Property Rights for Natural Resources and Sustainable Growth in a Two-Country Trade Model∗ Francisco Cabo†
Guiomar Martín-Herrán†
María Pilar Martínez-García‡§
Abstract We analyze a dynamic two-country trade model between a technological leading country and a natural resource dependent economy. The main goal of the paper is to analyze how the ownership and distribution of the exploitation rights upon the natural resource may affect the sustainable growth rate for the two trading economies, the resource conservation and the consumers welfare. The two trading economies are not symmetric. One country is a technological leader whereas its counterpart is a technological follower. In contrast to the leading country, the output production in the follower economy needs a renewable natural resource as an essential input. We analyze different property rights regimes depending on whether the resource is managed by a central authority ( a monopoly firm in the leading economy or the government in the follower country) or the exploitation rights are equally distributed among many harvesters in the follower country. Keywords: Property rights, international trade, renewable natural resources, endogenous growth, sustainability, technological diffusion. JEL Codes: C61, C62, Q20, F18. ∗
The authors have been partially supported by MEC under project SEJ2005-03858/ECON and by
JCYL under projects VA033B06 (first author) and VA045A06 (second and third authors). All projects are co-financed by FEDER funds. † Universidad de Valladolid. ‡ Universidad de Murcia. § Corresponding author. Dept. Métodos Cuantitativos para la Economía. Campus de Espinardo, 30100, Murcia, Spain ; E-mail:
[email protected]
1
1
Introduction
Developing countries are often rich in natural resources. Ownership and distribution of exploitation rights of these resources are at the core of many of the political conflicts that less developed countries have to face. Earlier colonization has meant foreign firms have for decades been the owners of the exploitation rights of profitable natural resources in these countries. After World War II, countries previously under colonial rules claimed the sovereignty over their natural resources. On the other hand, globalization of the world economy unties property rights of natural resources from the country authorities.1 This paper is mainly interested in the effect of the definition used for de facto property rights regime upon the economic growth rate, the resource conservation and the consumers’ welfare. The pattern of trade between developing and industrial countries is frequently characterized by the following property: developing countries export a natural resource intensive product in exchange for capital and technology intensive goods from industrial countries. Within this framework of technology diffusion through trade, this paper analyzes a dynamic two-country model of trade between a developing resource-dependent economy and an industrialized one.2 Assuming well defined property rights upon a renewable natural resource, we first compare the long-run equilibria when the resource is either managed by a foreign monopoly or by the domestic government. In addition, when the exploitation rights are not located abroad, they may be either in the hands of the government, or shared by many small harvesters. Trade relationships between different economies have rarely been taken into account in literature on sustainable growth. Among the exceptions, [6] studies a bilateral trade between a resource-dependent economy and a technological leader country. Although 1
As [4] note, ownership of forest and plans, concession rights and exploitation contracts are increas-
ingly held by foreign companies. 2 International trade on renewable resources and in value-added products based on these resources has been increasing during the last years. As [5] note, the exports of most forest products (wood pulp and paper, wood-based panels and furniture), as well as the fish production that is entering international trade, have expanded considerably over the past 25 years. Developing countries (e.g. Indonesia, Malasya, Brazil, Chile) are emerging as leading exporters in this international market (Barbier, 2001).
2
there is no trade in technological innovation, the growth in the resource-dependent economy comes from the innovative country, through international trade. Another exception is [10], who analyzes the sustainability of economic growth for a small open economy with a renewable natural resource. This economy is defined by two sectors, harvesting of the natural resource and production of the final output. In the former, the harvested resource is used to purchase imports of a consumption good from abroad. Economic growth comes from capital accumulation externalities in the final output sector. Under these assumptions neither the resource conservation nor the international trade are essential to generate sustainable growth. In [7], the idea of the international trade as a channel for economic growth merges with the two-sector economy modeled by [10]. In this work, the harvested resource is not traded abroad but becomes an essential factor in the final output sector. In this last sector, economic growth comes as an expansion in the varieties of intermediate inputs imported from a technological leader country. Therefore, technology diffusion through the trade of intermediate inputs is the transmission mechanism for growth from a technologically leading economy to a follower economy. The present paper inherits the main features of this model and analyzes how different ownership regimes for the property rights of the natural resource (foreign vs. domestic, single vs. multiple ownership) affect the long-run equilibria. There is a vast literature on property rights and natural resources exploitation, although only a small part considers economic growth or international trade aspects. The management of open-access resources and the differences between well- and illdefined property rights regimes are well studied in the literature. However, as far as we know, no attempt has been made to compare equilibria under different well-defined property rights regimes, taking into account foreign versus domestic, and single versus multiple ownership, which is our main goal. To place our work within this stream of the literature, we comment briefly one those works which are most closely related. The sustainability of the economic growth when the harvested resource is an essential input in the production of final output is also analyzed by [14] and [13]. The former compares the economic growth of a closed and an open economy, under the assumption of an open-access natural resource. In contrast [13] focuses on a small open economy 3
and studies the sustainability of growth under the assumption that the property rights belong to a central authority versus the open-access exploitation regime. They consider two types of final goods depending on whether or not the natural resource has been used as a productive input. Both final goods can be invested to increase the natural resource stock. This enables the resource extraction rate to be indefinitely higher than the natural regeneration along the sustainable growth equilibrium. In contrast, our assumptions of a single resource-based final good and no investment in natural capital make the environmental restriction stronger. In addition, our paper differs from [13] in the issue about the property rights of the natural resource. While they compare well- versus ill-defined property rights, we do this for three well-defined property rights regimes, in which the property rights belong to a monopolistic foreign firm, the central national authority, or they are shared by many private harvesters. The assumption of a priori well-defined property rights within the context of international trade is reinforced by [12]. They focus on the endogenous determination and evolution of a de facto property rights regime and show how the opening of trade may transform an open-access regime into a well-defined property rights regime. We show that, in the long-run, the stock of the resource is conserved at a lower level when the property rights are divided between many owners than when the resource is exploited by a single manager, either the domestic government or the foreign monopoly. We also show that when the trading economies have market power, and the terms of trade are thus determined through supply-demand decisions, the long-run growth rate of the two economies is greater when the property rights of the resource are in the hands of a foreign monopoly firm in the leading economy. On the contrary, in small economies with exogenous terms of trade, what matters is not the foreign or domestic ownership of the property rights but the degree of competitiveness in the harvesting sector. In other words, the foreign monopoly and the domestic government regimes lead to the same long-run growth rate, which may be either higher or lower than that attained under the shared-property regime. The comparison of consumers’ welfare in each country under the three different property rights regimes is not equivalent to the comparison of the growth rates. Welfare does not only depend on economic growth rate, but also on the terms of trade as well as the ratio of consumption per variety of intermediate input. 4
The rest of the paper is organized as follows. Section 2 presents the two-country trade model when the property rights for the natural resource belong to a single manager, either the monopoly firm in the leading country or the government in the follower country. Section 3 characterizes the sustainable growth path under both regimes and compares the results. Section 4 introduces the case of many harvesters of the natural resource, the so called shared-property regime, and compares the results under the three property rights regimes. Finally, section 5 offers conclusions.
2
The model
This section develops a two-country endogenous growth model of the type described by [2](Chapter 8). Using this model, we assume that one of the economies has two sectors, a final output and a natural resource sector, similar to the formulation in [10]. Technological improvements in this country are imported from a technologically leading country. Following [2], technological innovation carried out in the leading economy, L, flows to the follower country, F , through the trade of new varieties of productive inputs. Correspondingly, consumers in the leading country buy final goods produced in country F.
