]).

Note that the MRT p determines the common factor price ratio w/r and Sector i's inputs [k.sub.i] and [l.sub.i]. If p > H(p, k, [theta]) (p < H(p, k, [theta])), then factors will move toward Sector 2 (Sector 1), reducing (increasing) p and increasing (reducing) H because Sector 2 (Sector 1) pays relatively more. In equilibrium, factor prices are equalized and there is no further factor reallocation.

Let [p.bar] and [bar.p] be the prices under which the economy is specialized in Sectors 1 and 2, respectively. That is, for a given level of factor endowment k, p [greater than or equal to] [bar.p] (k) (p [less than or equal to] p(k)) means that the economy is specialized in the production of rent-seeking services (Sector 1's good). Note that [bar.p]'(k) [??] 0 and p'(k) [??] as [k.sub.1] [??] [k.sub.2].

PROPOSITION 2. Under No Reversal, a static equilibrium exists and is unique.

Proof See Appendix VI.

The proof shows that H(p, k, [theta]) begins at infinity and drops down to zero when [[alpha].sub.g] < 1, and when [[alpha].sub.g] > 1 it begins at zero and increases to infinity (as p varies from [p.bar] to [bar.p]). Hence the static equilibrium is stable when [[alpha].sub.g] < 1 and unstable when [[alpha].sub.g] > 1.

Although existence of a static equilibrium is crucial for logical consistency of the model, one can argue that uniqueness is not important, and even not desirable. In fact, because rent-seeking creates negative externalities in an economy, one should expect multiple equilibria arising from coordination failures. Moreover, as Engerman and Sokoloff (1997) argue, coordination failures, history dependence, and political economy issues help understanding why fundamental properties of economies differ. Inspecting the proofs of Propositions 1 and 2, we can extract the following:

PROPOSITION 3. A static equilibrium exists if either (1) [lim.sub.[theta]y [right arrow]0]g' ([theta]y) = [infinity] and [lim.sub.[theta]y [right arrow] [infinity]] [[alpha].sub.g] ([theta]y) < 1 or (2) [lim.sub.[theta]y [right arrow]0] g'([theta]y) = 0 and [lim.sub.[theta]y [right arrow] [infinity]] [[alpha].sub.g] ([theta]y) > 1.

In fact, [lim.sub.[theta]y [right arrow] 0] g'([theta]y) = [infinity] ([lim.sub.[theta]y [right arrow]0] g' ([theta]y)=0) ensures that H(p,k, [theta]) begins at infinity (zero) and [lim.sub.[theta]y [right arrow] [infinity]] [[alpha].sub.g]([theta]y) < 1([lim.sub.[theta]y [right arrow] [infinity]] [[alpha].sub.g] ([theta]y) > 1) ensures that it drops down to zero (increases to infinity). In particular, there can be many reversals in the nature of the returns to the aggregate rent-seeking activity. The case to avoid is the case of constant returns ([[alpha].sub.g] ([theta]y) = 1 for all [theta]y), as an equilibrium cannot be shown to exist.

In what follows we will assume implicitly either No Reversal and work with the unique static equilibrium, or the conditions in Proposition 3 are met and fix one static equilibrium throughout. That is, we will not consider dynamic effects of moving between two different static equilibria.

D. National Accounting in the Presence of Rent-Seeking

Observe that in this economy to produce one unit of good does not imply ownership of it. There are, therefore, three goods in this two-sector economy: the good, the rent-seeking service, and the good in somebody's hands. The price [p.sub.1] = 1 - g is the relative price of one unit of the good in units of goods at somebody's hands, and [p.sub.2] = g/y is the relative price of one unit of the rent-seeking service in units of goods at somebody's hands. Given the equilibrium that we constructed, it follows that

[y.sub.1] = [p.sub.1] [y.sub.1] + [p.sub.2][y.sub.2]

= [l.sub.1](w + [rk.sub.1]) + [l.sub.2](w + [rk.sub.2])=[p.sub.1] (w + rk), so one can view [p.sub.1] / [y.sub.1] + [p.sub.2] [y.sub.2] as total output of the economy in units of goods at somebody's hands. Moreover, the above equations show that in terms of national accounting, the equilibrium implies that total output can be computed as the sum of the value added in both industries. That is, if rent-seeking is an illegal activity, then total output would be computed as [y.sub.1] (what is produced and then captured by the outlaws). But if it is a legal activity, then its value added is [p.sub.2][y.sub.2] = g[y.sub.1] (that is the "output" of Sector 2), and the above equation shows that the usual accounting procedure (both from the value added and from the income perspectives) works. In other words, the static equilibrium takes into account the fact that rent-seeking activities might appear in GDP statistics. (5)

E. Consumers

At a point in time, that is, for the given values of k and [theta], the static model is solved yielding p, [y.sub.i](p,k), and the factor prices r and w. The representative household rents her capital and labor services to the firms. She chooses a consumption plan that solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

given k(0), where [rho] is the intertemporal discount rate, and [delta] is the physical depreciation rate.

This is the standard Ramsey problem that yields the following Euler equation

(12) c (t) = c(t)[gamma](c(t)) (r(t) - [rho] - [delta]),

where [gamma] is the intertemporal elasticity of substitution, and r = (1 - g([theta]y(p, k))[f'.sub.1] [([k.sub.1] (p))) as derived before.

From our discussion in Section III.D, it follows that

r(t)k(t) + w(t) = [p.sub.1](t)[y.sub.1](t) + [p.sub.2](t)[y.sub.2](t) = [y.sub.1] (t),

so that the dynamics are represented by the following dynamic system (omitting the variable t for ease of notation)

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

together with the initial condition for capital, k(0), and the terminal condition [lim.sub.t [right arrow]] [e.sup.-[rho]t] u' (c(t)) k (t) = 0, where p = p(k, [theta]) is the short-run equilibrium.

F. Long-Run Equilibrium

The long-run equilibrium is given by a capital stock and a relative price that satisfy the conditions of a steady state of the dynamic system (13). In other words, the following system of equations must hold in the long run:

(14) [[psi].sub.1] (p,k) = [g ([theta]y(p, k))/y(p, k)] [1/1- g ([theta]y(p, k))] -p =0

(15) [[psi].sub.2](p, k) = (1 - g([theta]y(p, k))) [f'.sub.1](([k.sub.1](p)) -([rho]+[delta])=0.