2.1
The technological follower
Two sectors coexist in this economy. The resource sector, which uses labor to harvest a renewable natural resource; and the final output sector with three productive inputs: labor, the extracted resource, and the intermediate goods developed in and traded from the leading economy. The stock of the renewable natural resource, S, evolves according to the natural reproduction of the resource minus the rate of harvest. The reproduction function, G(S), is assumed to be of the well-known logistic or Verlhust type (see, for example, [8]):
µ ¶ S G(S) = g˜S 1 − , C
where constants g˜ and C denote the intrinsic growth rate of the natural resource and the carrying capacity or saturation level of the resource, respectively. 5
The harvesting of the natural resource, H(v, S) depends on labor and on the abundance of the renewable resource (its stock). Thus, the harvest rate can be represented by H(v, S) = B [(1 − v)LF ]1−δ S θ ,
0 < δ < 1,
B > 0,
0 ≤ θ ≤ 1,
(1)
where LF is the constant labor force in country F , v is the portion of labor allocated to the final output sector (1−v to the resource sector). Parameter θ measures the elasticity of harvesting with respect to the stock of the resource. The decreasing marginal return to the stock of the natural resource comes as a result of the hypothesis of congestion, while the decreasing marginal return to labor is a consequence of ultimate gear saturation. One particular case is given by θ = 0, which implies that harvesting is independent of the stock size. Another particular case is θ = 1, where the harvest rate corresponds to the well-known Schaefer pattern used in many other models. In this case, the harvest is proportional to the stock of the renewable resource.3 To simplify notation, we shall name the harvest flow H, omitting arguments v and S. Thus, the dynamics of S is:4 ¶ µ S − B [(1 − v)LF ]1−δ S θ , S˙ = g˜S 1 − C
S(0) = S0 ,
(2)
where S0 denotes the initial stock of the resource. The final good production of a representative firm presents the following functional form: 1−α−β
YF = AF (vLF )
N X
XFαj H β ,
AF > 0,
0 < α, β, α + β < 1,
(3)
j=1
where XF j is the amount of nondurable input of type j ∈ {1, . . . , N }, traded from the leader to the follower country. This output-production function with an expanding variety of inputs has often been used in the literature and is based on [9], [11] and [15]. In addition to labor and intermediate goods, we consider the natural resource as an essential factor for production. This function presents diminishing marginal productivity for each input vLF , XF j and H, and constant returns to scale for all inputs taken together. 3
The hypothesis θ = 0 is appropriate for forests or fish living close to the surface; whereas, θ = 1 is
suitable for bottom-dwelling fish (see [10] and references therein). 4 The time argument is eliminated when no confusion can arise.
6
By perfect competition, net marginal productivities are equated to factor prices: Ã ! 1 1−α 1−α−β β αA YF YF F wF = (1−α−β) , pH = β , XF j = (vLF ) 1−α H 1−α , (4) F vLF H pj where wF is the wage rate in the final output sector, pH is the price of the natural resource and pFj is the number of units of YF that output producers pay for one unit of XF j to innovators in the technological leader country.
2.2
The technological leader
The production function of a representative firm in the final output sector is:5 YL = AL L1−α L
N X
α XLj .
(5)
j=1
By equating the marginal product to input prices, the wage rate, wL , and the total demand of intermediate good j by producers, XLj , can be written as: µ ¶ 1 αAL 1−α YL wL = (1 − α) , XLj = LL , LL pj
(6)
where pj are the units of output paid to innovators for one unit of the intermediate input j. As there are no innovators in the follower country, the production of intermediate goods is carried out entirely in the leading country. This situation applies as long as intellectual property rights are protected both domestically and internationally. The terms of trade that measure commerce between these two countries can be defined as the ratio between the units of YL and the units of YF paid for the same unit of intermediate good, that is, pF = pj /pFj . Once the innovator has invented the new intermediate good of type j, the unitary cost of production is one unit of YL , while the sale price equals pj units of YL . The monopolistic innovator decides the price pj to maximize its instantaneous profits from sales to final output producers in L and F : πj = (pj − 1) (XLj + XF j ) , 5
Subscript L denotes variables corresponding to the leading country. Parameters AL and LL may
differ from their corresponding parameters for country F . Differences between AF and AL could reflect differences in government policies. The gap between LF and LL would reflect the differences in scale between the two economies.
7
where XLj and XF j are given in equations (6) and (4). The optimal price for this problem is: pj = 1/α > 1. Once pj is known, the units of final output of country F paid for one unit of the intermediate good j can be written as a function of the terms of trade, that is, pFj = 1/(αpF ). Therefore, the amount of every intermediate good in each country is: 1
2
XLj = XL = LL AL1−α α 1−α , XF j = XF = (vLF )
1−α−β 1−α
(7) 1
2
β
(pF AF ) 1−α α 1−α H 1−α .
(8)
Note that although XL is constant, the quantity XF depends on pF , v and H (which is a function of v and S). Moreover, replacing (7) in (5), the following equality is obtained: α2 YL = N XL ,
(9)
The cost of inventing a new type of product is fixed at η units of output YL . Furthermore, an innovator must pay a cost to transfer and adapt its product for use in country F . This cost is fixed at ν units of output YL , with 0 < ν < η. We assume free entry into the business of being an inventor so that, in equilibrium, the present value of the profits for each intermediate good must equal η + ν, that is, η+ν =
Z
∞
t
where r¯(s, t) = [1/(s − t)]
Rs t
(pj − 1)(XL + XF )e−¯r(s,t)(s−t) ds,
(10)
r(w)dw is the average interest rate between times t and s.
Differentiating (10) with respect to t and taking into account that XF and r are time-dependent, it follows that: r=
2.3
(1 − α)(XL + XF ) . α(η + ν)
(11)
Property rights of the natural resource
Economic growth in the two trading economies will be driven by technological improvements made by innovators in the leading economy. The required investment is associated with the returns obtained by innovators, which depend on the demand for the intermediate goods made by final output producers in the leader and the follower countries. In 8
country F , the adoption of new technological goods developed abroad depends on the purchasing power of domestic firms. As we will show later, the location of the property rights of the natural resource (which can be either in the leader or in the follower country) influences the terms of trade between both countries. In this section we study two different property rights regimes for the natural resource located in the follower country. First, we assume that the property rights of the natural resource are in the hands of a foreign monopoly firm located in the technologically leading country. Second, we consider that the government in the follower country is the exclusive manager of the natural resource. Either the foreign monopolistic firm (in L) or the domestic government (in F ) harvests the resource and exploits its monopoly power, while selling the resource to final output producers in the follower country, to maximize the present value of their objective functions. When the natural resource is managed by a monopolistic agent, this knows the demand function of the resource made by final output producers in the follower country, given in (4). The resource demand can be written as: ∙
βAF (vLF )1−α−β N XFα H (pH , v) = pH d
1 ¸ 1−β
.
(12)
Furthermore, the market clearing condition ∙
βAF (vLF )1−α−β N XFα B(LF (1 − v))1−δ S θ = pH
1 ¸ 1−β
defines the price of the resource pH as a function of v and S, for given values of N and XF , pH (v, S) =
βAF (vLF )1−α−β N XFα [B(LF (1 − v))1−δ S θ ]
1−β
.
(13)
The monopolistic manager of the resource knows the process of price formation. With this information, when the manager decides the demand of labor, (1−v)LF , it determines the amount harvested (given S), and its price (given N and XF ). The monopoly in the leading country, given wF , chooses the labor demand in the resource sector to maximize its stream of profits, given by Π = pH H − wF (1 − v)LF ,
(14) 9
through an infinite time horizon, discounted at the average interest rate between 0 and Rt t, r¯(t, 0) = 1/t 0 r(w) dw. This firm is concerned with the dynamics of the natural
resource in (2). Thus, its maximization problem reads: max v
s.t.:
Z
∞
Π(v, S) e−¯r(t,0)t dt,
(15)
0
S˙ = G(S) − H(v, S),
S(0) = S0 ,
where Π(v, S) = pH (v, S)H(v, S) − wF (1 − v)LF . The first-order conditions for optimality are given by: H wF ∂H + (ξf − 1) = LF , v ∂v pH ¾ ½ ξ˙f 1 ∂H p˙H ∂H pH H 0 − + − εS = r − G (S) + , ξf ∂S ξf S ∂S pH
−εpvH
(16) (17)
where εpxH = (∂pH /∂x)(x/pH ) is the elasticity of pH with respect to x ∈ {v, S}, λf is the shadow value of the renewable resource and ξf = λf /pH is a modified shadow value, defined as the ratio between the shadow value of the stock of the resource and the price of the harvested resource in terms of domestic output. Subscript f stands for foreign monopoly. Considering a domestic government regime, the government in F does not maximize profits, but maximizes the present value of utility for the representative consumer. This consumer has one unit of labor per unit of time and receives a salary wF plus the revenues from harvesting, Π, that the government transfers to consumers (voters) in the form of lump-sum subsidies, at each instant. Since no innovative activity exists in country F and there is no international trade in financial assets, consumers in country F do not accumulate assets and they consume all their income. Therefore, its budget constraint is cF = wF +
Π pH H = + wF v, LF LF
(18)
where cF is the per capita consumption. Taking the salary wF as given, the government determines the demand of labor in the resource sector in order to maximize the present value of per capita utility, defined as u(cF ) = ln cF . The government dynamic problem
10
reads: max v
s.t.:
Z
∞
u(cF ) e−ρt dt,
(19)
0
S˙ = G(S) − H(v, S),
cF =
S(0) = S0 ,
pH (v, S)H(v, S) + wF v, LF
where ρ > 0 is the social discount rate. The first-order conditions for optimality are given by: wF H ∂H + (ξd − 1) = LF , v ∂v pH ¾ ½ ξ˙d 1 ∂H c˙F ∂H p˙H pH H 0 − + + εS = ρ − G (S) + − , ξd ∂S ξd S ∂S cF pH
−εpvH
(20) (21)
where λd is the shadow value of the resource, and ξd = λd LF /(u0 pH ) is the modified shadow value, defined as the ratio between the shadow value of the stock of the resource and the price of the harvested resource(this ratio is scaled by the population in the follower country)6 . Subscript d stands for domestic government. The elasticities εpvH and εpSH only depend on v and S. Likewise, the ratio wF /pH is also a function of these two variables,7 since from (4) it follows that: wF 1−α−β H = . pH β vLF Therefore, if the modified shadow values of the resource under the two property rights regimes, ξf and ξd , equate, then from equations (16) and (20) it immediately follows that, for a given stock of the resource, S, the foreign monopoly firm in country L and the government in country F would demand the same amount of labor to resource harvesting. As we will see, this is the case along the steady-state equilibria.
2.4
Consumers
Consumers in the leading country accumulate assets in the form of ownership claims on innovative firms. However, no asset accumulation is feasible for consumers in country 6
Notice that λd is measured in units of utility, while λd /u0 is the shadow value of the stock of the
resource in units of output YF . 7 Note that although wF and pH depend on N and XF , this is not the case either for the elasticities or for the quotient wF /pH .
11
F . A representative consumer in F decides on labor allocation in the final output sector or in the resource sector. Since consumers have no property rights on the resource, they decide regardless of the dynamics of the natural resource. With perfect mobility of labor, their static optimal decision equates wages in both sectors. The budget constraints for consumers in both, the leading and the follower countries, adopt different forms, depending on the resource property rights regime. If a foreign monopolistic firm located at country L exploits the resource, the budget constraint for a consumer in country L is:8 a˙ L = raL + wL + pF
Π − cL − pF cLF , aL (0) = aL0 , LL
(22)
where aL are the per capita assets, r is the rate of return on assets, cL is the per capita consumption of the final domestic good, and cLF is the per capita consumption of the final good imported from the technological follower country. In (22) we are assuming that consumers in country L equally share the monopolistic extractive firm and receive an equal portion of the dividends that it generates (pF Π/LL ). The price of the domestic final good is defined as the numeraire, pL = 1. Consequently, the terms of trade, pF , is also the price of the consumption good imported from F . A representative consumer maximizes the utility (a logarithmic function) of the streams of domestic and imported consumptions, discounted at rate ρ: max
cL ,cLF
Z
∞
[ln(cL ) + ln(cLF )] e−ρt dt,
(23)
0
subject to (22). With no international asset exchange, consumers in country F cannot invest in assets, but all their earnings are consumed. Therefore, at any time it holds that: cF = wF .
(24)
If the resource is not exploited by a foreign monopoly, but by the domestic government, per capita consumption is given by (18). Investment in assets in the leading country is independent of the monopolistic exploitation benefits, and its dynamics is given by a˙ L = raL + wL − cL − pF cLF , 8
aL (0) = aL0 .
(25)
The arguments of functions H(v, S) and Π(v, S) are generally skipped for notational simplicity.
12
A representative consumer in country L maximizes (23) subject to (25). The first order conditions for problems (23) subject to (22), and (23) subject to (25), lead to the same relationship between domestic and imported consumptions and their growth rates: cL = pF cLF ,
c˙L = r − ρ, cL
c˙LF p˙F =r−ρ− . cLF pF
(26)
With no international exchange of assets, total household assets in the leading economy, aL LL , are equal to the market value of the firms that produce these intermediate goods, (η + ν)N . Therefore, household assets run parallel to the number of varieties of intermediate inputs, N . The dynamics of the number of intermediate goods, N , can be obtained from the equality aL LL = (η + ν)N , taking into account the salary in the technologically leading country, given in (6), the relationship (9) and the assets dynamics. Therefore, taking into account (22) and (25), the growth rate of N for a foreign monopoly and for the domestic government regimes, is given by:
∙ ¸ YL ³ cL pF cLF ´ 1 1−α N˙ = − + LL −XL + XF +Φm (v, S) , N (0) = N0 , N η+ν N N N α
(27)
where m ∈ {f, d} stands for the foreign monopoly and the domestic government respectively, and Φf (v, S) =
v(1 − α) − (1 − α − β) pF YF pF Π(v, S) = , N v N
Φd (v, S) = 0.
Sustainable growth and property rights9
3
When the property rights of the resource belong to a foreign monopoly, the problem for the follower country, P Ff , is a static problem. At each moment consumers choose the share of labor devoted to each sector, and consume their earnings from salaries. They do not accumulate assets. Final output producers determine the static demands for inputs that maximize their profit at each moment (eq. (4)). Under this regime, the problem for the leading country, P Lf , is composed of two dynamic matters and 9
The proofs of the propositions in this section are presented in Appendix A.
13
a static question. Consumers in the leading country choose cL and cLF to maximize their discounted utility along an infinite time horizon (eq. (23)), subject to their budget constraint (eq.(22)) and taking the salary wL and the interest rate r as given (eqs. (6) and (11) respectively), which result from the maximization of the instantaneous profits of final output producers and innovators. At the same time, the monopoly firm that manages the resource determines the share of labor devoted to the extraction of the resource that maximizes the stream of discounted future profits (eq. (15)), subject to the dynamics of the natural resource and taking the salary wF (4) as given. The main difference between the foreign monopoly and the domestic government regimes is that the management of the resource moves from the monopoly in L to the government in F , which chooses the share of labor devoted to harvesting that maximizes the stream of discounted future per capita utilities (eq. (19)) subject to the dynamics of the natural resource (eq. (2)). Thus, the problem for the follower country P Fd incorporates a dynamic question, whereas the problem for the leader P Ld has consumption decisions as the only dynamic problem. The economies previously described face a problem of environmental shortage when agents have to restrict the optimal harvesting to rates which are below the harvesting rate that would be chosen under a myopic management which disregards the evolution of the resource. The following proposition shows the fraction of labor that the monopolistic agents employ in harvesting in the case of a myopic management. Proposition 1 When agents do not take into account the dynamics of the natural resource, the share of labor allocated to harvesting under a myopic management, be it the monopoly in the leader country or the government in the follower country, is 1 − v mm , with: v mm =
1+β ∈ (0, 1), 1 + βφ
where
φ=
1 − α − δβ > 1. 1−α−β
As we will see below, we are going to concentrate on equilibria with v > v mm .That is, we assume a scarce natural resource. Note that if v > v mm then the profit derived from the harvesting of the resource Π, given in (14), is positive when the factor prices are given by (4). We consider two scenarios, depending on the market power of the trading economies. 14
The first scenario assumes small open economies, with exogenously fixed terms of trade (unaffected by agents decisions) and assumed to be constant. For simplicity, we refer to this as small open economies (SOE) scenario. In the second scenario, named large open economies (LOE) scenario, the terms of trade, pF , are endogenously determined by equating the value of the final good traded from F to L, to the value of the intermediate goods sold from innovators in L to producers in F (the balanced trade equation): LL pF cLF = pj N XF .