In terms of stability, the condition for saddle point stability of Equation (13) is that the Jacobian of the linearized system be negative. That is,

(16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Appendix D.1 shows that [[alpha].sub.g] < [alpha].sub.1L] is a sufficient condition for the inequality above to hold. Accordingly, in what follows we will substitute No Reversal with the following:

ASSUMPTION 3. [[alpha].sub.g]([theta]y) < [[alpha].sub.1L] for all [theta]y [member of] [0, [infinity]].

In terms of existence and uniqueness, the second case in No Reversal, [[alpha].sub.g]([theta]y) > 1, is a sufficient condition (see Appendix C.3). It is the first case that needs to be strengthened by Assumption 3 above. On the other hand, because [[alpha].sub.g]([theta]y) > 1 generates unstable dynamics (and also unstable static equilibrium), we invoke Samuelson's Correspondence Principle and appeal to Assumption 3.

PROPOSITION 4. Under Assumption 3, the long-run equilibrium exists and is unique.

Proof See Appendix C.3.

The idea of the proof is shown by Figures 1 and 2. They represent the system [[psi].sub.1] (p, k) = 0 and [[psi].sub.2] (p, k)= 0 when the production functions are Cobb-Douglas and the aggregate rentseeking function is given by Equation (2). (The curve [[psi].sup.M.sub.1] = 0 refers to the monopoly solution of the models. See Section VI.) Figure 1 considers Sector 1 as capital intensive (the parameter values are {([[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.g], [theta], [delta], [rho]} = {1/3, 1/6, 1/8, 1, log(1.066), log(1.03)}) and in Figure 2, Sector 1 is labor intensive (the parameters are {1/6, 1/3, 1/8, 1, log(1.066), log(1.03)}). As it is clear from the figures, [[psi].sub.1](p, k) =0 and [[alpha].sub.2] (p, k) = 0 must intersect once, and only once.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

IV. PROPERTIES OF THE MODEL

A. Comparative Statics

From Equation (11), the effects of k and [theta] on the static equilibrium p are given by

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the long run, capital is endogenous and given by Equation (15). The only exogenous variable is [theta], the variable that captures the efficiency of the institutional background. When Sector 1 is capital intensive, the results are intuitive: a deterioration of institutional efficiency generates less capital and output in the long-run. But when the rent-seeking sector is capital intensive, then the reverse result cannot be ruled out. That is, it can be that as the institutional background becomes worse, long-run capital decreases and/or long-run income increases.

To understand this last point, observe that the return on capital is pinned down by the rate of time preference, so an increase in [theta] must lead to a reduction in [k.sub.1] and [k.sub.2]. When workers are reallocated from the productive to the rent-seeking sector, we may have an decrease in k whenever the rent-seeking sector is capital intensive. Moreover, we may still have a significant increase in [l.sub.i], which more than offsets the decrease in [k.sub.1], resulting in an increase in long-run output.

B. Features of the Dynamics

If the economy is not at its long-run equilibrium, it is at a transition path of capital accumulation. Along the transition path, in general, we have both capital accumulation and increasing output. When the rent-seeking sector is capital intensive, we can also have capital accumulation and decreasing output.

PROPOSITION 5. Along a transition path between steady states, both the relative output [y.sub.2]/[y.sub.1] and the relative value of Sector 2's output are monotone in k. That is, (dy/dk)[|.sub.[theta]] [??] 0 and (p[y.sub.2]/([y.sub.1] + p[y.sub.2]))[|.sub.[theta]] [??] 0 as [k.sub.1] [??] [k.sub.2].

Proof See Appendix D.2.

In particular, (d[y.sub.1]/dk) [|.sub.[theta]] > 0 if [k.sub.1] > [k.sub.2], and (d[y.sub.2]/dk) [|.sub.[theta]] > 0 if [k.sub.1] [less than or equal to] [k.sub.2]. We can only guarantee that output is increasing along the transition if Sector 1 is capital intensive.

Appendix D.2 also shows that (d[y.sub.1]/dk) [|.sub.[theta]] is indeterminate when [k.sub.1] < [k.sub.2]. Hence, when Sector 2 is capital intensive, we can have a situation where total output of the economy decreases while the capital stock increases.

Note that [k.sub.1] < [k.sub.2] is not only a theoretical possibility. Take a very underdeveloped economy, from sub-Saharan Africa, for instance. Its productive sector is the agricultural sector. Its rent-seeking sector is the army and armed bands. Because the agricultural sector in these countries is not capital intensive, it makes sense to consider the rent-seeking sector as the capital-intensive sector for this economy. Many sub-Saharan countries have been experiencing negative growth rates and positive investment. One way of explaining is that investment has been directed mainly to rentseeking activities. As an example, Appendix D.2 shows that, for the extreme case that the productive sector only employs labor ([alpha]1K = 0), the condition 1 > (1 - [[alpha].sub.g]) [[sigma].sub.2] is sufficient to ensure that (d[y.sub.1]/dk)[|.sub.[theta]] < 0, and this is the case as long as 0 [less than or equal to] [[sigma].sub.2] [less than or equal to] 1, which is not a strong assumption.

C. Scale Effects

The model was solved under the assumption of no technological progress. In one-sector exogenous growth models with technological progress, one can re-scale the variables to achieve a dynamic system with no autonomous part. With two sectors, this re-scaling only works when both sectors operate under the same technology. In other words, when [k.sub.1] [not equal to] [k.sub.2], our model does not deliver a balanced long-run solution. If [k.sub.1] > [k.sub.2] (which seems to be the case for developed economies), then the long-run solution implies that [y.sub.2]/[y.sub.1] increases over time (for a given level of institutional efficiency, [theta]). Likewise, if [k.sub.1] < [k.sub.2], then [y.sub.2]/[y.sub.1] decreases over time. The intuition behind this result is simple: with labor-saving technological progress, the labor-intensive industry becomes relatively more productive and attracts relatively more factors of production.