(28)
Definition 2 Given N(0) and S(0), and considering time paths for N , S, cL , cLF and v, such that problems P Ff − P Lf and P Fd − P Ld are solved, two types of equilibria may appear: • Small open economies equilibrium: pF is exogenously fixed in the international market and supposed constant at value pˆF . • Large open economies equilibrium: pF is endogenously determined from equation (28). In what follows, we concentrate exclusively on the steady-state equilibria, that is, equilibria with variables growing at constant rates. First we describe the behavior of the different variables along such steady-state equilibria. Proposition 3 If a steady-state equilibrium exists, along this path: • v, S, H, pF and r remain constant; • YL , YF , cL , cLF , cF , pH , wL , wF and N grow at rate r − ρ. Note that a steady-state equilibrium with the behaviour presented in Proposition 3 will be obtained if and only if variables v, S and c˜L = cL /N remain constant. If v and S remain constant, then the harvest rate, H, and the interest rate, r, given by (11), will also be constant. Moreover, output YF , given by (3), will grow at the same rate as the number of intermediate goods, N . This rate will be constant if and only if c˜L is also constant.10 Therefore, a steady-state equilibrium corresponds to a steady state 10
Along the steady-state equilibrium, the modified shadow price of the stock of the natural resource,
ξd and ξf , also remains constant under both property right regimes.
15
∗ , S ∗ and c˜ ∗ , of variables v, S and c˜L . Let us denote these steady-state values by vm Lm m
where the subscripts m can be f, if a foreign monopoly regime is considered, or d if the domestic government manages the resource. In the domestic government regime the revenues from harvesting, Π, are in the hands of consumers in F . Conversely, in the foreign monopoly regime, the revenues are transferred to the leading economy and its value in units of YL , pF Π, is consumed by consumers in this economy. These revenues determine the gap in the per capita consumption per unit of intermediate good between the foreign monopoly and the domestic government regimes, as shown in the following proposition. Proposition 4 The dynamics of the per capita consumption per unit of intermediate good in country L is: ½ ¾ ¾ ½ 1 . 2α 1 1−α 1−α 2˜ cL LL − (1 − α)LL AL α c˜L = c˜L − Φm (v, S) − ρ , η+ν
(29)
and the steady-state per capita consumption per unit of intermediate good reads: 1
c˜L ∗m
2α
∗ , S∗ ) ρ(η + ν) + LL AL1−α α 1−α (1 − α) + Φm (vm m = . 2LL
(30)
Corollary 5 Along a steady state equilibrium the per capita consumption per unit of intermediate good in L is larger under the foreign monopoly regime than under the domestic government regime, c˜L ∗f > c˜L ∗d . A problem of multiplicity of steady-state equilibria may arise, from the non-linearities in the production, harvesting and regeneration functions (Figure 1 illustrates an example ∗ , S ∗ , m ∈ {f, d} cannot with three steady-state equilibria). Explicit expressions of vm m
always be obtained for the two regimes. However, for the particular case when the harvesting function does not depend on the stock of the resource, θ = 0, these steadystate values can be easily found: Sf∗0
=
Sd∗0
C g˜ − ρ C< , = 2˜ g 2
vf∗0
=
vd∗0
1 =1− LF
∙
(˜ g 2 − ρ2 )C 4˜ gB
1 ¸ 1−δ
.
(31)
∗0 > 0, m ∈ {f, d} if and only if g ˜ > ρ, which also guarantees Remark 6 Note that Sm ∗0 < 1, m ∈ {f, d}. Moreover, when θ = 0, the assumption of environmental that vm
shortage (i.e. v > v mm ) requires the harvesting to be below the myopic extraction rate, 16
1
⋅ S=0
v
0.95
*
*
(Sm1,vm1) 0.9
(S* ,v* ) m2 m2 *
⋅ v=0
*
(Sm3,vm3)
0.85 0.5
1
1.5
2
2.5
3
3.5
S
4
C
Figure 1: Monopoly regime. Example three steady-state equilibria. + , H mm = B [(1 − (1 + β) / (1 + βφ)) LF ]1−δ , which is equivalent to condition g˜ < g˜mm
where + g˜mm
2H mm + = C
sµ
2H mm C
¶2
+ ρ2 .
(32)
That is, the intrinsic growth rate of the resource must be greater than the rate of temporal + , in order for a feasible interior steady state to exist. discount and lower than g˜mm ∗ , S ∗ , m ∈ {f, d} cannot be obtained. However, If θ > 0, explicit expressions of vm m
the following proposition guaranties the existence of a steady-state equilibrium. ∗ ∈ (v mm , 1), S ∗ ∈ (0, C) Proposition 7 If θ > 0, a steady-state equilibrium with vm m
and c˜L ∗m > 0, always exists. Furthermore, under sufficient condition: θ≥
2ρ , g˜ + ρ
(33)
the equilibrium is unique. Note that in (31) the steady-state values coincide when the resource is managed by a single owner, i.e. a foreign monopoly or the domestic government, Sf∗0 = Sd∗0 and vf∗0 = vd∗0 . This result is not exclusive for θ = 0, but applies for any value of θ ∈ [0, 1]. At the steady state, the growth rate of the consumption of the domestic good in country F equals r − ρ. Therefore, the dynamics of the modified shadow prices in (17) and (21) are the same. For a given v and S, the foreign monopoly and the domestic government 17
value the resource identically. As a consequence, equation (16) coincides with (20) and for a given S, both monopolistic managers present the same behaviour, choosing the same share of labor in the resource sector. Since equation (2) describes the dynamics of the resource under both regimes, the steady-state value of the labor demand in each sector and the stock of the resource are independent of the location of the property rights for the natural resource. The next proposition collects these results. Proposition 8 When the resource is monopoly-owned either by a foreign monopolistic firm in the leading country or by the government in the follower country, the steadystate values of the stock of the natural resource and the share of labor devoted to the final-output sector coincide under both property rights regimes. That is, Sf∗ = Sd∗ and vf∗ = vd∗ . A straightforward consequence of Corollary 5 is that the value of the constant demand for the imported consumption good in the leading country under the monopoly regime is above that under the domestic government regime (see the first equation in (26)). However, in the follower economy, from equation (8) and Proposition 8, the demand for each imported intermediate good as a function of the terms of trade is the same upward sloping and convex function under both regimes. Thus, the balanced trade equation (28) concludes that the steady-state value of the terms of trade, given by p∗F m =
(c˜L ∗m LL )1−α ∗ L )1−α−β A (H(v ∗ , S ∗ ))β αα+1 (vm F F m m
,
m ∈ {f, d}.
(34)
is higher under foreign monopoly than under domestic government management, as is stated in the next corollary. Corollary 9 Along the steady-state equilibrium, the terms of trade of large open economies are larger under the foreign monopoly regime than under the domestic government regime, p∗F f > p∗F d . When large open economies are considered, the better terms of trade for the follower country when a foreign monopoly manages the natural resource raise the demand for each intermediate good made by final output producers in this country. The rise in the demand for intermediate goods pushes up the rate of return on investment in the 18
leading country (see eq.(11)), which in turn pushes up the steady state growth rate of the economies. In the case of small open economies the terms of trade are, due to their lack of market power, the same exogenous constant for the two regimes. Therefore, the rate of return on investment is unaffected by the different property rights regimes, and so will be the growth rate. The results below state and compare the long-term growth rates for small and large open economies under the two property rights regimes considered. Proposition 10 Along a steady-state equilibrium the economies in both the technological leading and follower countries grow at the same rate given by: γ=
1−α [XL + XF ] − ρ, α(η + ν)
(35)
which for the different scenarios particularizes as follows: • Large open economies (LOE) " # 2α 1 (1 − α)α 1−α 1−α 1−α loe γm = (1 + α) LL AL − ρ + Φm (vf∗ , Sf∗ ), 2(η + ν) 2(η + ν) m ∈ {f, d}. • Small open economies (SOE) (1−α)α soe γm =
2 1−α
α(η+ν)
∙ ¸ 1 1 1−α−β β 1 1−α 1−α ∗ ∗ ∗ LL AL + (LF vm ) 1−α AF 1−α H(vm , Sm ) 1−α pˆF −ρ,
m ∈ {f, d}. Corollary 11 The long-run growth rates for SOE coincide under a foreign monopoly and the domestic government regimes, γfsoe = γdsoe . For LOE, the long-run growth rate under a foreign monopoly is higher than under the domestic government regime, γfloe > γdloe .