Using the predictions of our model, therefore, we expect an increase in the relative size of the rent-seeking sector in developed economies, for periods of time where the assumption of a given level of institutional efficiency is plausible. An indication of such phenomenon can be inferred from two papers that measure the aggregate burden of crime in the United States. Becker (1968) estimates that crime accounted for 4% of United States' GDP in 1965, while for Anderson (1999) this number increased to 10% in the 1990s. Therefore, in about 30 years, one part (the crime sector) of the rentseeking sector increased 150% as a fraction of GDP. A corollary of such prediction is that the efficiency of the institutions has to change eventually, otherwise a developed economy would become flooded with rent-seeking activities.

This provides one reason why institutions change over time. We can test the model with this prediction: if we have indication that over a long period of time [y.sub.2]/[y.sub.1] remained constant, and that [k.sub.1] > [k.sub.2] is a reasonable assumption, then, from the model, institutions must have become more efficient in protecting property rights. For the model to be consistent with these facts, we must also be able to ascertain that institutions did indeed become more efficient.

V. WELFARE ANALYSIS

The results above show that the dynamics of the model depend on factor intensity. That is, we have an indeterminacy. Such is not a concern when the welfare analysis is considered. There is a monotone relation between institutional efficiency and overall welfare in the economy. The worse the institutional background, the lower the welfare enjoyed by the representative consumer. The relevant criterion for evaluating economic performance is welfare, and under such criterion the variable [theta] does represent the "underlying determinants of economic performance," as Douglas North would put it.

A deterioration in the institutional set of the economy generates two effects. First, an increase in [theta] increases p (see Equation [18]) producing an inflow of factors toward the rent-seeking sector and a reduction in the productive sector's output. This is called Tullock effect. Second, from Equation (5), an increase in [theta] increases the distortion to capital accumulation (because it reduces 1 - g). This is called Harberger effect. Under the assumption that initially the economy is in long-run equilibrium, we show that (1) the welfare effect is composed of two components, which are the two abovementioned effects, (2) the marginal impact of a reduction on institutional efficiency is a reduction in welfare, and (3) if both sectors operate under the same technology, then Harberger effect is null.

Given that the economy is a representative agent economy, the intertemporal utility is the social welfare function. In order to evaluate the welfare impact of a marginal increase in [theta], taking into consideration the transitional dynamics, a technique developed by Judd (1982, 1987) is employed. Let W = [[integral].sup.[infinity].sub.0] [e.sup.-[rho]] u (c(t))dt be the welfare index. The impact of [theta] on W at steady state (denoted by an *) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [X.sub.[theta]]([??]) [equivalent to] [[integral].sup.[infinity].sub.0] [e.sup.-[??]t] (dx(t)/d[theta])dt is the Laplace transform of dx(t)/d[theta] for any function x(t). Hence, the effect on welfare is given by the Laplace transform (C[theta]([rho])) of (dc(t)/d[theta]) multiplied by marginal utility evaluated at [c.sup.*].

PROPOSITION 6. The impact of [theta] on W at steady state, (dW/d[theta])[|.sub.*], is negative. In particular, it is equal to the sum of two components, where the first is always negative and dominates the second. The first is called Tullock effect, and the second is called Harberger effect.

Proof Appendix E. 1 shows that

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [mu] is the positive eigenvalue associated with the matrix of the linearized dynamic system and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence, the impact of [theta] on welfare is given by the sum of two terms that are identified as Tullock and Harberger effects.

Now, under any configuration, Tullock effect is negative (see Equation [18]), Appendix E.2 shows that 0 [less than or equal to] AF [less than or equal to] 1, and Appendix E.4 shows that |(d [y.sub.1]/d[theta]) [|.sub.k,*]| > |((d[y.sub.1]/dk)[|.sub.[theta]],*]- [rho] - [delta]) (dk/d[theta])[|.sub.*] |. Hence

[d[y.sub.1]/d[theta]] [|.sub.k,*] + AF([d[y.sub.1]/dk] [|.sub.[theta],*] - [rho] - [delta]) [[d.sub.k]/d[theta]] [|.sub.*] < 0,

and we have proved.

Therefore, the effect of a change in the institutional background on welfare is unambiguous: welfare is reduced when institutions are less efficient. Of course, one has to take into account the costs of changing [theta] to determine whether it is worthwhile to undertake an institutional change. Because [theta] is exogenous to our model, we cannot assess the costs of changing it. Proposition 6 shows that it is always potentially welfare improving to make the institutional background more efficient in preventing rent-seeking activities.

Tullock effect is given by the instantaneous reduction on output, and consequently on consumption, resulting from the deterioration of the institutional set and the corresponding increase in the relative size of the rent-seeking industry.

Harberger effect is the composition of two terms. One is the net marginal impact of capital accumulation on output (net of physical depreciation and of the intertemporal opportunity cost of investment),

(20) (d[y.sub.1]/dk[|.sub.[theta],*] - [rho] - [delta]) [[d.sub.k]/d[theta]] [|.sub.*]].

The other is the attenuation factor (AF), which translates a change in the output because of capital accumulation into a change in welfare taking into consideration the transitional dynamics. The AF is increasing in the intertemporal elasticity of substitution [gamma] (c). Note that [lim.sub.[gamma] [right arrow]0] AF = 0 and [lim.sub.[gamma] [right arrow] [infinity]] AF = [lim.sub.[gamma] [right arrow] [infinity]] ([mu] - [rho])/[mu] = 1.

In Appendix D.2.2, we show that

d[y.sub.1]/dk [|.sub.[theta],*] - [rho] - [delta] [??] 0 as [k.sub.1] [??] [k.sub.2],

so, from Equation (20), Harberger effect is zero if [k.sub.1] = [k.sub.2]. That is, with rent-seeking, k is not only a good, it can also be a bad (when employed in Sector 2); as k increases, the economy gains because [k.sub.1] increases, and loses because [k.sub.2] also increases. Those two effects cancel out when technologies are equal in both sectors. One can argue, then, that Harberger effect on welfare is of second order. Not only is it always dominated by Tullock effect, it is only non-zero when technologies are not the same in both sectors.

Observe an analogy of Equation (19) with the Slutsky equation of consumer theory: a change in [theta] may be viewed as a change in the price of the consumption good. The total effect is then separated into the substitution effect (the Tullock effect, always of the right sign) and the income effect (the Harberger effect, which has ambiguous sign). What is shown is that the substitution effect always dominates the income effect making the consumption good an "ordinary" good.

This is the main qualitative result of the model. It makes a case for improving efficiency of institutions of property rights enforcement based on welfare grounds. Alternatively, it states that the main problem of a rent-seeking activity is that it employs productive resources that could have been employed socially valued activities.

VI. CONCLUDING REMARKS

We took an aggregate perspective to modeling rent-seeking in an economy. Our goal was to provide a simple formalization to help understanding the impact of rent-seeking activities on economic growth. It is our understanding that such a formalization has not yet been provided in the literature of economic growth. We have not provided new insights on the causes of rent-seeking activities. Rather, we summarized such causes in an aggregate rent-seeking technology g and a reduced-form parameter [theta] representing the efficiency of the institutional set in preventing such activities. Hence, our starting point was that rent-seeking activities exist and ought to be considered in models of economic growth. Such approach is analogous to the use of the cash-in-advance constraint in monetary models: money exists and ought to be considered in macroeconomic models. Likewise, our focus is on the implications for economic development of the existence of rent-seeking activities.

Let us now list a few interesting remarks on the model:

* Non-competitive rent-seeking sector and transition economies: With limited entry in the rent-seeking sector, we can show that less rentseeking will be generated and society will benefit. This fact can be used to describe some dynamics that ensued from three different historical events of the second half of last century: the end of European colonization in the 1960s and 1970s in many African countries, the end of political regimes based on military dictatorship in many Latin America countries in the 1980s, and finally the "fall of the wall" leading to the end of the communist regimes in east Europe in the late 1980s and early 1990s. These three recent episodes of the world history share one fundamental characteristic: there is a transition from a centralized political (economic) organization toward a more decentralized system. And such transitions were all accompanied by a period of economic recession. One rationale for that is provided by our monopoly result. That is, assuming that monopoly in rent-seeking takes place in either a colony (the European imperial power being the monopolist), or in a military dictatorship (the army being the monopolist), or a centralized economy (the communist party being the monopolist), a given level of institutional efficiency is associated with a better economic performance in the more centralized system. Also, the transition to a more open system of organizing either the politics or the economy means a lifting of the barriers to entry in the rent-seeking sector, so the economy is bound to experience a recession as productive resources are directed to rent-seeking activities.

* Foreign aid and rent-seeking: Consider now the issue of foreign aid in the presence of rent-seeking. There is a concern that aid, if the recipient economy does not have a good institutional set, is a waste of resources: it ends up as consumption, without any effect on the productive capacity of the economy (Burnside and Dollar 2000). The model shows that things might be even worse when rent-seeking is considered: the aid generates an increase in rent-seeking activities, so that the society would be better off if the aid was given directly as consumption goods to the households. To see this, consider the aid as a permanent flow, A, of resources per capita, so that per capita income becomes [y.sub.1] + A. The short-run equilibrium condition (11) is still valid, now with y = [y.sub.2]/([y.sub.1] +A). Consequently, in the short-run, an increase in A induces factors of production to move into the rent-seeking sector. Given the increase in the "pie," there are more resources to be transferred, and hence the transfer efforts increase. In the long-run, an increase in aid leads to an increase in y2 and a decrease in [y.sub.1]. Hence, there is an unambiguous increase in rent-seeking activities generated by the foreign aid.

* Endogenous TFP: The share of productive factors allocated in Sector 1 can be viewed as an endogenous part of the TFP. The output of an economy increases (decreases) as productive factors move from Sector 2 (1) to Sector 1 (2), and this happens for a given level of productive factors. So this is not accounted as a change in output because of a change in productive factors, hut because of a change in the productivity of the existing factors, i.e., a change in TFP. More formally, consider first the case with [k.sub.1] = [k.sub.2]. Then [y.sub.1] = [l.sub.1] f(k) = (1 - [l.sub.2])f(k), and the term 1 - [l.sub.2] can be viewed as (part of) the TFP. The bigger the rent-seeking sector, the less productive the economy. That is, for a given k, an increase in [l.sub.2] represents a decrease in TFP, because less output is produced by the same level of productive factors, k. The same intuition is also valid for generic values of [k.sub.1] and [k.sub.2]. In other words, let us assume that our model describes well two economies that are identical in every aspect but differ in the parameter [theta]. Then an observer looking at these economies from the viewpoint of a one-sector aggregate model would conclude that the economy with the smaller [theta] is the economy with higher TFP, although both economies operate under the same technology by assumption. In this sense, TFP (or part of it) is endogenously determined by the institutional efficiency.

Moreover, there is also a dynamic issue involved. A once-and-for-all change in institutional efficiency is given by a discrete jump in [theta]. This generates an immediate reallocation of factors between the sectors, and so an immediate change in TFP. But the economy enters in a transitory dynamic path toward its new steady state, and along this path further reallocations of factors take place. That is, along this path the share of the labor force allocated in the productive sector keeps changing and this is observationally equivalent to a continuous change of the TFP if a one-sector economy perspective is considered. Such dynamic behavior is to be contrasted with the usual exercises in the literature of considering once-and-for-all changes in TFP itself: the resulting transitory dynamics of capital accumulation does not include an associated transitory dynamics of TFP. Also, there is some evidence that TFP is indeed not constant. A simple look at the Summers and Heston data set reveals that several countries, like Venezuela, endured a process of reduction in TFP for the period of 1960-1990. Other countries, like Ireland in the 1990s, endured the opposite process. The model provides an immediate rationale for such facts, as one can easily argue that Venezuela and Ireland witnessed changes in institutional efficiency prior to (or during) the period in question. The issue of endogeneity of TFP is of great interest (see Prescott 1998) and ought to be studied further.

Finally, let us point out that rent-seeking activities cannot be viewed as the sole factor missing in the standard models of economic growth. A complete picture ought to include every factor that affects the accumulation as well as the quality of productive factors. Rentseeking activities (as modeled here) affect only the accumulation of capital, and are mute with respect to accumulation of human capital and technological change. A more general description of the impact of rent-seeking activities should include their effects on the accumulation of human capital, on technological change, and on the labor supply decision, among other factors. Such effects are likely to increase the power of our parameter [theta] in explaining the observed differences in per capita income among countries. Nevertheless, our simulation results show that rent-seeking activities are an important element in helping explaining long-run performance of economies, even without considering the effects mentioned above.