Shared property11
4
To avoid the “tragedy of the commons", the environmental literature has frequently relied on a central authority which strictly regulates the usage of the resource (this is 11
The proofs of the propositions in this section are presented in Appendix B.
19
the case in the two property rights regimes presented above). Next we consider the property rights are distributed among many equal agents who exploit their share of the resource. In this case of many managers of the resource, institutions who guarantee the exclusive access for users to their individual allotment, must be developed to avoid overexploitation. There exist real examples of this last case (see, for example, [3] and the real examples therein). In this section we assume that a representative consumer in country F manages an equal share of the natural resource s = S/LF . From (2) it evolves as: ³ s´ s˙ = g˜s 1 − − b(1 − v)1−δ sθ , κ
s(0) = s0 ,
(36)
and s0 = S0 /LF .12 where κ = C/LF , b = BLθ−δ F
The harvest rate of the representative agent reads: h(v, s) = b(1 − v)1−δ sθ ,
b > 0,
0 < δ < 1,
0 ≤ θ ≤ 1.
(37)
The aggregate harvest rate will be H(v, S) = LF h(v, s). A representative consumer receives a salary per unit of labor in the final output sector, and the income from the extraction of her share of the resource. As long as it does not accumulate assets, its budget restriction is: cF = vwF + pH h.
(38)
A representative consumer chooses the share of labor devoted to each sector that maximizes the present value of her utility, Z max v
∞
u(cF )e−ρt dt
(39)
0
subject to (38) and (36). The first-order conditions for optimality are given by:
12
wF ∂H = LF , ∂v pH 1 ∂h c˙F ∂h p˙H − + = ρ − g 0 (s) + − , ∂s ξsp ∂s cF pH
(ξsp − 1)
(40)
ξ˙sp ξsp
(41)
Note that κ = C/LF means that each share of the resource s has an equal carrying capacity and
the sum of all of them is the carrying capacity of the aggregate stock of the resource, S. In addition, means that each agent labor marginal productivity under the shared-property regime matches b = BLθ−δ F the global labor marginal productivity when the resource is monopoly-owned.
20
where λsp is the shadow values of the resource managed by a representative consumer, and ξsp = λsp /(u0 pH ) the modified shadow value, defined as the ratio between the shadow value of the stock of the resource (in terms of final output) and its sale price. To ensure that the resource is a real restriction on production and growth, we study first the case of an open-access resource, which may lead to the “tragedy of the commons". The following proposition shows the fraction of labor that a single agent employs for harvesting in this case of myopic management. Proposition 12 When agents disregard the dynamics of the natural resource, the share of labor allocated to harvesting the resource under open access is 1 − v oa , where: 0 < v oa =
1+β 1 < = vmm < 1. φ 1 + βφ
Therefore, the assumption of a scarce natural resource made on page 14, that is v > v mm , also guarantees scarcity when the resource is managed by many owners-users. Now, the matter in the leader country, P Lsp , is the problem of a representative consumer who has to choose cL and cLF to maximize the present value of its utility (eq. (23)), subject to its budget constraint (eq. (25)). The salary wL and the rate of return r will be given by (6) and (11), respectively. In a symmetric fashion, the problem for the follower country, P Fsp , considers a representative consumer who has to choose v to maximize the present value of its utility (eq. (39)) subject to the natural resource evolution (eq. (36)), and its budget constraint (eq. (38)). The wage rate, wF , and the price of the resource, pH , will be given by (4). When the property rights of the natural resource are located in the follower country, the revenues from harvesting do not flow to consumers in the leading economy. Therefore, consumption in the leading country is unaffected by harvesting revenues. This is true regardless of whether the resource belongs to the government or it is in the hands of many agents. Therefore, in the leading country per capita consumption of domestic good per unit of intermediate good at the steady state is equal under the shared-property and the domestic government regimes: c˜L ∗sp = c˜L ∗d . When the harvesting function does not depend on the stock of the resource, that is
21
θ = 0, explicit expressions for these steady-state values can be easily found: s∗0 sp
κ g˜ − ρ κ< , = 2˜ g 2
∗0 vsp
∙
(˜ g 2 − ρ2 )κ =1− 4˜ gb
1 ¸ 1−δ
.
(42)
Remark 13 As for the other two regimes, s∗0 ˜ > ρ, which also sp > 0 if and only if g ∗0 < 1. Moreover, when θ = 0, the assumption of a scarce natural resource guarantees vsp
(i.e. v > 1/φ) says that the harvesting h is below the open-access extraction, hoa = + , where b(1 − 1/φ)1−δ , which holds if and only if g˜ < g˜sp + g˜sp
2hoa + = κ
sµ
2hoa κ
¶2
+ ρ2 .
(43)
Thus, the intrinsic growth rate of the resource must be greater than the rate of temporal + , in order for a feasible interior steady state to exist. discount and lower than g˜sp
Note that, in this particular case in which harvesting is independent of the stock of ∗0 ∗0 ∗0 the resource, Sf∗0 = Sd∗0 = LF s∗0 sp and vf = vd = vsp . The main difference between the
monopolistic management of the resource (foreign monopoly or domestic government regimes) and the shared-property regime, is that, in the latter, the managers of the resource do not know the demand function of the resource (eq. (13)). However, when θ = 0 this demand is independent of the stock of the resource and so is harvesting. Thus per capita consumption (eq. (38)) in the shared property regime, and the instantaneous profits from harvesting (eq. (14)) in the case of a monopolistic manager, do not depend on the stock of the natural resource. As a consequence, regardless of who manages the ∗0 resource, there exist unique values s∗0 sp and vsp such that the stock of the natural resource
and its shadow value remain motionless. So, at the steady state, the three regimes give the same results for s and v. The differences in the optimal harvesting policies when the property rights of the resource belong to a single owner (the government), and when they are in the hands of many consumers are obtained by comparing systems (20)-(21) and (40)-(41). Contrary to the domestic government, the representative consumer does not know the demand function of the resource. In consequence, it does not discount the reduction in the price caused by an increment in the extraction, which increases the marginal gains of labor in the harvesting sector. Furthermore, when the manager of the resource knows the 22
demand function, and θ > 0, it acknowledges that a higher stock of the resource leads to higher harvesting and thus lower price. The time evolution of the modified shadow price of the natural resource in the case of a representative consumer disregards this negative effect. If θ > 0, this different behaviour induces at the steady-state equilibrium, the domestic government to follow a more conservationist policy, shown by a greater stock of the resource. However, the comparison of labor demand on the resource sector is not completely determined. Proposition 14 If θ > 0 and if there exists a unique and asymptotically steady-state ∗ < S ∗ = S ∗ , although equilibrium with v ∗ ∈ (v mm , 1), S ∗ ∈ (0, C) and c˜L ∗ > 0, then Ssp f d ∗ can be greater, lower or equal to v ∗ , with m ∈ {f, d}. However, condition (33) vsp m ∗ < v ∗ = v ∗ .13 assures that vsp f d
The previous proposition states that if the natural resource is managed by a monopoly firm in L, or by the government in F , then the natural resource is better conserved (in the sense of a higher stock in the long-run), than if the resource is exploited by many owners in the follower country. An immediate question arises: is this higher stock of the resource necessarily linked with a slower economic growth? To answer this question in a bilateral trade framework, first we have to look at the terms of trade. In the scenario of large open economies, the steady-state value of the terms of trade is obtained from the balanced trade equation (28): p∗F sp =
(c˜L ∗sp LL )1−α ∗ L )1−α−β A (L h(v ∗ , s∗ ))β αα+1 (vsp F F F sp sp
.