ABBREVIATIONS

AF: Attenuation Factor
CSF: Contest Success Function
GDP: Gross Domestic Product
MRT: Marginal Rate of Transformation
TFP: Total Factor Productivity

doi:10.1111/j.1465-7295.2010.00318.x

APPENDIX A THE STATIC MODEL: EXISTENCE AND COMPARATIVE STATICS PROPERTIES

A.1 The Two-Sector Model of Production

The following equations describe the 2 x 2 static model:

(A1) [y.sub.i] = [L.sub.i]/L [f.sub.i] ([K.sub.i]/[L.sub.i]) = [l.sub.i][f.sub.i]([k.sub.i]) i=1,2,

(A2) [l.sub.1] +[l.sub.2] = 1,

(A3) [k.sub.1] [l.sub.1] + [k.sub.2][l.sub.2] = k,

(A4) w = [p.sub.i]([f.sub.i] - [k.sub.i] [f'.sub.i]),

(A5) r = [p.sub.i] [f'.sub.i],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From Equations (A4) and (A5), we have

(A6) [omega] [equivalent to] w/r = [f.sub.i]/[f'.sub.i] - [k.sub.i],

or

(A7) [k.sub.i] = [k.sub.i] ([omega]), [d[k.sub.i]/d[omega]]= - [[([f'.sub.i]).sup.2]/[f.sub.i][f".sub.i]] > 0.

The relative price is the ratio of average costs:

p [equivalent to] [[p.sub.2]/[p.sub.1]] = [[(w[L.sub.2] + r[K.sub.2])/[L.sub.2][f.sub.2]] / [(w[L.sub.1]+r[K.sub.1])/[L.sub.1][f.sub.1]]] = [[omega]+[k.sub.2]/[omega]+[k.sub.1]] [[f.sub.1]/[f.sub.2]],

which is solved as

(A8)

[omega] = [omega] (p), [p/[omega]] [d[omega]/dp] = [([omega]+[k.sub.1])([omega]+[k.sub.2])/[omega]([k.sub.1]- [k.sub.2])] [??] 0 as [k.sub.1] [??] [k.sub.2].

Solving Equations (A2) and (A3) for [l.sub.i], after substituting into Equation (AI), we have the following supply functions:

(A9) [y.sub.1](p, k)= [l.sub.1][f.sub.1]([k.sub.1])= [[k.sub.2] ([omega] (p)) - k / [k.sub.2] ([omega] (p)) - [k.sub.1] ([omega] (p))] [f.sub.1] ([k.sub.1] ([omega] (p))),

(A10) [y.sub.2](p, k)= [l.sub.2][f.sub.2]([k.sub.2])= [k - [k.sub.1] ([omega] (p)) / [k.sub.2] ([omega] (p)) - [k.sub.1] ([omega] (p))] [f.sub.2] ([k.sub.2] ([omega] (p))).

Usually, we will write simply [k.sub.i] ([omega] (p)) = [k.sub.i] (p).

Finally, for a given factor endowment, there is a price [bar.p](k) such that the economy is specialized in the production of the rent-seeking service if p [greater than or equal to] [bar.p] (k), and there is a price [p.bar](k) such that the economy is specialized in the production of the first sector good if p [less than or equal to] p (k). Note that [bar.p](k) [??] 0 and [p.bar] k)[??] 0 as [k.sub.1] [??] [k.sub.2]. (6)

A.2 Comparative Statics

The following notation is employed from now on:

(A11) [[alpha].sub.iK] [equivalent to] 1 - [[alpha].sub.iL] = [k.sub.i], [[f'.sub.i]/[f.sub.i]],

(A12) [[sigma].sub.i] [equivalent to] [[omega]/[k.sub.i]] [d[k.sub.i]/d[omega]]= [[[alpha].sub.iL]/[[alpha].sub.iK]] [d[k.sub.i]/d[omega]]= - [[[alpha].sub.iL]/[[alpha].sub.iK]] [[([f'.sub.i]).sup.2]/[f.sub.i][f".sub.i]].

From Equation (A6), we have

(A13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whether [[alpha].sub.1K] [??] [[alpha].sub.2K].

The comparative statics for Equations (A9) and (A10) are:

(A14)

[k/[y.sub.2]] [[partial derivative][y.sub.1]/[partial derivative]k] [|.sub.p] = [k/k - [k.sub.2]] = [1/[l.sub.1]] [k/[k.sub.1] - [k.sub.2]],

(A15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

6. See Kemp (1969), chapter 1.

and

(A16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [l.sup.R] [equivalent to] [l.sub.2]/[l.sub.1]. In deriving these last two equations, we employed Equations (A3), (A6) (A8), and (All)-(A13). From Equations (A14) and (A15), we have:

(A18)

[1 - g/r] [[partial derivative][y.sub.1]/[partial derivative]k][|.sub.p] = [[[alpha].sub.2L]/ [[alpha].sub.1K] - [[alpha].sub.2K]],

(A19)

[1 - g/r] [[partial derivative][y.sub.2]/[partial derivative]k] [|.sub.p] = - [y/[l.sup.R]] [[[alpha].sub.2L]/ [[alpha].sub.1K] - [[alpha].sub.2K]] = - [1/p] [[[alpha].sub.1L]/[[alpha].sub.1K] - [[alpha].sub.2K]],

where in the last equation, we substitute Equations (11) and (18).

We will use the following results:

(A20) [[partial derivative][y.sub.1]/[partial derivative]p][|.sub.k] + p [[[partial derivative][y.sub.2]/[partial derivative]p] [|.sub.k] = 0,

(A21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation (A20) follows from Equations (A16), (A17), and the static equilibrium condition. Equation (A21) follows from Equations (A18) and (A19).

APPENDIX B ALTERNATIVE DERIVATION OF THE STATIC PROBLEM--SOME MICROSTRUCTURE

The approach above was an aggregate approach to the short-run supply side of the economy. Because the idea of the paper is to introduce rent-seeking activities in a macroeconomic framework, we view such approach as appropriate. Nevertheless, below we present one way of providing some microstructure to the rent-seeking sector. The underlying idea is that the static equilibrium defined above is compatible with a richer microstructure.