(44)
Taking into account (44), the next proposition states the long-run growth rates for small and large open economies under the shared-property regime. The proof follows the same pattern as the proof of Proposition 10. Proposition 15 Along a steady-state equilibrium the economies in both the technological leading and follower countries grow at the same rate given by: 13
Note that the steady-state equilibrium under shared property may not be unique. The sufficient
condition for uniqueness under monopolistic management (33), also applies under shared property.
23
• Large open economies (LOE) loe γsp
=
γdloe
"
# 2α 1 (1 − α)α 1−α 1−α LL AL − ρ . = (1 + α) 2(η + ν)
• Small open economies (SOE) 2 ∙ ¸ 1 1 1−α−β β 1 (1−α)α 1−α 1−α soe ∗ 1−α ∗ ∗ 1−α 1−α 1−α LL AL +(LF vsp) AF [LF h(vsp , ssp)] pˆF −ρ. γsp = α(η+ν) When the property rights of the resource are located in the follower economy the profits from resource harvesting do not reach consumers in the leading economy. In this country, the consumption of domestic good is the same constant regardless of whether the property rights are in the hands of many owners or the government in the follower economy. The same happens with the consumption of the imported good expressed in units of output in L, pF cLF (LHS of the balanced trade equation (28)). In the follower economy, the share of labor in each sector and the stock of the natural resource at the steady state do not necessarily coincide under domestic government and shared-property regimes (contrary to what happens when domestic government and foreign monopoly regimes are compared). Therefore, the final output producers in the follower economy may present a different demand function for intermediate goods, expressed as an upward slope function of the terms of trade. In the case of large open economies (LOE), a higher (resp. lower) demand curve under the shared-property regime than under the domestic government management, a lower value for the terms of trade (resp. higher) will balance the trade (see eq. (28)) and the imported amounts will coincide under both regimes. As a consequence, the rate of return on investment in new goods is the same under both regimes and so will be the growth rate of the economies. When the two countries are small (SOE), the terms of trade are an exogenous constant value. Therefore, a higher demand curve leads to a higher imported amount and thus, a higher growth rate. The demand curve and consequently the growth rate under shared property can be higher or lower than under monopolistic management. In the following corollary we summarize the comparisons between the three property rights regimes studied in this paper: foreign monopoly and domestic government regimes, already compared in Corollary 11, and the shared-property regime studied in this section. 24
Corollary 16 For LOE, the long-run growth rate under foreign monopoly is higher than under either the shared-property or the domestic government regimes, which share the loe = γ loe . For SOE, the long-run growth rates coincide under same growth rate, γfloe > γsp d
the foreign monopoly and the domestic government regimes, γfsoe = γdsoe , although they can be higher or lower than under the shared-property regime. When comparing the shared-property and the domestic government regimes, from Proposition 15 and Corollary 16 it follows that the gap in the terms of trade for large open economies has opposite sign to the gap in growth rates for small open economies. Resorting to numerical simulations, we compare both issues. Note that these comparisons can be done by evaluating the following expression for the different regimes: ∆(vk∗ , Sk∗ ) = vk∗ 1−α−β (1 − vk∗ )β(1−δ) Sk∗ βθ ,
k ∈ {m, sp}.
(45)
0.8
Monopoly 0.78
0.76
Shared−Property
0.74
0.72
0.7
0.68
0.66
0
0.1
0.2
0.3
0.4
0.5 θ
0.6
0.7
0.8
0.9
1
Figure 2: Comparison long-run growth rates SOE: Monopoly vs.
shared-property
regimes For this numerical analysis we assume α = 0.3, β = 0.3, δ = 0.1, LF = LL = 1, for which voa = 0.5970, and vmm = 0.8652. From the hypothesis LF = 1 it follows that B = b, C = κ and S = s. To establish the remainder of the parameters that define the regeneration and the harvesting functions, we focus on the extreme case in which myopic management leads to exhaustion. The next lemma collects the sufficient conditions for this behaviour: 25
Lemma 17 The stock of the resource harvested by a unique monopolistic and myopic manager is exhausted under sufficient conditions: 2−θ ¸1−δ ∙ β(φ − 1) (2 − θ) 1−θ LF , , C≤ g˜ ≤ B 1 + φβ 1−θ
(46)
where at least one inequality strictly holds. Notice that under conditions in Lemma 17, a non-myopic monopolistic manager will ∗ > v mm . behave less aggressively with the natural resource, that is, vm
When conditions (46) hold in equality, they describe a situation in which the harvesting effort under myopic administration lies on the edge of extinction. Let assume that this occurs for an intermediate value of the elasticity of harvesting with respect to the stock of the resource, θˆ = 0.5. Furthermore, we also assume that sufficient condition for uniqueness in (33) holds, for this value of θ. From these conditions, the parameters related to the natural resource sector, g˜, C and B can be derived. Figure 2 depicts the values ∆(vk∗ , Sk∗ ) as a function of θ, for the two monopolistic property rights regimes (a continuous line) and for the shared-property regime (a dashed line), all other parameters fixed. From this figure we can conclude that the long-run growth rates for SOE under the monopoly property rights regimes, γfsoe = γdsoe , can soe . Furthermore, for the be higher or lower than under the shared-property regime γsp
given parameters’ values, as θ increases “ceteris paribus", the more likely it is that the long-run growth rate under a monopolistic regime surpasses the growth rate under the shared-property regime, and the wider is the gap between them and vice versa. The opposite behaviour occurs as θ approaches 0. The terms of trade for LOE compare inversely to function ∆(vk∗ , Sk∗ ) in Figure 2. In consequence, the terms of trade for small values of θ are larger under monopolistic management than under shared property, and viceversa. The differences between the growth rates for SOE (or the terms of trade for LOE) under the different property rights regimes do not reproduce differences in the shares of labor in each sector. In particular, for the chosen parameter values, numerical solutions ∗ > v ∗ for any θ ∈ (0, 1). show that vm sp
The qualitative behaviour described in Figure 2 is robust to changes in all parameter values. Only for a very small θˆ do functions ∆(vk∗ , Sk∗ ), k ∈ {m, sp} not cross, but
26
the growth rate under monopolistic management is always higher than under shared property. Once the growth rates under the different property rights regimes have been compared, the question that arises is whether a higher growth inexorably leads to higher consumer welfare. In what follows, we denote by Vik the aggregate welfare along the steady-state equilibrium for a representative consumer in country i ∈ {L, F }, under the property rights regime k ∈ {f, d, sp}. We first present the results assuming that the resource is in the hands of a sole owner, either a foreign monopoly or the domestic government. Proposition 18 When the resource is in the hands of a monopolistic manager, then VLf > VLd regardless of the market power of the trading economies. Furthermore, VF f < VF d for SOE, while the comparison is unclear for LOE. The growth rate of the economies under the foreign monopoly is greater than under the domestic government regime for LOE, while they exactly match for SOE. Furthermore, in the leading economy the consumption of the domestic good per variety of intermediate is greater under foreign monopoly. This is also true for the imported good for SOE, although the comparison is unclear for LOE. Therefore, welfare in the leading economy is higher under foreign monopoly for SOE. The same result can be proved for LOE.14 Under the foreign monopoly regime, by equation (24), the consumption per variety of intermediate good in the follower country is given by the wage per variety. For SOE (with an exogenous constant pF ) this wage per variety exclusively depends on v ∗ and S ∗ , and hence remains the same under domestic government. Furthermore, under this regime, consumers also have access to the positive profits from the resource exploitation (see eq. (18)), which implies higher consumption per variety. As a consequence, given the same growth rate under the two regimes, welfare is higher under domestic government. 14
Since (vf∗ , Sf∗ ) = (vd∗ , Sd∗ ), note that c˜L ∗f cLF ˜ ∗f = (c˜L ∗f )1−α ∆(vf∗ , Sf∗ ) > c˜L ∗d cLF ˜ ∗d . Furthermore, taking
into account the utility function in (23) and the different growth rates for LOE under the two property rights regimes, it follows that the welfare in the leading country is greater under foreign monopoly than under domestic government.
27
For LOE the terms of trade are greater under the foreign monopoly, impeding the comparison of welfares. Next we compare welfares under the domestic government and the shared-property regimes. Proposition 19 When the property rights of the resource are located in the follower soe , which is equivalent to country then Vid > Visp , ∀i ∈ {L, F } if and only if γdsoe > γsp
p∗F d < p∗F sp . For SOE the growth rate under domestic government can be greater, equal or lower than under shared property, while for LOE the two growth rates equate. In the leading economy the consumption of the domestic good per variety of intermediate is the same under domestic government and shared-property regime. The consumption of imported goods per variety does not depend on the number of owners for SOE either, although this consumption will be greater under the regime with lower terms of trade for LOE. As a consequence, the welfare in the leading country compares similarly to the growth rate for SOE, and behaves contrarily to the terms of trade for LOE. As we have previously pointed out, the growth rate for SOE and the terms of trade for LOE compare in an opposite way. In the follower economy, the consumption per variety of intermediate goods can be greater under domestic government or shared property, mirroring the behaviour of the growth rates for SOE. Likewise, this consumption will be greater under the regime with smaller terms of trade for LOE. When the property rights for the resource are located in the follower country, the consumers’ welfares mimic the comparison of the growth rates for SOE, which is equivalent to the inverse-comparison of the terms of trade for LOE. The welfares in both, the leading and the follower country, move in the same direction leading to a win-win (or loss-loss) situation. When the resource is monopoly owned, the consumers’ welfares do not reproduce the behaviour of the growth rates. For SOE, although the growth rates equate, the welfare is higher under foreign monopoly in the leading economy, and lower in the follower, which implies a win-loss situation. For LOE, the result in the leading economy remains 28
valid, while welfares cannot be compared in the follower economy.
5
Concluding remarks
Considering a developing economy endowed with a renewable natural resource, used as an essential input in the final output sector, two main research questions are answered in this paper. First, assuming that this economy has no innovative activities, the trade of innovation from a technological leading country has proven to be an adequate mechanism to guarantee sustainable growth both in the leading and in the follower economies. The second question at stake has been the effect of the definition of different property rights regimes for the natural resource on the long-run growth rate of the economies, the consumers’ welfares and the conservation of the natural resource. Most of the literature centres on the comparison between well- and ill-defined property rights for the natural resource. Conversely three well-defined property rights regimes have been compared. In a first step, assuming sole ownership, the focus has been on the location of the property rights. Thus, results have been compared for a monopolistic firm located in the leading economy and the government in the follower country. In a second step, the attention has moved to the number of owners of the natural resource. Growth, resource conservation and welfare under the domestic government regime have been contrasted with the results under the shared-property regime, where the resource ownership is split among many consumers, also located in the follower economy. When the emphasis is on the location of the property rights, we have proved that the long-run stock of the natural resource and the share of labor devoted to each sector (final output and natural resource) coincide under the foreign monopoly and the domestic government regimes. Therefore, for small open economies, where the terms of trade are an exogenous constant, the growth rate is independent of the location of the monopolistic manager. However, when both economies are large, the terms of trade are greater when the owner of the natural resource is located in the leading economy. As a consequence, the long-run growth rate is also higher under this foreign monopoly regime. Throughout a steady state, consumers in the leading economy always achieve greater welfare when the natural resource is managed by a monopolistic firm located in this 29
economy than under domestic government. Likewise, one would predict that consumers in the follower country are better off when their government owns the resource. This result has been shown for small but not for large economies, where the change in the terms of trade prevents a clearcut comparison. Comparing single versus multiple ownership, we have seen that the natural resource is better conserved (in the sense of a higher stock in the long run), when managed by the domestic government than when exploited by many owners. This is true regardless of the market power of the economies, and irrespective of whether the growth rates under the two regimes match or diverge. In the case of small open economies, the long-run growth rates under these two regimes do not necessarily coincide. Indeed, numerical simulations conclude that the long-run growth rate under domestic government can be higher or lower than under shared property, depending on parameter values. When economies are small, the terms of trade are constant. Conversely, when economies are large the terms of trade adjust so that both regimes share the same growth rate. The property rights regime linked with a faster growth rate when economies are small is also the regime with lower terms of trade when economies are large. On comparing consumers’ welfare for property rights regimes with single or multiple ownership, we have shown that both in the leading and in the follower economy welfares are higher under the regime with faster growth (when economies are small), or equivalently with lower terms of trade (when economies are large). The paper can be extended in many ways. Growth, resource conservation and consumers’ welfare have been compared for three a priori well-defined property rights regimes. A first extension would focus on the endogenous determination of the property rights regime. Furthermore, up to now the analysis has been limited to long-run steady-state solutions. It would be worth studying the transitional dynamics towards these equilibria. Particular emphasis would be placed on the comparison of the different policies converging to the equilibria under the foreign monopoly and the domestic government regimes. These equilibria are characterized by the same resource stock and extraction effort. Two hypotheses regarding the natural resource sector could be relaxed in further research. In our model the harvested resource is first transformed into a final good, 30
exported later on. An immediate extension would be to allow the direct exportation of the raw resource. The second modification regards the technological progress, which could improve productivity not only in the final output sector but also in the resource sector. Another item in our research agenda deals with the effect on growth, conservation and consumers’ welfare of different property rights regimes when the follower economy is endowed with an exhaustible natural resource.
Appendix A Proof of Proposition 1. Under the myopic-management regime for a single owner if the resource is managed by a monopolistic firm in the leading country or the government in the following country, the fraction of labor allocated to the final-output sector coincides. Note that if the monopolistic firm in the leading country behaves myopically, it solves the maximization problem (15). Similarly, if the government in the follower country disregards the dynamics of the natural resource, it solves (19). From the necessary conditions for optimality, it is straightforward to show that the following expression must be satisfied in both single manager scenarios: wF LF + H
∂H ∂pH + pH = 0. ∂v ∂v
After some simplifications we obtain: v mm = (1 + β)/(1 + φβ). Proof of Proposition 3. Variables v and S cannot grow indefinitely at a non-zero constant rate because they are lower and upper bounded (v ∈ [0, 1] and S ∈ [0, C]). These variables must be constant on a steady-state equilibrium. Provided that H depends on v and S, which are motionless, the harvesting must also remain constant on a steady-state equilibrium. From (27) replacing the expression of the consumption of imported goods in the technologically leading country given in (26), the production function in this country (5), and taking into account (4), we can conclude that the growth rate of N is constant 31
along the steady-state equilibrium if the consumption of national good in the leader country, cL , grows at the same rate as the number of intermediate goods, N , and at the same time, the amount of each intermediate good used in the follower final-output sector, XF (v, S), is also stationary. From equation (8), since v and H are motionless, for XF to be constant, the terms of trade, pF , must also remain constant. Taking into account (11), provided that pF , v and H remain constant along the steady-state equilibrium, the interest rate r is also constant and equal to: ¸ ∙ 1 1−α−β β 2 1 1 − α 1−β α 1−α (1−α) LL AL +LF v (pF AF ) 1−α H 1−α r= . (η+ν)α
(47)
From (4) and (24) one gets that c˙F /cF = Y˙ F /YF . Taking into account (3), (8) and (1) the growth rate of consumption and final-output production of the follower country, it follows: c˙F N˙ βθ S˙ Y˙ F α p˙F (1 − α − β)(1 − φv) v˙ + + . = = + cF YF 1 − α pF (1 − α)(1 − v) v N 1−αS
(48)
Along the steady-state equilibrium, constants v, S and pF allow us to rewrite the growth rate of the final-output production, YF , and the consumption, cF , in the follower country, in (48), equal to the growth rate of the number of intermediate goods, N (which coincides with the growth rate of the national good consumption in the leading country, r − ρ). Furthermore, by (26), since pF remains constant along the steady-state equilibrium, the growth rate of the imported good, cLF , equals the growth rate of the national good consumption in the leading country. Since α2 YL = N XL , the growth rate of the production in the leading country also equals that of N , since XL is constant. Finally, provided that H remains constant, (4) shows that the price of the natural resource grows at the same rate as YF along the steady-state equilibrium. Proof of Proposition 4. From the definition of c˜L = cL /N and its dynamics in (26): .