B.1 Rent-Seeking Firm

Assume that the quantity of goods that a firm in Sector 2 expropriates from Sector 1 is a share of the total booty G in Equation (I). In particular, we assume that this share is of the additive CSF form] so that it can be written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. That is, firm i will fight for a share of G and the success of such a fight will be determined by the CSF. We assume that h(0) = 0 and h'(x) > 0, for any x [greater than or equal to] 0. In addition, we require that:

ASSUMPTION 2. There exists a unique [bar.x] such that arg [max.sub.x] (h(x)/x) = [bar.x].

7. Tullock (1980) introduced the CSF in the theory of rent-seeking. Hirshleifer (1989) named it and established its main properties and Skaperdas (1996) axiomatized the additive CSF.

In other words, there exists one, and only one, optimal scale for each firm in Sector 2. Observe that Assumption 2 does not require uniqueness of a point where marginal returns equal average returns, it only posits the existence of just one point that maximizes average returns.

Now assume that the analysis is made on the limit in which there are many firms in each sector so that the Dixit and Stiglitz (1977) assumption of discharging terms that depend on 1/[N.sub.1] or 1/[N.sub.2] from the first-order conditions can be made. That is, the optimal size [bar.x] of a firm in Sector 2 is small so that in equilibrium [N.sub.2] is large. Consequently, firm i's program becomes

(B1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the first-order conditions are

(B2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(B3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can think of firm i's program as a two-stage program: the aggregate proceeds of the rent-seeking industry are determined in the first stage, and then distributed among the firms in that industry in the second stage. Such a perspective reveals the lack of property rights on the proceeds of the rent-seeking industry: any given firm has the potential to access the whole aggregate booty. (8)

B.2 Free Entry

Equilibrium within each sector is achieved when each firm makes zero profit. From Equations (5) and (6), we have [[pi].sub.1i] = 0 for any i [member of] [N.sub.1] (hence [N.sub.1] is indeterminate). For Sector 2, substituting Equations (B2) and (B3) into Equation (B1) yields

(B4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is not necessarily zero. Here is where Assumption 2 plays a role. Setting [Y.sub.2i] = [bar.x]/[theta] in Equation (B4) yields [[pi].sub.2i] = 0 (because h'([bar.x]) = h([bar.x])/[bar.x]), so this level of [Y.sub.2i] for every firm in Sector 2 is an equilibrium with flee entry. It is unique by hypothesis.

Substituting the free entry condition [[pi].sub.2i] = 0 into Equations (B2) and (B3), it follows that a symmetric equilibrium is given by

(B5) [r.sub.2] = [g ([theta]y)/y] [f'.sub.2]([k.sub.2]),

(B6) [w.sub.2] = g [([theta]y)/y] [[f.sub.2]([k.sub.2]) - [k.sub.2] [f'.sub.2] ([k.sub.2])].

8. In the contest of rent-seeking, this formalization is equivalent to the idea of a competitive labor market: any firm and any worker potentially have access to the whole market. Such idea is quite useful, because one does not have to deal with the complicated dynamics of search and matching that is likely to describe real-world employer-employee relationships. We use the same principle here to describe the unproductive industry.

B.3 Equilibrium

With Equations (B5) and (B6), in the place of Equations (7) and (8), it is clear that the equilibrium condition remains (Equation [11]).

APPENDIX C EXISTENCE

C.1 Proof of Proposition 2

Let H:[p, [bar.p]] [right arrow] [R.sub.+] be the mapping defined by H(p) [equivalent to] H(p, k, [theta]) for given (k, [theta]), so that the static equilibrium p(k, [theta]) is given by a fixed point of H. Taking derivatives and rearranging:

[p/H(p)] [dH(p)/dp] = ([[alpha].sub.g] -1]) [p/y(p)] [dy(p)/dp] [??] 0 as [[alpha].sub.g] [??] 1,

because [dy(p)/dp] > 0. Assumption 1, therefore, ensures that H(p) is strictly monotone in p (decreasing if [[alpha].sub.g] < 1 and increasing if [[alpha].sub.g] > 1). Note that the effect of p on H is through its effect on y, so, for a given [theta] < [infinity], we can also write H as H(y). And note that [lim.sub.p [right arrow][p.bar]+] H(p)= [lim.sub.y [right arrow]0] H(y) and [lim.sub.p [right arrow][bar.p]] H(p) = [lim.sub.y [right arrow] [infinity]] H(y), so we will look at the limits as y varies in [0, [infinity]].

In what follows, Case 1 refers to the situation that [[alpha].sub.g] ([theta]y) < 1 for every [theta]y and Case 2 refers to [[alpha].sub.g]([theta]y) > 1 for every [theta]y. Note that in Case 1, there exists [[bar.[alpha]].sub.g] such that 1 > [[bar.[alpha]].sub.g] > [[alpha].sub.g] ([theta]y) for every [theta]y, and in Case 2 there exists [[[alpha].bar].sub.g] such that 1 < [[[alpha].bar].sub.g] < [[alpha].sub.g] ([theta]y) for every [theta]y. Finally, let [epsilon] > 0 be such that [[bar.[alpha]].sub.g] + [epsilon] < 1 < [[[alpha].bar].sub.g] - [epsilon].

Because g is strictly monotone and raises up from 0 to 1 as [theta]y moves from 0 to [infinity], for any given [theta] there exists [??],([theta]) > 0 such that g([theta][??]([theta])) = 1/2. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for Case 1 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for Case 2,

and note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the functions [bar.q](Y) and [q.bar] (y) are-strictly increasing functions with [bar.q] (0) = [q.bar](0) = 0. Also, because [[alpha].sub.g]([theta]y) < [[bar.[alpha]].sub.g] and [[alpha].sub.g] ([theta]y) > [[[alpha].bar].sub.g] [bar.q](y) is concave, and [q.bar](y) is convex (because the derivative is smaller (larger) than the average).