c˜L N˙ =r−ρ− . c˜L N 32
From this equation, the expression of r in (11), the dynamics of N in (27) and the first-order condition in (26), the dynamics of the consumption per unit of intermediate good in (29), it immediately follows. Proof of Proposition 7. At the steady-state equilibria, the stock of the resource has to be constant: S˙ = G(S) − H(v, S) = 0, which gives a definition of v as a function of S: Ã ¡ 1 g˜S 1−θ 1 − v˜(S) = 1 − LF B
S C
1 ¢ ! 1−δ
(49)
,
(50)
that satisfies v˜(0) = v˜(C) = 1, v˜(S) < 1 ∀S ∈ (0, C), and presents a minimum at S˜ = (1 − θ)C/(2 − θ). To completely determine the steady-state equilibria we must also consider the necessary conditions for optimality. Taking into account (4) and (13), after some calculus, the necessary condition (16) for the foreign monopoly (or (20) for the domestic government), can be rewritten as: −
1−α−β 1−δ 1 − α − β (1 − β)(1 − δ) − − (ξm − 1) = . v 1−v 1−v βv
At the steady state, equation (17) or (21) vanishes. Furthermore, if θ 6= 0, then ξm can be written as a function of v and S if S(ρ − G0 (S)) + θH(v, S) 6= 0: ξm (v, S) =
θβH(v, S) . S(ρ − G0 (S)) + θH(v, S)
Replacing this last expression in the previous equation, and considering that G(S) = H(v, S) at the steady state, then v can be written as a function of S if Γ(S)(1 + βφ) − θ(1 + β) 6= 0: vm (S) =
(Γ(S) − θ)(1 + β) , Γ(S)(1 + βφ) − θ(1 + β)
(51)
where Γ(S) = −
S )−ρ g˜(1 − 2 C (ρ − G0 (S))S . = S G(S) g˜(1 − C )
Function vm (S) is a hyperbola with an horizontal asymptote in vmm . Furthermore, differentiating in (51) and simplifying, we obtain: 0 (S) = vm
θ(φ − 1)β(1 + β)Γ0 (S) , [Γ(S)(1 + βφ) − θ(1 + β)]2
33
and since Γ0 (S) < 0 ∀S ∈ (0, C), vm (S) is a decreasing function of S. We are looking for steady-state equilibria with v ∗ ∈ (v mm , 1). Therefore, since the hyperbola presents a negative slope, we focus on the right branch. This right branch equals one at15 S ∗0 = (g − ρ)C/(2g). Since v˜(S) < 1 ∀S ∈ (0, C), v˜(C) = 1, then an equilibrium exists within the region of interest (v mm , 1) × (S ∗0 , C). From equation (30), the steady-state value of c˜L m follows. Condition S˜ ≤ S ∗0 implies that v˜(S) increases in the region of interest. Since v˜(C) = vm (S ∗0 ) = 1 and vm (S) decreases, then this condition is a sufficient condition to avoid multiple equilibria, which can be rewritten as (33). Proof of Proposition 10. The growth rate at the steady state can be obtained by the growth rate of the consumption of the national good in the technologically leading country in (26). This growth rate equals r − ρ, where r is given in (47). Therefore, denoting this growth rate by γ oe , it can be written as follows: γ oe =
1−α [XL + XF ] − ρ. α(η + ν)
(52)
The expressions of the long-run growth rates are obtained by replacing in (52) the expressions of the amount of intermediate good used in the leader and the follower countries, XL and XF , given in (8). To derive the growth rate of the economies in the SOE scenario, γ soe , the price is assumed to be constant and exogenously given, pˆF . Conversely, in the LOE scenario, the growth rate of the economies is obtained by replacing the terms of trade, pF , by their stationary value, p∗F m , m ∈ {f, d} given in (34). Substituting the value of p∗F m in loe in the statement of Proposition 10 (52) and simplifying, the different growth rates γm
follow.
Appendix B Proof of Proposition 12. 15
Recall that S ∗0 is the steady-state value of the stock of the resource when θ = 0.
34
Under open access, consumers do not take into account the dynamics of the natural resource. They solve the maximization problem (39) subject to their budget constraint given by (38). From the necessary conditions for optimality and the definition of h in (37), it follows that wF (1 − v) = (1 − δ)pH h. Taking into account expressions in (4), the fraction of labor allocated to the final-output sector, voa = 1/φ, immediately follows. Proof of Proposition 14. This proof compares the steady-state values of v and S under shared-property and domestic government (as already shown, the latter matches the results under foreign monopoly). When the natural resource is managed by a single owner, the steady-state equilibrium is defined by equating v˜(S) in (50) with vm (S) in (51), together with the expression of c˜Lm in (30). When the property rights belong to many agents, the same reasoning as for the characterization of vm (S), but considering the new necessary conditions in (40) and (41), can give function vsp (S): vsp (S) =
Γ(S) − θ , φΓ(S) − θ
(53)
where φΓ(S) − θ 6= 0. This function behaves similarly to vm (S), i.e. it is a hyperbola 0 (S) < 0 for all S ∈ (0, C) and v (S ∗0 ) = 1. with a horizontal asymptote in v oa , vsp sp
From expressions (51) and (53), and considering that Γ(S) < 0 ∀S ∈ (S ∗0 , C), it immediately follows that vsp (S) < vm (S) for any S ∈ (S ∗0 , C). Taking into account ∗ < S∗ . that within this region vsp (S) and vm (S) cross v˜(S) from above, then Ssp m ∗ can be higher equal or lower than Function v˜(S) is not monotonous, therefore, vm ∗ . However, under condition (33), v vsp ˜(S) is a growing function within region (S0∗ , C), ∗ < v∗ . and consequently, vsp m
Proof of Lemma 17. A myopic monopolist manager devotes an extraction effort to the resource sector equal to 1 − v mm . We look for the conditions that necessarily lead to extinction. For θ ∈ (0, 1), the steady state for the stock of the resource is given by equation: H(v mm , S) = G(S). 35
Note that only the right hand side of this equation is affected by parameters g and C, which increase the reproduction function. The equation above admits a trivial solution S = 0. Furthermore, depending on the parameter values, this equation may present one, two or none additional solutions. A unique interior solution would appear under tangency: ∂H mm (v , S) = G0 (S). ∂S From the two equations above, a unique interior solution appears when the following condition on g and C holds: gC
1−θ
∙
β(φ − 1) LF =B 1 + φβ
¸1−δ
(2 − θ)2−θ . (1 − θ)1−θ
(54)
Values of g and C, for which the LHS of (54) lies below the RHS, would imply the nonexistence of an equilibrium different from S = 0. In consequence, the resource would be depleted. When θ = 1, harvesting is linear in the stock of the resource. A sufficient condition for extinction is to consider the slope of the extraction function greater or equal to the slope of the regeneration function at S = 0, that is, ¸1−δ ∙ β(φ − 1) ∂H mm 0 (v , 0) ≥ G (0) ⇔ B LF ≥ g˜. ∂S 1 + φβ Taking into account the RHS of the last equivalence, the sufficient condition for the carrying capacity in (46), which guarantees the depletion of the resource (i.e. the LHS below the RHS of (54)) can be easily derived. For θ = 0 if conditions in Lemma 17 hold (at least one in strict inequality), then it immediately follows that H(vmm , S) > G(C/2), which means that the harvesting, independent of S, is greater than the maximum reproduction. Therefore the resource is depleted.
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