Consider y [right arrow] [infinity] first. For Case 1, compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because of concavity, if [bar.q"](y) = 0 for some y, then [bar.q"] (x) = 0 for all [greater than or equal to] y. So if there is a y such that [bar.q"] (y) = 0, we must have [([theta][[alpha]'.sub.g]([theta]x))/[[alpha].sub.g] ([theta]x)] = (1/x)([[bar.[alpha]].sub.g] - [[alpha].sub.g] ([theta]x))/[[bar.[alpha]].sub.g] for all x [greater than or equal to] y. That is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have a contradiction, and conclude that [bar.q"](y) < 0 for all y. Hence, eventually [bar.q](y) < y as y increases, and if [bar.q] (y) < y for some y then [bar.q] (x) < x for all x > y. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For Case 2, the argument is analogous. That is, compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because of convexity, if [q".bar] (y) = 0 for some y, then [q".bar] (x) = 0 for all x [greater than of equal to] y. So if there is a y such that [q".bar] (y) = 0, we must have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], That is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have a contradiction, and conclude that [q".bar](y) > 0 for all y. Hence, eventually [q.bar](y) > y as y increases, and if [q.bar] (y) > y for some y then [q.bar](x) > x for all x > y. Thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now consider y [right arrow] 0. For Case 1, if there exists a y such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Likewise, for Case 2, if there exists a y such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because g([theta][??]([theta])) = 1/2, we have [bar.q]([??]([theta])) = [q.bar] ([??]([theta])) = [??]([theta]), so for both cases we have y > 0 such that [bar.q](y) = y and [q.bar](y)= y. We showed above that [bar.q"] (y) < 0 and [q".bar](y) > 0 for all y. This, together with [bar.q](0) = [q.bar] (0) = 0 and [bar.q'] (y) > 0 and [q'.bar](y) > 0 for all v implies that for Case 1 there exists y [??]([theta]) such that [bar.q](y) > y and for Case 2 there exists y [??]([theta]) such that [q.bar] (y) < y, and we have proved.

Hence, either H(y) begins at infinity and drops down to zero (Case 1) or H(y) begins at zero and raises up to infinity (Case 2), as y varies from 0 to [infinity]. Existence and uniqueness are thus ensured.

C.2 Proof of Proposition 1

Note that, for a given [theta], [lim.sub.y[right arrow]0] H(y) = [lim.sub.y[right arrow]0] g'([theta]y) so the result follows from the proof of Proposition 2 above.

C.3 Proof of Proposition 4

Let [f'.sub.1]([k.sub.[rho]+[delta]]) = [rho] + [delta] and [[psi].sub.1] ([p.sub.[rho]+[delta]], [k.sub.[rho]+[delta]]) = 0. Given that on [p.bar] (k), the economy is specialized in the production of the productive good,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, we know that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In order to show that there is a point on [[psi].sub.1] = 0 such that r = [rho] + [delta] we show that [lim.sub.p[right arrow]0] r = [infinity]. Given Assumption 3, write [[alpha].sub.g] ([theta]y) [less than or equal to] [[alpha].sub.1L], for some [epsilon] > 0. From Equation (16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Integrating the last inequality yields

[r/[r.sub.0]] [greater than or equal to] [(p/[p.sub.0]).sup.-[beta]],

for any p [less than or equal to] [p.sub.0]. This shows existence. For uniqueness, notice that because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

so the curves intersect only once. (9)

Note that the second part of Assumption 1 is sufficient for the argument above, because [[alpha].sub.g]/(1-[[alpha].sub.g]) - [[alpha].sub.1L]/ (1 - [[alpha].sub.1L]) < 0 if [[alpha].sub.g] > 1.

APPENDIX D DYNAMICS AND LONG-RUN EQUILIBRIUM

D.1 Stability

The saddle point stability for the dynamic system (13) requires

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where in the last equality, Equation (17) was used. Using the definition of [[alpha].sub.g], Equations (A11), and (A12), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting Equations (A6)-(A8), one can have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where in this last equality we used Equation (A13). A sufficient condition for (k/r)(dr/dk)[|.sub.[theta]] < 0 when [[alpha].sub.1K] [[alpha].sub.2K] is Assumption 3 ([[alpha].sub.1L] > [[alpha].sub.g]), because

[[[alpha].sub.1L]1 - [[alpha].sub.2K] - [[alpha].sub.1L]] [greater than or equal to] [[[alpha].sub.1L]/1-[[alpha].sub.1L]] [greater than or equal to] [[[alpha].sub.g]/1 - [[alpha].sub.g]].

D.2 Features of the Dynamics

D.2.1 Behavior of [y.sub.1] and [y.sub.2]. Substituting Equation (17), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[DELTA] [equivalent to] 1 + (1 - [[alpha].sub.g]) p / y [partial derivative]y / [partial derivative]p [|.sub.k].

Using Equations (A20) and (A21), we have

(D3)[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Analogously, we have

(D4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally, it follows from Equation (D3) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting Equations CA14) and (AI7) in this last expression, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

D.2.2 Social Value of Capital. It follows from the short-run equilibrium condition, expressions (11), and (A20) that

(1 - g) p / [y.sub.1] [partial derivative][y.sub.1] / [partial derivative]p [|.sub.k] + g p / [y.sub.2] [partial derivative][y.sub.2] / [partial derivative]p [|.sub.k] = 0.

With the help of this last equation and Equation (D3), one can show that

(D5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

D.2.3 Behavior of y. Note that a consequence of these last two results is that, using Equation (A20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

D.3 Long-Run Capital Stock

Given that in the long run, the interest rate is fixed at [rho] + [delta], the effect of [theta] on k is given by ([theta]/k)(dk/d[theta])= -(([theta]/r)(dr/d[theta])[|.sub.k])/((k/r)(dr/dk)|[sub.[theta]). It follows from the stability, condition (16), that

dk / d[theta] [??] 0 as dr / d[theta] |[sub.k] [??] 0.

Following the same steps taken in Appendix B.1, recalling that Equation C18) implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

if [k.sub.1] [greater than or equal to] [k.sub.2].

From Equations (AI6) and (AI7), after substituting Equation (18), we have

(D6) -g p / y [partial derivative]y / [partial derivative]p [|.sub.k] = p / [y.sub.1] [partial derivative][y.sub.1] / [partial derivative]p [|.sub.k],

which, substituting in the previous equation, after some manipulations, follows

(D7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting Equations CA16) and 18), we have

(D8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

APPENDIX E WELFARE

E.1 Computation of [C.sub.[theta] ([rho])

Differentiating Equation (13) yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [increment of x](0) is the jump in the variable x right after the change in (10) [theta], we can write the system above as

(E1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where in the first line [increment of k] (0) = 0 was used (capital is the state variable). Rearranging,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

whose solutions are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The initial jump in consumption [increment of c](0) still has to be computed. From the second equation in (E1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [mu] is the positive eigenvalue of the linearized matrix. It must satisfy the characteristic equation:

(E2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence, from the first equation of (E1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting for [increment of c] (0) into the expression for C0(0), and rearranging terms yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where (dk/d[theta])[|.sub.*] = - (dr/d[theta])[|.sub.k,*]/(dr/dk)[|.sub.[theta],*], was used.

E.2 The AF

In this Appendix, we show that the AF satisfies 0 [less than or equal to ] AF [less than or equal to ] 1. Recalling that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

it follows that AF > 0 because [micro] - [rho] [??] 0 iff

[rho] + [delta] - d[y.sub.1] / dk [|.sub.[theta],*] + [c.sup.*][gamma]([c.sup.*]) / [rho] dr / dk [|.sub.[theta],*] [??] 0.

If (d[y.sub.1]/dk)[|.sub.[theta],*] - ([rho] + [delta]) [greater than or equal to] 0, then it is a direct consequence of the definition of AF such that AF < 1. If (d[y.sub.1]/dk[|.sub.[theta],*] - ([rho] + [delta])< 0, using the characteristic equation (E2), we can write AF as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(Recall that [micro] - ((d[y.sub.1]/dk[|.sub.[theta],*] - ([rho] + [delta])) - [rho] > 0 because AF > 0 and [micro] -((d[y.sub.1]/dk[|.sub.[theta],*] - ([rho] + [delta])) > 0 when (d[y.sub.1]/dk[|.sub.[theta],*] - ([rho] + [delta]) < 0.)

E.3 Outside the Steady State

The impact of 0 on welfare at any point in time is given

by

(E3) dW / d[theta] = [[integral].sup.[infinity].sub.0] [e.sup.[rho]t] u'(c(t)) dc(t) / d[theta] dt.

From the first equation of (13),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and hence

(E4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

But

(E5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so substituting Equations (E5) and (E4) into Equation (E3) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Euler equation (12), we can write the last result as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, given that (d[y.sub.1](t)/d[theta])[|.sub.k] = (d[y.sub.1](0)/d[theta])[|.sub.k], and that [lim.sub.t [right arrow] [infinity]](dk(t)/d[theta]) < (dk(t)/d[theta]) < 0 if [k.sub.1] [greater than or equal to] [k.sub.2] it follows that

(d[y.sub.1](t) / d[theta])[|.sub.k] + (d[y.sub.1] / dk)[|.sub.[theta]] - r) (dk(t) / d[theta]) < 0 for any t

and, consequently,

dW / d[theta] < 0 if [k.sub.1] [greater than or equal to] [k.sub.2].

E.4 Signing ((d[y.sub.1]/d[theta])[|.sub.k] + (d[y.sub.1] / dk)[|.sub.[theta] - r) (dk/d[theta]) When [k.sub.1] < [k.sub.2]

From Equation (18), we have that

d[y.sub.1]/d[theta][|.sub.k] = [partial derivative][y.sub.1]/[partial derivative]p[|.sub.k] [partial derivative]p/[partial derivative][theta][|.sub.k] = [partial derivative][y.sub.1]/[partial derivative]p[|.sub.k] p/[theta] [[alpha].sub.g] / [DELTA].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It follows from Equation (D5) that

-(d[y.sub.1]/dk[|.sub.[theta] - r) dk / d[theta] = [yu.sub.1]/k g/[DELTA] k/y [partial derivative]y/[partial derivative]k[|.sub.p] dk/d[theta].

From Equations (D2) and (D7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where we employed Equations (17) and (18).

Consequently, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

whether

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Given that

[[alpha].sub.L] / [[alpha].sub.2K] - [[alpha].sub.1K] = 1 - [[alpha].sub.1K] / [[alpha].sub.2K] - [[alpha].sub.1K] > 1 > g

the result follows.

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(1.) The Tullock-Harberger debate of the social costs of monopoly goes as follows: Harberger (1954) pointed out that the cost of the monopoly is the deadweight loss it generates, and found out that this loss is small. Tullock (1967) replied saying that the monopolist captures part of consumers' surplus, and hence that real resources would be employed to capture these economic rents, so the cost of a monopoly is much larger than the deadweight loss it generates. Hence, the Tullock effect measures the resources used to capture rents, and the Harberger effect measures the usual cost of inefficient institutions (the dynamic distortion). Posner (1975) evaluated empirically Harberger and Tullock effects for a monopoly in a partial equilibrium framework.

(2.) The following passage nicely summarizes his argument:

"Economic history may be thought of as a struggle between a propensity for growth and one for rent-seeking, that is, for someone improving his or her position, or a group bettering its position, at the expense of the general welfare. (...) Whenever conditions permitted, that is, when rent-seeking was somehow curbed, growth manifested itself." (Jones 1988)

(3.) The function G plays, in the context of rent-seeking, the role of the matching function in the equilibrium unemployment literature (see Mortensen and Pissarides 1994). Therefore, g is the rate that seekers of job position meet vacancies; here g is the rate that the seekers of rents exploit the productive sector. Although one activity, job search, is productive and the other, rentseeking, is not, the formal properties of the function g are the same.

(4.) Appendix A provides a short review of the static two-sector general equilibrium model. See chapter 1 of Kemp (1969) for a more complete presentation.

(5.) For example, consider a small town where there is one firm that offers protection against drug-related crimes, on its legal side, and provides drugs, on its illegal side. In effect, this company is a big scare: the fees paid for protection are just a transfer, because there would not be anything to be protected of if the firm did not exist. But because they are legal fees, they will appear in the national accounts. The equations above show that this is indeed the case in our model.

(9.) Recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(10.) That is, [DELTA]x(0) = [lim.sub.t[right arrow]0] + dx(t)/d(theta]).

Barelli: Department of Economics, University of Rochester, 214 Harkness Hall, Rochester, NY 14627. Phone 1-585-275-8075, Fax 1-585-256-2309, E-mail paulo.barelli@ rochester.edu; and Insper--Ibmec Sao Paulo

Pessoa: Centro de Crescimento Economico, IBRE, Fundacao Getulio Vargas, Rua Barao de Itambi, 60-80 andar, Botafogo, Rio de Janeiro, RJ, Brasil CEP: 22231-000. Phone 55-21-3799-6870, Fax 55-21-3799-6867, E-mail [email protected]